Adhesive Joining for Crashworthiness - Material Data and Explicit ...

THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Adhesive Joining for Crashworthiness

Material Data and Explicit FE-methods

THOMAS CARLBERGER

Department of Applied Mechanics

CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden, 2008

Adhesive joining for crashworthiness Material data and explicit FE-methods THOMAS CARLBERGER ISBN 978-91-7385-132-9 © THOMAS CARLBERGER, 2008 Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 2813 ISSN 0346-718X Department of Applied Mechanics Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone +46 (0)31 772 1000 Chalmers Reproservice Göteborg, Sweden, 2008

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Adhesive Joining for Crashworthiness Material Data and Explicit FE-methods

THOMAS CARLBERGER Department of Applied Mechanics Chalmers University of Technology

ABSTRACT

Today, crash simulations replace crash testing in the product development phase in the automotive industry. High quality simulations enable shorter product development time and higher competitiveness. However, increasing requirements regarding emissions and crashworthiness are demanding optimised material choice in the parts constituting the car body structure. Lightweight materials are becoming frequently used. Joining dissimilar materials is difficult using common joining techniques like spot welding. To this end, adhesive joining is currently gaining popularity not only due to the ability to join dissimilar materials, joint integrity and structural stiffness both increase by the use of adhesive joining. Moreover, the number of spot welds may be reduced in hybrid joints. In this thesis, adhesive joints are studied with respect to crashworthiness of automotive structures. The main task for the adhesive is not to dissipate the impact energy, but to keep the joint integrity so that the impact energy can be consumed by plastic work of the base materials. Fracture of adhesives can be accurately modelled by cohesive zones. The dynamic behaviour of finite element structures containing cohesive zones is studied using a simplified structure. An amplified strain rate is found in the adhesive as compared to the base material. The cohesive zone concept is used in the development of a 2D interphase element. The accuracy and time step influence of the interphase element is compared to solutions based on continuum element representation of the adhesive. The interphase element is found to predict fracture of the adhesive joint with engineering accuracy and has a small effect on the time step of the explicit FE method. The cohesive laws for use in the material models of the adhesive have been determined using dedicated test methods. The double cantilever beam specimen and the end notched flexure specimen are used with inverse methods to determine cohesive laws in peel and shear, respectively. The cohesive laws are determined for varying temperature, strain rate and adhesive layer thickness. A built up bimaterial beam is designed for testing and simulation of joints consisting of bolts, adhesives and combinations of bolts and adhesives, i.e. hybrid joints. The model of the hybrid beam developed was found to be able to predict results from impact tests, quantified as maximum load and deformed shape of the beam. Keywords: Adhesive joining; Dynamic fracture; Cohesive zone; Interphase Element; Bi-

material joining; Hybrid joints

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iii

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THESIS This thesis consists of five appended papers and a summary. Paper A Carlberger T, Stigh U. 2007. An explicit FE-model of impact fracture in an

adhesive joint. Engineering Fracture Mechanics 74 (2007) 2247-2262. Paper B Carlberger T, Alfredsson KS, Stigh U. Explicit FE-formulation of Interphase

Elements for Adhesive Joints. Accepted for publication in International Journal for Computational Methods in Engineering Science & Mechanics.

Paper C Carlberger T, Biel A. Influence of temperature and strain rate on cohesive

properties of a structural epoxy adhesive. To be submitted. Paper D Carlberger T. Influence of layer thickness on cohesive properties of an epoxy-

based adhesive – an experimental study. To be submitted. Paper E Carlberger T. Dynamic testing and simulation of hybrid joined bimaterial beam.

To be submitted.

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CONTRIBUTIONS TO CO-AUTHORED PAPERS Three of the papers were written in collaboration with co-authors. The contributions by the author of this thesis to these papers are listed below. Paper A Predominantly responsible for planning the paper. Responsible for performing the calculations. Wrote the paper in collaboration with the co-author. Paper B Predominantly responsible for planning the paper.

Developed the theory in collaboration with the co-authors Responsible for performing the derivations.

Wrote the paper in collaboration with the co-authors. Paper C Predominantly responsible for performing the experiments.

Predominantly responsible for evaluation of the experiments. Wrote the paper in collaboration with the co-author.

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CONTENTS ABSTRACT i

THESIS iii

CONTRIBUTIONS TO CO-AUTHORED PAPERS iv

CONTENTS v

PREFACE vii

REVIEW AND SUMMARY OF THESIS 1

1 Introduction and motivation 1

2 Adhesive layer model 4

3 Fracture of adhesive layers 5

4 Test methods 6

5 Parameters influencing adhesive joints 8

6 Manufacturing aspects 9

7 Summary of appended papers 10

8 Future work 11

References 13

APPENDED PAPERS

Paper A

Paper B

Paper C

Paper D

Paper E

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PREFACE The work presented in this thesis has been carried out during the years 2003-2008 at the Department of Engineering Science at the University of Skövde, Sweden. It is a part of a co-operation with the Mechanics of Materials group at the Department of Applied Mechanics at Chalmers University of Technology in Göteborg, Sweden. First, I would like to express my deepest gratitude towards my supervisor and co-author Professor Ulf Stigh, who has patiently guided me through these years. His good temper along with a genuine interest for the research subject inspires not only his researchers but all colleagues. I would also like to thank my co-authors Dr. Svante Alfredsson and Mr. Anders Biel for fruitful collaboration. Special thanks are directed to Dr. Kent Salomonsson for helping with computers and software. Further, I would like to extend many thanks to all my colleagues at the University of Skövde and at Chalmers University of Technology. Most of all, I would like to thank my family for supporting me and giving me the strength to fulfil this quest. Without your love and affection I would never have made it.

Sjuntorp in May 2008 Thomas Carlberger

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REVIEW AND SUMMARY OF THESIS References are made to Papers A through E in the thesis. 1. Introduction and motivation The need for safe and fuel efficient vehicles is rapidly increasing. Environmental concerns are continuously escalating. Focus is mainly on lowering of emissions. An issue, sharing this focus is safety. Concerning safety, WHO Europe report 127 000 persons killed and some 2.4 million persons injured every year in road accidents in the European region (Peden et al. 2004). Furthermore, WHO Europe report that road accidents account for killing more children and young people aged 5-29 than any other cause of death. WHO Europe has recently classed traffic injuries as a major health problem. Structural integrity of automotive car bodies is crucial for occupant safety. The strive to make cars lighter, and thus lowering the energy consumption, forces the automotive manufacturers to explore new materials with higher specific strength, so called “light weight materials” such as e.g. aluminium and magnesium alloys, composites and high strength steel. These materials are difficult to join using traditional methods like spot welding. A convenient method for joining dissimilar materials is adhesive joining. Adhesive joining is constantly gaining popularity in the automotive field. The reason is not solely the ability to join dissimilar materials. Adhesive joining will also enable reduced cost, improved strength, stiffness and fatigue properties. Since adhesives are mostly good electrical isolators, adhesive joining will improve corrosion resistance by separating materials from each other preventing galvanic corrosion and additionally seal out water from the structure. In modern product development, the engineers use simulations to verify the function of the product. Of special interest for the automobile industry is a rational treatment of adhesive joints in crash simulations. Effects of the special geometrical and material properties of adhesives have to be considered. Adhesives are used in thin layers; the thickness is typically a fraction of the thickness of the joined sheet metals. The sheet metals are modelled as shells and a structural model of the adhesive has to be properly adapted. Moreover, the adhesive consists of polymers. Generally, polymers are known to be temperature and strain rate dependent. In this thesis, these aspects of adhesive joining are studied. Adhesives behave as ductile materials when constrained between stiffer materials. A ductile fracture process may be described by void nucleation, void coalescence, macroscopic crack initiation, crack propagation and eventually catastrophic failure. These processes can be described more or less rigorously with constitutive models. A particularly simple model is given by a cohesive zone model. In such a model, the tractions exerted by the adhesive layer on the joined parts are given by the deformation of the adhesive layer as measured from the separation of the interfaces. Cohesive zone modelling is a common technique for capturing fracture of materials. This modelling technique has received an increasing interest during the last decades. This is partly due to the relative simplicity and numerical stability given by cohesive models. However, more importantly, cohesive models have proven to be able to accurately predict the fracture behaviour in many situations. This is the greatest merit of the model.

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If a cohesive zone representing an adhesive joint is subjected to dynamic loading, the joint will deform and eventually fracture depending on the shape and duration of the impacting stress wave. If the cohesive model is simplified, it is possible to determine an exact solution. It is then interesting to compare the exact solution with a finite element solution. This is studied in Paper A. A typical stress response for an engineering adhesive is given by the cohesive law in Fig. 1. In Paper A, this cohesive law is simplified with a saw-tooth shaped cohesive law capturing two of the most important properties of a cohesive law: the area under the curve equal to the fracture energy and the peak stress.

Figure 1. Cohesive law. Peel stress, σ, varies with the peel separation, w. The fracture energy, Jc, equals the area under the σ-w curve. Results from Paper C for the engineering adhesive DOW Betamate XW1044-3 with a 0.4 mm layer thickness and εɺ = 2.7⋅10-3 s-1. Structural adhesives are applied in the body shop to join structural parts before painting and assembling. Manufacturing aspects of adhesive joining in an automotive production line include wash out effects in the pre-paint process bath, initial joint strength needed to keep parts together before curing is achieved in the paint oven and working environmental issues in the body shop due to potentially hazardous chemical substances in the adhesives. A number of mechanical fasteners are needed to provide sufficient initial joint strength until the adhesive is fully cured in the paint process. During assembly in the body shop, the body structure is subjected to relatively low forces. Thus, only a minimum of mechanical fasteners are needed to provide the required strength. This addresses the task of assessing joint strength of combination joints comprising both adhesives and mechanical fasteners. This is the motivation for Paper E. Impact testing of complete car bodies during the development phase is dramatically decreasing thanks to the ability of computer simulations to reliably predict crash performance. Nowadays, testing is mainly performed to validate legal requirements. Prerequisites for successful crash simulation are among others, skilled personnel, reliable FE-code, reliable material models and sufficient computer performance. In the finite element code, material models are used to simulate the material response of each material, in this case, each adhesive material. New adhesives are continuously developed to meet the industrial demands. Material models have to be determined for each of these adhesives in order to enable computer simulation. Tests are performed to determine the material parameters of the material model used for modelling the adhesive. Three of the most important parameters influencing the properties of an adhesive are temperature, strain rate and adhesive thickness. This is the motivation for Paper C and Paper D. The temperature span during vehicle usage ranges roughly from -40°C to +125°C excluding the engine compartment. In structural important areas of the car body, the temperature ranges from -40°C to +60°C. Contrary to metals, polymers are relatively sensitive to temperature variation, changing fracture mode from brittle at low temperatures to ductile at high temperature. Adhesives for crash relevant joints are

σ σσ

w

Jc

wc

3

mostly epoxy based polymers. Since the polymer changes its properties with temperature, so does the adhesive. A relevant material parameter in this context is the glass transition temperature, Tg, below which the polymer behaves as glass, in contrast to the rubbery behaviour exhibited above Tg. For many epoxy-based adhesives, Tg ~ +100°C. Impact simulations are performed using explicit finite element codes. Explicit finite element codes are efficient for solving large dynamic problems, such as impact. The simple calculation scheme of the explicit FE-method enables solving very large models subjected to impact, cf. e.g. Hughes (2000). The explicit finite element method is numerically stable under the condition that the time step, ∆t, is less than the critical time step, the Courant limit (CFL-limit) cf. Courant et al. (1928). A simulation is progressed in small time steps and a large amount of time steps have to be used to accomplish the total simulation time. The response of a cohesive zone model, in the application of an adhesive joint, may be dependent on certain parameters of the explicit FE-code. Such parameters are e.g. Young’s modulus of the adhesive, adhesive thickness and minimum allowed time step size. This is the motivation for Paper A. The critical time step, ∆tc, is determined by 2/ωmax, where ωmax is the largest eigenfrequency of the structure. Determining the eigenfrequencies of a large FE-model is associated with large computational effort. It is therefore avoided in commercial explicit computer codes such as LS-Dyna and Abaqus/Explicit. Instead, an estimate of the critical time step is given by the time it takes an elastic wave to travel the smallest distance between two nodes in the structure. The estimation of the critical time step, ∆tc, is given by

minc

lt

c∆ = where

Ec

ρ= , (1a,b)

where c is the wave speed, lmin is the shortest distance between two nodes in an element, E is Young’s modulus and ρ is the density of the material. The element size and time step size are therefore intimately related in a crash simulation. There is a natural coupling between computer power and simulation complexity. The simulation complexity follows the computer development, such that results are achieved within 12 to 15 hours for a complete crash simulation of a car impact. Today, a typical element size for steel elements is about 5 mm, which, with Eq. (1), gives a time step size of roughly 1 µs. With a time step size of 1 µs, epoxy, which is the main constituent in crash modified adhesives, gives lmin = 1.2 mm. The small element size, necessary to model the adhesive thickness (typically 0.2 mm) requires the time step to be 1.2/0.2 = 6 times smaller, cf. Eq. (1a). This leads to a six times longer solution time, which is unacceptable. The same problem with the time step arises for simulations containing spot welds. A common technique used in the industry today to handle this time step size problem, is to use mass scaling: the mass of the spot weld is increased until the time step for the spot weld element matches the minimum time step size for the FE-model without spot welds. This technique has the advantage that the solution is reached in a reasonable time but it affects the accuracy of the simulation. It is necessary to estimate the effect of the added mass on the result. For instance, developers of airbag sensors avoid mass scaling of spot welds, since it destroys the accuracy of the calculated acceleration of the simulation, which is of great importance to airbag trigging. Mass scaling is often used together with stiffness reduction of the small elements in order to minimise the effect of locally added mass (DuBois, 2005). Since automotive body structures mainly consist of shell-shaped parts, the automotive body is modelled with structural shell elements using the displacement and rotation of the shell

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middle surface as the primary deformation variables. In a FE-context, the displacement of the interface between the shell surface and the adhesive layer is interpolated using the shape functions and nodal displacements and rotations. In general, it is possible to calculate the adhesive traction response from the adherend deformation by the cohesive law. The traction on the adhesive/adherend interface is transmitted to the shell nodes as forces and moments. To be specific, one interphase element takes up the volume from the mid-plane of one shell element, through the adhesive layer, ending at the mid-plane of the other shell, cf. Fig. 2. However, only the material within the adhesive layer contributes to the constitutive behaviour of the interphase element.

Figure 2. 2D cross section of adhesively joined sheets (adherends). (a): Adhesive and adherends. (b): Interphase element. The name interphase originates from the fact that the element models the adhesive phase between the adhesive/adherend interfaces. In Paper B it is shown that the interphase element presents a possibility to circumvent the time step problem, as well as simplifying the task of building the model. Several similar elements have been developed. As compared to the spring elements in Reedy and Mello (1997) and Borg et al. (2004), no consideration to the mesh size has to be done to input the constitutive behaviour to the interphase element. 2. Adhesive layer model Though adhesive joining is an ancient joining method, the methods for calculation of joint strength of adhesive joints are still improving. The constrained state of the adhesive, as a thin sheet between two much stiffer adherends, makes the adhesive behave quite differently than the bulk material. Geometrical aspects, like the thickness of the adhesive layer influences the fracture process, cf. e.g. Kinloch (1987) and Paper D. According to the adhesive layer theory, the deformation of an adhesive layer is dominated by the deformation modes peel and shear, cf. Fig. 2.

(b)

L

Adhesive

Adherend 1

Adherend 2

H1

H2

h

a7

a8 a9

a4

a5 a6

a10

a11 a12

a1

a2 a3

(a)

L

a4

a5 a6

a10

a11 a12

a7

a8 a9

h +

(H1+

H2)

/2

a3 a1

a2

5

Figure 2. Deformation modes of an adhesive layer with thickness h: peel, w, and shear, v. Conjugated stress components σ and τ. Schmidt (2007) shows in an asymptotic analysis that the behaviour of an elastic joint in small deformation is dominated by peel and shear modes under the condition that the layer thickness is much smaller than the thickness of the joined parts, the elastic modulus of the adhesive is much smaller than the modulus of the adherends and the in-plane length of the adhesive joint is much larger than the thickness of the adherends. Yang and Thouless (2001) show, by performing both simulations and experiments with in-elastic adhesive joints in large deformation, that the adhesive layer theory achieves good results in predicting the behaviour of real structures. 3. Fracture of adhesive layers Adhesives, confined to thin layers between much stiffer adherends, behave differently from adhesive bulk material. The reason for this is that the adherends prevent the adhesive from contracting in a natural way, in the elastic state this contraction is determined by Poisson’s coefficient. The cohesive law of an adhesive reveals that only a small part of the fracture energy is dissipated during the increasing part of the stress-elongation curve, cf. Fig. 1. The major energy dissipation is consumed during the softening part. To capture the softening part in a simple tension test, the well known problem of the stiffness relation between the tensile testing machine and the test specimen is brought up to date. The elastic energy loaded in the testing machine and the tensile test specimen is released when the slope of the cohesive law becomes negative. This may result in unstable breakage of the test specimen near the point of peak stress in the cohesive law. Engineering approaches are common in the search of adhesive cohesive laws. Adhesive joints are studied in tensile testing machines and force-displacement relations are recorded. To deduce the cohesive law, FE simulations are performed and compared with the test results. A problem with this approach is that different parameter settings may produce similar results and thus ambiguous properties are achieved. With this approach it is difficult to determine adhesive properties such as fracture energy and peak stress, and the capturing of the entire cohesive law is even more difficult. Yang et al. (2001) use a butt joint torsion test and succeed in predicting shear fracture of an ENF specimen of different adherend thicknesses.

Adhesive joints between plastically deforming thin adherends (automobile structures) may deform anticlastically, cf. e.g. Andersson and Biel (2006). This phenomenon increases the area moment of inertia of the adherend, which in turn makes the joint stronger. An adhesive with a high peak stress counteracts this phenomenon, compared to an adhesive with lower peak stress. A strong adhesive is therefore believed to have a smaller peak stress and a large fracture energy. Moreover, as the stiffness of the adhesive decreases, the stress distribution in an adhesive joint gets more homogeneous. That is, the peak stress decreases. To sum up, a

v τ

τ σ

w

σ

h

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successful adhesive should have large fracture energy, low Young’s modulus and a low peak stress. In this context it is also better to increase the thickness of the adhesive layer since this decreases the stiffness of the adhesive layer. 4. Test methods Andersson and Biel (2006) and Leffler et al. (2007) show convenient methods to measure the cohesive properties of adhesive layers by use of simple test geometries and inverse formulas. Throughout this thesis, the double cantilever beam (DCB) specimen, cf. Fig. 3, is used for pure peel deformation and the end notched flexure (ENF) specimen, cf. Fig. 4, for pure shear.

Figure 3. Double cantilever beam specimen.

The un-bonded part of the DCB specimen can be considered as a crack. That is, ap is the crack length, Fp is the applied force, θ is the adherend rotation at the loading point, δp is the deflection at the loading point, bp is the adherend width, Hp is the adherend height, Lp is the specimen length, and hp is the adhesive thickness.

Figure 4. End notch flexure specimen.

Fs

δs

as

Ls

Ls/2

h s

Hs

v

x

y

Hp

δ p+

hp

+ H

p

hp + w

hp

θ

Fp

Lp

Fp

θ

ap

x

y

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For the ENF specimen, Fs is the applied force, δs is the deflection at the loading point, Ls is the length between supports, bs is the adherend width, hs is the adhesive thickness, Hs is the adherend height and as is the un-bonded length. The latter is also referred to as the crack length.

The inverse formulas are based on the use of alternative integration paths for the evaluation of the J-integral, cf. Rice (1968).

( ), dx i i x

S

J Wn Tu S= −∫ (2)

Here, the integration path S can be chosen freely if the strain energy density W is independent of any explicit dependence of the x-coordinate. The outer unit normal to S is denoted n and the traction and displacement vectors are denoted T and u, respectively. Index notation is used with partial differentiation denoted by a comma and summation indicated by repeated indices. The strain energy density is given by

0

dij ijWε

σ ε= ∫ . (3)

Suppose an adhesive joint is arbitrarily loaded at one side as indicated in Fig. 5.

Figure 5. Alternative integration paths across an adhesive joint.

If the integration path AB is taken close to the adhesive tip, the J-integral is

B B

AB

A A

d dJ w vσ τ= +∫ ∫ . (3)

In the DCB experiment, v and τ equal zero due to symmetry; in the ENF experiment, w and σ equal zero. Equation (3) shows that cohesive laws are closely related to fracture mechanics. At the end of a fracture process, the cohesive stresses σ and τ are zero indicating that a crack has formed. After this moment, Eq. (3) shows a constant J. Thus, the maximum value of J can be identified with the fracture energy, Jc, and equals the area under the cohesive law. Now,

B

A

B´

A´

F2

F1

Adherend

Adherend

Adhesive

M1

M2

x

y

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taking the alternative integration path AA´B´B , the J-integral may be divided into three parts, JAA´ + JA´B´ + JB´B. The expressions are given in Eqs. (4a-c).

( )( )

( )( )

( ) ( )

A´

AA´ , ,

A

B´

A´B´ ,

A´

B

B´B , ,

B´

d ,

+d ,

d .

x x y x

x i i x

x x y x

J u u x

J Wn Tu y

J u u x

τ σ

τ σ

− −

+ +

= − + +

= −

= − + −

∫

∫

∫

(4a,b,c)

Here, the integral JA´B´ equals zero if the points A´ and B´ are placed sufficiently far from the adhesive tip, where the strain energy density, W, and the traction vector, T, are zero. If the material is (non-)linear elastic, the J-integral is shown to be path independent. The path independency of the J-integral applies also to elastic plastic materials, for monotonously increasing load. Path independency implies

AB AA´ A´B´ B´BJ J J J= + + (5)

( ) ( ) ( )A´ B

, , , ,

A B´

d -dx x y x x x y xu u x u u xτ σ τ σ− − + + = − + + − − + − ∫ ∫

( ) ( )R R

, , , ,

L L,,

d dx x x x y x y x

xx

u u x u u x

wv

τ σ+ − + −= − + −∫ ∫

, ,d dx xv x w xτ σ= +∫ ∫

( )

( )

dd d

dd

d dd

vv v x v x

xw

w w x w xx

= = = = =

d dv wτ σ= +∫ ∫ , (6)

where iu+ and iu− are the displacements on the upper and lower adherend respectively. Since

Eq. (6) equals Eq. (3), path independency is proven for this case. 5. Parameters influencing adhesive joints The parameters that influence adhesive joints may be categorised into two groups. • material related parameters, e.g:

o fracture energy o peak stress o Young’s modulus

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o Poisson’s ratio o strain rate o temperature o filler content o moisture content

• joint related parameters, e.g: o adhesive thickness o length and width of the adhesive joint o adherend stiffness o complementary fasteners o surface roughness

Some of the most important mechanical parameters are temperature, strain rate and adhesive thickness. The influences of temperature and strain rate on the cohesive law are studied in Paper C. The influence of adhesive thickness on the cohesive law is studied in Paper D. The interaction between adhesives and complementary fasteners is another area of growing interest, due to manufacturing requirements. Since the adhesive gains its full strength in the paint process oven, some additional means of fastener has to supply the initial joint strength through the body shop. A discrete, mechanical fastener, rivet or a spot weld, may complement the adhesive in this respect. The discrete fastener will decrease the peel and shear force on the surrounding adhesive, but it will also impose a stress concentration, which will increase stress near the fastener. This adhesive/discrete fastener interaction is the motivation for Paper E. A further motivation of Paper E is to verify the adhesive simulation technique in a more built up specimen subjected to impact. A bimaterial beam specimen is developed and analysed in different joint configurations. The specimen is developed with focus on multi-material use and the impact requirements for a bimaterial joint in the region bridging from the passenger compartment (steel) to the front or aft sections (aluminium) of the vehicle. The passenger compartment should resist deformation well and protect the passengers from the crumpling zones surrounding the passenger compartment. The impact energy should be consumed by plastic deformation of material in the crumpling zones. The adhesive joint is believed to be located between stiff steel parts and plastically deforming aluminium. This inflicts certain deformation requirements on the adhesive, like e.g. mixed mode loading, short process zone and anticlastic bending. 6. Manufacturing aspects Besides the strength aspects of adhesive joining, there are many practical aspects of joining to cope with before production start of an adhesively joined car body. The environmental issue of handling epoxy-based adhesives is a challenge. Un-cured adhesive must not get in contact with human skin and fumes are to be evacuated efficiently. Pre-cured adhesives in the form of tapes provide a convenient opportunity. Robot application of adhesives is necessary, both to ensure application precision and avoiding human contact. The car body structure consists mainly of deep-drawn sheet metal. The manufacturing tolerances are relatively large due to complications such as spring back. Joint tolerances are unlikely to be homogeneous. The adhesive thickness variation is an important parameter for the strength of the adhesive, cf. Paper D. However, an improvement of the thickness

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tolerance may be achieved by adhesive in tape form or with spacer balls added to the adhesive. Prior to curing, a limited joint strength is required. This may be provided by sparsely spaced fasteners such as rivets or bolts. Before the adhesive is cured in the paint oven, it has to pass the body pre-treatment baths without being washed out from the joint. On the finished vehicle, the adhesive shall withstand all environmental requirements, such as moisture, ultra-violet radiation, ozone, salt etc. without additional sealant required. 7. Summary of appended papers Paper A: An Explicit FE Model of Impact Fracture in an Adhesive Joint. A simplified structure containing a cohesive zone is studied under dynamic conditions. The structure represents some specific properties of an automotive structure and is simple enough to allow for closed form solutions obtained by the method of characteristics. These solutions are compared to results of explicit FE-analyses. The FE-solutions agree with the closed form solutions. Damage is included in the FE-model. Three constitutive models of the adhesive layer are used. It is shown that an amplification of the strain rate is achieved in the adhesive layer. It is also shown that an artificially increased flexibility of the adhesive in an aluminium structure gives only minor influences of the general behaviour. It is shown that in the cases of repeated loading and unloading, an explicit FE-analysis with a “large” time step is more prone to give immediate rupture. That is, it is expected that a simulation based on these principles will lead to a conservative prediction of the strength. Paper B: Explicit FE-formulation of Interphase Elements for Adhesive Joints. A 2D-interphase element formulation is developed for adhesive simulation, and implemented in an explicit FE-code. The interphase element is tested in a simplified joint example and compared with a straight-forward continuum approach. The time saving potential of the interphase element formulation is studied. Moreover, the interphase element formulation shows fast convergence and computer efficiency. Paper C: Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive. Effects of temperature and strain rate on the cohesive relation for an engineering epoxy adhesive are studied experimentally. Two parameters of the cohesive laws are given special attention: the fracture energy and the peak stress. Temperature experiments are performed in peel mode using the double cantilever beam specimen. The temperature experiments show monotonically decreasing peak stress with increasing temperature. The fracture energy is shown to be relatively insensitive to the variation in temperature. Strain rate experiments are performed in peel and in shear mode. In peel mode, the fracture energy increases slightly with increasing strain rate; in shear mode, the fracture energy decreases. The peak stresses in peel and shear mode both increase with increasing strain rate. In peel mode, only minor effects of plasticity are expected while in shear mode, the adhesive experiences large dissipation through plasticity. Paper D: Influence of layer thickness on cohesive properties of an epoxy-based adhesive – an experimental study. The adhesive layer thickness influence on the cohesive parameters is studied in peel and shear mode. A strong thickness influence of the fracture energy in peel mode is found. In the peel mode, the fracture energy has a maximum between 1.0 mm and 1.5 mm. In the shear mode, the thickness dependence is not as strong, but a similar trend is

11

evident. For both peel mode and shear mode, the peak stress shows little thickness dependence. A slight decrease in peak stress with increasing adhesive thickness is indicated. Paper E: Dynamic testing and simulation of hybrid joined bimaterial beam. A bimaterial beam specimen is developed for impact testing of joint configurations. The specimen is designed to give large loads in the joints. Thus, revealing the influence of the joints and the ability to accurately predict the strength using cohesive laws. The specimen is impacted in three point bending. Specimens are prepared with discrete joints and adhesives separately and in combinations (hybrid joint). Results show that the impact energy consumption depends on the joint integrity. A threshold value for the fracture energy of the adhesive seems to exist. Beneath this value, adhesive and discrete fastener work together increasing the impact energy. Above this value, the discrete fastener has a negative effect, and may be regarded as a stress concentration. Simulations are performed with adhesive cohesive elements in FE-models consisting of shells to predict impact energy values and the overall deformation mode. The simulations show that the impact resistance is predicted fairly well, but it is very important to get the fracture criteria right for the base materials. 8. Future work The strain rates in Paper C vary; for peel loading from about 10-4 s-1 to 10 s-1 and for shear loading from 10-3 s-1 to 1 s-1. Much higher strain rates have been predicted in Paper A, and are verified in simulations in Paper E. This leads to the conclusion that experiments should be performed up to 5000 s-1. An idea is to use high strength steel in the adherends and minimise the dimensions of the specimens to be able to increase the loading rate without inertial effects destroying the measurements. In Paper C, no temperature experiments with ENF specimens are performed. Shear deformation involves more plastic work of the adhesive, and thus it may be affected differently from the peel response. Smaller ENF specimens, as suggested above, are easier to study in temperature chambers. The experimental studies involve visual inspection of the fitted Prony approximations to experimental data. This process is sometimes rather difficult, since oscillations plague the Prony-series approximation. The Prony-series approximation is sensitive to noise in the measurement data, especially at the beginning of the measurement. Experimental data need to be trimmed correctly at the start of the measurement. The Prony-series approximation should cover the J values from zero to fracture, i.e. where J becomes constant (at maximum). Filtering of experimental data may be necessary. Moreover, for the Prony-series approximation, the derivative of J with respect to the deformation should be zero at J = 0 and subsequent values of J should always be non-zero. At fracture at the adhesive tip, J should theoretically have a maximum. Experimentally, this is not always the case. Sometimes we experience a non-zero derivative when fracture occurs. In Paper C, a method to deal with this is suggested. With the method we assume a bi-linear cohesive law. Obviously, the method may be further developed to be valid for other general cohesive laws. The thickness study in Paper D may to be completed with more thicknesses between 1 mm and 2 mm. This will help understand the complete shape of the thickness dependence curve.

