Microstructural Strengthening Mechanisms in Micro-Truss ... · Microstructural Strengthening Mechanisms in Micro-Truss Periodic Cellular Metals Brandon Andrew Bouwhuis Doctor of Philosophy,

Microstructural Strengthening Mechanisms in Micro-Truss

Periodic Cellular Metals

by

Brandon Andrew Bouwhuis

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Materials Science and Engineering University of Toronto

© Copyright by Brandon Andrew Bouwhuis, 2009

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Microstructural Strengthening Mechanisms in Micro-Truss Periodic Cellular Metals

Brandon Andrew Bouwhuis

Doctor of Philosophy, 2009

Graduate Department of Materials Science and Engineering

University of Toronto

Abstract This thesis investigates the effect of microstructural strengthening mechanisms on the

overall mechanical performance of micro-truss periodic cellular metals (PCMs). Prior to the

author’s work, the primary design considerations of micro-truss PCMs had been topological

issues, i.e. the architectural arrangement of the load-supporting ligaments. Very little attention

had been given to investigate the influence of microstructural effects within the cellular

ligaments. Of the four broad categories of strengthening mechanisms in metals, only solute and

second phase strengthening had previously been used in micro-trusses; the potential for

strengthening micro-truss materials by work-hardening or grain size reduction had not been

addressed.

In order to utilize these strengthening mechanisms in micro-truss PCMs, two issues

needed to be addressed. First, the deformation-forming method used to produce the micro-

trusses was analyzed in order to map the fabrication-induced (in-situ) strain as well as the range

of architectures that could be reached. Second, a new compression testing method was

developed to simulate the properties of the micro-truss as part of a common functional form, i.e.

as the core of a light-weight sandwich panel, and test the effectiveness of microstructural

strengthening mechanisms without the influence of typical high-temperature sandwich panel

joining processes, such as brazing.

The first strengthening mechanism was achieved by controlling the distribution of plastic

strain imparted to the micro-truss struts during fabrication. It was shown that this strain energy

can lead to a factor of three increase in compressive strength without an associated weight

penalty. An analytical model for the critical inelastic buckling stress of the micro-truss struts

during uniaxial compression was developed in terms of the axial flow stress during stretch

forming fabrication. The second mechanism was achieved by electrodeposition of a high-

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strength nanocrystalline metal sleeve around the cellular ligaments, producing new types of

hybrid nanocrystalline cellular metals. It was shown that despite the added mass, the

nanocrystalline sleeves could increase the weight-specific strength of micro-truss hybrids. An

isostrain model was developed based on the theoretical behaviour of a nanocrystalline metal

tube network in order to predict the compressive strength of the hybrid materials.

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Acknowledgements First, to my supervisor and mentor Professor Glenn D. Hibbard, I would like to express

my most sincere gratitude for his guidance, advice, and feedback, as well as his energy and

humour; with him this project was not only educational, but also tremendously enjoyable. I

would also like to extend my appreciation to my supervising committee for their valuable input.

I would like to thank Integran Technologies, Inc., for the processing and microstructural

characterization of electrodeposited nanocrystalline metal. The technical help of Sal Boccia,

John Calloway, and Dr. Dan Grozea at the University of Toronto was also extremely helpful in

characterizing the materials in this study. I would also like to extend my heart-felt thanks to the

Hybrid Materials Design Group at the University of Toronto, including Eral Bele for his many

helpful discussions and analysis, as well as Philip Egberts, Lily Cheng, Samson Ho, Evelyn Ng,

and Megan Hostetter, for their warmth, humour, and friendship.

Financial assistance from the Natural Sciences and Engineering Research Council of

Canada (NSERC), the University of Toronto Open Fellowship, the Dr. Burnett M. Thall

Graduate Scholarship in Science and Technology, and the Graduate Department of Materials

Science and Engineering is greatly appreciated.

Finally, this thesis would not have been possible without a great number of family and

friends whose constant help, support, and encouragement was invaluable throughout these past

years. In particular, I would like to thank my fiancée, Miranda Vandenberg, for her enduring

love, care, compassion, and understanding (as well as patience). Ik hou van jou.

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Table of Contents

ABSTRACT II

ACKNOWLEDGEMENTS IV

TABLE OF SYMBOLS VIII

1 INTRODUCTION: CELLULAR MATERIALS, ADVANTAGES, AND SCOPE 1

1.1 Cellular Materials and Material Property Space 1 1.1.1 Conventional Metal Foams 2 1.1.2 Periodic Cellular Metals 2

1.2 Functional Cellular Materials and the Sandwich Panel 3 1.2.1 Historical Application and Key Properties 3 1.2.2 Loading Conditions, Failure Modes, and Optimization 4 1.2.3 Assessment of Sandwich Panels with Different Cores 5

1.3 Scope of Thesis 6 1.3.1 Issues/Barriers to Implementing Microstructure Strengthening Mechanisms in PCMs 7 1.3.2 Directions for Microstructure Design 7 1.3.3 Thesis Outline 8

1.4 References 8

2 LITERATURE REVIEW 10

2.1 Architecture Classification 10 2.1.1 References 12

2.2 Fabrication Methods and Parent Materials 13 2.2.1 Casting Methods 13 2.2.2 Textile Lay-Up 15 2.2.3 Deformation-Forming 17 2.2.4 Additional Methods 19 2.2.5 References 21

2.3 Microstructural Strengthening of Cellular Metals 23 2.3.1 Microstructure Strengthening Mechanisms in Foams 23 2.3.2 Microstructure Strengthening Mechanisms in Micro-Truss PCMs 25 2.3.3 References 26

3 MICRO-TRUSS PCM FABRICATION BY PERFORATION-STRETCHING 28

3.1 Failure Mechanisms during PCM Fabrication 28 3.1.1 Introduction 28 3.1.2 Experimental Details 29 3.1.3 Results and Discussion 30

3.1.3.1 Perforation Stretching Force-Displacement Profile 30 3.1.3.2 Onset of Plasticity 32 3.1.3.3 Maximum Forming Force 34

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3.1.4 Conclusions 39 3.1.5 References 39

3.2 Modeling the Perforation Stretching Forming Process 41 3.2.1 Introduction 41 3.2.2 Materials and Experimental Details 41 3.2.3 Results and Discussion 43 3.2.4 Conclusions 48 3.2.5 References 48

4 TESTING METHODS 50

4.1 Testing the Micro-Truss Core as a Stand-Alone Material 50 4.1.1 Introduction 50 4.1.2 Experimental 51 4.1.3 Results 52

4.1.3.1 Free-Core Uniaxial Compression: Beam Bending on Steel Platens 52 4.1.3.2 Confined-Core Uniaxial Compression: Collapse Mechanisms 54

4.1.4 Discussion 57 4.1.4.1 Confinement Dependence 57 4.1.4.2 Effect of Collapse Mechanisms 58


4.2 Edge Effects in Compression Testing Periodic Cellular Metal Sandwich Cores 62 4.2.1 Introduction 62 4.2.2 Experimental 62 4.2.3 Results and Discussion 65

4.2.3.1 Simulating Sandwich Core Collapse 65 4.2.3.2 Edge Effects in Confined Core Compression 67


5 MICRO-TRUSS STRENGTHENING BY IN-SITU WORK HARDENING 73

5.1 Work Hardening as a Strengthening Mechanism in Periodic Cellular Materials 73 5.1.1 Introduction 73 5.1.2 Buckling Failure of AA3003 PCMs 74 5.1.3 Experimental 77 5.1.4 Results and Discussion 78 5.1.5 Conclusions 84 5.1.6 References 84

5.2 Relative Significance of In-Situ Work Hardening in Micro-Trusses 86 5.2.1 Introduction 86 5.2.2 Materials and Experimental Details 86 5.2.3 Results and Discussion 88 5.2.4 Conclusions 94 5.2.5 References 94

6 STRENGTHENING BY NANOCRYSTALLINE ELECTRODEPOSITION 96

6.1 Structural Nanocrystalline Ni Coatings on Periodic Cellular Steel 97 6.1.1 Introduction 97 6.1.2 Experimental 97 6.1.3 Results and Discussion 99

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6.2 Mechanical Properties of Hybrid Nanocrystalline Metal Foams 108 6.2.1 Introduction 108 6.2.2 Experimental 108 6.2.3 Results and Discussion 109 6.2.4 Conclusions 120 6.2.5 References 120

7 CONCLUSIONS 122

7.1 Fabrication and Testing of Micro-Truss Sandwich Cores 122

7.2 Microstructural Strengthening Mechanisms in Micro-Trusses 122 7.2.1 In-Situ Work Hardening 123 7.2.2 Electrodeposition of Nanocrystalline Metal Sleeves 124

8 FUTURE WORK 125

APPENDIX A: FABRICATING MICRO-TRUSS SANDWICH PANELS BY RESISTANCE BRAZING 127

APPENDIX B: PROPERTIES OF ELECTRODEPOSITED NANOCRYSTALLINE NICKEL 129

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Table of Symbols Precursor and Micro-Truss Architecture Parameters

l (lo) Strut length/reduced section length (initial)

t (to) Strut thickness (initial)

w (wo) Strut width (initial)

φ Perforated metal open area fraction

ρR (ρ) Truss relative density (absolute) FoamAlρ Foam absolute density (Section 6.2)

FoamAlNin /−ρ Nanocrystalline nickel foam absolute density (Section 6.2)

tf,Node Strut thickness at the node after forming

ω (ωM) Strut angle from horizontal (maximum)

h Truss height (foam height in Section 6.2)

I, A, r Strut moment of inertia, cross-sectional area, radius of gyration Foam

NinI − Foam ligament second moment of inertia (Section 6.2)

FoamNinH − Foam ligament moment for fully plastic hinging (Section 6.2)

λ Strut slenderness ratio

ATruss Truss compressive area

SA Truss surface area (Section 6.1)

nT Number of truss tetrahedral truss cells (Section 4.2)

nS Number of truss struts edgeTn , bulk

Tn Number of edge, bulk tetrahedral truss cells (Section 4.2)

S Foam specific surface area (Section 6.2)

V Foam volume (Section 6.2)

tn-Ni Nominal deposited thickness of nanocrystalline nickel

mn-Ni Deposited mass of nanocrystalline nickel nom

Nlnt − Nominal nanocrystalline Ni sleeve thickness (Section 6.2)

midNint − Measured mid-foam nanocrystalline Ni sleeve thickness (Section 6.2)

L Foam ligament length (Section 6.2)

d Foam ligament cylindrical diameter (Section 6.2)

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Precursor and Parent Material Parameters

σt, εt True stress, strain

ES, Et Elastic modulus, tangent modulus (∂σt/∂εt)

εU, εFail Uniform tensile strain, total failure strain

Δl (ΔlFail) Change in strut length during uniaxial tension (total failure displacement)

σYS, σUTS, τ Yield strength, ultimate tensile strength, ultimate shear strength

K, n Hollomon/Ludwik strength coefficient, strain-hardening exponent

m Strain-rate sensitivity

rd, td Precursor sheet rolling direction and transverse direction

R Sheet planar anisotropy ratio

ε0 Ramberg-Osgood Model plastic strain corresponding to σYS (0.002)

N Ramberg-Osgood Model strain-hardening exponent

σRev, σFor Bauschinger effect parameters: effective reverse, forward yield strength

λCo Wavelength of XRD cobalt Kα X-rays

ρNi Density of nickel

ENi Elastic modulus of nickel

Micro-Truss Forming Parameters

L Perforation center-to-center distance

R Rounded-square perforation corner radius

dForm (dM) Forming displacement (maximum)

dU Forming displacement coinciding with the uniform tensile strain εU

FForm (FO, FM) Forming force (plasticity onset, maximum)

Fb Force for plastic hinging in a single perforation strut during forming

Ft Force for tensile instability in a single perforation strut during forming

η Number of struts formed in the perforation stretching press (Sections 3.1 and 3.2)

ν Number of plastic hinges in the perforated metal precursor during forming

FLigament Uniaxial tensile force experienced by a ligament/strut

β, ξ, γ, ζ Fitting parameters for forming process (Section 3.1)

rpin Perforation-stretching press pin fillet radius

dpin, delectrode Diameter of perforation-stretching pin, resistance brazing electrode

x

Micro-Truss Mechanical Testing Parameters

σ Truss compressive strength (peak compressive strength in Section 6.1)

ε Truss compressive strain (foam compressive strain in Section 6.2)

F, d Truss compressive force, displacement

σP, σV Truss peak compressive strength, valley compressive strength

HV Vickers Microhardness

WH, FA Work-hardened, fully-annealed

Ω Truss strength ratio (σP,WH/σP,FA)

σF (FF) Strut failure stress (force)

σCR Strut critical buckling strength

FHinging Strut force for plastic hinging, loaded transversely

FInelastic Strut force for inelastic buckling, loaded axially

C, k Parameters describing end constraints for plastic hinging, strut buckling

E Truss compressive modulus (maximum loading modulus in Section 6.1)

FP, SL Truss peak compressive force, maximum loading stiffness

εD, JD Truss densification strain, densification energy density

κ, α Fitting parameters between σYS and σCR (Section 5.2)

ξ, φ Fitting parameters between FForm and FP (Section 5.2)

r Exponent in strength-density property space, i.e. σP ∝ ρr (Section 5.2)

α Knockdown factor for elastic modulus (Section 6.1)

Foam Mechanical Testing Parameters Foam

AlNinE /− Nanocrystalline nickel foam hybrid elastic modulus (Section 6.2)

FoamAlNin /−σ Nanocrystalline nickel foam hybrid peak strength (Section 6.2)

FoamAlE Aluminum foam elastic modulus (Section 6.2)

FoamAlσ Aluminum foam peak strength (Section 6.2)

f Nanocrystalline nickel foam hybrid thickness gradient factor (Section 6.2)

B Parameter describing end constraints for elastic beam bending (Section 6.2)

α, β Knockdown factor for foam modulus, strength (Section 6.2)

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1 Introduction: Cellular Materials, Advantages, and Scope This chapter is an introduction to cellular materials and their potential mechanical

properties. Section 1.1 reviews some of the basic characteristics of cellular materials and

addresses their advantages in terms of material property space. Comparisons are made between

conventional cellular materials (e.g. foams and honeycombs) and micro-truss periodic cellular

materials. Section 1.2 presents one of the common functional forms for cellular materials, the

sandwich panel, as well as their properties and optimization for particular loading conditions

and low-weight designs. Finally, Section 1.3 outlines the scope of this thesis: the investigation

of microstructural strengthening mechanisms in micro-truss periodic cellular metals.

1.1 Cellular Materials and Material Property Space

Cellular materials are considered hybrids of space (air) and solid material [1] that can

have a relative density (density of the cellular solid divided by that of the parent material)

significantly less than 1. From a design perspective, these materials have the potential to fill

holes in material property space at the low-density regime that are otherwise unattainable by

fully-dense materials [1,2] (Figure 1.1.1). Cellular materials possess effective or nominal

material properties which are a function of the cellular ligament arrangement (the architecture)

and the parent material making up the ligaments (the microstructure).

Figure 1.1.1. Modulus-density material property space. Cellular materials (e.g. foams and lattices) can occupy low-density regions unattainable by fully-dense materials. Figure from Ashby [2].

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1.1.1 Conventional Metal Foams

The architecture of foams can vary widely depending on the processing technique. One

of the key features is their stochasticity; their cell sizes and ligament dimensions are non-

uniform and instead exist over a statistical distribution. As a result, there can be significant

sample-to-sample property differences and structural gradients within a given sample. Foams

are generally open-celled or and closed-celled [3,4]. The first type contains a continuous

network of space channels that surrounds the metal ligaments, and the second type contains

isolated pockets of space enclosed by surrounding metal cell walls and webs (Figure 1.1.2).

Figure 1.1.2. Examples of commercial open-celled (left) and closed-celled (right) foams. Figures from Wadley [4].

More advanced types can be a combination of the two, such as open-cell foams with hollow

ligaments, e.g. [3], or brazed hollow metal spheres, e.g. [5]. Both Ashby et al. [3] and Banhart

[5] have conducted extensive reviews of the manufacture, characterization, and also application

of conventional cellular metals.

1.1.2 Periodic Cellular Metals

Periodic cellular metals (PCMs), in their simplest form, are a regular arrangement of

supporting ligaments. Some examples are shown in Figure 1.1.3 [6]. Periodic cellular metals can

be created with varying dimensions of open-cell connectivity. Completely closed-cell

architectures, such as egg-box and waffle-panel, e.g. [7,8], provide a regular closed arrangement

of support structures to reduce relative density without a great compromise in mechanical

properties. Other cores, e.g. prismatic (Figure 1.1.3) can be considered open-celled in one

dimension, but can be stacked to possess multiple open dimensions.

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Figure 1.1.3. Examples of periodic cellular metals. Figures and nomenclature from Wadley [6].

On the other hand, lattice or micro-truss PCMs are an example of fully-open cellular

materials. These architectures, including the pyramidal, tetrahedral, and Kagome structures, e.g.

[7,8], reduce the total material mass by retaining only that which has geometrically-high load-

bearing efficiency. The resultant relative density can be reduced to as low as 1%, e.g. [9], and

the weight-specific mechanical properties (e.g. strength-to-weight ratio) can be improved in

order to fill the low-density regions in material property space.

1.2 Functional Cellular Materials and the Sandwich Panel

Because of their structural efficiency, cellular materials are an attractive option for

weight-limited engineering applications. They can be used as a stand-alone material or, more

commonly, as part of a composite, i.e. the core of a sandwich panel; the latter results in the most

attractive utility of their cellular, light-weight properties, e.g. [7].

1.2.1 Historical Application and Key Properties

It is generally acknowledged that sandwich panels date back to the mid nineteenth

century [10], but it wasn’t until the Second World War with the British Mosquito aircraft that

sandwich panels were widely used [10,11]. The command module in the Apollo program also

used sandwich panels, which comprised two thin face sheets and a honeycomb core [11]. Since

then, sandwich panels have been created using stochastic and periodic cellular materials as their

cores; these sandwich panels have been used in high-performance environments such as

aerospace, automotive, and defense, e.g. [3,5]. For example, aerospace applications require

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materials with high bending stiffness and strength, while cost savings are achieved with less

overall material mass. The fabrication and performance of these panels must also be highly

predictable and uniform. To accomplish these goals, honeycomb panels have been used. On the

other hand, many defense applications require materials with superior energy absorption and

blast mitigation properties. Conventional foams have drawn interest in this area due to the shape

of their compressive stress-strain profile, which consists of a constant stress for an extended

range of strain. Good compressive performance attributes have been identified as being

necessary for structural applications, for example, that expose materials to large static and cyclic

loads, or require dimensional stability, e.g. [12,13], as well as aeronautical applications that

benefit from buoyancy-aid materials capable of supporting loads, e.g. [14]. Furthermore, high

compressive strength and stiffness are important for applications such as aircraft fuselages,

naval ship structures, truck storage tanks, and rail cars, e.g. [15].

1.2.2 Loading Conditions, Failure Modes, and Optimization

Cellular materials have been subjected to a wide range of loading conditions, including

compression, shear, bending, and fatigue. In particular, determining the compressive properties

of the sandwich core is a key step to mapping the failure mechanism maps for sandwich panels

under more complex loading conditions, such as bending. During bending of sandwich panels

with cellular material cores, there are typically four general failure modes (Figure 1.2.1): face

sheet yielding, face sheet buckling, core indentation, and core yield, e.g. [3,7].

Figure 1.2.1. Bending failure modes of a sandwich panel with a cellular core. Figures from Evans et al. [7].

In the case of a micro-truss panel, the main core failure mechanisms are strut yielding and

buckling, e.g. [8,16-18], which are both functions of the compressive properties of the truss

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ligaments. Moreover, optimal sandwich configurations are determined at the confluence point of

all failure mechanisms, e.g. [18] (Figure 1.2.2).

Figure 1.2.2. An example failure mechanism map for a sandwich panel with a micro-truss PCM core. The load capacity is represented by V/(EM)½. Figure from Zok et al. [18].

Finally, studies have also been conducted to determine minimum weight designs and

properties of cellular material core sandwich panels for use in light-weight environments, e.g.

[3,8,11,16-18]. For example, when seeking desirable weight-specific bending strength in a

panel, the material index characterizing beneficial performance is (σY)1/2/ρ, where σY is the yield

strength of the material and ρ its relative density [3]. A reduction of density alongside increases

in strength can therefore lead to great returns in performance. Furthermore, in cases where

density can not be decreased, microstructural design (e.g. increased σY) is an invaluable tool.

1.2.3 Assessment of Sandwich Panels with Different Cores

Each type of cellular material possesses its own scaling relationship for properties such

as strength and modulus as a function of relative density. To compare the performances between

different cellular material cores, scaling relationships for their failure modes can be evaluated.

To use a simplified case, foams typically possess an effective yield strength which scales as

(σY,Foam/σY,S) ∝ ρ3/2, where σY,S is the yield strength of the solid material [1,3]. On the other hand,

the effective yield strength of micro-trusses scales according to (σY,Truss/σY,S) ∝ ρ [1,3]; for a

given relative density, micro-trusses will outperform foams, and the magnitude of this difference

will become increasingly significant at the lowest densities.

The benefits of cellular materials depend on the intended application and governing

requirements. For example, when load capacity of the sandwich drives design, honeycomb

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panels are always lighter than foams for a given performance, e.g. [7]. Some environments can

also benefit from secondary, multifunctional characteristics such as active cooling made

possible by flow-through channels in prismatic materials, e.g. [19]. In general micro-truss panels

are nearly as light-weight as the most efficient competing designs, e.g. honeycombs; however,

when secondary considerations such as strut shape factors (e.g. hollow tube struts versus solid

cylindrical struts), multifunctionality, flexibility of application-specific design, and fabrication

costs are considered, lattice materials can become more attractive than competing cellular

material designs, e.g. [7]. Wicks and Hutchinson [17] have shown that in low-density regimes

(<4% relative density), micro-trusses can clearly outperform honeycombs for a given density

(Figure 1.2.3). Moreover, micro-trusses can be designed for many different configurations

depending upon the intended application [7].

Figure 1.2.3. Compressive strength of a regular tetragonal and hexagonal honeycomb core as a function of relative density. Figure and variables from Wicks and Hutchinson [17].

1.3 Scope of Thesis

Effective material properties of cellular materials are a function of their cellular ligament

shape and connectivity (the architecture) and the metallurgical state within the ligaments (the

microstructure). While architecture-property relationships have been called the ‘research

frontier’ [7], very little attention has been given to microstructural strengthening mechanisms in

micro-truss PCMs. This thesis investigates the potential for microstructural strengthening

mechanisms to further improve the weight-specific compressive properties of micro-truss

PCMs.

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1.3.1 Issues/Barriers to Implementing Microstructure Strengthening Mechanisms in PCMs

Before applying microstructure strengthening strategies to micro-truss PCMs, two issues

were identified that needed to be addressed. First, deformation-forming approaches are

attractive as micro-truss fabrication methods since they are based on simple sheet forming

methods and low relative densities can be fabricated from ductile precursors. These are also

potential routes by which work hardening may be used as a microstructural strengthening

mechanism. However, prior to this Ph.D. project, little attention had been given to the failure

mechanisms during deformation-forming micro-truss fabrication. In particular, given the

mechanical properties of the starting alloy sheet, the possible range of micro-truss architectures

which can be formed had not been studied.

The second issue concerns treating the micro-truss core as a stand-alone material. The

compressive properties of PCMs are important as a performance index, as well as a being a

component of more complex loading conditions (Section 1.2.2). PCMs in compression have

typically been tested as part of a sandwich panel wherein the PCM is a core material, joined to

external face-sheet materials. In this state, there are two shortcomings: the overall mechanical

response of the PCM can be a function of the face sheet dimensions, and conventional joining

processes (e.g. brazing) can in many cases irreversibly alter the microstructure of the as-

fabricated PCM. The latter shortcoming is particularly undesirable as there is potential for useful

information on the microstructure state and benefits of a particular strengthening mechanism to

be lost.

1.3.2 Directions for Microstructure Design

In fully-dense metals, there are typically four broad categories of strengthening

mechanisms available: solid solution or inclusion of particulates; second-phase precipitation;

deformation or work-hardening; and grain size refinement [20]. In cellular materials studied to

date, strengthening has been accomplished almost exclusively using the first two mechanisms.

Particular to micro-truss PCMs, the primary strengthening mechanism has been age-hardening,

whereby after fabrication the truss is given solutionizing then age-hardening heat treatments to

reach a final product. The reason for this, in part, is the use of aforementioned high-temperature

brazing heat treatments, which often erase prior microstructure history. However, fabrication

methods have been established which can incorporate work-hardening and grain size reduction

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into micro-truss PCMs. By considering these strengthening mechanisms as additional design

tools, the performance of micro-truss PCMs can be further improved.

1.3.3 Thesis Outline

This thesis is divided into the following seven chapters. Chapter 2 reviews the state of

the literature prior to the author’s thesis work, focusing on architecture definitions and general

microstructural strengthening techniques applied to cellular metals. Chapter 3 presents the work

done by the author to characterize stretch-bending fabrication of micro-truss cellular metals.

Chapter 4 introduces a new test method used to evaluate the micro-truss sandwich core as a

stand-alone material. Chapter 5 presents the author’s investigation into work hardening as a

strengthening mechanism in deformation-formed micro-trusses. Chapter 6 presents work done

by the author to electrodeposit high-strength nanocrystalline metal sleeves on cellular pre-forms.

Chapter 7 presents the conclusions of the present Ph.D. project, highlighting the key results of

microstructural strengthening mechanisms in micro-trusses. Finally, Chapter 8 describes some

directions for future work in microstructural strengthening and design of micro-truss PCMs.

1.4 References 1. MF Ashby, Philos. Trans. R. Soc. A 364 (2006) 15-30.

2. MF Ashby, Philos. Mag. 85 (2005) 3235-3257.

3. MF Ashby, A Evans, NA Fleck, LJ Gibson, JW Hutchinson, HNG Wadley, Metal Foams – A Design Guide,

Butterworth-Heinemann, Boston, 2000.

4. HNG Wadley, Adv. Eng. Mater. 4 (2002) 726-733.

5. J Banhart, Prog. Mater. Sci. 46 (2001) 559-632.

6. HNG Wadley, Philos. Trans. R. Soc. A 364 (2006) 31-68.

7. AG Evans, JW Hutchinson, NA Fleck, MF Ashby, HNG Wadley, Prog. Mater. Sci. 46 (2001) 309-327.

8. HNG Wadley, NA Fleck, AG Evans, Compos. Sci. Technol. 63 (2003) 2331-2343.

9. DT Queheillalt, HNG Wadley, Mater. Des. 30 (2009) 1966-1975.

10. HG Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Toronto, 1969.

11. JM Davies, Lightweight Sandwich Construction, Blackwell Science, Toronto, 2001.

12. SB Burns, SN Singh, JD Bowers, J. Cell. Plast. 34 (1998) 18-38.

13. LJ Gibson, MF Ashby, Cellular Solids, Structure and Properties, 2nd ed., Cambridge University Press,

Cambridge, 1997.

14. N Gupta, E Woldesenbet, S Sankaran, J. Mater. Sci. 36 (2001) 4485-4491.

15. S Aimmanee, JR Vinson, J. Sandwich Struct. Mater. 4 (2002) 115-139.

16. N Wicks, JW Hutchinson, Int. J. Solids Struct. 38 (2001) 5165-5183.

17. N Wicks, JW Hutchinson, Mech. Mater. 36 (2004) 739-751.

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18. FW Zok, SA Waltner, Z Wei, HJ Rathbun, RM McMeeking, AG Evans, Int. J. Solids Struct. 41 (2004) 6249-

6271.

19. J Whittenhauer, B Norris, J. Mater. 42 (1990) 36-41.

20. RW Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 4th ed., John Wiley & Sons,

New York, 1996.

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2 Literature Review This chapter provides a review of cellular materials. Section 2.1 reviews architecture

classifications for micro-trusses. Section 2.2 surveys micro-truss fabrication processes, as well

as material systems typically used in their fabrication. Section 2.3 then reviews the

microstructural strengthening mechanisms used in previous studies of cellular materials. This

section identifies a gap in the literature: neither strengthening by work hardening, nor grain size

refinement have previously been used as microstructural design tools to strengthen the load-

supporting ligaments of cellular materials.

2.1 Architecture Classification

The effective properties of cellular materials depend heavily on the structural

arrangement of their internal members, e.g. [1]. While architecturally simpler than their

conventional stochastic counterparts, there is still no single unifying method to classify the

architecture of periodic cellular materials.

Maxwell’s stability criterion [2] is the classical algebraic formulation to determine a

statically and kinematically determinate arrangement of struts. The condition for a three-

dimensional pin-jointed frame of b struts and j joints to be considered a rigid frame is

63 −= jb (2.1.1)

Tetrahedral (b = 6, j = 4) and octahedral/pyramidal (b = 12, j = 6) unit arrangements are among

examples which meet this criterion (Figure 2.1.1). On the other hand, foam materials are often

modeled as having tetrakaidecahedral (b = 36, j = 24) units, e.g. [3-4], which do not fit

Maxwell’s stability criterion, i.e. b < 3j – 6.

Figure 2.1.1. Examples of polyhedral cells which can make up a periodic cellular metal. Also shown are their evaluations against the Maxwell stability criterion. Figure from Ashby [1].

11

Since Maxwell’s paper, general Maxwell criteria have been proposed based on states of

self-stress and mechanisms for a of a single unit cell [5,6], and an arrangement of repeating unit

cells, e.g. [7]. Some of the more recent studies have also defined cellular architectures in terms

of bending-dominated mechanisms and stretching-dominated structures, e.g. [1,7,8]. In

conventional foams, for example, an external load is resolved transverse to the stochastic

internal load-supporting ligaments, thereby resulting in bending-dominated deformation. On the

other hand, in micro-truss PCMs an external load can be resolved axially along the supporting

ligaments, which demonstrates stretch-dominated deformation. As the load-bearing capacity of a

ligament is greater in (axial) stretching than in (transverse) bending, this results in trusses

having a greater weight-specific load-bearing potential compared to conventional metal foams,

e.g. [1,8-10] (Figure 2.1.2).

Figure 2.1.2. Example material property space map comparing compressive strengths between bending-dominated and stretching-dominated architectures. Figure from Ashby [1].

In some cases, however, other researchers, e.g. [10] have defined micro-trusses to

include bending-dominated architectures such as diamond textile, diamond collinear, and square

collinear. In the present study, the term micro-truss is reserved for stretching-dominated

architectures. A somewhat different approach considers the flow patterns possible within the

PCM. In this perspective PCMs possess periodic, fully open-cell, three-dimensional easy flow.

