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CAVS REPORT MSU.CAVS.CMD.2007-R0015 Confidential Printed September 2007

On Developing a Viscoelastic-Viscoplastic Model for Polymeric Materials E. B. Marin, R. Prabhu, M. F. Horstemeyer Prepared by Center for Advanced Vehicular Systems Mississippi State University Mississippi State, MS 39762 Web site: http://www.cavs.msstate.edu Not for public release.

September 2007

http://www.cavs.msstate.edu/

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Benton Gady (TARDEC) Yibin (Anna) Xue (MSU)

Theodore Currier (TARDEC) [email protected]

Mark Horstemeyer (MSU) Tom Lacy (MSU)

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On Developing a

Viscoelastic–Viscoplastic Model for

Polymeric Materials

Esteban B. Marin, Raj Prabhu, Mark F. HorstemeyerCenter For Advanced Vehicular Systems

Mississippi State UniversityP.O. Box 5405

Mississippi State, MS 39762

Abstract

This report presents the formulation of a three-dimensional, finite deformation, isothermal,viscoelastic-viscoplastic constitutive model for (thermoplastic) amorphous polymeric mate-rials. The formulation follows a widely used theory for amorphous polymers as describedby [Boyce et al., 1988], with the development framed in a thermodynamic setting as pre-sented by [Anand and Ames, 2006]. Conceptually, the main features of the model can bedescribed with a typical rheological model: a nonlinear Maxwell element in parallel witha nonlinear (hyperelastic) spring. The constitutive equations of the model are specializedto the one–dimensional case. The numerical integration of the one–dimensional equationsand corresponding implementation in MATLAB are described. This implementation is thenused to perform a parametric study of the model. This study shows that this initial versionof the model reproduces typical features experimentally observed in the stress response ofmany amorphous polymeric materials.

Keywords: viscoelasticity, viscoplasticity, finite deformation, constitutive integration, poly-meric materials.

1

Contents

1 Introduction 4

2 A Simple Viscoelastic Model 6

3 Constitutive Model for Amorphous Polymers 9

3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 One-Dimensional Constitutive Model 19

4.1 Reduced Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Numerical Integration of One–Dimensional Model . . . . . . . . . . . . . . . 214.3 Parametric Study based on One–Dimensional Model . . . . . . . . . . . . . . 24

5 Summary 31

6 References 33

7 Appendix 35

2

List of Figures

2.1 A simple viscoelastic model: the standard linear solid. . . . . . . . . . . . . . 63.1 Schematic representation of nonlinear rheological model for amorphous poly-

mers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1 A typical stress–strain response numerically determined from the one–dimensional

model: total stress σ and stresses in branches A, σA, and B, σB. The evolutionof the strength κ is also shown. . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Effect of applied deformation rate ε on the stress response. . . . . . . . . . . 264.3 Effect of the rate sensitivity exponent m on the overall stress response. . . . 274.4 Effect of the reference shear strain rate γ0 on the overall stress response. . . 274.5 Effect of hardening (softening) modulus h0 on the yield peak and post–yield

behavior of the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Effect of saturation value for the strength κS on the yield peak and post–yield

response of the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.7 Effect of initial value of the strength κ0 on the yield peak or macro–yielding

response of the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.8 Effect of network locking stretch λL on the large strain hardening response of

the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.9 Effect of rubbery modulus µR on the large strain hardening response of the

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3

1. Introduction

Polymeric materials have been extensively used in a broad range of applications; includingaerospace and automotive industries. Due to the increased use of polymers, much researchhas focused on developing continuum material models to understand their brittle, viscoelas-tic, viscoplastic and rubbery response under different loading conditions. At CAVS researchefforts in this direction have been recently initiated, work that have been triggered by anumber of research activities concerning the material behavior of polymeric–type materials.In this respect, this report documents the initial modeling efforts in this area.

In general, polymers exhibit a rich variety of material behavior which is very temper-ature and rate dependent. Such behavior is mainly due to their particular microstucture.Extensively intertwined, long molecular chains have backbone bonds which permit rotation,but very little extension. At temperatures well below the glass transition, backbone bondsare rigid and the network is stiffly interlocked, resulting in a brittle response. At highertemperatures, in a narrow range centered about the glass transition, backbone bonds areable to rotate to varying degrees, allowing molecules to partially disentangle and move (slip)relative to one another. This results in a variety of viscoelastic-viscoplastic behaviors. Attemperatures well above the glass transition backbone bonds rotate easily allowing completedisentanglement. Cross linked polymers retain some rigidity because of the bonding betweenmolecules, resulting in a rubbery behavior. On the other hand, uncross-linked polymers flowviscously. To capture such a wide range of responses, a number of constitutive models havebeen developed in the open literature. In particular, many physically-based constitutiveequations based on differential formulations using internal state variables have been pro-posed [Boyce et al., 1988; Anand, 1996; Bardenhagen et al., 1997; Reese and Govindjee,1998; Anand and Gurtin, 2003; Kontou, 2006; Anand and Ames, 2006].

In this report, we present the formulation of a constitutive framework focused on modelingthe response of polymers under isothermal deformations at temperatures below and close tothe glass transition temperature (thermoplastics with amorphous structure). As mentionedabove, in this regime the mechanical response of polymers present features of viscoelasticand viscoplastic behaviors that has been widely documented in the literature [Zairi, et al.,2007]. The presented formulation is based on a differential formulation using internal statevariables and follows closely a widely used theory for amorphous polymers [Boyce et al.,1988], with the development framed in a thermodynamic setting [Anand and Ames, 2006].Conceptually, the main features of the model can be described with a typical one-dimensionalrheological model: a nonlinear Maxwell element in parallel with a nonlinear (hyperelastic)

4

1. Introduction

spring.The presentation proceeds by describing the constitutive modeling of a simple one-

dimensional linear viscoelastic solid that will set the stage for the methodology used forformulating the constitutive framework. Next, the large deformation kinematics and ther-modynamics used to formulate the three-dimensional material model is presented. This isfollowed by the development of the constitutive equations of the model. These equationsare then simplified to the one-dimensional case whose numerical integration is described indetail. The numerical implementation of this integration scheme in MATLAB is then usedto carry out a parametric study of the model.

The work uses Gibb’s (direct) notation [Gurtin, 1981] to express tensor quantities andtheir mathematical operations. Consider for example the two second order tensors A and B.Tensor operations between the tensors A and B are indicated as AB for the inner product(a second order tensor), A⊗B for the dyadic product (a fourth order tensor), and A:B forthe scalar product (a scalar). Any other particular tensor notation used in the developmentwill be either clear from the context or noted in the text.

5

2. A Simple Viscoelastic Model

In order to set the stage for the development of the model, in this section we present theformulation of a simple one-dimensional rheological (spring-dashpot) model, the standardlinear solid. The standard linear solid consists of a linear spring and dashpot in series (aMaxwell element) in parallel with a linear spring, see Fig. 2.1. The formulation of theconstitutive equations of the model is carried out assuming small deformations and usinga thermodynamics formalism. The extension of this methodology to the three dimensionalfinite deformation case is presented in the next section, extension that will also account fornonlinear (inelasticty) effects on the material response, as presented by a number of authors[Boyce, et al., 1988; Reese and Govindjee, 1998; Kontou, 2006; Anand and Ames, 2006].

E

η E

ε ε

εe

1d

e

1

ε

Figure 2.1: A simple viscoelastic model: the standard linear solid.