12

Acknowledgements The author would like to thank the Swedish Consortium for Crashworthiness for funding this work. Special thanks are also directed to Svante Alfredsson, Anders Biel and Kent Salomonsson for fruitful discussions and help during the work of this thesis. Special thanks are directed to Mr. Stefan Zomborcsevics for enthusiastically and meticulously manufacturing test specimens.

13

References Andersson T, Biel A (2006) On the effective constitutive properties of a thin adhesive layer

loaded in peel. International Journal of Fracture 141: 227-246 Borg R, Nilsson L, Simonsson K (2004) Simulating DCB, ENF and MMB experiments using

shell elements and a cohesive zone model, Composites Science and Technology, Vol. 64, pp. 269-278.

Courant R, Friedrichs K, Lewy H (1928) Über die partiellen Differenzengleichungen der

mathematischen Physik, (On the Partial Difference Equations of Mathematical Physics) Mathematische Annalen 100, 32-74

DuBois PA (2005) Private communication.

Hughes TJR (2000) The Finite Element Method. New York: Dover, 2000. Kinloch AJ (1987) Adhesion and Adhesives – Science and Technology. Chapman and Hall,

London, 1987 Leffler K, Alfredsson KS, Stigh U (2007) Shear behaviour of adhesive layers. International

Journal of Solids and Structures 44:530-545 Peden M, Scurfield R, Sleet D, Mohan D, Hyder AA, Jarawan E, Mathers C (2004) World

Report on Road Traffic Injury Prevention. ISBN 92 4 156260 9. World Health Organization, Switzerland.

Reedy ED, Mello FJ (1997) Modeling the Initiation and Growth of Delaminations in

Composite Structures, Journal of Composite Materials, Vol. 31, No. 8, pp. 812-831. Rice JR (1968) A Path Independent integral and the Approximate Analysis of Strain

Concentrations by Notches and Cracks. Journal of Applied Mechanics 33:379-385 Schmidt P (2007) Computational Models of Adhesively Bonded Joints. PhD thesis,

Linköping University, ISBN/ISSN: 9789185715619 Yang QD, Thouless MD, Ward SM (1999) Numerical simulations of adhesively-bonded

beams failing with extensive plastic deformation. Journal of the Mechanics and Physics of Solids 47:1337-1353

Yang QD, Thouless MD (2001) Mixed-mode fracture analyses of plastically deforming

adhesive joints. International Journal of Fracture 110:175-187

Paper A

An explicit FE-model of impact fracture in an adhesive joint.

A1

Dynamic fracture of adhesive joints using explicit FE-code and the adhesive layer theory

T. Carlberger1 and U. Stigh2

1SAAB Automobile AB, SWEDEN and 2University of Skövde, SWEDEN

Abstract

Dynamic fracture of an adhesive layer in a structure is analysed. The structure represents some specific

properties of an automotive structure and is simple enough to allow for closed form solutions. These

solutions are compared to results of explicit FE-analyses. The FE-solutions agree with the closed form

solutions. Three constitutive models of the adhesive layer are used. It is shown that an amplification of

the strain rate is achieved in the adhesive layer. It is also shown that an artificially increased flexibility of

the adhesive gives only minor influences of the general behaviour. In some load cases, the adhesive layer

will experience repeated loading/unloading. It is shown that in these cases an explicit FE-analysis with a

“large” time step is more prone to give immediate rupture. Thus, the method is conservative.

Keywords: Adhesive joining; Dynamic fracture; Cohesive zone

1. Introduction

In the automotive industry increasing requirements on emissions, cost and crash performance is driving a

technology change from mono-material spot welded steel to multi-material adhesively joined car bodies.

Thus, the automotive industry is increasing its focus on the use of adhesive joints due to benefits in

strength, stiffness and the capability of joining dissimilar materials. One reason adhesive joining has not

reached general acceptance, is a lack of reliable and efficient simulation methods for adhesives in the

field of crash simulation. In the present work, an attempt is made to demonstrate a useful technique for

this purpose.

Most crash simulations are based on explicit FE-codes since these are capable of simulating fast events

within a reasonable execution time. Explicit FE-codes do not require the solution of systems of equations

as in implicit FE-codes. Instead, the equation of motion of each degree of freedom is solved individually,

cf. e.g. [1]. This allows for very large models such as complete and detailed automotive structures, to be

analysed for impact studies. Such a model typically consists of over one million degrees of freedom. The

limit on the degrees of freedom is due to limits in computer memory capacity and on the need to keep the

solution time reasonable. To this end, the Courant limit (CFL-limit) is essential; in order to achieve

numerical stability, the time step, ∆t, must be smaller than 2/ωmax, where ωmax is the largest

eigenfrequency of the structure. A useful estimate of the critical time step is given by the smallest time it

takes an elastic wave to travel the distance between two nodes in the structure. This estimated critical

time step ∆tc is given by

A2

minc

lt

c∆ = where

Ec

ρ= , (1a,b)

where c is the wave speed, lmin is the shortest distance between two nodes, c is the wave speed, E is the

Young’s modulus and ρ is the density of the material. If the time step is chosen larger than ∆tc the FE-

solution will fail due to numerical instability. With a typical time step ∆t of 1 µs, steel allows for a

minimum length lmin of about 5 mm and a typical adhesive material; epoxy has lmin ≈ 1.2 mm. This should

be compared to a typical adhesive layer thickness of 0.2 mm. Thus, to achieve a stable numerical solution

with an epoxy adhesive layer with 0.2 mm thickness, the time step has to be decreased to 0.165 µs. This

increases the simulation time by a factor of six. Today, a typical simulation time for a complete car crash

with the time step 1 µs is about 12 hours. With the smaller time step the simulation time increases to

about 72 hours. This is not acceptable. A trick, often used in the automotive industry, is “mass scaling”.

The material density is increased by a factor n for the “too small elements” until the Courant limit is

fulfilled with the desired time step. With Eq. (1) the scaling factor is

2c

min

tEn

lρ ∆

=

(2)

This procedure adds extra mass to the model, about 30 to 40 kg on a complete car body. This is obviously

a small extra weight as compared to the total mass of a car. However, the added mass is localized to the

joints which might create an abnormal stress distribution in the joint during the severe accelerations

during a crash simulation. Especially if the adhesive bond line is oriented in the crash direction, the added

mass can create too high stresses in the surrounding regions. This is a problem today, since spot welds are

simulated using this trick, [2]. In fact, when simulating spot welds, Young’s modulus is often reduced by

an order of magnitude to avoid excessive mass scaling, clearly indicating that the localized masses can be

a problem. Thus, the numerical problems encountered when joining sheet metal is not only limited to

adhesive joining but appears to be a general problem when using commercial explicit FE-codes. Since the

joints are not allowed to fracture during a crash, rigid links between the nodes at each sheet would appear

as a good approach. This would lead to a reduction of the number of degrees of freedom. However, the

risk for the joints to fracture cannot be determined using this approach.

A promising method to analyse adhesive joints is the use of the adhesive layer theory, cf. [3]. Based on an

asymptotic analysis it is concluded that two deformation modes dominate the behaviour of a thin layer

between stiffer adherends, cf. [4]. These deformation modes are here denoted peel deformation, w, and

shear deformation, v, cf. Fig. 1. The conjugated stresses are the peel stress, σ, and the shear stress, τ.

A3

Figure 1: Deformation modes of the adhesive layer with thickness h: peel, w, and shear, v. Conjugated

stress components σ and τ.

Experiments performed in pure peel and in pure shear on the commercial epoxy adhesive DOW Betamate

XW1044-3 show similar behaviour in peel and sheer, cf. [5,6] and Fig. 2. The shear curve essentially

appears as an enlarged version of the peel curve. As shown in the graphs, the traction-separation relations

start with a linearly increasing part corresponding to linear elasticity. After reaching a peak stress, the

stress decreases to zero stress after substantial deformation. As showed in [3,5,6], the area under the

traction-separation relation equals the fracture energy. Thus, the present adhesive is substantially stronger

in shear than in peel loading.

0

5

10

15

20

25

30

0 0,05 0,1 0,15 0,2

Deformation (mm)

Str

ess

(M

Pa

)

Peel

Shear

Figure 2: Constitutive behaviour in peel and shear for the engineering adhesive DOW Betamate

XW1044-3 with a 0.2 mm layer thickness, results from [5,6].

These results are confined to a low strain rate. A systematic analysis of the effect of strain rate is given in

[7], cf. Fig. 3. Although, the strain rate is very small as compared to the strain rates encountered in crash

simulations, the results indicate that the fracture energy varies with the deformation rate. Thus, a

constitutive model for commercial epoxy adhesives should be rate dependent. A nonlinear viscoelastic

model is suggested in [8].

v τ

τ σ

w

σ

h

A4

Figure 3: Fracture energy and maximum stress vs. rate of deformation in peel at w = 20 µm for DOW

Betamate XW1044-3 with a 0.2 mm layer thickness, results from [7].

Experiments conducted at high strain rates involving adhesive joints are scarce in the literature. In [9]

impact shear tests of adhesive layers are reported. In their setup they measure the average stress in an

adhesive layer and conclude that fracture at low stress levels are caused by tensile stress while fracture at

high stress levels are caused by a combination of shear and compression stress. In [10] the thick adherend

shear test is used to determine the dynamic shear modulus of structural adhesives. The authors conclude

that this method can be used for predicting the dynamic shear modulus if G/h > 1 GPa/mm, where G is

the shear modulus of the adhesive and h is the adhesive thickness. Since adhesive thicknesses are very

small, this ratio will usually be exceeded by most commercial structural adhesives. In [11] a review of the

standard ISO 11343, cf. [12] is performed. A correlation between results from the impacted wedge peel

test (IWP) and the fracture mechanically determined fracture energies, Gc, of some adhesives is

performed. It is shown that a finite-element analysis of the IWP test geometry may predict the failure

behaviour successfully. In [13] a two-parameter criterion for fracture is suggested, involving both a

critical limiting value of the stress, σmax, and the critical energy release rate, Gc. The authors suggest that

several parameters are involved in determining the fracture process. Including the aforementioned

parameters, the damage zone length and the critical displacement, are included in such a way that each

parameter is partly limiting.

Considering the similarities between adhesive joints and composite delamination, the field of search may

be enlarged by including the latter. Strain rates or crack speeds associated with composite testing and

simulation are generally much higher than for adhesives, due to the brittle nature of composites. In [14]

dynamic mode-I delamination of a carbon/epoxy composite is studied in a modified DCB specimen. It is

noted that the critical dynamic fracture energy is nearly constant and very close to the static fracture

toughness, GIc, of the composite. Numerical simulations performed by the authors show that the strain

energy release rate at the straight crack front is non-uniform and dips towards the free edges. In [15] an

investigation of the rate dependence of mode-I fracture toughness of carbon fibre epoxy laminates is

reported. The results show that the crack growth behaviour change from unstable to stable due to

wɺ (µm/s)

J c (

N/m

)

wɺ (µm/s)

σ ma

x (M

Pa

)

10-2 10-1 100 101 0

5

10

15

20

25

30

0

100

200

300

400

500

600

700

800

900

1000

10-2 10-1 100 101

A5

differences in static and dynamic fracture toughness, GIc. In [16] mode-II-dominated delamination of fibre

composites is studied achieving crack propagation speeds exceeding 1000 m/s.

The constitutive behaviour described by the graphs in Fig. 2 is only representative for a monotonically

increasing deformation of the adhesive layer. In a crash simulation, some parts of the adhesive joint may

experience such deformation. However, substantial parts of the joint will experience repeated loading and

unloading. Thus, a constitutive theory for the adhesive layer should be capable of treating unloading as

well as loading. In the present paper, we develop a continuum damage model for the adhesive layer. The

model is implemented in an explicit FE-code and a numerical example is studied. The problem analysed

is one which allows for a closed form and exact solution.

2. Simplified model

Studying the influence of a cohesive zone model during an impact situation in a complete vehicle body is

a very complicated task. During a crash event, the first contact between the vehicle and the obstructing

object results in an elastic wave being introduced in the car structure. This wave will quickly reach every

corner of the structure and reflect back. Since the waves propagate through the steel structure at roughly

5000 m/s, the entire car structure will have been reached by the elastic wave in less than 1 ms. This time

can be compared to the time for a crash, about 100 ms, cf. e.g. [17]. After the first passage of an elastic

wave, there will be reflected waves continuously passing through the structure and it will be virtually

impossible to follow each individual wave. Thus, all too many mechanisms are active at the same time to

make substantiate conclusions possible. In the present paper we will therefore study a simplified problem

that allows for some closed form solutions. In this way, numerical errors and potential problems can be

singled out.

In spite of its simplicity, the impacted one-dimensional bar illustrates many of the phenomena

encountered in an impact situation. Consider the system in Fig. 4. An elastic bar with length L and cross

sectional area A is loaded at its right end at t = 0 with a constant load F = σ0A, where different values of

σ0 will be analysed. The elastic modulus and density are E and ρ, respectively. At the left hand side, the

bar is connected to a rigid wall with a cohesive element that loads the bar with a force Fcoh = Aσ(w),

where σ(w) is the traction-separation law.

A6

Figure 4: Simplified model.

In order to simplify the real traction-separation relation shown in Fig. 2 the three step-wise linear traction-

separation relations of Fig. 5 are suggested. The model in Fig. 5a gives a good representation of the

behaviour of an adhesive joint during moderate loading. Both this model and the linearly softening model

in Fig. 5b allow for closed form solutions. However, the model in Fig. 5b is not useful for deformation

based FE-analyses since the model has an infinite initial stiffness, i.e. before the joint starts to soften. The

third model, cf. Fig. 5c, is in reasonable agreement with the experimental results in Fig. 2 and has been

used extensively in studies of adhesive layers.

Figure 5: Traction-separation relation for a) linear elastic, b) pure softening and c) linear elastic with

softening models, respectively.

One-dimensional elastic wave propagation is governed by the well known wave equation,

2 2

2 2 2

1u u

x c t

∂ ∂=∂ ∂

, (3)

where c is the elastic wave speed given by Eq. (1b). We will first use the classical method of

characteristics to develop two special solutions of the present problem, i.e. the solutions to the problem

with a linear elastic or a linear softening traction-separation relation, cf. Figs 5a,b respectively. The first

solution corresponds to a moderately loaded body, while the latter corresponds to an extensively loaded

body in the case where the elastic properties of the joint can be neglected.

In order to simplify the present problem, the solution of Eq. (3) is evaluated along the characteristics.

With v denoting the velocity of a material point, i.e. /v u t≡ ∂ ∂ , the characteristics are given in x-t-space

t

σ0

Fcoh = Aσ(w) F = σ0A E, A, ρ, L

w(t)

x

σ

wc w

ˆσ

w1

K 1

w

K 1

a)

wc w

ˆσ σ

b) c)

σ

A7

by straight curves determined by dx/dt = ±c. Along these curves, the solution of Eq. (3) is governed by

ordinary differential equations. The solutions are given by

(α): at d / d v E x t cσ ρ α− = = + (4a)

(β) at d / dv E x t cσ ρ β+ = = − (4b)

where α and β are constants along respectively characteristic. The curves are denoted α- and β-

characteristics. Thus, the partial differential equation (3) is reformulated into two algebraic equations

(4a,b). The solutions are conveniently separated in different regions in the x-t-space, cf. Fig. 6.

t

xL

σ = σ0

σ = σ(w)

T

3T

σ = v = 0

A

B

C

D

β

α

t

T

β

α

x

∆t∆t

Figure 6: Left: Regions in the x-t-space; Right: Details of region C

Along the boundary, t = 0, the initial conditions σ = v = 0 hold and along the boundary, x = L, the

boundary condition σ = σ0 holds. At the left boundary, x = 0, the stress is given by the traction separation

relation, σ = σ(w), where the relation between w and v is given by v(0,t) = dw/dt and w ≡ u(0,t). When the

load is applied at the right hand side at t = 0, an elastic wave with velocity c propagates from the right

hand side to the left. At time T = L/c the wave hits the adhesive layer. Thus, in the lower triangular region

in the x-t-space, i.e. the region A, both the stress and the velocity are zero. In region B, the behaviour is

governed by the stress free region below and the stressed boundary to the right. Along a α-characteristic

in region B, Eq. (4a) holds. The three un-known, σ, v and α, are determined by the conditions at the

boundaries to these two regions. At the boundary to the region A, both σ and v are zero and Eq. (4a) gives

the constant α = 0. At the right hand side, the stress equals σ0. Thus, Eq. (4a) gives the velocity

0

E

σνρ

= , (5)

A8

in region B; in region B, both the stress, σ = σ0 and the velocity are constant. Now, depending on the

specific traction-separation relation, specific solutions are derived after this point, i.e. in region C.

2.1 Linear elastic adhesive layer

In region C, the behaviour of the adhesive layer influences the results. With a low level of the applied

load, the adhesive layer behaves as a linear elastic medium. Thus, the boundary condition at x = 0 is

σ(0,t) = Kw(t), (6)

where K corresponds to the modulus of elasticity of the adhesive layer. A β-characteristic, Eq. (4b), in

region C is governed by the conditions at the boundaries to the adhesive layer at x = 0 and at the boundary

to region B. From the conditions at the boundary to region B, the constant is determined to β = 2σ0. The

conditions at the boundary to the adhesive layer yield the following ordinary differential equation for the

separation,

0

e

2d 1

d

ww

t K

στ τ

+ = (7)

where e /E Kτ ρ≡ is a characteristic time-parameter for the problem. Solving Eq. (7) with the initial

condition, w = 0 for t = T yields,

0

e

2( ) 1 exp

T tw t

K

στ

−= −

(8)

The corresponding velocity and stress are given by differentiation with respect to t and multiplication of

w(t) with K, respectively. The result is

( )00

e e

2(0, ) exp and 0, 2 1 exp

T t T tv t t

E

σ σ στ τρ

− −= = −

(9a,b)

To obtain the stress and velocity at (x,t) we now use the property of Eqs. (4a,b) to hold along a α- and β-

characteristic, respectively. Thus,

(α): ( ) ( ) ( ) ( )0, 0, , ,t t v t t E x t v x t Eσ ρ σ ρ− ∆ − − ∆ = − (10a)

(β): ( ) ( ) ( ) ( )0, 0, , ,t t v t t E x t v x t Eσ ρ σ ρ+ ∆ + + ∆ = + (10b)

A9

where ∆t is the time it takes a wave to travel to/from the adhesive layer to the point x, cf. Fig. 6. Thus, ∆t

= x/c. With Eqs. (9a,b), Eqs. (10a,b) constitute a linear system of equations for σ and v, the solution is

( ) 0e

/, 2 1 exp

T t x cx tσ σ

τ − += +

(11a)

0

e e

2 /( , ) exp

T t x cv x t

K

στ τ

− +=

(11b)

This way of stepping forward one region at a time can be used as long as desired, though the expressions

grow more and more complex as the solution proceeds.

2.2 Linear softening adhesive layer

With a large load and with a stiff adhesive layer, the softening behaviour of the layer is expected to

dominate the behaviour. In this case, a linear softening model is suitable, cf. Fig. 5b. The solution method

is similar to the one described for the elastic layer. As compared to the solution above, a softening

adhesive layer results in a change of Eq. (6) to

( )c

ˆ0, 1w

tw

σ σ

= −

, (12)

where the symbols are defined in Fig. 5b. Proceeding as described above, the solution in region C is given

by

( ) 00

c

/, 2 2 1 exp

t T x cx t

σσ σ σσ τ

− − = − −

(13a)

c 0

c c

/( , ) 2 1 exp

w t T x cv x t

στ σ τ

− − = − (13b)

where the characteristic time-parameter is now changed to c c /w Eτ ρ σ≡ . These closed-form solutions

give insight into the different length- and time-scales of a solution. The closed-form solutions will also be

compared to the results of FE-simulations.

A more realistic traction-separation law should have the characteristics indicated in Fig. 5c. Moreover,

during a crash simulation, multiple waves will travel back and forth through a structure. Thus, the models

of the adhesive joints must be able to handle unloading from a loaded state.

2.3 Unloading and damage

A10

Indications from in situ tensile tests in a SEM suggest that an adhesive layer will develop only small

amounts of plastic strain during peel loading, cf. [18]. Thus, most of the in-elastic deformation is

attributed to the development of microscopic cavities. These are expected to close upon unloading. A

simple constitutive model of this process is provided by the introduction of a damage variable, ω, cf. [19].

The damage parameter ω is defined through σ = (1-ω)Kw, cf. Fig. 7 . Suppose the specimen is loaded to

point A, with w1 < w < wc. At this point, unloading starts. Since we have exceeded w1, the stiffness at

unloading is (1-ω)K. At the next loading, the stiffness is still (1-ω)K until the point A is reached. After

this pint has been passed, softening proceeds until either next unloading occurs or wc is passed and the

cohesive zone fails completely.

σ

w

1

KA

wcw1

Figure 7: Cohesive zone model with damage ω and unloading.

2.4 Explicit FE-code

For comparisons an explicit FE-code is developed, cf. e.g. [1,20]. The code is developed in MATLAB

and provides an estimated critical time step, cf. Eq. (1a), and also the maximum eigenfrequency.

3. Analyses and results

In this section we will present some numerical results based on the closed form solution and the FE-

analysis. At moderate loading, the response of the adhesive layer is linear elastic and Eqs. (8,9) provide

the exact solution in region C where the behaviour of the adhesive layer is first influencing the results. A

typical result is given in Fig. 8. At x = 0, the wave first hits the layer at t = T. The stress and displacement

then gradually build up. At x = L/2, the elastic wave first passes on its way to the adhesive layer at t = T/2;

at t = 3T/2 the reflected wave returns. The sudden stress-dip at t = 3T/2 is due to the limited stiffness of

the adhesive layer.

A11

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,5 1 1,5 2

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,5 1 1,5 2

Figure 8: Stress (left) and displacement (right) response with a linear elastic adhesive layer. Solid line at x

= 0 and dashed line at x = L/2. Data for T = τe.

With an infinitely stiff layer, the stress in region C will be twice the applied load, i.e. σ = 2σ0. Indeed,

with K → ∞, e / 0E Kτ ρ≡ → , and Eq. (11b) yields σ = 2σ0 for T < t ≤ 3T. As indicated above, the

effect of the flexibility of the adhesive layer is to immediately reduce the stress at the left end to zero at

loading. However, with a large stiffness of the adhesive layer, the stress rapidly increases to 2σ0. Now if a

lower stiffness can be chosen without influencing the results in any substantial way, the critical time step

can be increased, cf. Eq. (1). As a reasonable criteria, the stress can be claimed to be only marginally

influenced by the flexibility of the adhesive layer if σ > 1.8σ0, at t = 1.2T, i.e. only a 10 % lower stress

than with an infinitely stiff layer attained within 10 % of the loading time. Evaluation of Eq. (9b) yields

the condition

12 12E E

KT L

ρ> = , (14)

As long as this condition is fulfilled, the stress state is virtually un-affected by the stiffness of the

adhesive layer. For a typical structural length, L = 0.2 m, the stiffness can be chosen as low as 12 TPa/m

(steel) and 4.2 TPa/m (Al). These values can be compared to K = 20 TPa/m for a typical epoxy adhesive

layer cf. e.g. [3]. Thus, the time step can be substantially increased with a decrease of the elastic stiffness

of the adhesive layer in an aluminium structure. However, the effect is only marginal for a steel structure.

The maximum stress in the adhesive layer is attained at t = 3T. Equation (11a) gives

max 0 0e

22 1 exp 2

Tσ σ στ

= − − <

. (15)

/t τ /t τ

0/σ σ 0/uE Lσ

A12

If maxσ σ≤ , no inelastic deformation takes place in the adhesive layer. However, if the loading is large

and if the stiffness of the adhesive layer is large, the linear softening model of Fig. 5b should provide a

good estimate. The solution Eq (13a) reveals that the adhesive layer will fracture during the first loading

period, T < t < 3T, if

( )( )

c0 0c c

c

exp 2 /1 1 if

2 2exp 2 / 1

TT T

T

τσ σ σ σ

τ≥ ≡ ≈ ≥

−

, (16)

where Tc ≈ τc/2, or equivalently, if c / 2 0.3 m (steel) to 0.1 m (Al)L Ew σ> ≈ where data for the

adhesive layer is chosen according to Fig. 2. If this condition is fulfilled, the applied load has to exceed

only half the adhesive strength for the layer to fail during the first loading period, i.e. about 10 MPa for

the adhesive in Fig. 2. The stress and displacement response for a load case where the adhesive layer fails

during the first loading period is given in Fig. 9. The first wave hits the adhesive layer at t = T and the

stress increases immediately to σ σ= . At the same time, the adhesive layer starts to elongate and the

stress falls off. At x = L/2, the first wave passes at t = T/2. At t = 3T/2, the wave returns after being

reflected by the adhesive layer.