In cases where the ligaments are themselves hollow, e.g. [11], three-dimensional easy flow can

be accompanied by closed-cell, one-dimensional flow through the ligaments themselves.

12

Another classification system considers PCMs in crystallographic terms. For example,

Hyun and Torquato [12] have described the Kagomé architecture (Figure 1.1.3) using unit-cell

and Bravais lattice analogies; Evans et al. [13] and Mines [14] have borrowed face-centered

cubic (FCC) and body-centered cubic (BCC) crystallographic terminology to describe the

architectures of PCMs; Jaconsen et al. [15] have used n-fold symmetry notation to help define

different types of polymer micro-truss architectures. These analogies consider both a point

group basis (i.e. the unit cell architecture) as well as lattice type (i.e. the periodic arrangement of

unit cells). Such analogies become particularly useful when characterizing multilayer PCM

structures, e.g. [14].

2.1.1 References 1. MF Ashby, Philos. Trans. R. Soc. A 364 (2006) 15-30.

2. JC Maxwell, Philos. Mag. 27 (1864) 294-299.

3. HX Zhu, JF Knott, NJ Mills, J. Mech. Phys. Solids 45 (1997) 319-343.

4. YW Kwon, RE Cooke, C Park, Mater. Sci. Eng. A 343 (2003) 63-70.

5. CR Calladine, Int. J. Solids Struct. 14 (1978) 161-172.

6. CR Calladine, Theory of Shell Structures, Cambridge University Press, Cambridge, 1983.

7. VS Deshpande, MF Ashby, NA Fleck, Acta Mater. 49 (2001) 1035-1040.


9. MF Ashby, YJM Bréchet, Acta Mater. 51 (2003) 5801-5821.


11. DT Queheillalt, HNG Wadley, Acta Mater. 53 (2005) 303-313.

12. S Hyun, S. Torquato, J. Mater. Res. 17 (2002) 137-144.

13. AG Evans, JW Hutchinson, NA Fleck, MF Ashby, HNG Wadley, Prog. Mater. Sci. 46 (2001) 309-327.

14. RAW Mines, Strain 44 (2008) 71-83.

15. AJ Jacobsen, W Barvosa-Carter, S Nutt, Acta Mater. 56 (2008) 2540-2548.

13

2.2 Fabrication Methods and Parent Materials

Metal micro-truss PCMs have been fabricated using a variety of techniques. Reviews

have been conducted recently by Wadley [1] and Mines [2]. Most fabrication techniques can be

grouped into three main types: casting, textile lay-up, and deformation forming.

2.2.1 Casting Methods

Figure 2.2.1 illustrates an investment casting approach, which produces the sandwich

panel in three steps.

Figure 2.2.1. Schematic of the investment casting fabrication method producing a micro-truss sandwich panel. Figure from Chiras et al. [5].

First, a polymeric pattern template is produced in the desired sandwich panel architecture. This

is often achieved using rapid prototyping methods. Second, this template is coated with a

ceramic shell. Third, molten metal is run inside the ceramic shell, melting and displacing the

polymer, casting the sandwich panel in a single piece. Benefits stem from rapid prototyping

design flexibility from the ligament to the panel as a whole. However, these advantages come

with trade-offs. For example, only alloy systems with high castability are suitable. Examples of

alloy systems used for cast micro-truss PCMs (and the resultant core relative densities) are given

in Table 2.2.1.

14

Table 2.2.1. Recent examples of PCM topologies, alloy systems, and core densities fabricated by investment casting.

Core Density Topology Alloy Totalb (Mg/m3) Relativea (%)

Reference

Tetrahedral Al-7Si-0.3Mg (LM25) 0.18 6.8 [3] Tetrahedral Cu-4Si-14Zn (MB1) 0.57 6.8 [3] Pyramidal Al-4Si-0.2Fe (A443) 0.15 5.6 [4] Tetrahedral Cu-2Be 0.20 2.3 [5] Pyramidal Ni-based superalloy (IN718) 0.46 5.6 [6] Kagomé Cu-2Be 0.18 2 [7] Pyramidal Al-3Mg-1Si-1Fe-1Mn (A516.1) 0.15 5.6 [8] Pyramidal Al-8Mg-1.8Fe-0.35Si (A518.0) 0.15 5.6 [8] Pyramidal Al-7Si-0.4Mg (A356) 0.15 5.6 [8] Pyramidal Ti-6Al-4V 0.25 5.6 [9] Pyramidal Ti-6Al-2Sn-4Zr-2Mo 0.26 5.6 [10] Pyramidal Mg-9Al-1Zn (AZ91) 0.19-0.29 11-16 [11] a – calculated using the relative density relationships in [3] unless explicitly stated b – calculated using the theoretical density of the alloy

Additional alloy systems are possible as well: Renauld et al. [12] have tested commercial

cast pyramidal micro-truss structures (trade name Lattice Block Material, core relative densities

between approximately 10-30%) made using, in addition to the common aluminum casting alloy

A356, also C95200 (copper), Hastelloy-X (nickel), Stellite 6 (cobalt), and 17-4 stainless steel

alloys. However, it is important to note that these cellular materials all have relatively high core

relative densities, e.g. greater than 5% [1]; casting smaller relative densities can result in a

number of casting defects, shown in Figure 2.2.2, from [5].

Figure 2.2.2. Casting defects existing in a micro-truss PCM fabricated by investment casting, having a core relative density of 2.3%. Figures from Chiras et al. [5].

15

2.2.2 Textile Lay-Up

PCM cores can also be produced using textiles. One example is shown in Figure 2.2.3;

by stacking woven metal screens upon each other in an alternating pattern, a periodic woven

core can be produced.

Figure 2.2.3. Textile lay-up fabrication method for PCMs using woven sheets as the precursor (left) and possible configurations (right). Figures from Tian et al. [14].

The primary benefit to this method is its relatively simple construction. However, as this method

depends on the availability and brazeability of commercial precursors, a somewhat limited

material choice is available; in most cases, woven copper [13,14], 304 stainless steel [13], 316

stainless steel [15], nichrome [16], and aluminum alloy 6061 [17] have been used. Moreover, it

is difficult to create cellular sandwich structures having relative densities less than about 10%

[17,18]. On the other hand, a more lightweight approach has been studied by Queheillalt and

Wadley [19] of stacking 304 stainless steel rods (solid or hollow) in collinear periodic

arrangements (see Figure 2.2.4), achieving relative densities as low as 3%. Moongkhamklang et

al. [20] fabricated a similar collinear PCM by stacking composite filaments of carbon-

fibre/silicon-carbide/Ti-6Al-4V, achieving relative densities as low as 5%.

16

Figure 2.2.4. Textile lay-up fabrication method for PCMs using solid or hollow rod precursors (left) and possible configurations (right). Figures from Queheillalt and Wadley [19].

Although simple to fabricate, stacking-type architectures have limited performance

because they do not satisfy Maxwell’s stability criterion. On the other hand, a number of studies

have developed textile lay-up-type methods to produce stretch-dominated micro-trusses and

sandwich panels. Fabrication of these types of structures has existed at least since the 1960’s,

e.g. [21,22], and some approaches have been patented, e.g. [23-26]. For example, Wallach and

Gibson [26] have described a method whereby wires are bent into a triangle-wave shape, then

cross-woven to produce a three-dimensional pyramidal micro-truss (Figure 2.2.5a). In a similar

fashion, Lim and Kang [27] have fabricated tetrahedral and Kagome truss cores by bending

single wires of low-carbon steel into a periodic triangle-wave pattern, then stacking them in a

junction-type arrangement (Figure 2.2.5b). Kang and coworkers [28-30] have also developed a

wire-woven bulk Kagome (WBK) structure (Figure 2.2.5c), which is assembled by first

applying torsion to wire precursors to give them a helical shape; second, placing the precursors

into a pre-set holding apparatus; third, weaving additional precursors individually into the

micro-truss matrix; and fourth, conducting a final brazing treatment to bond all wires. Similar to

the study of Queheillalt and Wadley [19], either solid or hollow wire precursors may be used to

reduce the total weight and increase the weight efficiency. Low relative-density WBK trusses

have been produced, e.g. from SAE 9254 spring steel and 6061 aluminum alloy (both 3.5%,

17

[29]) and 304 stainless steel (1.6 and 2.2%, [30]). Other studies have inserted 304 stainless steel

metal tubes through a face-sheet template to achieve a pyramidal micro-truss [1,31] (Figure

2.2.5d).

Figure 2.2.5. Textile fabrication methods to produce micro-truss PCMs: (a) from Wallach and Gibson [25]; (b) from Lim and Kang [26]; (c) from Lee et al. [27]; (d) from Pingle et al. [30].

2.2.3 Deformation-Forming

Micro-truss PCM architectures have been produced by plastically deforming perforated

sheet metal precursors using the bending brake (see Figure 2.2.6) and perforation-stretching

methods (see Figure 2.2.7).

18

Figure 2.2.6. Schematic of the bending brake method. Figure from Wadley [1].

Figure 2.2.7. Apparatus used and set-up for the perforation-stretching method (left) and formed tetrahedral unit cells (right). Figures from Sypeck and Wadley [43].

The first method passes sheet metal of elongated perforations through intersecting spur gears or

V-die mated pairs to alternately bend the perforated sheet metal. The second method places

planar, periodically-perforated sheet metal in an alternating-pin press. The pins apply force out-

of-plane, plastically deforming the sheet metal into a truss-like array of struts (i.e. metal

supports) and nodal peaks (i.e. strut intersections). In both cases, stacking multiple deformed

layers creates a multi-layer PCM structure. The bending brake method is simple, based on well-

known sheet forming processes, and conserves the ligament metal strut cross-section. However,

specific brake dies and perforation sizes are needed for different truss heights and strut angles.

Generally, the range of unit cells depends on the shape and size of the perforated sheet. As well,

the perforated precursors are limited to those with sufficient ductility, capable of achieving a

small inside radius of curvature during bending, see [32]. On the other hand, the perforation-

stretching method requires only a single press to produce PCM structures with a continuous

range of truss angles. In addition, the PCM nodes experience only vertical displacement during

fabrication. In general, this process possesses similarities to conventional stretch-bending. The

19

disadvantage is that the perforation-stretching process is more complex and materials are limited

to those with high formability. However, it has been established in conventional forming

processes that material limitations may be salvaged with iterative heat treatments or elevated

processing temperatures, e.g. [33].

Using the bending-brake approach, a number of micro-trusses have been fabricated.

Tetrahedral and pyramidal PCM architectures have been formed using elongated hexagon and

diamond perforations, respectively. Of these, only a limited set of precursor materials has been

used, e.g. aluminum alloy 6061 [34-37], stainless steel 304 [37-40], and titanium alloy Ti-6Al-

4V [41]. In these studies, most cell sizes and relative densities were on the orders of ~1 cm and

~2-8%, respectively, e.g. [34], although some reached relative densities as low as 1% [41].

Precursors have also been made without the associated waste from the perforation step.

For example, Brittain et al. [42] electroplated a silver grid consisting of holes shaped as

elongated diamonds, which was then formed using a bending-brake method into a pyramidal

meso-scale micro-truss (unit cell size ~1 mm) with 12% relative density. Alternatively, Kooistra

and Wadley [43] have used flattened expanded metal precursors to produce a pyramidal micro-

truss (unit cell size ~1 cm) with 5.7% relative density. On the same size scale, Queheillalt and

Wadley [44] have used the bending-brake approach to fabricate a tetrahedral micro-truss made

of hollow stainless steel 304 tubes; the relative density in this case was 2.8%.

The perforation-stretching approach has been used less frequently to produce micro-truss

PCMs. Prior to this thesis, only the tetrahedral type of architecture was successfully formed and

tested using one precursor material, stainless steel 304 [45,46]. However, Sypeck [18] suggested

attempts have been made to use this fabrication method with aluminum alloys, but that forming

was difficult and lead to early forming failure.

2.2.4 Additional Methods

Mines [2] has reviewed recent advances in selective laser melting (SLM) to fabricate

PCMs with a range of architectures. This process uses a laser to selectively melt and join metal

powder particles in a way similar to rapid prototyping. The laser moves layer-by-layer, heating

new particles and joining them to the underlying scaffold, thereby producing a monolithic

material (Figure 2.2.8). However, like investment casting, not all alloys systems can be used. To

be a suitable parent material, the powder precursor must meet the following requirements [47]:

it can be melted by laser heating, the average particle size is below ~40 μm, and it has suitable

flow characteristics. The precursor must also have appropriate thermal expansion and heat

20

conductivity coefficients [48]. Powder materials considered to date include stainless steel,

titanium, and nickel-based super-alloys [2,47], although in most SLM studies stainless steel 316

has been the precursor of choice, e.g. [47-49].

Figure 2.2.8. PCM fabricated by selective laser melting (left) and its unit cell (right). Figures from Mines [2].

Alternatively, micro-trusses can be fabricated by extrusion and electro-discharge

machining (EDM). In a recent study [50], a two-step process was employed (Figure 2.2.9). First,

a prismatic sandwich panel structure was extruded from a solid billet of aluminum alloy 6061,

creating a one-dimensional open-cell periodic cellular metal. Second, regular sections were

removed using EDM, cutting in the direction perpendicular to the extrusion; this produced a

three-dimensional open cellular structure with a 6.5% relative density. Again, the tradeoff for

this method is the range of alloy systems that can be used; for example, the authors in [50]

suggest an alloy system that can be extruded easily. Where extrusion is not possible, EDM may

be used to fabricate the entire micro-truss [50].

Figure 2.2.9. A micro-truss PCM fabricated using a combined extrusion and electro-discharge machining (EDM) approach. Figures from Queheillalt et al. [48].

21

2.2.5 References 1. HNG Wadley, Philos. Trans. R. Soc. A 364 (2006) 31-68.

2. RAW Mines, Strain 44 (2008) 71-83.

3. VS Deshpande, NA Fleck, Int. J. Solids Struct. 38 (2001) 6275-6305.

4. JC Wallach, LJ Gibson, Int. J. Solids Struct. 38 (2001) 7181-7196.

5. S Chiras, DR Mumm, AG Evans, N Wicks, JW Hutchinson, K Dharmasena, HNG Wadley, S Fichter, Int. J.

Solids Struct. 39 (2002) 4093-4115.

6. DL Krause, JD Whittenberger, PT Kantzos, MG Hebsur, Mechanical Testing of IN718 Lattice Block

Structures, NASA TM-2002-211325, 2002.

7. J Wang, AG Evans, K Dharmasena, HNG Wadley, Int. J. Solids Struct. 40 (2003) 6981-6988.

8. J Zhou, P Shrotriya, WO Soboyejo, Mech. Mater. 36 (2004) 723-737.

9. Q Li, EY Chen, DR Bice, DC Dunand, Metall. Mater. Trans. A 39 (2008) 441-449.

10. Q Li, EY Chen, DR Bice, DC Dunand, Adv. Eng. Mater. 10 (2008) 939-942.

11. SSC Ho, Lost Foam Casting of Periodic Cellular Materials with Aluminum and Magnesium Alloys, M.A.Sc.

Thesis, University of Toronto, Toronto, 2009.

12. ML Renauld, AF Giamei, MS Thompson, J Priluck, in: Porous and Cellular Materials for Structural

Applications, DS Schwartz, DS Shih, AG Evans, HNG Wadley (Eds.), Materials Research Society,

Warrendale, 1998, pp. 109-117.

13. DJ Sypeck, in: Processing and Properties of Lightweight Cellular Metals and Structures, AK Ghosh, TH

Sanders, TD Claar (Eds.), TMS, Philadelphia, 2002, pp. 35-45.

14. J Tian, T Kim, TJ Lu, HP Hodson, DT Queheillalt, DJ Sypeck, HNG Wadley, Int. J. Heat Mass Transfer 47

(2004) 3171-3186.

15. M Zupan, VS Deshpande, NA Fleck, Eur. J. Mech. A 23 (2004) 411-421.

16. DJ Sypeck, HNG Wadley, J. Mater. Res. 16 (2001) 890-897.

17. DJ Sypeck, Metall. Mater. Trans. B 36 (2005) 125-131.

18. DJ Sypeck, Appl. Compos. Mater. 12 (2005) 229-246.


20. P Moongkhamklang, DM Elzey, HNG Wadley, Composites Part A 39 (2008) 176-187.

21. JC Ladd, DL Allie, Prods. Finish. (Cincinnati, OH, US) 26(4) (1962) 50-56.

22. JC Ladd, DL Allie, in: Symposium on Electroforming – Applications, Uses, and Properties of Electroformed

Metals (ASTM STP 318), ASTM, Philadelphia, 1962, pp. 124-135.

23. JR Hale, US Patent 3,298,402, Jan. 17, 1967.

24. RF Artzer, US Patent 4,291,732, Sept. 29, 1981.

25. RF Artzer, US Patent 4,336,676, June 29, 1982.

26. JC Wallach, LJ Gibson, US Patent 6,644,535, Nov. 11, 2003.

27. JH Lim, KJ Kang, Int. J. Solids Struct. 43 (2006) 5228-5246.

28. YH Lee, BK Lee, I Jeon, KJ Kang, Acta Mater. 55 (2007) 6084-6094.

29. BK Lee, KJ Kang, Adv. Eng. Mater. 10 (2008) 835-839.

30. BK Lee, KJ Kang, Scripta Mater. 60 (2009) 391-394.

22

31. SM Pingle, NA Fleck, HNG Wadley, in: 13th European Conference on Composite Materials, Stockholm, 2008.

32. ASTM E290-04, American Society for Testing and Materials, Philadelphia, 2004.

33. GE Dieter, HA Kuhn, SL Semiatin, Handbook of Workability and Process Design, ASM International,

Materials Park, 2003.

34. GW Kooistra, VS Deshpande, HNG Wadley, Acta Mater. 52 (2004) 4229-4237.

35. X Tang, V Prakash, JJ Lewandowski, GW Kooistra, HNG Wadley, Acta Mater. 55 (2007) 2829-2840.

36. GW Kooistra, DT Queheillalt, HNG Wadley, Mater. Sci. Eng. A 472 (2008) 242-250.

37. CJ Yungwirth, HNG Wadley, J O’Connor, A Zakraysek, VS Deshpande, Int. J. Impact Eng. 35 (2008) 920-

936.

38. R Biagi, H Bart-Smith, Int. J. Solids Struct. 44 (2007) 4690-4706.

39. CJ Yungwirth, DD Radford, M Aronson, HNG Wadley, Composites Part B 39 (2008) 556-569.

40. HNG Wadley, KP Dharmasena, Y Chen, P Dudt, K Knight, R Charette, K Kiddy, Int. J. Impact Eng. 35 (2008)

1102-1114.

41. DT Queheillalt, HNG Wadley, Mater. Des. 30 (2009) 1966-1975.

42. ST Brittain, Y Sugimura, OJA Schueller, AG Evans, GM Whitesides, J Microelectromech. Syst. 10 (2001)

113-120.

43. GW Kooistra, HNG Wadley, Mater. Des. 28 (2007) 507-514.

44. DT Queheillalt, HNG Wadley, Mater. Sci. Eng. A 397 (2005) 132-137.

45. DJ Sypeck, HNG Wadley, Adv. Eng. Mater. 4 (2002) 759-764.

46. HJ Rathbun, Z Wei, MY He, FW Zok, AG Evans, DJ Sypeck, HNG Wadley, J. Appl. Mech. 71 (2004) 368-

374.

47. S McKown, Y Shen, WK Brookes, CJ Sutcliffe, WJ Cantwell, GS Langdon, GN Nurick, MD Theobald, Int. J.

Impact Eng. 35 (2008) 795-810.

48. O Rehme, C Emmelmann, Proceedings of SPIE 6107 (2006) 192-203.

49. M Santorinaios, W Brookes, CJ Sutcliffe, RAW Mines, WIT Trans. Built Env. 85 (2006) 481-490.

50. DT Queheillalt, Y Murty, HNG Wadley, Scripta Mater. 58 (2008) 76-79.

23

2.3 Microstructural Strengthening of Cellular Metals

While considerable effort has been spent considering the role of architecture on the

overall mechanical properties of cellular materials, comparatively little attention has been given

to the role of microstructural strengthening mechanisms.

2.3.1 Microstructure Strengthening Mechanisms in Foams

Conventional foam microstructures are generally limited by the available cast alloys, the

foaming agents, and the inclusions added to the melt. Cast foams are primarily based on Al, Mg,

Ni, or Ti alloys, but can also include Zn, Fe, Pb, Au, Cu, and Co alloys, e.g. [1,2]. In some

cases, alloys are chosen in order to apply age-hardening after foaming to increase the yield

strength of the foam ligaments, e.g. [3-9]. The foaming agent can also influence the resultant

foam microstructure. For example, a common foaming agent TiH2 may be replaced with CaCO3,

which adds Ca to the Al-rich eutectic structure of Al-Zn-Mg alloys and improves the uniformity

and strength of metal foam deformation, e.g. [10]. The foam strength can also be increased by

including dispersions or particle reinforcements into the melt. Particle additions to the melt

include SiC (e.g. [7,11-13]), Y2O3 (e.g. [14]), Al2O3/SiC (e.g. [15]), or reinforcing fibres (e.g.

[16-18]). Figure 2.3.1 displays an example of two commercially-available foams strengthened

using Si precipitates and SiC particle inclusions.

Figure 2.3.1. Microstructures of Alcan (left) and Alporas (right) foam ligaments strengthened by Si precipitates and SiC particle inclusions. Figures from Simone and Gibson [13].

24

Methods for strengthening metal foams by in-situ oxide particle formation has also been

proposed [19,20], but the specific oxygen-containing agent was not disclosed. Additions to the

melt can also produce intermetallic phases which increase the strength of the foam, e.g. steel

spheres in an Al-melt [21]. Some methods have also proposed adding scrap foamed aluminum

to increase the new foam strength, e.g. [22]. The concept of grain size strengthening in foams is

more complicated. Some studies test as-received foams against foams with similar architectures

given full annealing treatments (Figure 2.3.2), and attribute the differences in strength to a Hall-

Petch relationship, e.g. [23,24].

Figure 2.3.2. Grain size differences between large-grained sintered (left) and small-grained as-received (right) ligaments in an INCO nickel foam. Figures from Carpenter et al. [26].

Other studies propose more complicated strengthening mechanisms, combining aspects of grain

size, solid-solution strengthening, dislocations and nanopore networks, to compare properties of

foams having similar architectures but different microstructures, e.g. [25,26]. To the author’s

knowledge, only a single case exists where work-hardening was introduced as a strengthening

mechanism in cellular materials: Margevicius et al. [27] showed that mechanical working (i.e.

rolling) of age-hardened aluminum foams can improve their bending strength, so long as the

rolling temperature remains low enough to prevent dislocation recovery.

25

2.3.2 Microstructure Strengthening Mechanisms in Micro-Truss PCMs

While significant advances have been made in micro-trusses by incorporating stretch-

dominated architecture design, comparatively little attention has been given to PCM

microstructural strengthening mechanisms. As with metallic foams, the range of microstructures

is typically defined by the initial alloy system of the PCM. Some commercial micro-truss

sandwich panels have been produced using standard casting alloys based on Al, Cu, Co, Ti, Fe,

and Ni (e.g. JAMCORP [28,29]). In deformation-forming approaches, core joining processes

(e.g. brazing) expose the PCM core to high-temperatures, thereby removing much of the prior

microstructure developed during deformation forming. As a result, precipitation hardening is the

typical strengthening mechanism induced into the final sandwich panel, e.g. in a cast Cu-2%Be

alloy PCM [30] (Figure 2.3.3) and in a wrought AA6061 sheet formed into a PCM [31].

Figure 2.3.3. The dendritic microstructure of an as-cast Cu-2%Be micro-truss ligament. Figure from Chiras et al. [30].

However, there are very few studies comparing the performance of micro-trusses having the

same architecture but different microstructures. For example, Kooistra et al. [31] compared

compressive strengths of peak age-hardened and annealed micro-trusses formed with a number

of identical architectures; as expected, the age-hardening markedly improved the PCM

compressive strength, but there was little consideration of the microstructural mechanism by

which strengthening occurred. In another case, Zhou et al. [32] tested the compressive

performance of identical micro-truss PCMs architectures made from different standard Al-Si-

Mg-based casting alloys. However, all PCMs were tested only in their as-cast condition, which

allowed only a general investigation of the influence of microstructural strengthening

mechanisms. Li et al. [33] have similarly compared the compressive properties of micro-truss

26

PCMs with identical architectures from standard Ti-based casting alloys. The alloys were

selected on the basis of castability and a general high as-cast strength but with limited

consideration for the specific microstructure strengthening mechanisms. Therefore, in order to

fully explore and study the effects of microstructural strengthening, a fundamentally different

approach is needed.

2.3.3 References 1. MF Ashby, A Evans, NA Fleck, LJ Gibson, JW Hutchinson, HNG Wadley, Metal Foams – A Design Guide,


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8. SM Lee, K Ryu, Y Kwon, J-G Kim, W-S Cho, N-H Cho, CM Whang, Y-C Yoo, Mater. Sci. Forum 475-479

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11. I Jin, LD Kenny, H Sang, UT Patent 4,973,358, Nov. 1990.

12. O Prakash, H Sang, JD Embury, Mater. Sci. Eng. A 199 (1995) 195-203.

13. AE Simone, LJ Gibson, Acta Mater. 46 (1998) 3109-3123.

14. NO Zhao, B Jiang, XW Du, JJ Li, CS Shi, WX Zhao, Mater. Lett. 60 (2006) 1665-1668.

15. N Altinkok, J. Compos. Mater. 38 (2004) 1533-1543.

16. J Bjorksten, EJ Rock, US Patent 3,707,367, Dec. 1972.

17. LM Niebylski, CP Jarema, TE Lee, US Patent 3,940,262, Feb. 1976.

18. V Shapovalov, MRS Bull. 19 (2004) 24-28.

19. CB Berry Jr, US Patent 3,669,654, June 1972.

20. CB Berry Jr, RJ Fanning, US Patent 3,705,030, Dec. 1972.

21. LJ Vendra, A Rabiei, Mater. Sci. Eng. A 465 (2007) 59-67.

22. CB Berry Jr, AR Valdo, US Patent 3,725,037, Apr. 1973.

23. V Goussery, Y Bienvenu, S Forest, AF Gourgues, C Colin, JD Bartout, Adv. Eng. Mater. 6 (2004) 432-439.

24. K Lee, J Lewandowski, Mater. Sci. Technol. 21 (2005) 1355-1358.

25. OB Olurin, DS Wilkinson, GC Weatherly, V Paserin, J Shu, Compos. Sci. Technol. 63 (2003) 2317-2329.

26. GJC Carpenter, ZS Wronski, MW Phaneuf, Mater. Sci. Technol. 20 (2004) 1421-1426.

27. RW Margevicius, PW Stanek, LA Jacobson, Mater. Res. Soc. Symp. Proc. 521 (1998) 65-70.

27

28. ML Renauld, AF Giamei, MS Thompson, J Priluck, in: Porous and Cellular Materials for Structural

Applications, DS Schwartz, HNG Wadley, AG Evans (Eds.), Materials Research Society, Warrendale, 1998,

pp. 109-117.



Solids Struct. 39 (2002) 4093-4115.



33. Q Li, EY Chen, DR Bice, DC Dunand, Adv. Eng. Mater. 10 (2008) 939-942.

28

3 Micro-Truss PCM Fabrication by Perforation-Stretching This chapter focuses on the work done by the author to characterize micro-truss

fabrication by the perforation-stretching method. Section 3.1 is an initial study of perforation-

stretching fabrication. For the first time, the forming forces involved during fabrication were

used to model the onset and limit of useful plastic deformation. This section was published in

Metallurgical and Materials Transactions A [BA Bouwhuis, GD Hibbard, Failure mechanisms

during periodic cellular metal fabrication by perforation stretching, Metall. Mater. Trans. A 39

(2008) 3027-3033], reproduced with kind permission of Springer Science and Business Media.

Section 3.2 develops further an analytical approach to model the forming curve as a

whole and predict the accessible range of architectures, using the uniaxial tensile properties of

the starting precursor strut. This section is one half of a paper [BA Bouwhuis, GD Hibbard,

Relative significance of in-situ work-hardening in deformation-formed micro-truss materials],

published in Materials Science and Engineering A (doi:10.1016/j.msea.2009.08.015).

3.1 Failure Mechanisms during PCM Fabrication

3.1.1 Introduction

The range of perforation stretched PCM architectures spans from the undeformed

precursor to the maximum displacement depth of the press before sheet failure [1,2]. In

principle, any deformable precursor can be used to fabricate PCMs through this process.

However, the range of accessible PCM architectures will depend on the precursor’s formability.

For example, perforation stretching has been successful for comparatively formable metals such

as stainless steel 304 [3]. In contrast, early perforation stretching forming failure in aluminum

alloy PCMs limits the range of attainable architectures [1]. However, aluminum alloys are

particularly attractive in order to minimize the overall PCM mass. In order to apply the

perforation-stretching method to aluminum alloys and other materials with comparatively lower

formability, the failure mechanisms during fabrication need to be understood. The present study

is an initial examination of the forming and failure mechanisms associated with perforation

stretching as influenced by precursor microstructure and geometry.

29

3.1.2 Experimental Details

Pyramidal PCMs were fabricated from various thickness (to) square-punched aluminum

3003-H14 (half-hard) sheet, obtained from Woven Metal Products, Inc. (Alvin, TX). This alloy

is widely used in industry because of its low cost, ability to work harden, and good formability

[4]. Standard gauge thicknesses were tested (to in mm): 20 (0.80 ± 0.01), 18 (0.96 ± 0.01), 16

(1.26 ± 0.02) and 14 (1.56 ± 0.02) to investigate the effects of sheet thickness on formability.

The 90.7 mm2 (internal edge length = 9.5 mm) perforations were punched on a two-dimensional

square lattice of unit cell size 12.7 × 12.7 mm, resulting in an open area fraction φ of 0.56. The

resulting structure was a series of 4-strut nodes with arm cross-sections of width wo = 3.2 mm

and thickness to. The average microhardness (HV, 0.98 N, 20 measurements) of the starting

precursor sheets was 55 ± 3 (to = 0.80 mm), 52 ± 2 (to = 0.96 mm), 58 ± 4 (to = 1.26 mm), and

55 ± 3 (to = 1.56 mm); these values are typical of the H14 (half-hard) temper [4]. To survey the

effects of precursor microstructure, the sheets were tested as-received and following annealing

treatments of 1 hour at 250˚C, 350˚C, 415˚C, and 600˚C; these annealing temperatures were

chosen to span a range of hardened, recovered, and recrystallized microstructures for the

AA3003 alloy [5].

