Chapter 3

Constitutive Models $–$ Relations between Stress and Strain

3.10 Large Strain Viscoelasticity

This section describes constitutive equations that can be used to model large, irreversible deformations in polymers, and also to model biological tissue that is subjected to large shape changes.  Finite strain viscoelasticity is not as well developed as finite strain plasticity, and a number of different formulations exist. The model outlined here is based on Bergstrom and Boyce, J. Mech. Phys. Solids., Vol. 46, pp. 931-954, 1998.

The constitutive equation is intended to capture the following features of material behavior

1.      When the material is deformed very slowly (so that material behavior is fully reversible) it behaves like an ideal rubber, as described in Section 3.4;

2.      When deformed very quickly (so that there is no time for inelastic mechanisms to operate) it again behaves like an ideal rubber, but with different properties;

3.      At intermediate rates, the solid exhibits a rate dependent, hysteretic response.

The material is isotropic.

Material response to a pure volumetric strain ( ${\epsilon }_{11}={\epsilon }_{22}={\epsilon }_{33}=\delta V/V$ with all other ${\epsilon }_{ij}=0$ ) is perfectly elastic (with no time dependent behavior).

The material is nearly incompressible;

Hydrostatic stress has no effect on the deviatoric response of the solid.

.

The constitutive equations outlined in this section make use of many concepts from Sections 3.4, 3.6 and 3.7, so you may find it convenient to read these sections before the material to follow.

3.10.1 Kinematics for finite strain viscoelasticity.

The description of shape changes in polymers follows closely the approach outlined in 3.7.1.  Let ${x}_{i}$ be the position of a material particle in the undeformed solid. Suppose that the solid is subjected to a displacement field ${u}_{i}\left({x}_{k}\right)$, so that the point moves to ${y}_{i}={x}_{i}+{u}_{i}$.  Define

The deformation gradient and its jacobian

${F}_{ij}={\delta }_{ij}+\frac{\partial {u}_{i}}{\partial {x}_{j}}$       $J=\mathrm{det}\left(F\right)$

${L}_{ij}=\frac{\partial {\stackrel{˙}{u}}_{i}}{\partial {y}_{j}}={\stackrel{˙}{F}}_{ik}{F}_{kj}^{-1}$

The deformation gradient is decomposed into elastic and plastic parts as

${F}_{ij}={F}_{ik}^{e}{F}_{kj}^{p}$

The velocity gradient is decomposed into elastic and plastic parts as

${L}_{ij}={L}_{ij}^{e}+{L}_{ij}^{p},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{ij}^{e}={\stackrel{˙}{F}}_{ik}^{e}{F}_{kj}^{e-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{L}_{ij}^{p}={F}_{ik}^{e}{\stackrel{˙}{F}}_{kl}^{p}{F}_{lm}^{p-1}{F}_{mj}^{e-1}$

Define the elastic and plastic stretch rates and spin rates as

$\begin{array}{l}{D}_{ij}^{e}=\left({L}_{ij}^{e}+{L}_{ji}^{e}\right)/2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{ij}^{e}=\left({L}_{ij}^{e}-{L}_{ji}^{e}\right)/2\\ {D}_{ij}^{p}=\left({L}_{ij}^{p}+{L}_{ji}^{p}\right)/2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{ij}^{p}=\left({L}_{ij}^{p}-{L}_{ji}^{p}\right)/2\end{array}$

Define the Left Cauchy-Green deformation tensor for the total and elastic deformation gradients

${B}_{ij}={F}_{ik}{F}_{jk}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{B}_{ij}^{e}={F}_{ik}^{e}{F}_{jk}^{e}$

Define the invariants of B and ${B}^{e}$ as

$\begin{array}{l}{\overline{I}}_{1}=\frac{{B}_{kk}}{{J}^{2/3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{I}}_{2}=\frac{1}{2}\left({\overline{I}}_{1}^{2}-\frac{{B}_{ik}{B}_{ki}}{{J}^{4/3}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{I}}_{3}=\mathrm{det}B={J}^{2}\\ {\overline{I}}_{1}^{e}=\frac{{B}_{kk}^{e}}{{J}_{e}^{2/3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{I}}_{2}^{e}=\frac{1}{2}\left({\overline{I}}_{1}^{e2}-\frac{{B}_{ik}^{e}{B}_{ki}^{e}}{{J}_{e}^{4/3}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overline{I}}_{3}^{e}=\mathrm{det}{B}^{e}={J}_{e}^{2}\end{array}$

Denote the principal stretches for $B$ and ${B}^{e}$ by ${\lambda }_{i},{\lambda }_{i}^{e}$ (these are the square roots of the eigenvalues of $B$ and ${B}^{e}$ ) and principal stretch directions by ${b}_{i},{b}_{i}^{e}$.

