Simulation of Strength Difference in Elasto-plasticity for Adhesive Materials

17
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2005; 63:1461–1477 Published online 17 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1315 Simulation of strength difference in elasto-plasticity for adhesive materials Rolf Mahnken 1, , and Michael Schlimmer 2, 1 Chair of Engineering Mechanics (LTM), University of Paderborn, Warburger Str. 100, D-33100 Paderborn, Germany 2 Institute for Materials, University of Kassel, Mönchebergstr. 3, D-34109 Kassel, Germany SUMMARY Experimental evidence of certain adhesive materials reveals elastic strains, plastic strains and hardening. Furthermore, a pronounced strength difference effect between tension, torsion or combined loading is observed. For simulation of these phenomena, a yield function dependent on the first and second basic invariants of the related stress tensor in the framework of elasto-plasticity is used in this work. A plastic potential with the same mathematical structure is introduced to formulate the evolution equation for the inelastic strains. Furthermore, thermodynamic consistency of the model equations is considered, thus rendering some restrictions on the material parameters. For evolution of the strain like internal variable, two cases are considered, and the consequences on the thermodynamic consistency and the numerical implementation are extensively discussed. The resulting evolution equations are integrated with an implicit Euler scheme. In particular, the reduction of the resulting local problem is performed, and for the finite-element equilibrium iteration, the algorithmic tangent operator is derived. Two examples are presented. The first example demonstrates the capability of the model equations to simulate the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone. Copyright 2005 John Wiley & Sons, Ltd. KEY WORDS: adhesive materials; strength difference; elasto-plasticity; thermodynamic consistency; finite elements; parameter identification 1. INTRODUCTION In recent years, adhesion has gained growing attention for the bonding process of industrial products. Examples of particular application are the industrial branches of automotive industry, the aeroplane construction and the building industry. Compared to standard bonding processes, such as blind riveting, flow drill screwing, self-piercing riveting and clinching, adhesive Correspondence to: R. Mahnken, Chair of Engineering Mechanics (LTM), University of Paderborn, Warburger Str. 100, D-33100 Paderborn, Germany. E-mail: [email protected] E-mail: [email protected] Received 10 May 2004 Revised 12 October 2004 Copyright 2005 John Wiley & Sons, Ltd. Accepted 5 January 2005

Transcript of Simulation of Strength Difference in Elasto-plasticity for Adhesive Materials

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 63:1461–1477Published online 17 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1315

Simulation of strength difference in elasto-plasticity foradhesive materials

Rolf Mahnken1,∗,† and Michael Schlimmer2,‡

1Chair of Engineering Mechanics (LTM), University of Paderborn, Warburger Str. 100,

D-33100 Paderborn, Germany2Institute for Materials, University of Kassel, Mönchebergstr. 3, D-34109 Kassel, Germany

SUMMARY

Experimental evidence of certain adhesive materials reveals elastic strains, plastic strains and hardening.Furthermore, a pronounced strength difference effect between tension, torsion or combined loadingis observed. For simulation of these phenomena, a yield function dependent on the first and secondbasic invariants of the related stress tensor in the framework of elasto-plasticity is used in this work.A plastic potential with the same mathematical structure is introduced to formulate the evolutionequation for the inelastic strains. Furthermore, thermodynamic consistency of the model equations isconsidered, thus rendering some restrictions on the material parameters. For evolution of the strain likeinternal variable, two cases are considered, and the consequences on the thermodynamic consistencyand the numerical implementation are extensively discussed. The resulting evolution equations areintegrated with an implicit Euler scheme. In particular, the reduction of the resulting local problem isperformed, and for the finite-element equilibrium iteration, the algorithmic tangent operator is derived.Two examples are presented. The first example demonstrates the capability of the model equations tosimulate the yield strength difference between tension and torsion for the adhesive material Betamate1496. A second example investigates the deformation evolution of a compact tension specimen withan adhesive zone. Copyright � 2005 John Wiley & Sons, Ltd.

KEY WORDS: adhesive materials; strength difference; elasto-plasticity; thermodynamic consistency; finiteelements; parameter identification

1. INTRODUCTION

In recent years, adhesion has gained growing attention for the bonding process of industrialproducts. Examples of particular application are the industrial branches of automotive industry,the aeroplane construction and the building industry. Compared to standard bonding processes,such as blind riveting, flow drill screwing, self-piercing riveting and clinching, adhesive

∗Correspondence to: R. Mahnken, Chair of Engineering Mechanics (LTM), University of Paderborn, WarburgerStr. 100, D-33100 Paderborn, Germany.

