LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF … · LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF HETEROGENEOUS MATERIALS . METHOD INTRODUCTION PROBLEM A mechanical structure or structural

Institute of General Mechanics RWTH Aachen University, Germany

M. Chen, A. Hachemi, D. Weichert

PVP 2012, July 15-19, 2012, Toronto, Canada

LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF HETEROGENEOUS MATERIALS

METHOD

INTRODUCTION

PROBLEM

A mechanical structure or structural element made of composite materials operates beyond the elastic limit :

Determination of material properties

Variable loads with unknown evolution in time

Direct methods combined with homogenization technique

Direct methods give information on serviceability without calculating the evolution of mechanical field quantities.

Instantaneous collapse Limit Analysis Failure under variable loads Shakedown Analysis

Prediction of the global material properties by using homogenization theory

CONTENTS

Basic concepts • Static shakedown theorem

Direct methods applied to composites • Elements of homogenization theory • Boundary conditions • Consideration of kinematic hardening

Failure criterion of composites • Loci of yield strength fitting

Numerical examples • Periodic fiber reinforced metal matrix composites

• Porous material

Conclusions

BASIC CONCEPTS

ε

σ

σmax

ο ο

σ

ε ο

σ

ε ο

σ

ε

σmax σmax

σmax

σmin σmin

σmin σmin

Purely elastic Shakedown Low-cycle fatigue Ratcheting

𝜀𝜀𝑝𝑝(𝒙𝒙, 𝑡𝑡) = 0 lim𝑡𝑡→∞

𝜀𝜀�̇�𝑖𝑖𝑖𝑝𝑝 = 0 ∆𝜀𝜀𝑝𝑝(𝒙𝒙) = � �̇�𝜀𝑖𝑖𝑖𝑖

𝑝𝑝 (𝒙𝒙, 𝑡𝑡) d𝑡𝑡 = 0T

0 ∆𝜀𝜀𝑝𝑝(𝒙𝒙) = � �̇�𝜀𝑖𝑖𝑖𝑖

𝑝𝑝 (𝒙𝒙, 𝑡𝑡) d𝑡𝑡 ≠ 0T

0

Schematic illustration of different material behaviors

Definition of load domain*

BASIC CONCEPTS

* König, J.A. : Elsevier, Amsterdam (1987).

A finite number 𝑛 of types of loads:

𝑃 𝒙𝒙, 𝑡𝑡 = 𝑃 𝛽𝑠 𝑡𝑡 ,𝒙𝒙 𝑥 ∈ 𝑉 or SP; 𝑠 = 1,⋯ , 𝑟; 𝛽s− ≤ 𝛽s 𝑡𝑡 ≤ 𝛽𝑠+ 𝑠 = 1,⋯ ,𝑛

Loading domain 𝑃 can be described by a 𝑛-dimensional polyhedron:

𝑃(𝑥, 𝑡𝑡) = 𝑃|𝑃 = �𝜇𝑠 𝑡𝑡𝑛

𝑠=1

𝑃𝑠0 𝑥 , 𝜇𝑠(𝑡𝑡) ∈ 𝜇𝑖−,𝜇𝑖+

Two dimensional loading domain 𝓛 and α𝓛

Static shakedown theorem by Melan*

BASIC CONCEPTS

* Melan, E. : Sitzber. Akad. Wiss., Abt. IIA 145, 195-218 (1936 ).

If there exist a loading factor 𝛼 > 1 and a time-independent residual stress field 𝝆�(𝑥) whose superposition with elastic stresses 𝝈𝑬 does not exceed the yield condition 𝑭 ≤ 0 at any time 𝑡𝑡 > 0 and at all points 𝑥 ∈ 𝑉 in volume 𝑉 of the considered structure,

𝐹 𝛼𝝈𝑬 𝑥, 𝑡𝑡 + 𝝆� 𝑥 , 𝜎𝑌 𝑥 ≤ 0

where: 𝜎𝑌 𝑥 is yield stress.

𝝈𝑬 𝑥, 𝑡𝑡 is purely elastic stress reference.

then the system will shakedown under arbitrary load paths contained within given load domain 𝓛.

Numerical implementation

BASIC CONCEPTS

* Weichert, D., Hachemi, A. and Schwabe, F. : Mech. Res. Comm. 26, 309-318 (1999)

A purely elastic reference stress 𝝈𝑬 is calculated for each loading vertex by means of conventional FE-analysis

𝝈𝑬 and [C] are input data for the subsequent SD/LA-module.