Figure 9: Stress (left) and displacement (right) response with a linear softening adhesive layer. Solid line

at x = 0 and dashed line at x = L/2. Data for 0 c/ 1.425 and / 1.61Tσ σ τ= =.

We will now make comparisons with the results of FE-simulations. The simplified problem in Fig. 2 is

modelled with a linear elastic adhesive layer and simulated with an explicit FE-code. Data is chosen

according to L = 0.2 m, E = 210 GPa, ρ = 7800 kg/m3 and K = 20 TPa/m, respectively. The bar is divided

into 100 elements of equal length. Thus, the time it takes a wave to pass one element is 0.39 µs and the

time to pass the bar is T = 39 µs. The characteristic time-parameters are τ e= 2.0 µs and τc = 61 µs.

0 0.2

0.4

0.6

0.8

1 1.2

1.4

0 0.5 1 1.5 2 t/T

0

0.2 0.4 0.6 0.8

1

1.2 1.4

0 0.5 1 1.5 2

t/T

0/σ σ 0/uE Lσ

A13

Figure 10: Stress and displacement comparison between the FE-method (FE) and the closed form solution

(C.M.). The FE-simulation is evaluated with the time step ∆t = 0.367 µs, which is 95 % of the critical

time step for a system without a cohesive zone.

Figure 10 shows the stress and displacement history. As expected, the displacements of the two methods

agree excellently. In Fig. 10 another artefact becomes evident. At the end of the bar, subjected to the step

load, considerable numerical noise is induced in the simulation due to the steep gradient of the step load.

Thus, the FE-method is unable to reproduce abrupt transients. The conventional method to temper this

artefact is to gradually increase the applied load to its final value. Figure 11 shows the stress and

displacement history with an elastic linear-softening traction-separation law.

Figure 11: Stress and displacement history with an elastic linear-softening traction-separation law

including damage. The FE-simulation is evaluated with the time step ∆t = 0.367 µs, which is 95 % of the

critical time step for a system without an adhesive layer.

As evident from Fig. 11, the stress response resembles both the closed form solutions. At the time the

wave hits the adhesive layer, at 39 µs, the stress gradually builds up to the maximum stress, 20 MPa, at

about 120 µs. During this time interval, the solution resembles the solution with a linear elastic adhesive

0 100 0

10 20 30 40 50 60 70 80 90

100

Time (µs)

Dis

pla

cem

ent (µ

m)

x = 0 x = 0.5 L x = L

200 0 100 Time (µs)

Str

ess

(MP

a)

x = 0 x = 0.5 L x = L

200

30

25

20

15

10

5

0

-5

-10

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

Time (µs)

Dis

pla

cem

ent (µ

m)

C.M: x = 0 C.M: x = 0.5 L C.M: x = L FEM: x = 0 FEM: x = 0.5 L FEM: x = L

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

Time (µs)

Str

ess

(MP

a)

C.M: x = 0 C.M: x = 0.5 L C.M: x = L FEM: x = 0 FEM: x = 0.5 L FEM: x = L

A14

layer, cf. Fig. 8. After this point, the stress decreases similarly as in Fig. 9. The adhesive layer breaks at

about 160 µs and the bar accelerates as a free bar to the right.

In spite of these inherent drawbacks of the FE method, we consider these results successful. Excellent

agreement is achieved between the closed form solution, Eqs (11) and the numerical results for the elastic

layer. The conclusion is that the finite element model is capable of simulating the global characteristic

behaviour of the impacted bar with a satisfactory engineering accuracy.

Upon impacting of a car structure, stress waves are induced in the impacted area and transmitted

throughout the complete vehicle structure. Since the wave speed of steel exceeds 5000 m/s the whole

vehicle will be traversed by stress waves in less than one millisecond. A crash event has a duration of

around 100 ms. During this time, the induced and reflected stress waves will damage and fracture

different structural parts. It is plausible to assume that many adhesive joints will be repeatedly loaded and

unloaded during this event, causing an accumulation of damage in each adhesive region subjected plastic

deformation. This scenario may be simulated by adjusting the step load level such that rupture occurs

after several waves. If the cohesive force, Fcoh = σ(w)A, is plotted against the separation during this event,

a plot according to Fig. 12 results.

Figure 12: Force in adhesive layer versus separation during partial unloading with damage (left) and

damage as a function of time (right). The initial stiffness K is decreased in this simulation.

As evident in Fig. 12, the unloading follows the reduced stiffness (1-ω)K. Depending on the ratio between

the applied load and the cohesive strength, 0 /σ σ , the behaviour differs. With a large load, fracture

occurs during the first loading period. A smaller load leads to repeated loadings and unloadings until the

layer fractures. Numerical experiments show that a certain range of load levels exist during which

repeated loadings and unloadings occur without final fracture. Thus, the accumulation of damage stops

0 1 2 3 4 0

0.2

0.4

0.6

0.8

1.0

Time (ms)

Dam

age

5

0

4

8

12

16

20

0 10 20 30 40 50 60

Displacement (µm)

Co

hes

ive

force

(M

N)

A15

even though the loading continues. Numerical experiments also show that this behaviour is critically

dependent on the chosen time step, cf. Fig. 13.

Figure 13: Force in adhesive layer vs. separation during simulations with different time steps.

As shown in Fig. 13, a large time step leads to fracture after some loadings and unloadings. With a short

time step, giving a more accurate solution, the damage accumulation rests. Thus, a simulation of this

effect will be conservative; a longer time step will more likely predict fracture than a short time step.

In spite of these results, one should be aware of the difficulties for the explicit FE-method to deal with

constitutive relations with abrupt changes. The cohesive force is calculated from the deformation at the

previous time step, which results in a force delay depending on the time step size used. A smaller time

step gives a more accurate evaluation of the cohesive force than a large time step. Due to this, the

calculated dissipated energy from this cohesive zone behaviour will not be accurately predicted for

multiple unloadings and a larger time step.

Polymers, which constitute the bulk material in an adhesive, are known to be very strain rate dependent.

Crash analysts in the automotive industry observe strain rates in the base material around 300 s-1 in

simulations, [2]. Although very noisy, strain rates tend to be high in the sheet metal. This clearly justifies

the requirement of strain rate dependant constitutive models for the sheet metal. The strain rate

dependency increases the yield strength more than 100 % in low grade steel. High strength steel does not

show this strong dependency. Therefore, an interesting question is how high the strain rate in the adhesive

will be. The strain rate in the adhesive layer calculated in the FE-simulations is presented in Fig. 14. If the

load level is low, the joint will not fracture. But if the load exceeds a certain level, the joint will fracture

after several reflections of the stress wave. Further increase of the load leads to joint fracture after

consecutively fewer stress wave reflections, until fracture occurs after just one reflection. Note that the

time scales are equal in both graphs.

0 10 20 30 40 50 60 0

2

4

6

8

10

12

14

16

18

Displacement (µm)

∆t = 0.7 ∆tc

∆t = 0.3 ∆tc

∆t = ∆tc C

oh

esiv

e fo

rce

(MN

)

A16

Figure 14: Strain rate for joint fractured after multiple (left) or a single (right) stress wave reflection.

The strain rate is slightly above 2000 s-1 for the multiple wave rupture and around 4000 s-1 for the single

wave rupture. Test data is not generally available for adhesives or polymers up to these levels of strain

rate, but it is reasonable that this is of great importance in order to be able to simulate rupture of adhesive

joints accurately. The strain rate in the adhesive can be compared to the strain rate in the bar which is of

the order 1-100 s-1. Thus, the present geometry leads to a substantial amplification of the strain rate from

the structure to the adhesive layer. Note that the cohesive zone model used in this analysis is not strain

rate dependent. A strain rate dependent cohesive zone would influence the achieved strain rate.

4. Conclusions

The adhesive layer model with different traction-separation relations has been analysed with the method

of characteristics and also implemented in an FE-code to simulate an adhesive joint. The analyses show

that the relatively large stiffness of the adhesive layer gives a shortened critical time step. However, the

exact value of the stiffness has a minor effect on the solution if the stiffness is chosen larger than about

12E/L, cf. Eq. (14). This indicates a good possibility to adjust an FE-model to achieve a reasonable

execution time for aluminium structures. However, for steel structures additional measures have to be

taken in order to achieve a reasonable time step. If the applied load is larger than about half the strength

of the adhesive layer, the layer will start to soften, cf. Eq. (15). Furthermore, if the load is sufficiently

large, the adhesive layer will fracture during the first loading period, cf. Eq. (16). It is also showed that

the general behaviour can be reproduced with an explicit FE-code. The examples show that the

characteristics of the wave propagation and reflection behaviour are adequately handled by the cohesive

zone model. During repeated loading, the development of damage may stop. It is argued that this property

is dependent on the time step used in the simulations. Numerical examples do however show that a short

time step is more likely to predict a dormant damage. Thus, an explicit FE-simulation will be

conservative in this respect. Moreover, the simulations indicate a very large strain rate in the adhesive

Time (ms)

0

1000

2000

3000

4000

Str

ain

rat

e (s-1

)

.1 .2 .3 .4 .5 .6 0 .1 .2 .3 .4 .5 .6 -500

0

500

1000

1500

2000

2500

Time (ms)

Str

ain

rat

e (

s-1)

A17

layer. From the closed form solutions, Eqs (8,11,12,13), the ratio of the strain rate in the adhesive layer to

the strain rate in the bar can be derived. The ratios estimated to / 50E hK ≈ and c ˆ/ 1600Ew hσ ≈ for the

elastic and the softening layer, respectively; the numerical values corresponds to the adhesive layer

considered in this paper. Thus, the strain rate is expected to be several orders of magnitude larger in the

adhesive layer than in the bar for the present joint geometry. It is however concluded that a more realistic

traction-separation law should include strain-rate dependence.

Acknowledgements

The authors would like to acknowledge the Swedish Consortium for Crashworthiness for funding this

project. Special thanks are also directed to Kent Salomonsson and Svante Alfredsson for fruitful

discussions and help during the work of this paper.

References

[1] Belytschko T., Liu, W. K., Moran, B. Nonlinear Finite Elements for Continua and Structures.

Chichester: John Wiley & Sons, 2000.

[2] DuBois P. 2005; Private communication.

[3] Andersson T., Stigh U. The stress-elongation relation for an adhesive layer loaded in peel using

equilibrium of energetic forces. Int. J. Sol. Struct. 2004; 41, 413-434.

[4] Klarbring A. Derivation of a model of adhesively bonded joints by the asymptotic expansion method.

Int. J. Eng. Sci. 1991; 29 (4), 493-512.

[5] Andersson T., Biel A. On the effective constitutive properties of a thin adhesive layer loaded in peel.

2006; Accepted for publication in International Journal of Fracture.

[6] Leffler K., Alfredsson K. S., Stigh U. Shear behaviour of adhesive layers. 2006; Accepted for

publication in Int. J. Sol. Struct.

[7] Biel A. Constitutive behaviour and fracture toughness of an adhesive layer. 2005; Thesis for the

degree of licentiate of Engineering, Chalmers University of Technology, Sweden. (Available at

www.his.se/MechMat.)

[8] Popelar C. F., Liechti K. M. Multiaxial nonlinear viscoelastic characterization and modeling of a

structural adhesive. ASME J. Eng. Mat. Tachn., 119, 205-210.

A18

[9] Kihara K., Isono H., Yamabe H., Sugibayashi T. A study and evaluation of the shear strength of

adhesive layers subjected to impact loads. Int. J. Adh. Adh. 2003; 23 253-259.

[10] Maheri M. R, Adams R. D. Determination of dynamic shear modulus of structural adhesives in thick

adherend shear test specimens. Int. J. Adh. Adh. 2002; 22, 119-127.

[11] Blackman B. R. K., Kinloch A. J., Taylor A. C., Wang Y. The impact wedge-peel performance of

structural adhesives. J. Mat. Sci. 2000; 35 1867-1884.

[12] International Standard ISO 11343. Adhesives – Determination of dynamic resistance to cleavage of

high-strength adhesive bonds under impact conditions – Wedge impact method. 2003; ISO.

[13] Blackman B. R. K., Hadavinia H., Kinloch A. J., Williams J. G. The use of a Cohesive Zone Model

to study the fracture of fibre composites and adhesively-bonded joints. Int. J. Fract. 2003; 119: 25-46.

[14] Guo C., Sun C. T. Dynamic mode-I crack-propagation in a Carbon/Epoxy composite. Comp. Sci.

Techn. 1998; 58, 1405-1410.

[15] Kusaka T., Hojo M., Mai Y., Kurokawa T., Nojima T., Ochiai S. Rate Dependence of Mode-I

Fracture Behaviour in Carbon-fibre/Epoxy Composite Laminates. Comp. Sci. Techn. 1998; 58, 591-602.

[16] Tsai J. L, Guo C, Sun C. T. Dynamic delamination fracture toughness in unidirectional polymeric

composites. Comp Sci. Techn. 2001; 61, 87-94.

[17] Huang M. Vehicle Crash Mechanics. Boca Raton: CRC Press, 2002.

[18] Salomonsson K. Andersson T. Modeling and parameter calibration of an adhesive layer at the meso

level. Mechanics of Materials 2008: 40 ;1-2, 48-65.

[19] Alfredsson K. S., Stigh U. Continuum damage mechanics revised – A principle for mechanical and

thermal equivalence. Int. J. Sol. Struct. 2004; 41, 4025–4045.

[20] Carlberger T. Explicit FE-formulation of Interphase Elements for Adhesive Joints. 2006: Accepted

for publication in International Journal for Computational Methods in Engineering Science & Mechanics.

Paper B

Explicit FE-formulation of Interphase Elements for Adhesive Joints

B1

Explicit FE-formulation of Interphase Elements for Adhesive Joints

Thomas Carlberger1, K. Svante Alfredsson2 and Ulf Stigh2

1SAAB Automobile AB, SWEDEN and 2University of Skövde, SWEDEN Abstract The potential of adhesive bonding to improve the crashworthiness of cars is attracting the automotive industry. Large-scale simulations are time consuming when using the very small finite elements needed to model adhesive joints using conventional techniques. In the present work, a 2D-interphase element formulation is developed and implemented in an explicit FE-code. A simplified joint serves as a test example to compare the interphase element with a straight-forward continuum approach. A comparison shows the time saving potential of the present formulation as compared to the conventional approach. Moreover, the interphase element formulation shows fast convergence and computer efficiency. Keywords: Interphase element; Dynamic fracture; Adhesive joint

B2

Nomenclature

A metric matrix a1-a12 nodal degrees of freedom of interphase element aI(t) displacement and rotation of node I at time t B specimen width b volume load cI vector with arbitrary elements associated with node I. ∆t, ∆tc time step and critical time step Ω, ∂Ω body and outer surface δij Kronecker delta δ separation vector of adhesive layer E Young’s modulus f I force vector associated with node I F applied load vector G matrix in the relation between δ and a g essential boundary conditions h thickness of adhesive layer H rotational inertia compensation number H1, H2 thickness of lower and upper adherend, respectively I, J node numbers Jcons, Jlump moment of inertia from consistent (cons) and lumped (lump) mass matrix κ wave number ξ, ζ local coordinates of adhesive layer L, Lb, Ls element length, bonded length, and specimen length l0 distance between two zero-values of the peel stress M mass matrix m mass of adhesive in one element n outward normal vector n∆t, NDOF number of time steps, number of degrees of freedom NI shape function associated with node I. ωmax largest eigenfrequency of the structure ωɺ angular acceleration R virtual power associated with the adhesive layer r adherend number ρi , ρr material density for the interphase element (adhesive) and

the adherends respectively. 2,1∈r

S interface surface σσσσ Cauchy stress tensor Tc rise time Tr, Ts computer execution time and total time to simulate t time tev evaluation time for one degree of freedom and one time step

t prescribed traction vector Θ adherend cross-sectional rotation u displacement vector v, w shear and peel deformation of adhesive layer v displacement and rotation vector of adherend mid-line νννν weight function vector νa,i Poisson’s ratio of adherend (index a) and adhesive (index i) x coordinate

B3

1. Introduction Car bodies consist of large shell structures connected with some joining technique. Traditionally, the most frequent technique is spot-welding. Although this method has many advantages, it is essentially limited to mono-material joints. Increasing demands on fuel efficiency and lowered emissions increase the need for further optimisations. To this end, a multi-material car body is a promising possibility. Recently the potential to use adhesive bonding has been identified. With this method it is not only possible to join dissimilar materials but also to improve both stiffness and strength in mono-material structures.

In the product development phase, explicit FE-simulations are frequently used to evaluate the crashworthiness of prospective car bodies, cf. e.g. [1]. In these simulations, there is a need for efficient modelling technique of adhesive joints. Specifically it is necessary to model the structure in such a way that the simulation execution time, Tr, is kept as short as possible; typically not exceeding 12-15 hours. An inherent limit with the explicit FE-method is the Courant limit. In order for an explicit FE-simulation to be numerically stable, the time step ∆t must be smaller than 2/ωmax, where ωmax is the largest eigenfrequency of the structure, cf. e.g. [2]. The critical time step can be estimated as the shortest time it takes an elastic wave to travel the distance between two nodes. In practice, this critical time step sets a limit on the smallest element length for a given time step and total execution time. This critical element length is for an adhesive consisting mainly of epoxy about 1.2 mm with a given time step ∆t = 1 µs. This element length should be compared to the typical adhesive layer thickness, h = 0.2 mm. Thus, introduction of an adhesive layer modelled with continuum finite elements (three elements across the adhesive layer) into a structure with an original critical time step of 1 µs, will increase the execution time by a factor of about eighteen, cf. [1]. A trick, often used to circumvent the Courant limit, is mass scaling or a combination of mass scaling and a reduction of Young’s modulus, [3]; the material density is increased and/or Young’s modulus of the adhesive is decreased for the “too small elements” until the Courant limit is fulfilled. This method is to be used with caution since the added mass or decreased stiffness will influence the results. Especially if the adhesive bond line is oriented along the crash direction, added mass may give too large influences on the results. In a recent study of the butt-joint, a 40 % reduction of the stiffness of the adhesive layer is shown to be tolerable, [1]. These types of results should however be taken with some caution since they are confined to specific joint geometries and load cases.

One way to avoid the use of small time steps or mass-scaling is to model the adhesive layer as a cohesive zone rather than as a continuum. The use of cohesive laws to model adhesive joints was proposed in [4] and [5]. In this approach, the deformation of the adhesive layer is assumed to be dominated by two deformation modes viz. peel and shear, cf. Fig. 1.

Figure 1: Deformation modes of the adhesive layer with thickness h: peel, w, and shear, v. Conjugated stress components σ and τ.

This assumption is supported by asymptotic analyses, cf. [6]. A number of experimental and numerical studies show that the cohesive law model predicts fracture accurately, cf. e.g. [7,8,9] . In a cohesive law, the traction acting on the adherends depends on the deformation of the adhesive. At small deformation, the response is elastic. At some critical state, the adhesive deforms in-elastically. Eventually the traction tends to zero and the adhesive fractures. At this moment, a crack propagates in the adhesive layer. With a structural epoxy adhesive, the zone of in-elastic behaviour in front of the crack tip is very much larger than the thickness of the adhesive layer, cf. [8,10]. A recent review of cohesive models is given in [11].

τ

σ

w

σ

h

B4

In a finite element context, cohesive models are traditionally used to model fracture in a continuum model. However, in many adhesive joint applications, large structures are modelled with shell elements. In [12] and [13], FE-methods to analyse delamination of composite shells are developed. Nodes on two sides of the anticipated delamination are connected by a generalised spring. The force-deformation relation for the spring represents the action of the cohesive surface. That is, each spring represents the adhesive in an area around the spring. Thus, it is necessary to consider the mesh size when assigning constitutive properties to the spring.

In this paper a dedicated finite element for modelling adhesive bonds between shell elements is presented. The novel approach is to span the entire volume from the shell mid plane of the first adherend across the adhesive volume to the mid plane of the second adherend in one interphase element. As compared to the spring elements in [12,13], no consideration to the mesh size has to be done to input constitutive behaviour to the interphase element. The computational efficiency of the element is demonstrated by comparison with methods used in the automotive industry today, where continuum elements are used to model the adhesive layer.

2. Interphase formulation Figure 2 shows a part of an adhesive joint consisting of two shells. In this case the shells are two of the parts creating the B-pillar of a passenger vehicle. Typical dimensions are adhesive layer thickness h = 0.2 mm and thickness of shells H1 ≈ H2 = 0.8 mm. Thus, we often find h < H1 ≈ H2. The bonded sheets are usually modelled with shell elements in the explicit FE-formulation. This means that the displacement field in the shells is governed by nodal displacements and rotations in the FE-model, cf. e.g. [14,15]. The coupling of the rotational degrees of freedom to the deformation of the adhesive layer has to be considered in this case. For instance, assume a state of deformation where all displacements are zero but rotations exist, then the adhesive layer may be deformed in shear. Thus, in order to connect the thin adhesive layer correctly to the shells, the coupling of the rotational degrees of freedom of the shells to the deformation of the adhesive layer has to be considered. A simple method to achieve this coupling is to model the adhesive as a continuum using solid elements and to connect these to the shell elements by the use of rigid bars or stiff beam-elements, cf. Fig. 2.

Figure 2: 3D-model of adhesively joined sheets (adherends). The adhesive layer is represented by a solid element connected to the shell elements by rigid bars.

This approach does, however, introduce some disadvantages as will be demonstrated here. The adhesive element is connected to the adherends by rigid or stiff beams. Although this is a tedious work, the process may be automated. In this case, the use of rigid beams is preferred, since it does not add degrees of

rigid bar or stiff beam elements

mid surface of upper shell element

mid surface of lower shell element

solid element

H2

H1

B5

freedom to the system. The stiff beams will add degrees of freedom to the system, but more severely, they will negatively influence the critical time step, ∆tc, of the system. Since the adhesive is modelled as a continuum, the very thin thickness of the adhesive layer will, referring to the Courant limit, imply a correspondingly short time step.

An attractive alternative method to avoid this tedious work and not severely influencing the critical time step is to use an interphase element, cf. Fig. 3b.

Figure 3: 2D-model of adhesively joined sheets (adherends). (a): Adhesive and adherends. (b): Interphase element.

Obvious profits of modelling the adhesive with interphase elements are easier modelling and shorter execution time as will be exemplified in this paper. The interphase formulation uses relative displacements. It may be noted that the use of relative displacements reduce the condition number of matrices, cf. [16]. Next, we will derive the governing equations in the FE-problem and the consistent mass matrix.

2.1 Governing equations

Consider a body occupying a domain Ω with a boundaryΩ∂ . Balance of linear momentum in the body Ω is given by

0, =−+ iijji uρbσ ɺɺρ in Ω. (1)

where σσσσ is the Cauchy stress tensor, ρ the density, b the body force per unit mass, u the displacement vector. Dots are used to indicate differentiation with respect to time, t. Here, lowercase indices indicate Cartesian coordinates, i, j ∈ 1, 2, 3; all components are taken with respect to a common inertial frame. Moreover, summation is to be taken over repeated indices. On the parts of the boundary Ω∂ where the traction is prescribed the following equation applies

ijji tnσ = on ∂Ω. (2)

where t is the prescribed traction vector. On other parts ofΩ∂ , the displacement and velocity might be prescribed. To obtain the weak form of Eq. (1), we multiply with an arbitrary weight function νννν and use the divergence theorem, cf. e.g. [17]. Omitting some intermediate steps, the result is

( ), d d d d 0i j ji i i i i i iσ b t ρuν ν ρ ν νΩ Ω ∂Ω Ω

Ω − Ω − ∂Ω + Ω =∫ ∫ ∫ ∫ ɺɺ , (3)

which is known as the principle of virtual power. In order to FE-formulate this equation we introduce the conventional FE-ansatz

L

a4

a5 a6

a10

a11 a12

a7

a8 a9

H1/2

H2/2

h

(b)

a3 a1

a2

a10

a11 a12

a1

a2 a3

L

Adhesive

Adherend 1

Adherend 2

H1

H2

h

a7

a8 a9

a4

a5 a6

(a)

B6

u = ΝΙ aΙ , (4)

where the indices in capital letters indicate the node numbers, i.e. I = 1,2,…nnodes, and summation should be taken over repeated indices in capital letters. Here aI(t) is the displacement/rotation of node I and NI is the shape function associated with node I. The weight function νννν is approximated according to the Galerkin-method, i.e. using the same shape functions as are used for the displacements

νννν = ΝΙ cΙ , (5)

where cI is a vector with arbitrary elements associated with node I. Now, substituting Eq. (5) into the principle-of-virtual power, Eq. (3) and noting that the cI:s are arbitrary we arrive at

( ) 0ddd =Ω+Ω∂−Ω−Ω∂∂

∫∫∫∫ΩΩ∂ΩΩ

iIiIiIjij

I uρNtNdbNσx

Nɺɺρ . (6)

With Eq. (4) we arrive at

int extI I I− + =f f Ma 0ɺɺ , (7)

where

int dIiI ji

j

Nf σ

xΩ

∂≡ Ω∂∫

, ( )dextiI I i I if N b d N tρ

Ω ∂Ω

≡ Ω − ∂Ω∫ ∫ , (8a,b)

and

∫Ω

Ω≡ dJIijijIJ NρNM δ . (8c)

Here, δij is the Kronecker delta. Equation (7) is a set of second order ordinary differential equations,

kinIII

extI tt faMaafaaf ≡=− ɺɺɺɺ ),,(),,( int

, (9a)

where the elements of kinIf are described by

d dkini I I i I J iJf N ρu ρN N a

Ω Ω

≡ Ω = Ω∫ ∫ɺɺ ɺɺ . (9b)

With a diagonal mass matrix, Eq. (9a) constitutes the set of equations of the explicit FE-method. The procedure to diagonalise the generally non-diagonal mass matrix is known as lumping, cf. e.g. [2]. Essential boundary conditions complement Eq. (9a). These are imposed on nc degrees of freedom, i.e.

( , , ) 0, 1, 2, ...,I ct I n= =a aɺg , (10)

where the nodal degrees of freedom aI have been collected in the vector a. The elements of a are usually nodal displacements and, for beam and shell formulations, nodal displacements and rotations. Similarly, the elements of f are associated with nodal forces, and for beam and shell elements, with nodal forces and moments. These equations constitute the foundation for the development of an interphase element in the next subsection.

2.2 The interphase element

In this section we will derive the nodal forces and the mass matrix associated with the interphase element. We will also discuss the properties of damping associated with an adhesive layer. Consider an arbitrary point on the adherend r, 2,1∈r according to Fig. 4.

B7

Figure 4: Beam degrees of freedom, v(r) and displacements u(r) on the interface surfaces.

Here r = 1 denotes the lower adherend and r = 2 the upper one. Let a(r) denote the degrees of freedom of the adherend r. According to Fig. 3, the vectors are given by

[ ]T654321

)1( aaaaaa=a , and [ ]T121110987

)2( aaaaaa=a . (11)

Furthermore, let v(r) denote the displacement and rotation of a point along the mid line of adherend r. For a

beam element, this vector contains the two mid line displacements ( )rxu and ( )r

yu in x- and y-direction,

respectively and the cross-sectional rotation, i.e.)(rzΘ ,

T( ) ( ) ( ) ( )r r r rx y zu u = Θ

v . (12)

The local degrees of freedom v(r) are interpolated from the element nodal degrees of freedom, a(r), by

( ) ( ) ( )r r r=v N a . (13)

For a Mindlin beam element, the interpolations of the displacements and rotation of the middle line are independent, i.e.

