The Effect of Microstructure on Elastic-Plastic Models

For large deformations, the governing equations of elastic-plastic flow may lose their hyperbolicity and become ill posed at some critical values of the hardening modulus. This ill-posedness is characterized by uncontrolled growth of the amplitude of plane wave solutions in certain directions. To capture post-critical behavior, microstructure is built into the constitutive relations. Two types of microstructure are included: one accounts for intergranular rotation via Cosserat theory, and the other accounts for the formation of voids at the microscale by means of a new pressure term related to the gradient of the dilational deformation. Using both a linearized analysis and integral estimates, it is shown that the microstructure terms provide regularizing mechanisms that inhibit the occurrence of both shear band ill-posedness and flutter ill-posedness. Moreover, a local analysis shows that the problem can be reduced to two turning point singular Schrodinger equations in the neighborhood of points where the...