Chapter 3
Constitutive Models Relations between Stress and Strain
3.9 Large Strain, Rate Dependent
Plasticity
This section describes the constitutive
equations that are used to model large, permanent deformations in
polycrystalline solids. Representative
applications include models of metal forming; crash simulations, and various
military applications that are best left to the imagination. The constitutive equations are used mostly in
numerical simulations. It is usually
preferable to use a rate dependent,
viscoplasticity model for in computations, because they are less prone
to instabilities than rate independent models.
The rate independent limit can always be approximated by using a high
strain rate sensitivity.
The constitutive equations outlined in this
section make use of many concepts from Sections 3.6 and 3.7, so you may find it
convenient to read these sections before the material to follow.
3.9.1 Kinematics
of finite strain plasticity

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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
stretch rate and spin
Recall
that relates infinitesimal material fibers and in the deformed and undeformed solid,
respectively, as
To
decompose the deformation gradient into elastic and plastic parts, we borrow
ideas from crystal plasticity. The
plastic strain is assumed to shear the lattice, without stretching or rotating
it. The elastic deformation rotates and
stretches the lattice. We think of these two events occurring in sequence, with
the plastic deformation first, and the stretch and rotation second, giving
To
decompose the velocity gradient into elastic and plastic parts, note that
Thus the velocity gradient
contains two terms, one of which involves only measures of elastic deformation,
while the other contains measures of plastic deformation. We use this to decompose L into elastic and plastic
parts
Define
the elastic and plastic stretch rates and spin rates as
Constitutive equations must specify relations between the stresses (as
defined below) and the elastic and plastic parts of the deformation
gradient. The equations are usually
written in rate form, in which case the elastic and plastic stretch rates and
spin are related to the stress rate.
3.9.2 Stress
measures for finite deformation plasticity
Stress measures that appear in descriptions
of finite strain plasticity are summarized below:
The Cauchy
(“true”) stress represents the force per unit deformed area in the solid and is
defined by
Kirchhoff stress
Material
stress for intermediate configuration
Note that the material stress tensor is related to the Cauchy stress by
a function of ,
not F as in the usual
definition. This stress should be interpreted physically as a material stress
associated with the intermediate configuration.
This stress measure is introduced because the elastic constitutive equations
require an internal force measure that is work-conjugate to an appropriate
function of .
In addition, viscoplastic constitutive equations are often written in rate
form (as in 3.7), relating stain rate to stress and (for the elastic part)
stress rate. Stress rates are difficult
to work with in finite strain problems.
At first sight, it might appear that stress rate can be calculated by
simply taking the time derivative of the stress components ,
but in fact this is not a useful measure of stress rate. To see this, imagine applying a uniaxial
tensile stress to a material, and then rotating the entire test apparatus (so
the applied force and specimen rotate together). The time derivatives of the stress
components are nonzero, but the material actually experiences a time
independent force per unit area. As shown
below, the correct stress rate is the Jaumann
Rate with respect to the elastic spin, defined as
3.9.3
Elastic stress-strain relation for finite strain plasticity
Plastically deforming metals may experience large strains. The stresses remain modest, however, and are
usually substantially lower than the elastic modulus of the solid. The elastic strains are small, but the material may experience large rotations.
Under these conditions, the small-strain elastic constitutive equations of 3.1
cannot be used, but the simple generalized Hooke’s law described in Section 3.3
can be used. This law relates the
elastic part of the deformation gradient to stress, as follows
1. Define the Lagrangean elastic strain as
2. Assume that the material stress is proportional to
Lagrange strain, as ,
where are the components of the elastic stiffness
tensor (as defined and tabulated in Section 3.1), for the material with
orientation in the undeformed configuration.
3. For the special case of an elastically isotropic
material, with Young’s modulus E and
Poission ratio ,
the stress-strain law is
4. The elastic stress-strain law is often expressed in
rate form, as follows
where is the Jaumann rate of Kirchhoff stress; (this can be thought of as the components of
the elastic compliance tensor for material with orientation in the deformed
configuration), and is the elastic stretch rate. For the particular case of an isotropic
material with Young’s modulus E and
Poisson’s ratio ,
the stress rate can be approximated further as
Derivation of the rate form of the elastic
stress-strain law: Our goal is to derive the expression in (4) above, starting from the
stress-strain law in (2). To this end:
1. Take the time-derivative of
the constitutive equation:
2. Take the time derivative of
the formula relating material and Kirchhoff stress
3. Substitute for material
stress in terms of Kirchoff stress
4. Recall that ,
observe that ,
,
and substitute from (1)
5. Next, note that
so
6. Finally, assume that since the stresses are much less than the
modulus. This shows that
3.9.4 Plastic
constitutive law for finite strain viscoplasticity
Next, we turn to developing an appropriate
plastic constitutive law for finite deformations. The constitutive equations must specify a
relationship between work conjugate measures of stress and strain recall that is the rate of work done by stresses per unit
reference volume. Consequently, the
constitutive equations must relate ,
to and its rate.
Usually, plastic constitutive laws for finite
deformations are just simple extensions of small strain plasticity. For example, for a finite strain, rate dependent,
Mises solid with isotropic hardening power-law hardening we set
where and . The hardening rule is
where
Finite strain plasticity models disagree on the correct way to
prescribe . Many theories simply set .
Simple models of polycrystals give some support for this assumption, but it may
not be appropriate in materials that develop a significant texture. More complex models have also been
developed. For isotropically hardening
solids, predictions are relatively insensitive to the choice of ,
but any attempt to capture evolution of plastic anisotropy would need to specify
carefully.
Crystal
plasticity based models provide a way out of this difficulty, because they have
a clearer (but not completely unambiguous) definition of the plastic spin.