The Constitutive Models Suitable for Adhesives in … A J/PAJ Reports...The Constitutive Models Suitable for Adhesives in some ... The Constitutive Models Suitable for Adhesives in

NPL Report CMMT(B)130)

Project: PAJ1;Failure Criteria and their Application to Visco-Elastic/Visco-Plastic Materials

Report 3

The Constitutive Models Suitable for Adhesives in some Finite Element Codes and Suggested Methods of

Generating the Appropriate Materials Data

M. N. Charalambides

A Olusanya April 1997

NPL Report CMMT(B)130

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M. N. Charalambides A Olusanya

Centre for Materials Measurement and Technology National Physical Laboratory Teddington Middlesex, UK, TW11 0LW This report represents the deliverable for

Task 4. Milestone 15 Task 5. Milestone 19


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© Crown copyright 1997 Reproduced by permission of the Controller of HMSO ISSN 1361-4061 National Physical Laboratory Teddington, Middlesex, UK, TW11 0LW No extracts from this report may be reproduced without the prior written consent of the Managing Director National Physical Laboratory; the source must be acknowledged Approved on behalf of Managing Director, NPL, by Dr C Lea, Head, Centre for Materials Measurement and Technology


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ABSTRACT Commercial Finite Element Analysis (FEA) software packages are widely used for stress analysis of adhesive joints. This report is a survey of constitutive models implemented in various FEA packages which are suitable for adhesives. Elastic plastic models that take into account the pressure dependence of the plastic deformation are discussed. These models will be suitable for simulating behaviour of most structural adhesives, or adhesives whose operating temperatures are lower than their glass transition temperature. In addition, the stress/strain distributions in a bonded joint are such that the strain rate in the adhesive is not uniform. Therefore, material models where the strain rate dependence is taken into account are required. Various visco-plastic models, originally developed for metals are discussed. Material’s data that is required to define the elastic - plastic models are identified and experimental methods for their determination are outlined. Finally, large strain constitutive models that are suitable for simulating the behaviour of rubbery adhesives or for adhesives tested at temperatures above their glass transition temperatures are discussed. These are the linear viscoelastic, the hyperelastic and hyperfoam models. The hyperelastic and hyperfoam models are based on large strain constitutive theories developed for rubber. In ABAQUS a *VISCOELASTIC option exists which in combination with either the *HYPERELASTIC or the *HYPERFOAM materials option results in a non-linear, elastic, strain rate dependent material model.


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GLOSSARY c cohesion (ANSYS and LUSAS) Cij material parameters for hyperelastic model d cohesion (ABAQUS) Di material parameters for hyperelastic model E elastic modulus Eh hardening modulus from a yield stress vs. effective plastic strain plot Et hardening modulus from a yield stress vs. total strain plot

g i

P relative shear modulus of term i in Prony series

G limiting shear modulus at t®¥ G0 instantaneous shear modulus GR(t) shear relaxation modulus I1 first invariant of the stress tensor I1

' first invariant of the plastic strain tensor

I_

1 first invariant of deviatoric strain tensor for hyperelastic model

I_

2 second invariant of deviatoric strain tensor for hyperelastic model J volume ratio J2 second invariant of the deviatoric stress tensor J2

' second invariant of the deviatoric plastic strain tensor p pressure q von Mises equivalent stress W strain energy density function of hyperelastic material β friction angle (ABAQUS) γy shear plastic strain

ε.

e elastic strain rate ε p equivalent plastic strain

εcr

equivalent creep strain

ε.

p equivalent plastic strain rate

ε.

p plastic strain rate

ε. strain rate

ε.

p0 static equivalent plastic strain rate

ε. cr

equivalent creep strain rate

ε1 axial strain in uniaxial tension ε1e elastic component of ε1

ε1p plastic component of ε1 ε2 lateral strain in uniaxial tension ε2e elastic component of ε2 ε2p plastic component of ε2


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ε i engineering principal strains

ε ij component of the strain tensor εyc compressive plastic strain εyt tensile plastic strain (same as ε1p) ζ ratio of the yield stress in compression to the yield stress in tension corresponding to the same equivalent plastic strain

λ_

i deviatoric principal extension ratios λi principal extension ratios νe elastic Poisson’s ratio

νp plastic Poisson’s ratio νt Poisson’s ratio calculated from total strains νt

' Poisson’s ratio calculated from total engineering strains νi effective Poisson’s ratio, large strain case

σi principal stresses σij component of the stress tensor

σin stress at first yield σn hydrostatic stress (=I1/3) σy yield stress at arbitrary value of equivalent plastic strain rate σy0 static yield stress σyc yield stress in compression σyt yield stress in tension τi relaxation time of term i in Prony series τy yield stress in shear φ angle of internal friction (ANSYS and LUSAS) φf dilation angle (ANSYS) ψ dilation angle (ABAQUS)