The kinematics for this simple model can be written as

ε = εe = ε1 + εd (2.1)

where ε is the applied strain, (εe, ε1) are the elastic strains in the springs, and εd is theinelastic strain in the dashpot. On the other hand, the mechanical version of the dissipitationinequality can be expressed as [Coleman and Gurtin, 1967]

−ψ + σε ≥ 0 (2.2)

6


where σ is the total stress in the system, and ψ is the Helmholtz free energy associated withthe arrangement in Fig. 2.1. Here we assume that free energy depends on the elastic strains(εe, ε1), and hence its time derivative can be computed as

ψ = ψ(εe, ε1) → ψ =∂ψ

∂εeεe +

∂ψ

∂ε1ε1 (2.3)

where the rates of strain can be related by

ε = εe, ε1 = εe − εd (2.4)

Substituting Eqs.(2.3) and (2.4) into Eq.(2.2) and grouping terms, one obtains

(

− ∂ψ

∂εe︸︷︷︸

σe

− ∂ψ

∂ε1︸︷︷︸

σ1

+σ)

εe +∂ψ

∂ε1︸︷︷︸

σ1 = σd

εd ≥ 0 (2.5)

where σe and σ1 are the stresses in the springs, and σd is the (dissipative) stress in thedashpot. Note here that, in the literature, σe and σ1 are also called the equilibrium σEQ andnon-equilibrium σNEQ stresses [Reese and Govindjee, 1998]. Using standard arguments oneobtains from Eq.(2.5)

σ = σe + σ1 (2.6)

σd εd ≥ 0 (2.7)

where Eq.(2.6), derived from thermodynamic arguments, gives the balance between theapplied stress and the stresses in the springs, and Eq.(2.7) establishes that dissipation isnonnegative.

Specific constitutive equations for the springs can be obtained assuming the followingform for the free energy (Hookean springs)

ψ(εe, ε1) =1

2Eeε

2e +

1

2E1ε

21 (2.8)

from which the constitutive equations for the springs can be computed as

σe =∂ψ

∂εe= Eeεe (2.9)

σ1 =∂ψ

∂ε1= E1ε1 = E1(εe − εd) (2.10)

Also, assuming a linear–Newtonian– dashpot with a (constant) viscocity coefficient η, thedissipation inequality results

σd = ηεd → σdεd = η(εd)2 ≥ 0 (2.11)

7


which implies that η ≥ 0.Then, the constitutive equations for the simple viscoelastic model depicted in Fig. 2.1

can be summarized by

ε = εe = ε1 + εd

σ = σe + σ1 (2.12)

σe = Eeεe, σ1 = E1(εe − εd)

σ1 = σd = ηεd

Note that by eliminating the strains εe and ε1 from these equations, one can obtain thedifferential form of the model as

σ = E0ε− E1εd (2.13)

εd +1

τεd =

1

τε (2.14)

where E0 = Ee + E1 and τ = η/E1. The (hereditary) integral representation of the modelcan be derived by integrating Eq.(2.14) to obtain εd(t) and subtituting the resulting equationinto Eq.(2.13), to give

σ(t) =

∫ t

−∞

G(t− s)ε(s)ds, G(t) = Ee + E1 exp (−tτ

) (2.15)

where G(t) is called the relaxation modulus. Note that, based on the above methodology,the transformation from the differential to the integral form was facilitated by the linearityof the model (linear springs and linear dashpot). When nonlinearities are introduced in themodel (e.g. inelasticity), such transformation may not be possible. In such a case, one shouldfavor the differential representation of the model.

8

3. Constitutive Model for Amorphous

Polymers

In this section we derive a large deformation three-dimensional constitutive model forthe isothermal nonlinear viscoelastic-viscoplastic response of amorphous polymeric materi-als. The model follows a widely used theory for amorphous polymers as described by [Boyceet al., 1988], with the development framed in a thermodynamic setting as presented by[Anand and Ames, 2006]. Conceptually, the main features of the model can be described bythe rheological model presented in Fig. 3.1. As indicated by this figure, the model consistsof a nonlinear Maxwell element (branch A, non-equilibrium component) connected in par-allel with a Langevin spring (branch B, equlibrium component). The Maxwell componentinvolves a nonlinear elastic spring to represent elastic molecular interactions, and a nonlinearviscous dashpot to account for the non-newtonian viscoplastic flow arising from the motion ofpolymer segments (unkinking and sliding). On the other hand, the rubber-elastic Langevinspring, a concept derived from a non-Gaussian statistical mechanics theory of rubber elas-ticity [Arruda and Boyce, 1993], models the post-yield strain hardening at large strains dueto the alignment of the long-chain polymer molecules.

spring

τF,

nonlineardashpot spring

nonlinear

A

B

nonlinear Langevin

Figure 3.1: Schematic representation of nonlinear rheological model for amorphous polymers.

In what follows, we describe the large deformation kinematics and the thermodynamicsetting of the model, aspects that are used to develop the three-dimensional constitutiveequations. In a latter section, we present the one-dimensional version (uniaxial extension)

9

3. Constitutive Model for Amorphous Polymers

of the equations and the corresponding integration scheme. The corresponding numericalimplementation in MATLAB is then used to perform a parametric study of the model.

3.1 Kinematics

The arrangement of the model elements presented in Fig. 3.1 indicates that the totaldeformation gradient F is applied to both branches (deformation–driven problem), i.e.,

F = F A = F B (3.1)

Branch B, which describes the rubbery behavior of the material by means of a Langevinspring, is modeled using hyperelasticity, and hence, its deformation is fully defined by F .However, the response of branch A, which should capture inelasticity effects on the materialresponse, is represented using the multiplicative decomposition of F into elastic and plastic(inelastic) components

F = FeF

p, J = detF = JeJp = Je > 0, Je = detF e, Jp = detF p = 1 (3.2)

As described by [Anand and Ames, 2006], Fp represents a local plastic (isochoric) defor-

mation of the material due to plastic mechanisms such as stretching, rotation and relativeslippage of the molecular chains, while F

e represents elastic mechanisms such as strectchingand rotation of the intermolecular structure. Note that decomposition (3.2) suggests thatthere exist an intermediate configuration between the undeformed B0 and the current Bconfigurations, which is denoted here by B. Hypothetically, B is obtained from B by unload-ing through F

e−1 to a zero stress state (a relaxed configuration). Using Eq.(3.2), we cancompute the velocity gradient l in B as

l = F F−1 = l

e + FeL

pF

e−1, Lp

= FpF

p−1 (3.3)

where le = F

eF

e−1. Note that le and L

pcan be decomposed into their symmetric and

skew parts, i.e., le = d

e + we and Lp

= Dp

+ Wp. Here we take W

p= 0, a reasonable

assumption for solids that are isotropic-viscoplastic or amorphous [Gurtin and Anand, 2005].Hence, L

p= D

p.

As an aside, we note that the Cauchy stresses in branches A and B, denoted here as σA

and σB, respectively, can be written as [Holzapfel, 2000]

σA = Je−1τA = Je−1

FeSAF

eT , σB = J−1τB = J−1

FSBFT , (3.4)

where τA and τB are the Kirchhoff stresses in branches A and B and SA and SB arethe corresponding second Piola-Kirchhoff stresses expressed in configurations B and B0,respectively. Note that τ A is defined by a force per unit volume in B and τB is a force perunit volume of B0. However, since J = J e, Eq.(3.2), both volumes should be the same. Also,as shown below, the Kirchhoff stresses in each branch are related to the overall Kirchhoffstress τ by τ = τ A + τB, which translates to σ = σA + σB.

