EN222: Mechanics of Solids

  

 Division of Engineering

    Brown University

 

 

Finite Deformation Plasticity

 

The constitutive model developed in the preceding section is valid only for infinitesimal deformations.  The main ideas can be extended to finite deformation, but some care is required in selecting appropriate measures of deformation and stress.  As before, our focus is plastic flow of metals, although the same approach is often used to model other materials.

 

For infinitesimal deformations we decomposed a plastic strain rate into elastic and plastic parts

and then devised constitutive laws relating stress and stress rate to the elastic and plastic strain rates.

 

We will use a similar approach for finite deformations.  In this case we will attempt to relate the velocity gradient

to appropriate finite deformation measures of stress and stress rate. To begin, we need to find a suitable way to decompose a general deformation into elastic and plastic parts.  To do this, we will review some ideas in kinematics.

 

Kinematics of finite strain plasticity

 

Consider an infinitesimal volume element within a plastically deforming solid.  Following the usual argument, the deformations of all infintesimal volume elements are homogeneous.  A general homogeneous deformation can be represented by a tensor F that maps a line element dX in the reference configuration to dx in the deformed configuration

To decompose this 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.  As a constitutive assumption, we think of these two events occurring in sequence, with the plastic deformation first, and the stretch and rotation second, giving

 

We now proceed to calculate the velocity gradient L  Recall that

Decomposing F into elastic and plastic parts gives

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

 

To account properly for rigid rotations, we also decompose L and its elastic and plastic parts into symmetric and skew terms, as follows

 

 

Kinetics for finite deformation plasticity

 

The Cauchy stress

is the most intuitive measure of stress in any finite deformation problem.  It is not always the most convenient stress measure for specifying constitutive laws, however.  This is because in devising constitutive laws, we must ensure that our measures of deformation and stress are work conjugate (we can regard the deformation measure as a set of generalized coordinates, which must have appropriate work-conjugate generalized forces.  Our constitutive law will only make sense if we specify relationships between work conjugate variables).  The Cauchy stress is not work conjugate to any convenient deformation measure.

 

Instead, we will adopt the Kirchhoff stress as our stress measure.  Recall that

where J=det(F).  Recall also that Kirchhoff stress is work conjuagate to the stretch rate tensor, so that  gives the stress power per unit reference volume.

 

Our objective in formulating a constitutive law will be to set up an appropriate relationship between D and .

 

 

Constitutive Laws

 

We now return to the issue of formulating appropriate constitutive laws for finite deformation plasticity.  Recall that for infinitesimal deformations, we decomposed strain as

and then derived relationships between stress and  and .  We need to do the same for finite deformation.

 

Elastic part

 

For infinitesimal deformations, we wrote the elastic part of the constitutive law in rate form as

For finite deformations we will find that the appropriate constitutive law (for metals, at least, where the elastic stretches are small) is

,

where  are the elastic moduli in the deformed configuration,  are the components of the elastic part of the rate of deformation tensor, and  are the components of the Jaumann stress rate, defined by

This law is sometimes adopted (out of thin air) as a constitutive assumption in itself – but then it’s hard to see why the Jaumann rate is the appropriate objective measure of stress rate.  We will proceed more carefully. 

 

In metal plasticity, the elastic part of the deformation gradient  represents the distortion of the crystal lattice under the action of stresses.  The stresses are generally not large (deviatoric stresses aren’t greater than the yield stress) and most crystals have a very stiff elastic response, so we anticipate that  will in general describe a small elastic stretching of the lattice, and may also include a finite rotation.  Because of the finite rotation we can’t use a linear elastic stress-strain relation, but a generalized Hooke’s law works well for situations where stretches are small but rotations may be large. To set up such a constitutive law, we define the elastic Lagrange strain

as the deformation measure.  The appropriate work-conjugate stress measure is the Material stress

Hooke’s law then gives a linear relationship between stress and strain

where  is the usual tensor of elastic moduli – exactly the same set of material constants we use in the linear elastic constitutive law  - and the two dots denote the appropriate product between a 4 tensor and 2 tensor.  Unlike the linear elastic law, however, the generalized Hooke’s law works for finite rotations, because  for a rigid rotation.

 

We now need to manipulate this constitutive law to obtain an expression relating Kirchhoff stress rate to elastic stretch rate . This is a tedious but straightforward exercise.  Differentiating with respect to time

We now need to manipulate this into a relationship between  and .   Recall that

Rearranging

The term involving Lagrange strain rate may be rearranged into a form that may be interpreted physically

(it is easiest to show the second line using index notation).  We define

and note that  may be interpreted as the spatial elastic moduli (the main effect of  is to rotate the elastic moduli to the deformed configuration).

 

We also identify

as the Jaumann rate of Kirchoff stress. 

 

To summarize, then we have

For metals, the elastic moduli are large, so that the first term on the right hand side is much greater than the other two.  In this case we may write

 

 

Plastic constitutive law

 

Next, we turn to developing an appropriate plastic constitutive law for finite deformations.  In this case we need to find a way to relate ,  to  and its rate.

 

Often, plastic constitutive laws for finite deformations are just simple extensions of small strain plasticity.  For example, for a finite strain, rate independent, Mises solid we set

where

is the deviatoric Kirchoff stress

is the effective stress (or Mises stress) and h is the slope of the true stress –v- true plastic strain curve.

 

Kinematic hardening laws can be developed in much the same way, although some care is required to devise an appropriately objective measure of the rate of translation of the yield surface.

 

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 true 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.  There used to be heated arguments on this topic, but these have largely died out now that funding in the area has dried up…  In any case, crystal plasticity models provide a way out of this difficulty, because they have an unambiguous definition of the plastic spin (and no, it is not zero).