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CONTENTS

1. Introduction 8

2. Elastic - Plastic Models 8

3. Viscoplastic Models 14

4. Determination Of Material Parameters 16

4.1. Elastic Properties 16

4.2. Hardening Curves 16

4.3. Calculation Of The Dilation Angle 18

4.4. Determination Of Strain Rate Dependence Of Yield 19

5. Constitutive Models For Large Strain Materials 20

5.1. Linear Viscoelasticity 20

5.2. Hyperelasticity 21

5.3. Combination Of Hyperelastic And Viscoelastic Material Models In Abaqus 24

5.4. Elastomeric Foam Behaviour 25

6. Conclusions 27

7. References 27

Appendix 1:The Creep‘ Power Law’ 28

Appendix 2 Evaluation Of The Parameter ζ From Tensile And Shear Tests 29

Appendix 3 Evaluation Of νP When The Latter Varies With Strain 31

Appendix 4 Evaluation Of Plastic Strain Rate For A Constant Strain Rate Test 32


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1. INTRODUCTION Finite Element Modelling (FEM) or Finite Element Analysis (FEA) is used extensively in stress analyses of adhesive joints. This is due to the complex stress/strain distributions that exist in such joints and the inability of analytical methods to cope with such complexities. When using commercial FEA software packages, an important step in the analysis is to define the material’s constitutive behaviour. The basis of this report is to identify constitutive models which are suitable to model adhesive behaviour. Adhesives are viscoelastic materials. As a result, a differences in the stress - strain curve will be noted for changes in the applied strain rate. This effect is most pronounced when the adhesive is operating at temperatures close to its glass transition temperature; where changes in the stress - strain curve are observed for small changes in the strain rate. For glassy, structural adhesives, this effect is less noticeable due to the fact that their glass transition temperature is above the usual operating temperatures. In addition, the stress - strain curves of adhesives are usually highly non-linear with plastic flow possibly occurring at high strains. Research on conventional, i.e. metallic materials, has been performed for a considerably longer time compared to polymeric materials. From this research, extensive constitutive theories exist for metals that take into account plasticity and the strain rate dependence of yield. Another ‘material’ that has been studied extensively is soil, for which ‘plasticity’ theories have been developed that account for a pressure dependence for plastic deformation. These models have been implemented in FEA software packages. This report summarises the constitutive models which could be utilised to model adhesives, in addition to those that have been developed specifically for polymeric materials. In this report, sections 2 and 3 discuss the plastic and viscoplastic constitutive models and section 4 describes the experiments required to determine the various material parameters that appear in the models. Section 5 describes the constitutive theories developed for rubbers and similar materials which can be used to model the flexible polyurethane, polybutadiene types of adhesive. Section 6 is a brief summary of the main conclusions and recommendations. 2 ELASTIC - PLASTIC MODELS In this section, it is assumed that the material is elastic - plastic. In the initial, linear region, the theory of elasticity using Hooke’s law is valid. In the non-linear region, plastic deformation is assumed to take place. The stress analysis when plastic deformation is occurring is described below. Rate independent plasticity is based on three fundamental concepts: i) the yield criterion; determines the stress state necessary for plastic deformation ii) the hardening rule; describes how the material’s resistance to further yield changes with

increasing strain


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iii) the flow rule; defines the incremental plastic strains as a function of the stresses. For metals, the classic von Mises yield criterion is usually assumed. This produces a yield criterion independent of the hydrostatic component of stress, Equation 1:

( ) ( ) ( )[ ]σ σ σ σ σ σ σ1 2 2 1yt 3 3 2J= − + − + − =12

32 2 2

(1)

where σi are the principal stresses σyt is the tensile yield stress and J2 is the second invariant of the deviatoric stress tensor. In general, the tensile yield stress need not be constant, e.g. the material could strain harden. Plastic deformation is assumed to take place under constant volume conditions, e.g. the ‘plastic’ Poisson’s ratio, νp, is equal to 0.5. The independence of yield on hydrostatic stress implies that the tensile and compressive yield stresses are the same. However, studies on polymers reveal that, the chain structure generally leads to higher yield stress in compression than in tension. Hence, a deviation from the classical von Mises yield criterion is observed. Two main forms of pressure dependent criteria appear in the literature related to the yield of adhesives 1,2. They are usually referred to as the Drucker-Prager and Raghava’s criteria and are given by Equations 2 and 3 respectively:

( ) ( )σ

ζζ

ζζyt 2 1

+12

J-1

2I= +

3 (2)

( )ζσ ζ −1 σyt

22 yt 1J I= +3 (3)

where: I1 is the first invariant of the stress tensor and ζ is the ratio of the yield stress in compression to the yield stress in tension corresponding to the same equivalent plastic strain, ε p . The equivalent plastic strain and the equivalent stress are scalar quantities that allow the analysis of situations where multi-axial loading is present. For uniaxial tension they reduce to the tensile plastic strain and tensile stress, respectively. Equation (2) has originated from deformation theories of soils 3 and it has been subsequently used for polymers4. Equation (3) was proposed specifically for polymers 5. Both equations revert to equation (1), for ζ=1. Experimental methods for determining ζ are discussed in section 4.


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Equations 2 and 3 can be found in various FEA packages. For example, ABAQUS 6 has a general form of the Drucker-Prager criterion, called the ‘linear extended Drucker-Prager’ model, Equation 4: d = t - ptanβ (4)

where p is the pressure and is defined as -I1/3 β is the friction angle of the material d is the cohesion of the material and t is defined as, Equation 5:

t =12

q 1+1K K

rq

− −

11

3

(5)

where r is the third invariant of the deviatoric stress tensor q is the von Mises equivalent stress which from equation 1, is equal to 3J2 .

For K=1, t is equal to q and yielding is independent of the third deviatoric stress invariant. Therefore Equation 4 can be rewritten as:

d = 3JI

2 + tanβ 1

3 (6)

Multiplying Equation 6 by the factor ζ

ζ+

12

and comparing with Equation 2, the following identities

are derived:

tanβζζ

=−+

3

11

(7)

d =2

ytζ

ζσ

+

1 (8)

The data that is required to be defined for this ‘linear’ model are β , K and the yield stress in tension, compression or shear. The options of perfect plasticity, constant yield stress, or isotropic hardening, yield stress is a function of von Mises equivalent plastic strain, are available.


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The von Mises equivalent plastic strain, ε p , is equal to, Equation 9:

ε p 2'J=

23

(9)

where J2

' is the second invariant of the deviatoric plastic strain tensor. Note: in deriving Equation 9, it is assumed that the ‘plastic’ Poisson’s ratio is 0.5. Tensile, compressive or shear data can be used to define isotropic hardening. In all three cases, the data have to be converted to corresponding data of von Mises equivalent stress, Equation 1 and strain, Equation 9. A further parameter that is required to be defined is the dilation angle, ψ. This parameter appears in the flow rule and determines whether the flow rule is associated or non-associated. Associated flow implies that the plastic strain increments are normal to the yield surface and under such conditions ψ = β . The tangent of ψ is defined as the ratio of the volumetric plastic strain to the von Mises equivalent plastic strain, Equation 10:

tan =I

23

J

1'

2'

ψ (10)

where I1

' is the first invariant of the plastic strain tensor and J2

' is the second invariant of the deviatoric plastic strain tensor. Under uniaxial tensile loading conditions, the invariants I1

' and J2' are given by:

( )I1'

p yt= −1 2ν ε and ( )J 2'

p yt2= +

13

12

ν ε (11)

where εyt is the plastic strain in the loading direction and νp is the ‘plastic’ Poisson’s ratio, e.g. the ratio of the transverse to the axial plastic strains. Substitution of Equation 11 into Equation 10 results in the following relationship between ψ and νp,

Equation 12:


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( )( )tan = p

p

ψν

ν

3 1 2

2 1

−

+ (12)

For νp=0.5, e.g. yield at constant volume, ψ=0. When ψ is greater than zero, the material expands during plastic deformation and νp is smaller than 0.5. Section 4 describes an experimental method for determining νp and hence, ψ. The Drucker-Prager yield criterion (Equation 2) is also implemented in LUSAS 7 and ANSYS 8, in the form shown in Equation 13:

( ) ( )63 3

6ccos(3 3

0sin(

sin( sin(φ)σ

φ)φ)

φ)n

2J−

+ −−

= (13)

where φ is the angle of internal friction c is the cohesion and σn is the hydrostatic stress (=I1/3). By rearranging and comparing Equations 2 and 13, the following relationships between c, φ, σyt and ζ are obtained:

( ) ( )sin φ =

−+

3 13 1ζζ

(14)

c =6

ytζσ

ζ − 2 (15)

The parameters that need to be defined for this model are the initial values of c0 and φ0, when the equivalent plastic strain is zero. The option of isotropic hardening is available in LUSAS and it is defined by giving the gradients of the cohesion and the friction angle as a function of the equivalent plastic strain. The ANSYS FEA package allows the definition of the dilation angle, φf, as an option.


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Raghava’s yield criterion is implemented in LUSAS, in exactly the same form as Equation 3. The parameters that are required to be defined are the elastic properties and the initial values of the tensile and compressive yield stresses. Isotropic hardening is optional and an assumption of associated flow is made. Raghava’s criterion also exists as an option in ABAQUS under the terminology the ‘exponent Drucker-Prager’ model, Equation 16: aq - p = pb

t (16) where a and b are material parameters and pt is a hardening parameter. Assuming the special case of b=2, rearranging and comparing with Equation 3, the relationships for a and the hardening parameter pt, can be derived, Equations 17 and 18:

( )a =1

3 yζ σ− 1 (17)

( )p ty=

−ζσζ3 1

(18)

The data that is required to be defined for the ‘exponent’ model are a, b, ψ and the yield stress in tension, compression or shear. The options of perfect plasticity, or isotropic hardening are once again available. The flow for this model is assumed to be non-associated. A less commonly used pressure dependent yield criterion is the Mohr-Coulomb criterion. This has also been developed from constitutive theories developed for soils 3. It is available in LUSAS, Equation 19: ( ) ( )σ σ σ σ φ φ1 3 1 3 2ccos 0− + + − =sin (19) where σ1 and σ3 are the maximum and minimum principal stresses. In order to define c and φ in terms of ζ and σyt, the loading conditions corresponding to tension (σ1

= σyt, σ2 = σ3=0) and compression (σ1 = σ2=0, σ3 = -ζσyt) are substituted into Equation 19 yielding simultaneous equations that can be solved for c and φ, Equation 20:

sin =φζζ

−+

11

and c = ytσ ζ2

(20)

This model has similar data requirements to the Drucker-Prager model, Equation 13.


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3. VISCOPLASTIC MODELS For some adhesives, strain rate dependence can not be ignored. The deformations in a typical joint are such that the strain rate is not uniform throughout the adhesive. Hence, if any of the constitutive theories described in section 2 are to be used, options have be available in the FEA software which will account for the extra variable, i.e. strain rate. There are two ways of introducing a strain rate dependent yield. One is to separate the strain and strain rate effects and the other is to combine both dependencies into the material’s parameters. Here, the material parameter that is assumed to vary with strain rate is σyt. The Cowper and Symonds model is a commonly used strain rate dependent yield model that separates the strain rate from the strain effects. This model assumes that the hardening curves corresponding to the various strain rates are ‘similar’ and hence can be defined by a single ‘static’ hardening curve and a scaling factor. For example, in LS-DYNA 3D 9, material model 24, assumes von Mises yield with the following relationship for the static yield stress, σyt0:

σ σ εyt0 in pf= +

_ (21)

where, σin is the stress at first yield and f pε_

is a hardening function that can be defined either in

tabular form or in terms of linear hardening (bilinear model). The strain rate dependence is taken into account by the Cowper and Symonds model that modifies the ‘static’ yield stress, σyt0, with the scaling factor:

σ

σεyt

yt0

p

P

C= +

1

1.

(22)

where ε.

p is the equivalent plastic strain rate and C and P are material parameters that need to be defined. A very similar model also exists in ABAQUS. Here the scaling factor is defined in terms of the ‘power law’:

εσσ

.p

yt

yt0

m

F= −

1 for σyt/σyt0 > 1.0 (23)

where F and m are material properties that need to be defined. Equation (23) can be used in conjunction with the ‘linear’(equation 4) and ‘exponent’ (equation 16) models.


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An alternative option which is available in ABAQUS and LS-DYNA 3D is the capability to define

the scaling factor directly as a function of the equivalent plastic strain rate, ε.

p . This requires

numerical data in the form of (σyt/σyt0) vs. ε.

p . Another example of separable strain and strain rate dependent yield model is material model 15 of LS-DYNA 3D. This assumes von Mises yield (equation 1) and strain rate dependence is defined by:

σσ

ε

ε

yt

yt0

p

p

bln= +

10

.

. (24)

where b is a material constant that needs to be defined and ε.

p0 is the equivalent plastic strain rate of the ‘static test’, that is when σyt is equal to σyt0. Equation (24) is known as the Eyring model. A yield model where the strain and strain rate effects are not separable is given by material 19 in LS-DYNA 3D. The von Mises yield criterion is assumed and hardening is defined as:

σ σ ε ε εyt in p h p pE=

+

. . _

(25)

where Eh is the hardening modulus from a yield stress vs. effective plastic strain plot and σin is the

initial yield stress. Both Eh and σin are functions of the effective plastic strain rate, ε.

p . Therefore, there is no separation of the strain and strain rate effects. LS-DYNA 3D calculates Eh from E (elastic modulus) and Et (hardening modulus from a yield stress vs. total strain plot) using Eh=Et/(1-Et/E). Definition of E and Et as functions of effective plastic strain rate is required to lead to a strain rate

dependent Eh. Defining E as a function of ε.

p is a very useful option when modelling ductile adhesives, especially where changes in strain rate lead to changes in the whole of the stress - strain curve and not just in the ‘plastic’ part of the curve. However, the disadvantage of this specific model of LS DYNA 3D, is that it does not take into account the pressure dependence of yield. Another option for defining a non separable, strain and strain rate dependent yield, is to give complete hardening curves as numerical stress - strain data for various strain rates. This may lead to a large amount number of data having to be entered. It can be used in both ABAQUS and LS-DYNA 3D, with the Drucker-Prager and von Mises models respectively.