The forming load was applied using a 50-kN capacity Shimadzu screw-driven

compression platform. Deformation was conducted at a constant linear crosshead displacement

rate of 5 mm/min. The pins of the perforation-stretching die had a 3.2 mm diameter and a fillet

radius of 1 mm at the point of sheet metal contact (Figure 3.1.1a). Forming force (FForm)-

displacement (dForm) curves were obtained to track the deformation profile with out-of-plane pin

displacement (Figure 3.1.1b). The resultant PCM architecture was characterized by the relative

density ρR (i.e. PCM density divided by that of the fully dense material) and truss angle ω (i.e.

angle between the horizontal and inclined strut). Three samples were tested for each of the

forming conditions and all tests were conducted at room temperature. Microstructural

characterization of the as-fabricated structures was performed on cross sections taken along the

transition from strut to node. The sections were prepared using standard metallographic

practices (anodized using Barker’s reagent) and imaged using polarized light. Fracture surfaces

were characterized by scanning electron microscopy (SEM).

30

Figure 3.1.1. Schematic diagrams of a pyramidal unit formed by perforation stretching (a) prior to and (b) after forming. (c) Also shown is a schematic diagram of the nonstandard tensile coupon cut from the perforated sheet precursor.

In order to directly measure the mechanical properties of the struts in the perforated

metal precursors, non-standard (miniature) coupons having the geometry shown in Figure 3.1.1c

were tested in uniaxial tension. The reduced section length was the internal edge length of a

single perforation, 9.5 mm. A similar approach has been used to determine tensile properties of

single ligaments in metal foams [6]. In the present study, tensile testing was conducted using a

10-kN capacity bench-top Shimadzu screw-driven test platform at a crosshead displacement rate

of 1 mm/min. Machine compliance effects complicate the displacement-based estimates, e.g.

[7], but are not expected to affect the measured force-based properties such as yield strength and

ultimate tensile strength.

3.1.3 Results and Discussion

3.1.3.1 Perforation Stretching Force-Displacement Profile

Figure 3.1.2 presents typical tensile test curves (in force-displacement and equivalent

stress-strain) for the to = 1.26 mm starting sheet in the as-received condition and after 1 hour

annealing treatments at 250˚C, 350˚C, 415˚C, and 600˚C. A typical forming curve for an

aluminum alloy 3003-H14 precursor (to = 0.96 mm) is displayed in Figure 3.1.3.

31

Figure 3.1.2. Tensile force-displacement diagram (with nominal stress-strain representation) of the nonstandard coupons shown in Figure 3.1.1c. Coupons were tested in the as-received condition and after 1 h annealing treatments at 250˚C, 350˚C, 415˚C, and 600˚C (curves shown are for the to = 1.26 mm starting sheet).

Figure 3.1.3. Perforation-stretching force-displacement diagram for a 3003-H14 precursor (to = 0.96 mm), showing the plasticity onset force FO, maximum forming force FM, and maximum stretch displacement dM. Inset shows a typical PCM sample after forming.

Following a perforation-stretching pin displacement dForm, the PCM relative density can be

approximated by [2]:

( )φρ −+

= 1, FormNodef

oR dt

t (3.1.1)

The variable tf,Node denotes the final sheet thickness at the nodes after forming. The truss angle ω

can be approximated by [2]:

⎟⎠⎞

⎜⎝⎛=

Ld Formarctanω (3.1.2)

32

where L is the center-to-center distance between oppositely traveling press pins (L = 12.7 mm).

The maximum displacement dM attainable before sheet failure determines the range of PCM

architectures that can be fabricated from a given precursor; the minimum achievable relative

density is ρR,Min = ρR(dM), while the maximum truss angle is ωM = ω(dM).

The forming forces contain the deformation history of the precursor material and can be

used to study the forming and failure mechanisms. The onset force FO indicates the start of

plastic deformation, and can be measured using a method similar to ASTM F 1575 [8] by taking

FO as the intersection of the forming curve with an offset line parallel to the initial slope of

elastic bending. An offset of 5% of the starting sheet thickness was chosen (after ASTM F 1575)

and ranged from 0.04 mm (for to = 0.80 mm) to 0.08 mm (for to = 1.56 mm). The forming force

continued to increase until one or more of the struts had failed. The end of useful plastic

deformation (and the limit of formability) was defined by the maximum force FM and its

corresponding maximum displacement dM (Figure 3.1.3).

3.1.3.2 Onset of Plasticity

Figure 3.1.4 summarizes the measured FO for the four sheet thicknesses and range of

pre-forming annealing temperatures. Overall, FO spans more than an order of magnitude from

2.50 ± 0.05 kN (to = 1.56 mm, as-received) to 0.20 ± 0.01 kN (to = 0.80 mm, 600˚C for 1 hour)

and generally decreases with increasing annealing temperature. However, the rate at which FO

decreases as well as the temperature at which the transition is complete differs for each sheet

thickness.

Figure 3.1.4. (left) Plasticity onset force FO and (right) precursor yield strength σYS as a function of performing anneal temperature. Note that a temperature of 25˚C was used to plot the values for the H14 (as-received) material.

33

This variance can be seen more clearly when the measured precursor σYS values are plotted as a

function of annealing temperature in Figure 3.1.4. The as-received precursors have yield

strengths ranging from approximately 127 MPa to 143 MPa, which is consistent with hardened

AA3003 sheet [4]. On the other hand, the greatest annealing treatment (600˚C, 1 hour) reduces

all measured precursor σYS values to a range between approximately 46 MPa and 51 MPa; this

range is consistent with that of softened AA3003 [4]. The typical strain relief thermal treatment

to return a worked 3003 aluminum alloy to its softened state is 415˚C for 1 hour [4]; however,

some sheets (e.g. to = 1.26 mm) have not reached their fully softened conditions by 415˚C for 1

hour, whereas others (e.g. to = 0.96 mm) have softened after annealing at 350˚C for 1 hour. This

range of treatments to fully anneal the as-received AA3003 sheet precursors is perhaps not

surprising given that recovery and recrystallization during the pre-forming anneal will have been

influenced by the processing history of the particular sheet from cast ingot to wrought alloy

[2,5,9-11].

It was observed during the early stages of forming that the precursor sheet would

plastically deform around the press pins. The mid-span point force at which a pair of fixed-

end/fixed-end beams (i.e. Figure 3.1.1a) begins to plastically deform in bending is given by, e.g.

[12]:

226

82

×⎟⎟⎠

⎞⎜⎜⎝

⎛=

Ltw

F YSoob

σ (3.1.3)

where σYS is the precursor yield strength. The overall force at the onset of plastic bending

becomes FO ∝ νFb, where ν is the number of press pins (for the present case, ν = 12); this gives

the following relationship for the bending force:

YSoYSoo

O tL

twF σξ

σβν 2

2

226

8 =⎥⎥⎦

⎤

⎢⎢⎣

⎡×⎟⎟

⎠

⎞⎜⎜⎝

⎛= (3.1.4)

The coefficient β is a fitting parameter between FO and νFb, whereas ξ represents the product of

all constants. The value in using a conventional beam-bending approach to model the initial

stage of plastic deformation is shown by plotting the experimentally measured FO values against

σYS×to2 in Figure 3.1.5. In agreement with Eq. (3.1.4), a linear fit for all AA3003 precursors is

obtained with a value of ξ = 7.68 ± 0.17 (R2 = 0.974), or β = 1.92 ± 0.04 (using L = 12.7 mm).

Over the range in strut cross-section aspect ratio (from wo/to = 2.0 to wo/to = 4.0) and

microstructures (H14 half-hard temper to fully annealed), the initial stage of plastic deformation

34

is bending-dominated and a simple plastic bending cantilever model is able to account for the

nearly order of magnitude range in FO.

Figure 3.1.5. Plasticity onset force FO plotted as a function of the varying precursor material (yield strength, σYS, measured from nonstandard coupons) and geometric (initial sheet thickness, to) parameters in the plastic bending mode, Eq. (3.1.4).

3.1.3.3 Maximum Forming Force

Figure 3.1.6 displays FM for the range of thicknesses and pre-forming annealing

temperatures. Generally, FM increases with increasing pre-forming anneal temperature. This is

perhaps initially unexpected considering that annealing typically results in softening, decreasing

both σYS (and FO, Figure 3.1.4) and the ultimate tensile strength σUTS (Figure 3.1.6 displays the

measured σUTS values from the precursor tensile coupons).

Figure 3.1.6. (left) Maximum forming force FM and (right) precursor ultimate tensile strength σUTS as a function of the performing anneal temperature. Note that a temperature of 25˚C was used to plot the values for the H14 (as-received) material.

35

The as-received precursors have σUTS ranging from approximately 150 MPa to 162 MPa, which

is consistent with H14 cold-rolled AA3003 sheet [4]. On the other hand, annealing at 600˚C for

1 hour reduces σUTS values to a range between approximately 104 MPa and 109 MPa, consistent

with that of an AA3003 O-temper [4]. Sypeck reported a pin punch through perforation-

stretching failure mechanism for aluminum alloys [1]. The force required to punch a hole

through sheet material is proportional to its ultimate shear strength τ, e.g. [13]. The ultimate

shear strength is generally proportional to the ultimate tensile strength σUTS (see [4] for AA3003

and [14] for other materials). Figure 3.1.7 plots FM against the respective σUTS measured from

the non-standard tensile test coupons. If pin punch through is the dominant failure mechanism,

then FM is expected to increase with σUTS for each sheet thickness. However, the softened

conditions have the highest FM despite having the lowest σUTS, suggesting that forming failure in

the present case is not solely due to a pin punch through mechanism.

Figure 3.1.7. Maximum forming force FM plotted as a function of the ultimate tensile stress σUTS (measured from nonstandard tensile coupons) for the thickest (to = 1.56) and thinnest (to = 0.80 mm) sheet precursors, showing the generally decreasing trend of FM with increasing σUTS.

Figure 3.1.8 shows optical micrographs of cross-sections through the failed regions of

PCMs (to = 1.56 mm) that were formed from the H14 condition and after pre-annealing at 415˚C

for 1 hour.

36

Figure 3.1.8. Optical microscopy cross sections of failed struts (to = 1.56 mm) fabricated from the (a) hardened (as-received) and (b) softened (415˚C, 1 h) state.

Macroscopically, the sheet failure is similar for both cases, occurring near the edge of contact

between the press pins and the metal precursor, and necking is observed at each node. However,

there is a significant difference in the extent of necking (Figure 3.1.8). The H14 precursor fails

with relatively little necking and strain localization can be seen to arise from an unstable shear

band (Figure 3.1.8a). The deformation pattern in the shear band shows grain rotation, which

suggests the presence of tensile stress components [15,16]. In addition, the fracture surface of

the hardened precursor appears to have a generally elongated dimple morphology (Figure

3.1.9a) suggesting influence by shear stresses, e.g. [17]. In contrast, the annealed (415˚C, 1

hour) precursor fails after extensive necking, and the larger, equiaxed grains undergo elongation

37

in and near the necked region (Figure 3.1.8b). Furthermore, the fracture surface has a deep,

equiaxed dimple morphology (Figure 3.1.9b) similar to microvoid coalescence under tensile

loading, e.g. [17]. The fracture surfaces in Figure 3.1.9 mirror those observed in conventional

tensile or stretch forming failure of aluminum sheet [5,15,16,18,19], i.e. instability via shear

banding is dominant for hardened materials whereas necking instability is dominant for softened

materials.

Figure 3.1.9. Fracture surfaces of the failed struts that were fabricated from the (a) hardened (as-received) and (b) softened (415˚C, 1 h) precursors.

The maximum resolved force carrying capacity of a single precursor strut Ft occurs at

the point of tensile plastic instability and can be approximated by

MUTSoot twF ωσ sin××= (3.1.5)

The maximum PCM forming force is then FM = ζFt where ζ = γη (η is the number of struts

stretched), and γ is a constant, expected to be on the order of unity. Figure 3.1.10 displays FM

plotted against Ft for all precursors. The data shows that Eq. (3.1.5) (using γ = 1) is in good

agreement to the perforation stretching FM values. However, the tension model appears to agree

best with the fully-softened precursors, which have fracture surfaces indicative of tensile

instability (Figure 3.1.9b). In contrast, the tension model had the least agreement for the

hardened precursors, which exhibited some evidence of shearing instability (Figure 3.1.9a).

38

Figure 3.1.10. Maximum forming force FM plotted as a function of the maximum resolved force carrying capacity of a single precursor strut Ft, Eq. (3.1.5).

For all precursors, the best linear fit is found using ζ = 45.86 ± 0.75 mm, or γ = 1.04 ± 0.02 (R2

= 0.935). Previous failure observations [1] suggested that materials with high shear strength

could better resist pin punch through and thus reach greater ranges of perforation stretching

formed architectures. In contrast, the results of the present forming conditions (Figures 3.1.7-

3.1.10) show that general perforation-stretching failure can be attributed largely to tensile

instability mechanisms.

In conventional stretch forming processes, a greater forming limit can be achieved with

materials exhibiting greater tensile failure strains [20,21]. This suggests that dM can be increased

by both uniform and non-uniform strain components. The uniform strain component is increased

with the strain-hardening coefficient n, which is larger for materials with a greater ratio of

(σUTS/σYS), e.g. [22]. The non-uniform strain component can be increased with precursor

geometry, e.g. Barba’s Law [23] in which the non-uniform strain is proportional to ( )( ) 1−ooo ltw ,

where lo is the initial reduced section length. Evidence of this has been shown [2] for perforated

metal precursors with similar microstructures and equal values of wo and lo; precursors with

greater to were found to have greater tensile failure strains and, in turn, greater values of dM. In

addition, the non-uniform strain component can be increased by the strain-rate sensitivity index

m (see [24]). An increased value of n extends the uniform plastic deformation by improving the

ability to work harden, whereas increasing m will extend the post-uniform deformation by

prolonging the transition from diffuse to localized necking, e.g. [19,24]. Kwag and Morris [5]

have performed stretch forming studies on AA3003 and shown that the parameters n and m can

be related to the microstructure of the AA3003 precursor: smaller values of n (e.g. 0.02) could

be attributed to dislocation pile-up, reduced grain size, and retained deformation texture [5]. In

39

contrast, the larger values of n in recrystallized materials (e.g. 0.20) can be related to more

operative slip systems, increased grain size, and/or a more random texture. On the other hand, m

values are generally smaller in recrystallized materials (e.g. 0.007 to 0.002) as their

microstructures are able to relax following additional strain more easily [5]. The consideration

of precursor microstructure and geometry in terms of the available uniform and non-uniform

plastic strain is therefore important to the processing stage of PCM cores fabricated using

perforation-stretching methods.

In addition to the material properties of the precursor sheet, the forming pin geometry

should have an effect on the perforation stretch forming limits. Experimental results of

aluminum alloy and steel sheets subjected to stretch bending tests have shown that the

formability improves significantly with increasing punch curvature for a constant sheet

thickness [25,26]. This increase is due to the increased uniformity of the strain distribution

around the punch shoulder, which increases the limit strains by modifying the instability

criterion and delaying the formation of a local neck [25,26]. By decreasing a stretch-bending

ratio to/rpin past a certain critical value, it has been shown that the site of fracture shifts from near

the punch contact, i.e. where combined bending and stretching strains occur, to the side-wall

region undergoing pure stretching [27]. This failure mode may permit the greatest range of

perforation stretched PCM architectures.

3.1.4 Conclusions

This study has characterized the perforation stretching deformation process as a method

to fabricate periodic cellular metal truss sandwich core materials. It is shown that with the

present forming conditions this PCM fabrication method can be modeled after conventional

bending and tension mechanisms, and that forming limits occur largely via tensile instability.

Models for the onset of plasticity and the maximum forming force were developed to describe

the beginning and end of useful plastic deformation, respectively, and were found to hold for a

range of aluminum alloy precursor thicknesses and microstructures.

3.1.5 References 1. DJ Sypeck, Appl. Compos. Mater. 12 (2005) 229–246.

2. BA Bouwhuis, GD Hibbard, in: Porous Metals and Metallic Foams, LP Lefebvre, J Banhart, D Dunand (Eds.),

DEStech Publications, Pennsylvania, 2008, pp. 91–94.

3. DJ Sypeck, HNG Wadley, Adv. Eng. Mater. 4 (2002) 759–764.

4. JR Davis (Ed.), Aluminum and Aluminum Alloys, ASM International, Materials Park, 1993.

40

5. Y Kwag, JG Morris, Mater. Sci. Eng. 77 (1986) 59–74.

6. J Zhou, S Allameh, WO Soboyejo, J. Mater. Sci. 40 (2005) 429–39.

7. MA Meyers, KK Chawla, Mechanical Metallurgy – Principles and Applications, Prentice-Hall, Inc.,

Englewood Cliffs, 1984.

8. ASTM F1575-03, American Society for Testing and Materials, Philadelphia, 2003.

9. WC Setzer, JG Morris, Trans. ASM 57 (1964) 589–602.

10. JG Morris, Trans. ASM 59 (1966) 1006–1009.

11. SP Chen, NCW Kuijpers, S Van Der Zwaag, Mater. Sci. Eng. A 341 (2003) 296–306.



13. EV Crane, Plastic Working of Metals and Non-Metallic Materials in Presses, 3rd ed., John Wiley & Sons, New

York, 1944.

14. AW Spaink, Report No. SM-63, Royal Netherlands Aircraft Factories Fokker, The Netherlands, 1971.

15. Q Zhou, T Wierzbicki, Int. J. Mech. Sci. 38 (1996) 303–24.

16. M Li, Int. J. Mech. Sci. 42 (2000) 907–923.

17. RW Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 4th ed., John Wiley & Sons,

New York, 1996.

18. JL Duncan, JE Bird, Sheet Met. Ind. 55 (1978) 1015–1025.

19. JD Embury, JL Duncan, Ann. Rev. Mater. Sci. 11 (1981) 505–521.

20. R Stevenson, J. Appl. Metalwork. 3 (1984) 272–280.

21. J Kaneko, M Sugamata, M Takeshima, J. Jpn. Inst. Light Met. 35 (1985) 12–19.

22. WA Backofen, Deformation Processing, Addison-Wesley Publishing Co., Reading, 1972.

23. GE Dieter, Mechanical Metallurgy, 3rd ed., McGraw-Hill, Inc., Toronto, 1986.

24. AK Ghosh, Metall. Trans. A 8 (1977) 1221–1232.

25. MY Demeri, J. Appl. Metalwork. 2 (1981) 3–10.

26. MY Demeri, J. Appl. Metalwork. 2 (1983) 288–292.

27. S Sriram, C Wong, M Huang, B Yan, J. Mater. Manuf. 112 (2003) 641–649.

41

3.2 Modeling the Perforation Stretching Forming Process

3.2.1 Introduction

The distribution of plastic strain during fabrication plays a critical role in determining

the range of accessible perforation-stretched micro-truss cellular architectures; using

conventional perforation geometries, the largest strains have been observed near the hinge

region of the struts, which limits the maximum truss angle and minimum relative density that

could be formed [1,2]. It was shown that by modifying the starting sheet perforation geometry,

the plastic strain distribution could be controlled; by using a rounded-square perforation pattern,

a broader range of cellular architectures could be fabricated and the forming strains could be

concentrated in the mid-strut region, where they are needed most to increase the inelastic

buckling resistance of the micro-truss struts [3].

The present section develops an analytical forming model for the perforation-stretching

process. Micro-truss sandwich cores were fabricated from stainless steel (SS304),

electrolytically-pure copper (Cu110), and aluminum alloy (AA3003). These alloys possess a

considerable range of strength, ductility, and strain hardening characteristics. The tensile

properties of each perforated precursor material are used to determine the range of micro-truss

architectures that can be formed and the extent of plastic strain introduced during perforation

stretching.

3.2.2 Materials and Experimental Details

Perforation-stretching precursors were designed as rounded-square punched metal sheet

in standard gauge thicknesses (to); the perforation pattern was a square of size 11.11 mm with

corner radius of R = 2.78 mm and ligament width of wo = 1.59 mm. The squares centres were L

= 12.70 mm apart, creating a reduced section length lo = 5.56 mm, Figure 3.2.1. The as-received

precursor metals were non-precipitation hardenable aluminum alloy 3003 H14 (1/2-hard) (to =

0.77 mm), electrolytic tough-pitch copper 110 H00 (1/8-hard) (to = 0.79 mm), and austenitic

stainless steel 304 (to = 0.86 mm). These materials were chosen to represent technologically

significant alloys possessing a range of strength, ductility, and work-hardening capacity.

42

Figure 3.2.1. Rounded-square perforation pattern in precursors. Parameters include: corner radius R, ligament width wo, square centre-to-centre distance L, and reduced section length lo.

The aluminum and copper as-received sheets were tested in their as-received condition and in an

annealed condition; these annealing treatments followed typical schedules to return the metals to

their fully softened states: 415˚C for 1 hour (AA3003) [4] and 500˚C for 1 hour (Cu110) [5].

The stainless steel was found to be sufficiently formable in the as-received condition, and did

not require an annealing treatment prior to testing.

The tensile properties of the precursor struts were measured directly from miniature

coupons cut from the rolling and transverse direction (rd and td, respectively) of the perforated

sheets, after [2]. The rounded square perforation pattern ensured the reduced section length lo

lay outside of the test machine grips. Figure 3.2.2 displays typical rolling direction (rd) tensile

curves for each precursor tested, illustrating the significant property differences between the

various alloys and tempers; values of yield strength, ultimate tensile strength, uniform strain,

and strain to failure are summarized in Table 3.2.1.

Figure 3.2.2. Typical rolling direction (rd) tensile curves of the precursor alloys.

43

Table 3.2.1. Tensile properties of precursor material coupons in the rolling direction (rd) and transverse direction (td): yield strength σYS, ultimate tensile strength σUTS, uniform tensile strain εU, and total failure strain εFail. Strains are measured using ε = Δl/lo.

Precursor Direction σYS (MPa) σUTS (MPa) εU (mm/mm) εFail (mm/mm) SS304

As-Received rd td

433.6 ± 4.2 443.3 ± 12.6

744.6 ± 6.5 783.5 ± 1.1

0.689 ± 0.027 0.660 ± 0.033

0.785 ± 0.029 0.744 ± 0.027

Cu110 As-Received

rd td

188.4 ± 4.6 191.1 ± 8.0

229.1 ± 1.3 242.2 ± 2.0

0.172 ± 0.004 0.161 ± 0.014

0.313 ± 0.017 0.310 ± 0.017

AA3003 As-Received

rd td

140.7 ± 2.3 135.3 ± 2.3

151.9 ± 1.2 155.9 ± 0.5

0.022 ± 0.002 0.014 ± 0.002

0.122 ± 0.003 0.110 ± 0.004

Cu110 Annealed

rd td

71.2 ± 7.9 65.9 ± 9.1

213.7 ± 1.5 216.3 ± 2.3

0.514 ± 0.014 0.490 ± 0.009

0.650 ± 0.026 0.593 ± 0.029

AA3003 Annealed

rd td

49.0 ± 2.0 49.8 ± 4.8

104.7 ± 0.4 99.6 ± 0.2

0.283 ± 0.003 0.290 ± 0.007

0.410 ± 0.010 0.403 ± 0.014

Micro-truss sandwich core PCMs were fabricated using a modified stretch-bend forming

technique, after [2,6] . For each precursor, a forming force (FForm)-displacement (dForm) curve

was obtained using a screw-driven universal testing machine. A schematic diagram illustrating

the forces and geometry involved is shown in Figure 3.2.3. Failure was observed by a sudden

drop in forming force which corresponded to the fracture of one or more ligaments, after [2].

Figure 3.2.3. Cross-section schematic of the perforation-stretching process. Parameters include: precursor unit cell length (and pin-centre-to-pin-centre distance) L, perforation-stretching forming depth dForm with force FForm, and ligament force FLigament.


Figure 3.2.4 presents forming force – displacement (FForm – dForm) curves for each alloy

in the as-received condition and the AA3003 and Cu110 alloys after annealing treatments. In all

cases, forming failure occurred by a tensile instability at mid-strut (Figure 3.2.4, inset).

44

Figure 3.2.4. Forming force – displacement (FForm – dForm) curves for the precursors used in the present study. Forming failure via tensile instability is observed at the mid-strut (inset).

This failure mode is typical of rounded square perforation geometries where the plastic

deformation introduced during forming is concentrated along the reduced cross-section length of

the micro-truss strut [3]. The following analysis uses the projected tensile forces and

displacements to model the stretching-dominated stage of the micro-truss forming curve. First,

the force experienced by the ligament during uniaxial tension FLigament = σe×A (where σe is the

engineering stress, Figure 3.2.2, and A is the initial ligament cross-sectional area) was resolved

into a forming force FForm. Second, the change in length during uniaxial tension Δl was

converted to a forming displacement dForm.

The perforation-stretching displacement dForm can be related to the tensile increase in

length Δl by the idealized relation:

( ) 22 LLld Form −+Δ= (3.2.1)

In the case of tensile deformation, the material stress-strain behaviour is typically represented by

a relationship of the form σ = f(ε). This general constitutive behaviour can be transformed into a

specific relationship for the perforation-stretching sheet geometry by considering instead the

tensile force and change in ligament length, i.e. FLigament = g(Δl). The transformation between Δl

and dForm is accomplished by considering FLigament as a function of dForm instead of Δl, i.e.

FLigament = g(dForm), or

( ) ⎟⎠⎞⎜

⎝⎛ −+Δ= 22 LLlgFLigament (3.2.2)

The transformation from FLigament to FForm is made using a force balance (Figure 3.2.3):

Lld

FF FormLigamentForm +Δ

= η (3.2.3)

45

The parameter η accounts for the number of struts formed in tension (in the present case, η =

48). Combining Eq. (3.2.1) and Eq. (3.2.3) results in the transformation for force:

( )Ll

LLlF

Lld

FF LigamentForm

LigamentForm +Δ−+Δ

=+Δ

=22

ηη (3.2.4)

Together, both transformations result in:

( ) ( ) ( )FormForm dhLl

LLlLLlgF =

+Δ−+Δ

×⎟⎠⎞⎜

⎝⎛ −+Δ=

2222 η (3.2.5)

For each material, the g(Δl) data is collected from the miniature coupon, then transformed to the

respective forming curve FForm = h(dForm). Figure 3.2.5 displays the transformed forming curve

for the as-received stainless steel precursor using Eq. (3.2.5). The early experimental forming

curve possesses a non-linear region which deviates from the tensile model. However, as the

tensile forming component becomes dominant, there is good agreement between the analytical

and experimental forming curves.

Figure 3.2.5. Conversion of rolling direction (rd) and tensile direction (td) tensile curves plotted against experimental forming data for the SS304 precursor.

Three tensile curves for each material were transformed into forming curves, and the

corresponding average maximum forming force FM and corresponding maximum displacement

dM values were determined. Figure 3.2.6 plots the experimental values of maximum forming

force (FM,Expt.) and displacement (dM,Expt.) against the respective model values (FM,Model and

dM,Model). These results illustrate the important point that the simple analytical model (Eq. 3.2.5)

can account for the range of forming forces and instances of failure between the widely different

precursor materials.

46

Figure 3.2.6. Comparison of model forming failure parameters against experimental results. Model maximum force FM,Model (left) and maximum displacement dM,Model (right) were determined from the forming curves generated using the tensile projection model of Eqs. (3.2.1-3.2.5) for coupons along the rolling direction and transverse direction. The corresponding experimental FM,Expt. and dM,Expt. data were determined from forming tests.

An important extension of this analytical model concerns the planar anisotropy of the starting

sheet material, which can arise in the final product following processing stages such as rolling

[7,8]. Within a given sheet material, the planar anisotropy ratio R (see [7]) describes the range

of strain capacities between different in-plane directions; it is ideally equal to one for a material

with planar isotropy. For most steels, and aluminum and copper alloys, R ranges from

approximately 0.5 to 1.5, while for titanium alloys R can span between 4 to 6 [7]. In the present

study, the most significant planar anisotropy was seen for the annealed Cu110 sheets; the

transverse direction exhibited a smaller strain to failure and thus controlled the overall

formability of the micro-truss architecture (Figure 3.2.6).

The maximum forming displacement is a particularly important parameter, used to

define the upper limit of architectures that can be fabricated [1,2]. A simplified model for dM can

be made using the change in length in uniaxial tension at failure ΔlFail = (εFail × lo) and Eq.

(3.2.1). Figure 3.2.7 presents dM plotted against ΔlFail for conventional square perforated

AA3003 with different work-hardened precursor microstructures and sheet thicknesses [1] and

AA6061 with different age-hardened precursor microstructures [9]. While the present approach

works well for the rounded square perforations when failure is tensile-dominated, it over-

predicts the upper forming limit by approximately 33% when fracture occurs by a combination

of bending and tensile deformation, as in the case for conventional square-perforated sheet [2].

The former perforation type is more desirable in order to localize the majority of the forming

47

strains to the mid-strut region. Furthermore, the good agreement in Figure 3.2.7 for the rounded-

square perforated sheet is significant because the upper forming limit and range of architectures

achievable can be determined a priori knowing only ΔlFail from precursor coupons and L from

the forming apparatus.

Figure 3.2.7. A simplified model for predicting the maximum forming displacement dM as a function of the tensile displacement at failure ΔlFail. The model (Eq. (3.2.1)) agrees well for the rounded-square perforations, and over-predicts dM by ~33% for the conventional square perforations in AA3003 [1] and AA6061 [9].

While it is useful to know the upper forming limit dM, it is also desirable to make

samples that have neck-free struts. A safe forming limit can be defined as the point when

uniform plastic deformation ends; beyond this point, tensile instability results in neck formation

within the micro-truss strut. Eq. (3.2.5) was used to obtain the forming displacement dU,Model

which coincides with the uniform tensile strain εU. The values of dU,Model for each precursor as

well as the uniform forming ratio dU,Model/dM,Model are summarized in Table 3.2.2. The SS304,

annealed Cu110, and annealed AA3003 materials have small non-uniform tensile strain, and so

a safe forming limit may be chosen as approximately 90-95% of their respective dM. On the

other hand, the as-received AA3003 (H14) material can be safely formed to only 1.42 mm, or a

maximum truss angle of <10˚, while the as-received Cu110 (H00) has a safe limit of 4.85 mm,

which leads to a maximum forming angle of only ~20˚.