†E-mail: [email protected]‡E-mail: [email protected]

Received 10 May 2004Revised 12 October 2004

Copyright � 2005 John Wiley & Sons, Ltd. Accepted 5 January 2005

1462 R. MAHNKEN AND M. SCHLIMMER

processing has several advantages: firstly, in particular due to automation, adhesive bonding iseconomical. Furthermore, as a technical aspect, adhesive is able to make seams tight whereotherwise an additional sealing would be necessary. In this way, structural adhesive bonds invehicle construction permit a stiffness increase of the car body as well as an improved crashperformance at a nearly constant weight. Moreover, the adhesion technique offers ideal featuresfor mixed construction.

Despite the increasing range for application of structural adhesives in vehicle construction,the demand of the vehicle industry for reliable and efficient numerical design methods toconsider the adhesive bonds in the construction process has not yet been satisfied. However, inview of reducing development times for new model ranges, calculation and simulation becomesabsolutely necessary. Therefore, in this publication, a simulation procedure related to the groupof so-called crash optimized adhesives is presented.

In particular, we are concerned with the adhesive material Betamate 1496. Experimentalobservations of this material have been published by Schlimmer in Reference [1] for combinedtension torsion tests. The main conclusions of the results are as follows:

1. There exist an almost linear relation in the initial regime of the stress strain diagram upto a certain threshold (Phase I).

2. Above the threshold, the stress strain relation increases non-linearly with a lower slopeas in Phase I up to a maximum point (Phase II).

3. Increasing loading above the maximum point renders a decreasing stress strain relation(Phase III).

4. The stress strain relation for unloading in the Phase II regime follows approximately theslope of the initial stress strain relation of Phase I.

5. The stress strain relations in Phase II and Phase III are dependent on the tension torsionloading relation. In particular, the yield strength is largest for pure tension and smallestfor pure torsion (strength difference effect).

In this way, the macroscopic stress strain relations for adhesive materials exhibit similar charac-teristics as observed in metals. In particular, the effects of elastic strain, remaining plastic strainand hardening can be interpreted in the observations. However, contrary to the micromechanicaldevelopment of dislocations in metals, the underlying microscopic phenomena for adhesivematerials are not fully understood so far and deserve further investigations.

The purpose of this work is to concentrate on the observed macroscopic effects of adhesivematerials. This encompasses the formulation and the efficient numerical treatment of a consti-tutive model, thus enabling to simulate the above-mentioned macroscopic stress strain relationswith a general finite element programme.

The above macroscopic phenomenon of strength difference is, from a more general pointof view, an example of so-called asymmetric effects or as denoted by Altenbach et al. [2]non-classical effects. These are defined by the observation that a certain type of experiment,such as a tension test, is not sufficient in order to characterize the material for different loadingscenarios. Instead, additional independent types of experiments, such as compression, shear andhydrostatic tests, are necessary in order to get a more comprehensive (although in general stillnot complete) characterization of the material. It should be emphasized, that the strengthdifference effect examined in this paper is restricted to isotropic materials, and thereforecarefully has to be distinguished from the effects of initial (material) anisotropy and inducedanisotropy (such as a Bauschinger effect or damaged induced anisotropy).

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SIMULATION OF STRENGTH DIFFERENCE 1463

Several publications can be found in the literature for simulation of inelastic materialbehaviour with strength difference effects. Many of the approaches are based on a stresspotential dependent on the stress tensor and further state variables, e.g. which describe thestate of hardening, softening or damage, respectively. Typically, the potential is formulatedwith polynomials in terms of invariants of the related stress tensor. One of the earlier formu-lations of this type is given by Schlimmer [1] in 1974. Along this line, further constitutiveequations have been formulated, e.g. in Spitzig et al. [3], Zolochevskii [4], Altenbach et al. [2],Ehlers [5], Betten et al. [6] and Mahnken [7, 8] among others.

A yield criterion for an isotropic elastic/plastic continuum can be written in terms of thefirst invariant of the Cauchy stress tensor and the second and third basic invariants of thedeviatoric Cauchy stress tensor, respectively. As discussed extensively in Ehlers [5], a generalyield function of this type allows simultaneously for the two effects of pressure stress stateand deviatoric stress state dependence, where the latter can be different, e.g. in tension andcompression.

In order to simulate the experimental data currently available from Schlimmer [1] for adhesivematerials, the third invariant is disregarded in the approach of this paper. Instead a yield functiondependent on the first invariant of the stress tensor and the second invariant of the deviatoricstress tensor according to Schlimmer [1, 9] is considered. Additionally, a plastic potential withthe same mathematical structure is introduced to formulate the evolution equation for theinelastic strains. Two different approaches (subsequently referred as ‘Case 1’ and ‘Case 2’)are postulated for evolution of a strain like internal variable. Upon considering thermodynamicconsistency of the model equations, for both Cases 1 and 2 some restrictions on the materialparameters are derived. Furthermore, Case 2 gives a restriction on evolution of the strain likeinternal variable.