Finite element discretization

Principle of virtual work

� 𝛿𝜺 𝑇 𝛼𝝈𝐸 + 𝝆�𝑉

d𝑉 = � 𝛿𝒖 𝑇 𝐩∗𝜕𝑉

d𝑆 + � 𝛿𝒖 𝑇 𝐟∗𝑉

d𝑉

Equilibrium conditions for 𝝆� are satisfied by principle of virtual work * :

� 𝛿𝜺 𝑇

𝑉𝝆� d𝑉 = 𝛿𝒖 � 𝑩𝑇 𝝆�

𝑉d𝑉 = 0

⇒ � 𝑩𝑇 𝝆� d𝑉𝑉𝑒

= � 𝑩𝑇 𝝆� 𝐽 d𝑟 d𝑠 d𝑡𝑡1

−1= � 𝐽 𝑩𝑇 𝝆�

𝑁𝑁𝐸

𝑗=1

= 0

⇒

� � 𝐽 𝑩𝑇 𝝆�𝑁𝑁𝐸

𝑗=1

𝑁𝐸

𝑘=1

= 𝑪 𝝆� = 0

Numerical implementation

BASIC CONCEPTS

Mathematic formulation of shakedown problem

Objective function Load factor 𝛼 Variables 𝛼 and residual stress field 𝝆� Linear equality constraints Self-equilibrated condition Nonlinear non-equality constraints Yield condition

• Algorithm Augmented Lagrangian method Sequential quadratic programming Interior Point Methed ……

• Software Packages LANCELOT *, … SNOPT, NPSOL,NLPQL IPDCA**, IPOPT*** … ……

Large-scale optimization

* Conn, A.R., Gould, N.I.M. and Toint, Ph.L. : Berlin Heidelberg, Springer-Verlag (1992).

** Pham Dinh Tao; Le Thi Hoai An. : SIAM J. Opt. 8, 476-505 (1998 ).

*** Wächter, A. and Biegler, L. T. : Math. Program.106(1), 25-57 (2006).

with NGS is the total number of Gauss Points; 𝑛 is the number of independent loads; 𝑃𝑘 is the load vertex. 𝑘 =1 corresponds to limit analysis; 𝑘 = 2𝑛 corresponds to shakedown analysis.

max𝛼

� 𝑪 𝝆� = 0𝐹 𝛼𝝈𝑖𝐸 𝑃𝑘 + 𝝆�𝑖 ,𝜎𝑌𝑖 ≤ 0𝑖𝑖 ∈ 1,𝑁𝑁𝑆 , 𝑘 ∈ 1, 2𝑛

Objective

Micro-/Mesoscopic Level --------------------------------------- Numerical Model of limit and shakedown analysis of RVE

Numerical Solution of limit and shakedown problem (FEM & Optimization)

Macroscopic Level ---------------------------------------- Comparison of loading carrying capacity of composites with different fiber distributions

Prediction of elastic and plastic material properties

Homogenization

DIRECT METHODS APPLIED TO COMPOSITES

Assumptions

• Periodic composites

• At least one ductile phase


* Suquet, P. : Doctor Thesis, Pierre & Marie Curie, (1982)

Heterogeneous Material RVE Homogenized Material

Localisation Globalisation

𝜉𝜉 = 𝑥 𝜃𝜃⁄ , θ: a small parameter

𝜮𝜮(𝑥) =1𝑉� 𝝈(𝜉𝜉) d𝑉 = ⟨𝝈(𝜉𝜉)⟩𝑉

𝑬(𝑥) =1𝑉� 𝜺(𝜉𝜉) d𝑉 = ⟨𝜺(𝜉𝜉)⟩𝑉

Concept of representative volume element (RVE)

Average field quantities *

Homogenization theory


* Weichert, D., Hachemi, A. and Schwabe, F. : Mech. Res. Comm. 26(3), 309-318 (1999). ** Suquet, P. : Doctor Thesis, Pierre & Marie Curie, (1982).