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In addition to all the models that were described above, most FEA packages also include classical metal creep models. These are also viscoplastic models where the inelastic strains are the product of two independent processes: plastic strains that are time independent and creep strains that are time dependent. The usual von Mises yield criterion is used to define the onset of plastic deformation whereas creep occurs when the body is stressed. The use of such models involves the definition of a ‘creep law’, e.g. a function that defines the equivalent creep strain rate in terms of the von Mises equivalent stress, temperature and time (or strain). These are derived from uniaxial experiments. Some of the most widely used models (for example the power creep law) are implemented in FEA packages. In addition, in some packages an option exists which enables the definition of a user’s own creep law. The main difference between the creep constitutive models and the models described above, is in the time frame over which the deformation occurs; in creep, the deformation takes place over a much larger time frame. Appendix 1 give an example of a creep law and the experimental determination of its parameters is given in. 4. DETERMINATION OF MATERIAL PARAMETERS This section describes experimental methods for determining the material parameters required in the constitutive models described in sections 2 and 3. 4.1 ELASTIC PROPERTIES The elastic parameters that are required are Young’s modulus, E and the elastic Poisson’s ratio, νe. Young’s modulus may be determined from a uniaxial tensile test and Poisson’s ratio may be determined during the same test, by taking simultaneous measurements of the lateral contraction of the specimen. An extensometer or any other device for measuring the displacement accurately is required for both measurements. Note that if the shear modulus, G, is known or measured, νe can be calculated from (E-2G)/2G. 4.2 HARDENING CURVES The tensile stress at first yield and the complete tensile hardening curve are obtained from a tensile test. The hardening data are represented according to the limitations of the specific FEA software. For example, some packages allow only bi-linear representation, others allow for multi-point definitions. For many adhesives, the hardening curve will extend to strain values in excess of 10% and the cross sectional area of the sample will significantly reduce with increasing strain.


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In such cases the engineering stress, σ ' , should be converted to true stress, σ, and the engineering strain, ε ' , should be converted to true or logarithmic strain, ε, using:

( )σ

σ

ν ε=

−

'

' '12

t

(26)

and

( )ε ε= ln 1 + ' (27)

where νt

' is the ratio of the lateral to tensile total engineering strains obtained from a uniaxial tensile test. The parameter ζ appears in all forms of pressure dependent criteria. It is defined as the ratio of the yield stress in compression, σyc, to the yield stress in tension, σyt, corresponding to the same equivalent plastic strain, ε p . In the discussion that follows it is assumed that the stress - strain curves in tension and compression, engineering stress and strain are converted to true stress - true strain plots. In cases where tensile and compressive data are available, ζ can be determined as follows. The yield point in both tensile and compressive stress - strain curves is determined and plots of σyc vs. εyc (compressive plastic strain) and σyt vs. εyt (tensile plastic strain), are constructed. Note that the plastic strains are calculated by subtracting the elastic strains (equal to current stress divided by the elastic modulus) from the total strains. The parameter ζ is then calculated by dividing values of σyc and σyt that correspond to the same slope, that is σyc/εyc= σyt/εyt. This will ensure that the ε p

corresponding to σyc and σyt are the same 2 . This procedure results in values of ζ that are generally functions of ε p . If the dependence of ζ on ε p is not very strong, an average value, valid for the range of plastic strains of interest, can be used. Compression tests can lead to inhomogeneous stress - strain states due to frictional effects between the sample and the loading platens. Evidence of such an occurrence is the ‘barrelling’ of cylindrical samples in compression, with the diameter half way through the length of the compressed sample being larger than the diameter at the top and bottom surfaces. This effect may be reduced by the use of suitable lubricants but it is usually extremely difficult to eliminate it completely. The use of longer specimens will also reduce the frictional effects, however these might lead to buckling. For this reason, instead of performing tensile and compressive tests, a more reliable method is to perform tensile and shear tests. With plots of σyt vs. εyt and τy (shear yield stress) vs. γy (shear plastic strain) now available, the shear plastic strains are calculated by subtracting the elastic strain components (equal to current stress divided by the shear modulus) from the total strains.


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Using equations (2) and (3), the relationships between τy and σyt corresponding to the same ε p are (see Appendix 2):

( )στ

+ 12

yt

y

=3 ζ

ζ (28)

στ

yt

y

3=

ζ (29)

Equations (28) and (29) are for the Drucker-Prager and Raghava criteria, respectively. The procedure for calculating ζ is described below. The proof of the method is given in Appendix 2. i) From corresponding σyt and εyt values calculate the ‘tensile’ slope, St = σyt/εyt. ii) Calculate the corresponding ‘shear’ slope, Ss, from Ss=St/2(1+νp), where νp is the ‘plastic’ Poisson’s ratio (see Section 4.3). iii) Determine the values of τy and γy such that τy/γy=Ss. This ensures that the shear and tensile data correspond to the same ε p (see Appendix 2). iv) Calculate ζ by substituting the corresponding σyt and τy in equation (28) or (29). This procedure should be repeated for various points on the σyt vs. εyt curves. It is important to realise that depending on the type of adhesive tested, ζ may remain constant or vary with strain. If ζ is constant and the tensile and shear stress - strain data show plateau stress values at large strains, the determination of ζ is greatly simplified. The shear and tensile plateau values are substituted in equations (28) or (29) to calculate ζ. 4.3 CALCULATION OF THE DILATION ANGLE The last parameter that needs to be defined in the rate independent elastic - plastic models, is the dilation angle, ψ. This can be calculated using equation (12), if the Poisson’s ratio, νp, in the plastic region, i.e. the ratio of the lateral to tensile plastic strains, is known. In the following discussion, it is assumed that the stresses and strains are converted to true stresses and true strains. A procedure for the determination of νp is given below. A uniaxial tensile test is performed during which the tensile extension and lateral contraction are measured. A plot of νt vs. tensile strain is constructed, where νt is the ‘total’ Poisson’s ratio, i.e. the ratio of the lateral to tensile total strains. The initial portion of such a plot, corresponds to the elastic region and will be a horizontal line, i.e. νt will be constant and equal to the ‘elastic’ Poisson’s ratio, νe. Thereafter, depending on whether the νp is higher, equal or lower than the νe, the ‘total’ Poisson’s ratio, νt, will increase, remain constant or decrease, with strain.