( ) ( ) ( )

(1) 11 2

1 1 12(1)1 2

(1) 1 26

0 0 0 0

0 0 0 0:

0 0 0 0

x

y

z

au N Na

u N N

N Na

= = = Θ

v N a (14)

where the nodal shape functions are given by N1 = 1 – ξ and N2 = ξ with ξ given in Fig. 4. Now, let

( ) T( ) ( )r r rx yu u =

u be the displacement on the interface between the adherend (r) and the adhesive.

Figure 4 shows

u(2)

v(2)

u(1)

v (1)

Adherend 2 mid line

Adherend 1 mid line

Reference configuration

Current configuration

ξ node

interface surface S(2)

interface surface S(1)

x

y

z

B8

( ) ( )( )r rr=u A v (15)

where

−=

010

2/01 1)1( HA and

=

010

2/01 2)2( HA , (16)

are metric matrices. The displacement of the interface boundary, u(r), is obtained from Eqs. (13) and (15):

)()()()( rrrr aNAu = . (17)

The separation of the interface surfaces S(1) and S(2), i.e. the deformation of the adhesive layer, δ, is obtained from Eq. (17) as

[ ] GaaNANAuuδ ≡−=−= )2()2()1()1()1()2( , where [ ]TT)2(T)1( aaa = (18a,b)

whereby the matrix G is defined. The total separation of the adherends is given by the components δx = deformation of the adhesive layer in the x-direction and δy = deformation in the y-direction. This forms the

separation vector T

x yδ δ = δ . By use of the orientation of the adhesive layer, the separation vector

can be decomposed in the peel and shear components, respectively.

Comparing with Eq. (17), weight functions, νννν(r), are chosen according to the Galerkin method as

νννν(r)=A(r)N(r)c(r) (19)

where c(r) contains arbitrarily chosen nodal values. Now, form the virtual power associated with the action of the adhesive on the adherends, R, by

( ) SRJS

d)(

T)1()2( tνν∫ −−= (20)

where t is the traction acting on the adhesive on the adhesive/adherend interface S(J), cf. Eq. (3). With Eq. (19)

Gcνν =− )1()2( where [ ]TT)2(T)1( ccc = , (21a,b)

and we arrive at

∫ −≡−=)(

TTT dJS

SR FctGc (22)

where the integral is identified as the contribution to the nodal force vector from the adhesive layer. We may express the integrand in this expression by the use of Eq. (18a) as

tAN

ANtG

−= T)2(T)2(

T)1(T)1(T (23)

leading to

[ ]TT)2(T)1( FFF −= , where ( )∫=)(

dT)()()(

IS

III StNAF (24a,b)

which are the forces used in the explicit FE-formulation. Up to this point, the derivation is applicable for general, non-linear cohesive laws. For simplicity, we now assume linear elastic behaviour of the adhesive in two dimensions. If we assume that the adhesive layer is oriented along the x-axis, the traction is given by

B9

/ 0

0 /

G h

E h

=

t δ (25)

where E and G are the elastic modulii in pure uniaxial straining and shear, respectively. A non-zero value of the normal traction, t2, implies that the lateral normal stress in the adherends, σ22, is non-zero, cf. Eq. (2). Strictly speaking, this stress leads to deformations of the adherends. However, most beam element formulations do not account for lateral normal stresses and corresponding deformations. Thus, when implementing the interphase element into an existing FE-code, this effect cannot be accounted for. Fortunately, due to the large mismatch in stiffness between the adherends and the adhesive, the thickness change of the adherends will only have a minor influence on the results in terms of the adhesive deformations. Hence the effect of adherend thickness change is neglected is the present study. The stable time step is calculated according to ∆tc ≤ 2/ωmax, where ωmax is the maximum eigenfrequency of the structure. It should be noted that the calculation of ωmax is not performed in commercial explicit codes due to the amount of work associated with this task and the requirements of computer memory. Instead, the approximation

minc

lt

c∆ = where

Ec

ρ= (26a,b)

and lmin is the shortest element length in the model, is used. However, the approximation does not take care of contact stiffness, such as cohesive zone stiffness. Care has to be taken in choosing e.g. contact penalty stiffness or cohesive zone stiffness, such that the influence on the time step does not exceed the scale factor.

We will now derive the mass matrix, Eq. (8c), associated with the interphase element. Let ζ denote a through the thickness coordinate in the adhesive layer with ζ = 0 at the lower interface and ζ = 1 at the upper interface, cf. Fig. 5.

Figure 5: Local thickness- and length- coordinate systems.

The displacement u in the adhesive at a point ζ is written

(1) (1) (2) (2)el(1 )ζ ζ = − =

u A N A N a N a , (27)

and the element mass matrix, Eq. (8c), is derived in a straightforward manner. The result is

∫∫

==

1

0(2)(2)(2)T(2)T(1)(1)(2)T(2)T

(2)(2)(1)T(1)T(1)(1)(1)T(1)T

iel

Teli d

2

2

6d ξρρ

NAANNAAN

NAANNAANNNM

BhLV

V

(28)

where ρi is the density of the adhesive. Evaluation gives the consistent element mass matrix

L

h ξ

ζ

B10

−−−−

−−−−−−

−−

=

22

2

222

22

22

2112112

1

221

2112112

112

1

2211

i

4

016

8016

2044M.YS

080016

4088016

022044

040080016

2044088016

204022044

080040080016

4082044088016

144

H

H

HHH

HH

HHHHHHH

HHH

HHHHHHHHH

HHHH

hLBρM (29)

Since we will use the element mass matrix in an explicit FE-code, this matrix has to be transformed to a diagonal matrix to allow the simple numerical scheme in the explicit FE-code. In most instances, this is a well-established procedure; the literature provides several methods for lumping, cf. e.g. [2,17]. In the present case, the mass of the adhesive layer has an offset to the nodes of the interphase element. Thus, if we simply distribute the mass of the adhesive layer, m = ρBhL, to the nodes, the moment of inertia will be too large . In a crashworthiness analysis, essentially rigid body accelerations are anticipated in large parts of the structure. Thus, we aim at developing a method to lump the element mass matrix such that a rigid body motion gives the correct momentum. Accordingly, the adhesive mass is evenly distributed to the four nodes. By this procedure m/4 is placed at each of the four nodes and on the diagonal of the lumped mass matrix at the positions associated with the translational degrees of freedom. This procedure provides the correct inertia for rigid body translational motion. To determine a suitable diagonal element for the rotational degrees of freedom, we write the lumped element mass matrix as

diag22

22

21

21

ilumped 11111111

4HHHH

hLB ααααρ=M (30)

where α is a number to be determined to give an appropriate moment of inertia.

Now, we assume a rigid body rotation of the adhesive layer about its mass centre, cf. Fig. 6.

Figure 6: Adhesive, as rigid body, rotating around centre.

This motion gives the kinetic energy

( ) 222i24

1 ωρ LhBhLT += , (31)

where ω is the angular velocity. With the suggested lumped element mass matrix, Eq. (30), the kinetic energy for the interphase element is

( ) ( ) 222

21

22

22

1iT 222282

1 ωααρ

−−+

++

+== HHL

hHhHBhLT lumpedaMa ɺɺ . (32)

h L

ωɺ

B11

With ( ) ( ) ( )

( )2 2 2 21 2 1 2

2 21 2

3 6 4

12

H H h H H L h

H Hα

+ + + + += −

+, the lumped mass matrix gives the same kinetic energy

as the adhesive layer. It may be noted that α is negative, i.e. the d.o.f. corresponding to rotations of the nodes are given negative values for the moment of inertia to compensate for the excessive rotational inertia caused by the lumping.

The need for an improvement of the element mass matrix is dependent on the relation between the mass matrix of the adherend and the mass matrix of the adhesive. The lumped element mass matrix for a Mindlin beam element is

2 2

diag12 12 12 12 , 1,2

24r r rMindlin r r

BH LH H r

ρ= ∈M . (33)

This beam element mass matrix can be compared to the mass matrix of the interphase element. To this end, form the “collected” element mass matrix of the two beam elements connecting to the interphase element, i.e.

1Mindlin

Mindlin 2Mindlin

=

M 0M

0 M. (34)

To study the importance of modelling the interphase mass matrix correctly, we form the ratio between the inertia of the adherend and the interphase. With typical values the relations in Table 1 result.

1.4consadhes.

lumpedadhes. =

J

J

adhesive thickness h = 0.2 mm

adherend thickness H1= H2 = 0.8 mm

adhesive density ρi = 1350 kg/m3

adherend density ρ1= ρ2 = 7800 kg/m3

element length L = 5 mm

46lumpedadhes.

lumpedadher. =

J

J

Table 1: Ratio between lumped rotational inertia,lumpedadhes.J , and consistent rotational inertia, cons

adhes.J ,

for the interphase element and ratio between adherend lumped inertia, lumpedadher.J , and interphase

element lumped inertia, lumpedadhes.J , with typical data for an automotive application.

As Table 1 clearly shows, the elements of the mass matrix of the interphase element are much smaller than those of the mass matrix of the beam element. For numerical stability of the explicit method, increased mass is positive, and thus we will not ignore the adhesive mass completely. Therefore we will use the lumped mass matrix of Eq. (30) for the interphase element.

Material damping is negative for the numerical performance of the explicit FE-method, since it decreases the stable time step, cf. e.g. [17]. Damping in structures is due to mechanisms such as hysteresis in the material and slip in connections. These mechanisms are not well understood. A popular assumption is damping due to viscous mechanisms. However, the amount of dissipated energy due to viscous damping is generally negligible as compared to the energy “dissipated” by the plastically deforming adherends in a crash analysis. The main task for the adhesive is not to dissipate energy, but keep the adherends together.

B12

In this way, the structure is able to dissipate a much larger amount of energy. Since this is a frequent question regarding fracture of adhesives, let us do the following observation. Assume a vehicle with the mass 1000 kg is travelling at the velocity 20 m/s. The kinetic energy is then 200 000 J. The length of all joints in a car is approximately 400 m. If all joints were adhesively bonded and broke at the moment of impact with a typical joint width 15 mm and typical fracture energy 1000 J/m2, the total fracture energy would be 6000 J. Thus, it is obvious that the kinetic energy is dissipated by other, more important, mechanisms than the fracture energy of the adhesive. The most important energy dissipation mechanism is plastic work by the adherends.

3. Numerical studies

The interphase formulation is tested in a 2D explicit FE-model of a Double Cantilever Beam, DCB, structure subjected to symmetric dynamic loading, cf. Fig. 7.

Figure 7: (a) DCB modelled with interphase elements, and (b) DCB modelled with continuum adhesive and rigid connections. The layer thickness, h, is 0.2 mm.

The purpose of the study is to evaluate the convergence of the simplified interphase formulation. The beams are modelled with linear elastic, linearly interpolated Mindlin beam elements and the adhesive with linear elastic, linearly interpolated interphase elements, cf. Eqs. (24,30) and analysed using an explicit Matlab® FE-code. This model, shown in Fig. 7a, is denoted the ‘beam/interphase model’ in the following. Due to the constraint imposed by the stiff adherends, an effective Young’s modulus,E , of the adhesive layer is used in the interphase element,

( )( )( )

i i

i i

1

1 1 2

EE

νν ν

−=

+ −, (35)

cf. [18]. For comparison, an alternative DCB-model is developed using conventional linearly elastic, linearly interpolated Mindlin beam elements for the adherends and linear elastic, four-node rectangular continuum elements for the adhesive, cf. Fig. 7b. The continuum elements of the adhesive are connected to the beam elements of the adherends by means of rigid connections, cf. Fig. 7b. The adhesive layer is modelled with three elements over the thickness to provide a possibility for some stress distribution through the thickness of the adhesive layer. This model, which is denoted the ‘beam/continuum model’, is evaluated with the commercial FE-software ABAQUS/Explicit, v. 6.5-1. Additionally, we compare these

continuum element

(a)

(b)

interphase element

beam elements, H1 = H2 = 0.8 mm

rigid connection

Ls = 30 mm

Lb = 15 mm

F

F

F

F = 10 N

B13

two approximate models to a fully converged 2D-model where both the adherends and the adhesive layer are modelled using continuum elements, cf. Fig. 8.

Figure 8: Continuum 2D model for comparison. The adhesive starts at x = 0 and the adhesive middle layer at y = 0.

This model is denoted the ‘continuum model’. All models are in plane deformation. The DCB structure is given typical automotive dimensions: specimen length, Ls = 30 mm, specimen width, B = 30 mm, adherend height H1 = H2 = H = 0.8 mm, adhesive layer thickness h = 0.2 mm and the bonded length Lb = 15 mm, cf. Fig 7. The material of the adherends is steel with Young’s modulus Er = 210 GPa, density ρr = 7800 kg/m3, and Poisson’s ratio νr = 0.3. The adhesive material is epoxy with Young’s modulus Ei = 2.0 GPa, density ρi = 1350 kg/m3, and Poisson’s ratio νi = 0.4. A symmetric force F = 10 N is applied in a smooth manner, such that F = 5(1-cos(π(t/Tc)) N for t < Tc, where Tc corresponds to a fraction of the primary eigenfrequency in bending; the corresponding period is T = 400 µs. For t ≥ Tc, the force F = 10 N, is held constant. Depending on the rise time of the applied force, the deformation response at the adhesive tip will be influenced by wave propagation. This is shown in Fig. 9 for the beam/interphase model.

x

y

B14

Figure 9: Load rise time, Tc, influence on peel stress response at adhesive tip, calculated with the interphase model.

With a long rise time, longer than about 0.1T, the response is mainly characterized by the lowest eigenmode and the influence of Tc is marginal. With a gradually increasing rise time, the response approaches that of a quasi-static solution. For relevance in comparison, the rise-time, Tc, is here chosen to 0.1T, since this gives both a dynamic and a relatively smooth stress response. The comparison is designed to reveal whether or not the interphase element formulation works in the explicit formulation. In this sense, the comparison is believed to be relevant. In an explicit FE-analysis, the elastic properties are crucial for the critical time step. Hence, we focus attention to the linear elastic part of the solution. However, the linear elastic part represents only a minor part of the cohesive law, cf. Fig. 10.

Figure 10: Traction-separation behaviour for the engineering epoxy adhesive DOW Betamate XW1044-3 with layer thickness h = 0.2 mm. Data from [10].

The solution in the linear region will affect the initiation of non-linear response, but will hardly influence the response in the non-linear region. The variation of the peel deformation, δy, along the adhesive layer is evaluated at the moment the first maximum deflection of the loading points is passed. The total deformation of the adhesive layer is used since, for the beam/interphase model it suffices to capture the deformation in the part of the adhesive layer where non-linear deformation takes place. Converged results of all three models are shown in Fig. 11.

0

5

10

15

25

0 10 20 30 40 50 60

20

σ (

MP

a)

w (µm)

0

5

10

15

20

25

30

0 0.25 0.5 0.75 1.0normalised time ( t/Τ )

pee

l str

ess

(MP

a) interphase, Tc = 0.001T

interphase, Tc = 0.01T interphase, Tc = 0.1T interphase, Tc = 0.5T

interphase, static

B15

Figure 11: Displacement at adhesive tip vicinity. Results from continuum, beam/interphase and beam/continuum models.

The results show minor differences between the different modelling techniques. Convergence is studied as the mesh size is decreased. A relevant length parameter in this context is the wave

number, 4 3/6 hHEE i≡κ , that emerges from a quasi-static analysis, cf. e.g. [19]. The wave number has

a simple interpretation; with l0 denoting the distance between two neighbouring zero-values of the peel stress, κl0 = π. Thus, it is expected that an element length, L, smaller than l0 = π/κ ≈ 3 mm should be necessary for convergence. That is, a number of elements span l0. The deformation profiles obtained with different element sizes are plotted in Fig. 12 for both the beam/interphase and the beam/continuum models.

Figure 12: Displacement profiles acquired with consecutively smaller elements with (a) beam/interphase formulation and (b) beam/continuum formulation. Captions correspond to element size κL.

The convergence is similar for the two models. Figure 13a shows that the converged value of the maximum peel deformation is achieved already with κL = 2.4 with the beam/interphase model.

4.92.41.2

0.310.150.076

-1.0

-0.5

0

0.5

1.0

1.5

0 5 10 15 distance from adhesive tip (mm)

sep

arat

ion (µm

)

0.59

(b)

-1.0

-0.5

0

0.5

1.0

1.5

0 5 10 distance from adhesive tip (mm)

sep

arat

ion (µm

)

4.92.41.20.610.310.150.076

(a)

15

-0.4

0

0.4

0.8

1.2

0 1 2 3 4 5

distance from adhesive tip (mm)

def

lect

ion

(µm

) continuum

beam-continuum beam-interphase

B16

Figure 13: Convergence study. (a) Peel displacement at the adhesive tip. (b) Peel stress at the adhesive middle layer for the beam/continuum model.

Thus, with the element length 2.3 mm a correct maximum peel deformation is achieved. However, although the deformation profile resembles the converged one, it appears too coarse. Using half that element length, κL = 1.2 gives a result that appears more adequate. Thus, using more than about three elements to resolve the deformation profile between two zero-values of the peel deformation appears adequate for many applications.

For the beam/interphase model, convergence of the deformation implies that also the stress is converged since the stress is proportional to the deformation, cf. Eq. (25). However, for the beam/continuum model the stress in the adhesive layer is computed from the strain field which converges considerably slower than the displacement field. Convergence of stress at the middle layer of the adhesive tip is shown in Fig. 13b for the beam/continuum model. The beam/continuum model requires an element length κL = 0.31, for convergence of the stress at the middle of the adhesive tip. That is one fourth of the element length required by the beam/interphase model, which for the 3D case means that the number of elements required for simulation of an adhesive joint with continuum elements is 16 times larger.The material data needed for simulation depends on the theory used for the modelling. Thus, if a beam/continuum model is to be used, the experiments used to measure the material behaviour should be evaluated considering the full field of stress and deformation in the adhesive layer. On the other hand, if the beam/interphase model is to be used, only the peel and shear stress and deformation of the layer should be considered. This latter method appears adequate and preferable, cf. e.g. [7]. The simulations using the beam/continuum model suffer from longitudinal stress waves overlaying the peel stress and also, to some extent, from hour-glassing. Neither of these deficits plague the beam/interphase model, since the interphase element is fully integrated and lacks stiffness in the longitudinal direction.

Apart from providing an easier geometrical modelling as compared to the beam/continuum model, the beam/interphase model gives a shorter execution time. With Ts denoting the total time to be simulated, the number of time steps n∆t = Ts/∆t = Tsc/L, where the time step is related to the wave speed c and the element length according to the estimated Courant limit, cf. e.g. [1]. Thus, the smaller elements which are necessary using the beam/continuum model, lead to an increased number of time steps. With a specific object to be analysed, the length of the elements and the number of nodes are related to the size of the object. For the models of the DCB-specimen analysed here, the number of pairs of beam elements along the bond line equals Lb/L, cf. Figs. 7b and 14. With the beam/interphase model, each pair of beam element nodes contributes with six degrees of freedom (DOF) to the total number of DOF. With the beam/continuum model, using three elements through the thickness of the adhesive layer, each pair of beam element nodes contributes with ten DOF, cf. Fig. 14.

Relative element size, κL, mm

0

5

10

15

20

0.0 0.1 1 10

Pee

l str

ess

(MP

a)

(b)

0.4

0.6

0.8

1.0

1.2

0.01 0.1 1 10

Relative element size, κL, (mm)

Ad

hes

ive

tip d

isp

lace

men

t (µm) beam/interphase

beam/continuum continuum

(a)

B17

Figure 14: Principal sketches of 2D adhesive joints modelled by (a) beam/interphase elements and by (b) beam/continuum elements.

The total number of DOF for an analysis with the beam/interphase model is NDOF = 6(Lb/L+1)≈ 6Lb/L. The factor 6 is changed to 10 for the beam/continuum model. The element length L for a converged analysis for the beam/continuum model is one fourth of that for the beam/interphase model, which implies the same relation between the required time steps for the two models. With tev denoting the evaluation time for one degree of freedom and for one time step, the total execution time for an analysis Tr = 6tevTsc/L(Lb/L+1) ≈6tevTsc Lb /L

2, where the factor 6 is changed to 10 for the continuum model. Thus, the relation between the execution time with the beam/continuum model to the execution time for the beam/interface model is about 27. A similar analysis for a 3D shell/interphase model gives Tr ≈ 12tevTsc(Lb)

2/L3 cf. Fig. 15.

Figure 15: Principal sketches of 3D adhesive joints modelled by (a) interphase elements and by (b) continuum elements.

The factor 12 is changed to 18 for a shell/continuum model. Thus, the relation between the execution time with a shell/continuum model to the execution time for a shell/interface model is about 96. These numbers apply to structures constituting the adhesive joint. In automotive structures where the adhesive joints constitute a minor part of the complete structure, the numbers are smaller. Considering a structure where one percent of the total degrees of freedom (DOF) connect to adhesive joints modelled with interphase

interphase elements

shell adherends 6

6

L

(a)

continuum elements shell adherends

6

6

L

18 degrees of freedom

(b)

3 3

rigid connections

12 degrees of freedom

rigid connections

interphase elements

beam adherends

3

3

L

6 degrees of freedom (a)

continuum elements

beam adherends

3

3

L

10 degrees of freedom (b)

2 2

B18

elements. If we substitute the interphase elements for continuum elements, the total number of DOF will have to increase due to the slower convergence of this element type. In the part of the structure constituting adhesive joints, the number of DOF will increase by a factor of 24, i.e. the total number of DOF will increase by 23 %. The time step is four times smaller when continuum elements are used instead of interphase elements. This smaller time step is used for the complete structure. Altogether this renders an execution time which is almost five times the execution time when interphase elements are used. This is not acceptable for engineering use.

4. Results and conclusions The governing equations for impact simulation and the finite element formulation are derived for adhesively bonded structures. An interphase formulation is presented for the 2D case of two beams joined by a thin adhesive layer. The interphase element mass matrix is derived and lumped. It is argued that the element material damping matrix is of little importance relative other dissipative mechanisms. Finally, a 2D verification simulation is performed on a pure peel DCB specimen using the interphase elements representing adhesive joining of the two beam adherends. This model is compared to an identical structure where the adhesive is modelled with continuum elements connected to the two beams by rigid connections. The comparison not only involves the convergence of the two techniques but also their respective total execution times. The simulation using the interphase element formulation converges faster and can be evaluated using larger time steps and fewer degrees of freedom, thus rendering a significantly reduced execution time. The interphase formulation is an attractive technique of modelling adhesive joints during impact of shell structures.

Acknowledgements The authors are grateful to Mr. Anders Biel and Dr. Kent Salomonsson for fruitful discussions during this study. The authors would also like to acknowledge the Swedish Consortium for Crashworthiness for funding this project.

References:

[1] T Carlberger, and U. Stigh, An explicit FE-model of impact fracture in an adhesive joint, Engineering Fracture Mechanics, vol. 74, Issue 14, pp. 2247-2262, 2007.

[2] R.D. Cook, D.S. Malkus, M.E. Plesha, and R.J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., John Wiley & Sons, 2002.

[3] duBois PA. Engineering consultant. Private communication, 2005. [4] P.J. Gustafsson, Analysis of generalized Volkersen joints in terms of non-linear fracture mechanics, Mechanical behaviour of adhesive joints, Proceedings of European Mech. Colloquium 227, pp. 323-338, 1987. [5] U. Stigh, Damage and crack growth analysis of the double cantilever beam specimen, International Journal of Fracture, Vol. 37, pp. R13-R18, 1988. [6] P. Schmidt, Modelling of Adhesively Bonded Joints - Asymptotic Analysis, Submitted for publication [7] T. Andersson, and U. Stigh, The stress-elongation relation for an adhesive layer loaded in peel using equilibrium of energetic forces, International Journal of Solids and Structures, Vol. 41, pp. 413-434, 2004. [8] K. Leffler, K.S. Alfredsson, and U. Stigh, Shear behaviour of adhesive layers, International Journal of Solids and Structures. Vol. 44, pp. 530-545, 2007. [9] S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, and P.D. Zavattieri, Mixed-mode cohesive-zone models for fracture of an adhesively bonded polymer–matrix composite, Engineering Fracture Mechanics, Vol. 73, pp. 64-78, 2006. [10] T. Andersson T, and A. Biel, On the effective constitutive properties of a thin adhesive layer loaded in peel, International Journal of Fracture, Vol. 141, pp. 227-246, 2006.

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[11] R. de Borst, J.J.C Remmers, and A. Needleman, Mesh-independent discrete numerical representations of cohesive-zone models, Engineering Fracture Mechanics, Vol. 73, pp. 160-177, 2006. [12] E.D. Reedy, and F.J. Mello, Modeling the Initiation and Growth of Delaminations in Composite Structures, Journal of Composite Materials, Vol. 31, No. 8, pp. 812-831,1997. [13] R. Borg, L. Nilsson, and K. Simonsson, Simulating DCB, ENF and MMB experiments using shell elements and a cohesive zone model, Composites Science and Technology, Vol. 64, pp. 269-278, 2004. [14] ABAQUS Analysis User’s Manual, Version 6.5, ABAQUS Inc. Providence, USA, 2004. [15] LS-Dyna Keywords User’s Manual, Version 970, Livermore Software Technology Corporation, April 2003. [16] K.J. Bathe, Finite element procedures, Prentice Hall, 1996. [17] T. Belytschko, W.K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, Chichester, 2000. [18] U. Edlund, and A. Klarbring, Analysis of elastic and elastic-plastic adhesive joints using a mathematical programming approach, Computer Methods in Applied Mechanics and Engineering, Vol. 78, Issue 1, pp. 19-47, 1990 [19] K.S. Alfredsson, and J.L. Högberg, A Closed-Form Solution to Statically Indeterminate Adhesive Joint Problems - Exemplified on ELS-specimens, International Journal of Adhesion and Adhesives, to appear. DOI:10.1016/j.ijadhadh.2007.10.002, 2007

Paper C

Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive

C1

Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive

Thomas Carlberger1 and Anders Biel2

1SAAB Automobile AB, SE-461 80 Trollhättan, SWEDEN

2University of Skövde, PO Box 408, SE-541 28 Skövde, SWEDEN Abstract Effects of temperature and strain rate on the cohesive relation for an engineering epoxy adhesive are studied experimentally. Two parameters of the cohesive laws are given special attention: the fracture energy and the peak stress. Temperature experiments are performed in peel mode using the double cantilever beam specimen. The temperature varies from -40°C to +80ºC. The temperature experiments show monotonically decreasing peak stress with increasing temperature from about 50 MPa at -40ºC to about 10 MPa at +80ºC. The fracture energy is shown to be relatively insensitive to the variation in temperature. Strain rate experiments are performed in peel mode using the double cantilever beam specimen and in shear mode, using the end notch flexure specimen. The strain rates vary; for peel loading from about 10-4 s-1 to 10 s-1 and for shear loading from 10-3 s-1 to 1 s-1. In the peel mode, the fracture energy increases slightly with increasing strain rate; in shear mode, the fracture energy decreases. The peak stresses in the peel and shear mode both increase with increasing strain rate. In peel mode, only minor effects of plasticity are expected while in shear mode, the adhesive experiences large dissipation through plasticity. Rate dependent plasticity, may explain the differences in influence of strain rate on fracture energy between the peel mode and the shear mode. Keywords: Cohesive laws, Strain rate, Temperature dependence, Experimental, DCB-specimen, ENF-specimen, Crashworthiness. 1. Introduction

Important design goals for modern cars are to reduce the emissions and to increase the crashworthiness. On account of this, the use of structural adhesives has become more frequent and accordingly the mechanical behaviour of adhesive joints is important. Adhesives in the form of thin layers between stiffer adherends behave differently from the adhesive as a bulk material. This is due to the constrained state of deformation. Thus, the fracture properties will depend on geometrical aspects like the thickness of the adhesive layer; cf. e.g. Kinloch (1987). It can be shown that the deformation of an adhesive layer is dominated by two deformation modes, namely peel and shear, cf. Fig. 1. Schmidt (2007) shows in an asymptotic analysis that these deformation modes dominate the behaviour of an elastic structure in small deformation provided the layer thickness is much smaller than the thickness of the joined parts, i.e. the adherends; the elastic modulus of the adhesive is much smaller than the modulus of the adherends; and the in-plane length of the adhesive joint is much larger than the thickness of the adherends. A model of the adhesive based on these deformation modes is known as an adhesive layer theory. Experiences from simulations and experiments with in-elastic adhesive joints in large deformation show that the adhesive layer theory has a good potential to capture the behaviour of real structures, cf. e.g. Yang and Thouless (2001). In three dimensions, the shear component is split into two orthogonal shear components. These deformation modes correspond to modus I (peel) and modus II and III (shear) of fracture

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mechanics. In the literature, the relations between the peel stress, σ, and the shear stress, τ, and the conjugated deformation measures, w, and v, are denoted cohesive laws, traction-separation relations or stress-deformation relations. We will here use the term “cohesive laws” and remember that these laws, for the case of an adhesive layer, reflects the behaviour of a material volume. For instance it reflects the elastic properties of the adhesive at small strains. It can also be noted that in-plane straining of the adhesive layer has a substantial influence on the behaviour of the adhesive joint provided the adherends deform substantially, cf. Salomonsson and Stigh (2008).