3.10.2 Stress measures for finite strain viscoelasticity.

Usually stress-strain laws are given as equations relating Cauchy stress (`true’ stress) ${\sigma }_{ij}$ to left Cauchy-Green deformation tensor.  For some computations it may be more convenient to use other stress measures.  They are defined below, for convenience.

Cauchy (“true”) stress represents the force per unit deformed area in the solid and is defined by

${n}_{i}{\sigma }_{ij}=\underset{dA\to 0}{Lim}\frac{d{P}_{j}^{\left(n\right)}}{dA}$

Kirchhoff stress  $\tau =J\sigma \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\tau }_{ij}=J{\sigma }_{ij}$

The constitutive model must specify relations between stress, the total deformation gradient F, the elastic part of the deformation gradient ${F}^{e}$, and the plastic part of the deformation gradient.

3.10.3 Relation between stress, deformation measures and strain energy density

Just as for hyperelastic materials, the instantaneous stress in a hyperviscoelastic solid is calculated from a strain energy density function $U$.  For viscoelastic materials, the strain energy density is separated into two parts

$U\left({\overline{I}}_{1},{\overline{I}}_{2},J,{\overline{I}}_{1}^{e},{\overline{I}}_{2}^{e}\right)={U}_{\infty }\left({\overline{I}}_{1},{\overline{I}}_{2},J\right)+{U}_{T}\left({\overline{I}}_{1}^{e},{\overline{I}}_{2}^{e},{J}_{e}\right)$

Here

1.      ${U}_{\infty }\left({\overline{I}}_{1}^{},{\overline{I}}_{2}^{},J\right)$ specifies the strain energy density in the fully relaxed material.  It is represents the effect of a set of polymer chains in the solid which can only accommodate deformation by stretching to follow the total extension.  ${U}_{\infty }=0$ for a material that exhibits steady-state creep.

2.      ${U}_{T}\left({\overline{I}}_{1}^{e},{\overline{I}}_{2}^{e}\right)$ is an additional, transient contribution to the total strain energy.  This contribution gradually relaxes with time.   It represents a set of polymer chains which initially stretch with the solid, but with time are able to relax towards their preferred configuration.

The stress is related to the energy density by

$\begin{array}{l}{\sigma }_{ij}=\frac{2}{J}\left[\frac{1}{{J}^{2/3}}\left(\frac{\partial {U}_{\infty }}{\partial {\overline{I}}_{1}}+{\overline{I}}_{1}\frac{\partial {U}_{\infty }}{\partial {\overline{I}}_{2}}\right){B}_{ij}-\frac{{\overline{I}}_{1}}{3}\frac{\partial {U}_{\infty }}{\partial {\overline{I}}_{1}}{\delta }_{ij}-\frac{1}{{J}^{4/3}}\frac{\partial {U}_{\infty }}{\partial {\overline{I}}_{2}}{B}_{ik}{B}_{kj}\right]+\frac{\partial {U}_{\infty }}{\partial J}{\delta }_{ij}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{2}{{J}_{e}}\left[\frac{1}{{J}_{e}^{2/3}}\left(\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}+{\overline{I}}_{1}^{e}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}\right){B}_{ij}^{e}-\frac{{\overline{I}}_{1}^{e}}{3}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}{\delta }_{ij}-\frac{1}{{J}_{e}^{4/3}}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}{B}_{ik}^{e}{B}_{kj}^{e}\right]+\frac{\partial {U}_{T}}{\partial {J}_{e}}{\delta }_{ij}\end{array}$

You can use any of the hyperelastic strain energy density potentials listed in Section 3.4 to describe a particular material.  It is sensible to choose ${U}_{T},{U}_{\infty }$ to have the same functional form (but with different material constants).   Note also that since the inelastic strains are assumed to be volume preserving (see below), ${J}_{e}=J$, and therefore once can take $\partial {U}_{T}/\partial {J}_{e}=0$ without loss of generality.