Finally, it is important to note that there is a trade-off between the forming range and

therefore the in-situ work hardening possible during forming, and the strength (or work-

hardened state) of the precursor prior to forming. Full annealing prior to forming is desirable

when the benefits of decreased relative density, increased truss angle, and greater work-

48

hardening index outweigh the benefit of higher initial strength. For the SS304 precursor,

annealing prior to forming was not necessary since the as-received state, which was between an

1/8 and 1/4 hard condition [10], already possessed a good combination of strength and ductility.

On the other hand, for AA3003, annealing the starting H14 temper prior to forming was

necessary in order to attain lower relative densities and higher micro-truss strut angles.

Table 3.2.2. Forming displacement dU,Model corresponding to the uniform engineering tensile strain εU, and the ratio of dU,Model to dM,Model found using Eqs. (3.2.1-3.2.5).

Precursor Direction dU,Model (mm) dU,Model/dM,Model SS304

As-Received rd td

10.58 ± 0.24 10.32 ± 0.29

0.96 ± 0.04 0.96 ± 0.05

Cu110 As-Received

rd td

5.02 ± 0.07 4.85 ± 0.22

0.81 ± 0.03 0.80 ± 0.06

AA3003 As-Received

rd td

1.75 ± 0.09 1.42 ± 0.10

0.49 ± 0.03 0.42 ± 0.04

Cu110 Annealed

rd td

8.99 ± 0.14 8.75 ± 0.08

0.97 ± 0.03 0.88 ± 0.03

AA3003 Annealed

rd td

6.51 ± 0.05 6.60 ± 0.10

0.88 ± 0.02 0.89 ± 0.04

3.2.4 Conclusions

This study has investigated the relative formability of different materials in the

perforation-stretching process. A simple tensile deformation based model was used to predict

the attainable range of cellular architectures and the extent of plastic strain in the cellular

ligaments. Of the materials tested in this study, the stainless steel 304 precursor was shown to

reach the highest truss angle and lowest relative density, which is primarily a result of its high

ductility and tensile strain capacity.

3.2.5 References 1. BA Bouwhuis, GD Hibbard, in: Porous Metals and Metallic Foams, LP Lefebvre, J Banhart, D Dunand (Eds.),

DEStech Publications Inc., Pennsylvania, 2008, pp. 91-94.

2. BA Bouwhuis, GD Hibbard, Metall. Mater. Trans. A 39 (2008) 3027-3033.

3. E Bele, BA Bouwhuis, GD Hibbard, Composites Part A 40 (2009) 1158-1166.


5. JR Davis (Ed.), Copper and Copper Alloys, ASM International, Materials Park, 2001.


7. GE Dieter, Mechanical Metallurgy, 3rd ed., McGraw-Hill, Toronto, 1986.

49

8. FJ Humphreys, M Hatherly, Recrystallization and Related Annealing Phenomena, 2nd ed., Elsevier, Boston,

2004.

9. BA Bouwhuis, L Cheng, GD Hibbard, in: Processing and Fabrication of Advanced Materials XVI, M Gupta,

TS Srivatsan, SK Thakur (Eds.), Research Publishing Services, Singapore, 2008, pp. 480-489.

10. ASM Handbook, vol. 1: Properties and Selection – Irons, Steels, and High Performance Alloys, ASM

International, Materials Park, 1990.

50

4 Testing Methods This chapter presents the author’s work to develop a new test method which evaluates

the properties of the micro-truss core as a stand-alone material, in an effort to preserve the

microstructural artifacts and strengthening components from the fabrication stage. Section 4.1

introduces the new test method which uses confinement plates to mechanically lock every PCM

node in place, thereby restricting lateral movement and inducing the same collapse mechanism a

PCM core would experience as part of a fully-bonded sandwich panel. Results show this new

test method allows direct testing of PCM microstructural design strategies, independent of the

face-sheet materials and joining processes. This section is part of a paper published in

Metallurgical and Materials Transactions B [BA Bouwhuis, GD Hibbard, Compression testing

of periodic cellular sandwich cores, Metall. Mater. Trans. B 37 (2006) 919-927], reproduced

with kind permission of Springer Science and Business Media.

The author has also used this new test method to investigate specimen size-dependent

property measurements, which are a well-known problem in the field of cellular materials.

Section 4.2 presents a study of size effects in the uniaxial compression testing of PCMs. This

was the first study of its type and was published in the Journal of Materials Science [BA

Bouwhuis, E Bele, GD Hibbard, Edge effects in compression testing periodic cellular metal

sandwich cores, J. Mater. Sci. 43 (2008) 3267-3273], reproduced with kind permission of

Springer Science and Business Media.

4.1 Testing the Micro-Truss Core as a Stand-Alone Material

4.1.1 Introduction

PCMs are typically tested as part of a sandwich panel in which the PCM core is fixed

(i.e. confined) between solid face sheets [1-4], imposing a support-thickness-dependence as well

as a confinement-dependence [5-7]. During compression testing failure typically occurs by

plastic buckling of the PCM struts [2,3,5]. When a strut buckles, the rotation of its ends (i.e.

hinges) is opposed by their bending stiffness [3,6]; the magnitude of this opposition depends on

the fully plastic moment (i.e. plastic hinging) of the beam. Therefore, both strut and node

properties contribute to the overall PCM performance.

In order to investigate these failure mechanisms, this study examines the collapse

mechanisms in PCM sandwich cores by compression testing in two limiting conditions. In the

51

first case (i.e. confined compression), the PCM nodes are laterally confined, which induces

ideally axial compressive stresses into the struts and leads to strut failure by buckling. In the

second case (i.e. free compression), the PCM cores are placed between smooth compression

platens, where the nodes are restricted only by interfacial friction; this induces transverse

bending stresses into the struts and leads to failure by plastic hinging. In this sense, free

compression represents an absence of rigid strut confinement.

4.1.2 Experimental

Pyramidal truss core PCM was fabricated from a 0.81 mm thick (to) square punched

aluminium 3003-H14 sheet, purchased from Woven Metal Products, Inc. (Alvin, TX). The

90.82 mm2 (internal edge length = 9.53 mm) perforations were punched on a 2D square lattice

of unit cell size 12.7 mm × 12.7 mm. The resulting structure is a series of 4-rayed nodes with

arm cross-sections of 3.18 mm (wo) × 0.81 mm (to), having 56% open area. The sheet was

strain-relief annealed at 415˚C for 1 hour to obtain an O temper [8].

A perforation-stretching process was used similar to that described by Sypeck and

Wadley [4] to fabricate the PCM cores. The linear deformation rate was ~5mm/min and a final

pyramidal height of 7.3 mm was achieved. The result is a truss angle ω (see [6]) of 30˚ and

relative density ρR of 5% (0.13 Mg/m3), pyramid shown in Figure 4.1.1.

Figure 4.1.1. Pyramidal PCM unit cell. Strut angle ω traverses between the horizontal and the strut formed with height (h), thickness (t), width (w), and length (l).

After forming, the truss struts had cross-section w = 3.08 ± 0.02 mm and t = 0.70 ± 0.02 mm

over seven measurements, and length l = 11.00 mm. As the regularity of the cellular material

increases, bulk properties can be determined with smaller ratios of specimen size to cell size [9].

In the limiting case, single PCM unit-cells analysis have been used to model bulk PCM

52

performance [10-12]. For this study, PCM truss cores were cut into 36-node squares for an

effective compression-surface area of 44.4 cm2 (i.e. the projected 2-dimensional area enclosed

by the 6×6-node pyramidal core).

Uniaxial compression testing was performed at a crosshead displacement rate of 1

mm/min. Five test samples were used for each compression test condition, after ASTM-C365

[13]. Nominal strains were measured from crosshead displacement [9,14-17]. Two types of

compression testing were studied: (1) free and (2) confined compression.

In the free-core uniaxial compression mode, a single layer of pyramidal PCM truss core

is placed between tempered steel compression platens. This compression method illustrates

force resistance by bending of the truss core struts. In the confined-core uniaxial compression

mode, a single layer of pyramidal PCM truss core is fixed within confinement plates, shown in

Figure 4.1.2, and placed between the compression test platens.

Figure 4.1.2. Confinement plate design for confined-PCM compression testing. Pyramidal PCM test specimen has nodes fixed in one-sided confined-compression test condition.

The confinement plates were machined from AISI 1020 mild steel with 1.2 mm recessed

channels. These plates laterally confine the PCM nodes through the duration of compression,

simulating the behaviour in a PCM sandwich panel; this method illustrates force resistance by

buckling of the PCM struts. The degree of confinement is varied between two test sub-groups:

1) one-side and 2) two-sides confined. In the former case, a confinement plate was used on one

side and a steel compression platen was used on the other.

4.1.3 Results

4.1.3.1 Free-Core Uniaxial Compression: Beam Bending on Steel Platens

Figure 4.1.3 presents the result of a typical PCM core on steel platen in free

compression, shown as a uniaxial compression stress-strain (σ-ε) curve.

53

Figure 4.1.3. Typical free (i.e. absence of rigid confinement) compression test result for PCM on steel platens. Inset gives the effective elastic modulus measured from the slope of the unloading/loading curve at ~75% of peak compressive stress (left). Also included is an example of the collapse mechanism during free compression (right).

The peak compressive strength, σP, of a PCM was taken as the initial peak stress [1,3], the same

as for metallic foams [9]. Additionally, we can define a valley (i.e. densification) compressive

strength, σV, which is the minimum stress value after σP during truss collapse. Peak compressive

strength values averaged 0.207 MPa with a standard deviation of 0.005 MPa. Valley

compressive strength values averaged 0.142 MPa with a standard deviation of 0.003 MPa

(summarized in Table 4.1.1).

Table 4.1.1. PCM compression tests with 1) free (zero-sided) confinement, 2) single-sided simulated confinement, and 3) double-sided simulated confinement. Tabulation of: peak compressive strength (σP); valley compressive strength (σV); effective elastic modulus (E); densification strain (εD); and respective energy densities for densification strains (JD for εD and J55 for ε55). ε55 is defined as a strain value of 0.55.

Test Value σP (MPa)

σV (MPa)

E (MPa)

εD (mm/mm)

JD (kJ/m3)

J55 (kJ/m3)

Average 0.207 0.142 6.15 0.81 147.4 86.3 Std. Dev. 0.005 0.003 0.47 0.01 1.4 1.9

Free PCM

% Error 2.3 1.6 7.7 1.0 1.0 2.2 Average 0.767 0.431 42.33 0.62 470.1 283.7 Std. Dev. 0.023 0.039 2.20 0.02 42.6 20.7

Single-Sided Confined PCM

% Error 3.0 9.0 5.2 3.0 9.1 7.3 Average 0.817 0.490 47.31 0.65 551.0 289.8 Std. Dev. 0.014 0.046 1.08 0.01 8.3 7.8

Double-Sided Confined PCM

% Error 1.7 9.4 2.3 1.6 1.5 2.7

54

PCM modulus is considered the slope of the loading curve [1], ideally dependent only on

the PCM architecture (i.e. ρR and ω) and the stiffness of the parent material [6,18]. However,

modulus of the loading curve is reduced during early stages of deformation due to initial

bedding-in of the PCM specimen [3,5] and fabrication imperfections [19]. As a result,

measurements of an elastic modulus E should be made from the slope of an unloading/reloading

curve. For this study, modulus is taken as the slope of an unloading/reloading curve, unloaded

from ~75% of the peak compressive strength (Figure 4.1.3, inset), the same method used for

conventional metallic foams [9,20]. Applying this method to PCMs, the elastic modulus values

averaged 6.15 MPa with standard deviation of 0.47 MPa (Table 4.1.1).

Several definitions have been used in the literature for the densification strain, εD. For

conventional metallic foams, εD can be generalized to scale with (1 – relative density) [9].

Elsewhere, εD has been chosen as the strain at which the compressive stress equals twice the

peak value [21], or at a strain where the stress is equal to 1.5 times the stress at 50% strain [20],

or as the strain at which the stress-strain curve begins to rise (i.e. ε at σV) [22]. For PCMs the

densification strain has been selected at the strain where the stress re-attains its initial peak value

[3], at values between 0.5 and 0.6 [23], and at strains exceeding approximately 50% [7]. In

Table 4.1.1, ε55 is defined as a strain of 0.55, and εD is taken after Olurin et al. [21] to remove

bedding-in effects: εD = εσP×2 – εσP/2 (for this subgroup, εD = 0.81 ± 0.01).

The definition of densification strain is important because it directly factors into the

measured energy absorption, J, during truss collapse [9]. J is related to the area under the σ-ε

curve; the upper-bound integral can be approximated using successive strain intervals (i.e. i – 1,

i) multiplied by the stress at the leading strain (i.e. σ(εi)). The calculation of J as a Riemann sum

can then be accomplished over the whole strain interval i using (e.g. [24]):

( )∑ −−⋅=i

iiiJ 1εεσε (4.1.1)

Applying this method to PCMs, the values for energy absorption are markedly different

between JD (using εD) and J55 (using ε55) (Table 4.1.1). For example, there is a 71% difference in

calculated J55 and JD values for truss cores in free compression.

4.1.3.2 Confined-Core Uniaxial Compression: Collapse Mechanisms

The effect of nodal confinement on the truss core buckling stiffness and strength was

investigated with the same PCM structure as the previous sections. In the fully-confined test

condition, both top and bottom sets of truss core nodes were placed in confinement plates. The

55

mechanical properties for this subgroup of 5 samples are summarized in Table 4.1.1 and a

typical stress-strain curve is shown in Figure 4.1.4. Also shown in Figure 4.1.4 is a typical

stress-strain curve for PCM-steel free compression as a reference. It is worth noting that neither

the average peak strength (0.817 ± 0.014 MPa) nor modulus (47.31 ± 1.08 MPa) are

significantly larger than the strength and modulus of metal foams possessing similar relative

densities, e.g. [14,20]. This low performance is due to the sub-optimal (i.e. rectangular) micro-

truss strut cross-sectional area as well as the low truss angle used in this initial study, and in

addition may be caused by the non-idealities listed above.

Figure 4.1.4. Typical confined-compression test result for PCM truss core in two-sided nodal confinement. Also plotted is a typical curve for free compression of PCM-steel platens.

To characterize the collapse mechanisms of truss cores having nodes confined, a single-

sided confined PCM sample was compressed to increasing strains and the number of collapse

mechanisms counted. The unconfined PCM nodes are grouped into one of two groups: 1) nodes

on the outer edge of the PCM (i.e. with 3 support struts), denoted by O; and nodes internal to the

PCM (i.e. with 4 support struts), denoted by I. Collapse mechanisms were divided into four

groups, illustrated schematically and structurally in Figure 4.1.5: A) first-mode buckling of

PCM struts (primary weakening); B) contact of buckled PCM struts with compression surfaces

(secondary strengthening); C) second-mode buckling of PCM struts (secondary weakening); and

D) secondary contact of buckled PCM struts (tertiary strengthening). Collapse mechanisms are

counted separately for the two node groups (O and I), giving a total of 8 groups: A, B, C, D,

each for O and I nodes. Figure 4.1.6 displays the count as a percentage of the total number of

struts for O (24) and I (32) nodes. As a reference, the compression stress-strain curve for the as-

tested one-sided confined sample is also included for this specific node count.

56

Figure 4.1.5. Schematics with examples of collapse mechanisms during confined compression. Mechanisms occur for each PCM strut in order of (a) to (d) for increasing compressive strain, seen in Figure 4.1.6.

Figure 4.1.6. Single-side confinement compression and collapse mechanism profile. The mechanism count depicts the percentage of total struts counted in failure mechanisms A through D (Figure 4.1.5): (a, left) node groups on the outside of the PCM (i.e., 3 support struts) are denoted by O and (b, right) node groups internal to the PCM (i.e., four support struts) are denoted by I. Total O struts are 24, and total I struts are 32.

57

4.1.4 Discussion

4.1.4.1 Confinement Dependence

External supports to sandwich cores incur a degree of confinement and local

strengthening of the core edges [9,13]. This is simulated in the present study by the presence of

confinement plates which restrict nodal spreading. In this case, compression can be thought of

as an array of pyramids, where their bases are held confined. In ideal compression conditions,

all pyramidal units will have their four struts respond identically and simultaneously to loading

through yield and post-yield plastic buckling and hinging. The pyramidal peak will then be

without lateral displacement to densification. However, constructed PCM unit cells are not

ideal; there are slight strength differences between the struts resulting in non-uniform collapse.

Following failure of the weakest strut, a single pyramid will be structurally indeterminate and

pyramidal peak forces will have an unbalanced lateral component. From this point until

complete pyramidal collapse (i.e. densification) the struts undergo a mixture of beam buckling

and nodal bending.

In Figure 4.1.6 there is a clear difference between the performance of weaker outer (O)

nodes (Figure 4.1.6a) and stronger internal (I) nodes (Figure 4.1.6b); pyramids with uneven

supports (i.e. O) fail before those with full support (i.e. I). For example, the peak compressive

stress coincides with the onset of primary buckling of the outer nodes (at ε ≈ 0.17). The decrease

following σP is observed with the continued increase of edge primary buckling as well as the

onset and increase of internal primary buckling (at ε ≈ 0.24). Similarly, the increase in strength

beyond the valley compressive stress coincides with the onset of secondary strengthening of the

outer nodes in which buckled struts make contact with the flat surface (Figure 4.1.5) (at ε ≈

0.28). The strength increase following σV continues with increasing edge node secondary

strengthening as well as the onset and increase of internal node secondary strengthening. The

rate of increase in strength after σV levels off at the onset of secondary weakening (at ε ≈ 0.56-

0.65) and again rises with tertiary strengthening (at ε ≈ 0.80), increasing to the point of

densification. Note that the percent count of both O and I nodes levels off before 100% (Figure

4.1.6); this is due to the non-ideal and non-uniform buckling conditions, where pyramidal

‘tipping’ permits some struts to simply bend without progressing through mechanisms A to D.

The lower percent count of O struts buckling is a direct consequence of their greater structural

indeterminacy.

58

4.1.4.2 Effect of Collapse Mechanisms

The effect of confinement degree in compression on the measured collapse mechanism

mechanical properties (σP, σV, E, and JD) is summarized in Figure 4.1.7.

Figure 4.1.7. Comparison of free- and confined-PCM compression test results for test methods. Top to bottom: energy density upon densification (JD), effective elastic modulus (E), valley compressive strength (σV), and peak compressive strength (σP).

The widest range in properties is between the free compression and the two-sided confined

conditions. Between these two testing limits, σP, σV, E, and JD increase by 295%, 245%, 670%,

and 275%, respectively. These initial comparisons highlight the differences of bending (i.e. free

compression) and yielding/buckling (i.e. fully confined compression) deformation detailed by

the scaling relations of Ashby et al. [9,18]: for a relative density of 0.05, the stretching-governed

structure is expected to be about 4 times as strong and an order of magnitude stiffer than the

bending structure.

Furthermore, the sample-to-sample variability in compression test behaviour of the

PCMs in this study is lower than what is typically observed for conventional metallic foams. For

example, the standard deviation in foam elastic modulus has been noted as typically between 5

and 30% of the mean, while that in the compressive strength is typically between 5 and 15%

[9,14,20,21]. Due to their highly regular structure, PCMs are postulated to have lower standard

deviations than their stochastic foam counterparts [1]. The mechanical property data in Table

59

4.1.1 supports this assertion, with 2.3 to 7.7% elastic modulus and 1.7 to 3.0% peak strength

standard deviation.

The results from the current study can be compared to existing models for collapse

performance of PCM truss cores. First, the measured σP values can be related to the plastic

hinging and buckling forces of PCM struts in free and confined compression, respectively. For

the PCM specimen in Figure 4.1.2 undergoing plastic collapse, there are 120 plastic hinges

during free compression and 60 struts undergoing plastic buckling in confined compression. The

plastic hinging mechanism follows the well-known beam-failure relation (e.g. [9]):

( )2/4

2

lwtCF YS

Hingingσ

= (4.1.2)

From Figure 4.1.1, (w), (t), and (l) are strut dimensions and C is a coefficient describing

the failure mode. A beam supported on a single side has C = 1 for full plasticity. Using modulus

ES (69 GPa) and σYS = 42 MPa for AA3003-O [25] as a lower bound, and assuming all hinges

collapse simultaneously, Eq. (4.1.2) calculates FHinging = 2.8 N (PCM σP = 0.08 MPa). From

Table 4.1.1, the measured value of FHinging = 6.0 N (σP, free = 0.207 MPa) is greater than its

theoretical value. The difference between the two can be attributed to the work hardening

induced during perforation stretching, which can increase the local yield strength near the node.

It has been shown [3,6] that for a PCM core made from an elastic-strain hardening

material (i.e. AA3003), the struts collapse by buckling at an inelastic bifurcation force:

12

3

2

22 wtl

EkF t

Inelasticπ

= (4.1.3)

Here, the coefficient k describes the rotational connectivity of the struts: an upper bound of k = 2

represents full nodal rotational confinement, and a lower bound of k = 1 represents pin-jointed

nodes. The true stress (σt)-true strain (εt) tangent modulus Et = ∂σt/∂εt replaces the parent elastic

modulus ES in the well-known Euler beam-buckling relation (e.g. [9]). In the case of a linear

strain hardening material, Et is a constant. In the case of AA3003-O, the tensile true stress-strain

curve follows a power-law relationship: ntt Kεσ = (4.1.4)

where the strength coefficient K and the strain-hardening exponent n can be approximated by

0.2 GPa and 0.25, respectively [26-28]. The tangent modulus Et is then approximated as a

function of strain:

60

75.01 50 −− ≈=∂∂

= tnt

t

tt KnE εε

εσ

(4.1.5)

In the post-yielding range of the tensile stress-strain curve for AA3003-O (i.e. true strain

0.001 < εt < 0.1) the tangent modulus Et ranges from 8.9 GPa to 0.3 GPa. Using these values in

Eq. (4.1.5) gives a general range of inelastic buckling force (and strength) from 0.3 kN (σP =

3.45 MPa) to 2.8 N (σP = 0.04 MPa); this range encompasses the measured value of FInelastic = 60

N (σP, two-sided confined = 0.82 MPa). The experimentally measured σP corresponds to Et at a

true strain εt = 0.007 (Et = 2.1 GPa). This inelastic buckling model is analyzed more thoroughly

for micro-truss PCMs in Chapters 5 and 6.

The present data can be used to illustrate the different strengths between node-bending

(free-PCM) and strut-buckling (fully-confined-PCM) failure mechanisms; conditions of

confined and free compression can produce mixtures of these failure mechanisms. In practice,

any internal joint of a PCM core behaves in a fashion intermediate to bending and buckling

conditions [4]. Further, the effective mechanical properties of both periodic and stochastic open-

cell cores are largely dictated by the nodal connectivity at the junctions between struts [7].

Depending on the presence of confinement, the periodic framework in this study is not

necessarily rigid, but instead can collapse through a range of bending and buckling responses.

Compression testing of PCM cores in the limiting nodal confinement conditions outlined in this

study permits the direct measurement of bending and buckling performance of a given PCM

architecture or microstructure, facilitating the development of higher stiffness and higher

strength truss structures.

4.1.5 Conclusions

This study has investigated the independent deformation mechanisms of identical PCM

truss core materials in a range of confinement-degree compression testing conditions. Increases

in PCM compressive strength (295%), valley compressive strength (245%), elastic modulus

(670%), and energy absorbed at densification (275%) were measured between the limiting

conditions of free compression and two-sided (i.e. fully confined) compression. In free

compression, the performance is determined by the bending stiffness and strengths of the truss

core struts, and collapse deformation occurs solely by plastic hinging. In confined compression,

the performance is determined by the axial stiffness and buckling strengths of the struts as well

as plastic hinging in the post-peak collapse. Successive weakening and strengthening

mechanisms during truss collapse have been followed and correlated to the stress-strain curve.

61

4.1.6 References 1. HNG Wadley, NA Fleck, AG Evans, Compos. Sci. Technol. 63 (2003) 2331-2343.


Solids Struct. 39 (2002) 4093-4115.



5. VS Deshpande, NA Fleck, MF Ashby, J. Mech. Phys. Solids 49 (2001) 1747-1769.


7. M Zupan, VS Deshpande, NA Fleck, Eur. J. Mech. A 23 (2004) 411-421.

8. ASM Handbook, vol. 4: Heat Treating, ASM International, Materials Park, 1991.




11. S Hyun, AM Karlsson, S Torquato, AG Evans, Int. J. Solids Struct. 40 (2003) 6989-6998.


374.

13. ASTM C365-06, American Society for Testing and Materials, Philadelphia, 2006.


15. B Kriszt, B Foroughi, K Faure, HP Degischer, Mater. Sci. Technol. 16 (2000) 792-796.

16. EW Andrews, G Gioux, P Onck, LJ Gibson, Int. J. Mech. Sci. 43 (2001) 701-713.

17. U Ramamurty, A Paul, Acta Mater. 52 (2004) 869-876.


19. HNG Wadley, Philos. Trans. R. Soc. London Ser. A 362 (2006) 31-68.

20. E Andrews, W Sanders, LJ Gibson, Mater. Sci. Eng. A 270 (1999) 113-124.

21. OB Olurin, NA Fleck, MF Ashby, Mater. Sci. Eng. A 291 (2000) 136-146.

22. A-M Harte, NA Fleck, MF Ashby, Acta Mater. 47 (1999) 2511-2524.


24. S Salas, E Hille, GJ Etgen, Calculus – One and Several Variables, 8th ed., John Wiley & Sons, Inc., Toronto,

1999.

25. ASM Handbook, vol. 2: Properties and Selection – Nonferrous Alloys and Special-Purpose Materials, ASM


26. Atlas of Stress-Strain Curves, 2nd ed., ASM International, Materials Park, 2002.

27. GE Dieter, HA Kuhn, SL Semiatin, Handbook of Workability and Process Design, ASM International,

Materials Park, 2003.

28. N Abedrabbo, F Pourboghrat, J Carsley, Int. J. Plast. 22 (2006) 314-341.

62

4.2 Edge Effects in Compression Testing Periodic Cellular Metal Sandwich Cores

4.2.1 Introduction

The following section examines the significance of sample size in confined compression

testing of PCMs. Sample size effects are a potentially significant complication in the mechanical

testing of cellular sandwich core materials because the size of the cell structure is typically on

the same scale as the sample itself. For example, there must be a minimum number of cells in

metallic foam samples to reflect a statistical distribution of random imperfections [1]. In

addition, fixing foams between face sheets (creating a sandwich panel) constrains the cells

closest to the sheets, resulting in a through-thickness strain gradient during loading [2,3]. Lastly,

cells near the edges of the specimen are subjected to boundary-layer effects and constraint

effects [4-8]; as a result, edge cells typically contribute less to the overall performance than cells

in the middle of the specimen. To date, edge effects have not been systematically studied in

PCM cores. The present study examines the mechanical testing of stand-alone tetrahedral PCM

truss cores with a focus on the nature of edge effects present during compression testing.

4.2.2 Experimental

Tetrahedral PCM truss cores were fabricated from a 0.79 ± 0.01 mm thick (to)

hexagonal-punched aluminum 3003-H14 sheet, purchased from McNichols Perforated Products

(Atlanta, GA). The 34.92 mm2 (internal edge length lo = 3.67 mm) base metal perforations were

arranged in a 2D hexagonal lattice, unit cell in Figure 4.2.1. The planar perforated structure is an

array of 3-rayed nodes with strut cross-sections of 0.79 mm (width wo) × thickness to, having

79% open area. A modified perforation-stretching process, based on the method described by

Sypeck and Wadley [9], was used to fabricate the PCM cores. Struts were stretched at room

temperature by applying out-of-plane force at their intersections (nodes) using hardened steel

dowel pins of 1.1 mm diameter. The forming load was applied using a Shimadzu screw-driven

compression platform at a constant linear deformation rate (1 mm/min) to achieve a PCM

relative density (PCM density divided by AA3003 density) of 6.0% (0.16 Mg/m3). After

fabrication, the mid-strut cross-section decreased from 0.79 mm × 0.79 mm to 0.76 mm × 0.76

mm. A schematic diagram of a PCM tetrahedral unit is shown in Figure 4.2.1.

63

Figure 4.2.1. Schematic diagrams of the perforated precursor dimensions prior to forming (left), a single PCM tetrahedron (centre), and a sample of 33 tetrahedral units (sample S2 in Table 4.2.1) (right). The shaded region in (c) illustrates a 2-D hexagonal unit cell, while bulk and edge tetrahedral units are indicated by solid and dashed lines, respectively.

A range of sample sizes were fabricated and are summarized in Table 4.2.1. Note that

sample geometries were selected to contain only whole supporting members. Sample size was

defined by the total number of ‘edge’ ( edgeTn ) and ‘bulk’ ( bulk

Tn ) tetrahedral units, Table 4.2.1.

Bulk tetrahedral units had three struts, with each strut intersecting adjacent struts at a three-

rayed node. Edge units had only two struts and/or two-rayed node connectivity. Figure 4.2.1c

shows a schematic diagram of the S2 sample geometry having 19 edge and 14 bulk tetrahedral

units. Of the 19 tetrahedral units making up the sample perimeter, 11 had only two struts, while

8 had three struts, but only two-rayed node connectivity. The total number of tetrahedra ranged

from 8 to 52, while the fraction of bulk tetrahedra ranged from 0.03 to 0.52. The total number of

struts per sample ranged from 19 to 142 (Table 4.2.1).

Table 4.2.1. List of PCM truss core specimens tested and summary of architectural features: sample name (with number of repeats); total number of tetrahedra nT; number of edge edge

Tn and bulk tetrahedra bulk

Tn (with fraction of total); and total number of struts nS.

Name nT edgeTn bulk

Tn nS S1 (3) 52 25 (0.48) 27 (0.52) 142 S2 (3) 33 19 (0.58) 14 (0.42) 88 S3 (4) 15 11 (0.73) 4 (0.27) 38 S4 (6) 8 7 (0.88) 1(0.12) 19 X (3) 29 28 (0.97) 1(0.03) 71

Uniaxial compression testing of PCM truss cores was performed at a crosshead

displacement rate of 1 mm/min. Between three and six test trials were conducted for each

sample size (Table 4.2.1); a greater number of trials were conducted for smaller test specimens,

after [4]. Compressive strains were estimated from crosshead displacement [5,6,10,11]. The

64

confined-core uniaxial compression mode was used to simulate truss core collapse in a

sandwich panel. Single layer tetrahedral PCM truss cores were fixed between confinement

plates, which laterally confine the nodes resulting in an inelastic buckling failure mechanism.

The confinement plates were machined from AISI 1020 mild steel and had three sets of parallel

recessed channels (0.8 mm deep) offset by 120˚ in order to confine the tetrahedral PCM truss

core nodes, Figure 4.2.2.

