The aim of this research is the development and implementation of effective numerical methods for the solution of partial differential equations which model the nonlinear mechanics of solids. The principal investigator will use numerical methods to study quasilinear partial differential equations governing large motions of nonlinearly elastic, elastoplastic and viscoplastic materials. The equations governing the motions of such materials are generally systems of hyperbolic conservation laws which are ideally suited for numerical study. In particular, the PI will utilize an efficient numerical scheme based on finite-difference approximations and inspired by numerical methods from gas dynamics in order to study antiplane and radial motions of elastoplastic materials, with the goal of characterizing the formation of shocks. Additional effort will be directed toward the study of the formation of shear bands in viscoplastic materials. The principal investigator will study the effects of strain-gradient regularization on the development and propagation of shear bands in two-dimensional nonlinear viscoplasticity. This will include studies of problems where the solutions exhibit narrow shear bands, across which rapid variations in strain occur. The PI will develop adaptive pseudo-spectral methods to resolve spatial regions where rapid variations occur during plastic deformation.
The behavior of materials such as rubber, steel, or concrete can be described by complicated systems of partial differential equations which are in general difficult to analyze. However, there are certain special situations in which these equations simplify to a point where they can be both analyzed theoretically and computed numerically. Examples include the motion of a material block between a fixed and a moving plane and the compression of a solid ball of material. As a focus of this project, the principal investigator will investigate the nature of shock formation and of permanent plastic deformation on the formation and propagation of shocks for these types of motions. A second area of research is the formation of shear bands. These are localized regions of intense shear that form when ductile solids undergo large deformations. Shocks and shear bands correspond to possible regions in which material failure occurs, and their accurate determination is therefore very important. The principal investigator will take advantage of the special form of the equations to apply high-performance numerical methods that have been developed for applications in gas dynamics.
