Why gradient flows of some energies good for defect equilibria are not good for dynamics, and an improvement

Straight screw dislocations are line defects in crystalline materials and wedge disclinations are line defects in nematic liquid crystals. In this talk, I will discuss the development and implications of a single pde model intended to describe equilibrium states and dynamics of these defects. These topological defects are classically treated as singularities that result in infinite total energy in bodies of finite extent that behave linearly in their elastic response. I will explain how such singularities can be alleviated by the introduction of an additional 'eigendeformation' field, beyond the fundamental fields of the classical theories involved. The eigendeformation field bears much similarity to gauge fields in high- energy physics, but arises from an entirely different standpoint not involving the notion of gauge invariance in our considerations. It will then be shown that an (L2) gradient flow of a 'canonical', phase- field type (up to details) energy function coupling the deformation to the eigendeformation field that succeeds in predicting the defect equilibrium states of interest necessarily has to fail in predicting particular types of physically important defect dynamics. Instead, a dynamical model based on the same
energy but involving a conservation statement for topological charge of the line defect field for its evolution will be shown to succeed. This is joint work with Chiqun Zhang, graduate student at CMU.