Fig. 1 Deformation modes of the adhesive layer with thickness h: peel, w, and shear, v. Conjugated stress components σ and τ. A number of experimental methods exist to determine cohesive properties for adhesive layers at quasistatic loading in room temperature. For peel loading the double cantilever beam specimen (DCB) is used, cf. Fig. 2. Olsson and Stigh (1989) derive a method to measure the cohesive law in peel using a DCB-specimen loaded by transversal forces; Suo et al. (1992) derive a similar method where the DCB-specimen is loaded by bending moments. For shear loading, the end notch flexure specimen (ENF) is used, cf. Alfredsson (2004) and Fig. 3. Recently, methods for mixed mode loading have been developed; cf. Sørensen et al. (2006) and Högberg et al. (2007). The methods have been used extensively to characterize structural adhesives, cf. e.g. for peel loading Stigh and Andersson (2000), Sørensen (2002), Andersson and Biel (2006) and for shear loading Alfredsson et al. (2003), Leffler et al. (2007). Blackman et al. (2003) study the use of a cohesive zone in fracture analyses of fibre composites and adhesively bonded joints. They suggest a two-parameter criterion for fracture instead of the conventional one-parameter, governed by the critical fracture energy solely. This second parameter is suggested to be a critical limiting value of stress, σmax. This can be viewed as a simplified version of a full cohesive law description of the adhesive layer. Biel and Stigh (2008) also conclude that the fracture energy and the peak stress are the two most important parameters of cohesive laws.

v τ

τ σ

w

σ

h

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Fig. 2 Double cantilever beam specimen.

Figure 2 shows the DCB specimen in an exaggeratedly deformed state. The un-bonded part of the DCB specimen can be considered as a crack. That is, ap is the crack length, bp the adherend width, Hp the adherend height, Lp the specimen length, and hp the adhesive thickness; subscript p indicates peel.

Fig. 3 End notch flexure specimen.

Figure 3 shows the ENF specimen. The geometry is defined by the length between supports, Ls, the adherend width, bs, the adhesive thickness hs, the adherend height, Hs, and the un-bonded length, as. The latter is also referred to as the crack length. Subscript s indicates shear.

Two essential aspects of cohesive properties of an adhesive layer are dependences on temperature and strain rate. An automotive car body is subjected to a significant temperature range during its usage. Excluding the engine compartment, measurements show temperatures in the range -40°C to 125°C. Crash requirements of automotive structures are supposed to be fulfilled throughout the temperature range. Crash analysts in the automotive industry observe strain rates in the base material up to about 300 s-1 in simulations (e.g. DuBois, 2005); Carlberger and Stigh (2007) show that even larger strain rates can be expected in adhesive layers. Adhesives, which mainly consist of polymers, are temperature and strain rate

Hp δ

p+ h

p +

Hp

hp + w

hp

θ

Fp

Lp

Fp

θ

ap

x

Fs

δs

as

Ls

Ls/2

h s

Hs

v

x

y

y

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dependent. Thermo-set epoxy adhesives of current interest show a glass transition temperature, Tg, between 85°C and 120°C. Thus, the temperature effect is expected to be significant in the temperature range. Static fracture of adhesive joints is frequently studied. However, only few results are available in the open literature concerning effects of temperature and strain rate on the fracture properties. Some temperature studies have been performed. For instance Ashcroft et al. (2001) show a transition in fracture mode from brittle at -50ºC, stick-slip at 22ºC to ductile at 90ºC for epoxy-bonded panels of CFRP. The failure locus transfers from the composite substrate (matrix/fibre debonding failure parallel to the fibres) to the adhesive (cohesive failure) with increasing temperature. In a study of an epoxy adhesive, Chai (2004) finds that the mode II yield stress decreases linearly with increasing temperature while increasing logarithmically with increasing strain rate. Furthermore, the author shows that the mode II fracture energy decreases in the region 0.7 < T/Tg < 1.0. Guo and Sun (1998) study dynamic mode I delamination of a CFRP and note that the critical dynamic fracture energy is nearly constant and very close to the static fracture toughness of the composite. Kusaka et al. (1998) investigate the rate dependence of mode I fracture toughness of CFRP. They find a region of rate dependency for the fracture toughness GIc surrounded by regions of rate independency below and above this transition region. In and below the rate dependent region, two distinct fracture areas are observed. The first area is a whitened zone corresponding to initiation of unstable fracture. The second area is a dark area corresponding to propagation of unstable fracture. The fracture surface in the whitened zone shows highly damaged matrix resin and fibre/matrix debonding. The dark region shows smooth matrix failure with little fibre/matrix debonding. None of these studies catch the entire cohesive behaviour. An initial experimental study to determine the cohesive relation at varied temperatures is performed by Biel and Carlberger (2007). This study shows a strong temperature dependence of the cohesive law. Carlberger and Stigh (2007) show that the strain rate in an adhesive is amplified compared to the joined material, implying that dynamic adhesive material parameters are better determined in adhesive joints and not through bulk material testing. To this end, it is noticeable that recent developments of crashworthy adhesives are focusing on increasing fracture energy rather than ultimate strength. This new behaviour is created through alteration of the chemical mixture e.g. by adding elastomeric prepolymers, cf. Lutz and Schneider (2006). Such chemical modifications are likely to influence the temperature dependency of the adhesive, stressing the need for renewed temperature studies. Blackman et al. (2000) perform a critical review of the standard ISO 11343 for impact wedge-peel testing of adhesives. This method is used to evaluate crash resistant adhesives for the automobile industry. A correlation between the impact wedge peel test results and the fracture mechanically determined fracture energies, Jc, of the adhesives is performed. It is shown that a correlation exists. However, the material of the adherends influences the correlation. Moreover, the linear relation between the fracture energy and the evaluated load does not extrapolate through the origin. The value of this test to determine the rate dependence of cohesive laws is questionable. In this paper experiments are performed in order to achieve the cohesive relation for different temperatures and strain rates. The adhesive studied is an engineering epoxy, DOW-Betamate XW1044-3. The curing condition is 180ºC for 30 minutes. At room temperature the adhesive has a viscosity of 4 kPas (similar to toffee). It is preheated to about 50ºC before application. After curing, the glass transition temperature, Tg ≈ 90ºC. The plan of the paper is first to give a short background to the experimental methods and then to present the results from the temperature experiments and the experiments with varied strain rate. The paper ends with a summary of major results and conclusions.

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2. Experimental methods

Andersson and Biel (2006) and Leffler et al. (2007) show convenient methods to measure the cohesive properties of adhesive layers by use of simple test geometries and inverse formulas. In the present work, the double cantilever beam (DCB) specimen, cf. Fig. 2, is used for pure peel deformation and the end notched flexure (ENF) specimen, cf. Fig. 3, for pure shear.

The inverse formulas are based on the use of alternative integration paths in the evaluation of the path-independent J-integral,

( ), dx i i x

S

J Wn Tu S= −∫ (1)

Here, the integration path S can be chosen freely if the strain energy density W is independent of any explicit dependence of the x-coordinate, cf. Rice (1968). The outer unit normal to S is denoted n and the traction and displacement vectors are denoted T and u, respectively. Index notation is used with partial differentiation denoted by a comma and summation indicated by repeated indexes. Taking an integration path encircling the start of the adhesive layer gives,

d dJ w vσ τ= +∫ ∫ (2)

For the DCB-specimen, loading is antisymmetrical and no shear deformation is possible. Thus, only the first term on the r.h.s. of Eq. (2) remains; for the ENF-specimen, loading is symmetric and only second term on the r.h.s. of Eq. (2) remains. This relation shows that cohesive laws are closely related to fracture mechanics. At the final end of a fracture process, the cohesive stresses, σ and τ, are zero indicating that a crack has formed. After this moment, Eq. (2) shows a constant J. Thus, the maximum value of J can be identified with the fracture energy, Jc, and equals the area under the cohesive law. Now, taking an alternative integration path at the exterior boundary of the specimens gives,

p

p

2FJ

b

θ= ,

2 2s s s

2 3s s s s s

9 3

16 8

F a F vJ

E b H b H= + (3a,b)

for the DCB- and ENF-specimens, respectively. By equalling the alternative expressions for J, the evolution of J with the deformation of the adhesive layer can be measured experimentally. It may be noted that Young’s modulus of the adherends, Es, only enters Eq. (3b). Detailed derivations of Eqs (2) and (3a,b) are given by Andersson and Stigh (2004) and Alfredsson (2004). During a peel experiment (DCB), the applied force, Fp, the rotation of the loading point, θ, and the elongation w of the adhesive at the tip of the adhesive layer are measured. With Eq. (3a), the evolution of J during the experiment is derived. Using the alternative expression for J, cf. Eq. (2), and differentiate with respect to w, the cohesive law in peel is derived. During a shear experiment (ENF), the applied force, Fs, is measured together with the shear deformation at the tip of the adhesive layer, v. With Eq. (3b), the evolution of J is derived. Using Eq. (2) and differentiate with respect to v gives the cohesive law in shear. However, differentiation of experimental data elevates errors in the measured data. A useful method is to start with a least square adaption using a Prony-series to the J-data. Inspection of the success of the adaption to each experiment determines the number of terms to use in the

C6

Prony-series. The Prony-series is finally differentiated to produce the cohesive law, cf. Andersson and Biel (2006). Experimental data and a Prony-series adaption are shown in Fig. 4. As observed, the Prony-series adapts well to the experimental data.

Fig. 4 J vs. separation relations for the epoxy adhesive DOW Betamate XW1044-3 at T = 20°C, adhesive thickness 0.2 mm and εɺ = 1.7⋅10-3 s-1 (peel) εɺ = 5.0⋅10-3 s-1(shear). Thick curves correspond to raw data and thin curves to Prony series adaption. The J-separation relations give after differentiation the cohesive laws. An example is shown in Fig. 5.

Fig. 5 Constitutive behaviour in peel and shear for the engineering adhesive DOW Betamate XW1044-3 with a 0.4 mm layer thickness and εɺ = 2.7⋅⋅⋅⋅10-3 s-1 (peel) γɺ =

6.0⋅⋅⋅⋅10-4 s-1 (shear). Equations (2,3) are based on assuming linear elastic behaviour of the adherends during the experiments. This has to be checked both in the design of the specimens and in the evaluation of the experiments. It might need to be stressed that the adhesive is not assumed linearly elastic. An alternative expression to Eq. (3a), based on linear elastic fracture mechanics, is given by Tamuzs et al. (2003),

2/32p p p p

p p p p

3

2

F E IJ

E I b F

δ =

(4)

0

10

20

30

0 50 100 150

Peel Shear

deformation (µm)

stre

ss (

MPa

)

0

1000

2000

3000

0 50 100 150

separation (µm)

J (N

/m) Jshear

Jshear (Prony) Jpeel Jpeel (Prony)

C7

where Ep is Young’s modulus of the adherends, Ip is the adherend cross sectional moment of inertia and δp is the displacement of the loading points. In the derivation of this equation, the adhesive is assumed rigid.

Fig. 6 Evaluation of the alternative expression. A: Stress elongation relation B: Energy release rate. Equation (3a) and Eq. (4) give almost identical results for the DCB-specimen. Figure 6 shows results of a FE-simulation where the adherends are represented with 8800 beam elements and the adhesive layer with 8001 non-linear springs using realistic material and geometrical data. The cohesive law of the spring model is adapted to experimental results presented by Andersson and Biel (2006). The cohesive law used as input to the model is shown by the thick grey curve in Fig. 6A. Evaluating the simulation according to Eq.’s (3a) and (4) gives the energy release rates shown in Fig. 6B. Use of Eq. (2) gives the cohesive laws shown in Fig. 6A. Thus, evaluation of the simulation with Eq. (3a) gives the identical cohesive law as used as input. The alternative method gives a slightly larger peak stress and slightly larger fracture energy. This result is in accordance with Biel and Stigh (2008) where the error introduced by the use of Eq. (4) is shown to be less than 4 % during crack propagation. Thus, Eq. (4) can be used as an alternative to Eq. (3a). Determining the critical fracture energy, Jc, from experiments is in theory easy. The critical fracture energy is obtained by reading the J-value at the point on the J-w curve where the tangent becomes horizontal corresponding to zero stress, cf. Eq.’s (2) and (3). However, in some cases we find the J-curve to have a slightly positive tangent. This may be explained by e.g. non-homogeneous properties along the adhesive, pre-damaged adhesive at the tip or strain rate effects during loading. If the measured J-w curve does not have a zero tangent, a special method is used. A saw tooth approximation of the cohesive law is made by adapting the peak stress,σ , and the initial stiffness, kp, to the experimental results. An approximate guess of the fracture energy, Jc, is made by choosing a value of wc, cf. Fig. 7.

σ (M

Pa)

0 10 20 30 40 50 60 70 800

100

200

300

400

500

600

700

800

data1data2

0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

data1data2data3

Eq. (3a) Eq. (4)

FE-model Using Eq. (3a) Using Eq. (4)

J (N

/m)

w (µm) w (µm)

A B

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Fig. 7 Solid curve: cohesive law obtained by differentiation of the J-w relation. Dotted curve: a saw tooth approximation. Assuming the saw tooth model to be a good approximation of the cohesive law, a simulation of the F–δ relation is made using a closed form solution (Stigh, 1988). Now, if the simulated F–δ relation does not agree with the experimental data, the critical separation, wc, is changed until the saw tooth model returns an F–δ relation in good agreement with the measured F–δ relation. This method is valid even if the saw tooth model is not a good approximation of the cohesive law since the F–δ relation is only slightly sensitive to the details of the cohesive law in these cases, cf. Biel and Stigh (2008). 2.1 Study of influence of temperature

Here we study the influence of the temperature on the cohesive law in peel mode using the DCB-specimen. The temperature range is -40°C to 80°C in intervals of 20ºC. Special consideration is given the variation of fracture energy and peak stress with temperature.

A test machine, especially designed for testing of DCB-specimens, is used, cf. Fig. 8. The specimen is oriented vertically in the machine. Both crossheads are moving symmetrically around the centre of the machine. Two horizontally working ball screws powered by an electric motor control the motion. A load cell measures the force Fp. The angle, θ, at the loading point is measured with an incremental shaft encoder. The position of the crossheads is measured with a linear potentiometer. Two linear variable differential transducers (LVDT) are applied at the outside of the adherends to measure the elongation of the adhesive at the tip of the adhesive layer (not shown in Fig. 8).

The specimens are designed to secure elastic behaviour of the adherends. These are made of tool steel, Rigor Uddeholm, with a minimum yield-strength 500 MPa. The geometry is given by ap = 80 mm, bp= 5 mm, Hp = 6.6 mm, hp = 0.2 mm and Lp = 160 mm. In the preparation of the specimens, two plates are joined with the adhesive. The initial crack length is created by a 0.2 mm thick PTFE-film that also works as a distance giving the correct layer thickness. Before applying the adhesive, the plates are thoroughly cleaned with heptane and acetone. The adhesive is cured for 30 minutes at 180ºC. Slow cooling is allowed in order to minimise the influence of residual stresses. After curing, the plates are cut into specimens by use of band saw. Finally, the specimens are machined to specified dimensions.

σ (M

Pa)

50 100 0 0

5

10

15

20

w (µm)

kp

1

Jc

w1 wc

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Fig. 8 DCB test machine without LVDTs and specimen.

The specimens and the entire tensile test machine, except the control box and the computer for data acquisition, are placed in a climate chamber. The test equipment is calibrated after the desired temperature is stabilised. A specimen is mounted, and after an additional temperature check, the experiment is performed. The temperature is measured individually for each experiment using a thermocouple attached to a dummy specimen in the climate chamber. A temperature variation of ±2ºC is allowed.

2.2 Study of influence of strain rate

Both DCB- and ENF-specimens are used to determine cohesive parameters for the adhesive at moderately high strain rates.

A nominal strain (ε, γ) is taken as the deformation w (peel) or v (shear) at the adhesive tip, divided by the initial adhesive layer thickness, h. The thickness is measured individually on each specimen.

,w v

h hε γ= = . (5a,b)

Strain rate is taken as the time derivative of the measured strain. In order to avoid excessive noise in the differentiating operation of numerical data, a polynomial is first fitted to the experimental data and subsequently differentiated with respect to time. The number of terms used in the polynomial is determined by visually observing the raw data and the adapted polynomial.

For the lower strain rates in peel, the DCB-test machine described above is used and the experiments are evaluated using Eq. (3a), cf. Fig. 8. For higher strain rates in peel, the DCB-specimen is used with a 90° lever mechanism to achieve the desired displacement of the loading points, cf. Fig. 9. The 90° levers are mounted on a cross bar and placed in a servo-hydraulic testing machine in such a manner that the DCB-specimen is oriented vertically. The 90° levers have 1:1 ratio, enabling direct measurement of deflection, δp, and force, Fp, by the crosshead force and movement. The 90° levers are small and light and are considered insignificant for the loading rates in this study. For the ENF-experiments in shear, a clip gauge is used to measure the shear deformation v at the crack tip. The ENF specimen is

Gearbox Fast rate motor Slow rate motor Shaft encoder

Ball screw Load cell Crossheads Potentiomenter

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simply placed in a servo-hydraulic testing machine equipped with supports spaced at 1000 mm and mounted on stiff beams.

The same geometry is chosen for the DCB-specimens as in the temperature study. For the ENF-specimens, the design is more complicated; cf. Leffler et al. (2007). Three criteria have to be met: 1. the adherends should behave linearly elastic during the experiments; 2. the specimens should be mechanically stable; and 3. the distance between the crack tip and the loading point, Ls/2-as, should be large enough to capture the process zone. The geometry is given by: Ls=1000 mm, bs = 32.8 mm, and hs = 0.2 mm. Three different values for the adherend height Hs are used, 16.6 mm, 25.6 mm, and 32.6 mm. Two different values for the un-bonded length, as, are used, 300 mm and 350 mm.

Fig. 9 DCB-specimen placed vertically in a servo-hydraulic testing machine with a clip gauge attached to it for measurement of the adhesive tip deformation. The use of the DCB- and ENF-specimen under dynamic conditions has to be validated to assure that the measured results are not affected by inertia. This is done by simulating the behaviour of the specimens quasi-statically and dynamically using anticipated cohesive laws. The application of the DCB- and ENF-specimens is considered valid if there are no significant differences between the results of dynamic and quasi-static simulations. Figure 10 shows results of simulations for both the DCB- and ENF-specimens at different loading rates. For the DCB-specimen the results agree well with the quasi-static results for a crosshead speed up to 200 mm/s. For the ENF-specimen a crosshead speed up to 100 mm/s is considered valid.

A number of details have been omitted in this validation such as dynamic effects caused by inertia in mounted equipment, e.g. the force gauge and contact plate. To be on the safe side, a reduction of the maximum crosshead speed is done. In these experiments, the crosshead speed 50 mm/s ( pδɺ = 100 mm/s) is used for the DCB experiments, and the crosshead speed 100

mm/s ( sδɺ = 100 mm/s) is used for the ENF experiments. Since the total crosshead movement is only 2.5 mm in the DCB experiment, and 10 mm in the ENF experiment, the measurement time is less than 0.1 s.

crosshead

90° arm levers

DCB specimen

cross bar

pivot point

adhesive tip

2Fp

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Fig. 10 Validity simulations. Force - displacement curves for a) peel, b) shear. The data is filtered using the function smooth in Abaqus.

3. Experimental results and discussion

The strain rate varies considerably during an experiment. Due to the softening of the adhesive, a constant crosshead velocity results in an increasing strain rate during an experiment, cf. Fig. 11a and Leffler et al. (2007). It is not obvious at which point a representative strain rate is to be chosen. One relevant point to consider is the time the peak stress is passed. This occurs relatively early in an experiment. Another point is when half the fracture energy is consumed, which is a later point in time with higher strain rate. The latter point is preferred since it is believed to be less sensitive to the differentiation of the J-w data. It is also more close to an average of the strain rate during the test.

Fig. 11 (a) Strain rate variation with the deformation, w, during one DCB-experiment. Triangle marks peak stress, circle marks half the consumed fracture energy, Jc. (b) Strain rate at half consumed fracture energy varies with temperature due to machine friction.

0

1·10-3

2·10-3

3·10-3

-40 -20 0 20 40 60 80

T (°C)

stra

in r

ate

(s-1)

(b)

0 50 100 1500

2·10-3

4·10-3

6·10-3

8·10-3

10·10-3

12·10-3

14·10-3

w (µm)

stra

in r

ate (

s-1)

εɺ

εɺ at Jc/2

εɺ at peak stress

(a)

0

2

4

6

8

0 5 10 15

δ (mm)

Fs (

kN)

100 mm/s 200 mm/s

quasi-static

(b)

0

40

80

120

0 1 2 3 4 5

δ (mm)

Fp

(N)

500 mm/s 200 mm/s

1000 mm/s

(a)

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3.1 Influence of temperature

The DCB-experiments are performed with a nominally constant crosshead velocity, pδɺ = 10

µm/s. Due to friction in the machinery, the crosshead velocity varies with the temperature. This is indicated in Fig. 11b. A total of 49 experiments are performed with about seven experiments at each temperature, cf. Table 1.

Temperature -40ºC -20ºC 0ºC 20ºC 40ºC 60ºC 80ºC Number of specimens 5 6 7 7 8 8 8

Table 1 Number of evaluated specimens at each temperature.

For each experiment a Prony series is adapted. The number of terms used is between 15 and 25; the number is determined by visually inspecting the adaption. In Fig. 12, representative J vs. w and σ vs. w curves are shown; J is evaluated using Eq. (3a). As shown in Fig. 12b, the peak stress and the critical elongation are strongly dependent on the temperature. The elongation at the peak stress occurs at about 2-6 µm for all experiments.

Fig. 12 (a) J as a function of w; (b) σ as a function of w.

The average value and standard deviation of the fracture energy and the peak stress at each temperature are given in Fig. 13.

0 10 20 30 40 50 60 70 80 90 100 1100

100

200

300

400

500

600

700

800

900

1000

0 10 20 30 40 50 60 70 80 90 100 1100

10

20

30

40

50

60

T = -40ºC

T = 0ºC

T = -20ºC

T = 20ºC

T = 40ºC T = 60ºC T = 80ºC

w (µm)

σ (M

Pa)

w (µm)

J (N

/m)

T = -40ºC

T = 0ºC

T = -20ºC

T = 20ºC

T = 40ºC T = 60ºC T = 80ºC

C13

Fig. 13 (a) Jc vs. temperature; (b) σ vs. temperature.

At lower temperatures, the fracture energy seems independent of the temperature. For the experiments performed at 60ºC and 80ºC the fracture energy is slightly lower indicating the vicinity to the glass transition temperature. The maximum temperature for automotive applications is around 120ºC in areas close to the roof and windscreen, but in the areas where crash energy is absorbed; the maximum temperature reaches around 70°C. Figure 13b shows how the peak stress, σ of the adhesive varies with temperature. The peak stress decreases monotonically with increasing temperature.

Figure 14 shows two fracture surfaces placed side-by-side and photographed. The fracture surface at 80ºC shows more stress-whitening than the fracture surface at -40ºC. The variation of the stress-whitening is gradual and if all fracture surfaces are placed consecutively side-by-side, the variation would not be apparent.

Fig. 14 Fracture surfaces from experiments performed at -40ºC and 80ºC.

3.2 Influence of strain rate

-40 ºC 80 ºC

-40 0 40 80 0

20

40

60

T (°C)

σ (M

Pa)

±1 s

(b)

-40 0 40 80 0

200

400

600

800

1000

1200

T (°C)

J c (

N/m

)

±1 s

(a)

C14

This part of the work includes both peel (DCB) and shear experiments (ENF). The experiments are performed at room temperature. A total of 38 experiments are performed with at least three experiments at each strain rate, cf. Table 2. Specimen type DCB ENF

εɺ (s-1) 1.5·10-4 1.4·10-3 4.7·10-3 4.2·10-2 11.3 2.9·10-3 5.0·10-3 3.0

Jc (N/m) 780 860 800 920 1200 3300 1900 2700

ˆ ˆ,σ τ (MPa) 25 23 23 28 33 28 22 64

Number of specimens

3 7 4 4 3 3 8 6

Table 2 Number of evaluated test specimens at each strain rate. Values for εɺ , Jc and peak stress are averaged for each group.

The DCB experiments are evaluated with Eq. (3a) up to strain rate 4.2·10-2 s-1, while the DCB experiments at the higher strain rate are evaluated with Eq. (4). The crosshead speed 50 mm/s is used in the high-speed DCB-experiments. It should be noticed that the validation procedure of the dynamic DCB study showed a crosshead speeds up to about 200 mm/s is possible without appreciable effects of inertia, cf. Fig. 10. In Fig. 15, the energy release rate, J, is plotted as a function of the displacement w for DCB-experiments and v for the ENF-experiments, at different strain rates.

Fig. 15 Experimentally determined energy release rate, J vs. deformation for varying strain rates. (a) peel, (b) shear. The fracture energy and peak stress varies with strain rate as shown in Fig. 16a and b, respectively.

