Stability of strain-gradient plastic materials

A formulation of Fleck and Willis (2009a,b) for strain-gradient plasticity has been adapted to provide possible descriptions for materials that initially strain-harden but eventually soften. In the absence of gradient terms, such material is unstable for any wavelength and subject to localization in the softening regime. Gradient terms do not mitigate the basic (infinite wavelength) material instability but they do inhibit the development of short-wavelength disturbances; they prevent localization but still may permit the development of narrow shear bands. In this work, the basic stability problem is studied via consideration of a small, generally time dependent, perturbation of an initially uniform state of deformation. The linearized problem for the perturbation is formulated for the general case of rate-dependent gradient plasticity but special attention is paid to the rate-independent limit. An interesting feature is that a qualitative difference is found between the effects of “energetic” and “dissipative” strain-gradient terms in this limit: energetic gradient terms permit the unbounded growth of any disturbance with wavelength larger than a critical value, whereas a disturbance of any finite wavelength in a medium with dissipative gradient terms can become unbounded only when the yield strength tends to zero.