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
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 workconjugate 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 stressstrain 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 smallstrain 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 stressstrain law is
4. The elastic stressstrain 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
stressstrain law: Our goal is to derive the expression in (4) above, starting from the
stressstrain law in (2). To this end:
1.
Take the timederivative 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 powerlaw
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.