10


3.2 Thermodynamics

The mechanical version of the global dissipitation (Clausius-Duhem) inequality assertsthat temporal changes in the total free energy be no greater than the rate at which totalwork is performed. Mathematically this statement can be written as [Gurtin and Anand,2005] ∗

− D

Dt

∫

B

Ψdv = −∫

B

J−1 ˙Ψdv ≥∫

B

σ:ldv (3.5)

where Ψ and Ψ denote the Helmholtz free energy per unit volume in the current B andrelaxed B configurations, respectively. Note here that, if we denote Ψ0 the free energy perunit volume in B0, one can write Ψ = Ψ0 since Jp = 1. In Eq.(3.5), the stress power perunit volume in B can be expressed as

σ:l = σ:le + J−1F

eSF

eT :F eL

pF

e−1

= σ:le + J−1C

eS:L

p

= σ:de + J−1M :D

p(3.6)

where M = CeS is the Mandel stress, with C

e= F

eTF

e being the elastic Cauchy-Greentensor. In this last equation we have used Eq.(3.3), the decomposition l

e = de + we, the

symmetry of σ, the relationship σ = J e−1τ = J−1FeSF

eT , and the fact that Lp

= Dp

(Wp

= 0). Substituting Eq.(3.6) into Eq.(3.5), one obtains∫

B

J−1(− ˙Ψ + τ :de + M :Dp)dv ≥ 0 (3.7)

The local form of the dissipitation inequality can be obtained by localizing the above integralequation, resulting in

τ :de + M :Dp − ˙Ψ ≥ 0 (3.8)

∗Elemental volumes in configurations B0 (dV ), B (dV ), and B (dv) are related by dv = JedV = JdV . Inaddition, dV = JpdV = dV since Jp = 1. Hence, denoting Ψ the free energy per unit volume in B.

∫

B

ΨdV =

∫

B

ΨJe−1dv =

∫

B

ΨJ−1dv =

∫

B

Ψdv

Therefore,

D

Dt

∫

B

Ψdv =D

Dt

∫

B

ΨdV =D

Dt

∫

B0

ΨJpdV =

∫

B

( ˙Ψ + Ψtr(Lp))dV =

∫

B

˙ΨdV

where tr(Lp) = Jp/Jp = 0. Finally, since dV = Je−1dv = J−1dv, then

D

Dt

∫

B

Ψdv =

∫

B

˙ΨJ−1dv

11


Following existing theories [Reese and Govindjee, 1998; Anand and Ames, 2006], we assumethat the Helmholtz free energy, typically an isotropic function of its arguments, depends onthe independent variables C

eand the Cauchy-Green tensor C = F

TF , i.e.,

Ψ = ˆΨ(Ce,C) (3.9)

and, hence, its time derivative can be computed as

˙Ψ =∂ ˆΨ

∂Ce :

˙C

e+∂ ˆΨ

∂C:C (3.10)

The first term of this equation can be evaluated as

∂ ˆΨ

∂Ce :

˙C

e=∂ ˆΨ

∂Ce :(F

eTF

e + FeT

Fe) = 2F e ∂

ˆΨ

∂Ce F

eT : FeF

e−1︸︷︷︸

le

= 2F e ∂ˆΨ

∂Ce F

eT :de (3.11)

while the second term can be reduced to

∂ ˆΨ

∂C:C =

∂ ˆΨ

∂C:(F

TF + F

TF ) = 2F

∂ ˆΨ

∂CF

T : F F−1

︸︷︷︸

l

= 2F∂ ˆΨ

∂CF

T :de + CeF

e−1(2F∂ ˆΨ

∂CF

T )F e−T :Dp

(3.12)

where we have used Eq.(3.3) together with le = d

e + we and the symmetry of ∂Ψ/∂Ce

and∂Ψ/∂C. Substituting Eqs.(3.10) together with Eqs.(3.11) and (3.12) into the dissipitationinequality, Eq.(3.8), we obtain

[

τ−Fe 2

∂ ˆΨ

∂Ce︸︷︷︸

SA

FeT −F 2

∂ ˆΨ

∂C︸︷︷︸

SB

FT]

:de+[

M−CeF

e−1(F 2∂ ˆΨ

∂C︸︷︷︸

SB

FT )F e−T

]

:Dp ≥ 0 (3.13)

Using standard arguments [Coleman and Gurtin, 1967], we obtain from the first term

τ = τA + τB, τA = FeSAF

eTτB = FSBF

T (3.14)

What is left for the dissipitation inequality is then[

M − CeF

e−1τBF

e−T]

:Dp ≥ 0 (3.15)

Replacing the expression fot the Mandel stress, M = CeS = C

eF

e−1τF

e−T , in the aboveequation, one gets for the reduced dissipitation inequality

CeF

e−1

τ A︷︸︸︷

(τ − τB) Fe−T

︸︷︷︸¯SA

:Dp ≥ 0 → MA:D

p ≥ 0 (3.16)

where MA = CeSA is the Mandel stress in branch A. Note that Eqs.(3.14) and (3.16) are

the three-dimensional counterparts of Eqs.(2.6) and (2.7), respectively.

12


3.3 Model Equations

Without loss of generality, we additively decomposed the free energy as

ˆΨ(Ce,C) = ˆΨe(C

e) + ˆΨC(C) (3.17)

Then the 2nd Piola-Kirchhoff stresses in Eq.(3.14) can be computed using

SA = 2∂ ˆΨe

∂Ce SB = 2

∂ ˆΨC

∂C(3.18)

To develop the expression for the stress in branch A, we start by using the polar decompo-sition of F

e

Fe = R

eU

e, Ce= F

eTF

e = Ue2 (3.19)

and by writing the spectral representation of Ue and C

eas

Ue =

3∑

I=1

λeInI ⊗ nI , C

e= U

e2 =3∑

I=1

λe2I nI ⊗ nI , (3.20)

with λI being the positive eigenvalues and nI being the orthonormal eigenvectors of Ue.

Note that UenI = λe

InI and C

enI = λe2

InI . Here, we also introduce the principal elastic

logarithmic strains EeI

= lnλeI, and consider the isotropy of ˆΨe(C

e) to equivalently express

it as: ˆΨe(Ce) → ˘Ψe(λ

eI) → ˜Ψe(E

eI). Therefore, the stress SA, Eq.(3.18)1, can be written as

SA = 2∂ ˆΨe

∂Ce = 2

3∑

I=1

∂ ˘Ψe(λeI)

∂λeI

∂λeI

∂Ce = 2

3∑

I=1

∂ ˜Ψe(EeI)

∂EeI

∂EeI

∂λeI

∂λeI

∂Ce (3.21)

Noting that ∂λeI/∂C

e= nI ⊗ nI/2λ

eI

and ∂EeI/∂λe

I= λe

I−1, we can simplify Eq.(3.21) to

SA =3∑

I=1

1

λe2I

∂ ˜Ψe(EeI)

∂EeI

nI ⊗ nI (3.22)

Therefore the Mandel stress MA and kirchhoff stress τ A can be written as

MA = CeSA =

3∑

I=1

∂ ˜Ψe(EeI)

∂EeI

nI ⊗ nI , τA = FeSAF

eT = ReMAR

eT (3.23)

Note that MA is symmetric. To obtain an explicit expression for MA, one can consider thefollowing simple generalization of the classical strain energy function of infinitesimal isotropicelasticity which uses a logarithmic measure of finite strain,

