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 ${x}_{i}$ 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 stretch rate and spin

${D}_{ij}=\left({L}_{ij}+{L}_{ji}\right)/2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{ij}=\left({L}_{ij}-{L}_{ji}\right)/2$

Recall that ${F}_{ij}$ relates infinitesimal material fibers $d{y}_{i}$ and $d{x}_{i}$ in the deformed and undeformed solid, respectively, as

$d{y}_{i}={F}_{ij}d{x}_{j}$

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

$d{y}_{i}={F}_{ij}d{x}_{j}={F}_{ik}^{e}{F}_{kj}^{p}d{x}_{j}$

To decompose the velocity gradient into elastic and plastic parts, note that

${L}_{ij}={\stackrel{˙}{F}}_{ik}{F}_{kj}^{-1}=\left({\stackrel{˙}{F}}_{ik}^{e}{F}_{kl}^{p}+{F}_{ik}^{e}{\stackrel{˙}{F}}_{kl}^{p}\right)\left({F}_{lm}^{p-1}{F}_{mj}^{e-1}\right)={\stackrel{˙}{F}}_{ik}^{e}{F}_{kj}^{e-1}+{F}_{ik}^{e}{\stackrel{˙}{F}}_{kl}^{p}{F}_{lm}^{p-1}{F}_{mj}^{e-1}$

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

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

Define the elastic and plastic stretch rates and spin rates as

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

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

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

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

Material stress for intermediate configuration  ${\Sigma }_{ij}=J{F}_{ik}^{e-1}{\sigma }_{kl}{F}_{jl}^{e-1}$

Note that the material stress tensor is related to the Cauchy stress by a function of $\text{\hspace{0.17em}}{F}^{e}$, 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 $\text{\hspace{0.17em}}{F}^{e}$.

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 $d{\sigma }_{ij}/dt$, 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

$\stackrel{\nabla e}{{\sigma }_{ij}}=\frac{d{\sigma }_{ij}}{dt}-{W}_{ik}^{e}{\sigma }_{kj}+{\sigma }_{ik}{W}_{kj}^{e}$

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 ${E}_{ij}^{e}=\left({F}_{ki}^{e}{F}_{kj}^{e}-{\delta }_{ij}\right)/2$

2.      Assume that the material stress is proportional to Lagrange strain, as ${\Sigma }_{ij}={C}_{ijkl}{E}_{kl}^{e}$, where ${C}_{ijkl}$ 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 $\nu$, the stress-strain law is

${\Sigma }_{ij}=\frac{E}{1+\nu }\left\{{E}_{ij}^{e}+\frac{\nu }{1-2\nu }{E}_{kk}^{e}{\delta }_{ij}\right\}$

4.      The elastic stress-strain law is often expressed in rate form, as follows

$\stackrel{\nabla e}{{\tau }_{ij}}\approx {C}_{ijkl}^{e}{D}_{kl}^{e}$

where $\stackrel{\nabla e}{{\tau }_{ij}}$ is the Jaumann rate of Kirchhoff stress; ${C}_{ijkl}^{e}={F}_{in}^{e}{F}_{jm}^{e}{C}_{nmpq}{F}_{kp}^{e}{F}_{lq}^{e}$ (this can be thought of as the components of the elastic compliance tensor for material with orientation in the deformed configuration), and ${D}_{ij}^{e}$ is the elastic stretch rate.  For the particular case of an isotropic material with Young’s modulus E and Poisson’s ratio $\nu$, the stress rate can be approximated further as

$\stackrel{\nabla e}{{\tau }_{ij}}\approx \frac{E}{1+\nu }\left\{{D}_{ij}^{e}+\frac{\nu }{1-2\nu }{D}_{kk}^{e}{\delta }_{ij}\right\}$

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: $\frac{d{\Sigma }_{ij}}{dt}={C}_{ijkl}\frac{d{E}_{kl}^{e}}{dt}$

2.      Take the time derivative of the formula relating material and Kirchhoff stress

$\begin{array}{l}{\tau }_{ij}={F}_{ik}^{e}{\Sigma }_{kl}{F}_{jl}^{e}\\ ⇒\frac{d{\tau }_{ij}}{dt}=\frac{d{F}_{ik}^{e}}{dt}{\Sigma }_{kl}{F}_{jl}^{e}+{F}_{ik}^{e}\frac{d{\Sigma }_{kl}}{dt}{F}_{jl}^{e}+{F}_{ik}^{e}{\Sigma }_{kl}\frac{d{F}_{jl}^{e}}{dt}\end{array}$