Figure 4.2.2. PCM truss core placed in the recessed channels of the bottom confinement plate used to simulate the nodal confinement in a sandwich panel.

Reference sandwich structures were made by resistance brazing the tetrahedral truss-

cores to the same perforated sheet precursors used in forming; details can be found in [12]. Note

that the diameter of the resistance welding electrodes (1.6 mm) was larger than the diameter of

the steel dowel pin (1.1 mm) used for the perforation stretching process. This flattened the truss

nodes at the expense of strut length; see schematic diagrams in Figure 4.2.3. The change in strut

length was measured from cross-sections of the as-formed truss core and resistance brazed

sandwich panel; strut length decreased from 3.95 ± 0.15 mm in the as-formed PCM core to 3.48

± 0.13 mm in the resistance brazed sandwich panel. This change in strut length, accompanied by

a negligible change in truss height h (2.72 ± 0.02 mm for the as-formed core and 2.71 ± 0.06

mm in the sandwich), meant that the truss angle ω increased from 28.5 ± 0.1˚ in the as-formed

core to 32.8 ± 0.1˚ in the resistance brazed sandwich.

65

Figure 4.2.3. Schematic cross-section of the as-formed truss-core (left) and brazed sandwich panel (right) showing the truss angle ω, truss height h, strut length l, thickness t, and pin/electrode diameter (dpin and delectrode).


4.2.3.1 Simulating Sandwich Core Collapse

The force-displacement curve for uniaxial compression of the reference sandwich panel

(with stress-strain axes also shown) is presented in Figure 4.2.4.

Figure 4.2.4. Load (F)-displacement (d) and stress (σ)-strain (ε) curves of a confined-core PCM and a sandwich structure. Inset shows the load, unload, reload path used to fully seat the truss core in the confinement plate. Note that the strain was estimated from the cross-head displacement.

Post compression analysis showed that all struts failed by an inelastic buckling mechanism and

that none of the joints between truss core and facing sheet had failed. The same tetrahedral truss

cores used to fabricate the reference sandwich panel were also compression tested using the

confinement plates shown in Figure 4.2.2. Samples were loaded until the struts failed by

inelastic buckling, i.e. until just after the initial peak force in the force-displacement curve (FP).

66

A typical confined core force-displacement curve is shown with the reference sandwich panel in

Figure 4.2.4. Note that truss cores tested in the confinement plates showed an initial bedding-in

effect; however, once the truss core was fully seated in the confinement plate, e.g. by applying a

small preload of 0.33FP (see inset in Figure 4.2.4), it exhibited nearly the same elastic force-

displacement profile as the reference sandwich panel.

The relative peak strengths measured for the confined cores (1.88 ± 0.04 kN) and the

reference sandwich panels (2.07 ± 0.09 kN), can be considered in terms of the changes in truss

geometry caused by resistance brazing (Figure 4.2.3). Recent PCM compression studies that

have observed overall failure by inelastic buckling of supporting members (e.g. [13,14]) have

modeled the inelastic response of the PCM struts after Shanley [15]. These studies investigated

the axial load to invoke inelastic buckling, FInelastic, of a single ideal uniform column with

tangent modulus ET = ∂σ/∂ε:

12

3

2

22 wtl

EkF t

Inelasticπ

= (4.2.1)

where t and l are geometric parameters (Figure 4.2.3), w is the strut width, and k is a constant

reflecting the end constraints of the column. In practice, the inelastic buckling strength is

reduced by fabrication defects and geometric imperfections (e.g. [14,16]). This inelastic

buckling force per strut can be incorporated into PCM analysis to give a predicted upper limit

for the PCM peak force, FP, measured during compression testing as:

InelasticS

P Fn

Fωsin

= (4.2.2)

Here, ω is the truss angle and nS is the total number of struts in the sample. Figure 4.2.3

illustrates that resistance brazing with a larger diameter electrode than the forming dowel pin

had the effect of reducing the strut length, l, and increasing the truss angle, ω (i.e. effectively

strengthening the sandwich panel). Collecting the changing geometric factors in Eqs. (4.2.1) and

(4.2.2) gives the following ratio:

( )( )

2sinsin

P S C C

P S SC

F lF l

⎛ ⎞=⎜ ⎟⎝ ⎠

ωω

(4.2.3)

where subscripts S and C refer to the sandwich and confined core structures, respectively. The

observations of the present study agree well with the predictions of Eq. (4.2.3):

( ) ( ) 1.11 0.07P PS CF F = ± and ( )2 sin sin 1.14 0.09C S C Sl l ⋅ = ±ω ω . While it is expected that

there may be small differences in the end constraints for struts that have been brazed versus

67

struts that have been locked in place by the confinement plates, the present results suggest that

the measured difference in peak strength is largely determined by the changes in truss length

and truss angle during the joining step.

The elastic properties of the reference sandwich and confined truss cores can be

similarly compared. The loading stiffness SL was taken as the maximum value of the tangent to

the loading curve prior to FP. The measured SL values can be related to the axial stiffness of the

PCM struts. Assuming all struts are axially compressed simultaneously under the applied force,

the overall PCM stiffness per strut can be calculated using the work-energy method, e.g. [17]:

lwtE

S SL ω2sin

1= (4.2.4)

where ES is the Young’s modulus of the parent material. The geometrical factors controlling the

stiffness can be collected from Eq. (4.2.4) to give:

( )( )

2

2

sinsin

L S C C

L S SC

S lS l

⎛ ⎞=⎜ ⎟⎝ ⎠

ωω

(4.2.5)

where (SL)S and (SL)C refer to the loading stiffness of sandwich and confined core trusses

respectively. Using the experimentally measured strut lengths and truss angles, 2

2

sin 1.03 0.12sin

C C

S S

ll

⎛ ⎞= ±⎜ ⎟

⎝ ⎠

ωω

, which is consistent with the observation that there was no

significant difference in the elastic region of the load-displacement curves of Figure 4.2.4.

4.2.3.2 Edge Effects in Confined Core Compression

The effect of sample size and fraction of edge tetrahedra was examined by compression

testing a range of sample geometries all having the same truss core architecture. Samples were

loaded until failure occurred by the same inelastic buckling mechanism as seen for the reference

sandwich structure. The peak force values ranged from 3.15 ± 0.07 N for sample S1 (nS = 142)

to 0.38 ± 0.03 kN for sample S4 (nS = 19). This is shown in Figure 4.2.5 where the peak force

FP is plotted as a function of the total number of struts nS; a fitted linear relationship of FP/nS =

21.8 ± 0.3 N/strut is seen over the range from 19 to 142 total struts. Similarly, the maximum

loading slope before the peak strength increased from 0.81 ± 0.14 kN/mm for sample size S4 to

7.73 ± 0.56 kN/mm for sample size S1 and can be seen in Figure 4.2.6 giving a fitted linear

relationship of SL/nS = 51.5 ± 2.5 N/mm·strut. Note that these linear relationships occur despite

the significant differences in the relative fractions of edge to bulk tetrahedra between the

68

samples (Table 4.2.1). The relationships are also consistent with the prediction that edge effects

become less significant as the regularity of the cellular architecture increases [5].

Figure 4.2.5. Measured PCM peak force, FP, for the confined core samples as a function of the total number of PCM struts, nS.

Figure 4.2.6. Measured PCM maximum loading stiffness, SL, for the confined core samples as a function of the total number of PCM struts, nS.

In order to investigate the significance of edge effects further, special X-shaped samples

(Figure 4.2.7) were fabricated. These samples had 71 total struts, of which 68 belonged to edge

tetrahedral units. Of the 28 edge tetrahedral units making up the X-shaped sample perimeter, 16

had only two struts, while eight had three struts, but only two-rayed node connectivity. The

measured peak strength and loading stiffness of the X-shaped coupon was 1.40 ± 0.01 kN (19.7

± 2.0 N/strut) and 3.07 ± 0.23 kN/mm (43.3 ± 3.3 N/mm·strut); the peak is within one standard

deviation, and the stiffness is within two standard deviations from the linear relationships shown

in Figures 4.2.5 and 4.2.6. This small effect of edge tetrahedra fraction is in significant contrast

69

to conventional metallic foams where edge cells are less constrained and can cause a significant

decrease in measured values.

Figure 4.2.7. Schematic diagram of an X-shaped sample with high fraction (0.97) of edge tetrahedral (indicated by dashed lines).

For example, Andrews et al. [6] studied edge effects in a closed-cell Alporas foam with an

average cell size of 4.5 mm, as well as an open cell Duocel foam with cell size between 3.0 mm

and 4.5 mm. In both cases, a 26% decrease in peak stress was found from larger samples

(approximately 8 × 8 cells in area) to small samples (approximately 3 × 3 cells in area).

Similarly for both foams, a 20-29% decrease in unloading modulus was found for the same

decrease in ratio of specimen size to cell size. The difference between foam edge effects and

PCM edge effects may be understood more clearly through the analytical models of Onck et al.

[7] for honeycomb materials: for whole (i.e. integer number) of cells comprising the specimens,

the load-bearing capability is increased as the less capable cells (i.e. cut cells on the edges of the

specimen) are not present. The ideal model of Onck et al. [7] predicts that for specimens with an

integer number of cells along its length, the value of FP/FP,Bulk should remain equal to unity for

specimen length-to-cell size ratios down to 1; this appears to agree well with the current results.

In contrast, the ideal value of SL/SL,Bulk is expected to decrease to below 0.8 for specimen length-

to-cell size ratios smaller than 3. The results of the present study suggest that the reduction in SL

is not as significant for PCMs as suggested for honeycombs by Onck et al. [7]. However, this is

primarily because the confinement plates act to remove the non-load-carrying cells and maintain

uniform compressive deflection across the cells present. Due to the edge effects for conventional

metallic foams, minimum ratios for the number of cells per sample dimension have been given

70

in the range of 5-7 [5-8]. For PCMs, a similar (albeit smaller) minimum ratio can also be

suggested where incomplete edge cells are included in the specimen [5].

Size effects can also be considered in terms of the sample-to-sample variability for a

given test geometry. Figure 4.2.8 presents the standard deviation in the peak force

measurements as a function of the number of total struts.

Figure 4.2.8. Percent variation in confined-compression peak force FP (standard deviation divided by average FP) as a function of the total number of PCM struts within the specimens.

The % variation in FP (standard deviation divided by average FP) decreases from ~7% per strut

for the smallest samples (S4, 19 struts) to below 3% for the larger samples (S1, 142 struts). For

the limiting case of uniaxial compression testing of single columns, Perry and Chilver [18]

found 5% to 10% variation for inelastic buckling failure, which is consistent with the increasing

percent variation per strut with decreasing sample size found in the present study. Once

buckling is initiated in the weakest member, the load is redistributed to the remaining supports

which continue to buckle in a sequential fashion, e.g. [19-21]. On the other hand, the smaller

percent variation for larger PCM samples is consistent with previous studies of conventional

metallic foams, e.g. [6]. Furthermore, the % variation in SL decreases from ~17% (S4) to ~7%

(S1) with increasing specimen size. However, it should be noted that the percent variation

measured for PCMs in the present study is lower than the values typically seen for foams; in the

limit of testing a sufficiently large foam specimen to neglect size effects, the percent variation

ranges from 5% to 15% for strength and 5% to 30% for stiffness [5,6]. It has been shown that as

the number of cells in a cellular material increases, the statistical distribution of cell properties

becomes more continuous, e.g. [22]. Further, the load capacity decreases quickly with the

increasing percentage of failed cells [1]; therefore, the performance of samples with smaller

71

numbers of cells will rely heavily on the performance distribution of single cells. Overall, as the

dispersity of cell properties in a cellular material increases, its stability increases; this results in a

decrease in deformation localization and a decrease in the scatter between test specimens [21].

The present results suggest that the confinement plates effectively provide the same

periodically-rigid boundary conditions throughout the sample. FEM models have been used to

simulate this type of boundary condition as well as non-periodic conditions (e.g. rigid boundary

conditions applied only to edge cells) and found that cell collapse occurs at a higher stress for

the periodically-rigid constraints, e.g. [1,23]. The importance of periodic boundary conditions

can also be seen from the mechanical test data of Zupan et al. [24], in which two confinement

approaches were used. In the first, specimens were placed in a hollow box which restricted

nodal spreading at the outer edge (i.e. rigid but non-periodic boundary conditions). In the second

approach, nodes were bonded to face sheets using polyurethane adhesive providing a

periodically-rigid boundary condition. The non-periodic boundary conditions imposed by the

hollow box resulted in plastic hinge-like deformation and a peak stress similar to the unconfined

specimen (on steel platen interface). In contrast, the adhesively bonded specimen showed

buckling-like deformation and a marked increase in both peak stress and stiffness,

corresponding to a typical bulk PCM sandwich panel response. Overall, PCM cores can be

tested in compression as stand-alone materials provided that the appropriate boundary

conditions are applied to each truss core node. Furthermore, the confinement plates largely

eliminate edge effects, allowing the PCM mechanical properties to be determined from

relatively small scale samples.

4.2.4 Conclusions

This study developed a test method for evaluating the compressive performance of PCM

truss cores. Edge effects, which are widely observed in conventional cellular foam materials,

were examined in PCMs for stretching-dominated response (i.e. failure by inelastic buckling).

Periodically-rigid boundary conditions were applied uniformly to each node of the PCM truss

core using confinement plates, which simulates the behaviour in a sandwich panel. This test

method determines the buckling strength on a per-strut basis. No significant edge effects are

seen over an order of magnitude range in strut number. It was found that meaningful

compression data can be obtained from truss cores with specimens as small as 2 × 2 unit cells.

72

4.2.5 References 1. C Chen, TJ Lu, NA Fleck, J. Mech. Phys. Solids 47 (1999) 2235-2272.

2. A-F Bastawros, H Bart-Smith, AG Evans, J. Mech. Phys. Solids 48 (2000) 301-322.

3. C Chen, NA Fleck, J. Mech. Phys. Solids 50 (2002) 955-977.

4. R Brezny, DJ Green, J. Mater. Sci. 25 (1990) 4571-4578.



6. EW Andrews, G Gioux, P Onck, LJ Gibson, Int. J. Mech. Sci. 43 (2001) 701-713.

7. PR Onck, EW Andrews, LJ Gibson, Int. J. Mech. Sci. 43 (2001) 681-699.

8. PR Onck, MRS Bull. 29 (2003) 279-283.



11. U Ramamurty, A Paul, Acta Mater. 52 (2004) 869-876.

12. E Bele, The Effect of Architecture, Joint Strength and Joint Density on the Structural Performance of

Resistance Brazed Periodic Cellular Metals, B.A.Sc. Thesis, University of Toronto, Toronto, 2006.



15. FR Shanley, J. Aeronaut. Sci. 14 (1947) 261-268.

16. HNG Wadley, NA Fleck, AG Evans, Compos. Sci. Technol. 63 (2003) 2331-2343.

17. HNG Beer, ER Johnston, Mechanics of Materials, 2nd ed., McGraw-Hill, Toronto, 1992.

18. SH Perry, AH Chilver, Proc. Instn. Civ. Engrs. Part 2 – Res. Theory 61 (1976) 109-125.

19. LC Schmidt, PR Morgan, A Hanaor, J. Struct. Div. 108 (1982) 1324-1335.

20. A Hanaor, C Marsh, GAR Parke, J. Struct. Eng. 115 (1989) 1021-1037.

21. MW Schraad, Mech. Mater. 39 (2007) 183-198.

22. C Tekoglu, PR Onck, J. Mater. Sci. 40 (2005) 5911-5917.

23. A Öchsner, K Lamprecht, Mech. Res. Comm. 30 (2003) 573-579.

24. M Zupan, C Chen, NA Fleck, Int. J. Mech. Sci. 45 (2003) 851-871.

73

5 Micro-Truss Strengthening by In-Situ Work hardening This chapter presents work done by the author to investigate fabrication-induced or in-

situ work hardening as a strengthening mechanism in micro-truss PCMs. This strengthening

mechanism had not previously been considered for micro-truss materials; however, it may be

the only strengthening mechanism available to some precursors, e.g. the non-heat-treatable

aluminum alloys. By utilizing the plastic strain introduced during deformation-forming (e.g.

perforation-stretching) this chapter demonstrates that the PCM compressive strength can be

markedly improved. Section 5.1 presents an initial study conducted on a commercial aluminum

alloy. This section was part of a paper published in Materials Science and Engineering A [E

Bele, BA Bouwhuis, GD Hibbard, Work hardening as a strengthening mechanism in periodic

cellular materials, Mater. Sci. Eng. A 489 (2008) 29-37]. Section 5.2 investigates the relative

significance of in-situ work hardening in different precursor materials having a considerable

range of strength, ductility, and work-hardening capacity, where each uses work-hardening as a

primary microstructural strengthening mechanism. The results from this section show that new

regions in material property space can be achieved by retaining the strain energy introduced

during fabrication. This section is the second half of a paper [BA Bouwhuis, GD Hibbard,

Relative significance of in-situ work-hardening in deformation-formed micro-truss materials],

published in Materials Science and Engineering A (doi:10.1016/j.msea.2009.08.015).

5.1 Work Hardening as a Strengthening Mechanism in Periodic Cellular Materials

5.1.1 Introduction

Due to the nature of conventional foaming methods, work hardening is generally not

applicable as a strengthening mechanism in stochastic metal foams. However, it may be possible

to usefully apply work hardening in certain types of periodic cellular materials. For example,

periodic cellular metals can be fabricated by sheet forming operations [1-4], in which perforated

metal sheets are plastically deformed such that alternate nodes are displaced above and below

the starting plane, thus creating a three-dimensional truss structure. Sandwich structures are

subsequently assembled by joining the PCM cores to face sheets; deformation-formed PCM

cores have been joined using brazing (typically for aluminum and copper alloys [3,5–7]),

diffusion bonding (for titanium alloys [8]) or transient liquid phase bonding (for stainless steels,

74

superalloys and copper alloys [4,6,7,9]). While each of these approaches can produce

sufficiently strong joints, they subject the entire assembly to elevated temperatures that are often

high enough to remove the effects of prior strain hardening. An important example is the case of

non-precipitation hardenable aluminum alloys, such as AA3003, that are typically strengthened

by cold working; brazed structures (e.g., PCM core sandwiches) of non-heat treatable aluminum

alloys are expected to have the properties of the fully annealed material, e.g. [10]. The following

section examines the potential significance of work hardening as a strengthening mechanism in

deformation formed AA3003 PCMs.

5.1.2 Buckling Failure of AA3003 PCMs

PCM truss cores are composed of a series of supports (struts) arranged in a periodic

architecture; the failure of these struts marks the peak strength achievable by the PCM. The

idealized mechanical performance of tetrahedral and pyramidal PCM cores in compression can

be given by (e.g. [11]):

ωρσσ 2sinRFP = (5.1.1)

where ρR is the relative density, ω is the truss angle, and σF is the failure strength of the

columnar support. In the limiting case of stocky columns (i.e., small slenderness ratio), the struts

will fail by yielding and σF from Eq. (5.1.1) is replaced by the yield strength σYS. In contrast,

PCM struts having medium-to-high slenderness ratios will fail by buckling, in which case the

failure strength is determined by the critical buckling stress σCR after Shanley [12], which can be

expressed as [3,11,13]:

( )2

22

2

22

/ rlEk

AlIEk tt

CRππ

σ == (5.1.2)

where Et is the tangent modulus (Et = ∂σt/∂εt), A is the cross-sectional area of the supporting

member, I is the moment of inertia, and k describes the rotational stiffness of the column. The

critical buckling stress can also be expressed in terms of the slenderness ratio (l/r) where r is the

radius of gyration (in the present case r = t/√12). Note that in the case of very high slenderness

ratios, Et ≡ ES (Young’s modulus) and elastic buckling occurs.

The Ramberg–Osgood model (ROM) [14] is a simple constitutive relationship that can

be used to describe the elastic to plastic stress–strain behaviour and can be expressed as [20]: N

YS

t

S

tt E ⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

σσ

εσ

ε 0 (5.1.3)

75

where ES is the Young’s modulus (ES = 70 GPa [16]), N is a strain hardening exponent, σYS is

the yield strength from the 0.2% offset method (e.g. [15,17]), and ε0 is the plastic strain

corresponding to σYS (ε0 = 0.002). This constitutive model is useful in modeling inelastic

buckling behaviour [18] and has been used to model the behaviour of both hardened and

softened aluminum alloys, e.g. [15,19]. True stress–strain curves from published ASM data can

be used to determine approximate Ramberg–Osgood parameters for AA3003 in the annealed (O

temper), quarter hard (H12 temper), half-hard (H14 temper), three-quarter hard (H16 temper)

and fully hard (H18 temper) conditions [20]. In addition, the stress–strain behaviour of a lightly

rolled (one cold-roll pass) condition (H111 temper) has been investigated [21]. Table 5.1.1

summarizes the ROM parameters for the five AA3003 tempers; the fitted parameters are similar

to those typically reported for hard and soft aluminum alloys, e.g. [18,19].

Table 5.1.1. Summary of AA3003 Ramberg–Osgood fitting parameters: Young’s modulus (ES), 0.2% offset yield strength (σYS), plastic strain at σYS (ε0), and strain-hardening exponent N.

Condition ES (GPa) σYS (MPa) ε0 (mm/mm) N O [20] 70 36 0.002 3.6 H111 [21] 70 67 0.002 7.0 H12 [20] 70 119 0.002 13 H14 [20] 70 147 0.002 17 H18 [20] 70 195 0.002 18

The tangent modulus can be obtained from the Ramberg–Osgood relationship as:

11

0

11

−−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=N

YS

t

YSSt

tt N

EE

σσ

σε

σε

(5.1.4)

The slenderness ratio at which σt = σCR can then be expressed as [18]:

t

teff Er

lσ

π=⎟⎟⎠

⎞⎜⎜⎝

⎛ (5.1.5)

where leff is the effective length, which takes into account both the strut geometry and end

constraints, i.e., leff = (l/k). Figure 5.1.1 presents the critical buckling stress as a function of the

slenderness ratio for the five AA3003 tempers summarized in Table 5.1.1 along with their

respective yield strengths. Several different regimes can be identified with respect to the

slenderness ratio. At very high slenderness ratios (leff/r >~350), each of the tempers (ranging

from fully hard (H18) to fully annealed (O)) fail by elastic buckling and there is no performance

advantage to be gained by incorporating work hardening as a strengthening mechanism.

76

Figure 5.1.1. Column curves giving the slenderness ratio as a function of the critical stress for the Ramberg-Osgood constitutive relationships of AA3003 (see Table 5.1.1).

However, for the range of ~80 < leff/r <~350, the softer tempers successively deviate from the

elastic buckling curve due to their lower yield strength and fail instead by an inelastic buckling

mechanism. For this slenderness regime, incorporating work hardening increases the strut yield

strength and maintains elastic buckling to smaller slenderness ratios. Finally, for slenderness

ratios of leff/r <~80, the fully hard (H18) temper deviates from the elastic buckling curve,

placing all conventional AA3003 tempers within the inelastic buckling range. In this regime, the

benefit to work hardening as a strengthening mechanism is fully realized: struts with greater

yield strength will buckle inelastically at higher critical buckling strengths. For example, at a

slenderness ratio of 40, a yield strength equivalent to H18 will have a fourfold increase in

inelastic buckling strength (and strength-per-weight ratio) over a fully annealed strut. At still

smaller slenderness ratios, i.e., as σCR approaches σYS, there is a transition to failure by yielding.

There is therefore significant potential for using work hardening as a strengthening

mechanism when PCM failure occurs by inelastic buckling. In the following study, pyramidal

AA3003 truss cores are fabricated by a perforation-stretching method and the mechanical

properties of work-hardened and annealed truss cores are compared. Next, an alternative joining

method (resistance brazing) is developed with the objective of localizing the heat-affected zone

to the truss core nodes such that work hardening can be usefully retained in the struts of the final

sandwich structure. Finally, PCM sandwich prototypes are developed and their mechanical

properties evaluated.

77

5.1.3 Experimental

The starting material was a perforated AA3003-H14 sheet (of thickness to = 0.81 mm)

purchased from Woven Metal Products Inc. (Alvin, TX). The 90.82 mm2 square perforations

were punched on a 2D square lattice of unit cell size 12.7 mm × 12.7 mm, creating an open area

fraction of φ = 0.56. The as-received sheet was strain relief annealed at 415˚C for 1 h [22] to

improve the formability. The pyramidal cores (Figure 5.1.2) were fabricated by deforming

alternating nodes above and below the starting plane using a modified perforation-stretching

process, similar to that described by Sypeck and Wadley [2]. The cores had a final height (h) of

8.0 mm, a truss angle (ω) of 37.5° (Figure 5.1.2), and a relative density of 5.0%. PCM truss

cores were cut into 36-node squares for a projected 2D area of 4440 mm2. A subset of five as-

fabricated structures were subjected to a typical conventional brazing schedule (5 min at 595˚C)

that has been used to join PCM truss cores of the same AA3003 alloy [5]. Details on the

resistance brazing methodology can be found in [23] and Appendix A.

Figure 5.1.2. Schematic diagram of pyramidal unit (left) and cross-section of a constituent strut (right) having dimensions of strut length l, truss height h, sheet thickness t, and truss angle ω.

Microhardness profiles were measured along the struts of the strain relief annealed, as-

formed, heat-treated, and joined truss cores. Samples were mounted longitudinally in epoxy to

expose the strut length cross-section and prepared using standard metallographic techniques.

Measurements were taken using an MHV 2000 microhardness tester with a 0.49 N applied load

and 10 s dwell time. Uniaxial compression testing was performed at a cross-head displacement

rate of 1 mm/min. Five test samples were used for each condition, after ASTM-C365 [24].

Resistance brazed sandwich panels were loaded directly between the tempered steel

compression platens. Stand-alone PCM cores were tested using confinement plates (i.e.,

recessed channels in steel compression plates that rigidly lock the truss core nodes in place –

78

details in [25]). This test method can be used to simulate the mechanical performance that stand-

alone truss cores would exhibit in a sandwich structure [25].


The potential of work hardening as a strengthening mechanism in AA3003 pyramidal

PCM cores was examined by characterizing as-fabricated (work-hardened) and annealed

(simulated brazing heat treatment of 5 min at 595˚C) trusses. Figure 5.1.3 presents an optical

micrograph showing the microhardness profiles that were recorded along each side of a

longitudinal strut cross-section, designated as ‘inner’ and ‘outer’ profiles. Each profile was

subdivided into three regions based on the hardness increase and relative strut thickness

reduction introduced during the perforation-stretching forming step (Figure 5.1.3).

Figure 5.1.3. Optical micrograph showing profiles of inner and outer microhardness indentations on a work-hardened (as-fabricated) pyramidal truss core and the subdivision into node (I), hinge (II), and strut (III) regions.

Region II has the largest reduction in thickness and is the plastic hinge that separates the node

(I) and strut (III) regions. Hardness values for both profiles are plotted in Figure 5.1.4 as a

function of position from the node centre. For reference, the microhardness profiles of the

starting sheet (i.e., after strain relief annealing but before deformation forming) are also shown

in Figure 5.1.4; the average microhardness of the strut before PCM fabrication was 36.7 ± 1.1

HV. During perforation stretching the hardness of the strut (Region III) increased to 44.0 ± 1.4

HV, while the plastic hinge (Region II) experienced the greatest work hardening with an as-

fabricated hardness of 48.5 ± 0.6 HV.

79

Figure 5.1.4. Microhardness profiles along the inner (left) and outer (right) paths. Regions I (node), II (hinge), and III (strut) are defined in Figure 5.1.3. The midpoint of the deformation formed strut occurs at a profile distance of ~8 mm.

On the other hand, essentially all of the work hardening has been removed during the simulated

brazing heat treatment and the hardness was reduced to 35.8 ± 1.0 HV in Region III and 37.5 ±

0.7 HV in Region II (i.e., to essentially the same value as the starting material).

Figure 5.1.5 presents typical stress–strain curves for the as fabricated and annealed truss

cores. While both sets of materials undergo the same type of inelastic buckling failure (Figure

5.1.5), the brazing heat treatment has a significant effect on the mechanical performance.

Figure 5.1.5. Compressive stress-strain curves of work-hardened and heat-treated (5 min at 595˚C) cores (left) and strut deformations corresponding to position 1 (top right) and position 2 (bottom right) of the stress-strain curves. The peak strength (σP) and densification energy (JD) is shown for the heat-treated core.

80

First, the peak strength (σP), defined as the maximum stress supported by the struts prior to

buckling, decreases from 0.98 ± 0.04 MPa for the work-hardened case to 0.50 ± 0.02 MPa for

the heat-treated cores. In other words, nearly half of the potential peak strength of the truss core

is lost during the simulated brazing heat treatment. In addition, the compressive modulus

(maximum slope of the stress–strain curve in the pre buckling region) decreases from E = 21.5 ±

2.1 to 16.6 ± 2.8 MPa and the densification energy (energy absorbed between 1/2σP and 2σP

[26]) decreases from JD = 588 ± 37 to 225 ± 11 kJ/m3 after the simulated brazing heat treatment.

For the two sets of macroscopically identical PCM truss core architectures in the present

study, the peak strength ratio can be expressed as

( )( )

( )( )HTCR

WHCR

HTP

WHP

σσ

σσ

= (5.1.6)

where the subscripts WH and HT refer to the work-hardened and heat-treated conditions,

respectively. The as-formed strut length and thickness in the present study are 11.0 mm and 0.8

mm, respectively. A rigid joint assumption (where k = 2 (e.g. [27])) gives a reasonable upper

estimate for the effective length (where leff/r = 48), while a pin joint assumption (where k = 1

(e.g. [27])) provides a reasonable lower estimate with leff/r = 24. The critical buckling stress for

these two slenderness ratios are plotted in Figure 5.1.6 as a function of the material yield

strength, i.e., a plot of the intersection of the horizontal lines for slenderness ratios of 24 and 48

with the column curves in Figure 5.1.1. In general, the trend shows that the critical buckling

strength increases with strut yield strength. The yield strength in the as-fabricated and heat-

treated struts can be estimated from microhardness measurements using empirical correlations

for aluminum alloys developed by Cahoon et al. [28]. The yield strength can be given by

( )nYS

HV 1.03

=σ (5.1.7)

where HV is the Vickers microhardness in MPa and n is the Holloman strain hardening

exponent [27]. Holloman exponents for AA3003 range from 0.24 for the annealed O-temper to

~0.06 for the H12, H14, and H18 work-hardened tempers [20]. The PCM cores given the

simulated brazing heat treatment had a mean strut hardness of 36 HV (Region III – Figure

5.1.4), which is approximately the same as the hardness of the H111 temper (34 HV calculated

using Eq. (5.1.7) and n = 0.215 [21]). The strain hardening exponent for H111 can be used to

provide a reasonable estimate of 72 MPa for the heat-treated PCM strut yield strength. In

contrast, the as-fabricated truss core had an average strut hardness of 44 HV, which is

approximately equal to the hardness of the H12 temper (42 HV using n = 0.06 [20]). Using the

81

H12 strain hardening exponent of n = 0.06 gives an estimate of σYS = 125 MPa for the work-

hardened strut. These yield strength estimates can be used in Figure 5.1.6 to give the critical

buckling stress for the two limiting slenderness ratios; values range from 60 to 74 MPa for the

heat treated core and from 105 to 120 MPa for the work-hardened core. The difference in these

ranges is consistent with the factor of two increase in buckling strength achieved through work

hardening.