Static direct methods for periodic composites *

Localization problem – Boundary conditions **

Strain method

Stress method

Periodicity

Uniform strain is imposed on 𝜕𝑉 : 𝒖 = 𝑬 ∙ 𝝃 on 𝜕𝑉

Uniform stress is imposed on 𝜕𝑉 : 𝝈 ∙ 𝒏 = 𝜮𝜮 ∙ 𝒏 on 𝜕𝑉

Anti-periodicity of stress : 𝝈 ∙ 𝒏 anti-periodic on 𝜕𝑉 Decomposition of local strain : 𝜀𝜀 𝒖 = 𝑬 + 𝜀𝜀 𝒖per = 𝑬 + 𝜺per Average of 𝜺∗ over the RVE : 𝜺per = 0

Under thermal loading

𝜮𝜮 =1𝑉� 𝛼𝝈𝑬 + 𝝆� d𝑉𝑉

=1𝑉� 𝛼𝝈𝑬 d𝑉𝑉

+1𝑉� 𝝆� d𝑉𝑉

with 1𝑉� 𝝆� d𝑉𝑉

= 0


* Magoariec, H., Bourgeois, S. and Débordes, O. : Int. J. Plasticity. 20, 1655-1675 (2004). ** Chen, M., Hachemi, A. and Weichert, D. : Proc. Appl. Math. Mech. 106, 405-406 (2010).

Before deformation after deformation symmetrical part

Periodic composites – strain method *

Finite element discretization using non-conforming element ** • Material Model: elastic-perfectly plastic • von Mises yield criterion

max𝛼

� 𝑪 𝝆� = 0𝐹 𝛼𝜎𝑖𝐸 𝑃�𝑘 + 𝝆�𝑖 ,𝜎𝑌𝑖 ≤ 0𝑖𝑖 ∈ 1,𝑁𝑁𝑆 , 𝑘 = 1,⋯ , 2𝑛

Nr. Vriables: 6NGS+1 Nr. Equality Constraint: 3NK+9NE Nr. Inequality Constraint: NL*NGS NL=1, limit analysis; NL=2𝑛, shakedown analysis

𝓟𝐬𝐬𝐬𝐬𝐬𝐬

div 𝝈𝐸 = 0 in 𝑉 𝝈𝐸 = 𝒅: 𝑬 + 𝜺per in 𝑉 𝝈𝐸 ∙ 𝒏 anti − periodic on 𝜕𝑉 𝒖per periodic on 𝜕𝑉 𝜺 = 𝜠

𝓟𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐫𝐬 �div 𝝆� = 0 in 𝑉

𝝆� ∙ 𝒏 anti − periodic on 𝜕𝑉

Unlimited kinematic hardening


Consideration of kinematic hardening

Sup { 𝛼 } s.t. 𝐹 𝛼𝝈𝐸 + 𝝆� − 𝝅,𝝈𝒀 ≤ 0

Limited kinematic hardening

* Weichert D., Gross-Weege J. Int. J. Mech. Sci., 30, 757-767 (1988). ** Stein E., Zhang G. and König J.A. Int.J.Plast., 8,1-31 (1992).

Sup { 𝛼 } s.t. 𝐹 𝛼𝝈𝐸 + 𝝆� − 𝝅,𝝈𝒀 ≤ 0 𝐹 𝛼𝝈𝐸 + 𝝆�,𝝈𝑼 ≤ 0*

OR 𝐹 𝝅,𝝈𝑼 − 𝝈𝒀 ≤ 0**

where : 𝝅 is back stress.

where : 𝝈𝑼 is ultimate stress.

FAILURE CRITERION OF COMPOSITES

Failure : Every material has certain strength, expressed in terms of stress or strain, beyond which the structures fracture or fail to carry the load.

For heterogeneous material, consisted of two or more than two phases materials, how to determine the Failure Criterion?

Why Need Failure Criterion for Composites? • To determine weak and strong directions • To guide local design of composites • To guide global design of composites-structures

Macromechanical Failure Theories in Composites ?? • Maximum stress theory • Maximum strain theory • Tsai-Hill theory (Deviatoric strain energy theory) • Tsai-Wu theory (Interactive tensor polynomial theory) • …… • Loci of yield strength fitting based on limit macroscopic stress domain

* Hosford, W.F. : Oxford University (1993)

𝜎12 + 𝜎2

2 − � 2𝑅𝑅𝑅𝑅+1

�𝜎1𝜎2 − 𝑋𝑋2 = 0

where 𝑅𝑅 = 2 �𝑍𝑍𝑋𝑋�

2− 1

with 𝑋𝑋 = 1√𝐹+𝐻𝐻

and 𝑍𝑍 = 1√2𝐹

R : is a measure of the plastic

anisotropy of a rolled metal

Hill’s yield criterion in plane stress*


-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5R=1R>1R<1

The plastic strain ratio R is a parameter that indicates the ability of a sheet metal to resist thinning or thickening when subjected to either tensile or compressive forces in the plane of the sheet.