A further part of the paper is devoted to numerical aspects. An implicit Euler backwardscheme is used for integration of the evolution equations, thus resulting in a (local) non-linearproblem of dimension 7. It is shown that this problem can be reduced to an equivalentone-dimensional problem for Case 2, whereas the dimension for Case 1 is two. The result-ing non-linear equation is solved with a Newton algorithm. Additionally, the correspondingalgorithmic tangent operator for the (global) finite element equilibrium iteration of the finiteelement structure scheme is derived.

An outline of this work is as follows: Section 2 introduces the yield function and the plasticpotential to formulate the evolution equation for the inelastic strains. Furthermore, consequencesof thermodynamic consistency are investigated. Section 3 describes the integration scheme andthe implementation into a finite element programme. For illustrative purposes, two examplesare presented in Section 4: firstly, the model is applied to simulate the yield strength differenceobserved experimentally by Schlimmer [1] for adhesive materials. A second example investigatesthe deformation evolution of a shear–tension specimen.

1.1. Notations

Square brackets [•] are used throughout the paper to denote ‘function of’ in order to dis-tinguish them from mathematical groupings with parenthesis (•). The three basic invariantsI1[A], I2[A], I3[A] of an arbitrary second-order tensor A are defined as

I1[A] = 1 : A, I2[A] = 12 1 : A2, I3[A] = 1

3 1 : A3 (1)

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1464 R. MAHNKEN AND M. SCHLIMMER

Here, 1 is a second-order unit tensor, defined as 1 ·u = u for arbitrary vector (first-order tensor)u. Consequently, the deviatoric/volumetric additive decomposition of A can be written as

A = Adev + Avol where Adev = Idev : A, Avol = 13 I1[A]1, Idev = I − 1

3 1 ⊗ 1 (2)

where I is a fourth-order unit tensor, defined as I : A = A.

2. CONSTITUTIVE EQUATIONS

2.1. Strain decomposition and Hooke’s law

Upon using standard notation within a geometrically linear theory, the following relations arevalid:

1. � = �el + �pl

2. �el = C−1 : �

3. C = 2GIdev + K1 ⊗ 1

(3)

Equation (3.1) expresses the additive decomposition of the total strain tensor � into an elasticpart �el and a plastic part �pl, and in Equation (3.2), the stress tensor � is related to theelastic strain tensor �el by the fourth-order elasticity tensor C. For the case of isotropic linearelasticity, C is given in Equation (3.3), where G and K denote the shear and bulk modu-lus, respectively, and the fourth-order projection tensor Idev is defined in Equation (2). Theplastic strain tensor �pl of Equation (3.1) is obtained from evolution equations as discussednext.

2.2. Yield function and plastic potential

A yield criterion for an elastic/plastic continuum in terms of the symmetric second-order stresstensor � can be written in the form �[�] = 0. For an isotropic material, the yield functionreduces as �[I1, I2, I3] = 0, where the three basic invariants I1, I2, I3 of the Cauchy stresstensor are determined according to Equation (1). Alternatively, the yield function can be recastinto the form �[I1, I

′2, I

′3] = 0, where the prime refers to invariants of the deviatoric stress

tensor �dev, i.e.

I1 = 1 : �, I ′2 = 1

2 1 : �2dev, I ′

3 = 13 1 : �3

dev (4)

As discussed extensively in Schlimmer [9] and Ehlers [10], a general yield function of thistype allows simultaneously for the two effects of pressure stress state and deviatoric stressstate dependence, where the latter can be different, e.g. in tension and compression. However,the experimental observations of these effects require sophisticated and careful measurementtechniques. This has been carried out, e.g. Spitzig et al. [3] for martensite materials, and theresulting experimental observations are simulated in Mahnken [7].

In order to simulate the experimental data currently available for adhesive materials, thethird invariant I ′

3 is disregarded, thus restricting our attention to the following yield

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SIMULATION OF STRENGTH DIFFERENCE 1465

function:1. � = I ′

2 − 13 �

2. � = Y 2 − a1Y0I1 − a2I21

3. Y = Y0 + R[ev]4. R[ev] = q(1 − exp[−bev]) + Hev

(5)

Remarks

1. The above yield function basically has the mathematical structure formulated by Schlimmer[9] in 1974. Let us recall from References [5, 9] that it reduces to certain well-knowncriteria. Then for the case a1 = a2 = 0, the von Mises cylinder is obtained. For Y = Y0,� = a1/2 �= 0, a2 = −a2

1/4 the right circular cone√

3I ′2 = Y0 − �I1 of Drucker and

Prager is retrieved. Furthermore, a1 = 0, a2 �= 0 establishes Green’s ellipsoid [10].2. The stress like internal variable R[ev] of Equation (5.4) plays the role of a hardening

function dependent on the strain like internal ev defined in the next section.3. The related material parameters characterizing � are Y0, a1, a2, q, b, H .