˜Ψe(EeI ) = µ(Ee2

1 + Ee22 + Ee2

3 ) +1

2(K − 2

3µ)(Ee

1 + Ee2 + Ee

3)2 (3.24)

13


where µ and K are the elastic shear and bulk moduli modeling the elastic behavior of branchA. Using this expression in Eq.(3.23)1, one can write for the Mandel stress MA,

MA = 2µEe+ (K − 2

3µ)Tr(E

e)1, E

e=

3∑

I=1

lnλeI nI ⊗ nI = ln U

e (3.25)

where 1 =∑3

I=1 nI ⊗ nI is the second order identity tensor, and Tr(Ee) = E

e:1. Equations

(3.25) and (3.23)2 completely define the nonlinear elastic response of branch A.To determine the stress in branch B, we model the elastic response of the Langevin

spring using finite strain elasticity with uncoupled, volumetric / deviatoric responses. Forthis purpose, we introduce the folowing volumetric / deviatoric multiplicative split of F ,

F = J1/3F

∗, detF ∗ = 1, J = detF (3.26)

which results in the following expression for the right Cauchy-Green tensor C

C = FTF = J2/3

C∗, C

∗ = F∗T

F∗, J = (detC)1/2 (3.27)

To align with decomposition (3.26), one can use the following additive split for ˆΨC(C)

ˆΨC(C) = ˆΨC∗(C∗) + ˆΨJ(J) (3.28)

and introduce an effective distorsional stretch ratio defined by †

λ =1√3

√

Tr(C∗) (3.29)

Then, using the isotropy of ˆΨC∗(C∗), one could express ˆΨC∗(C∗) → ˆΨλ(λ) and, hence, derivethe following expression for the 2nd Piola-Kirchhoff SB

SB = 2∂ ˆΨC

∂C(C) = 2

∂ ˆΨλ(λ)

∂λ

∂λ

∂C+ 2

∂ ˆΨJ(J)

∂J

∂J

∂C(3.30)

where, from Eq.(3.29),

∂λ

∂C=

1

6λtr(∂C∗

∂C

)

(3.31)

†Consider the polar decomposition of F∗ = R

∗U

∗. The spectral decomposition of U∗ and C

∗ = U∗2

can be written as

U∗ =

3∑

I=1

λIN I ⊗ N I , C∗ =

3∑

I=1

λ2

IN I ⊗ N I

where λI are the positive eigenvalues and N I are the orthonormal eigenvectors of U∗. Noting that tr(C∗) =

∑3

I=1λ2

ITr(N I ⊗ N I) =∑

3

I=1λ2

I , one can defined an average or effective stretch λ as

λ2 =1

3

3∑

I=1

λ2

I → λ =1√3

√

Tr(C∗)

14


Expressions for ∂C∗/∂C and ∂J/∂C can be obtained from Eqs.(3.27) as [Simo and Hughes,1998]

∂C∗

∂C= J2/3(I − 1

3C ⊗ C

−1),∂J

∂C=

1

2JC

−1 (3.32)

By substituting these expressions into Eq.(3.30) and performing some algebraic manipula-tions, we can write for the stress SB,

SB = µB C−1DEVC

∗ + κB C−1 (3.33)

where DEV(•) = (•) − 1/3Tr(•)1, Tr(•) = (•):1. In this equation, the distorsional andvolumetric elastic properties of the Langevin spring (rubbery behavior of the material) arerepresented by the shear modulus function µB and the bulk modulus function κB, respec-tively. These material functions are given by

µB = µB(λ) =1

3λ

∂ ˆΨλ(λ)

∂λ, κB = κB(J) = J

∂ ˆΨJ(J)

∂J(3.34)

By pushing forward Eq.(3.33) to the current configuration, we obtain the expression for theKirchhoff stress τ B acting on the hyperelastic rubbery spring as

τB = FSBFT = µB(λ) devb

∗ + κB(J)1, (3.35)

where, as above, dev(•) = (•) − 1/3tr(•)1, tr(•) = (•):1, and the effective stretch λ is nowexpressed as

λ =1√3

√

tr(b∗), b∗ = F

∗F

∗T (3.36)

As mentioned before, the hyperelastic stress response given by Eq.(3.33) or Eq.(3.35) mod-els the strain hardening at finite strains due to chain-alignment of the polymer molecularnetwork. This internal network stress, also called entropic (rubbery) restorative stress orback stress [Arruda and Boyce, 1993], will be fully specified once specific functional forms

for ˆΨλ(λ) and ˆΨJ(J) are prescribed. In this respect, based on statistical mechanics modelsof rubber elasticity, a number of authors, e.g. [Anand and Ames, 2006], has considered forˆΨλ(λ) the functional form

ˆΨλ(λ) = µRλ2L

[ λ

λL

X + ln( X

sinhX

)

− 1

λL

Y − ln( Y

sinhY

)]

(3.37)

X = L−1( λ

λL

)

, Y = L−1( 1

λL

)

(3.38)

where L−1 is the inverse of the Langevin function L(•) = coth(•) − (•)−1, and µR and λL

are two material parameters called the rubbery modulus and the network locking stretch,

15


respectively. On the other hand, for ˆΨJ(J) we use the functional form [Simo and Hughes,1998]

ˆΨJ(J) =1

2KB

[1

2(J2 − 1) − ln J

]

(3.39)

where KB is an elastic bulk modulus. Hence, using Eqs.(3.34), (3.37) and (3.39), we obtainfor the shear µB(λ) and bulk κB(J) modulus functions

µB = µB(λ) = µRλL

3λL−1

( λ

λL

)

, κB = κB(J) =1

2KB(J2 − 1), (3.40)

expressions that together with Eq.(3.35) fully define the hyperelastic response of branch B.To complement the constitutive description of amorphous polymers, we need to describe

the response of the nonlinear viscous dashpot. This element should be able to capture theinelastic mechanisms present during the yield behavior of the polymeric material, mecha-nisms that are typically associated with the localized slip or viscous flow processes resultingfrom the permanent displacement of long-chain molecules with respect to each other. Tocapture these effects, the behavior of the dashpot is represented with the flow rule [Boyce,et al., 1998; Anand and Ames, 2006]

Fp

= DpF

p, Dp

=1√2γp

Np, (3.41)

where γp =√

2‖ Dp ‖1/2

, the equivalent plastic shear-strain rate, is given by the kineticequation

γp = Φ(τA, πA, κ), (3.42)

and Np, a unit deviatoric tensor defining the direction of plastic flow, is defined by the

symmetric deviatoric tensor DEVMA, i.e.,

Np

=DEVMA

‖ DEVMA ‖(3.43)

In Eq.(3.42), τA and πA are the effective shear stress and effective pressures, respectively,defined by

τA =1√2‖ DEVMA ‖, πA = −1

3Tr(MA), (3.44)

and κ is a stress-like internal state variable representing aspects of the molecular resistanceto plastic flow associated with the inelastic mechanisms and whose evolution equation canin general be represented by

κ = Θ(κ, γp) (3.45)

16


Note that Eq.(3.42) includes a pressure-dependence of the inelastic resistance to plastic flow,as espect that has been observed experimentally for isotropic amorphous polymeric materialssubjected to moderately large hydrostatic pressures. Typical functional forms for Φ and Θused in the literature are a power law form and a Voce’s type hardening (softening) rule,respectively,

Φ(τA, πA, κ) = γ0

( τAκ+ απA

)1/m

, Φ(κ, γp) = h0

(

1 − κ

κS

)

γp (3.46)

where γ0 is a reference strain rate, m is a strain rate sensitivity parameter, α is a pressuresensitivity parameter, h0 is a hardening (softening) modulus and κS is a saturation valueof the molecular resistance to plastic flow. It is important to note here many polymericmaterials show a strain–softening region in their stress–strain response just after yielding[Boyce, et al., 1988]. Such an effect on the model response is incorporated using Eq.(3.46)2

with an appropriate choice of h0 while keeping κ0 > κS. Here, κ0 is the initial deformationresistance (shear yield strength) of the material.