Hill’s yield criterion in plane strain (general stress state)


“y-convention” ,i.e.(Z,Y’,Z’’):

• Rotate the 𝑥1𝑦𝑦1𝑧𝑧1-system about the 𝑧𝑧1-axis (Z) by angle Ψ;

• Rotate the current system about the new 𝑦𝑦-axis (Y’) by angle Θ;

• Rotate the current system about the new 𝑧𝑧 –axis (Z’’) by angle Φ.

�𝑥2𝑦𝑦2𝑧𝑧2

� = 𝑇𝑇−1 �𝑥1𝑦𝑦1𝑧𝑧1

�

Rotation matrix is:

𝑇𝑇𝑥1𝑦𝑦1𝑧𝑧1 = �cosΨ − sinΨ 0sinΨ cosΨ 0

0 0 1��

cos Θ 0 sin Θ0 1 0

−sin Θ 0 cos Θ��

cosΦ − sinΦ 0sinΦ cosΦ 0

0 0 1�

with

Ψ =𝜋𝜋4

; Θ= tan−1�√2� ; Φ = −𝜋𝜋4



• Calculation of principle stress

• Projection in 𝛑𝛑-plane

𝐹(𝜎2 − 𝜎3)2 + 𝑁(𝜎3 − 𝜎1)2 + 𝐻𝐻(𝜎1 − 𝜎2)2 − 1 = 0

𝐹,𝑁,𝐻𝐻 are defined as:

𝐹 =12 �

1𝑌𝑌2 +

1𝑍𝑍2 −

1𝑋𝑋2� ; 𝑁 =

12 �

1𝑍𝑍2 +

1𝑋𝑋2 −

1𝑌𝑌2� ; 𝐻𝐻 =

12 �

1𝑋𝑋2 +

1𝑌𝑌2 −

1𝑍𝑍2�

Let: �𝜎1𝜎2𝜎3

� = 𝑇𝑇 �𝛾𝛾1𝛾𝛾2𝛾𝛾3

�

(𝐹 + 1.866𝐻𝐻 + 0.134𝑁)𝛾𝛾12 + (𝐹 + 0.134𝐻𝐻 + 1.866𝑁)𝛾𝛾2

2 +(𝑁 − 2𝐹 + 𝐻𝐻)𝛾𝛾1𝛾𝛾2 = 1

Stress Invariants 𝜎𝑛 Independent on coordinate system

𝜎𝑖𝑖𝑖𝑖 Dependent on coordinate system



For homogenous material, 𝑋𝑋 = 𝑌𝑌 = 𝑍𝑍, i.e. 𝐹 = 𝑁 = 𝐻𝐻:

𝛾𝛾12 + 𝛾𝛾2

2 = 𝐶𝐶 ; here: 𝐶𝐶 =1

3𝐹 =23𝜎𝑌𝑌

2

For transversely homogenous material, assume 𝑌𝑌 = 𝑍𝑍 , i.e. 𝑁 = 𝐻𝐻

(𝐹 + 2𝐻𝐻)𝛾𝛾12 + (𝐹 + 2𝐻𝐻)𝛾𝛾2

2 + (2𝐻𝐻 − 2𝐹)𝛾𝛾1𝛾𝛾2 = 1

(𝐹 + 1.866𝐻𝐻 + 0.134𝑁)𝛾𝛾12 + (𝐹 + 0.134𝐻𝐻 + 1.866𝑁)𝛾𝛾2

2 +(𝑁 − 2𝐹 + 𝐻𝐻)𝛾𝛾1𝛾𝛾2 = 1

NUMERICAL EXAMPLES

E (MPa) 𝜐 𝜎𝑌 (MPa) 𝜎𝑈 (MPa)

Matrix (Al) 70e3 0.3 80 120

Fiber (Al2O3) 370e3 0.3 2000 ---

Material properties:

Periodic composites: Perfect bounding Different fiber distribution Different volume fraction (0 ~ 50%)