Figure 5.1.6. The critical buckling strength, determined by the intersection of the slenderness ratios leff/r = 24 and leff/r = 48 with the AA3003 column curves in Figure 5.1.1, plotted against the yield strength of each AA3003 temper.

One approach that may be useful in preserving work hardening in the PCM struts would

be to localize the heat input to the truss core nodes. Resistance spot welding is an obvious

candidate to localize the heat input. A window of resistance brazing process parameters was

developed for joining the AA3003 micro-truss sandwich panels in [23]. Prototype sandwich

panels were joined using two limiting brazing conditions of high current/short time (Setting 1)

and low current/long time (Setting 2). Figure 5.1.7 presents an example of a pyramidal PCM

core that was resistance brazed to the perforated facing sheets.

Figure 5.1.7. Resistance brazed sandwich panel with pyramidal PCM core.

82

Sections were cut from sandwiches that were joined using both settings, mounted in cross-

section, and prepared using standard metallographic methods. Microhardness profiles were

recorded along both sides of the strut and were divided into node (I), hinge (II), and strut (III)

regions (Figure 5.1.8). Figure 5.1.9 summarizes the hardness profiles for the two resistance

brazing settings and gives reference hardness profiles for the as-fabricated and heat-treated

cores. There was little difference between the hardness profiles of the two resistance brazed

sandwiches.

Figure 5.1.8. Optical micrograph showing profiles of inner and outer microhardness indentations on a resistance brazed PCM sandwich panel and the subdivision into node (I), hinge (II), and strut (III) regions.

Figure 5.1.9. Microhardness profiles along the inner (left) and outer (right) paths. Regions I (node), II (hinge), and III (strut) are defined in Figure 5.1.8. The midpoint of the deformation formed strut occurs at a profile distance of ~8 mm.

83

Similar to the case for the simulated conventional brazing treatment, resistance brazing

decreases the hardness in the hinge region from 49 HV in the as-fabricated condition to 36 HV

after joining. However, for the case of resistance brazing, this heat-affected zone only extends

~1.2 mm into the 12.4 mm strut. In fact both the as-fabricated core and the resistance brazed

sandwiches have approximately the same mid-strut hardness of 44 and 43 HV, respectively.

While conventional brazing and resistance brazing result in essentially the same hardness

reduction in the heat-affected zone, resistance brazing has had the desired effect of localizing

the thermal exposure to the near-node region.

The overall mechanical performance of the resistance brazed sandwich structures was

measured in compression. Figure 5.1.10 presents typical stress-strain curves for the prototype

PCM sandwich panels as well as reference stress-strain curves for the as-fabricated and heat-

treated cores. It was observed that none of the external corner or edge joints failed by shear

through the filler metal (i.e., no failure in those joints subjected to unsymmetrical loading).

Instead, the resistance brazed struts collapsed by the same inelastic buckling mechanism (Figure

5.1.10) that was observed for the as-fabricated and heat-treated cores tested in confined

compression (Figure 5.1.5).

Figure 5.1.10. Compressive stress-strain curves of resistance brazed sandwiches (left) and strut deformations corresponding to position 1 (top right) and position 2 (bottom right) of the stress-strain curves. Also shown for reference in (a) are stress-strain curves for the work-hardened and heat-treated cores.

The effect of localizing the heat-affected zone to the truss core nodes can be clearly seen. The

peak strengths of the resistance brazed sandwich panels were 0.95 ± 0.04 MPa (Setting 1) and

0.92 ± 0.05 MPa (Setting 2), showing a 90 and 84% improvement on the peak strength of the

84

truss cores subjected to a conventional brazing heat treatment (0.50 ± 0.02 MPa). Furthermore,

the compressive modulus of the resistance brazed sandwiches, 22.3 ± 1.7 MPa (Setting 1) and

21.9 ± 3.1 MPa (Setting 2), were approximately the same as that of the as-fabricated core (21.5

± 2.1 MPa) and 33% higher than for the cores subjected to the conventional brazing treatment

(16.6 ± 2.8 MPa). The fact that resistance brazed peak strengths are only ~5% lower than the

peak strength of the as-fabricated structure, means that much of the work hardening induced

during the perforation-stretching process has been usefully retained in the final PCM assembly.

5.1.5 Conclusions

The present results indicate that there is significant potential in using work hardening to

improve the overall performance of PCM truss cores. Work-hardened cores exhibited a factor of

two increase in peak strength over heat-treated cores having the same architecture and it was

found that this difference was consistent with the expected increase in the critical inelastic

buckling strength of the struts. In order to preserve work hardening in the joined PCM core

sandwich panels, a resistance brazing method was developed to localize the heat-affected zone

to the truss nodes. Microhardness profiles of prototype PCM sandwich panels showed that much

of the work hardening was preserved in the struts and compression testing showed only a ~5%

decrease in peak strength after resistance brazing.

5.1.6 References 1. HNG Wadley, NA Fleck, AG Evans, Compos. Sci. Technol. 63 (2003) 2331-2343.




374.


6. T Wen, J Tian, TJ Lu, DT Queheillalt, HNG Wadley, Int. J. Heat Mass Trans. 49 (2006) 3313-3324.

7. DJ Sypeck, Metall. Mater. Trans. B 36 (2005) 125-131.

8. RV Safiullin, OA Sudenko, FU Enikeev, RY Lutfullin, Mater. Sci. Forum 243–245 (1997) 769-774.

9. J Tian, T Kim, TJ Lu, HP Hodson, DT Queheillalt, DJ Sypeck, HNG Wadley, Int. J. Heat Mass Trans. 47

(2004) 3171-3186.

10. Metals Handbook, vol. 6: Welding, Brazing and Soldering, ASM International, Materials Park, 1993.

11. VS Deshpande, NA Fleck, Int. J. Sol. Struct. 38 (2001) 6275-6305.

12. FR Shanley, J. Aeronaut. Sci. 14 (1947) 261-268.

13. S Chiras, DR Mumm, AG Evans, N Wicks, JW Hutchinson, S Fichter, K Dharmasena, HNG Wadley, Int. J.

Solids Struct. 39 (2002) 4093-4115.

85

14. W Ramberg, WR Osgood, Description of Stress–Strain Curves by Three Parameters, Technical Note 902,

NACA, Washington, 1943.

15. A de Martino, R Landolfo, FM Mazzolani, Mater. Struct. 23 (1990) 59-67.

16. ASM Handbook, vol. 2: Properties and Selection – Non-ferrous Alloys and Special-Purpose Materials, ASM


17. AS Niles, JS Nevell, Airplane Structures, 4th ed., John Wiley & Sons, New York, 1954.

18. FR Shanley, Strength of Materials, McGraw-Hill, New York, 1957.

19. KJR Rasmussen, J Rondal, Eng. Struct. 23 (2000) 1505-1517.

20. Atlas of Stress–Strain Curves, 2nd ed., ASM International, Materials Park, 2002.

21. N Abedrabbo, F Pourboghrat, J Carsley, Int. J. Plast. 22 (2006) 314-341.


23. E Bele, The Effect of Architecture, Joint Strength and Joint Density on the Structural Performance of


24. ASTM C365-06, American Society for Testing and Materials, Philadelphia, 2006.

25. BA Bouwhuis, GD Hibbard, Metall. Mater. Trans. B 37 (2006) 919-927.

26. OB Olurin, NA Fleck, MF Ashby, Mater. Sci. Eng. A 291 (2000) 136-146.



28. JR Cahoon, WH Broughton, AR Kutzak, Metall. Trans. 2 (1971) 1979-1983.

86

5.2 Relative Significance of In-Situ Work Hardening in Micro-Trusses

5.2.1 Introduction

The present section investigates the relative significance of in-situ work-hardening in

micro-truss sandwich cores fabricated from stainless steel (SS304), electrolytically-pure copper

(Cu110), and aluminum alloy (AA3003). As outlined in Section 3.2, these alloys possess a

considerable range of strength, ductility, and strain hardening characteristics; each of these

alloys also uses work-hardening as their primary microstructural strengthening mechanism. In

this section, an analytical model is developed which uses the tensile flow stress of the individual

struts during forming to predict the inelastic buckling resistance of the micro-truss struts. With

the results from Section 3.2, this approach can be used to determine the regions of strength-

density material property space accessible for a perforation-stretched micro-truss, using simply

the tensile properties of the starting material.

5.2.2 Materials and Experimental Details

For the most formable precursor materials: SS304 (as-received), Cu110 (annealed), and

AA3003 (annealed) – the respective upper forming limits were used to determine a safe limit for

fabricating PCMs for compression testing. The forming displacement dForm was chosen from a

minimum of 5.35 mm to the maximum uniform forming displacement (see Section 3.2, Table

3.2.2), i.e. to ~95% dM for SS304, and ~90% dM for Cu110 and AA3003. Table 5.2.1

summarizes the architectures and their respective forming parameters. All micro-truss

specimens were compression-tested in the work-hardened state and in a post-forming annealed

state. For AA3003 PCMs in the latter, the heat schedule followed previous AA3003 micro-truss

brazing treatments (595˚C for 5 minutes) [1]. Similarly, for the annealed SS304 PCMs, heat

treatment was conducted at 1100˚C for 1 hour to follow previous SS304 micro-truss brazing

schedules [2]. For the Cu110 PCM truss cores, annealing repeated the initial softening treatment

(500˚C, 1 hour [3]).

87

Table 5.2.1. Summary of micro-truss architectures and properties for SS304, Cu110, and AA3003 PCMs formed to a range of architectures. Parameters include: forming displacement dForm, truss angle ω (after [4]), micro-truss height h, relative density ρR, total density ρ, strut slenderness ratio λ, forming force FForm, measured compressive strength σP, and the strength ratio Ω = σP,WH/σP,FA. WH and FA denote work-hardened and annealed states, respectively.

Micro-Truss

# dForm (mm)

ω h (mm)

ρR (%)

ρ (Mg/m3)

λ FForm (kN)

σP,WH (MPa)

σP,FA (MPa) Ω

1 5.35 22.8 5.42 ± 0.05 4.36 0.345 34.5 ± 1.3 17.49 ± 0.07 2.06 ± 0.02 0.96 ± 0.02 2.15 ± 0.06 2 6.05 25.5 6.09 ± 0.07 3.89 0.307 35.9 ± 1.3 19.74 ± 0.06 2.67 ± 0.04 1.10 ± 0.01 2.42 ± 0.06 3 6.69 27.8 6.68 ± 0.04 3.54 0.280 36.9 ± 1.2 22.09 ± 0.08 2.88 ± 0.03 1.20 ± 0.01 2.39 ± 0.05 4 7.47 30.5 7.44 ± 0.04 3.18 0.251 38.1 ± 1.3 24.54 ± 0.10 3.27 ± 0.07 1.26 ± 0.01 2.60 ± 0.08 5 8.04 32.3 7.94 ± 0.07 2.98 0.235 39.2 ± 1.3 26.48 ± 0.07 3.36 ± 0.06 1.28 ± 0.01 2.63 ± 0.07 6 8.95 35.2 8.83 ± 0.06 2.68 0.212 41.3 ± 1.4 29.24 ± 0.11 3.58 ± 0.02 1.28 ± 0.03 2.80 ± 0.09

SS304

7 10.05 38.4 9.85 ± 0.06 2.40 0.190 44.9 ± 1.3 31.88 ± 0.12 3.52 ± 0.05 1.19 ± 0.03 2.96 ± 0.12 1 5.35 22.8 5.78 ± 0.08 3.76 0.336 37.2 ± 1.4 4.42 ± 0.03 0.74 ± 0.02 0.30 ± 0.00 2.51 ± 0.08 2 6.05 25.5 6.58 ± 0.07 3.30 0.295 38.2 ± 1.4 5.45 ± 0.01 0.80 ± 0.01 0.31 ± 0.00 2.54 ± 0.06 3 6.79 28.1 7.22 ± 0.05 3.01 0.269 39.9 ± 1.4 6.16 ± 0.03 0.90 ± 0.03 0.32 ± 0.00 2.84 ± 0.13 4 7.30 29.9 7.63 ± 0.07 2.85 0.255 41.2 ± 1.5 6.71 ± 0.01 1.02 ± 0.02 0.36 ± 0.01 2.86 ± 0.11 5 8.00 32.2 8.34 ± 0.07 2.60 0.233 43.3 ± 1.7 7.38 ± 0.01 1.01 ± 0.05 0.35 ± 0.02 3.19 ± 0.19

Cu110

6 8.40 33.5 8.78 ± 0.05 2.47 0.221 45.0 ± 1.7 7.66 ± 0.03 0.97 ± 0.02 0.33 ± 0.01 3.09 ± 0.10 1 5.35 22.8 5.84 ± 0.07 3.63 0.099 36.2 ± 1.4 2.71 ± 0.01 0.36 ± 0.01 0.23 ± 0.00 1.61 ± 0.06 2 6.05 25.5 6.54 ± 0.09 3.24 0.088 38.7 ± 1.5 3.04 ± 0.01 0.42 ± 0.01 0.26 ± 0.00 1.63 ± 0.05 AA3003 3 6.69 27.8 7.16 ± 0.07 2.96 0.081 42.4 ± 1.6 3.31 ± 0.01 0.46 ± 0.02 0.26 ± 0.00 1.78 ± 0.11

88

Optical microscopy was used to characterize the microstructure of the fabricated micro-

trusses; samples were prepared using standard metallographic techniques and etched with

Barker’s reagent (AA3003) and acetal glyceria (SS304). A confined compression method was

used to test the mechanical properties of the fabricated truss-cores [5,6]. This method laterally

confines the PCM nodes, simulating the behaviour the core would experience as part of a

sandwich panel. The measured peak strength was taken as the maximum value after the initial

loading slope, measured using a specimen compression area of 29.1 cm2 (2 × 2 pyramidal cells).


Figure 5.2.1 presents typical PCM compression stress-strain curves for work-hardened

and fully annealed AA3003 PCMs (architecture type #3, Table 5.2.1) as well as for SS304

micro-trusses (architecture type #7, Table 5.2.1). In all cases, the measured initial peak strength

was controlled by inelastic buckling of the micro-truss struts. This strength was higher for the

work-hardened samples than the post-forming annealed samples.

Figure 5.2.1. Measured compression stress-strain curves for work-hardened and annealed #3 3003 PCMs (left). The peak strength was controlled by inelastic buckling of the micro-truss ligaments (Inset). Stress-strain curves for work-hardened and annealed #7 SS304 micro-trusses are shown in (right) with reference curves for #3 AA3003 micro-trusses. Tests performed on the SS304 micro-trusses were stopped following the initial peak strength.

The improvement in peak strength is represented in Table 5.2.1 by the micro-truss strength ratio

Ω = σP,WH/σP,FA. This ratio increased with forming displacement: in AA3003, for example, a

relatively small dForm of 5.35 mm lead to Ω of 1.61, whereas increasing dForm to 6.69 mm raised

Ω to 1.78. The value of Ω also varied considerably between the precursor materials. At a

constant forming displacement of dForm of 6.69 mm, Ω increases from 1.78 in AA3003 to 2.39 in

89

SS304 to 2.84 in Cu110. The largest value, Ω = 3.09 ± 0.10, was obtained for Cu110 that had

been formed to dForm = 8.40 mm.

The significant difference in strut microstructure between the work-hardened and

annealed states can be seen in Figure 5.2.2. It is important to note that the annealed

microstructures (created after following typical AA3003 [1] and SS304 [2] micro-truss

sandwich panel brazing schedules) possess a grain size that approaches the thickness dimension

of the micro-truss struts (Figure 5.2.2): there was an average of six grains through the AA3003

strut thickness an average of eight grains through the SS304 strut thickness. These values fall

within the range of critical sample size to grain size ratios, e.g. [7], which can lead to decreases

in strength, larger statistical variation in tensile characteristics, smaller fatigue endurance limit,

and reduced ductility, e.g. [7,8]. The relative scale of annealed microstructures with respect to

the micro-truss dimensions may therefore become an issue in the design of these cellular

materials and it would be expected to become increasingly important as the dimensions of the

micro-truss architectures are scaled down further.

Figure 5.2.2. Mid-strut microstructures of AA3003 (left) and SS304 (right) for the micro-truss compression curves shown in Figure 5.2.1. Post-fabrication annealing (following typically-used micro-truss sandwich panel brazing schedules) results in a grain size approaching the thickness of the micro-truss struts.

Micro-truss strut failure leads to the initial peak load. Using a force balance, the ideal

peak force of a micro-truss PCM in compression can be given by:

Ll

dFnF Form

FSP +Δ= (5.2.1)

90

where FP = σP×ATruss, FF is the failure force of a columnar support, and the parameter nS

accounts for the number of struts in the micro-truss (in the present case, nS = 44). For the case of

inelastic buckling, FF becomes the critical force FCR and can be expressed as [9]:

Ligamentt

LigamentCRCR AEk

AF 2

22

λπ

σ == (5.2.2)

where Et is the tangent modulus (Et = ∂σt/∂εt, where σt and εt are the true stress and strain, resp-

ectively) and k describes the rotational stiffness of the column. The slenderness ratio λ is defined

as l/r, where l is the reduced section length of the formed strut and r its radius of gyration.

A link can be made between the forming forces and the micro-truss compressive

properties. On one hand, the terminal forming force FForm (Section 3.2, Eq. (3.2.3)) contains the

ligament flow stress and can be used as a measure of the work-hardened strut yield strength σYS.

On the other hand, the truss collapse force FP depends on the critical ligament strength σCR (Eq.

(5.2.1)). It should be expected that σCR is close to σYS: it is in the early plastic yielding region

(i.e. near σYS) where Et = ∂σt/∂εt decreases most quickly from the elastic modulus to satisfy Eq.

(5.2.2). In order to develop a relationship between σYS and σCR for a range of FCC materials

having widely different strength, ductility, and work-hardening capabilities, additional data was

analyzed as follows: ASM published tensile stress-strain curves [10] were collected for

aluminum AA1100 (O, H12, H14, H18, and H24 tempers), AA3003 (O, H12, H16, H18, and

H26 tempers), copper Cu110 (5 work-hardened states), nickel Ni200 (one state), as well as for

stainless steel SS304 (two states); each curve was converted to a column curve, after [9]; the

critical buckling strengths were determined for slenderness ratios of 30, 40, and 50 in order to

bracket the slenderness ratios used in the present study (Table 5.2.1); and the critical buckling

stress was plotted as a function of the material yield strength (Figure 5.2.3). For each

slenderness ratio and all materials, the data can be fit to a power-law, σCR = κ(σYS)α; specific

values of κ, α, and R2 are summarized in Table 5.2.2. In stocky columns (e.g. λ = 30), κ and α

are closest to 1; as the slenderness ratio increases, κ and α increasingly deviate from 1 due to the

shape of the column curve, e.g. [9]. Therefore, as a first approximation, a correlation between

forming and compression can be represented by a similar power-law relationship, FP =

ξ(FForm)φ, and compared to the ideal σCR = κ(σYS)α correlation for the average slenderness ratio (λ

= 40) of the PCM architectures and precursor materials in this study (Table 5.2.1). The

parameter α undergoes the smallest changes over the slenderness ratio range of 30 ≤ λ ≤ 50

(Table 5.2.2), and as a first approximation φ was set equal to α, while the coefficient ξ was used

as a fitting parameter for the FP and FForm data.

91

Figure 5.2.3. Correlation between yield strength and critical buckling stress based on published ASM tensile curves [10] for columns having slenderness ratios of λ = 30, 40, and 50.

Table 5.2.2. Power-law fitting parameters κ and α in the relationship σCR = κ(σYS)α for slenderness ratios λ bracketing those fabricated in the present study (Table 5.2.1).

λ κ α R2 30 1.05 ± 0.06 0.98 ± 0.01 0.999 40 1.17 ± 0.05 0.94 ± 0.01 0.999 50 1.28 ± 0.12 0.92 ± 0.02 0.996

Figure 5.2.4 presents a strength – density (σP – ρ) material property space map for all

experimental samples and the respective model trends. The work-hardening trends were

obtained by taking the forming curves modelled after tensile tests (see Section 3.2, Figure

3.2.5), then converting the forming force FForm to the theoretical collapse force FP at every dForm

using the correlation FP = ξ(FForm)α. Similar values of ξWH were found to fit the experimental

work-hardened micro-truss data for each of the three precursor materials: ξWH = 0.50 (SS304),

0.54 (Cu110), and 0.53 (AA3003). Predicted trends for the annealed micro-trusses were

obtained by modelling the forming curve after an ideally plastic material, i.e. FLigament = g(Δl) =

σYS×A. A good fit to the experimental annealed micro-truss data was found for each of the the

fully-annealed micro-trusses: ξFA = 0.55 (SS304), 0.53 (Cu110), and 0.61 (AA3003).

92

Figure 5.2.4. The strength – density (σP – ρ) material property space map for the micro-trusses in this study, showing the experimental data points and analytical trendlines. Increasing the forming displacement corresponds to a property space traversal of increasing σP with decreasing ρ.

Some insight into the physical significance of the fitting parameters ξWH and ξFA may be

found by considering their relationship to the ideal column fitting parameter κ. For example,

strut imperfections reduce the inelastic buckling resistance and can lead to the sequential rather

than simultaneous failure of the micro-truss struts [11-13]. In the annealed samples, the average

ξFA was approximately half the ideal value of κ, which corresponds to the same order of strength

knockdown factor as those measured in previous micro-truss studies, e.g. ~0.6 to 0.8 [14-17].

It is also important to consider the potential significance of the Bauschinger effect, since

the reversal of stress direction between forming and compression may introduce a secondary

knockdown effect. In the simplest case, it has been shown that changing strain from a forward

(tensile) direction to a reverse (compression) direction can reduce the effective yield strength in

stainless steel, electrolytic copper, and dispersion-hardened aluminum alloys, e.g. [18-25]. For

example, the ratio of effective reverse yield strength to forward yield stress σRev/σFor can range

from approximately 0.53 to 0.77 for stainless steels [18-20], from 0.50 to 0.65 for

polycrystalline copper [21,22], and from 0.50 to 0.88 for dispersion-hardened aluminum alloys

[23-25]. However, the most important concept for the critical buckling stress under a

Bauschinger effect is the shape of the reloading curve, i.e. Et in Eq. (5.2.2). On one hand, it has

been shown [26] that purely elastic behaviour in compression is rarely observed following

tensile pre-deformations, and the reverse tangent modulus at zero reverse strain is reduced, e.g.

Et(εRev = 0)/ES as low as 0.3 [26]. This would reduce the critical buckling stress satisfying Eq.

(5.2.2), as well as the conventional 0.2% yield strength. On the other hand, there is often a

93

pronounced rounding of the reverse loading curve, in some cases leading to a larger reverse

strain-hardening index n than in an un-worked counterpart in forward loading, e.g. [25,27,28];

this will alter the value of Et as well as the critical inelastic buckling stress satisfying Eq. (5.2.2).

Grabowski [27,28] studied the compressive inelastic buckling stress of PA2N aluminum alloy

(equivalent to the dispersion-hardened AA5052 alloy [29]) columns following tensile (in-situ

work-hardening) pre-strains. There it was found that the shape of the reverse stress-strain curve

plays a large role in determining the load-bearing response, and that following only 3.5% tensile

pre-strain, the critical buckling stress could be increased by ~10%. However, this type of

improvement is not always seen. For example, while un-strained plain carbon steel columns

demonstrate a sharp buckling peak stress, columns given a tensile pre-strain tend to buckle more

gradually with a smaller and more rounded peak stress [26]. Despite the complicating effects of

load reversal, Figure 5.2.4 illustrates that in-situ work hardening can improve the strength of the

present micro-trusses, and that the simple approach taken in the present study is able to define

the boundaries in strength-density material property space for all three precursor materials over

a range of work-hardened and fully-annealed states.

Finally, Figure 5.2.4 shows that for a given strut precursor, the micro-truss strength

increases with decreasing density, i.e. σP ∝ ρr where r < 0. This is significant because it is

different from the typical relationships for foams (where r is ideally 1.5) and lattices (where r is

ideally 1) [30]. However, there are two competing mechanisms: the increased forming

displacement results in a larger truss angle and corresponding truss strength by way of increased

structural efficiency and axially-resolved loads; but the larger slenderness ratio of the micro-

truss ligaments results in a smaller critical buckling stress, e.g. [9,15]. At large enough

displacements, the benefits from the first mechanism will be overcome by the effects of the

second. In addition, each precursor traverses the σP – ρ space differently. This relative

significance of in-situ work-hardening may be understood by considering the strength ratio Ω as

a function of the forming displacement. Figure 5.2.5 plots the experimental Ω values as well as

the predicted ratio of σP,WH/σP,FA (from Figure 5.2.4) as a function of dForm (see Section 3.2, Eq.

(3.2.1)). The trends can be understood by fitting the average experimental tensile curves for the

three starting materials to a standard constitutive model such as the Ludwik relationship (σt = σYS

+ Kεtn [31]): as-received SS304 (K = 1336 MPa, n = 0.70), annealed Cu110 (K = 463 MPa, n =

0.57), and annealed AA3003 (K = 124, n = 0.31). The ratio of work-hardened-to-annealed flow

stress (σt/σYS) can be used as a measure of the relative significance of work hardening as a

function of strain; at a given strain (or dForm) this ratio scales with (K/σYS), which increases from

94

~2.5 (AA3003) to ~5.3 (SS304, annealed σYS of ~250 MPa) to ~6.8 (Cu110). Furthermore, the

derivative of (σt/σYS) with respect to strain describes the in-situ relative hardening rate; at a given

strain this scales with (nK/σYS), which increases from ~0.8 (AA3003) to ~3.7 (SS304) to ~3.8

(Cu110). It is significant that this simplified tensile based deformation approach is able to

provide reasonable agreement to the relative inelastic buckling resistance of the work-hardened

and annealed micro-truss cellular architectures.

Figure 5.2.5. Experimental micro-truss strength ratio Ω as a function of in-situ forming displacement dForm for the annealed AA3003, annealed Cu110, and as-received SS304 precursors. Trendlines are formed using the analytical σP,WH and σP,FA from Figure 5.2.4.

5.2.4 Conclusions

This study has investigated the relative influences of in-situ work hardening in

deformation-formed micro-truss PCMs. Uniaxial compression testing illustrated that the plastic

deformation accumulated during fabrication could be used to improve the peak and specific

strength of PCMs made from work-hardenable precursors. In contrast, post-forming high-

temperature treatments, e.g. the thermal exposure seen in conventional brazing can result in the

loss of work-hardening and the reduction in the overall structural performance. Of the materials

tested in this study, the stainless steel 304 was shown to reach the highest weight-specific

strength after work-hardening due to its comparatively larger initial strength, good ductility, and

high work hardening rate.

5.2.5 References 1. GW Kooistra, HNG Wadley, Mater. Des. 28 (2007) 507-514.


95

3. JR Davis (Ed.), Copper and Copper Alloys, ASM International, Materials Park, 2001.



6. BA Bouwhuis, E Bele, GD Hibbard, J. Mater. Sci. 43 (2008) 3267-3273.

7. T Connolley, PE McHugh, M Bruzzi, Fatigue Fract. Eng. Mater. Struct. 28 (2005) 1119-1152.

8. MGD Geers, WAM Brekelmans, PJM Janssen, Int. J. Solids Struct. 43 (2006) 7304-7321.


10. Atlas of Stress-Strain Curves, 2nd ed., ASM International, Materials Park, 2002.


12. ZP Bažant, Z. Angew. Math. Mech. 80 (2000) 709-732.

13. MW Schraad, Mech. Mater. 39 (2007) 183-198.

14. JH Lim, KJ Kang, Int. J. Solids Struct. 43 (2006) 5228-5246.

15. E Bele, BA Bouwhuis, GD Hibbard, Mater. Sci. Eng. A 489 (2008) 29-37.

16. BA Bouwhuis, SK Tang, GD Hibbard, Composites Part A 39 (2008) 1556-1564.

17. LM Gordon, BA Bouwhuis, M Suralvo, JL McCrea, G Palumbo, GD Hibbard, Acta Mater. 57 (2009) 932-939.

18. D Lee, F Zaverl Jr, J. Eng. Mater. Technol. 101 (1979) 59-67.

19. MC Mataya, MJ Carr, G Krauss, Mater. Sci. Eng. 57 (1983) 205-222.

20. Y Xia, Z Wang, X Du, Acta Metall. Sin. Ser. A (Engl. Lett.) 2 (1989) 1-6.

21. I Gokyu, T Kishi, H Wada, J. Jpn. Inst. Met. 34 (1970) 154-161.

22. T Hasegawa, T Yakou, UF Kocks, Trans. Jpn. Inst. Met. 27 (1986) 425-433.

23. DV Wilson, Acta Metall. 13 (1965) 807-814.

24. T Kishi, R Horiuchi, J. Jpn. Inst. Met. 39 (1975) 92-98.

25. J Grabowski, J. Test. Eval. 16 (1988) 328-332.

26. B Scholtes, O Vöhringer, E Macherauch, Mater. Forum 10 (1987) 54-57.

27. T Hasegawa, T Yakou, Scripta Metall. 14 (1980) 1083-1087.

28. J Grabowski, Archiwum Inżynierii Lądowej 32 (1986) 401-419.

29. WF Keller (Ed.), Handbook of International Alloy Compositions and Designations Volume III – Aluminum,

Metals and Ceramics Information Center Battelle, Columbus, 1980.