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For all these cases, the value of νt corresponding to large strain values, will be equal to νp, as the deformation will essentially be plastic, i.e. the ‘total’ Poisson’s ratio will tend to a value corresponding to the ‘plastic’ Poisson’s ratio, for large strains. The above is summarised by the these relationships: νt = νe for ε≤ε t0 where ε t0 is the total strain at first yield νt < νe for ε>ε t0 and νp < νe νt = νe for ε>ε t0 and νp = νe νt > νe for ε>ε t0 and νp > νe

νt ≈ νp for large values of ε. This procedure is not suitable for the case where νp varies with strain. Most FEA packages do not allow a definition of varying ψ or νp, however, the procedure for calculating the plastic νp in such cases, as a function of strain, is described in Appendix 3. Finally, the ‘total’ Poisson’s ratio calculated from engineering strains, νt

' , is also required to convert engineering stresses to true stresses. The ‘plastic’ Poisson’s ratio, νp, is required for the calculation of ζ (Section 4.2). 4.4 DETERMINATION OF STRAIN RATE DEPENDENCE OF YIELD To determine the effect of the strain rate on the material’s behaviour, additional tests, usually uniaxial tensile tests, are required for a number of strain rates. For the case of uniaxial tension, the equivalent

plastic strain ε p and equivalent plastic strain rate ε.

p are equal to the tensile plastic strain, εyt, and

tensile plastic strain rate, ε.

yt . As already discussed in section 3, there are three popular methods for defining the strain rate dependence of yield in FEA packages. These are:

i)direct entry of numerical data of σyt vs. εyt for various ε.

yt ,

ii)entry of the corresponding data of scaling factor (σyt/σyt0) vs. ε.

yt , iii)approximation of the data with one of the models described in section 3. For example, if the

‘power law’ of equation (23) is to be used, plots of log ε.

yt vs. log (σyt/ σyt0 -1) should be constructed. The material parameters F and m can then be calculated from the intercept and the gradient of the line that best fits the experimental data.


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All of the above methods require the experimental data to be defined in terms of various constant

values of ε.

yt . However, the tests are usually performed under constant strain rate, ε.

. Appendix 4

describes a method for determining ε.

yt for a constant strain rate (ε.

) test. 5. CONSTITUTIVE MODELS FOR LARGE STRAIN MATERIALS The plasticity models described in sections 2 and 3 are believed to be the most suitable for modelling impact of glassy, structural adhesives that are operating at temperatures below their glass transition temperature. For adhesives operating close to or above their transition temperature, the non-linear appearance of the stress - strain curve might not be predominantly due to plastic flow but due to dissipative losses caused by viscous effects or to non-linear elastic effects. In such cases, it might be more precise to use a viscoelastic constitutive model. 5.1. LINEAR VISCOELASTICITY Small strain, linear, viscoelastic models have been developed and are implemented in ABAQUS and ANSYS. These can be used to model adhesives whose stress - strain curves are essentially linear and depend on rate of deformation as well as temperature. However, linear viscoelasticity will generally lead to non-linear stress - strain curves, due to relaxation effects occurring during the loading phase of the test. Using the terminology of ABAQUS, the time dependent material behaviour is approximated with the Prony series for the shear relaxation modulus, GR(t):

( ) ( )G tG

g eR

0i

P -t/

i=1

Ni= − −∑1 1 τ (30)

where N (number of terms in Prony series), g i

P (relative modulus of term i) and τi (relaxation time of

term i) are material parameters that need to be defined. The maximum allowable value of N in this

model is 13. G0 is the instantaneous shear modulus and it is related to g i

P through:

G G + G g0 0i=1

N

i

P= ∑ (31)

where G is the limiting shear modulus at t→∞. The instantaneous shear modulus G0 has to ‘correspond’ to the Young’s modulus and the Poisson’s ratio entered in the elastic definition of the material, i.e. E and ν should be obtained from a high rate test. An independent relaxation function for the bulk modulus, similar in form to equation (30), is an option.


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Establishing GR(t) from equation (31), enables σ(t) as a function of strain history or the stress - strain curves at specified strain rates to be calculated. For example for a shear test, in which a time varying strain γ(t) is applied, the response τ(t) is given by:

( ) ( )τ

γ= ∫G t - s

d sds

dsR0

t

(32)

Equation (32) is based on the Boltzmann superposition principle for evaluating the response for a number of simple loading programmes.

The constants g i

P and τi in equation (30), may be entered directly or calculated by ABAQUS from

experimental shear relaxation or creep curves. In the latter case, the experimental data has to be entered as ‘normalised shear modulus (GR(t)/G0) or compliance (JR(t)/J0) vs. time’ data, G0 and J0 being the instantaneous shear modulus and compliance respectively. This option is not available in ANSYS, where material constants have to be entered directly. In ABAQUS and ANSYS, the shift function describing the effects of temperature on the viscoelastic material behaviour by the reduced or pseudo time concept can be defined. In most cases linear relaxation effects alone are not enough to explain the amount of non-linearity present in the stress - strain curve and non - linear viscoelastic models should ideally be used. However, these are complex models where the Boltzmann superposition principle used in linear viscoelasticity is no longer valid in its original, unmodified form. This means that the assumption that, each increment in stress makes an independent contribution to the overall strain, is not correct. Because of their complexity, non-linear viscoelastic models have not yet been implemented in the widely available FEA packages. However, some rubbery adhesives could be modelled in ABAQUS which allows the combination of the linear viscoelastic model with a non-linear elastic or a hyperelastic model 5.2. HYPERELASTICITY Hyperelastic models exist in both ABAQUS and ANSYS packages. These are large strain, non-linear, elastic models based on constitutive theories of rubbers. They can be used to model non-linear materials that extend to large, recoverable strains and are almost incompressible, i.e. the Poisson’s ratio is in the range of 0.45 to 0.5. When this model is used on its own, the assumption is made that the material is strain rate independent. Hyperelastic materials are described in terms of an elastic potential scalar function W (or strain energy density function) whose derivative with respect to a strain component, ε ij, determines the corresponding stress component, σij. This can be expressed by:

