Resolution of a Conjecture in Nonlocal Strain-gradient Plasticity

Strain-gradient theories of plasticity have been successful in modeling the behavior of complex materials. However, the traditional formulation of these theories lacks a material length scale, and is thus incapable of capturing experimentally observed size effects that play an important role in the behavior of nano structures. As a result, a modified theory was proposed which incorporates an intrinsic dissipative length scale. The theory predicts that the solutions to the flow rule are global minimizers of the functional for energy dissipation. We prove that there are no global minimizers of the functional, thus resolving a previously unsolved conjecture. Our result shows that the variational formulation of the theory is unviable. The non-existence of a global minimizer appears to be related to the formation of infinitely fine plastic boundary layers.