A summary of the constitutive model for the viscoelastic-viscoplastic response of amor-phous polymers is presented in Box 1.

17


Box 1. Constitutive Model for Glassy Polymers

Branch A:

F A = F = FeF

p, Fe = R

eU

e

τA = ReMAR

eT , MA = 2µ ln Ue + (K − 2

3µ) Tr(ln U

e)1

Fp

= LpF

p

Lp

= Dp+ W

p, D

p=

1

2γp

Np, W

p= 0

γp = Φ(τA, πA, κ), κ = Θ(κ, γp)

where

Np

=DEVMA

‖ DEVMA ‖, τA =

1√2‖ DEVMA ‖, πA = −1

3Tr(MA)

Branch B:

F B = F = J1/3F

∗, det F∗ = 1, det F = J

τB = µRλL

3λL−1

( λ

λL

)

devb∗ +

1

2KB(J2 − 1)1

where

b∗ = F

∗F

∗T , λ =1√3

√trb∗

Total Kirchhoff stress and Cauchy stress:

τ = τA + τB, σ = J−1τ

Material parameters: µ,K; γ0, α,m;h0, κS;µR, λL, KB.

18

4. One–Dimensional Constitutive

Model

4.1 Reduced Constitutive Equations

In this section we simplify the foregoing constitutive equations to the case of uniaxialloading. For this particular loading, the deformation is prescribed by

x1 = λ1X1, x2 = λ2X2, x3 = λ3X3, λ2 = λ3 (4.1)

where XI and xi are the coordinates of the undeformed and deformed body configurations,respectively, and λI are the principal stretches. The deformation gradient F can then becomputed as

F =∂x

∂X=

λ1 0 00 λ2 00 0 λ2

, J = detF = λ1λ22 > 0, λ1 > 0 (4.2)

Note that for the deformation (4.2), we can write the polar decomposition of F as F =RU = U since R = 1. Also, here we simplify the problem by assuming a volume-preservingdeformation, i.e., J = 1 (the reduced one-dimensional equations will not capture pressureeffects on the response). This specific assumption implies that the stretches are related byλ2

2 = 1/λ1, and that, from Eq.(3.26), F = J 1/3F∗ = F

∗. Introducing the notation λ1 = λ,one can then express the velocity gradient l = d + w, with w = 0, and the tensor b

∗,Eq.(3.36)2, as

l = d = F F−1 =

λ

λD, D =

1 0 00 −1

20

0 0 −12

(4.3)

and

b∗ = F

∗F

∗T =

λ2 0 00 1/λ 00 0 1/λ

(4.4)

19

4. One–Dimensional Constitutive Model

This last expression implies that

tr(b∗) = λ2 +2

λ, devb

∗ =2

3

(

λ2 − 1

λ

)

D (4.5)

Therefore, the stress in branch B given by Eq.(3.35), with τ B = JσB = σB, can be computedas

σB =2

3µB(λ)

(

λ2 − 1

λ

)

D, λ =1√3

(

λ2 +2

λ

)

(4.6)

with µB(λ) obtained from Eq.(3.40)1. To compute the stress in branch A, we express theelastic F

e and plastic Fp parts of F , with R

e = 1, as

Fe = U

e =

λe1 0 0

0 λe2 0

0 0 λe2

Fp =

λp1 0 0

0 λp2 0

0 0 λp2

(4.7)

Here, we let λe1 = λe, and since Je = J = 1, we obtain λe2

2 = 1/λe. Therefore the logarithmicstrain, Eq.(3.25)2, and the corresponding stress in branch A, Eq.(3.23)2 with σA = JeτA =τA = MA, are

Ee= ln U

e = lnλeD → σA = MA = 2µ lnλe

D (4.8)

To determine the expression for the flow rule, Eq.(3.41)1, we denote λp1 = λp. Hence, since

Jp = 1, we can also write λp22 = 1/λp. Therefore, using Eq.(4.7)2 we can write the flow rule

as

FpF

p−1 = Dp

=λp

λpD (4.9)

The expression for the plastic flow direction Np

and the corresponding one for Dp

can beobtained from Eqs.(3.43) and (3.41)2. For this purpose, we denote σA = 2µ lnλe. Then,using Eq.(4.8)2 one obtains

Np

=

√

2

3sign(σA) → D

p=

1√3γpsign(σA)D (4.10)

where γp is given by Eq.(3.42) with

τA =

√3

2|σA|, πA = 0 (4.11)

A summary of the one-dimensional equations of the model is given in Box 2, where wehave used Eqs.(3.46) and the notation d11 = d = ε and Dp

11 = Dp.

20


Box 2. One-Dimensional Constitutive Model

Applied Load: λ = ελ

Branch A: λ = λeλp

σA = 2µ ln(λe)

λp = Dpλp

Dp =1√3γpsign(σA), γp = γ0

(√

3

2

|σA|κ

)1/m

κ = h0(1 − κ

κS

)γp

Branch B: σB =2

3µB(λ) (λ2 − λ−1), λ =

1√3(λ2 + 2λ−1)1/2

Cauchy Stress: σ = σA + σB

Material Parameters: µ, γ0, m, h0, κS, µR, λL.

4.2 Numerical Integration of One–Dimensional Model

In this section we develop the numerical integration procedure for the one–dimensionalconstitutitve equations given in Box 2. This integration proceeds by discretizing the defor-mation history in time and numerically integrating the equations over each time step ∆t.Denoting a general time step interval as [tn, tn+1] with tn+1 = tn + ∆t, and using the sub-cripts n and n+1 to represent variables evaluated at tn and tn+1, we can write the integratedone–dimensional equations as

Applied load:

λn+1 = exp(∆t ε)λn, or λn+1 = exp(ε tn+1) (4.12)

Branch A:

λn+1 = λen+1λ

pn+1

σA n+1 = 2µ ln(λen+1)

λpn+1 = exp(∆tDp

n+1)λpn ≈ (1 + ∆tDp

n+1)λpn

Dpn+1 =

1√3γp

n+1sign(σA n) (4.13)

γpn+1 = ˆγ

p(σA n+1, κn+1) = γ0

(√

3

2

|σA n+1|κn+1

)1/m

21


κn+1 = κn + h0

(

1 − κn

κS

)

∆tγpn+1

Branch B:

σB n+1 =2

3µB(λn+1)(λ

2n+1 − λ−1

n+1), λn+1 =1√3(λ2

n+1 + 2λ−1n+1)

1/2 (4.14)

Cauchy Stress:

σn+1 = σA n+1 + σB n+1 (4.15)

where we have used a semi-implicit integration procedure [Moran, et al., 1990] for the equa-tions defining the behavior of branch A. This semi-implicit scheme is characterized for beingimplicit in the incremental plastic shear strain ∆γp and explicit in the flow direction and othervariables. We note here that the above integration scheme assumes (i) that the deformationpath represented by ε is given, (ii) that the variables (σA n, λ

pn, κn) describing the response

of branch A are known, and (iii) that the material parameters (µ, γ0, m, h, κS, µR, λL)are input. Also note that while the response of branch B is fully determined once λn+1

is known, computing the response of branch A requires an iterative procedure to computethe updated quantities (σA n+1, λ

pn+1, κn+1). To develop this iterative scheme, we proceed by

writing ∆γpn+1 as

∆γpn+1 = ∆tˆγ

p(σA n+1, κn+1) (4.16)

Using Eqns.(4.13), we can readily establish the functional dependence σA n+1 = σA(∆γpn+1)

and κn+1 = κ(∆γpn+1). Then, from Eq.(4.16) we can write a residual as

R∆γp, n+1 = R(∆γpn+1) = ∆γp

n+1 − ∆t ˆγp(σA(∆γp

n+1), κ(∆γpn+1)) (4.17)

which is used to devise a Newton-Raphson iterative scheme. The linearization of this residualgives

[ dR(∆γpn+1)

d∆γpn+1

︸︷︷︸

∂∆γpR∆γp, n+1

]

d∆γpn+1 = −R∆γp, n+1 (4.18)

where the Jacobian ∂∆γpR∆γp, n+1 can be computed as

dR(∆γpn+1)

d∆γpn+1

= 1 − ∆t( ∂γp

n+1

∂σA n+1

dσA n+1

d∆γpn+1

+∂γp

n+1

∂κn+1

dκn+1

d∆γpn+1

)

(4.19)

The derivative dσA n+1/d∆γpn+1 can be obtained from Eqs.(4.13)1−4 as

dσA n+1

d∆γpn+1

= − 2µ√3

[

1 +∆tγp

n+1√3

sign(σA n)]−1

sign(σA n) (4.20)

22


To obtain the expressions for the partial derivatives ∂γpn+1/∂σA n+1, ∂γ

pn+1/∂κn+1 and the

total derivative dκn+1/d∆γpn+1 one needs to have the specific functional forms for the kinetic

equation and hardening rule, respectively. For the particular case given by Eqs.(4.13)4−5,these derivatives are determined as

∂γpn+1

∂σA n+1

=γp

n+1sign(σA n+1)

m|σA n+1|,

∂γpn+1

∂κn+1

= − γpn+1

mκn+1

(4.21)

dκn+1

d∆γpn+1

= h0(1 − κn

κS

) (4.22)

Once ∆γpn+1 is computed by solving iteratively Eq.(4.18), from Eqs.(4.13) one can update

the plastic λpn+1 and elastic λe

n+1 stretches, and then the stress σA n+1 and hardness κn+1.Finally, one can compute the overall Cauchy stress σn+1 using Eq.(4.15). A summary of thisintegration scheme is presented in Box 3, and the corresponding numerical implementationin MATLAB is documented in the appendix.

Box 3. Integration Procedure for One-Dimensional Model

1. Known quantities:

ε; σA n, λpn, κn; material properties : µ; γ0,m;h, κS;µR, λL.

2. Compute total stretch λn+1:

λn+1 = exp(ε tn+1)

3. Compute Cauchy stress in branch B:

σB n+1 = 2/3 µB(λn+1)(λ2n+1 − λ−1

n+1)

where λn+1 and µB(λn+1) can be computed from Eqs.(4.14)2 and (3.40)1.

4. Compute Cauchy stress in branch A:

(a) Compute ∆γpn+1 by solving iteratively (iteration (i)):

∆γ(i+1)n+1 = ∆γ

(i)n+1 −

[

∂∆γpR(i)∆γp, n+1

]−1

R(i)∆γp, n+1

where R∆γp, n+1 and ∂∆γpR∆γp, n+1 are given by Eqs.(4.17) and (4.19).

(b) Update the plastic and elastic stretches:

λpn+1 =

[

1 + ∆γpn+1sign(σA n)/

√3]

λpn, λe

n+1 = λn+1λp−1n+1

(c) Update stress and hardness:

σA n+1 = 2µ ln(λen+1), κn+1 = κn + h0(1 − κn/κS)∆γp

n+1

5. Update total Cauchy stress:

σn+1 = σA n+1 + σB n+1

23


4.3 Parametric Study based on One–Dimensional Model

In this section we carry out a parametric (sensitivity) study of the one–dimensional con-stitutive model to understand what salient features of the stress-strain response are affectedwhen changing the values of the material parameters. This study assumes an isothermaluniaxial extension with an applied strain rate of ε = 3 × 10−4 s−1. The nominal values ofthe material parameters chosen for this study are [Anand and Ames, 2006]

µ = 1.58 × 103 MPa

γ0 = 1 × 10−4 s−1

m = 0.265

h0 = 250.0 MPa

κS = 24.0 MPa

κ0 = 44.0 MPa

λL = 1.65

µR = 14.0 MPa

Recall that the parameter µ defines the instantaneous elastic response of branch A, theconstants for the flow rule (γ0, m) and hardening law (h0, κS, κ0) determine the viscousresponse of the dashpot, with κ0 being the initial deformation resistance, and the parameters(λL, µR) describe the entropic hardening response (hyperelastic behavior) at large strainsof branch B. Note that the combined response of the springs and dashpot will give theviscoelastic–viscoplastic behavior of the system. In what follows, we focus our attention onstudying the effect of the parameters (γ0,m; h, κS, κ0; µR, λL) on the strain-stress response.Note here that due to the assumptions imposed when deriving the reduced constitutiveequations, i.e., J = 1, the one-dimensional model will not be able to capture pressure effectson the material response, and hence, the computed stress will mainly be deviatoric.

A typical true stress versus true strain curve in monotonic simple extension to a strain of0.5 is shown in Fig. 4.1. This curve was computed with the nominal values of the parameters.After an approximately linear initial region, the stress-strain curve becomes nonlinear priorto reaching a peak in the stress at a strain of about 0.035. The material then strain-softens until a minimum in stress is reached at a strain of approximately 0.16. After this,the material exhibits a broad region of rapid strain hardening, as the stress once again risesbecause of the alignment and locking of the polymer chains. Hence, typical features observedexperimentally in the mechanical behavior of many thermoplastic amorphous polymers aredisplayed by the computed response of the model: an initial elastic-viscoelastic response,followed by a nonlinear viscoelastic-viscoplastic behavior, a region of strain softening afteryielding and finally subsequent strain hardening. Of course, the predominance of each ofthese features can be adjusted by varying the material parameters, as shown in the nextfigures.

The contribution of the stress in each branch to the total stress response is also displayedin Fig. 4.1. As shown, the stress in branch A dominates the overall response in the pre– and

24


post–yield regions until a minimum stress is reached during strain softening. The subsequentincrease in stress level is due to the contribution of the hyperelastic stress response of branchB. This figure also shows the evolution of the strength resistance to plastic flow κ. Asshown, the strength decays after a short transient from its initial value of κ0 = 44 MPa tothe saturation value of κS = 24 MPa. In fact, for the particular evolution law chosen for κ,Eq.(3.46)2, the strain softening effect in the model is induced by keeping κS < κ0.

The effect of the applied strain rate on the stress-strain curve is presented in Fig. 4.2,where the response of the model has been computed at two strain rates: 3 × 10−4 s−1 and3 × 10−5 s−1. For the chosen nominal values of the parameters, we observe a strong ratedependence of the model response. In particular, increases in the applied strain rate resultin increases of both the yield–peak and the level of post–yield strain softening. It is notedhere that this rate dependent effect is tied to the parameters of the power–law flow rule, therate sensitivity exponent m and the reference shear strain rate γ0. Figures 4.3–4.4 show theeffect of changing the values of these parameters on the stress-strain response. As depictedin these figures, the rate dependence of the model response increases with increases in mand decreases in γ0, with the shape changes of the stress-curve curve being similar to theones obtained when changing the applied strain rate.