σ∂∂εij

ij

W= (33)


22

The strain energy density in ABAQUS is given by:

( )W = C I I1D

Jiji+ j=1

N _

1

i _

2

j

iel

i=1

N 2i

∑ ∑−

−

+ −3 3 1 (34)

where N is a material parameter that defines the number of terms in the strain energy function and Cij and Di are temperature dependent material parameters. The maximum value of N is 6.

I_

1 and I_

2 are the deviatoric strain invariants and they are equal to:

I_

1 = + +λ λ λ12

22

32 (35)

I_

2 = + +− − −

λ λ λ12

22

32 (36)

where λ_

i are the deviatoric principal extension ratios (λ i = J-1/3iλ ), J is the volume ratio (equal to

λ1λ2λ3) and λi are the principal extension ratios, i.e. the ratio of the deformed length to the original length. The relationship between the extension ratio, λi and the engineering strain, ε i, is: λi = +1 ε i (37) The last term of equation (34) represents the hydrostatic (volumetric) work. Jel is defined as:

( )JJ

Jelth th

= =λ λ λ

λ λ λ1 2 3

1 2 3

(38)

where ( )λ λ λ1 2 3 th

is the volume change due to temperature changes.

For incompressible behaviour,λ λ λ1 2 3 =1, Di are set to zero and equations (35) and (36) can be written as:

I_

1 = + +λ λ λ12

22

32 (39)

I_

2 = + +λ λ λ λ λ λ12

22

12

32

22

32 (40)

Assuming incompressible behaviour and N=1, W becomes:

( ) ( )W = C I C I10 1 01− + −3 32 (41)

Equation (41) is the classical Mooney-Rivlin law.


23

For a uniaxial tension test, λ1 = λ, λ2 = λ3= λ−1/2, substitution into equation (41) and differentiating with respect to λ yields the following relationship for the uniaxial engineering stress:

σ λλ λ

= 2 C C10 01−

+

1 12

(42)

For N=2, there are five terms in W, for N=3, nine terms etc. ANSYS also supports hyperelastic materials with a model similar to the polynomial form of ABAQUS, however the maximum value of N is limited to 3. The material parameters for up to N=2 can be determined by ABAQUS and ANSYS from a linear regression fit of experimental data. For good characterisation of the test material it is recommended that three experiments are performed, these are uniaxial, equibiaxial and planar (pure shear) tests. A uniaxial tension test is usually preferred over a uniaxial compression test as the latter usually suffers from an inhomogeneous stress - strain state due to frictional effects between the sample and the loading platens. The tension test is performed by pulling a dumbbell shaped sample. Note that if both compression and tension test data are used, difficulties might arise in the least squares fit, leading to inaccurate calculations of the model’s parameters. Tension tests are also usually preferred in equibiaxial experiments over the compression tests due to practical difficulties with the experimental set-up. In addition, it can be shown that for a fully incompressible material, the compressive equibiaxial test is equivalent to the tensile uniaxial test. The equibiaxial tension test is performed on square sheets with gripping tabs in the shape of a cruciform, in a biaxial testing machine. Planar (or pure shear) tests are performed on thin, short and wide rectangular samples. Wide loading grips are used to grip the sample along its wide edges and they are moved apart either in tension or compression. The planar tension and compression tests can be shown to be equivalent for an incompressible material. When it is necessary to allow some compressibility in the material model, the coefficients Di must also be given. For a general case, these can be calculated by the FEA package if volumetric test data as pressure vs. volume ratio are defined.


24

Where volumetric test data are not available but the Poisson’s ratio, ν, is known from uniaxial test data, the following procedure is suggested: i) Choose N=1. This should be a sufficiently accurate model for strains less than 100%. ii) Enter the uniaxial, equibiaxial and planar test data in a preliminary run and allow the FEA

package to calculate the constants C10 and C01, i.e. assume incompressibility. iii) Calculate the initial Young’s modulus from:

( )E C + C0 10 01= 6 (43) This should agree with the slope at the origin of the uniaxial nominal stress - strain curve. iv) From E0 and ν calculate the initial bulk modulus, K0:

( )K =E

31- 200

ν (44)

v) Calculate D1 from:

DK1

0

=2

(45)

vi) Define the material’s hyperelastic behaviour by entering directly C10, C01 and D1 and

continue with the analysis. Non zero values of Di affect the uniaxial, equibiaxial and planar stress results. However, since the material is assumed to be only slightly compressible (ν = 0.45 to 0.5), the techniques described for obtaining the Cij should give sufficiently accurate values even though they assume that the material is fully incompressible. 5.3. COMBINATION OF HYPERELASTIC AND VISCOELASTIC MATERIAL MODELS IN ABAQUS In ABAQUS, it is possible to combine the hyperelastic and the linear viscoelastic models. This results in non-linear, strain rate dependent model that may be applicable for some rubbery adhesives (or adhesives tested above their glass - transition temperature). As mentioned earlier, no plasticity exists in this model. The deformation is elastic and is recovered upon unloading with time.