Periodic fiber reinforced metal matrix composites

Unidirectional fiber reinforced periodic composites

Representative volume element

Finite element model

Square pattern

Rotated pattern

Hexagonal pattern

0.200

0.220

0.240

0.260

0.280

0.300

0.320

0 10 20 30 40 50

Poiss

on R

atio

Fiber Volume Fraction (%)

Homogenized Poisson Ratio

Square-X,Y

R-Square-X,Y

Hexagonal-X

Hexagonal-Y

40

60

80

100

120

140

160

0 10 20 30 40 50

Youn

g's

Mod

ulus

(G

Pa)

Fiber Volume Fraction (%)

Homogenized Young's Modulus

Square-X,Y

R-Square-X,Y

Hexagonal-X

Hexagonal-Y

NUMERICAL EXAMPLES

Homogenized elastic material properties • Elastic-perfectly plastic

Matrix: • Elastic-perfectly plastic (SD) • Unlimited kinematic hardening(SDH-UN) • Limited kinematic hardening(SDH)

Admissible shakedown displacement domain

Admissible limit displacement domain

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4U2/

U0

U1/U0

SD

SDH

SDH-UN

AP -6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6U2/

U0

U1/U0

LMLMH

NUMERICAL EXAMPLES

Fiber: Square pattern distributed Elastic-perfectly plastic Volume fraction 40%

Displacement domain for kinematic hardening

Admissible macroscopic shakedown stress domain

Admissible macroscopic limit stress domain

NUMERICAL EXAMPLES

Macroscopic stress domain for kinematic hardening

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3Σ22/σm

Σ11/σm

SD

SDH

SDH-UN -5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -3 -1 1 3 5Σ22/σm

Σ11/σm

LM

LMH

Matrix: • Elastic-perfectly plastic (SD) • Unlimited kinematic hardening(SDH-UN) • Limited kinematic hardening(SDH)

Fiber: Square pattern distributed Elastic-perfectly plastic Volume fraction 40%

-500 0 500

-800

-600

-400

-200

0

200

400

600

800

Admissible Stress Domain

LM

-200 -100 0 100 200-200

-150

-100

-50

0

50

100

150

200Admissible Stress Domain-Projected in Pi-plane

LM

Major axis: a=241.83 Minor axis: b=67.78

X = 296.18 Mpa = 3.70 𝜎m Y = Z= 98.5 MPa = 1.23 𝜎m

NUMERICAL EXAMPLES

Prediction of plastic material properties • Elastic-perfectly plastic

NUMERICAL EXAMPLES

E (MPa) 𝜐 𝜎𝑌 (MPa) Steel (Outside) 200e3 0.30 360

Aluminum (Inside) 72e3 0.33 100

Material properties:

Porous material

Items Dimensions (mm) Width 8 Height 6

Radius of the hole 1.5 Thickness of single layer 2 Thickness of two layers 4

Dimensions of RVE

NUMERICAL EXAMPLES

Porous material

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8U2/U

0

U1/U0

SD_SteSD_AluSD_2Mat

-8

-6

-4

-2

0

2

4

6

8

-8 -6 -4 -2 0 2 4 6 8U2/U

0

U1/U0

LM_SteLM_AluLM_2Mat

-400

-200

0

200

400

-400 -200 0 200 400Σ22(

MPa)

Σ11(MPa)

SD_SteSD_AluSD_2Mat

-400

-200

0

200

400

-400 -200 0 200 400

Σ22(

MPa)

Σ11(MPa)

LM_SteLM_AluLM_2Mat

• Displacement domain • Macroscopic stress domain

CONCLUSIONS

Summary

Perspectives

• Application of direct methods to composites. Three boundary conditions are discussed.

• Prediction of transverse elastic material properties of unidirectional continuous fiber reinforced metal matrix composites based on homogenization theory.

• Consideration of the hardening enlarged the shakedown and limit domain.

• Definition of yield loci for periodic composites and the prediction of plastic material properties by using yield surface fitting.

• Apply direct methods to other composites types and more complex problems.

• Consider thermal loads, as well as the temperature dependent yield strength.

• The presented results for the fiber-reinforced composite are only numerical, and a future effort has to be made in order to compare with experimental results quantitatively.

Thanks for

your attention!

LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF … · LIMIT LOADS FOR STRUCTURAL ELEMENTS MADE OF HETEROGENEOUS MATERIALS . METHOD INTRODUCTION PROBLEM A mechanical structure or structural

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