The parameters defining the evolution law for the internal strength, the hardening–softening modulus h0 and the deformation resistance saturation value κS, affect importantlythe post–yield strain softening response, and to a lesser degree the yield peak or macroyield-ing. This can be observed from Figs. 4.5–4.6 that shows that the level of strain softeningdecreases as the value of h0 decreases or the value of κS increases, while, at the same time,these changes affect slightly the transient peak associated with yielding. On the other hand,the initial deformation resistance κ0 defines the shear yield strength of the material. As such,changing its values will mainly affect the yield peak or macro–yielding response, as shownin Fig. 4.7.

At large strains, the model response shows a rapid strain–hardening behavior. Physi-cally, this hardening behavior is induced by the alignment of the macro–molecular networkbuilt of entangled polymer molecules, and it is typically associated with a decrease in theconfigurational entropy of the material. As mentioned before, such hardening effect is in-troduced in the model through a Langevin spring. Hence, the material parameters of thisspring will define the level of strain hardening, as shown in Figs. 4.8–4.9, with the trend thatthis large-strain hardening effect gets more pronounced as the magnitude of λL is decreasedor the value of µR is increased.

25


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

strain

stre

ss(M

Pa)

κ

σ

σB

σA

Figure 4.1: A typical stress–strain response numerically determined from the one–dimensional model: total stress σ and stresses in branches A, σA, and B, σB. The evolutionof the strength κ is also shown.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

strain

stre

ss(M

Pa)

ε=3x10-5 s-1

ε=3x10-4 s-1

.

.

Figure 4.2: Effect of applied deformation rate ε on the stress response.

26


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

strain

stre

ss(M

Pa)

m=0.265

m=0.465

m

m=0.065

Figure 4.3: Effect of the rate sensitivity exponent m on the overall stress response.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

strain

stre

ss(M

Pa)

γ0=1x10-5 s-1

γ0

γ0=1x10-4 s-1

γ0=1x10-3 s-1

Figure 4.4: Effect of the reference shear strain rate γ0 on the overall stress response.

27


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

strain

stre

ss(M

Pa)

h0=5.0x102 MPa

h0

h0=2.5x102 MPa

h0=0.5x102 MPa

Figure 4.5: Effect of hardening (softening) modulus h0 on the yield peak and post–yieldbehavior of the model.

28


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

strain

stre

ss(M

Pa)

κS=24.0 MPa

κS=34.0 MPa

κS

κS=14.0 MPa

Figure 4.6: Effect of saturation value for the strength κS on the yield peak and post–yieldresponse of the model.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

strain

stre

ss(M

Pa)

κ0=44.0 MPa

κ0=54.0 MPa

κ0

κ0=34.0 MPa

Figure 4.7: Effect of initial value of the strength κ0 on the yield peak or macro–yieldingresponse of the model.

29


0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

strain

stre

ss(M

Pa)

λL=2.05

λL

λL=1.65

λL=1.25

Figure 4.8: Effect of network locking stretch λL on the large strain hardening response of themodel.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

strain

stre

ss(M

Pa)

µR=24 MPa

µR

µR=14 MPa

µR=4 MPa

Figure 4.9: Effect of rubbery modulus µR on the large strain hardening response of the model.

30

5. Summary

A preliminary version of a three-dimensional viscoelastic-viscoplastic constitutive modelfor the isothermal, finite deformation of glassy polymers has been presented. The formula-tion has been developed in a thermodynamic framework and follows a widely used theoryfor amorphous polymers. Conceptually, the main components of the model consists of anonlinear Maxwell element in parallel with a hyperelastic spring, an arrangement typicallyused to describe the constitutive behavior of amorphous polymers. A parametric study us-ing the reduced one–dimensional constitutive equations has shown that major features ofthe stress-strain response of these thermoplastic polymeric materials are captured by themodel: the plastic flow process, the post yield–strain softening behavior and the subsequentstrain hardening at large strains. No attempt has been made to fit the model parameters toexperimental data, as aspect that will be covered in a future report.

This work represents the initial steps at CAVS towards building a robust continuum in-ternal state variable material model for predicting the response of polymeric–type materialsfor a broad range of temperatures and strain rates. The success of such modeling endeavourswill depend, of course, to a large extent on the combination of theoretical and experimen-tal efforts. From the theoretical viewpoint, a number of tasks can currently be identified.Among these are: (i) to develop a constitutive integration scheme for the three-dimensionalequations and the corresponding numerical implementation in finite element codes, (ii) todevise a systematic parameter identification procedure in MATLAB using optimization tech-niques, and possibly, analytical sensitivity coefficients, and (iii) to identify the experimentaltests needed for a complete evaluation of the model parameters. On the other hand, theexperimental aspects of our modeling thrust should focus on building a material databasethat could be used to adapt the constitutive framework to fit the response of particularamorphous polymeric materials. Specific characterization experiments to be performed mayinitially include low–strain rate tests such as monotonic strain–controlled compression tests,cyclic strain–controlled compression–tension tests, and stress-controlled compression creeptests. In additon microindentation experiments may very well complement the experimentaldatabase and be used as a verification/validation experiment for the model predictions.

Finally, future work on polymers modeling should enhance and/or modify the presentedconstitutive framework to address (a) the effect of damage, temperature, pressure andanisotropy as well the behavior of mixed amorphous–crystalline polymers, and (b) the me-chanical behavior of soft and hard biological tissues such as bones, ligaments, tendons andmuscles. Such a continuum modeling framework will have important practical applications

31

5. Summary

to current research projects at CAVS which include modeling the behavior of polyurethane(a polymer used in run-flat insert tire designs) and predicting the response of human beings(biological tissues) in crash simulations.

32

6. References

Anand, L. (1996), “A Constitutive Model for Compressible Elastomeric Solids,” Computa-

tional Mechanics , Vol. 18, pp. 339–355.

Anand, L. and Gurtin, M. E. (2005), “A Theory of Amorphous Solids Undergoing LargeDeformations, with Application to Polymeric Glasses,” International Journal of Solids and

Structures , Vol. 40, pp. 1465–1487.

Anand, L. and Ames, N. M. (2006), “On Modeling the Micro-Indentation Response of anAmorphous Polymer,” International Journal of Plasticity , Vol. 22, pp. 1123–1170.

Arruda, E. M., and Boyce, M. C., (1993), “A Three-Dimensional Constitutive Modelfor the Large Stretch Behavior of Rubber Elastic Materials”, Journal of the Mechanics and

Physics of Solids, 41:389–412.

Bardenhagen, S. G., Stout, M. G., and Gray, G. T., (1997), “Three–Dimensional,Finite Deformation, Viscoplastic Constitutive Models for Polymeric Materials”, Mechanics

of Materials, 25:235–253.

Boyce, M. C., Parks, D. M., and Argon, A. S., (1988), “Large Inelastic Deformation ofGlassy Polymers. Part I: Rate Dependent Constitutive Model,” Mechanics of Materials ,7:15–33.

Coleman, B., and Gurtin, M., (1967), “Thermodynamics With Internal State Variables”,J. Chem. Phys., 47:597–613.

