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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.
In addition, we assume
 The material is isotropic.
Material response to a pure volumetric
strain ( with all other ) 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 be the position of a material particle in
the undeformed solid. Suppose that the solid is subjected to a displacement
field ,
so that the point moves to . Define
The deformation gradient and its jacobian
The
velocity gradient
The
deformation gradient is decomposed into elastic and plastic parts as
The
velocity gradient is decomposed into elastic and plastic parts as
Define the elastic and plastic stretch rates and spin rates as
Define the Left Cauchy-Green deformation
tensor for the total and elastic deformation gradients
Define
the invariants of B and as
Denote the principal stretches for and by (these are the square roots of the
eigenvalues of and ) and principal stretch directions by .
3.10.2 Stress
measures for finite strain viscoelasticity.
Usually stress-strain laws are given as equations relating Cauchy
stress (`true’ stress) 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
Kirchhoff stress
The
constitutive model must specify relations between stress, the total deformation
gradient F, the elastic part of
the deformation gradient ,
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 . For viscoelastic materials, the strain
energy density is separated into two parts
Here
1. 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. for a material that exhibits steady-state
creep.
2. 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
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 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), ,
and therefore once can take 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 and plastic spin 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 as
1. Define the effective stress
2. The plastic strain rate is then
3. Here, is the magnitude of the plastic strain rate,
which is a function of temperature T,
the effective stress 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
where
,
,
and are temperature dependent material
properties.
Additional constitutive
equations must specify . This has not been studied in detail:
usually we just take .
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
Material behavior is
therefore characterized by values of the two shear moduli ,
the bulk modulus K, the coefficients
,
,
and the parameters ,
,
and ,
as outlined in the preceding section.
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Material
Parameters for a Nitrile Rubber (after Bergstrom & Boyce, 1998)
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0.29MPa
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6
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0.73MPa
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4
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100MPa
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7 sec-1
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1MPa
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-0.6
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5.0
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