In order to account for a non-associated flow rule, additionally a plastic potential �∗ is intro-duced with the same mathematical structure as the yield function defined in Equation (5)

1. �∗ = I ′2 − 1

3 �∗

2. �∗ = Y 2 − a∗1Y0I1 − a∗

2I 21

(6)

RemarkCompared to the yield function � additional material parameters related to the plastic potential�∗ are a∗

1 and a∗2 .

2.3. Rate equations

For evolution of the plastic strain tensor, the following rate equation is derived from the plasticpotential as:

1. �pl = ���∗

��= ��dev + �t1

2. t = 13 (a∗

1Y0 + 2a∗2I1)

(7)

and where the plastic multiplier is obtained from the loading and unloading conditions

� � 0, �� = 0, � � 0 (8)

In order to formulate a rate equation for the strain like internal variable ev two differentapproaches are considered: for both cases, the following equivalence of dissipated powerexpressions are postulated:

Case 1. evY = � : �pl ⇒ ev = 1

Y� : �pl = �

Y(tI1 + 2I ′

2)

Case 2. evY0 = � : �pl ⇒ ev = 1

Y0� : �pl = �

Y0(tI1 + 2I ′

2)

(9)

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1466 R. MAHNKEN AND M. SCHLIMMER

RemarkFor the case a1 = a∗

1 and a2 = a∗2 the flow rule (7) is associative; otherwise it is non-associative.

2.4. Thermodynamic consistency

A thermodynamic argument requires the dissipation for arbitrary stress and strain states to benon-negative, see e.g. Truesdell and Noll [11]. Without going into details of the thermodynamicframework, the (reduced) dissipation as a consequence of the inelastic strain rate (7) is calculatedaccording to D = � : �pl − Rev � 0. Upon considering Cases 1 and 2 in Equation (9), thefollowing relations are obtained:

Case 1. D = (Y − R)ev = Y0ev � 0

Case 2. D = (Y0 − R)ev � 0(10)

Since Y0 is always non-negative, the requirement for Case 1 in Equation (10) is equivalent tothe following inequality:

� : �pl = �� : (�dev + t1)

= �� : �dev + �

3� : 1a∗

1Y0 + �

3� : 12a∗

2I1

= 2�I ′2 + �

3I1a

∗1Y0 + 2�

3a∗

2I 21

� 0

In the above derivations, additionally Equation (4) for the first stress invariant I1 and thesecond deviatoric stress invariant I ′

2 has been used. Several conclusions can be envisagedin order to satisfy the above inequality. Since, due to Equation (8), the plastic multiplier �and, furthermore, I ′

2 and I 21 are always non-negative, we choose the restrictions a∗

1 = 0,

a∗2 � 0 on the material parameters, thus satisfying inequality (10) for all stress and strain

states.Concerning Case 2 in Equation (10), additionally the term (Y0 − R) must be non-negative.

For the specific hardening function (5.4) with Y0, H, q � 0, this is satisfied by the restrictionev � (Y0 − q)/H . To summarize the following restrictions on the material parameters and thestrain like internal variable ev are sufficient for thermodynamic consistency of the constitutiveequations:

Case 1. Y0 � 0, a∗1 = 0, a∗

2 � 0

Case 2. Y0 � 0, a∗1 = 0, a∗

2 � 0, H � 0 q � 0 ev � Y0 − q

H

(11)

RemarkAs will be seen in Section 4.1, simulation of the experimental data for adhesive materials withstrength difference requires a1 > 0. From the remark in Section 2.3, it follows that the resultingflow rule is non-associative.

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SIMULATION OF STRENGTH DIFFERENCE 1467

3. NUMERICAL IMPLEMENTATION

3.1. Integration scheme for the evolution equations

Following standard integration procedures in finite element techniques a strain-driven algorithmis considered, where the total strain tensor (n+1)� and initial values (n)�pl, (n)ev are given at arepresentative time step (n)t . Then, it is the object to find the corresponding quantities (n+1)�pl,(n+1)ev at time (n+1)t = (n)t + �t consistent with the constitutive equations of the previoussections; see e.g. Reference [12].

For numerical integration of the rate equations (7) and (9), an Euler backward rule rendersthe following update scheme for the plastic strain tensor:

1. (n+1)�pl = (n)�pl + ��pl

2. ��pl = ��(n+1)�dev + ��(n+1)t1

3. (n+1)t = 13 (a∗

1Y0 + 2a∗2(n+1)I1)

(12)

and the strain like internal variable.