25

This is not a non-linear viscoelastic model, the viscoelasticity is still linear and its governing equations are similar to the ones described in section 5.1, i.e. the time dependent part of the material’s response, is not affected by strain. The difference is that the relaxation coefficients of equation (34), are applied to the constants that define the hyperelastic behaviour, i.e., GR(t)/G0 is replaced by

( )C t CijR

ij0/ (polynomial form) or ( )µ µi

Ri0t / (Ogden form). When this combined model is used, it is

only the behaviour at long times that is dependent on strain, i.e. hyperelastic, a separation of strain and time effects is made. The proposed procedure for use of this model is described below. This procedure is only valid for the case where the hyperelastic behaviour is the classical Mooney-Rivlin formulation with ν=0.5. Similar procedures can be followed for more general cases. i)Define C10 and C01. This can be done directly or by providing the appropriate experimental data, (section 5.2). The strain rate for these tests should be sufficiently high to ensure that no relaxation effects are allowed to take place as the instantaneous response of the material will be calculated from these parameters. ii)From the above definitions the ‘instantaneous’ response i.e. the initial shear modulus is calculated from: G C C0 10 01= + = +2 1 2( ) µ µ (46)

iii)The time dependent behaviour is defined through g i

P and τi. The shear moduli g i

P are normalised

with respect to the ‘instantaneous’ modulus calculated from equation (46). The alternative is to provide experimental data, e.g. shear normalised moduli vs. time (section 5.3). The value of the normalised shear modulus at t→∞, should also be defined. ABAQUS will then use a linear regression procedure to calculate the viscoelastic parameters. 5.4. ELASTOMERIC FOAM BEHAVIOUR Elastomeric foams are cellular materials that have the following mechanical characteristics: 1) They can deform elastically up to 90% compression and 2) Their porosity allows very large volumetric deformations i.e. the effective Poisson’s ratio is

less than 0.45-0.5. Elastomeric foam materials are modelled in ABAQUS using the *HYPERFOAM option, Section 8.9.26 of the User's manual, which is a non-linear elastic model.


26

The elastic behaviour of the foams is based on a modified Hill strain energy function.

where

and λi are the principal stretches. The elastic and thermal volume ratios Jel and Jth are

where J is the total volume ratio (current volume divided by original volume), and the thermal strain ε th follows from the temperature and the isotropic thermal expansion coefficient defined in the ABAQUS *EXPANSION material option The coefficients µi are related to the initial shear modulus µo,

the initial bulk modulus K0 follows from

and β i is related to the effective Poisson’s ratio νi,

52

In ABAQUS the test data are specified as nominal stress-nominal strain pairs using combinations of uniaxial, equibiaxial, planar, simple shear and volumetric test data. Time or frequency dependent elastic behaviour can be modelled by using the *VISCOELASTIC option in conjunction with the *HYPERFOAM material option.

U = 2 $ + $ + $ - 3 + 1 (J - 1) i=1

Ni

i2 1 2 3

i

el-i i i i i∑

µα λ λ λ β

α α α α β 47

i th-

i 1 2 3 el$ = J $ $ $ = J λ λ λ λ λ_ → 48

el

th

th th3J =

JJ

and J = (1 + ) ε 49

oi+1

N

i = µ µ∑ 50

0i=1

N

i iK = 213

+ ∑

µ β 51

i

i

i

β νν=

−1 2


27

6. CONCLUSIONS This report has described a number of constitutive models that can be used to model adhesives. The suitability of each of these models, depends on the type of adhesive, loading regime and operating conditions. Elastic - plastic, pressure dependent, models can be used for structural adhesives when the operating temperatures are low in comparison to their glass transition temperature. If these are to be used in situations where a varying strain rate will be present, viscoplastic i.e. strain rate dependent plasticity models can be used. In order to define the material parameters appearing in the above constitutive models, stress-strain curves under two different states of stress are required. It is recommended that these should be uniaxial tension and shear tests. The tensile tests need to be performed for a range of strain rates. Procedures for determining the various material parameters from the experimental data have been suggested. For more rubbery adhesives the hyperelastic model or hyperfoam model if the effective Poisson’s ratio is less than 0.45 may prove to be more suitable. If viscoelastic behaviour needs to be taken into account the hyperfoam model can be combined with a linear viscoelastic model within the ABAQUS FEA programme. All of these models have been described and some of the experiments that are required to define the relevant material parameters have been outlined. 7. REFERENCES 1 C.K. Lim, M.A. Acitelli and W.C. Hamm, “Failure criterion of a typical polyamide cured

epoxy adhesive”, J. Adhesion, Vol. 6, pp. 281-288, 1974. 2) S. Gali, G. Dolev and O. Ishai, “An effective stress/strain concept in the mechanical

characterisation of structural adhesive bonding”, Int. J. Adhesion and Adhesives, Vol. 1, pp. 135-140, 1981.

3) D.C. Drucker and W. Prager, “Soil mechanics and plastic analysis or limit design”, Quarterly of Applied Mathematics, Vol. 10, pp. 157-165, 1952.

4) S.S. Sternstein and L. Ongchin, American Chemical Society Polymer Preprints, Vol. 10, pp. 1117-., 1969.

5) R. Raghava, R. Caddell and G.S.Y. Yeh, “The macroscopic yield behaviour of polymers”, J. Materials Science, Volume 8, pp. 225-232, 1973.

6) ABAQUS User’s and Theory Manuals, Version 5.5, Hibbit, Karlsson & Sorensen Inc., USA, 1995.

7) LUSAS User’s and Theory Manuals, Version 11, FEA Ltd., UK. 8) ANSYS Theory Manual, Version 5.2, Swanson Analysis Systems Inc., USA. 9) LS-DYNA 3D User’s and Theory Manuals, Version 936, Ove Arup & Partners, UK.