0

1000

2000

3000

0 50 100 150

v (µm)

J (N

/m)

3.0 s-1

2.9·10-3 s-1

(b)

0 10 20 30 40 50 60 70 0

200

400

600

800

1000

1200

w (µm)

J (N

/m)

εɺ =11 s-1 εɺ =0.042 s-1 εɺ =0.0047 s-1

εɺ =0.0014 s-1

εɺ =0.00015 s-1

(a)

C15

Fig. 16 (a) fracture energy vs. strain rate, (b) peak stress vs. strain rate for peel and shear. *Experiments marked are performed with constant shear deformation rate, vɺ . The strain rate study spans five decades, from 10-4 s-1 to 101 s-1. The strain rate dependence seems linear in a lin-log plot for peel; both Jc and σ increase with increasing strain rate. For shear loading, the peak stress increases with strain rate while the fracture energy decreases. The shear study includes experiments performed with constant shear deformation rate. Constant deformation rate results in accelerating strain rate. Thus, a constant strain rate would require a decelerating deformation rate. This is believed to be the reason for the large difference in fracture energy between the ENF experiments performed with constant deformation rate (γɺ= 2.9·10-3 s-1) and the ENF experiments performed with constant strain rate (γɺ= 5.0·10-3 s-1). 4. Discussion

Both variations in temperature and strain rate give small effects on the fracture energy. The fracture energy decreases about 10 % with an increase in temperature from -40oC to +80oC in peel. Or, if the temperature is related to the glass transition temperature, the temperature span corresponds to an increase in relative temperature from about 0.64 to 0.97, i.e. an increase with about 50 %. A similar increase in fracture energy is noted with an increase in strain rate with five decades in peel. A limited effect of strain rate on fracture energy is also observed in shear. However, in this case, the fracture energy decreases with increasing strain rate. These results can be viewed in the light of analyses performed by Salomonsson and Andersson (2008). In a detailed model of the adhesive layer, it is revealed that peel loading results in no plastic deformation of the adhesive. On the other hand, for shear dominated loading, large scale plasticity occurs in the adhesive. A recent experimental study confirms minor plastic deformation in peel (Biel, 2008). It is plausible to assume that the plastic strain is rate dependent and, at high strain rates in shear, less time is available to give large plastic strain. Thus, the decrease in fracture energy with strain rate in shear is a result of strain rate dependent plasticity of the adhesive; at peel, only minor plasticity occurs and the strain rate dependence is small. Another support for this idea is given by the results of Chai (2004). In his study of the influence of temperature on the fracture energy in shear, he shows a much larger influence of temperature than we find in peel. That is, if rate and temperature dependent plasticity is the source of high fracture energy in shear, we would expect a larger temperature influence in shear than in peel. The influence of strain rate on the fracture energy in peel appears to be similar as the influence of crack speed on the fracture energy in modus I for

0

10

20

30

40

50

60

70

80

10-4 10-3 10-2 10-1 1 101 102

strain rate (s-1)

pea

k st

ress

(M

Pa)

peel shear shear* lin. appr.

(b)

peel shear shear* lin. appr.

0

1000

2000

3000

10-4 10-3 10-2 10-1 1 101 102

strain rate (s-1)

J c (

N/m

) (a)

C16

delamination of a CFRP reported by Guo and Sun (1998). Though, in their case the fracture energy is only about 20 % of the fracture energy of the present adhesive.

The peak stress in peel and shear is influenced by strain rate. The influence is much larger in shear than in peel. This is in accordance with the rate dependent plasticity model discussed above. When comparing our results with Chai (2004), we find a much larger influence of strain rate on the peak stress in shear than he finds on the influence on the yield strength. Apart from the difference between the peak stress and the yield strength, Chai (2004) studies thinner adhesive layers as compared to the present study. It might be argued that a thicker adhesive layer results in more rate dependent plasticity. Thus, a larger influence of strain rate on the peak stress for a thicker adhesive layer.

Increasing temperature gives a smaller peak stress in peel. Moreover, an increasing strain rate gives larger peak stresses both in peel and shear. The result is in accordance with a time dependent plasticity model of the adhesive. This has also been deduced from other experiments with the same adhesive (Andersson and Biel, 2006). The result can in some respect be compared to findings by Chai (2004). In his study of an epoxy adhesive, the yield strength in shear decreases by about 90 % with an increase in relative temperature from 0.7 to 1.0 T/Tg. In our study the maximum peel strength decreases by about 80 % in the temperature range 0.64 to 0.97 T/Tg. In many load cases, the fracture energy is the most important property of the adhesive. However, the peak stress gives in some cases large influences on the structural behaviour, cf. e.g. Andersson and Biel (2006).

The strain rate study covers five decades in peel (DCB) and four decades in shear (ENF). Both the fracture energy and the peak stress in peel increase slightly with strain rate. The results indicate

Ic Ic00

00

log

ˆ ˆ log

JJ J

σ

εε

εσ σε

= =

ɺ

ɺ

ɺ

ɺ

(6a,b)

with JIc0 = 34 N/m, 0Jε =ɺ 3.0 10-14 s-1, 0σ = 0.9 MPa and 0σε =ɺ 3.8 10-15 s-1. These equations

are dashed in Figs. 16a and b.

The fracture energy in peel is significantly smaller than that in shear. Moreover, the fracture energy in peel increases slightly with strain rate. By this reason, it may be argued that simulations using quasi-static data from room temperature are conservative. A dynamic simulation using quasi-static data for the cohesive law of the adhesive, will thus under-estimate the joint strength.

Often when studying the viscoelasticity of polymers, a time-temperature shift is discovered. That is, the influence of a change in temperature is the same as a shifting of the time scale. With polymeric adhesives such a time-temperature shift would be very profitable. If this shift could be secured, the extremely expensive and complicated dynamic experiments reported here could be exchanged with simpler quasistatic experiments in a climate chamber. Indications of such properties have been reported (cf. Chai, 2004). The present study indicates however no such relation for the present epoxy adhesive.

C17

5. Conclusions

Experiments have been performed to evaluate the temperature dependence and the strain rate dependence of the cohesive laws for an engineering epoxy adhesive. The relevant temperature range for vehicle usage conditions is covered. It is shown that the fracture energy, i.e. the area under the cohesive law is virtually independent of temperature and strain rate. At higher temperatures, Jc decreases slightly with T; at the highest temperature, 80ºC, Jc has decreased about 10 % in peel. The fracture energy in peel increases slightly with increasing strain rate while it decreases slightly in shear. The peak stress decreases monotonically for all temperatures considered and increases with strain rate. Moreover, the stress whitening increases gradually with increasing temperature.

The study shows that the quasi-static methods for evaluation of the DCB- and ENF-specimens can be used at moderately large strain rates; with the present geometries up to 40 s-1 for peel (DCB) and 6 s-1 for shear (ENF). It also appears possible to optimize the specimens for even higher strain rates. It is also shown that Eq. (4) can be used instead of Eq. (3a) with only minor error. Equation (4) is in some cases easier to use since there is no need to measure the rotation of the loading point. It is regrettable to conclude that a time-temperature shift does not work with the present adhesive. Acknowledgements The authors wish to thank Professor Ulf Stigh for fruitful discussions and help during the work presented in this paper. Special thanks are also directed to Stefan Zomborcsevics for help with manufacturing of the specimens and Mattias Widmark at Volvo Material Technology for performing the high strain rate experiments. The authors also thank the Swedish Consortium for Crashworthiness for funding this project.

References

Alfredsson KS, Biel A, Leffler K. (2003) An experimental method to determine the complete stress-deformation relation for a structural adhesive layer loaded in shear, In Proceedings of the 9th International Conference on The Mechanical Behaviour of Materials, Geneva, Switzerland 2002

Alfredsson KS (2004) On the instantaneous energy release rate of the end-notch flexure adhesive joint specimen. International Journal of Solids and Structures 41:4787–4807

Andersson T, Biel A (2006) On the effective constitutive properties of a thin adhesive layer loaded in peel. International Journal of Fracture 141: 227-246

Andersson T, Stigh U (2004) The stress-elongation relation for an adhesive layer loaded in modus I using equilibrium of energetic forces. International Journal of Solids and Structures 41:413-434

Ashcroft IA, Hughes DJ, Shaw SJ (2001) Mode I fracture of epoxy bonded composites joints: 1. Quasi-static loading. International Journal of Adhesion and Adhesives 21:87-99

Biel A, Carlberger T (2007) Influence of temperature on cohesive parameters for adhesives. In: Sørensen BF, Mikelsen LP, Lilholt H, Goutianos S, Abdul-Mahdi FS (Eds) Procceedings of 28th Risø international symposium on materials science 2007

Biel A, Stigh U (2007) An analysis of the evaluation of the fracture energy using the DCB-specimen. Archives of Mechanics 59:311-327

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Biel A, Stigh, U (2008) Effects of constitutive parameters on the accuracy of measured fracture energy using the DCB-specimen. Engineering Fracture Mechanics, Engineering Fracture Mechanics 75, 2968-2983

Biel A (2008) Cohesive laws for adhesives at repeated loading – an experimental method. Manuscript in preparation.

Blackman BRK, Kinloch AJ, Taylor AC, Wang Y (2000) The impact wedge-peel performance of structural adhesives. Journal of Materials Science 35:1867-1884

Blackman BRK, Hadavinia H, Kinloch AJ, Williams JG. (2003) The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. International Journal of Fracture 119:25-46

Carlberger T, Stigh U (2007) An explicit FE-model of impact fracture in an adhesive joint. Engineering Fracture Mechanics 74:2247-2262

Chai H (2004) The effects of bond thickness, rate and temperature on the deformation and fracture of structural adhesives under shear loading. International Journal of Fracture 130:497-515

DuBois PA (2005) Engineering consultant. Private communication

Guo C, Sun CT (1998) Dynamic Mode-I crack-propagation in a Carbon / Epoxy Composite. Composites Science and Technology 58:1405-1410

Högberg JL, Sørensen BF, Stigh U (2007) Constitutive behaviour of mixed mode loaded adhesive layer. International Journal of Solids and Structures 44:8335-8354

Kinloch AJ (1987) Adhesion and Adhesives – Science and Technology. Chapman and Hall, London, 1987

Kusaka T, Hojo M, Mai Y, Kurokawa T, Nojima T, Ochiai S (1998) Rate Dependence of Mode-I Fracture Behaviour in Carbon-fibre/Epoxy Composite Laminates. Composites Science and Technology 58:591-602

Leffler K, Alfredsson KS, Stigh U (2007) Shear behaviour of adhesive layers. International Journal of Solids and Structures 44:530-545

Lutz A, Schneider D (2006) Toughened epoxy adhesive composition. USPTO Applicaton #: 20060276601 - Class: 525528000 (USPTO), Dow Chemical Company - Midland, MI, US

Olsson P, Stigh U (1989) On the determination of the constitutive properties of the interphase layers – an exact solution. International Journal of Fracture 41:71-76

Rice JR (1968) A Path Independent integral and the Approximate Analysis of Strain Concentrations by Notches and Cracks. Journal of Applied Mechanics 33:379-385

Salomonsson K, Andersson T (2008) Modeling and parameter calibration of an adhesive layer at the meso level. Mechanics of Materials, 40: 48-65

Salomonsson K, Stigh U (2008) An adhesive interphase element for structural analyses. International Journal for Numerical Methods in Engineering. DOI: 10.1002/nme.2333. To appear.

Schmidt P (2007) Computational Models of Adhesively Bonded Joints. PhD thesis, Linköping University

Suo Z, Bao G, Fan B (1992) Delamination R-curve phenomena due to damage. Journal of the Mechanics and Physics of Solids 40:1-16

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Stigh U (1988) Damage and crack growth analysis of the double cantilever beam specimen. International Journal of Fracture 37:R13-18

Stigh U, Andersson T (2000) An experimental method to determine the complete stress-elongatioin relation for a structural adhesive layer loaded in peel, In Fracture of Polymers, Composites and Adhesives (Eds. Williams J.G. and Pavan A.), ESIS publication 27, pp. 297-306

Sørensen BF (2002) Cohesive law and notch sensitivity of adhesive joints. Acta Materialia 50:1053-1061

Sørensen BF, Jørgensen K, Jacobsen TK, Østergaard RC (2006) DCB-specimen loaded with uneven bending moments. International Journal of Fracture 141:163–176

Tamuzs V, Tarasovs S, Vilks U (2003) Delamination properties of translaminar-reinforced composites. Composites Science and Technology 63:1423-1431

Yang QD, Thouless MD (2001) Mixed-mode fracture analyses of plastically deforming adhesive joints. International Journal of Fracture 110:175-187

Paper D

Influence of layer thickness on cohesive properties of an epoxy-based adhesive –an experimental study

D1

Influence of layer thickness on cohesive properties of an epoxy-based adhesive

– an experimental study

T. Carlberger

SAAB Automobile AB, SWEDEN

Abstract The influence of thickness of the adhesive layer on the cohesive parameters is studied in peel and shear mode for the engineering adhesive Dow Betamate XW1044-3. The adhesive thickness influences the fracture energy strongly. The fracture energy in peel mode increases monotonically as the thickness is increased from 0.1 mm to about 1.0 mm. At 1.5 mm, the fracture energy is slightly lower than for 1.0 mm, indicating that the fracture energy has a maximum between 1.0 mm and 1.5 mm. In shear mode, the thickness dependence is not as strong, but an increase of the fracture energy with increasing thickness is evident. For both peel mode and shear mode, the peak stress shows little thickness dependence. A slight decrease in peak stress with increasing adhesive thickness is indicated.

Keywords: Adhesive joining; Thickness dependence; Cohesive zone model, Experimental, Fracture

Nomenclature a initial crack length b specimen width ∅ diameter ε,γ peel strain, shear strain

,ε γɺ ɺ strain rate (peel, shear)

F force δ deflection at loading point E Young’s modulus h adhesive layer thickness H adherend height I moment of inertia J energy release rate Jc critical energy release rate, fracture energy KIC fracture toughness in mode I L specimen length n normal vector

ˆ,σ σ peel stress, peak peel stress

S integration path ˆ,τ τ shear stress, peak shear stress

T traction vector θ rotation of the adherend at the loading point u displacement vector v shear deformation at adhesive tip W strain energy density w peel deformation at adhesive tip x length coordinate starting at adhesive tip

D2

1. Introduction Emission of carbon dioxide is driving the automotive industry to find methods of reducing fuel consumption. An efficient way to do this is to reduce the weight of the vehicle; the car body structure accounts for approximately one fifth of the total vehicle weight. This can be minimized by the use of multi-material structures leading to a joining challenge. In the car industry, spot welding is the traditional joining technique for mono-material joining. That is, to join steel to steel or to join aluminium to aluminium, etc. Although this method has many advantages, it is essentially limited to mono-material joints. Adhesive joining is a well-established joining technique in many industrial areas, but in the car industry, until recently this technique has almost exclusively been used in load bearing structures for extreme carbon fibre reinforced race cars. Recent development of adhesive joining techniques has enabled mass production of adhesively bonded car body structures. With adhesive joining, similar or dissimilar materials may not only be joined, improvements of both stiffness and strength are achieved simultaneously. During product development, computer simulations are required to give reliable information for the engineers. Efforts are made to reduce testing to a minimum, due to the slow and expensive process of manufacturing test objects and evaluating tests. The testing, that is performed today, is mainly done to fulfil legal requirements and requirements from independent organisations as e.g. euro NCAP. Computer simulations are much more time and cost efficient, on the essential condition that the simulations are reliable. Crash simulation is done by use of the explicit FE-method, cf. e.g. [1]. A possibility, for modelling adhesives with finite elements, is given by the adhesive layer theory, cf. e.g. [2] and a cohesive zone model. The parameters of the cohesive zone model are determined by performing physical tests. In [3] the authors present an experimental method for determining the cohesive zone properties for the adhesive, based on equilibrium of energetic forces and the adhesive layer theory. The method is shown to be capable of predicting the strength of the adhesive with good precision. Determining material parameters and validating simulation models is essential to the reliability of this methodology, cf. [4]. In [5,6] cohesive parameters are determined for different modes of deformation and for one nominal adhesive thickness. The difficulty of obtaining a fixed adhesive thickness in a car body structure, gives rise to the demand of understanding how the adhesive thickness influence the strength properties. The car body structure is assembled in the manufacturing process, and subsequently painted. The paint and the adhesive are hardened together in the paint oven. Until this moment, the adhesive has very limited strength. To secure the car body integrity prior to hardening, some means of additional joining must be provided. This may be riveting or bolting for multi-material joints; for mono-material joints, spot welding is often used. The car body structure is built up by joining large preformed parts mostly consisting of sheet metal. Manufacturing tolerances give rise to uneven gap width that should be taken up by the adhesive. The mechanical joints will keep the parts firmly together but will allow the gap width to vary between the fasteners.

A promising method, when performing simulation of adhesive joints, is the use of cohesive finite elements, cf. e.g. [7]. The cohesive element presented by [8] is an element for efficient modelling and simulation of adhesive joints between shell elements. This special element uses a cohesive zone model for determining the response on the connected shell elements. The fracture characteristics of the adhesive are known to change with the adhesive thickness, cf. e.g. [9-11]. Thus, the thickness influence of the parameters of the cohesive zone parameters must be determined. The scope of this work is to determine the thickness influence on the cohesive law for an epoxy-based structural adhesive.

In [9] the simple peel test for flexible substrates is used together with the T-peel test for rubbery

adhesives of thickness between 0.2 mm and 6 mm. It is shown that the fracture energy, Jc, increases with increasing adhesive thickness, h, until the thickness reaches 1 mm, for a polyethy1ene terephthalate adhesive. For values above 1 mm, the fracture energy is not affected by the increase in thickness. In [11] shear experiments are performed with the napkin ring specimen and the end notched flexure (ENF) specimen using a toughened epoxy-based adhesive. The author finds a logarithmically increasing ultimate shear strain with decreasing bond thickness and monotonically increasing normalised mode II fracture energy with decreasing bond thickness under a certain threshold value. Above this threshold value, the mode II fracture energy is insensitive to bond thickness. In [12] experiments and numerical studies of effects of the bond thickness are performed using the compact tension specimen. They find a maximum KIC for the bond thickness 0.8 mm. The reason for this, they report, is that the crack tip stress field is affected by the adhesive thickness through the restriction of plastic deformation. In [13] a double cantilever beam

D3

(DCB) specimen is used to determine the effects of the bond thickness on Jc for a rubber-modified epoxy adhesive. They find a linearly increasing Jc extending to a maximum value followed by a rapid asymptotic decline towards a value representing the bulk adhesive material, with increasing adhesive thickness. The adhesive thickness, for which the maximum of Jc occurs, is shown to be slightly less than 1 mm, which is in accordance with the findings of [12]. In [14] a lap shear joint with an L-shaped profile to stiffen the joint is tested for three different adhesive thicknesses. Adherends are manufactured of glass-fibre-reinforced vinylester composite laminates by resin infusion and bonded with an epoxy adhesive. The fracture load decreases monotonically with increasing adhesive thickness. The lap-shear test does generally produce a mixed mode stress field, and in this case, with the stiffener un-symmetrically placed, the stress field is non-trivial.

In the present work, cohesive zone parameters are determined for peel (corresponding to mode I in fracture mechanics) and shear (mode II) for varying adhesive thickness. The specimens used are the DCB and ENF specimens. The adhesive used in this work is an epoxy-based structural adhesive (Dow Betamate 1044-3). 2. Theoretical background and design of experiments According to the adhesive layer theory, cf. [2], deformation of the adhesive is dominated by two deformation modes, namely mode I (peel) and mode II (shear), cf. Fig 1.

Fig. 1. Deformation modes of the adhesive layer with thickness h: peel, w, and shear, v. Conjugated stress components σ and τ.

In the adhesive layer theory, it is assumed that the adhesive layer is thin and flexible compared to the adherends. With metal adherends, Young’s modulus of epoxy adhesives is typically less than 5 % of that of that of the adherends. For a nominal adhesive thickness h = 0.2 mm, the cohesive laws in peel, σ(w), and in shear, τ(v), for the epoxy-based adhesive, Dow Betamate XW1044-3, are presented in Fig. 2.

Fig. 2. Cohesive laws in peel and shear for the engineering adhesive DOW Betamate XW1044-3 with a 0.2 mm layer thickness.

0

10

20

30

0 50 100 150 separation (µm)

stre

ss (

MP

a)

shear peel

v τ

τ σ

w

σ

h

D4

Fig. 3. Double cantilever beam (DCB) specimen. The un-bonded part of the specimen can be considered as a crack. That is, ap is the crack length (ap = 80 mm). Adherend width, bp= 5 mm, adherend height, Hp = 6.6 mm, specimen length, Lp = 160 mm, adhesive thickness, hp. Deformation is exaggerated for clarity.

Fig. 4. ENF specimen. Un-bonded length, as = 300 mm, length between supports, Ls=1000 mm, adherend height, Hs = 16 mm, adherend width, bs = 32.8 mm, and adhesive thickness hs.

In the present work, the DCB specimen, cf. Fig 3, is used for pure peel deformation. In [15], Olsson and Stigh develop an inverse method for pure peel using the DCB specimen. For pure shear, [16] Alfredsson develops an inverse method using the ENF specimen, cf. Fig. 4. The inverse formulas are based on the use of alternative integration paths for the evaluation of the J-integral

( ), dx i i xS

J Wn T u S= −∫ (1)

Here, the integration path S can be chosen freely if the strain energy density W is independent of any explicit dependence of the x-coordinate, cf. [17]. The outer unit normal to S is denoted n and the traction vector and displacement vectors are denoted T and u, respectively. Index notation is used with partial differentiation denoted by a comma and summation indicated by repeated indexes. Taking an integration path encircling the start of the adhesive layer gives

Fs

δs

as

Ls

Ls/2

h s

Hs

v

x

δ p+

hp

+ H

p hp + w

hp

Hp

θ

Fp

Lp

Fp

θ

ap

x

D5

d dJ w vσ τ= +∫ ∫ (2)

for peel deformation and shear deformation. These relations show that cohesive laws are closely related to fracture mechanics. At the end of a fracture process, the cohesive stress, σ and τ, are zero indicating that a crack has formed. After this moment, Eq. (2) shows constant J. Thus, the maximum value of J is identified with the fracture energy, Jc. Moreover, Jc equals the area under the cohesive law.

Now, taking an alternative integration path at the exterior boundary of the specimens give

p

p

2FJ

b

θ= ,

2 2s s s

2 3s ss s s

9 3

816

F a F vJ

b HE b H= + (3a,b)

for the DCB- and ENF-specimens, respectively. Subscripts p and s indicates variables connected to peel (DCB) and shear (ENF) experiments, respectively. The geometries of the specimens are given by the height of the adherends, H, the width, b, and the crack length, a. Young’s modulus is denoted E and only enters Eq. (3b). During a peel experiment (DCB), the applied force, F, and the rotation of the loading point, θ, are measured as functions of the elongation w of the adhesive at the tip of the adhesive layer. During a shear experiment (ENF), the applied force, F, is measured together with the shear deformation at the tip of the adhesive layer, v. By equalling the alternative expressions for J, the evolution of J with the deformation of the adhesive layer is derived from the measurements. Differentiation with respect to deformation, w or v, gives the cohesive law, cf. Eq. (2). However, differentiation of experimental data elevates errors in the measured data. A useful method is to start with a least square adaption of a Prony-series to the J-data. Inspection of the success of the adaption to each experiment determines the number of terms to use in the Prony-series. Finally, the Prony-series is differentiated to produce the cohesive law, cf. [3].

Equations (2,3) are based on assuming linear elastic behaviour of the adherends during the experiments. This has to be checked both in the design of the specimens and in the evaluation of the experiments. It might need to be stressed that the adhesive is not assumed linearly elastic. An alternative expression to Eq. (3a) based on linear fracture mechanics is given by [18].

2/32p p p p

p p p p

3

2

F E IJ

E I b F

δ =

(4)

where Ep is Young’s modulus of the adherends, Ip is the adherend cross sectional moment of inertia and δp is the displacement of the loading points. In the derivation of this equation, the adhesive is assumed rigid. In [19] the authors show that Eqs. (3a) and (4) give almost identical results for the DCB specimen geometry in this case. The error in fracture energy in using Eq. (4) is typically less than 4 %.

These inverse methods have recently proven successful in determining the cohesive zone parameters for adhesives, cf. e.g. [7] and [6]. The cohesive laws presented in Fig. 2 are determined by differentiation of the J-separation relations given in Fig. 5.

D6

Fig. 5. J – deformation relation. Epoxy adhesive DOW Betamate XW1044-3 at T = 20°C, adhesive thickness 0.2 mm, εɺ = 1.7 ·10-3 s-1 (peel) and γɺ = 5.0 ·10-3 s-1 (shear).

The critical fracture energy is the energy released per unit crack surface when the crack starts to

propagate. In theory, with homogeneous adhesive properties along the adhesive this occurs when the derivative of J with respect to the adhesive deformation at the adhesive tip equals zero, cf. Eq. (2). Although theoretically simple, this method to measure Jc for an adhesive may encounter practical problems. In some experiments no horizontal asymptote is found in the J-curves. A straightforward method to identify the fracture energy is to observe the crack tip region and determine the moment of crack propagation visually. However, the fracture process of engineering adhesives involves nucleation, growth and coalescing of micro cracks. That is, it is virtually impossible to identify the moment of crack propagation by this method, cf. [6]. An alternative method used in this paper is described in the Appendix.

The material in the adherends in both the DCB- and ENF specimens is Rigor Uddeholm tool steel with a minimum yield-strength 500 MPa. With the chosen geometries, this ensures elastic behaviour of the adherends throughout the experiments. The initial crack length is created by a 0.2 mm thick PTFE-film that also works as a distance giving the correct layer thickness. Prior to applying the adhesive, the adherends are thoroughly cleaned with heptane and acetone. The adhesive is a toughened engineering epoxy, DOW-Betamate XW1044-3. The curing condition is 180ºC for 30 minutes, corresponding to the hardening process for the paint hardening process in the automotive manufacturing. At room temperature, the uncured adhesive has a viscosity of 4 kPas. The DCB specimens are produced from plates of the adherend material that are bonded and subsequently cured in the curing oven. After curing is achieved, slow cooling is allowed in order to minimise the influence of residual stresses. Next, the DCB plates are cut into specimens with band saw. Finally, the DCB specimens are machined to specified dimensions. The ENF specimens are bonded individually. The single added process in the manufacturing of the ENF specimens is a removal of excess adhesive after curing. The geometries are given by ap = 80 mm, bp= 5 mm, Hp = 6.6 mm, and Lp = 160 mm, for the DCB-specimens and as = 300 mm, Ls=1000 mm, Hs = 16 mm, and bs = 32.8 mm, for the ENF-specimens. The following nominal adhesive layer thicknesses are considered: hp = 0.1, 02, 0.4, 0.6, 0.8, 1.0, 1.6 mm (DCB) and hs = 0.1, 02, 0.4, 0.6, 0.8, 1.0 mm (ENF). The thickness is measured individually for each specimen.

3. Test setup 3.1 DCB experiments A specially designed test machine is used for the testing of DCB-specimens, cf. Fig. 6. The DCB specimen is oriented vertically in the centre of the machine. Both crossheads move symmetrically around the centre of the machine. Two horizontally working ball screws powered by an electric motor control the displacement, δp. A load cell measures the force Fp. The rotation, θ, at the loading point is measured with an incremental shaft encoder. The position of the crossheads is measured with a linear potentiometer. Two linear variable differential transducers (LVDT) are applied at the outside of the adherends to measure the elongation of the adhesive layer at the tip of the adhesive layer (not shown in Fig. 6).

0

500

1000

1500

2000

2500

0 50 100 150 separation (µm)

J (N

/m)

Jshear Jpeel (Prony) Jpeel Jpeel (Prony)

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Fig. 6. DCB test machine without LVDT:s and specimen.

The crosshead speed is constant during an experiment and chosen such that no effects of inertia will affect the results. The test is quasi-static. 3.2 ENF experiments The ENF experiments are performed using a tensile test machine LLOYD LR10k plus, cf. Fig. 7.