1. (n+1)ev = (n)ev + �ev

2. �ev =

⎧⎪⎪⎨⎪⎪⎩

��

Y((n+1)t (n+1)I1 + 2(n+1)I ′

2) for Case 1

��

Y0((n+1)t (n+1)I1 + 2(n+1)I ′

2) for Case 2

(13)

respectively, and where Case 1 and Case 2 refer to the distinction introduced in Equation (9).In order to simplify the notation, the index n + 1 referring to the actual time step will beomitted subsequently.

3.2. Volumetric/deviatoric split of the integration scheme

The integration scheme (12) can be written in terms of the volumetric and deviatoric stresstensor as follows: upon inserting Equation (12.1) into Equation (3.1), solving for �el andinserting into Equation (3.2) renders

� = �tr − C : ��pl, where �tr = C : (� − (n)�pl) (14)

Furthermore, using the relation

C : ��pl = 2G���dev + ��3tK1 (15)

the deviatoric/volumetric split of Equation (14) is obtained as

1. �vol = �trvol − K��3t1, �tr

vol = 13 I tr

1 1, I tr1 = 3K tr[� − (n)�pl]

2. �dev = �trdev − 2G���dev, �tr

dev = 2G(�dev − (n)�pldev)

(16)

where tr[� − (n)�pl] = 1 : (� − (n)�pl). Note that the trial stress tensors �trdev and �tr

vol aredetermined directly from the given quantities (n+1)� and (n)�pl; however, the final stress tensors�dev and �vol are obtained iteratively as discussed next.

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1468 R. MAHNKEN AND M. SCHLIMMER

3.3. Reduction of the integration scheme to a single scalar equation for Case 2

In the following, it will be shown that Equation (16) can be reduced to a single scalar equationanalogously to the radial-return scheme of elasto-plasticity, e.g. described in Reference [12].For reasons explained below, we will concentrate on Case 2 introduced in Equation (9).

Inserting Equation (12) into Equation (16.1) implies the scalar equation

13 I1 = 1

3 I tr1 − K��3t = 1

3 I tr1 − K��(a∗

1Y0 + 2a∗2I1)

which can be solved for I1 as

I1 = I tr1 − 3K��a∗

1Y0

1 + 3K��2a∗2

(17)

Upon rewriting, Equation (16.2) yields

��dev = �trdev, where � = (1 + 2G��) (18)

which means that �dev is coaxial to �trdev. Taking the inner product of both sides in Equation (18)

renders

�2�dev : �dev = �trdev : �tr

dev ⇒ �2I ′2 = I

′tr2 (19)

where analogously to Equation (4) the second invariant of the deviatoric trial stress tensorI

′tr2 = 1

2 �trdev : �tr

dev has been introduced.Using Equation (12.2) and Equation (19), the strain like internal variable of Equation (13.1)

can now explicitly be written as

ev = (n)ev + ��

Y0

(tI1 + 2

�2 I′tr2

)(20)

Note that this explicit formulation is not possible for Case 1 introduced in Equation (9), since,according to Equation (5) the hardening variable Y = Y [ev] is dependent on the strain likevariable ev .

Next, using the loading and unloading conditions (8) and furthermore Equation (5.1), itfollows that � = I ′

2 − 13 H = 0 for the case of loading, which allows the following non-linear

residual of Equation (19) in terms of the plastic multiplier

1. r[��] = 3I′tr2 − ��2 where

2. � = Y 2 − a1Y0I1 − a2I21

3. I1 = I tr1 − 3K��a∗

1Y0

1 + 3K��2a∗2

4. � = 1 + 2G��

5. Y = Y0 + R

6. R = q(1 − exp[−bev]) + Hev

7. ev = (n)ev + ��

Y0

(tI1 + 2

�2 I′tr2

)

(21)

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SIMULATION OF STRENGTH DIFFERENCE 1469

The above non-linear equation is solved iteratively with a Newton method

��(k+1) = ��(k) − 1

J (k)r[��(k)], k = 0, 1, 2, . . . , where J = �r

���(22)

From the relations (21), the following result is obtained for J :

1. J = −(

�2 ��

���+ 2��2G

)where

2.��

���= 2YY ′ ��e

���− (a1Y0 + 2a2I1)

�I1

���

3.�I1

���= −3K

I12a∗2 + a∗

1Y0

1 + 3K��2a∗2

4.��e

���= 1

Y0

(tI1 + 2

�2 I′tr2

)+ ��

Y0

(2a∗

2

3

�I1

���I1 + t

�I1

���− 8G

�3 I′tr2

)(23)

Note that additionally the notation Y ′ = dY/dev has been used.