28

APPENDIX 1: THE CREEP ‘POWER LAW’ An experimental procedure is described which allows the determination of the material constants that appear in the creep ‘power law’. The terminology of ABAQUS 6 will be used:

ε. cr

n kAq t= (1.1)

where ε. cr

is the equivalent creep strain rate, q is the von Mises equivalent stress, t is time and A, n and k are material constants to be determined. Integrating equation (1) leads to:

εcr n k+1A

k + 1q t= (1.2)

where εcr

is the equivalent creep strain. Equation (1.2) can be converted to a logarithmic form:

( ) ( ) ( )log logA

k +1nlogq k + 1 logt logB nlogq k +1 logt

crε =

+ + = + + (1.3)

where B=A/(k+1). Equation (1.3) can be used in conjunction with uniaxial creep experimental data to determine A, n and k. The proposed procedure is as follows: i) Perform a uniaxial creep test where the applied load is constant and record the creep strain as a

function of time, i.e. keep q constant and vary t. For a uniaxial tensile test, εcr

is equal to the creep

strain and q is equal to the applied stress. From this test, a plot of logεcr

vs. logt should be constructed. The gradient of the line that best fits the experimental data, will be equal to (k+1), hence k can be evaluated. The intercept of the line will be equal to (logB+nlogq). ii) Perform several creep tests at various constant loads and measure the resulting creep strain after a

constant time period, i.e. keep t constant and vary q. From this, a plot of log εcr

vs. logq should be constructed, whose gradient will be equal to n. Its intercept will be equal to {logB+(k+1)logt)}. Since k is determined, B and hence A can be assessed. A cross check on accuracy should be performed by calculating B via the intercept obtained in step i), as well. Note that in all of the above, it was assumed that a constant applied load leads to a constant stress, q, therefore, the above procedure is valid only if the reduction in the specimen’s cross section during creep, is negligible.


29

APPENDIX 2 EVALUATION OF THE PARAMETER ζ FROM TENSILE AND SHEAR TESTS

For pure shear loading conditions where a shear stress τy is applied, the stress invariants are equal to: I1 = 0, J2 = τ y

2 (2.1) Substituting equation (1.1) in equations (2) and (3), expressions relating the tensile yield stress to the shear yield stress corresponding to the same equivalent plastic strain, are obtained:

( )στ

+ 12

yt

y

=3 ζ

ζ (2.2)

στ

yt

y

3=

ζ (2.3)

Equations (2.2) and (2.3) are for the Drucker-Prager and Raghava criteria respectively. In order to proceed further, the equivalent plastic strain relationships are required, i.e. the yield criteria need to be re-written in terms of strain invariants. Equations (2.4) and (2.5) correspond to the Drucker-Prager and Raghava criteria respectively:

( )( )

( )( )ε

ζp

p2'

p1' =

3J I

+12ζ ν

ζ −12ζ ν

11

11 2+

+−

(2.4)

( )( )

( )( )

( )( )

εpp

1'

2

p

1'

p

2'= I

1- 2I J

ζ − 12ζ ν ζ

ζ − 1

ν

ζ

1+ ν

11 2

12

122

2

2−+ + (2.5)

The expressions (2.4) and (2.5) are such that, for uniaxial tensile loading, εp is equal to the tensile

plastic strain, εyt. For pure shear loading conditions, the strain invariants I1' and J 2

' are equal to:

I1' = 0, J 2

' = γ4y2

(2.6)


30

Substitution of (2.6) into equations (2.4) and (2.5), yields the equivalent plastic strains in a shear test for Drucker-Prager and Raghava, equations (2.6) and (2.7) respectively:

( )( )ε εp

y

pyt=

3 +1

4 1+

γ ζ

ν ζ= (2.7)

( )ε εpy

pyt=

3γ

2 1+ ν ζ= (2.8)

Equations (2.7) and (2.8) relate the tensile plastic strain to the shear plastic strain corresponding to the same equivalent stress. Using equations (2.2) and (2.7) the following relationship is derived:

( )[ ]σε

τγ

νyt

yt

y

yp= +2 1 (2.9)

or

( )[ ]S St s p= +2 1 ν (2.10)

Equation (2.10) was used in Section 4.2 in order to determine the parameter ζ.


31

APPENDIX 3 EVALUATION OF ν P WHEN THE LATTER VARIES WITH STRAIN Section 4 described a method that can be used to determine νp when the latter is constant. In cases where νp varies with strain or when it is desirable to examine the validity of the assumption that νp is constant, the following procedure can be used. Uniaxial tensile experimental data are required which should be converted to true stress - true strain for large strains. i) For any value of axial strain, ε1, calculate the elastic, ε1e, and plastic, ε1p, components, using:

εσ

1e =Ε

(3.1)

ε ε ε1p 1e= −1 (3.2)

where E is the elastic modulus and σ is the stress corresponding to the stain ε1. ii) Calculate the elastic, ε2e, and plastic, ε2p components of the lateral strain, ε2, which corresponds to ε1, from: ε ν ε2e e e= 1 (3.3) ε ε ε2p 2e-= 2 (3.4)

where νe is the elastic Poisson’s ratio. iii) Calculate the plastic Poisson’s ratio, νp, from:

νεεp

2p

1p

= (3.5)

The above procedure should be repeated for a range of ε1 values in order to determine the variation of νp with ε1. In many cases an average value might have to be defined as most widely available FEA packages do not allow for a varying νp.


32

APPENDIX 4 EVALUATION OF PLASTIC STRAIN RATE FOR A CONSTANT STRAIN RATE TEST The tensile yield stress, σyt, is a function of the plastic strain, εyt:

( )σ εyt ytf= (4.1)

where f is determined experimentally from a tensile test at constant strain rate, ε..

From the usual assumption regarding the additive decomposition of the total strain, the following is obtained:

ε ε ε. . .

= +e yt (4.2)

where ε.

e and ε.

yt are the elastic and plastic strain rates respectively. The elastic strain rate can be written as:

εσ..

eyt

E= (4.3)

where E is the elastic modulus. Combining equations (4.1), (4.2) and (4.3):

( )ε ε εε

ε ε. . .

= = +1 1Ε

f.

+.

Edf

dyt yt

yt

yt yt (4.4)

or, solving for ε.

yt :

εε..

ythE

E

=+1

(4.5)

where Eh is df/dεyt, i.e. the slope of the tensile yield stress against tensile plastic strain curve.


33

For a linear elastic - perfectly plastic material, Eh is zero and from equation (4.5), ε.

yt is constant and

equal to ε.

. For linear hardening (bi-linear stress - strain curve), Eh is constant and ε.

yt is again

constant but smaller than ε.

. For the general case where the function f in equation (4.1) is neither a

constant nor linear, ε.

yt will vary throughout the test. Since the ratio Eh/E will generally decrease with

increasing strain, ε.

yt will increase during the test and will approach a value equal to ε. at large

strains. The above observations need to be considered when defining the strain rate dependence of yield.

The representation of the hardening curve will have consequences on the validity of constant ε.

yt tests that most FEA packages assume.

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