30. MF Ashby, Trans. R. Soc. London Ser. A 364 (2006) 15-30.

31. L Radke, PI Welch, Z. Metallkd. 74 (1983) 226-232.

96

6 Strengthening by Nanocrystalline Electrodeposition Electrodeposition has emerged as a technologically and economically viable route to

produce nanostructured metals and alloys. It is a non-line-of-sight deposition technique, and

material can be produced in the form of coatings, continuous thin foils, thick plate, or can be

electroformed to produce small- to large-scale net shape components or microelectromechanical

system (MEMS) devices. In addition, large strength increases can be achieved by grain size

reduction to the nanometer scale. In the present chapter, high-strength nanocrystalline sleeves

were electroformed at Integran Technologies. Integran is a Toronto-based company that is a

world leader in metallurgical nanotechnologies. Characterization of the as-deposited

nanocrystalline Ni grain sizes by X-ray diffraction (XRD) was conducted by Integran staff.

This chapter presents the author’s work done to study the potential of nanocrystalline

electrodeposition as a means of structurally reinforcing cellular metals. Sections 6.1 investigates

the effect of deposited metal thickness on plain carbon steel micro-truss sandwich cores, and

uses an isostrain model to predict the effective strength and stiffness of the nanocrystalline

metal-steel micro-truss hybrid. This section has been published in Composites Science and

Technology [BA Bouwhuis, T Ronis, JL McCrea, G Palumbo, GD Hibbard, Structural

nanocrystalline Ni coatings on periodic cellular steel, Compos. Sci. Technol. 69 (2008) 385-

390]. Next, Section 6.2 investigates the effect of nanocrystalline metal deposited on a

commercially-available open-cell aluminum foam, and again uses an isostrain model to predict

the effective compressive properties of the foam. The key result is that the structural

reinforcement provided by the network of nanocrystalline Ni tubes is significantly more

efficient when arranged in a stretch-dominated architecture compared to a bending-dominated

one. This section has been accepted for publication in Acta Materialia [BA Bouwhuis, JL

McCrea, G Palumbo, GD Hibbard, Mechanical properties of hybrid nanocrystalline metal

foams, Acta Mater. 57 (2009), 4046-4053].

97

6.1 Structural Nanocrystalline Ni Coatings on Periodic Cellular Steel

6.1.1 Introduction

Electroplated coatings such as nickel, chromium, zinc, and tin are widely used to

improve the corrosion resistance of steels [1]. An electrodeposited nanocrystalline Ni coating on

periodic cellular steel may be able to provide more than just corrosion protection; it may also

provide enhanced mechanical stability because of the nearly order of magnitude strength

increase that can be obtained in Ni by grain size reduction to the nm-scale [2]. In this scenario,

the coating would act as an ultra-high strength skin encapsulating the periodic cellular steel

micro-truss, effectively creating a composite micro-truss. This approach may be most effective

in cases where failure occurs by inelastic buckling, since the nanocrystalline Ni would be

optimally located away from the neutral bending axis of the strut. The present study is a first

examination of the potential of multifunctional structural coatings on plain carbon periodic

cellular steel.

6.1.2 Experimental

Perforated AISI-SAE 1006 low carbon steel (sheet thickness to = 0.66 ± 0.01 mm) was

purchased from McNichols Perforated Products (Atlanta, GA). The 25.81 mm2 square

perforations were punched on a two-dimensional square lattice of unit cell size 6.35 mm × 6.35

mm, creating an open area fraction of φ = 0.64. The pyramidal cores (strut geometry shown in

Figure 6.1.1) were fabricated by deforming alternating nodes above and below the starting plane

using a modified stretch-bending process, details in [3]. Miniature test coupons were also cut

from the as-received sheet and measured in tension, after [3]. For the as-received perforated

steel sheet, strut failure occurred adjacent to the forming pin at a maximum displacement of dM

= 3.86 ± 0.03 mm, which corresponded to a relative density of 5.3%. A conservative upper

forming limit of 0.95dM (dForm = 3.66 mm) was selected for making all the micro-trusses in the

present study. This corresponded to an as-fabricated relative density of 5.9% and a truss angle

of ω = 35°, with the struts having a width w = 1.13 ± 0.02 mm, thickness t = 0.63 ± 0.01 mm,

and length l = 5.79 ± 0.09 mm (over 10 measurements each). The mid-strut hardness increased

by ~36% from 115 ± 3 HV in the as-received perforated steel sheet to 156 ± 14 HV after stretch-

bend fabrication.

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Figure 6.1.1. Schematic diagram of a micro-truss strut, showing truss height (h), strut length (l), strut thickness (t), truss angle (ω), and deposited nickel coating thickness (tn-Ni).

Nanocrystalline Ni was pulse current electrodeposited around the struts and nodes of the

steel micro-trusses. Electrodeposition was conducted using a modified Watt’s bath containing

nickel sulfate, nickel chloride, boric acid, and saccharin with a consumable Ni anode at a pH of

2.5, after [4-6]. Nucleation of new grains is favoured by pulsed deposition, with pulse-on times

typically in the range of 2.5 to 10 ms and pulse-off times in the range of 10 to 100 ms [4-6].

Reference coupons (electroplated using the same deposition conditions as the steel micro-truss)

were characterized by X-ray diffraction (XRD) using cobalt Kα X-rays (λCo = 0.179 nm). A

nominal thickness for the electrodeposited nickel was calculated based on the increase in mass

during the plating process by assuming a uniform thickness of nickel over the entire surface area

of the micro-truss, i.e. tn-Ni = mn-Ni/SAρNi, where mn-Ni is the electrodeposited mass, SA is the

electroplated surface area, and ρNi is the density of nickel. A range of nominal coating

thicknesses, from 19.5 ± 0.3 μm to 60.1 ± 0.2 μm, was obtained by adjusting the deposition

time.

Microhardness measurements were made on strut cross-sections that had been mounted

in epoxy and prepared using standard metallographic methods. Measurements were taken using

an MHV 2000 microhardness tester with a 0.49 N applied load and 10 second dwell time.

Uniaxial compression testing was performed at a cross-head displacement rate of 1 mm/min.

Stand alone PCM cores were tested using confinement plates (i.e. recessed channels in steel

compression plates that rigidly lock the truss-core nodes in place – details in Refs. [7,8]). This

test method can be used to simulate the mechanical performance that stand-alone truss cores

would exhibit in a sandwich structure [7,8]. A subset of samples were loaded to strains of up to

ε ≈ 0.7 and characterized by scanning electron microscopy (SEM).

99


A cross-section of an electrodeposited plain carbon steel strut (tn-Ni = 60 μm) is shown in

Figure 6.1.2; microhardness indentations indicated a hardness of 475 ± 14 HV, which is typical

of pulse current deposited nanocrystalline Ni [6].

Figure 6.1.2. SEM micrograph of a plain carbon steel strut with electrodeposited nanocrystalline Ni coating (tn-Ni = 60 μm).

Figure 6.1.3 presents reference XRD patterns of electrodeposited nanocrystalline Ni and a Ni

powder standard; an average grain size of 16 nm was determined from the diffraction peak

broadening using the Scherrer relationship, e.g. [9]. In addition, the electrodeposit has a strong

(200) texture component.

Figure 6.1.3. XRD pattern of electrodeposited nanocrystalline coating (λCo = 0.179 nm) and polycrystalline Ni powder standard.

100

The mechanical properties of the as-fabricated (uncoated) and electroplated micro-

trusses were evaluated in uniaxial compression. Figure 6.1.4 presents stress-strain curves for the

as-fabricated periodic cellular steel micro-trusses.

Figure 6.1.4. Typical stress-strain compression curves for the as-fabricated plain carbon steel micro-trusses (average peak stress of σ = 5.95 ± 0.02 MPa). The inset shows the inelastic buckling failure of a sample loaded to a strain of ~0.3.

The initial elastic region had a slope of E = 60.1 ± 1.4 MPa (Table 6.1.1). The idealized

compressive modulus of a pyramidal micro-truss can be expressed as [10]:

ωρ 4sinRSEE = (6.1.1)

where ES is the elastic modulus of the individual struts, ρR is the relative density, and ω is the

strut angle. Similar to previous studies [e.g. 11-15], the measured compressive modulus is

significantly lower than the value of E ≈ 1350 MPa predicted from the idealized model (in this

case by a knockdown factor of α = 0.045). Significant overpredictions in the elastic modulus

have been attributed to factors such as bedding-in of truss nodes, geometric imperfections, and

uneven loading during the initial stages of testing [11-15]. In particular, the idealized model

assumes that the strain energy is only transferred into the struts as axial deformation [10],

whereas a real micro-truss will always contain eccentricities such that the truss members

undergo a combination of axial and transverse loading.

After the initial elastic region, the uncoated samples collapsed at a peak strength of σ =

5.95 ± 0.03 MPa (Table 6.1.1). The idealized strength of pyramidal PCM truss cores in

compression can be given by [10]:

ωρσσ 2sinRF= (6.1.2)

101

where σF is the failure strength of the strut. For stocky columns having a small slenderness

ratio, the struts will fail by yielding and σF from Eq. (6.1.2) will be replaced by the yield

strength σYS. In contrast, struts having medium-to-high slenderness ratios will fail by buckling,

in which case the failure strength is determined by the critical buckling stress σCR. The struts in

the present study failed by an inelastic buckling mechanism (Figure 6.1.4 inset shows a sample

loaded to ε ≈ 0.3), similar to what has been reported previously for aluminum alloy micro-

trusses [e.g. 7,8,12,14]. The measured peak strength of σ = 5.95 MPa, corresponds to a critical

buckling strength of σCR = 300 MPa. While this is somewhat above the yield strength of the

starting precursor sheet (270 ± 10 MPa), it is not likely above the work hardened strut yield

strength (the mid-strut hardness increased from 115 HV to 156 HV after stretch-bend

fabrication).

Table 6.1.1. Mechanical properties of the as-fabricated (uncoated) and electroplated plain carbon steel micro-trusses summarizing the nominal nanocrystalline Ni coating thickness (tn-Ni), density (ρ), elastic modulus (E), and peak strength (σ).

tn-Ni (μm)

ρ (Mg/m3)

E (MPa)

σ (MPa)

0 0.465 ± 0.005 60.1 ± 1.4 5.95 ± 0.02 19.5 ± 0.3 0.640 ± 0.003 71.1 ± 1.8 8.11 ± 0.15 29.8 ± 0.2 0.658 ± 0.005 76.1 ± 3.1 9.42 ± 0.09 34.8 ± 0.9 0.678 ± 0.005 75.6 ± 2.2 9.94 ± 0.24 40.9 ± 0.7 0.687 ± 0.002 80.9 ± 1.4 10.8 ± 0.10 45.0 ± 0.1 0.694 ± 0.008 82.5 ± 1.2 11.5 ± 0.09 60.1 ± 0.2 0.744 ± 0.010 88.2 ± 5.3 13.1 ± 0.29

Figure 6.1.5 presents representative stress-strain curves for the electroplated micro-

trusses at several different coating thicknesses. Overall, the electroplated samples underwent

the same general failure mechanism as the uncoated samples (Figure 6.1.5 inset shows a tn-Ni =

60 μm sample loaded to ε ≈ 0.4) and both the initial elastic slope and the peak stress increased

with coating thickness. For example, a 60 μm thick coating resulted in a 120% increase in peak

strength and a 40% increase in specific strength (σ/ρ). It is worth noting that a ~50 μm thick

nanocrystalline Ni coating would provide the same inelastic buckling resistance as the 1.13 mm

× 0.63 mm cross-section plain carbon steel struts.

102

Figure 6.1.5. Typical stress-strain compression curves for nanocrystalline Ni coated plain carbon steel micro-trusses having structural coating thicknesses ranging from tn-Ni = 20 μm to 60 μm. Inset shows a coated micro-truss (tn-Ni = 60 μm) having undergone inelastic buckling failure after loading to a strain of ~0.4.

The increase in strength and modulus provided by the structural coating can be analyzed

in terms of the predicted mechanical behaviour of a hollow tube nanocrystalline Ni micro-truss

of varying wall thickness. This approach effectively treats the electroplated samples as a two

component composite micro-truss. A reference tensile curve for free-standing nanocrystalline

Ni (electrodeposited using the same deposition conditions and having the same hardness [16])

was fit to a Ramberg-Osgood constitutive model [17]. This is a simple relationship that can be

used to describe the elastic to plastic stress-strain behaviour and can be expressed as: N

YS

t

S

tt E ⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

σσσ

ε 002.0 (6.1.3)

where ES is the Young’s modulus (ES = 145 GPa), N is a strain-hardening exponent (N = 4.66),

and σYS is the yield strength from the 0.2% offset method (σYS = 825 MPa) (Appendix B).

Unlike the case for fibre or particle reinforced composites, where the addition of the

reinforcing phase necessarily displaces an equivalent volume fraction from the matrix, the

components of the present composite are interpenetrating structures in which the effective area

fraction of each component is essentially constant regardless of the coating thickness. An initial

estimate for the mechanical performance can then be taken by using an iso-strain approach in

which the plain carbon steel micro-truss and the nanocrystalline Ni micro-truss shell suffer the

same strain during loading. Figure 6.1.6 plots the measured increase in elastic modulus

( electroplated uncoatedE E EΔ = − ) as a function of electroplated Ni coating thickness.

103

Figure 6.1.6. Measured increase in compressive modulus as a function of nanocrystalline Ni coating thickness. Also shown is the predicted curve for a hollow tube nanocrystalline Ni micro-truss from Eq. (6.1.1).

Also shown is the predicted compressive modulus for a hollow tube nanocrystalline Ni micro-

truss (from Eq. 6.1.1). In general, there is reasonable agreement between the model and the

measured modulus increase when the same knockdown factor measured for the uncoated steel

micro-truss (α = 0.045) is used for predicting the behaviour of the conformal hollow tube

nanocrystalline Ni micro-truss.

A similar approach can be taken for the peak strength. The failure strength of struts

having medium-to-high slenderness ratios will be given by the critical buckling stress σCR which

can be expressed as (after Shanley [18]):

2

22

AlIEk t

CRπ

σ = (6.1.4)

where Et is the tangent modulus (Et = ∂σt/∂εt), A is the cross-sectional area of the supporting

member, I is the moment of inertia, and k describes the rotational stiffness of the column. The

predicted buckling strength of the nanocrystalline sleeves can be seen in Figure 6.1.7 for the

limiting boundary conditions of pin jointed (k = 1, [e.g. 11,19]) and rigid (k = 2, [e.g. 12,14])

end constraints. Also plotted in Figure 6.1.7 are the experimentally measured strength increases

( uncoatedtedelectropla σσσ −=Δ ) as a function of the electroplated Ni thickness; the experimentally

measured values are generally close to or below the k = 1 boundary condition. In practice, a

micro-truss joint would be expected to behave in a fashion intermediate to these two end

constraint conditions, e.g. [10,20]. However, the critical buckling stress of Eq. (6.1.4) only

provides an upper estimate. Struts that are not straight, uniform, or otherwise ideal will have a

reduced critical inelastic buckling stress [18]. In addition, there are likely residual stresses in

104

the electrodeposited nanocrystalline Ni [21], which would also affect the critical buckling

strength [18] and measured grain size (Figure 6.1.3). While the failure of composite micro-

trusses is a complex problem, to a first approximation the inelastic buckling strength of hollow

tube nanocrystalline micro-trusses can reasonably account for the experimentally measured

strength increase.

Figure 6.1.7. Measured increase in peak compressive strength as a function of nanocrystalline Ni coating thickness. Also shown are the predicted curves for a hollow tube nanocrystalline Ni micro-truss from Eqs. (6.1.2) to (6.1.4) in the limiting case of pin-joint (k = 1) and rigid-joint (k = 2) end constraints.

Finally, a potentially significant issue is the coating integrity through the elastic

deformation of the composite micro-truss. For effective corrosion protection, barrier coatings

such as Ni on steel require continuous and unbroken coverage since the coating is more cathodic

than the plain carbon steel it is protecting [1,22]. A previous study of aluminum micro-trusses

that were reinforced with high-strength nanocrystalline Ni-Fe sleeves reported early crack

formation in the hinge region of the struts prior to inelastic buckling collapse [23]. Early crack

formation in the present system would be a significant concern because it would provide

locations for preferential corrosion to occur [22]. Crack formation in the nanocrystalline Ni-

Fe/aluminum struts resulted in a series of small load drops, which could be clearly seen as sharp

valleys in a plot of the tangent modulus as a function of strain [23]. There were no obvious load

drops in the stress strain curve of the electroplated steel micro-trusses (Figure 6.1.5) and there

was essentially no difference in the rate of local slope change between the uncoated and

electroplated micro-trusses (Figure 6.1.8).

105

Figure 6.1.8. Tangent modulus of the uniaxial compression stress-strain curve as a function of strain for the uncoated and tn-Ni = 60 μm thick electroplated samples.

This was confirmed by SEM characterization of samples loaded to a series of prescribed strains;

even after inelastic buckling had occurred (e.g. ε ≈ 0.4, Figure 6.1.9a) no cracks could be

detected in the electroplated coating. It was only after loading beyond the minimum in the

compressive stress strain curve (e.g. ε ≈ 0.6, Figure 6.1.9b) that cracks could be detected. This

is significant because it means that the mechanical integrity of the coating is maintained even

after inelastic buckling of the struts has occurred.

Figure 6.1.9. SEM micrographs of samples loaded to ε ≈ 0.4 (a) and ε ≈ 0.4 (b). Mechanical integrity of the structural coating was maintained beyond the inelastic buckling collapse (a) and coating fracture did not occur until after the buckled strut made contact with the compression platen (b).

106

6.1.4 Conclusions

The present results show that stretch-bending is a very simple approach to produce open-

celled plain carbon steel architectures having relative densities down to ~5%. Electroplated

nanocrystalline Ni acted as a structural coating, increasing both the compressive modulus and

the inelastic buckling strength. A coating thickness of only ~50 μm was needed to double the

inelastic buckling strength of the starting plain carbon steel struts. The performance

enhancement in the composite micro-trusses is due to the combined effects of the starting

nanocrystalline grain size in the coating and the optimally positioned structural coating away

from the neutral bending axis. It was found that a composite micro-truss approach was able to

account for both the compressive modulus and inelastic buckling strength increases. Finally, it

was found that the mechanical integrity of the nanocrystalline coating was maintained through

the initial stages of inelastic buckling collapse.

6.1.5 References 1. ASM Handbook, vol. 5: Surface Engineering, ASM International, Materials Park, 1994.

2. N Wang, Z Wang, KT Aust, U Erb, Mater. Sci. Eng. A 237 (1997) 150-158.


4. U Erb, AM El-Sherik, US Patent 5,353,266, October 1994.

5. U Erb, AM El-Sherik, CKS Cheung, MJ Aus, US Patent 5,433,797, July 1995.

6. AM El-Sherik, U Erb, J. Mater. Sci. 30 (1995) 5743-5749.


8. BA Bouwhuis, E Bele, GD Hibbard, J. Mater. Sci. 43 (2008) 3267-3273.

9. BD Cullity, Elements of X-ray Diffraction, 2nd ed., Addison-Wesley, Don Mills, 1978.


11. VS Deshpande, NA Fleck, MF Ashby, J. Mech. Phys. Solids 49 (2001) 1747-1769.




15. P Moongkhamklang, DM Elzey, HNG Wadley, Composites Part A 39 (2008) 176-187.

16. Integran Technologies, Inc., unpublished research.

17. W Ramberg, WR Osgood, Description of stress–strain curves by three parameters, Technical Note 902,

NACA, Washington, 1943.




21. AM El-Sherik, J Shirokoff, U Erb, J. Alloys Compd. 389 (2005) 140-143.

107

22. BA Bouwhuis, T Ronis, GD Hibbard, JL McCrea, G Palumbo, in: Surface Modification Technologies XXII,

TS Sudarhsan, P Nylen (Eds.), VALAR Docs, Edgewood, pp. 349-356.

23. M Suralvo, B Bouwhuis, JL McCrea, G Palumbo, GD Hibbard, Scripta Mater. 58 (2008) 247-250.

108

6.2 Mechanical Properties of Hybrid Nanocrystalline Metal Foams

6.2.1 Introduction

In contrast to micro-truss materials, conventional metal foams typically fail by bending-

dominated deformation mechanisms [1]. Nanocrystalline electrodeposition can also be used to

reinforce these cellular materials. For example, a recent study reported on the strength increase

that could be obtained by the electrodeposition of nanocrystalline Ni-W on an open-cell

aluminum foam [2]. Overall, the mechanical performance increase that can be obtained through

electrodeposition depends on a complex interaction between the starting cellular architecture

and the material properties of the nanocrystalline sleeves. The present study begins to address

these issues by examining the mechanical performance and failure mechanisms that occur

during the uniaxial compression of conventional open cell aluminum foams that have been

reinforced with electrodeposited nanocrystalline Ni.

6.2.2 Experimental

Open cell AA6101 aluminum foam having a relative density of ρ ≈ 7.2%, nominal pore

size of 1.27 mm (20 pores per inch), and specific surface area of S = 1.19 mm2/mm3 [3] was

purchased from ERG Aerospace (of Oakland, CA). Nanocrystalline nickel/aluminum foam (n-

Ni/Al Foam) hybrids were fabricated by pulse current electroplating 76.2 × 76.2 × 12.7 mm

sections of the starting aluminum foam with nanocrystalline Ni. Electrodeposition was

conducted using a modified Watt’s bath, after [4-6], and the same deposition conditions used in

Section 6.1. The thickness of the high strength nanocrystalline sleeves was controlled by the

electrodeposition time; the density of the hybrid foams ranged from ρ = 0.485 ± 0.016 to 0.984

± 0.051 Mg/m3, Table 6.2.1. Coupons with dimensions 10.0 × 10.0 × 12.7 mm were cut from

the coated and uncoated foams by electrical discharge machining (EDM) and tested in uniaxial

compression at a cross-head displacement rate of 1 mm/min; the long axis of the cells was

oriented longitudinally to the loading direction. Five samples were tested for each thickness

group. Recorded cross-head displacements, adjusted for machine compliance based on the

measured load, were used to estimate the compressive strain. Vickers micro-hardness

measurements (0.49 N load, 10 s dwell time) were conducted on the nanocrystalline sleeves

(472 ± 26 HV) and aluminum foam cores (81 ± 11 HV).

109

Table 6.2.1. Mechanical properties of the as-received AA6101 metal foams and electrodeposited n-Ni/Al foam hybrids summarizing the density ( Foam

AlNin /−ρ ), nominal n-Ni sleeve thickness ( nom

Nint − ), mid-foam n-Ni thickness ( midNint − ), compressive modulus ( Foam

AlNinE /− ), and peak strength ( Foam

AlNin /−σ ). Foam

AlNin /−ρ (Mg/m3)

nomNint −

(μm)

midNint −

(μm)

FoamAlNinE /−

(MPa)

FoamAlNin /−σ

(MPa) Al 0.196 ± 0.006 0 0 125 ± 18 1.26 ± 0.05 B 0.485 ± 0.016 25.6 ± 1.8 13.0 ± 2.6 258 ± 52 2.87 ± 0.19 C 0.602 ± 0.039 39.0 ± 4.8 15.5 ± 2.9 338 ± 58 4.16 ± 0.21 D 0.712 ± 0.036 47.3 ± 3.9 20.2 ± 4.9 376 ± 37 4.22 ± 0.24 E 0.790 ± 0.027 56.7 ± 4.0 22.1 ± 2.6 385 ± 90 4.95 ± 0.56 F 0.984 ± 0.051 72.3 ± 6.0 28.8 ± 2.4 455 ± 137 6.54 ± 0.46

The nanocrystalline Ni grain size was determined by X-ray diffraction (XRD)

characterization using Co-Kα radiation (λCo = 1.79 nm) on aluminum coupons electroplated

using the same deposition conditions as the n-Ni/Al hybrid foams. An average grain size of 16

nm was measured from the diffraction peak broadening (using the Scherrer relationship [e.g. 7]),

which is typical of n-Ni produced by pulse current electrodeposition, e.g. [6,8]. The n-Ni

thickness distribution on the electrodeposited foams was characterized by scanning electron

microscopy (SEM) of polished cross-sections that had been cut by EDM. Failure mechanisms

of the electrodeposited hybrids were investigated by SEM characterization of samples pre-

loaded to characteristic strain values.


A nominal thickness for the n-Ni sleeves ( nomNint − ) can be calculated by assuming a

uniform distribution of electrodeposited Ni mass (mn-Ni) over the specific surface area of the

starting aluminum foam (S), i.e.:

Ni

NinnomNin SV

mt

ρ−

− = (6.2.1)

where ρNi is the density of Ni, and V is the volume of the Al foam specimen. Nominal n-Ni

thicknesses ranged from 25.6 ± 1.8 μm to 72.3 ± 6.0 μm, values summarized in Table 6.2.1.

This simple parameter, however, does not reflect the distribution of electrodeposited

nanocrystalline sleeve thicknesses through the foam height, h, which can be seen in Figure 6.2.1

(sample type F, Table 6.2.1). For all samples, the thickness of the n-Ni sleeves was smallest and

most uniform within the middle region of the foam. This structural gradient is not surprising

110

given the complex cellular architecture and the electromagnetic shielding induced by the outer

ligaments of the foam, as well as a likely reduced electrolyte flow within the foam [e.g. 9].

Sleeve thicknesses from the middle of the foam (i.e. for ligaments located within the band of

~0.25h ≤ z ≤ ~0.75h) were measured for each sample ( midNint − ) and are summarized in Table 6.2.1

with typical strut cross-sections presented in Figure 6.2.2.

Figure 6.2.1. Overview SEM micrograph of a n-Ni/Al foam hybrid (sample type F in Table 6.2.1). The back-scattered electron atomic number contrast distinguishes the darker aluminum core from the lighter nanocrystalline sleeve and shows the gradient in n-Ni sleeve thickness.

111

Figure 6.2.2. Typical ligament cross-sections from the mid-height region of the electrodeposited n-Ni/Al foam hybrids, showing thin (left) and thick (right) sleeve reinforcement ( mid

Nint − = 13.0 ± 2.6 μm and mid

Nint − = 28.8 ± 2.4 μm, respectively).

Figure 6.2.3 plots the nomNint − and mid

Nint − thicknesses as a function of the n-Ni/Al hybrid foam

density. Like the nominal thickness, the thicknesses of the middle sleeves increased linearly

with overall sample density, but were only ~40% of the nominal value. Note that while the

ligaments located in the middle of the foam have the least structural reinforcement (and will

consequently be a zone of weakness during subsequent uniaxial compression testing), some

degree of thickness non-uniformity may in fact be beneficial.

Figure 6.2.3. Nominal n-Ni sleeve thickness ( nom

Nint − , calculated from Eq. 6.2.1) and mid-sample sleeve thickness ( mid

Nint − , measured from strut cross-sections) as a function of hybrid foam density ( Foam

AlNin /−ρ ).

112

For example, if the structural objective of the hybrid foam is to resist externally applied bending

loads, then having the majority of the nanocrystalline reinforcement located near the outer

surfaces may provide enhanced weight specific bending resistance compared to uniformly

coated foam cores.

Figure 6.2.4 presents typical uniaxial compression stress-strain curves for the reference

(uncoated) aluminum foam and the n-Ni/Al hybrid foams. Significant increases to both the peak

strength and compressive modulus were seen with increasing electrodeposit sleeve thickness,

summarized in Table 6.2.1. Even the thinnest nanocrystalline sleeves (sample type B

corresponding to nomNint − = 25.6 μm and mid

Nint − = 13.0 μm) more than doubled the average modulus

and peak strength. For the thickest n-Ni sleeves (sample type F corresponding to nomNint − = 72.3

μm and midNint − = 28.8 μm), the average compressive modulus increased by a factor of 3.6 (from

125 to 455 MPa), while the peak strength increased by a factor of 5.2 (from 1.26 to 6.54 MPa).

The sleeves also changed the overall form of the compression profile.

Figure 6.2.4. Typical stress-strain curves of uncoated foam samples and with increasing coating thickness.

The uncoated samples exhibited the expected ductile metal foam compression behaviour in

which the initial compressive modulus was followed by a nearly constant plateau stress as the

ligaments plastically deformed in bending [1,10]. In contrast, the hybrid foams exhibited a

decrease in stress following the peak towards a local minimum (valley strength) before

increasing again as densification progressed. Note that a comparatively sharper post-peak

decrease was seen in the study of n-NiW/Al hybrid foams with a similar Al foam core [2] and

113

may be related to the lower tensile ductility seen in nanocrystalline Ni-W alloys [e.g. 11]

compared to nanocrystalline Ni and Ni-Fe alloys [12,13].

In order to investigate the deformation mechanisms progressing from the peak to valley

strength, hybrid samples were pre-loaded to a range of compressive strains and examined in the

SEM. There were no obvious signs of macroscopic structural change in the samples pre-loaded

to just before the peak stress. On the other hand, both cracks and wrinkles could be seen in a

subset of the n-Ni sleeves for samples pre-loaded to just after the initial peak. These failed

sleeves were all located within the middle of the foam where the ligaments had the least

structural reinforcement. An example of a failed hybrid ligament in a type F foam loaded to ε ≈

0.035 is presented in Figure 6.2.5; cracks in the sleeves were seen on the tensile side while some

evidence of sleeve wrinkling could be seen on the compression side of the rotated ligament. For

ductile foams that fail by bending-dominated mechanisms, the onset of plasticity in the

ligament-beams occurs when the stress at their outer surfaces reaches the yield stress, while the

maximum load-carrying capacity occurs by the formation of fully plastic hinges [1,10].

Figure 6.2.5. SEM micrographs of ligaments loaded to immediately after peak in an F foam (left), and to the valley strength (right). Cracks could be seen on the tensile sides of ligaments (*), and wrinkles on the compression sides (**), both near the ligament ends.

Similarly, the transition from initial elastic deformation through to the peak strength in the n-

Ni/Al hybrid is likely controlled by bending-dominated plastic deformation in the composite

ligaments. However, since the ductility of electrodeposited nanocrystalline Ni and Ni-Fe alloys

114

(up to ~10% elongation to failure [12,13]) is less than that of AA6101 in the T6 condition

(~20% strain to failure [14]), and because the n-Ni is positioned on the outside of the ligaments,

there is likely a critical plastic strain above which crack formation in the n-Ni sleeve occurs.

Crack propagation would contribute to the decrease in overall load carrying capacity past the

peak. This can be seen in Figure 6.2.5, which shows a type F sample pre-loaded to a strain of ε

≈ 0.085 (i.e. near the minimum valley strength). Sleeve fracture, however, did not fully

progress through to the compression side of the rotated strut even for samples loaded to just

beyond the valley strength, which may explain why this stress was greater in the hybrids than

the plateau stress for the uncoated Al core.