3.4. Final stress and strain determination

After having determined the plastic multiplier ��, the final stress tensor is obtained as

� = �dev + �vol where �dev = 1

��tr

dev, �vol = 1

3I11 (24)

and the plastic strains are updated according to relations (11) and (13), respectively.

3.5. Algorithmic tangent modulus

The algorithmic tangent modulus necessary for applying a Newton method for iterative solutionof the global equilibrium problem requires the derivative of the stress tensor � defined inEquation (24) with respect to the total strain tensor �, i.e.

C = d�

d�(25)

Straightforward differentiation renders the following result:

C = 2G

�Idev +

(1

3

�I1

���1 − 2G

�2 �trdev

)⊗ ���

��+ B

1 + 3K��2a∗2

1 ⊗ 1 (26)

where �I1/��� is given in Equation (23.3). The term ���/�� is obtained as follows: weconsider the local problem (21) as an implicit function and conclude

r[�, �[��]] = 0 ⇒ dr

d�= �r

��+ �r

���

���

��⇒ ���

��= −J−1 �r

��(27)

where J is already defined in Equation (22). The result for �r/�� is as follows:

�r

��=

(a1Y0 + 2a2I1 − 2YY ′��t

1

Y0

)�23K

1 + 3K��2a∗2

1 +(

3 − 4YY ′��

Y0�2

)2G�tr

dev (28)

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1470 R. MAHNKEN AND M. SCHLIMMER

The above results for the stress tensor Equation (24), the plastic strains Equations (12) and (13)and the algorithmic tangent operator can be implemented into a general finite element pro-gramme. The example shown below has been obtained with the programme ABAQUS [13].

4. REPRESENTATIVE EXAMPLES

4.1. Simulation of experimental data

Experimental measurements for macroscopic characterization of strength difference of adhesivematerials are difficult to obtain and require advanced testing procedures. In the following, wedescribe briefly the procedure used by Schlimmer in Reference [1].

For preparation of the experiments, cylindrical specimen were glued as shown in Figure 1(a).The materials are aluminium for the cylindrical specimen and Betamate 1496 for the adhesivematerial, respectively. Then the specimens are subjected to combined loading in tension andshear as shown Figure 1(a). An experimental measurement technique according to Figure 1(b)is used in order to obtain the axial average deformation v = (v1 +v2)/2 and the circumferentialdeformations u and uF for the upper and lower part, respectively. Based on these quantities,axial strains and shear strains

�x = ln(

1 + v

d

), �xy = arctan

(u − uF

d + v

)(29)

are determined, where d = 2 mm denotes the thickness of the adhesive zone.It is obvious, that the above quantities are not fully consistent with the geometric linear

theory assumed in this paper. Therefore, the extension of the constitutive equations to a geo-metrically non-linear theory is of major importance in the ongoing project.

Figure 1. (a) Combined loading in tension and torsion for cylindrical specimen glued with Betamate1496; and (b) experimental measurement of axial and circumferential displacements.

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SIMULATION OF STRENGTH DIFFERENCE 1471

Table I. Shear/axial strain ratios in four different experiments.

Shear/axial strain ratioNo. Designation � = 1

2 �xy/�xx

1 ‘Tension’ 02 ‘Torsion’ ∞3 ‘G3E6’ 0.54 ‘G6E3’ 2.0

00 0.05 0.1 0.15

5

10

15

20

25

30

35

40

Axial strain [-]

Axial stress vs Axial strain

Axi

al s

tres

s [M

Pa]

Tension, Exp.

G3E6, Exp.

G6E3, Exp.

Tension

G3E6

G6E3

Figure 2. Axial stress versus axial strain for three different shear/axial strain ratios (cf. Table I).Dashed lines with symbols represent experiment, whereas solid lines represent simulation.

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4

Shear strain [-]

Shear stress vs shear strain

She

ar s

tres

s [M

Pa]

Torsion, Exp

G3E6, Exp.

G6E3, Exp.

Torsion

G3E6

G6E3

Figure 3. Shear stress versus shear strain for three different shear/axial strain ratios (cf. Table I).Dashed lines with symbols represent experiment, whereas solid lines represent simulation.

Concerning further strain variables, additional assumptions are needed for the combinedaluminium/adhesive specimen. In this way, it is assumed that due to the higher stiffness of thealuminium cylinder the additional axial strains are zero, i.e. �yy = �zz = 0. Consequently, in

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1472 R. MAHNKEN AND M. SCHLIMMER

Table II. Resulting material parameters for Betamate 1496 obtained from least-squares minimization.