Fig. 7. Tensile testing machine LLOYD LR10K plus, used for ENF experiments. An ENF specimen is mounted in the machine.

Gearbox Fast rate motor Slow rate motor Shaft encoder

Ball screw Load cell Crossheads Potentiometer

load cell

specimen

supports

D8

The supports are solid ∅ 20 mm cylinders placed on stiff I-beams at ±500 mm distance symmetrically from the machine centre line. The measurement of the adhesive tip deformation is done by an LVDT fixed to the adherends by a mechanical attachment (not shown in Fig. 7). The force, Fs, is measured with a load cell mounted on the crosshead, and the displacement, δs, is given by the crosshead position 4. Experimental results 4.1 DCB experiments Though the crosshead speed is constant, the strain rate varies during a DCB experiment, cf. Fig. 8a. When the adhesive softens at the tip of the adhesive layer, the strain rate accelerates. Since the adhesive is polymer based, and polymers are known to be strain rate dependent, it is plausible to assume a strain rate depending constitutive law. The strain rate measured at the moment the peak stress occurs is relatively low, cf. triangle in Fig. 8a. The strain rate at the point when half the fracture energy is consumed is higher and may be considered as a mean value of the strain rate over the duration of the experiment, cf. circle in Fig. 8a. Thus, the strain rate is measured in this way throughout the paper. The strain rate in each experiment is plotted against the adhesive thickness, h, in Fig. 8b.

Fig. 8. (a) Strain rate variation with the deformation, w, during one experiment. Triangle marks peak stress, circle marks half the consumed fracture energy, Jc. (b) strain rate variation with adhesive thickness, h.

The crosshead speed is the same for all thicknesses except 0.2 mm, where it is lower. The strain rate measurement shows inverse thickness dependence. One significant difference is noticed at thickness 0.2 mm. These experiments were from a different batch, tested previously using the crosshead speed 0.61 mm/min, cf. [6]. All other experiments are performed with the crosshead speed 1.80 mm/min. In a parallel paper, the strain rate dependence in peel for the adhesive Dow Betamate XW1044-3 is studied, cf. [20]. There is a slight strain rate dependence of both the fracture energy Jc and the peak stress, but it does not change the results noticeably. This is likely to be the reason why, in Fig. 9, the fracture energy at h = 0.2 mm does not differ significantly from the overall trend.

0

2

4

6

8

10

0 0.5 1 1.5h (mm)

stra

in r

ate

(s-1)

x10-3 b

0 50 100 150 200 250 3000

2

4

6

8

10

12

w (µm)

stra

in r

ate

(s-1)

a x10-3

D9

Fig. 9. (a) Fracture energy, (b) Peak stress vs. adhesive thickness, h, for peel mode.

The results in Fig. 9a show increasing fracture energy with increasing adhesive thickness up to about

1.0 mm, and then declining fracture energy above this value. These results confirm the findings in [9,12,13] for the adhesive thicknesses between 0.2 and 1.6 mm, although in [9] a different adhesive is used. The fracture energy for the adhesive thickness 0.1 mm fits in an extrapolated manner between 0.2 mm and zero thickness. Figure 9b shows the peak peel stress vs. the adhesive thickness, h. As shown, the peak stress appears independent of the layer thickness. 4.2 ENF experiments The strain rate varies also during the ENF experiments. One example is shown in Fig. 10a. When the adhesive softens at the tip of the adhesive layer, the strain rate accelerates. The strain rate in each experiment is plotted vs. the adhesive thickness, h, in Fig. 10b.

Fig. 10. (a) Strain rate variation with deformation, v, during one experiment. Triangle marks peak stress, circle marks half the consumed fracture energy, Jc. (b) strain rate variation with adhesive thickness, h.

The strain rate for the ENF experiments declines with increasing thickness in the same manner as for the DCB experiments. Fracture energy and peak stress are given in Figs. 11a and b.

aaaa

0

500

1000

1500

2000

0 0.5 1 1.5

h (mm)

J c (

N/m

)

10

20

30

40

00.5 1 1.5

h (mm)

pea

k str

ess

(MP

a)

bbbb

1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

h (mm)

stra

in r

ate

(s-1)

x10-3 bbbb

0 20 40 60 80 1000

1

2

3

4

5

6 x 10-3

v (µm)

stra

in r

ate (

s-1)

aaaa

D10

Fig. 11. (a) Fracture energy, (b) Peak stress vs. adhesive thickness, h, for shear mode.

The fracture energy increases with increasing adhesive thickness between 0.1 mm and 1.0 mm. The fracture energy for the specimens with adhesive thickness 0.2 mm are a bit too high to fit nicely into the overall trend in Fig. 11a. The results for shear mode at the thickness 0.2 mm are taken from a different experiment, performed earlier and are manufactured using an adhesive from a different batch, cf. [5]. The behaviour of the fracture energy in shear differs significantly from the findings in [11] although the adhesives are toughened epoxy in both studies. The peak shear stress seems to be relatively independent of the adhesive thickness. 5. Conclusions The adhesive thickness influences the fracture energy in peel mode strongly. The fracture energy increases monotonically as the thickness is increased from 0.1 mm to about 1.0 mm. At 1.5 mm, the fracture energy is slightly lower than for 1.0 mm adhesive thickness, indicating that the fracture energy has a maximum between 1.0 mm and 1.5 mm. In the shear mode, the thickness dependence is not as strong, but a similar trend is evident. The larger scatter in the shear study makes the interpretation less clear, although a similar behaviour is seen. The shear study lacks coverage of adhesive thickness values above 1 mm. For both peel mode and shear mode, the peak stress shows little thickness dependence, although a slight decrease in peak stress with increasing adhesive thickness is indicated. Acknowledgements The author thanks the Swedish Consortium for Crashworthiness for funding this project. Many thanks are also directed to Professor Ulf Stigh for fruitful discussions. Special thanks are directed to Mr. Stefan Zomborcsevics for preparing specimens and Mr. Anders Biel for helping with the measurements.

References [1] Belytschko T, Liu WK, Moran B. Nonlinear finite elements for continua and structures. Chichester:

John Wiley and Sons; 2000.

[2] Schmidt P, Edlund U. Analysis of adhesively bonded joints: a finite element method and a material model with damage. Int. J. Numer. Meth. Engng 2006;66:1271–1308

[3] Andersson T, Stigh U. The stress-elongation relation for an adhesive layer loaded in modus I using equilibrium of energetic forces. International Journal of Solids and Structures 2004;41:413-434

0

1000

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3000

4000

0 0.2 0.4 0.6 0.8 1

h (mm)

J c (

N/m

) aaaa

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40

50

0 0.2 0.4 0.6 0.8 1

h (mm)

pea

k st

ress (

MP

a)

bbbb

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[4] DuBois PA. Crashworthiness Engineering, Course Notes. Livermore: Livermore Software Technology Corporation; 2004.

[5] Leffler K, Alfredsson KS, Stigh U. Shear behaviour of adhesive layers. International Journal of Solids and Structures 2007;44:530-545

[6] Andersson T, Biel A. On the effective constitutive properties of a thin adhesive layer loaded in peel. International Journal of Fracture 2006;141: 227-246

[7] Reedy ED, Mello FJ. Modeling the Initiation and Growth of Delaminations in Composite Structures. Journal of Composite Materials, 1997;vol. 31, No. 8:812-831

[8] Carlberger T, Alfredsson KS, Stigh U. Explicit FE-formulation of Interphase Elements for Adhesive Joints. Accepted for publication in International Journal for Computational Methods in Engineering Science & Mechanics.

[9] Dillard, D.A.; Pocius, A.V.; Chaudhury, M. Adhesion Science and Engineering Volume 1. ISBN-13: 9780444511409a , 806 pp. Elsevier; 2002.

[10] Kinloch, A.J., Adhesion and Adhesives: Science and Technology. London, Chapman and Hall; 1987.

[11] Chai H. The effects of bond thickness, rate and temperature on the deformation and fracture of structural adhesives under shear loading. International Journal of Fracture 2004;130: 497–515.

[12] Choi JY, Kim HJ, Lim JK, and Mai Y-W. Numerical Analysis of Adhesive Thickness Effect on Fracture Toughness in Adhesive-Bonded Joints. Key Engineering Materials 2004;270-273:1200-1205.

[13] Duan K, Hu X, Mai Y-W. Substrate constraint and adhesive thickness effects on fracture toughness of adhesive joints. J. Adhesion Sci. Technol. 2004; 18: 1:39–53 [14] Taib AA, Boukhili R, Achiou S, Gordon S, Boukehili H. Bonded joints with composite adherends. Part I. Effect of specimen configuration, adhesive thickness, spew fillet and adherend stiffness on fracture. International Journal of Adhesion & Adhesives 2006;26:226–236

[15] Olsson P, Stigh U (1989) On the determination of the constitutive properties of the interphase layers – an exact solution. International Journal of Fracture 41:71-76

[16] Alfredsson KS (2004) On the instantaneous energy release rate of the end-notch flexure adhesive joint specimen. International Journal of Solids and Structures 41:4787–4807

[17] Rice JR. A Path Independent integral and the Approximate Analysis of Strain Concentrations by Notches and Cracks. Journal of Applied Mechanics 1968;33:379-385

[18] Tamuzs V, Tarasovs S, Vilks U. Delamination properties of translaminar-reinforced composites. Composites Science and Technology 2003;63:1423-1431

[19] Biel A, Stigh U. Effects of constitutive parameters on the accuracy of measured fracture energy using the DCB-specimen. Engineering Fracture Mechanics, To appear. DOI:10.1016/j.engfracmech.2008.01.002.

[20] Carlberger T, Biel A. Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive. To be submitted.

Paper E

Dynamic testing and simulation of hybrid joined bimaterial beam

E1

Dynamic testing and simulation of hybrid joined bimaterial beam

Thomas Carlberger

SAAB Automobile AB, SE-461 80 Trollhättan, SWEDEN [email protected], telephone: +46 520-84 119

Abstract A bimaterial beam specimen is developed for real like low velocity impact testing of joint configurations. The specimen is subjected to three point bending. Specimens are prepared with discrete joints and adhesives, separately and in combination (hybrid joint). Results show that the impact energy consumption depends on the joint integrity. A threshold value for the fracture energy of the adhesive seems to exist. Beneath this value, adhesive and discrete fastener work together increasing the impact energy. Above this value, the discrete fastener has a negative effect, and may be regarded as a stress concentration. Simulations are performed with adhesive cohesive elements in FE-models consisting of shells to predict impact energy values and the overall deformation mode. 1. Introduction In the automotive industry, environmental concerns are driving vehicle weight reduction since weight is largely responsible for fuel consumption. At the same time, vehicle safety is gaining importance rapidly, as traffic accidents are identified as a major cost for society. These requirements for weight reduction and increased safety, lead to the requirement of optimal material choice for each component in the car body structure. The use of different materials in the structure leads to bi-material joints, requiring a change in joining technique from spot welding to riveting, bolting, adhesive joining, or a combination of these techniques, known as hybrid joining. The traditional joining technique in automotive structures is spot welding. Weight saving requirements drives the material optimisation of each part. It may therefore come to a structure with dissimilar materials in adjoining parts. One advantage of adhesive joining is the ability of joining dissimilar materials. Thus, it is natural to manufacture the built-up test object of different materials, e.g. aluminium and steel. The use of different materials in one structure makes it necessary to adopt joining techniques which allow bimaterial joints. Mechanical joining (bolting or riveting) and adhesive joining fulfil this criterion. Hybrid joining (mechanical joining combined with adhesive joining) is interesting from both economical and strength point of view. In a hybrid joint, the spacing between the mechanical fasteners is a parameter of special interest. Joining by means of discrete fasteners, such as bolts, self piercing rivets or clinching, is time consuming and thus, costly. The number of mechanical fasteners is therefore to be minimised. The use of adhesive joining represents a possibility to both reduce assembly time, by reducing the number of discrete fasteners, and increase vehicle structural integrity by providing a stronger joint. Adhesive joining is most likely to be implemented in the body assembly shop. The adhesive gains strength in the hardening process initiated by the high temperature in the paint shop oven. Prior to the hardening of the adhesive in

E2

the paint shop oven, a limited number of mechanical fasteners are inevitably required to provide the necessary strength to keep the body structural integrity through the body assembly shop. Shortened vehicle development plans leave no time for manufacturing of prototypes for development testing. The testing performed today is mainly done to fulfil legal requirements. Thus, computer crash simulations are required to be reliable enough to eliminate development testing. Crash simulation of adhesively joined structures has been very computational time consuming. In crash simulation, the explicit Finite Element Method (FEM) is the most commonly used simulation tool. The simple calculation scheme of the explicit FE-method allows for solving extremely large models subjected to dynamic loading such as impact, cf. e.g. [1]. A simulation is progressed in small time steps and a large amount of time steps have to be used to accomplish the total simulation time. However, the explicit FE-scheme is only conditionally stable. The condition is that the time step may not exceed a certain critical time step, called the Courant limit or the CFL-limit, cf. [2]. This time step is inversely proportional to the largest eigenfrequency of the structure. A simple estimate is given by the shortest time it takes an elastic wave to pass any element in the structure. In [3] the authors suggest an interphase element for efficient crash simulation of adhesive joints between shell substrates. In the present paper, the suggested modelling technique is used. To verify the adhesive simulation technique, a built-up specimen is used showing that global and local failure modes may be predicted with engineering accuracy. Studying built-up structures subjected to impact is very common, and many different structures are used. In [4-13] various thin-walled profiles are studied regarding impact energy absorption, folding pattern, trigger functions and joint configurations during axial crushing. The axial crush test is applicable to structures in the crumpling zone. Bending specimens, in varying shapes and sizes, are very common in testing and simulation, cf. e.g. [14-17]. In [15] the authors use an off-centre axial crushing of an “S”-shaped beam to study the joint configuration and cross-sectional properties undergoing large deformation. A bimaterial joint is likely to be located in the region surrounding the passenger compartment. Since the passenger compartment is not supposed to deform during crash, the bending beam specimen is considered to be a more relevant test specimen than the axial crush specimen in the present study. The objective of the present paper is to study how different joining techniques affect the impact properties of a built up structure. Many studies are devoted to adhesive bonding in combination with spot welding or bolting, cf. e.g. [18-25]. Dedicated adhesive joint test specimens are tailored to allow for deduction of the properties of the adhesive. Such specimens tend to be geometrically simple and far from a more realistic, built-up structure. A common misunderstanding, regarding adhesives and adhesive fracture energy in crashworthy design, is that the impact energy is consumed by fracturing the adhesive during the impact. This is not true, cf. [3]. The main task for the adhesive in a crashworthy structure is to keep the adjoined parts together and forcing them to deform plastically in a controlled manner. Thus, the impact energy is consumed by plastic deformation work of the adherends. Structures, optimised for absorbing impact energy,

E3

are designed to crumple in a predefined way. Regions with high material strength are deformed in a controlled manner to absorb the impact energy by plasticity work in the material. One purpose of the present study is to examine the influence of the joint type on the absorbed energy capacity of a built-up, bi-material structure. Another purpose is to verify a recently developed numerical model for the adhesive, cf. [3]. In the present paper we study dynamic loading of a beam structure build up of aluminium alloy shells and steel plates. A number of different joining techniques are studied: mechanical fasteners, adhesive joining, and hybrid joints. Two different adhesives are used: Dow Betamate XW1044-3 and Dow Betamate 1496V. The latter adhesive is marketed by the manufacturer as a crash resistant adhesive. The cohesive laws, in peel and shear, of the adhesives are determined using the DCB and ENF specimen. For the simulations in the present study, the cohesive law of the adhesive is approximated with a bilinear relation with linear damage evolution using Abaqus cohesive elements. 2. Design of test In designing a structure for impact, one main rule is to avoid high stress in the joints. This is a well known design criterion. A structure for load bearing purposes would thus avoid subjecting the joint to excessive stress, and failure would be initiated by plastically deforming the base material. A test specimen for studying the influence of the joint configuration should instead be designed to give large loads in the joints. This allows the joints to partly fracture in addition to plastically deforming the base material. Thus, the joint geometry will reveal the ability of the joint to transmit large loads and also the ability of the simulation technique to accurately simulate the joints. A measure for determining the impact efficiency of a structure, is the impact resistance W. The impact resistance W is calculated as

∫= δdFW , (1)

where F is the impact force and δ the distance travelled by the force. With a failure mode involving buckling, the impact resistance W will vary with the buckling wavelength a, cf. [5]. During an impact test, the impact force and displacement of the impacting body are measured simultaneously. Legal impact requirements for vehicle manufacturers, such as e.g. [26, 27] are based on these parameters. Typically, the requirement stipulates a force level to be exceeded during a prescribed displacement, in essence corresponding to an impact resistance requirement. In our case, a test specimen should show a relation between the joint configuration and the impact resistance. Traditional adhesive joint testing is based on pulling joined parts apart during energy absorption. It is believed to be more relevant in crash situations to subject a structure to compressive impact, since it will involve more complicated buckling and folding of the base materials involved. In the sequel, W evaluated at δ = 65 mm is denoted impact resistance. This parameter is used to discriminate between different joint configurations and to compare the experiments to simulations.

E4

2.1 Specimen geometry The most common built-up test specimen for impact testing is believed to be the axial column crush specimen, cf. e.g. [4-13]. This specimen is relevant for base material tests and testing of crash boxes. A bimaterial structure is likely to be located between the passenger compartment and the surrounding structure. In this location such severe deformation, as in the axial crush test, is not fully relevant. Instead, another common test specimen for impact testing which seems more adequate is the three (or four) point bending beam test specimen, cf. Fig. 1 and [14-17]. The beam shaped specimen is placed on two supports and transversally impacted by a drop head weight.

Figure 1. Typical hat profile beam for three-point or four-point bending.

The typical distance between the supports is Ls = 350 mm. The hat profile beam consists of a profile, bent to shape from sheet metal, and a flat bottom plate attached to it as shown in Fig. 1. The structure is relatively simple to manufacture, and using aluminium for the hat profile and high strength steel for the bottom plate a simple bimaterial beam is created. For the profile part, an aluminium alloy with Rm = 140 MPa of thickness t = 2 mm is used and for the bottom plate a standard automotive steel with Rm = 400 MPa of thickness t = 2 mm. The single hat profile cross section is shown in Fig. 2.

Figure 2. Cross section of the single hat profile beam.

The bend radii R = 4 mm, flange width, c = 23 mm, profile height, h = 25 mm and the overall width of the profile, b = 96 mm. The overall length is L = 500 mm. Simulations of

R

c b

h R R

c

t

t

mechanical fastener

mechanical fastener

supports

drop head weight

spot welds or mechanical fasteners and/or adhesive

hat profile

bottom plate

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the hat profile beam in Fig. 1 reveal a severe cross-sectional collapse, in both three-point bending and four-point bending. As the drop head impacts the profile, the profile deforms in the lateral plane and decreases the moment of inertia of the beam. Depending on sheet thickness, Young’s modulus and yield strength, the profile may then collapse, reducing the moment of inertia to a minimum, which in turn reduces the shear stress in the joint. Such a collapse reduces the stresses in the joint between the hat profile and the bottom plate. These findings are in accordance with previous work [16, 17]. The cross-sectional collapse may be less severe if the beam is inverted, letting the bottom plate (steel) receive the impact. In the inverted position, the beam is bent in the opposite direction. Instead of compressive force, as for the original position, the load on the aluminium hat profile is now a tensile load. The steel plate is subjected to compressive force, and the joint is subjected to shear stress, which is desired. The transversal force on the top of the profile is in this configuration reduced to one half compared to the force impacting the specimen in the original configuration. Simulations show that this still gives cross sectional collapse and the shear stresses in the joint are not critical. Since the stress in the joint is of main interest, the test specimen needs to be modified in order to achieve high stresses in the joints. Using the simple beam specimen, a modification of the geometry is suggested to achieve the joint configuration-impact resistance relation. It is well known from axial column crush testing that the folding pattern of the sheet metal is decisive for the impact energy absorption capacity. A shorter folding distance involves more plastic work of the material and thus a larger amount of impact energy may be consumed, cf. [5]. To achieve compression in the flange of the single hat profile beam, the flanges should be oriented vertically, i.e. the flat plate on the bottom of the profile is to be aligned with the impacting direction of the impact body. This orientation of the single hat profile beam is unstable. A straight forward solution to this is to double the simple beam specimen by mirroring it across the profile top surface, creating a double symmetric profile beam, here referred to as a H-beam, cf. Fig. 3.

Figure 3. Modified profile beam, named H-beam. Vertical orientation.

The H-beam consists of two identical single hat profile beams connected to each other by mechanical joining method (bolting) along the mid symmetry line on the top surfaces of each single profile beam, cf. Fig. 4.

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Figure 4. Cross section of the H-beam. Vertical orientation.

The H-beam may be used in two orientations, horizontal or vertical. Simulations with the H-beam in the horizontal orientation, prevents the instability induced buckling of the flange. Moreover, the horizontal configuration shows problems with cross-sectional collapse, but not as severe as with the single profile beam. Furthermore, the joint between the aluminium profile and the steel plate is near the outermost fibre of the beam which reduces the shear stress in the joint. The large ratio of Young’s modulus between steel and aluminium work in favour of increasing the shear stress. Simulations of both the horizontal and vertical configuration show a more complex loading of both the adhesive and the base material, aluminium and steel, than in the single hat profile beam. The stress state in the impact H-beam is a superposition of bending stresses and shear stresses due to transverse forces. 2.2 Joint configuration For the H-beam specimen to show discrepancy in impact energy between joining techniques, the flanges should be subjected to compressive force. Compressive force in combination with discrete fasteners, may lead to local buckling of the flanges between the fasteners and a characteristic wave pattern with wavelength, a, is produced, cf. Fig. 5.

Figure 5. Buckling between discrete (symbolic) fasteners.

To trigger a pure axial buckling mode, the flanges are required to be flat and not subjected to transverse bending. The reason for the single hat profile beam to

mechanical fasteners

symmetry plane

a

discrete fasteners

F F

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malfunction, in this sense, is that the compressive buckling mode triggers a collapse of the cross section. In the case of discrete fasteners, the wavelength, a, equals the fastener spacing distance, s. Similar to axial instability, the buckling load increases as the free length between fasteners decreases. In an axial compression configuration, it is possible to determine the minimum wavelength for discrete fasteners. Beneath the minimum distance, the buckling failure mode changes to compressive yield, and the failure load does not increase any further as the fastener spacing, a, decreases. Prior to manufacturing of the test specimens, all test configurations are simulated using the commercial FE-software package ABAQUS/Explicit v. 6.7. A finite element model with shell elements representing the profile and plate parts, is used, cf. Fig. 6. Shell element S4R is used.

Figure 6. FE-model of the H-beam.

In the automotive industry, common mechanical fasteners for dissimilar materials are self-piercing rivets or clinching, cf. e.g. [28, 29]. The process to introduce these joining techniques to a material/thickness combination is tedious. The strength of the joint is dependent on several variables, both geometrical (sheet thicknesses) and material (ductility, ultimate strength etc.). For simplicity, in this study, we simply use M5 8.8 bolts and nuts as mechanical fasteners. The location of the mechanical fasteners is indicated in Fig. 7.

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Figure 7. Drill pattern for mechanical fasteners. (a) dense fastener spacing. (b) sparse fastener spacing. The starting offset along the beam, d1 = 25 mm, flange offset, d2 = 10 mm, centre spacing, d3, and the hole diameter, ∅ = 5.1 mm. The placement of the mechanical fasteners is done in two ways. The first uses fastener spacing s = 50 mm, cf. Fig. 7a. This is considered comparable to a spot weld spacing in an automotive structure without adhesive. The sparse spacing distance is s = 100 mm, cf. Fig. 7b. The sparse spacing distance is used to show that the use of adhesive in a joint may allow a sparser placing of the mechanical fasteners. In the sparse pattern, the holes in the flanges are placed alternately on the opposite side of the flat plate. In this way the sensitivity to the impact position, relative to the mechanical fasteners, is less pronounced. In the finished H-beam specimen with sparse spacing, the hole patterns are placed anti-symmetrically, cf. right and left side in Fig. 3. The dense fastener spacing is only used for the pure mechanical joint. The joint configurations considered are given in Table 1. With Bolt 50 and Bolt 100, the traditional joining method using discrete mechanical fasteners is tested. With Adhesive 1 and Adhesive 2, adhesive joining without mechanical fasteners is tested. These specimens correspond to a joint design where the mechanical integrity of the car body is secured with a fixture during manufacturing; the fixture is dismantled after the paint baking process. Hybrid 1 and Hybrid 2 specimen correspond to hybrid joining where the mechanical integrity during manufacturing is given by discrete fasteners that remain in the structure after manufacturing. joint configuration Bolt 50 Bolt 100 Adhesive 1 Adhesive 2 Hybrid 1 Hybrid 2

fastener spacing (mm)

50 100 - - 100 100

mechanical fastener

bolted bolted - - bolted bolted

Adhesive - - XW1044-3 1496V XW1044-3 1496V

number of specimens

2 2 3 8 3 3

Table 1. Joint configurations.

d 3

d 2

s d1

∅

s/2

s

d 2

∅

(b) (a)

d 3

d1

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2.3 Material data The adhesives are epoxy-based structural adhesives from Dow Automotive, Dow Betamate XW1044-3 and Dow Betamate 1496V. Both adhesives are used in the automotive industry today. Betamate XW1044-3 is being replaced by e.g. Betamate 1496V in many cases, due to the improved crash performance. The adhesive thickness, h = 0.2 mm is considered typical in the automotive industry. It is achieved by placing a 0.2 mm diameter wire along the joint midline before applying the adhesive and subsequently clamping. Both adhesives are cured according to the supplier recommendation at 180°C for 30 minutes. The adhesive properties are determined by performing tests with the double cantilever beam specimen (DCB) and the end notched flexure specimen (ENF). Both quasi-static and dynamic tests are performed. Results for the quasi-static case for the adhesive Betamate XW1044-3 are presented in Fig. 8.

Figure 8. Constitutive behaviour in peel and shear for the engineering adhesives DOW Betamate XW1044-3 (left) and DOW Betamate 1496V (right) with 0.2 mm layer thickness. Results for Betamate XW1044-3 from [30,31]. The test specimens for the two curves in Fig. 8 are DCB for pure peel and ENF for pure shear. These tests are for pure modes and there are other tests for mixed mode, cf. [32]. These types of tests are well suited for determining the quasi-static parameters of the cohesive law for an adhesive. In a recent study the DCB and ENF tests are used to achieve semi-dynamic results, i.e. strain rates in the order of 1 s-1 are achieved, cf. [33]. The shear results for the adhesive DOW Betamate 1496V are taken from such a test. The constitutive behaviours in Fig. 8 is implemented into a 3D cohesive element in ABAQUS/Explicit as a bilinear (saw-tooth) model, cf. Fig. 9.

0

5

10

15

20

25

30

0 0.05 0.1 0.15

deformation (mm)

stre

ss (

MP

a)

Peel

Shear

XW1044-3

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2

Peel

Shear

deformation (mm)

stre

ss (

MP

a)

1496V

E10

Figure 9. Approximation of the constitutive behaviour with a linear elastic, linear softening model. Solid line: experimentally determined traction-separation relation. Dotted line: saw tooth approximation. The parameters for the cohesive element are given for pure peel (mode I) and pure shear (mode II). Mixed mode is governed by Eq. (2), cf. [34]: the mixed mode fracture energy, JTC, under any combination of modes I and II is assumed to be given by

( ) IITC IC IIC IC

I II

mJ

J J J JJ J

= + − +

(2)

where JIC and JIIC are the critical strain energy release rates in peel and shear respectively and JI and JII are the energy release rates in peel and shear respectively and m is a characteristic parameter of the material considered, cf. [34]. Softening is obtained by introducing a damage parameter ω as indicated in Fig. 9. Material data for the base materials is obtained from simple tensile tests. The exponent m = 1 is chosen. Tabulated approximations of the stress-strain curves in Fig. 10 are used in the FE simulations. The stress-strain data are given in Fig. 10.

