At certain critical values of the hardening modulus, the governing equations of elasto-plastic how may lose their hyperbolicity and exhibit two modes of ill-posedness: shear-band and flutter ill-posedness. These modes of ill-posedness are characterized by the uncontrolled growth of modes at infinitely fine scales, which ultimately violates the continuum assumption. In previous work [L. An and A. Peirce, SIAM J. Appl. Math., 54(1994), pp. 708-730], a continuum model accounting for microscale deformations was built. Linear analysis demonstrated the regularizing effect of the microstructure and provided a relationship between the width of the localized instabilities and the microlength scale. In this paper a weakly nonlinear analysis is used to explore the immediate postcritical behavior of the solutions. For both one-dimensional and anti-plane shear models, post-critical deformations in the plastic regions are shown to be governed by the Boussinesq equation (one of the completely integrable PDEs having soliton solutions), which describes the Essential coupling between the focusing effect of the nonlinearity and the dispersive effect of the microstructure terms. The soliton solution in the plastic region is patched to the solution in the elastic regions to provide a special solution to the weakly nonlinear system. This solution is used to derive a relation between the width of the shear band and the length scale of the microstructure. A multiple scale analysis of the constant displacement solution is used to reduce the perturbed problem to a nonlinear Schrodinger equation in the amplitude functions-which turn out to be unstable for large time scales. Stability analyses of more complicated special solutions show that the low wave number solutions are unstable even on the fast time scales while the high wave numbers are damped by the dispersive microstructure terms. These theoretical results are corroborated by numerical evidence. This pervasive instability in the strain-softening regime immediately after failure, indicates that the material will rapidly move to a lower residual stress state with well-defined shear bands.