� b a1 a2 a∗1 a∗

2E [dimen- Y0 q [dimen- H [dimen- [dimen- [dimen- [dimen-

[MPa] sion less] [MPa] [MPa] sion less] [MPa] sion less] sion less] sion less] sion less]

Value 1588.7 0.34 36.11 8.08 101.1 33.96 0.643 0.122 0.0 0.279

full detail, the coefficients of the strain tensor for axial and torsional loading are as follows:

�Axial =⎡⎢⎣

�xx 0 0

0 0 0

0 0 0

⎤⎥⎦ , �Torsion =

⎡⎢⎣

0 12 �xy 0

12 �xy 0

0 0 0

⎤⎥⎦ (30)

Experiments have been performed with four different ratios of �xy and �xx as summarized inTable I. Note that the first and second experiments refer to the special cases of pure tensionand pure torsion, respectively.

The final results of the experiments are summarized in Figures 2 and 3, respectively. Inparticular, the results demonstrate the dependence of the threshold value on the loading ratio �in Table I.

For simulation of the experimental effects, material parameters Y0, q, b, a1, a2, a∗1 , a∗

2introduced in Section 2 have to be determined by taking into account the constraints forthe strains of Equation (30). Furthermore, the restriction a∗

1 = 0 in Equation (11) is taken intoaccount. Several strategies can be envisaged for the task of parameter identification; see e.g.Reference [14]. For the problem at hand, a non-linear least-squares functional is formulatedconsidering the experimental data of Reference [1]. Then a Simplex Nelder algorithm isused for minimization; see e.g. Reference [15]. The resulting parameter set is summarized inTable II. From relation (13), we conclude that thermodynamic consistency of the constitutiveequations is guaranteed as long as the strain like internal variable satisfies

ev � Y0 − q

H= 0.82 (= 82%)

which is very much beyond a small strain theory. Therefore, for the purpose of this work,Case 1 is justified.

In addition to the experimental data, the result of the simulation is also shown in Figures 2and 3 for the axial stress and the shear stress, respectively. The comparison demonstrates thecapability of the model to simulate the strength difference including the hardening behaviourfor the different loading cases with accuracy sufficient for practical purposes.

4.2. Compact tension specimen with adhesive zone

In this example, the constitutive equations including the material parameters of Table I areused in order to simulate a compact tension specimen with an adhesive zone. The geometry ofthe specimen is shown in Figure 4. The specimen consists of two parts made out of steel withYoung’s modulus E = 210 GPa and Poisson ratio � = 0.3. These are assumed to be glued withthe adhesive material Betamate 1496 with a thickness of d = 5 mm. A total load of 2800 N is

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1461–1477

SIMULATION OF STRENGTH DIFFERENCE 1473

Figure 4. Compact tension specimen: geometry of the sample.

applied to the holes in vertical direction, thus resulting in a Mode I scenario for the crack tip.The finite element mesh is shown in the top left part of Figure 5, and the top right part ofFigure 5 shows the resulting deformed mesh with a scale deformation factor of 3. The bottompart of Figure 5 defines the location of the ‘Center Node’ for the results of Figure 7. Figure6 depicts the von Mises stress, the pressure (which is the negative first stress invariant dividedby three) and the strain like internal variable defined in Equation (9) in the adhesive zone.Figure 7 exhibits some load dependent quantities in the ‘Center Node’ of the adhesive zonenear the crack tip. Figure 7 (top) shows the von Mises stress and the pressure and Figure 7(bottom) the development of the strain like internal variable.

5. SUMMARY AND OUTLOOK

This contribution has concentrated on the numerical implementation of a constitutive model pro-posed by Schlimmer [1, 9], thus enabling to simulate the strength difference in elasto-plasticity

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1461–1477

1474 R. MAHNKEN AND M. SCHLIMMER

Figure 5. Compact tension specimen. Top left: undeformed mesh; top right: deformed mesh of adhesivezone with deformation scale factor equal to 3; bottom: location of ‘Center Node’.

for adhesive materials. Two different approaches (referred as ‘Case 1’ and ‘Case 2’) are pos-tulated for evolution of a strain like internal variable. Upon considering thermodynamic con-sistency of the model equations, for both Cases 1 and 2, some restrictions on the materialparameters are derived. Furthermore, Case 2 gives a restriction for evolution of the strain likeinternal variable. For the specific material Betamate 1496, this limit becomes 82%, which,however, is not significant for small strain problems.

An implicit Euler backward scheme is used for integration of the evolution equations, thusresulting in a non-linear local problem of dimension 7. It is shown that for Case 2, this problemcan be reduced to a one-dimensional problem, which is solved with a Newton algorithm. The

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1461–1477

SIMULATION OF STRENGTH DIFFERENCE 1475

Figure 6. Compact tension specimen. Top: von Mises stress; middle: pressure; bottom: strain likeinternal variable in the adhesive zone. The deformation scale factor is equal to 3.