σ (M

Pa)

50 100 0 0

5

10

15

20

w (µm)

k

1

Jc

ω = 0 ω = 1

w1 wc

E11

Figure 10. Experimentally determined stress-strain curves for the sheet materials aluminium and steel. The stress-strain curve for aluminium shows softening of the material, which is possible, since the stiffness of the specimen is sufficiently low in comparison to the tensile test machine stiffness. Localisation (necking) causes large local strains, which are not possible to capture with this method. The ultimate plastic strain, p

cε , is given by

0lnpc

A

Aε =

, (3)

where A0 is the initial cross sectional area and A is the cross sectional area at fracture. Nominal initial cross section is 16 mm2 (8mm by 2mm). All tensile test specimens fracture in a ductile manner, cf. Fig. 11.

Figure 11. Tensile test specimens. Left: steel; Right: aluminium.

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

true strain (%)

tru

e st

ress

(MP

a)

Aluminium

Steel

E12

The tensile test specimens are cut from the tested H-beam specimens after completed impact test. The strain rate dependence of the material may be of importance in dynamic studies. In [2] it is shown that the strain rate in the adhesive may be many times larger than in the base material due to the ratio of Young’s modulii between the adherend and the adhesive. In the open literature only limited data is available concerning strain rate effects for the specific adhesives. Polymers are known to show positive strain rate sensitivity. For the H-beam, typical strain rates in the simulations are given for the base materials and for the adhesive in Fig. 12a and 12b.

Figure 12. Typical strain rate in the base material (a) and the adhesive (b) during a simulation. Although the strain rate is important for the simulation results, it is difficult to determine experimentally. Some data is available for the base materials, but limited data is available for the adhesive. In a recent study of the adhesive Betamate XW1044-3, the strain rate sensitivity of the fracture energy, Jc, in peel is shown to be slightly positive and in shear to be slightly negative, cf. [33] and Fig. 13.

-20

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9

time (ms)

stra

in r

ate

(s-1)

Aluminium

Steel

(a)

0

1000

2000

3000

4000

0 1 2 3 4 5

time (ms)

stra

in r

ate

(s-1)

(b)

peel shear

Adhesive

E13

Figure 13. (a) fracture energy vs. strain rate, (b) peak stress vs. strain rate for peel and shear. Data from [33]. Both the fracture energy and the peak stress in peel increase slightly with strain rate. The results in peel loading indicate

Ic Ic00

00

log

ˆ ˆ log

JJ J

σ

εε

εσ σε

= =

ɺ

ɺ

ɺ

ɺ

(4a,b)

with JIc0 = 34 N/m, 0Jε =ɺ 3.0 10-14 s-1, 0σ = 0.9 MPa and 0σε =ɺ 3.8 10-15 s-1. These

equations are dashed in Figs. 13a,b.

An increase of strain rate by 1 decade gives 78 N/m increase in fracture energy in peel; an increase in strain rate by 1 decade gives an increase by 2.1 MPa in peak stress in peel. Thus, the strain rate effect is small and not considered in the simulations. Data corresponding to quasistatic loading is used throughout the simulations. During curing, the difference in thermal expansion of steel, αs = 11·10-6 K-1 and aluminium, αa = 23·10-6 K-1 cause residual stresses in the base materials and in the adhesive. Since the strength of both the aluminium and the steel is much higher than the strength of the adhesive, the residual stress in the adhesive is considered important. The curing process, recommended by the adhesive manufacturer, involves (rapid) heating of the specimen to the prescribed curing temperature 180°C; retaining this temperature for 30 minutes and subsequently cooling the specimen slowly to room temperature. This process should minimise the residual stresses. Curing is considered fulfilled after 30 minutes at 180°C. At this stage the adhesive has obtained its solid properties. An important parameter in this context is the glass transition temperature, Tg; both for Betamate XW1044-3 and 1496V, Tg ≈ 90-100°C. It may be assumed that no residual

peel shear lin. appr.

0

10

20

30

40

50

60

70

80

10-4 10-3 10-2 10-1 1 101 102

strain rate (s-1)

pea

k st

ress (

MP

a)

(b)

peel shear lin. appr.

0

1000

2000

3000

10-4 10-3 10-2 10-1 1 101 102

strain rate (s-1)

J c (

N/m

)

(a)

E14

stresses develop in the adhesive above Tg. A simulation of the thermally induced residual stresses is performed of a temperature drop from Tg to room temperature, i.e. 80°C. The simulation is based upon the assumption that no relaxation of the residual stress occurs below Tg. Residual stresses are analysed in the adhesive along the line marked in Fig. 14a, which is the most stressed part of the adhesive.

Figure 14. Residual stresses in the adhesive from curing process. (a) location of the measurement line. (b) adhesive peel stress (σyy) and shear stress (τyx). The simulation shows high shear stress and moderate peel stress near the ends of the profile. In the mid section, both shear stress and peel stress are relatively low. Since the impact test shows fracture of the adhesive in the mid section of the beam, the residual stresses are neglected in this study, although the general conclusion must be that residual stresses due to the curing process must be considered. The assumption that no relaxation occurs between Tg and room temperature may also be questioned. 3. Simulation The FE-model in Fig. 6 consists of shell elements representing the aluminium profiles and the steel plates, cohesive elements representing the adhesive and discrete elements representing the bolts and nuts. The shell elements are modelled with S4R-shell elements and an offset of half the shell thickness towards the adhesive, such that Abaqus cohesive elements, COH3D8, can connect directly to the shell nodes without any rigid coupling elements. The metals are modelled as elastic-plastic material models with a fracture criterion based on plastic strain to failure. This is a very crude fracture criterion, but since limited material parameters are available for the used materials, this criterion is used. For the cohesive elements representing the adhesives, cohesive laws (saw-tooth) are used where the peak stress, initial stiffness and the fracture energy, Jc, are specified. These material parameters are determined for the adhesives by using the DCB and ENF specimens.

(a)

z x

y -20

-10

0

10

20

0 100 200 300 400 500

x (mm)

Str

ess

(MP

a)

shear peel

(b)

E15

The discrete fasteners (bolt and nut) are modelled as CONN3D2-elements in Abaqus /Explicit with a simple maximum force fracture criterion. Fracture of discrete fasteners and base material is calibrated against the experimental fracture appearance of the Bolt 50 specimen. During deformation of the elements, some elements are compressed, with decreasing element size as a consequence. This in turn reduces the time step. To avoid excessive solution times, variable mass scaling is chosen such that elements that become compressed are mass scaled when the element size requires a time step beneath a prescribed value. This value is chosen as a fraction of the initial critical time step, preferably between 80% and 95%. By mass scaling of the compressed elements, it is secured that the added mass is localised only to the elements that need mass scaling caused by compression of the element. Thus, the total added mass is small enough not to severely influence the overall behaviour. For the models without adhesive, the initial time step is 214 ns. For these simulations, a time step limit to 200 ns is chosen. For the models including adhesive, the adhesive elements are the limiting elements, and the initial time step is 8 ns. This time step gives a total solution time of 130 hours. Mass scaling of the adhesive elements by a factor 100 increases the critical time step to 80 ns, reducing the total solution time to 13 hours. The adhesive mass increases from 9g for the total model to 900g. The model weight is 3.3 kg with mass scaled adhesive. Thus, the added mass constitutes 27% of the total model weight. This is generally not negligible, but since the adhesive is located perpendicularly towards the impact direction, it is considered tolerable in this particular case. For the variable mass scaling, the threshold is chosen to 70 ns. The added mass affects the impact energy capacity and has to be considered in the evaluation of the simulations. The simulations run decomposed into 16 domains on a personal computer with 8 processors. 4. Experimental The loading is supplied by an Instron VHS 100/20 servo hydraulic testing machine with capacity of dynamic loading up to 25 m/s, measurement of impact force up to 100 kN, and displacement up to 200 mm. Data acquisition is available up to 5 Ms/s in four channels. The H-beam specimen is placed on a fork shaped crosshead with two cylindrical supports of diameter 50 mm spaced at 350 mm distance, cf. Fig. 15.

E16

Figure 15. Overview of impact testing machine with mounted specimen.

The crosshead is accelerated to a specified impact speed vertically upwards. The specimen is accelerated by the supports and is impacted by a rigidly mounted impact body on the top surface. The impact body is a cylinder with a diameter of 50 mm and length 35 mm, such that it impacts on the aluminium profile between the vertical flanges of the steel sheets, cf. Fig. 6. This results in relatively gently introduction of buckling forces on the steel parts and subjects the joints to mixed mode stress. For comparison between different joint configurations of the H-beam in 3-point impact bending, the impact resistance is evaluated at δ = 65 mm, cf. Eq. (1). This value is empirically determined as an appropriate distance depending on the geometry of the beam and distance between the supports. To assess the experiments, the overall deformation mode is also evaluated and the extent of fractured adhesive is given special attention. The H-beam specimens are arranged in increasing impact resistance order in Fig. 16.

impactor specimen

crosshead

E17

Figure 16. Impact energy absorption for different joint configurations.

As expected, the mechanically joined H-beam, (screw and nut) Bolt 100, is the specimen with the smallest impact resistance. Figure 17 shows a Bolt 100 specimen after test.

Figure 17. Bolt 100 specimen after test.

As visible, the flanges have buckled between the fasteners. Some of the mechanical fasteners have also been torn out of the aluminium alloy. The experimentally determined impact resistance is 930 J which is slightly overestimated by the simulation to 980 J (5%). The torn out fasteners reduce the impact resistance significantly. The fold pattern is determined by the placement and distance between the bolts near x = L/2, cf. Fig. 5. Bolt 50 shows a denser pattern of buckling of the steel plates than Bolt 100, cf. Fig. 18.

Simulation

Experiment

0

500

1000

1500

2000

2500

Bo

lt 1

00

Bo

lt 5

0

Ad

hes

ive

1

Hyb

rid

1

Ad

hes

ive

2

Hyb

rid

2

joint configuration

Imp

act

resi

sta

nce (J)

98

0

93

0 1

23

0

11

40

13

40

11

70

15

70

16

40

16

70

17

60

232

0

E18

Figure 18. Bolt 50 specimen after test. The specimen is deformed manually after the test in order to free the specimen from the impactor.

The impact resistance is slightly higher than for Bolt 100, W = 1140 J in the experiment. The simulation again overestimates this slightly by 8% to 1230 J. The current design goal of avoiding local buckling between discrete fasteners, appears correct, since it improves the impact resistance. The improvement from Bolt 100 to Bolt 50 is about 23% (experimental values). Further improvement is expected for decreased fastener spacing. The fold pattern is changed from the Bolt 100 specimen, which is a direct consequence of the reduced fastener spacing. Even in this case, we find a number of fasteners torn out of the aluminium alloy. Adhesive 1 shows slightly larger impact resistance than Bolt 50 which is noticeable since DOW Betamate XW1044-3 is not supposed to be used for crash applications. In the experiment, the impact resistance is 1170 J and the simulation predicts an impact resistance 1340 J, which is an overestimation of 15%. It is also observed that the steel plates are completely separated from the aluminium profile at the end of the experiments, cf. Fig. 19.

E19

Figure 19. Adhesive 1 specimen after test.

The simulations show large adhesive fracture area, but not total fracture. This might be due to the fact that the total displacement in the experiments exceeds 150 mm but the simulation only covers 120 mm displacement. Catastrophic failure, as in the Adhesive 1 experiment, is obviously not allowed in a car structure. However, for the first part of the experiment, with δ < 65 mm, we may assume that the adhesive joints have not failed completely. The experiment shows severe fracture of the adhesive, but no material fracture in the metals. The Hybrid 1 specimen, cf. Fig. 20, has both discrete mechanical fasteners spaced at 100 mm distance and the adhesive Dow Betamate XW1044-3.

Figure 20. Hybrid 1 specimen after test.

The impact resistance is 1640 J in the experiment compared to 1570 J in the simulation. The Hybrid 1 experiment is unfortunately performed at a displacement velocity of 0.1

E20

m/s as compared to 9 m/s for the other experiments. This was discovered late and has not been attended to. Despite this, the impact resistance shows good agreement and is only underestimated by 4%. In Fig. 24 there are principal differences to be observed between simulation and experiment, due to this. Nevertheless, the results show that the mechanical fasteners work together with the adhesive since the impact energy is significantly increased compared to the Bolt 100 and the Adhesive 1 specimen. The Bolt 50 impact resistance is also significantly exceeded. This shows that the fastener spacing may be increased if the discrete mechanical joint is complemented with adhesive. Figure 21 shows Adhesive 2 (Dow Betamate 1496V) after the test.

Figure 21. Adhesive 2 specimen after test.

In this case, the adhesive is so strong that the impactor penetrates the aluminium alloy. Several different pressure plates are used to reduce the local stresses beneath the impact body in order to eliminate penetration. Four different plates of varying thickness and length are tried, but all specimens are penetrated in the Adhesive 2 case. It seems that the strength of the aluminium alloy is too low in comparison to the steel. A larger impact body might have been sufficient, but this was discovered too late in the test series. No useable experimental impact resistance values are determined for the Adhesive 2 specimen. The simulated impact resistance is achieved by increasing the plastic strain to failure used in the aluminium shell material model. The simulations render an impact resistance 1670 J. The Hybrid 2 specimen, cf. Fig. 22, which has both bolts at 100 mm spacing and the adhesive Dow Betamate 1496V, shows neither base material fracture nor adhesive fracture.

E21

Figure 22. Hybrid 2 specimen after test. Considerable effort had to be spent in order to free the specimen from the impactor after the test. The experimental impact resistance of the Hybrid 2 specimen is W = 2320 J, which is underestimated 24% in the simulation to 1760 J. This is a substantial improvement compared to the Hybrid 1 specimen. The impact resistance for this specimen is believed to be relevant also for the Adhesive 2 specimen, since the Adhesive 2 specimen was penetrated in all experiments, and thus only small strength variations in the aluminium material are likely to have caused the Hybrid 2 specimen not to penetrate. An immediate conclusion is that this configuration is close to the ultimate strength of this beam. The adhesive Dow Betamate 1496V has about 2.5 times the fracture energy of Dow Betamate XW1044-3. The Adhesive 1 specimen shows that the adhesive Dow Betamate XW1044-3 does not fulfil the requirements for the joint without mechanical fasteners. The adhesive Dow Betamate 1496V has the potential strength to fulfil the joint integrity requirement by itself. To verify this, further tests are required. A suggestion is that, for each joint configuration, a threshold value exists for the adhesive fracture energy where no mechanical fasteners are needed. The cohesive law has an importance when the adherends are able to deform plastically since the anticlastic deformation of the adherends is influenced by the initial stiffness/ peak stress, cf. [30]. To elucidate the quality of the simulations, force-displacement curves are shown in Fig. 23.

E22

Figure 23. Experimental force – displacement curves. Note: No dynamic results for Hybrid 1 nor for Adhesive 2. *The curve for Hybrid 1 is achieved for quasistatic loading.

The impact resistance is mostly dependent on the amount of plastic deformation induced in the metal parts. The impact load is transmitted via the aluminium part over the joint to the steel part. If the joint fractures, the transmission is less effective and the total impact energy is considerably reduced. The force-displacement curves are in principal dominated by three parts: 0→A, A→B and B→δ = 65mm. Studying the simulations reveals the reasons for the transitions. The simulations show qualitative agreement with the experiments, cf. Fig. 24.

0

10

20

30

40

50

0 10 20 30 40 50 60

displacement (mm)

forc

e (k

N)

Bolt 100

Bolt 50

Adhesive 1

Hybrid 2 A

B

Hybrid 1*

E23

Figure 24. Comparison between simulations and experiments. * For Hybrid 1, there are only quasistatic experiments to compare with. All simulations seem to underestimate the breakpoint A corresponding to initial yield of the aluminium part beneath the impactor. Simulation and experimental result show relatively good agreement for the Bolt 50 and Adhesive 1 specimens, in spite of this deficiency. Other results are not as good, but some resemblance is recognised. The considerable mass scaling may have an influence on the simulation results. The initial part, 0→A, corresponds to the linear elastic behaviour of the upper surface of the aluminium profile as it is subjected to the impacting body. At point A the top surface of the aluminium profile buckles and the response follows the plastically deforming part of the aluminium profile until point B is reached. Depending on the joint integrity, point B is reached early or late. The slope of the part of the curve between point A and point B depends on the plastic work dissipated in the aluminium profile and the friction between the impacting body and the aluminium surface. An increase of the general friction in the model (same friction coefficient for all contact surface pairs) moves the point B leftwards in Figs. 23-25. The slope of the part A→B varies in the experiments due to the different improvised pressure plates used to prevent the impactor from penetrating the aluminium

0

10

20

30

40

50

0 10 20 30 40 50 60

Hybrid 1*

forc

e (k

N)

displacement (mm)

0

10

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30

40

50

0 10 20 30 40 50 60displacement (mm)

forc

e (k

N)

Adhesive 1

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40

50

0 10 20 30 40 50 60 displacement (mm)

forc

e (k

N) Bolt 50

0

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0 10 20 30 40 50 60 displacement (mm)

forc

e (k

N) Bolt 100

0

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0 10 20 30 40 50 60displacement (mm)

forc

e (k

N)

Hybrid 2

E24

profile. Low joint integrity may cause point B to be reached before point A. At point B the steel flanges reach the buckling load and the total force is reduced. The fold pattern influences the slope of the curve after point B is reached. The impact resistance of the joint configurations are compared in Fig. 16. A comparison of the force – displacement curves is done in Fig. 25.

Figure 25. Simulated force - displacement curves.

All force – displacement curves, except Bolt 100 and (arguably) Bolt 50, clearly show the discussed characteristic points A and B. The curves follow basically the same pattern. The initial part 0→A, is similar for all curves. The intermediate part, A→B, also follows the same pattern in all simulations though point B is reached later as the joint integrity is increased. Part B→δ = 65mm is the post buckling part and follows a declining trend, determined by the buckling pattern. The improvement in impact resistance, W, between Adhesive 1 and Hybrid 1 is clearly visible but the rather small improvement in impact resistance, W, between Adhesive 2 and Hybrid 2 appears more significant when the force – displacement curves are examined. One aspect to keep in mind when evaluating the results is that the discrete fasteners do not subtract any adhesive area in the simulations. This is not realistic and may have an influence on the impact resistance of the hybrid beams. In many of the tests, the aluminium alloy fractures in a ductile manner. There are many possibilities of describing material fracture in commercial FE-codes. Ductile fracture requires different criteria than brittle fracture. Ductile fracture is observed both in the aluminium alloy and in the steel. Simple criteria like e.g. maximum stress at failure or plastic strain to failure are easy to use but do not predict ductile fracture well. Other criteria are based on non-linear fracture mechanics. The fracture criterion for the cohesive element, used for the modelling the adhesive is based on the J-integral, which in turn is based on non-linear fracture mechanics. In experiments where fracture only occurs in the adhesive, good agreement is generally found between experiments and simulations, cf. e.g. [30,31].

0

10

20

30

40

50

0 10 20 30 40 50 60

displacement (mm)

forc

e (k

N)

Hybrid 2

Adhesive 1 Hybrid 1

Adhesive 2

Bolt 50 Bolt 100

E25

Conclusions The design and testing of the H-beam specimen focuses on revealing limits in the strength and modelling of the joints. In impact of optimally designed structures, the impact energy should be dissipated mainly through plastic deformation of the steel plates. In order for this to become true, the joints shall distribute the load from the impactor hitting the aluminium profile to the steel plates. The study suggests that a limit exists for the adhesive strength, above which no further strengthening of the adhesive bonding improves the impact resistance of the H-beam. Below this strength limit, the adhesive is unable to secure the integrity of the joints without additional discrete mechanical fasteners. It is suggested that the most important strength property of the adhesive is its fracture energy, i.e. the area under the cohesive law. The cohesive law has an influence on the anticlastic behaviour of the adherends as they are plastically deformed. High peak stress and/or high initial stiffness oppose anticlastic behaviour. Further, the mismatch in thermal expansion coefficient between steel and aluminium may give stress problems in the manufacturing process of the specimens; residual stresses in the adhesive are not generally negligible. A general conclusion is that even though only quasistatic material data is used, the simulation accuracy seems in many cases to be acceptable. For better accuracy it is recommended to spend resources on attaining material strain rate dependency for both base materials and adhesives. It is also advisable to use better fracture criteria than plastic strain to failure. Acknowledgements The author thanks the Swedish Consortium for Crashworthiness for funding this project. Many thanks are also directed to Professor Ulf Stigh for fruitful discussions. Special thanks are directed to Mr. Stefan Zomborcsevics for help with manufacturing of the curing rig, Dr. Kent Salomonsson for helping with the simulations and computers and Mr. Mattias Widmark at Volvo Materials Techniques for performing the impact tests.

E26

References [1] Hughes TJR. The Finite Element Method. New York: Dover, 2000. [2] Courant R, Friedrichs K, Lewy H. Über die partiellen Differenzengleichungen der mathematischen Physik, (On the Partial Difference Equations of Mathematical Physics) Mathematische Annalen 100, 32-74 (1928) [3] Carlberger T, Alfredsson KS, Stigh U. Explicit FE-formulation of Interphase Elements for Adhesive Joints. Submitted for publication. [4] Rossi A, Fawaz Z, Behdinan K. Numerical simulation of the axial collapse of thin-walled polygonal section tubes. Thin-Walled Structures, v 43, n 10, Oct. 2005, p 1646-61 [5] Marsolek J, Reimerdes H-G. Energy absorption of metallic cylindrical shells with non-axisymmetric folding patterns. International Journal of Impact Engineering, v 30, n 8-9, Sept.-Oct. 2004, p 1209-23 [6] Xiong Zhang, Gengdong Cheng, Hui Zhang. Theoretical prediction and numerical simulation of multi-cell square thin-walled structures. Thin-Walled Structures, v 44, n 11, Nov. 2006, p 1185-91 [7] Tarigopula V, Langseth M, Hopperstad OS, Clausen AH. Axial crushing of thin-walled high-strength steel sections. International Journal of Impact Engineering, v 32, n 5, May 2006, p 847-82 [8] Hong-Wei Song, Zi-Jie Fan, Gang Yu, Qing-Chun Wang, Tobota A. Partition energy absorption of axially crushed aluminum foam-filled hat sections. International Journal of Solids and Structures, v 42, n 9-10, May 2005, p 2575-600 [9] Yujiang Xiang, Qian Wang, Zijie Fan, Hongbing Fang. Optimal crashworthiness design of a spot-welded thin-walled hat section. Finite Elements in Analysis and Design, v 42, n 10, June 2006, p 846-55 [10] Belingardi G, Goglio L, Rossetto M. Impact behaviour of bonded built-up beams: Experimental results. International Journal of Adhesion and Adhesives, v 25, n 2, April, 2005, p 173-180 [11] DiPaolo BP, Tom JG. A study on an axial crush configuration response of thin-wall, steel box components: The quasi-static experiments. Int J Sol Struct 43 (2006) 7752-7775 [12] Han DC, Park SH. Collapse behavior of square thin-walled columns subjected to oblique loads. Thin-Walled Structures 35 (1999) 167–184

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[13] Hanssen AG, Langseth M, Hopperstad OS. Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler. International Journal of Impact Engineering 24 (2000) 475-507 [14] Jianmei He, Martin Y. M. Chiang, Donald L. Hunston. Assessment of Sandwich Beam in Three-Point Bending for Measuring Adhesive Shear Modulus. Transactions of the ASME, Vol. 123, JULY 2001 [15] Kim H-S, Wierzbicki T. Effect of the cross-sectional shape of hat-type cross sections on crash resistance of an "S"-frame. Thin-Walled Structures, v 39, n 7, July, 2001, p 535-554. [16] Chen W. Experimental and numerical study on bending collapse of aluminum foam-filled hat profiles. International Journal of Solids and Structures, v 38, n 44-45, Oct 12, 2001, p 7919-7944 [17] Santosa S, Banhart J, Wierzbicki T. Experimental and numerical analyses of bending of foam-filled sections. Acta Mechanica, v 148, n 1-4, 2001, p 199-213 [18] Baohua Chang, Yaowu Shi, Liangqing Lu. Studies on the stress distribution and fatigue behavior of weld-bonded lap shear joints. Journal of Materials Processing Technology 108 (2001) 307-313 [19] Darwish SMH, Ghanya A. Critical assessment of weld-bonded technologies. Journal of Materials Processing Technology 105 (2000) 221-229 [20] Darwish SM, Al-Samhan AM. Peel and shear strength of spot-welded and weld-bonded dissimilar thickness joints. Journal of Materials Processing Technology 147 (2004) 51–59 [21] Darwish SM, Al-Samhan A. Design rationale of weld-bonded joints. International Journal of Adhesion & Adhesives 24 (2004) 367–377 [22] Gonçalves VM, Martins PAF. Joining stainless steel parts by means of weld bonding. Int J Mech Mater Des (2006) 3:91–101 [23] Kelly G. Quasi-static strength and fatigue life of hybrid (bonded/bolted) composite single-lap joints. Composite Structures 72 (2006) 119–129 [24] Kunc V, Erdman D, Klett L. Hybrid Joining in Automotive Applications. International SAMPE Symposium and Exhibition (Proceedings), v 49, SAMPE 2004 Conference Proceedings - Materials and Processing Technology - 60 Years of SAMPE Progress, 2004, p 3926-3939

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[25] Santos IO, Zhang W, Gonçalves VM, Bay N, Martins PAF, Weld bonding of stainless steel. International Journal of Machine Tools & Manufacture 44 (2004) 1431–1439 [26] Federal Safety Standards MVSS 214. Motor Vehicle Safety Standard No. 214 Side Impact Protection. Effective Date January 1, 1973 [27] Federal Safety Standards MVSS 216. Motor Vehicle Safety Standard No. 216 Roof Crush Resistance Passenger Cars & Light Trucks, (GVWR 6.000 Pounds or Less). Effective – September 1, 1973 [28] Porcaro R, Hanssen AG, Langseth M, Aalberg A. The behaviour of a self-piercing riveted connection under quasi-static loading conditions. International Journal of Solids and Structures 43 (2006) 5110–5131 [29] Varis J. Economics of clinched joint compared to riveted joint and example of applying calculations to a volume product. Journal of Materials Processing Technology 172 (2006) 130–138. [30] Andersson T, Biel A. On the effective constitutive properties of a thin adhesive layer loaded in peel. International Journal of Fracture 141 (2006) 227-246

[31] Leffler K., Alfredsson KS, Stigh U. Shear behaviour of adhesive layers. International Journal of Solids and Structures, 44:2 (2007) 530-45 [32] Högberg JL, Stigh U. Specimen proposals for mixed mode testing of adhesive layer. Engineering Fracture Mechanics 73 (2006) 2541–2556. [33] Carlberger T, Biel A. Influence of temperature and strain rate on cohesive properties of a structural epoxy adhesive. To be submitted. [34] Benzeggagh ML, Kenane M. Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus. Composites Science and Technology 56 (1996) 439-449.