3.10.4 Strain relaxation

The strain rate dependence and irreversibility of a viscoelastic material can be modeled using the framework described in Section 3.8 for finite strain viscoplasticity.  The constitutive equations must specify the plastic stretch rate ${D}_{ij}^{p}$ and plastic spin ${W}_{ij}^{p}$ as a function of stress.   The expressions given here follow Bergstrom and Boyce, J. Mech. Phys. Solids., Vol. 46, pp. 931-954, 1998.

Define the deviatoric Kirchhoff stress resulting from the elastic part of the deformation gradient ${F}_{ij}^{e}$ as

${{\tau }^{\prime }}_{ij}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\left[\frac{1}{{J}_{e}^{2/3}}\left(\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}+{\overline{I}}_{1}^{e}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}\right){B}_{ij}^{e}-\frac{{\overline{I}}_{1}^{e}}{3}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{1}^{e}}{\delta }_{ij}-\frac{1}{{J}_{e}^{4/3}}\frac{\partial {U}_{T}}{\partial {\overline{I}}_{2}^{e}}{B}_{ik}^{e}{B}_{kj}^{e}\right]$

1.      Define the effective stress

${\tau }_{e}=\sqrt{3{{\tau }^{\prime }}_{ij}{{\tau }^{\prime }}_{ij}/2}$

2.      The plastic strain rate is then

${D}_{ij}^{p}={\stackrel{˙}{\epsilon }}_{e}\left({\tau }_{e},{I}_{1}^{e},{I}_{2}^{e},T\right)\frac{3{{\tau }^{\prime }}_{ij}}{2{\tau }_{e}}$

3.      Here, ${\stackrel{˙}{\epsilon }}_{e}$ is the magnitude of the plastic strain rate, which is a function of temperature T, the effective stress ${\tau }_{e}$ and the elastic strain.  This function must be calibrated experimentally.  Bergstrom & Boyce suggest that the following function should describe approximately the relaxation dynamics of long-chain molecules

${\stackrel{˙}{\epsilon }}_{e}={\stackrel{˙}{\epsilon }}_{0}{\left(\sqrt{{I}_{1}^{e}}-\sqrt{3}\right)}^{n}{\left(\frac{{\tau }_{e}}{{\tau }_{0}}\right)}^{m}$

where ${\stackrel{˙}{\epsilon }}_{0}>0$, $-1, ${\tau }_{0}>0$ and $m>0$ are temperature dependent material properties.

Additional constitutive equations must specify ${W}_{ij}^{p}$.   This has not been studied in detail: usually we just take ${W}_{ij}^{p}=0$.

3.10.5 Representative values for material parameters in a finite-strain viscoelastic model

Bergstrom and Boyce, J. Mech.Phys. Solids., Vol. 46, pp. 931-954, 1998 give experimental data for the rate dependent response of several rubbers, and fit material properties to their data.   They use the Arruda-Boyce potential for both ${U}_{\infty },{U}_{T}$

${U}_{\infty }={\mu }_{\infty }\left\{\frac{1}{2}\left({\overline{I}}_{1}^{}-3\right)+\frac{1}{20{\beta }_{\infty }^{2}}\left({\overline{I}}_{1}^{2}-9\right)+\frac{11}{1050{\beta }_{\infty }^{4}}\left({\overline{I}}_{1}^{3}-27\right)+...\right\}+\frac{K}{2}{\left(J-1\right)}^{2}$

${U}_{T}={\mu }_{T}\left\{\frac{1}{2}\left({\overline{I}}_{1}^{}-3\right)+\frac{1}{20{\beta }_{T}^{2}}\left({\overline{I}}_{1}^{2}-9\right)+\frac{11}{1050{\beta }_{T}^{4}}\left({\overline{I}}_{1}^{3}-27\right)+...\right\}$

Material behavior is therefore characterized by values of the two shear moduli ${\mu }_{T},{\mu }_{\infty }$, the bulk modulus K, the coefficients ${\beta }_{\infty }$, ${\beta }_{T}$, and the parameters ${\stackrel{˙}{\epsilon }}_{0}$, $n$, ${\tau }_{0}$ and $m$, as outlined in the preceding section.

 Material Parameters for a Nitrile Rubber (after Bergstrom & Boyce, 1998) ${\mu }_{\infty }$ ${\beta }_{\infty }$ ${\mu }_{T}$ ${\beta }_{T}$ $K$ ${\stackrel{˙}{\epsilon }}_{0}$ ${\tau }_{0}$ $n$ $m$ 0.29MPa 6 0.73MPa 4 100MPa 7 sec-1 1MPa -0.6 5.0