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1461–1477

1476 R. MAHNKEN AND M. SCHLIMMER

-40

-30

-20

-10

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Load factor

stre

sses

[M

Pa]

von MisesPressure

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.4 0.6 0.8 1Load factor

inte

rnal

var

iab

le [

-]

Int. Variable

0.2

Figure 7. Compact tension specimen: Development of load dependent quantities ver-sus load factor in the ‘Center Node’ of Figure 5. Top: von Mises stress and pressure;

bottom: strain like internal variable.

dimension of the non-linear problem for Case 1 is two. Furthermore, the associated algorithmictangent operator for the global FE-equilibrium iteration scheme is derived.

The comparison of calculated and experimental results demonstrates the capability of themodel to simulate the related characteristic for a Betamate 1496 of strength difference andhardening effects.

The resulting algorithm has been obtained in the beginning of a project on simulation ofadhesive materials. In this respect, it has to be noted that the experimental quantities are notfully consistent with the geometric linear theory assumed in the model formulation. Therefore,the extension of the constitutive equations to a geometrically non-linear theory is of majorimportance in the ongoing work.

Furthermore, it has to be remarked that principally a different behaviour in tension andcompression can be simulated by incorporation of odd power terms for the first invariant of thestress tensor into the yield function. However, as observed, e.g. by Spitzig [3] for martensitematerials, the relation between the first and second invariant may not be identical in tension andcompression tests, if these are superposed by a hydrostatic stress state. Similar effects might

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SIMULATION OF STRENGTH DIFFERENCE 1477

occur for adhesive materials, which would require also the third invariant of the deviatoric stresstensor in future work; see e.g. Reference [7]. Further envisaging aspects are rate-dependenceand damage effects.

ACKNOWLEDGEMENTS

For providing the finite element topology of the compact tension specimen, Dr. Jendrny from theLaboratorium für Werkstoff- und Fügetechnik, University of Paderborn is gratefully acknowledged.

REFERENCES

1. Schlimmer M. Grundlagen zur Berechnung des mechanischen Verhaltens von strukturellen Klebverbindungendes Fahrzeugbaus. In Proceedings of Mechanisches Fügen und Kleben, Hahn O (ed.), 2003.

2. Altenbach H, Altenbach J, Zolochevsky A. Erweiterte Deformationsmodelle und Versagenskriterien derWerkstoffmechanik. Deutscher Verlag für Grundstoffindustrie: Stuttgart, 1995.

3. Spitzig WA, Sober RJ, Richmond O. Pressure dependence of yielding and associate volume expansion intempered martensite. Acta Metallurgica 1975; 23:885–893.

4. Zolochevskii AA. Modification of the theory of plasticity of materials differently resistant to tension andcompression for simple loading processes. Soviet Applied Mechanics (transl. Prikladnaya Mekhanika) 1989;24(12):1212–1217.

5. Ehlers W. A single-surface yield function for geomaterials. Archive of Applied Mechanics 1995; 65:246–259.6. Betten J, Sklepus S, Zolochevsky A. A creep damage model for initially isotropic materials with different

properties in tension and compression. Engineering Fracture Mechanics 1998; 59(5):623–641.7. Mahnken R. Strength difference in compression and tension and pressure dependence of yielding in elasto-

plasticity. Computer Methods in Applied Mechanics and Engineering 2001; 90:5057–5080.8. Mahnken R. Theoretical, numerical and identification aspects of a new model class for ductile damage.

International Journal of Plasticity 2002; 18:801–831.9. Schlimmer M. Fließverhalten plastisch kompressibler Werkstoffe. Dissertation, RWTH Aachen, 1974.

10. Green RJ. A plasticity theory for porous solids. International Journal of Mechanical Sciences 1972; 14:215–224.

11. Truesdell C, Noll W. The Nonlinear Field Theories of Mechanics (2nd edn). Springer: Berlin, 1965.12. Simo JC, Hughes TJR. Computational inelasticity, interdisciplinary applied mathematics. Mechanics and

Materials, vol. 7. Springer: Berlin, 1998.13. ABAQUS Finite Element Program, Version 6.3. Hibbit, Karlsson, Sorensen, Inc.: Providence, RI, USA, 2002.14. Mahnken R. Identification of material parameters for constitutive equations. In Encyclopedia of Computational

Mechanics, Stein E, de Borst, Hughes (eds), vol. 2. Wiley: Chichester, 2004.15. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes in Fortran. Cambridge University

Press: Cambridge, 1992.

Copyright � 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1461–1477