3.      Substitute for material stress in terms of Kirchoff stress

$\frac{d{\tau }_{ij}}{dt}=\frac{d{F}_{ik}^{e}}{dt}{F}_{kl}^{e-1}{\tau }_{lj}+{F}_{ik}^{e}\frac{d{\Sigma }_{kl}}{dt}{F}_{jl}^{e}+{\tau }_{il}{F}_{lk}^{e-1}\frac{d{F}_{jk}^{e}}{dt}$

4.      Recall that ${\stackrel{˙}{F}}_{ik}^{e}{F}_{kj}^{e-1}={L}_{ij}^{e}={D}_{ij}^{e}+{W}_{ij}^{e}$, observe that ${W}_{ij}^{e}=-{W}_{ji}^{e}$, ${D}_{ij}^{e}={D}_{ji}^{e}$, and substitute from (1)

$\frac{d{\tau }_{ij}}{dt}=\left({D}_{ik}^{e}+{W}_{ik}^{e}\right){\tau }_{kj}+{F}_{ik}^{e}{C}_{klmn}\frac{d{E}_{mn}^{e}}{dt}{F}_{jl}^{e}+{\tau }_{ik}\left({D}_{kj}^{e}-{W}_{kj}^{e}\right)$

5.      Next, note that

$\frac{d{E}_{mn}^{e}}{dt}={F}_{pm}^{e}{D}_{pq}^{e}{F}_{qn}^{e}$

so

$\begin{array}{l}\frac{d{\tau }_{ij}}{dt}=\left({D}_{ik}^{e}+{W}_{ik}^{e}\right){\tau }_{kj}+{F}_{ik}^{e}{F}_{jl}^{e}{C}_{klmn}{F}_{pm}^{e}{F}_{qn}^{e}{D}_{pq}^{e}+{\tau }_{ik}\left({D}_{kj}^{e}-{W}_{kj}^{e}\right)\\ ⇒\frac{d{\tau }_{ij}}{dt}+{\tau }_{ik}{W}_{kj}^{e}-{W}_{ik}^{e}{\tau }_{kj}={C}_{ijpq}^{e}{D}_{pq}^{e}+{D}_{ik}^{e}{\tau }_{kj}+{\tau }_{ik}{D}_{kj}^{e}\end{array}$

6.      Finally, assume that ${\tau }_{ik}{D}_{kj}^{e}<<{C}_{ijpq}^{e}{D}_{pq}^{e}$ since the stresses are much less than the modulus. This shows that

$\stackrel{\nabla e}{{\tau }_{ij}}\approx {C}_{ijkl}^{e}{D}_{kl}^{e}$

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 ${\tau }_{ij}{L}_{ij}$ is the rate of work done by stresses per unit reference volume.  Consequently, the constitutive equations must relate ${D}^{p}$, ${W}^{p}$ to $\tau$ 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

${D}_{ij}^{p}={\stackrel{˙}{\epsilon }}_{0}{\left(\frac{{\tau }_{e}}{{\sigma }_{0}}\right)}^{m}\frac{3{{\tau }^{\prime }}_{ij}}{2{\tau }_{e}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}_{ij}^{p}=0$

where ${{\tau }^{\prime }}_{ij}={\tau }_{ij}-{\tau }_{kk}{\delta }_{ij}$ and ${\tau }_{e}=\sqrt{3{{\tau }^{\prime }}_{ij}{{\tau }^{\prime }}_{ij}/2}$.  The hardening rule is

${\sigma }_{0}=Y{\left(1+\frac{{\epsilon }_{e}}{{\epsilon }_{0}}\right)}^{1/n}$

where  ${\epsilon }_{0}=\int {\stackrel{˙}{\epsilon }}_{0}{\left({\tau }_{e}/{\sigma }_{0}\right)}^{m}dt$

Finite strain plasticity models disagree on the correct way to prescribe ${W}^{p}$.  Many theories simply set ${W}^{p}=0$. 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 ${W}^{p}$, but any attempt to capture evolution of plastic anisotropy would need to specify ${W}^{p}$ 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.