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 equations reach the critical value of the hardening modulus. Using matched asymptotics and Wentzel-Kramers-Brillouin (WKB) theory, a relation is derived between the thickness of the localization (internal layer) and the internal length scale of the material introduced by the microstructure terms