The post peak behaviour (i.e. beyond a strain of ε ≈ 0.1) was also very different between

the starting AA6101 foam and the n-Ni/Al hybrid. In the case of the AA6101 foam, the

resistance to collapse is more or less uniform through the thickness, resulting in a nearly

constant plateau stress. On the other hand, the composite foam has a n-Ni structural

reinforcement gradient because of electromagnetic shielding effects. This results in early

densification of the weakest (middle) section of the foam followed by progressive fracture

within the stronger, outer regions of the hybrid leading to the overall trend of increasing stress

beyond the initial valley. Some parallels can be drawn to the case for conventional metal foams

having a relative density gradient in the through thickness direction, e.g. [15]: it was shown that

these foams possessed a pronounced increasing compressive stress following the initial

transition to plasticity; this was attributed to deformation initiating in the weakest (lowest

relative density) section of the foam and progressing into the stronger (higher relative density)

sections of the foam [15].

In a previous study of n-Ni reinforced steel micro-trusses, a two component composite

cellular material approach was taken to model the modulus and strength increase seen after

nanocrystalline electrodeposition [16]. In terms of the n-Ni/Al foam hybrids of the present

study, this approach effectively considers the measured increase in modulus and strength in

terms of the predicted behaviour of a hollow-tube nanocrystalline Ni foam. Note that unlike the

case for typical rule-of-mixtures composite models of fully-dense materials, reinforcement by a

given amount of nanocrystalline sleeve in the cellular composite does not displace an equivalent

volume fraction of the starting aluminum foam. Using an isostrain assumption, the

contributions from each cellular component are therefore simply additive [16].

115

Figure 6.2.6 plots the experimental data showing the increase in compressive modulus

( FoamAl

FoamAlNin EEE −=Δ − / ) against the increase in density ( Foam

AlFoam

AlNin ρρρ −=Δ − / ) due to the

electrodeposited nanocrystalline sleeves.

Figure 6.2.6. Experimentally-measured modulus increase ΔE plotted against hybrid density increase Δρ. Model Foam

NinE − curves are obtained using Eqs. (6.2.2) to (6.2.4) and are plotted for thickness gradients corresponding to f = 0, f = 0.50, and f = 0.75, where f is the middle fraction of the foam possessing mid

Nint − .

A predictive model for the compressive modulus of a hollow tube nanocrystalline foam can be

made starting from the well-established compressive modulus of a metal foam using the

relationship, after [1,10]

42LIBE

EFoam

NinNiFoamNin

−− = α (6.2.2)

where B is a parameter describing the end conditions of the ligament (B = 192 for a fixed end-

fixed end ligament point loaded at mid-span [10]), ENi is the elastic modulus of nickel (207 GPa

[17]), L is the ideal cell size, I is the second moment of area, and α is a knockdown factor. The

ligaments of ERG foams are often approximated as cylindrical rods during modeling [18-22],

and a cylindrical tube approximation was also used as the basis for describing the conformal

nanocrystalline foam. The moment of inertia for a hollow cylindrical nanocrystalline sleeve can

be expressed as [1,10],

( )( )64

44 dtdI NinFoam

Nin−+

= −−

π (6.2.3)

where d is the uncoated foam ligament diameter.

116

The predicted FoamAlE of the uncoated foam provides a basis for comparison between the

idealized model (Eq. 6.2.2 and 6.2.3) and the experimentally-measured values. Using EAl = 69.6

GPa for AA6101 [14], and cylindrical ligaments of diameter d ≈ 0.2 mm and L = 1.0 mm for an

ERG foam having 1.27 mm nominal pore size and 7.2% relative density (after [18]), the

modulus of the idealized foam is ~305 MPa. For an experimentally-measured FoamAlE = 125

MPa, this corresponds to a modulus knockdown factor of α ≈ 0.41, which is comparable to what

has been seen in conventional metal foam studies, e.g. [10,23].

Predicted FoamNinE − values were calculated for each of the n-Ni/Al hybrid foam densities

using the nominal thickness nomNint − and are plotted with the experimental data (ΔE) in Figure

6.2.6. Simply using the nominal thickness significantly over-predicts the measured value. This

is largely because the structural gradient within the n-Ni/Al hybrid foams (Figure 6.2.1) causes

the middle of the foam to have a much lower effective modulus than the outer regions which

received greater n-Ni sleeve reinforcement. One approach to account for this structural gradient

is to determine an effective modulus for the middle fraction f of the foam (based on the

measured midNint − value) and an effective modulus for the outer regions based on a calculated

average thickness endNint − that conserves the total electrodeposited mass. This approach can be

expressed as: 1

,,

1−

−−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟

⎠

⎞⎜⎜⎝

⎛= endFoam

NinmidFoam

Nin

FoamNin E

fE

fE (6.2.4)

where midFoamNinE ,

− is the modulus in the middle foam section (calculated using midNint − ) and endFoam

NinE ,−

is the modulus in the foam ends (calculated using endNint − ). The parameter f essentially partitions

the nanocrystalline reinforcement into a compliant inner section, which is bounded by more

rigid outer sections. Note that using the nominal coating thickness to predict FoamNinE − is

equivalent to the case when f = 0 in Eq. (6.2.4). Better agreement to the experimentally

measured values is seen with increasing f (Figure 6.2.6). This suggests that the overall

compressive modulus of the composite foam is dominated by the mid-height region, which has

received the least structural reinforcement. Based on the electrodeposited n-Ni thickness

distribution (Figure 6.2.1), a reasonable upper value for the parameter f is on the order of ~0.75.

Finally, it should be noted that while a constant modulus knockdown factor (α = 0.41) was used

to estimate the modulus increase provided by the nanocrystalline Ni, the actual value of α may

117

depend on the degree of structural reinforcement. For example, it has been suggested that in

foams with relative density gradients, the knockdown factor may vary as a function of density

[15].

A composite foam approach can also be taken to model the strength increase provided by

the nanocrystalline sleeves. Figure 6.2.7 plots the experimentally-measured strength increase

( FoamAl

FoamAlNin σσσ −=Δ − / ) against the increase in density (Δρ). Using the idealized foam model,

the initial peak foam strength can be expressed as [1,10]:

3,

LHC Foam

NinNinYSFoamNin

−−− =

σβσ (6.2.5)

where the parameter C describes the end condition (8 for a fixed end-fixed end ligament point

loaded at the mid-span [10]), σYS,n-Ni is the yield strength of the electrodeposited nanocrystalline

Ni (900 MPa [24]), and FoamNinH − is the moment measuring the resistance to fully plastic bending.

Using a hollow cylindrical nanocrystalline sleeve, this becomes

( )6

33 dtdH NinFoam

Nin−+

= −− (6.2.6)

Some insight into the significance of the strength knockdown factor β can be obtained from the

predicted strength of the uncoated foam. A ligament yield strength of 124 MPa has been used in

[25] for ERG AA6101 foams having the same architecture (20 pores per inch and ~7% relative

density) and comparable ligament hardness (71 ± 5 HV) as the present study (81 ± 11 HV over

20 measurements). Using σYS,Al = 124 MPa in Eq. (6.2.5) results in a predicted strength of the

uncoated foam of ~1.3 MPa, which is effectively the same as the measured uncoated foam

strength. This suggests that β ≈ 1, which is comparable to what has been seen in conventional

metal foam studies, e.g. [10,23]. The effect of sleeve uniformity on the potential significance of

nanocrystalline reinforcement can be seen when FoamNin−σ values are calculated using the nominal

thickness nomNint − and the measured mid-foam sleeve thickness mid

Nint − (Figure 6.2.7). The nominal

thickness significantly over-predicts the strength while the mid-foam thickness provides good

agreement and is consistent with the experimental observation that the first composite struts to

fail were located within the middle band of the electrodeposited foam.

118

Figure 6.2.7. Experimentally-measured strength increase Δσ plotted against hybrid density increase Δρ. Model Foam

Nin−σ curves were calculated using Eqs. (6.2.5) to (6.2.6) with either the nominal thickness nom

Nint − or mid-foam thickness midNint − .

While coating uniformity is one factor in determining the effectiveness of the

nanocrystalline sleeves at reinforcing cellular materials, a more important factor is the strength-

limiting failure mechanism. The peak strength of the n-Ni/Al foam hybrids was largely

controlled by bending-dominated plastic deformation of the conformal nanocrystalline sleeves.

In contrast, hybrid nanocrystalline micro-truss cellular materials failed primarily by axial

deformation mechanisms, such as inelastic buckling [16,26-28]. Whether bending-dominated or

stretching-dominated deformation will occur during cellular material collapse is largely

controlled by the inter-connectivity of the struts or ligaments [1,29]. In other words, while

nanocrystalline electrodeposition can be used to significantly reinforce both conventional foams

and micro-truss materials, whether the failure mechanism of the interconnected network of

nanocrystalline tubes is bending-dominated or stretching-dominated depends on the architecture

of the cellular core. The relative significance of this architectural effect can be seen in Figure

6.2.8 where the increase in strength (Δσ) is plotted against the corresponding density increase

(Δρ) for the hybrid foams of the present study and the hybrid micro-trusses of previous studies

[16,26-28].

119

Figure 6.2.8. Experimentally-measured strength increase Δσ versus density increase Δρ of hybrid nanocrystalline metal foams and micro-trusses.

When electrodeposited on an open-cell foam, the specific strength of the nanocrystalline Ni

network of tubes ranged between ~5.6 and ~7.1 MPa/(Mg/m3). Note that depending on the

geometry, a truly hollow nanocrystalline foam may be subject to additional types of failure

mechanisms such as local buckling. Over a similar range of density increase the specific

strength in micro-truss form was between ~19 to ~52 MPa/(Mg/m3) [26-28]; the highest values

were obtained in hybrids with the greatest truss angle, where the composite struts had clearly

failed by inelastic buckling. Note that the electrodeposited sleeve thickness is not a meaningful

point of comparison between the hybrid foam and micro-truss studies. In the case of the

aluminum micro-truss studies, the sleeve thickness was on the order of 75 to 400 μm [26,27],

which is much greater than the nominal thicknesses of the present study. However, the specific

surface area (S) of the starting aluminum pre-form architecture was an order of magnitude

smaller for the micro-trusses (~0.10 to ~0.20 mm2/mm3 [26,27]) than for the foams (1.19

mm2/mm3 in present study). This means that even though up to ~400 μm thick nanocrystalline

sleeves were deposited in [26], the actual density increase was nearly the same (0.347 Mg/m3)

as the thinnest coatings of the present study (where nomNint − = 25.6 μm).

Finally, while the nanocrystalline Ni sleeves had the effect of controlling the overall

strength of the foam hybrids, they did not significantly increase the specific strength of the

starting Al foam, i.e. FoamAl

FoamAl ρσ / = 6.43 ± 0.45 MPa/(Mg/m3). For example, the thickest

sleeve coating (Type F, Table 6.2.1) resulted in a strength increase of Δσ = 5.28 MPa and a

density increase of Δρ = 0.788 Mg/m3, giving an effective overall specific strength of ~6.7

MPa/(Mg/m3) for the sleeve. Had the coating thickness been uniformly distributed over the

120

starting aluminum core, a strength increase of Δσ = ~14.6 MPa could have been obtained

(Figure 6.2.7), which would have corresponded to an effective specific strength of ~18.5

MPa/(Mg/m3) for the sleeve and 16.1 MPa/(Mg/m3) for the overall n-Ni/Al foam hybrid. While

improving the coating uniformity would increase the weight specific compressive strength of the

hybrid foams, there is much greater opportunity for enhancing the performance of the hybrid

nanocrystalline micro-trusses. For example, simply increasing the truss angle can have a large

effect on the performance of the nanocrystalline reinforcement; for a constant nominal sleeve

thickness of ~255 μm, increasing the truss angle from 22.4˚ to 32.6˚ to 40.4˚ increased the

nanocrystalline sleeve ρσ ΔΔ / from 19 to 45 to 52 MPa/(Mg/m3) [27].

6.2.4 Conclusions

This study has investigated the potential for electrodeposited nanocrystalline Ni to be

used as a structural reinforcement in open-cell aluminum foams. Models for the hybrid

compressive modulus and strength were derived based on the predicted behaviour of a hollow

nanocrystalline Ni foam. A significant issue was the uniformity of the electrodeposited coating;

the nanocrystalline reinforcement was smallest within the mid-height region of the foam. This

band of least structural reinforcement controlled the peak compressive strength and played a

significant role in determining the overall compressive modulus. Improving the deposition

uniformity would increase the weight specific strength of the hybrid nanocrystalline foam, but

the potential for strengthening is limited by the bending-dominated failure mechanism of the

starting cellular architecture.

6.2.5 References 1. MF Ashby, Philos. Trans. R. Soc. A 364 (2006) 15-30.

2. Y Boonyongmaneerat, CA Schuh, DC Dunand, Scripta Mater. 59 (2008) 336-339.

3. ERG Aerospace website: www.ergaerospace.com.

4. U Erb, AM El-Sherik, US Patent 5,353,266, October 1994.

5. U Erb, AM El-Sherik, CKS Cheung, MJ Aus, US Patent 5,433,797, July 1995.

6. AM El-Sherik, U Erb, J. Mater. Sci. 30 (1995) 5743-5749.

7. BD Cullity, Elements of X-ray diffraction, 2nd ed., Addison-Wesley, Don Mills, 1978.

8. AM El-Sherik, U Erb, G Palumbo, KT Aust, Scripta Metall. Mater. 27 (1992) 1185-1188.

9. M Schlesinger, M Paunovic (Eds.), Modern Electroplating, 4th ed., Wiley Interscience, New York, 2000.

10. MF Ashby, AG Evans, NA Fleck, LJ Gibson, JW Hutchinson, HNG Wadley, Metal Foams – A Design Guide,


11. A Giga, Y Kimoto, Y Takigawa, K Higashi, Scripta Mater. 55 (2006) 143-146.

121

12. HQ Li, F Ebrahimi, Appl. Phys. Lett. 84 (2004) 4307-4309.

13. HS Wei, GD Hibbard, G Palumbo, U Erb, Scripta Mater. 57 (2007) 996-999.


15. AH Brothers, DC Dunand, Mater. Sci. Eng. A 489 (2008) 439-443.

16. BA Bouwhuis, T Ronis, JL McCrea, G Palumbo, GD Hibbard, Compos. Sci. Technol. 69 (2009) 385-390.

17. JR Davis (Ed.), Nickel, Cobalt, and Their Alloys, ASM International, Materials Park, 2000.

18. TJ Lu, HA Stone, MF Ashby, Acta Mater. 46 (1998) 3619-3635.

19. J Zhou, C Mercer, WO Soboyejo, Metall. Mater. Trans. A 33 (2002) 1413-1427.

20. F Scheffler, R Herrmann, W Schwieger, M Scheffler, Microporous Mesoporous Mater. 67 (2004) 53-59.

21. Ozmat B, Leyda B, Benson B. Mater. Manuf. Processes 19 (2004) 839-862.

22. EN Schmierer, A Razani, J. Heat. Transfer 128 (2006) 1194-1203.

23. B Vamsi Krishna, S Bose, A Bandyopadhyay, Mater. Sci. Eng. A 452-453 (2007) 178-188.

24. Integran Technologies, Inc., unpublished research.

25. E Amsterdam, JThM De Hosson, PR Onck, Scripta Mater. 59 (2008) 653-656.

26. M Suralvo, B Bouwhuis, JL McCrea, G Palumbo, GD Hibbard, Scripta Mater. 58 (2008) 247-250.

27. BA Bouwhuis, T Ronis, JL McCrea, G Palumbo, GD Hibbard, in: Cellular Metals for Structural and

Functional Applications, G Stephani, B Kieback (Eds.), Fraunhofer IFAM, Dresden, 2009, pp. 159-164.

28. LM Gordon, BA Bouwhuis, M Suralvo, JL McCrea, G Palumbo, GD Hibbard, Acta Mater. 57 (2009) 932-939.


122

7 Conclusions The author has shown in this Ph.D. thesis that microstructural strengthening mechanisms

can be useful tools to increase the load-bearing capacity of the micro-truss ligaments, allowing

new regions of strength-density property space to be reached.

7.1 Fabrication and Testing of Micro-Truss Sandwich Cores

Micro-truss PCMs were fabricated using a modified perforation-stretching technique.

For this process the author developed a new model based simply on the tensile properties of the

precursor material, which can a priori (1) predict the range of accessible micro-truss

architectures, (2) model the regions of strength-density material property space attainable by

micro-trusses, and (3) evaluate the significance of in-situ work hardening in different material

systems.

A new compression test method was developed which enabled the micro-truss to be

tested as a stand-alone material, simulating the response as part of a sandwich core. This method

also preserves important microstructural information, such as the existence of strengthening

mechanisms, which are often lost during conventional sandwich panel joining processes, e.g.

high-temperature brazing. This new test method was subsequently used to test the effect of in-

situ work hardening and electrodeposition on the compressive properties of micro-truss

sandwich cores.

7.2 Microstructural Strengthening Mechanisms in Micro-Trusses

The results of the present study can be summarized in a strength-density material

property space map, Figure 7.2.1. In general, the two strengthening mechanisms traverse very

different paths through strength-density material property space. While each strengthening

mechanism has been considered separately, they are complimentary approaches; at any given

forming displacement, a micro-truss can be used as a cellular pre-form upon which

nanocrystalline metal can be deposited.

123

Figure 7.2.1. Strength-density material property space map for the cellular materials studied in this thesis.

7.2.1 In-Situ Work Hardening

In-situ work hardening was achieved by controlling the distribution of plastic strain

imparted to the micro-truss struts during fabrication. It was shown that this strain energy can

lead to a factor of three increase in compressive strength without any associated weight penalty.

In particular, annealed copper and stainless steel were found to be good candidates, where both

materials possessed good combinations of work-hardening ability and ductility; work hardening

increased the compressive strength for these micro-trusses by up to a factor of three. The largest

gain was found for the stainless steel precursor, which was able to reach a weight-specific

strength of nearly 19 MPa/(Mg/m3).

An analytical model for the critical inelastic buckling stress of the micro-truss struts

during uniaxial compression was developed in terms of the axial flow stress during stretch

forming fabrication. During work hardening, increasing the forming displacement results in a

reduced relative density; at the same time, a larger truss angle (corresponding to an increased

structural efficiency) and fabrication-induced plastic strain leads to an increase in micro-truss

strength. On the other hand, increased forming displacement also results in a larger slenderness

ratio of the micro-truss ligaments and therefore a smaller critical buckling stress. At large

enough displacements, the increased truss angle and plastic strain will be overcome by the

reduction in critical buckling stress, leading to an overall decline in micro-truss strength.

124

7.2.2 Electrodeposition of Nanocrystalline Metal Sleeves

High-strength nanocrystalline metal sleeves were electrodeposited around the load-

supporting ligaments of cellular metal pre-forms, producing new types of hybrid nanocrystalline

cellular metals. It was shown that despite the added mass, the nanocrystalline sleeves could

increase the weight-specific strength of the micro-truss hybrids. For example, the largest coating

thickness (60 μm) improved the weight-specific strength of the hybrid steel micro-truss over the

uncoated micro-truss nearly 40%, from 12.8 to 17.7 MPa/(Mg/m3). In addition, a ~50 μm thick

nanocrystalline Ni coating was able to provide the same inelastic buckling resistance as the 1.13

mm × 0.63 mm cross-section plain carbon steel struts.

An isostrain model was developed based on the theoretical behaviour of a

nanocrystalline metal tube network in order to predict the compressive strength of the hybrid

materials. This model was developed for micro-truss hybrids, and then applied to open-cell

foam hybrids; for both cellular networks, the added load capacity of the sleeve agreed well with

its theoretical load capacity. However, the micro-truss struts serve as an example of a trade-off:

the area bending moment of the nanocrystalline sleeve increases with greater sleeve thickness,

improving the strut critical buckling stress; but as the sleeve thickness becomes large, the

overall weight-specific benefits diminish due to the added mass of nanocrystalline metal. A

trade-off between load capacity and weight can also be seen between the two hybrid types: the

benefits in the foam are limited due to the bending-dominated capacity (a function of thickness

to the third power), while the stretching-dominated capacity (a function of thickness to the

fourth power) results in a greater increase in micro-trusses specific strength. In other words, the

effectiveness of the nanocrystalline structural reinforcement depends on the connectivity of the

starting cellular pre-form.

125

8 Future Work This chapter outlines directions for future study of microstructural strengthening

mechanisms in micro-truss periodic cellular metals. In this section are issues that were raised

during the present thesis and would benefit from further study; and additional projects which

may be undertaken in an effort to optimize the performance of micro-truss materials

strengthened using microstructural mechanisms.

In the case of work hardening, further analysis is needed to determine the extent to

which the Bauschinger effect changes the critical buckling stress of the struts following forming.

Future work may also be useful to further improve the micro-truss mechanical performance

through work hardening and plastic strain. For example, it may be desirable to minimize the

Bauschinger effect in work-hardened micro-trusses for improved strength, and may be

accomplished using low-temperature annealing treatments following forming without a great

loss in dislocation density. Alternatively, supplemental strengthening mechanisms may aid not

only in the prevention of a Bauschinger effect but also the further improvement of the ligament

critical buckling stress. For example, small tensile pre-strains followed by a low-temperature

aging treatment in aluminum alloy 2024 has shown an increased strength (greater than 600

MPa) by way of a dislocation pinning mechanism, e.g. [A Latkowski, Aluminium 62 (1986)

113-115]. Alternatively, materials which utilize different plasticity mechanisms may be studied;

for example, high-manganese or twin-induced plasticity (TWIP) steel possesses good

formability characteristics, including high ratio of ultimate tensile stress-to-yield stress, high

ductility, high flow stress at large strains (between 1.3 and 1.8 GPa), and high work-hardening

ability, e.g. [S Allain, JP Chateau, O Bouaziz, S Migot, N Guelton, Mater. Sci. Eng. A 387-389

(2004) 158-162]; these characteristics are advantageous for deformation-formed micro-trusses

to reach low relative densities and high truss angles.

Forming limitations due to localization of plastic strain were also observed in the present

study. To improve the distribution of strain imparted, more work is needed to design in parallel

the many parameters of the forming process, including the precursor material and shape as well

as the forming press geometry. Furthermore, while the current forming apparatus was useful in

imparting the plastic strain, there was less focus on the final shape of the micro-truss core. For

instance, non-ideal effects, such as rounding at the nodes or imperfectly straight struts, were a

by-product of the press used in the present study. Additional study is needed to determine these

effects on the performance of the micro-truss.

126

In the case of electrodeposition of nanocrystalline metal sleeves, it was observed that the

non-uniformity of nanocrystalline Ni on the commercial aluminum foam resulted in a structural

reinforcement gradient. While the micro-truss cellular architecture is far simpler, it is still

susceptible to non-uniformity in the deposited sleeve thickness. Further work is therefore

needed to optimize the distribution of nanocrystalline metal. In addition, the nanocrystalline

foam hybrid was found to behave in a brittle manner during loading, where cracks were found in

the coated ligaments. On the other hand, the micro-truss ligaments were found to remain intact

through the initial failure mode. These differences may be due in part to the strength of the core-

sleeve bond established during electrodeposition; future work should investigate the role of the

inter-layer bond strength, as well as the influence of improved sleeve ductility on the overall

hybrid performance. More detailed work will also be needed to investigate the effect of strut

imperfections in the cellular pre-form, including surface roughness, out-of-straightness, and

node geometry in an effort to further improve the hybrid properties. In addition, fabricating and

testing hybrids with identical architectures but different electrodeposited grain sizes would

prove useful to develop a multi-scale model, spanning from Hall-Petch strengthening at the

grain size to inelastic buckling at the ligament size. Residual stresses in the deposited metal

should also be investigated to determine their role in the performance of the cellular hybrids. A

more broad study may investigate the electrodeposition of a range of nanocrystalline alloys to

map the properties attainable by hybrid micro-trusses. In these cases, the underlying architecture

is also of primary importance; concurrent design of a suitable micro-truss pre-form would be

required.

Last, work is needed in order to optimize the manufacturing of micro-trusses towards

more complex loading conditions, such as bending. Some initial topological optimization

protocols have already been suggested, e.g. [FW Zok, HJ Rathbun, Z Wei, AG Evans, Int. J.

Solids Struct. 40 (2003) 5707-5722; FW Zok, SA Waltner, Z Wei, HJ Rathbun, RM

McMeeking, AG Evans, Int. J. Solids Struct. 41 (2004) 6249-6271]. A full process will include

the influence of the manufacturing process, such as changes in ligament dimensions and

Bauschinger effects (e.g. in deformation-forming); it will also include non-ideal components,

such as strut imperfections and metal distributions (e.g. electrodeposition).

127

Appendix A: Fabricating Micro-Truss Sandwich Panels by Resistance Brazing In this appendix, the details surrounding the resistance brazing joining technique are

presented with respect to the aluminum alloy 3003 (AA3003) micro-truss sandwich panels

fabricated in Section 5.1. More detail can be found in Reference [1].

Resistance brazing was performed by a resistance welding machine comprised of a 2.5

kVA UNITEK 1-140 power supply and 2-037 weld head. To join the core micro-truss and face

sheets, a force of 83.4 N was imparted on two molybdenum capped electrodes with a diameter

of 3.16 mm. The brazing paste had an 80% KAlF4 – 15% K2AlF5 – 5% K3AlF6 liquid flux and

an 88% Al – 12% Si filler metal powder (Omni Technologies Corp., Brentwood, NH). For this

brazing paste, there is a window of at least 60˚C between the upper melting point of the filler

metal (582˚C [2]) and the solidus of the AA3003 aluminum alloy (643˚C [3]).

A window of operative resistance brazing times for square-perforated AA3003 was

determined for a range of current settings (% of maximum heat input) in [1,4] (Figure A1). Low

applied current or brazing time caused insufficient melting of the filler metal, preventing its

flow and consolidation in a homogeneous region. The result was a weak joint that failed by

shear through the filler metal. Conversely, an excessively high current or brazing time generated

enough heat to melt the base metal, resulting in undesirable electrode indentation that weakened

the joint by reducing the cross-sectional area. Between these extremes, there existed a window

of brazing settings which produced a desirable joint (defined by failure in the base metal during

lap shear tensile tests). Figure A1 also shows the high current/short time (Setting 1) and low

current/long time (Setting 2) brazing conditions used in Section 5.1, which defined the limits of

the adequate processing window.

128

Figure A1. Resistance brazing process window for AA3003. Joints brazed with insufficient heat fail by shear in the filler metal. Excessive heat causes melting of the parent metal and electrode punch-through. Adequate heat is defined as the heat input which creates failure outside the joint. The limiting brazing settings for the micro-truss sandwich panels are also shown.

References 1. E Bele, The Effect of Architecture, Joint Strength and Joint Density on the Structural Performance of


2. Omni Technologies Corporation: www.omnibraze.com/brazing.html.


4. E Bele, BA Bouwhuis, GD Hibbard, Mater. Sci. Eng. A 489 (2008) 29-37.

129

Appendix B: Properties of Electrodeposited Nanocrystalline Nickel

In this appendix, the tensile properties of electrodeposited nanocrystalline nickel

fabricated by Integran Technologies, Inc., are presented along with their parametric

representation in the form of the Ramberg-Osgood Model (ROM):

N

YS

t

S

tt E ⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

σσσ

ε 002.0 (B.1)

Tensile stress-strain (σt – εt) property data for nanocrystalline nickel (n-Ni)

electrodeposited using the same conditions as the n-Ni/steel micro-truss hybrids (sample A033-4

[1]) was provided by Integran Technologies, Inc. (Figure B1). The specimen had dimensions

according to the ASTM E8 subsize coupon geometry [2]. The elastic modulus ES in the early

strain range (0 ≤ εt ≤ 0.002) was measured as 145 GPa, and the 0.2% offset yield strength σYS

was measured as 825 MPa. Also shown in Figure B1 is the ROM representation of this curve, fit

using the strain-hardening index N.

Figure B1. Typical true stress – true strain representative tensile curve for electrodeposited nanocrystalline nickel. Also shown is the ROM parameterized representation.

The quality of this ROM fit is similar to other studies which have parameterized micro-truss

ligament material properties after the ROM, e.g. [3,4]. It is important to note that the early

region of plasticity is most critical for inelastic buckling and therefore requires the best curve fit;

the tangent modulus Et = (∂σt/∂εt) decreases quickly with increasing stress, satisfying the

Shanley model for critical buckling stress σCR:

130

2

22

λπ

σ tCR

Ek= (B.2)

where k is a coefficient describing the rigidity of the joints, and λ is the strut slenderness ratio.

It is also important to note that the measured nanocrystalline nickel elastic modulus (145

GPa) is lower than the typically reported value for polycrystalline nickel (207 GPa [5]). This

phenomenon is still under study, but may be related to crystallographic texture effects and the

increased intercrystalline volume fraction in nanocrystalline metals, e.g. [6,7]. However, there is

relatively little influence from the elastic modulus during inelastic buckling. This can be seen in

Figure B2 which plots Et as a function of stress for n-Ni using the fitting parameters from Figure

B1 (ES = 145 GPa, σYS = 825 MPa, N = 4.66) and the hypothetical case where ES = 207 GPa, σYS

= 825 MPa, and N = 4.66. The critical inelastic buckling stress range observed in Section 6.1

(900 ≤ σCR ≤ 1000) corresponds to a tangent modulus in the knee of the curve, in the

approximate range 0.010 ≤ εt ≤ 0.012. In this range, using ES = 207 MPa instead of ES = 145

GPa will change the tangent modulus by less than 10%.

Figure B2. True stress – tangent modulus plot using experimental data and Et calculated using the ROM. The tangent modulus decreases quickly with increasing stress. The influence of the elastic modulus is small, as evidenced by the near overlap of the curves using ES = 145 GPa and 207 GPa in the critical inelastic buckling stress range.

References 1. Integran Technologies, Inc., unpublished research.

2. ASTM E8-04, American Society for Testing and Materials, Philadelphia, 2004.



Solids Struct. 39 (2002) 4093-4115.

131

5. JR Davis (Ed.), Nickel, Cobalt, and Their Alloys, ASM International, Materials Park, 2000.

6. TD Shen, CC Koch, TY Tsui, GM Pharr, J. Mater. Res. 10 (1995) 2892-2896.

7. Y Zhou, U Erb, KT Aust, G Palumbo, Z. Metallkd. 94 (2003) 1157-1161.

Microstructural Strengthening Mechanisms in Micro-Truss ... · Microstructural Strengthening Mechanisms in Micro-Truss Periodic Cellular Metals Brandon Andrew Bouwhuis Doctor of Philosophy,

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