Gurtin, M., (1981), “An Introduction to Continuum Mechanics” Academic Press.

Gurtin, M. E. and Anand, L. (2005), “The Decomposition F = FeF

p, Material Sym-metry, and Plastic Irrotationality for Solids that are Isotropic-Viscoplastic or Amorphous,”International Journal of Plasticity , Vol. 21, pp. 1686–1719.

Holzapfel, G. A., (2000), “Nonlinear Solid Mechanics. A Continuum Approach for En-gineering”, J. Wiley & Sons, Ltd.

Kontou, E., (2006), “Viscoplastic Deformation of an Epoxy Resin at Elevated Tempera-tures”, Journal of Applied Polymer Science, 101:2027–2033.

Moran, B., Ortiz, M., and Shih, C. F., (1990), “Formulation of Implicit Finite ElementMethods for Multiplicative Finite Deformation Plasticity,” Int. J. for Numerical Methods in

33

6. References

Engng., 29:483–514.

Simo, J. C., and Hughes, T. J. R., (1998), “Computational Inelasticity”, Springer.

Reese, S., and Govindjee, S., (1998), “A Theory of Finite Viscoelasticity and NumericalAspects”, Int. J. of Solids and Structures, 35:3455–3482.

Zairi, F., Nait-Abdelaziz, M., Woznica, K., and Gloaguen, J.-M., (2006), “Elasto-Viscoplastic Constitutive Equations for the Description of Glassy Polymers Behavior atConstant Strain Rate”, Journal of Engineering Materials and Technology, 129:29–35.

34

7. Appendix

This appendix presents the MATLAB implementation of the numerical integration schemedescribed by Box 3. This implementation has been used to perform the parametric study ofthe one-dimensional constitutive model.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Implementation of preliminary version of one-dimensional (1-D)

% viscoelastic-viscoplastic model. The model follows closely

% Anand’s formulation. The integration of the 1-D equations -

% branch A - is performed using a semi-implicit scheme. The

% reduced equations do not capture pressure effects.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Material properties

mu = 1.58d3; % shear modulus (MPa)

gamma0 = 1d-4; % reference shear strain rate (1/s)

m = 0.265; % strain-rate sensitivity exponent

h_0 = 2.5d2; % hardening modulus (MPa)

kappa_sat = 24.d0; % saturation level of strength (MPa)

kappa_0 = 44; % initial strength (MPa)

lambda_L = 1.65d0; % rubbery modulus (MPa)

mu_R = 14.0d0; % network locking stretch

% Some numerical constants

SMALL = 1.d-16; % a small number

RTOLER = 1.d-8; % rel. tolerance for Newton’s iters

ATOLER = 1.d-8; % abs. tolerance for Newton’s iters

% Deformation Path

e_dot = 3.d-4; % strain rate

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7. Appendix

e_max = 0.5; % max strain

Nincrs = 800; % number of increments

N = Nincrs;

t = zeros(N,1);

e = zeros(N,1);

dt = zeros(N,1);

t_max = abs(e_max/e_dot);

delt_t = t_max / N;

delt_e = e_dot * delt_t;

for i = 1:N

e(i) = i*delt_e;

dt(i) = delt_t;

end

% Initializations

tauA_n = SMALL;

kappa_n = kappa_0;

lambda_p_n = 1.0;

dgam_n = SMALL;

fullData1 = [];

% Integrate model for given deformation history

time = 0.d0;

numIters = 100;

for i = 1:N

% Time step and total time

dtime = dt(i);

time = time + dt(i);

t(i) = time;

% Total stretch at time t

lambda = exp(e(i));

36

7. Appendix

% Calculate stress due to B system

lambda_bar = sqrt(lambda*lambda + 2/lambda)/sqrt(3);

arg = lambda_bar / lambda_L;

if ( 0 < abs(arg) < 0.84136d0 )

invLang = 1.31446d0 * tan(1.58986d0*arg) + 0.91209d0 * arg;

elseif (abs(arg) < 1.0 )

invLang = 1.d0/(sign(arg) - arg);

else

’error in inverse langevin argument’

end

mu_B = mu_R * lambda_L / (3.d0*lambda_bar) * invLang;

tauB = 2./3. * mu_B * (lambda*lambda - 1/lambda);

% Calculate stress due to A system

%---- Initial guess for dgam

dgam = dgam_n;

%---- Compute initial residual

lambda_p = (1 + dgam*sign(tauA_n)/sqrt(3)) * lambda_p_n;

lambda_e = lambda / lambda_p;

tauA = 2*mu*log(lambda_e);

kappa = kappa_n + h_0*(1-kappa_n/kappa_sat)*dgam;

gammaDot = gamma0 * ( sqrt(3)/2 * abs(tauA)/kappa )^(1/m);

resid = dgam - dtime * gammaDot;

resid0 = resid;

res_0 = resid;

%---- Newton’s iteration to compute incremental plastic shear strain

converged = 0;

iters = 0;

while ( (converged == 0) & (iters < numIters) )

iters = iters + 1;

dgam0 = dgam;

37

7. Appendix

%-------- Compute Jacobian

DgamdotDtauA = gammaDot * sign(tauA) / (m * abs(tauA));

DgamdotDkapp = - gammaDot / (m * kappa);

DtauADdgam = -2/sqrt(3) * mu * sign(tauA_n) / ...

( 1 + dtime*gammaDot*sign(tauA_n)/sqrt(3) );

DkappDdgam = h_0 * (1 - kappa_n/kappa_sat);

jac = 1 - dtime * ( DgamdotDtauA * DtauADdgam ...

+ DgamdotDkapp * DkappDdgam );

if (abs(jac) < SMALL)

’Warning: Jacobian < SMALL’

end

%-------- Compute incremental dgam and update

ddgam = - resid / jac;

search = 1;

dgam = dgam0 + search*ddgam;

if (dgam < 0.0)

dgam = SMALL;

end

%-------- New residual

lambda_p = (1 + dgam*sign(tauA_n)/sqrt(3))*lambda_p_n;






%-------- Simple line search

while ( abs(resid) > abs(resid0) )

search = search*0.5;

if (search < SMALL)

’Warning: search < SMALL’

end

dgam = dgam0 + search*ddgam;

if (dgam < 0.0) dgam = SMALL; end

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7. Appendix







end

resid0 = resid;

if (abs(resid) < abs(res_0)*RTOLER | abs(resid) < ATOLER)

converged = 1;

end

end

if (iters >= numIters)

’Warning: iters >= numIters’

end

%---- Update variables





gammaDot = dgam / dtime;

tauA_n = tauA;

kappa_n = kappa;

lambda_p_n = lambda_p;

dgam_n = dgam;

tau = tauA + tauB;

%---- Save variables

fullData1(i,1) = e(i);

fullData1(i,2) = tau;

fullData1(i,3) = tauA;

fullData1(i,4) = tauB;

fullData1(i,5) = kappa;

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7. Appendix

end

save -ascii visco_stress_semi fullData1;

figure(10)

clf

set(10, ’Position’, [60 470 560 420])

xlabel(’strain’)

ylabel(’stress’)

hold on

plot((fullData1(:,1)), fullData1(:,2),’r-’)

plot((fullData1(:,1)), fullData1(:,3),’b-’)

plot((fullData1(:,1)), fullData1(:,4),’m-’)

plot((fullData1(:,1)), fullData1(:,5),’rx’)

legend( ’tau’, ’tauA’, ’tauB’, ’kappa’)

title(’stress response - semi-implicit integration.’)

hold off

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