Nonlinear Mechanics of Defects in Solids

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This award supports research towards the application of modern tools of differential geometry to the mechanics of defects in solids. The formulation would improve the accuracy of failure predictions for nonlinear materials such as metals, soft materials, and biological tissues. Defects in solids play a crucial role in determining the mechanical and physical properties of many engineering materials. However, the analysis of defects has been restricted to linear elasticity in almost all the existing theoretical works. Many physical theories are most naturally expressed in terms of geometric notions. The research outcomes of this project will also be summarized in a new textbook for an advanced course on differential geometry applied to mechanics of materials.This research program aims to use techniques from modern differential geometry to formulate a fully nonlinear geometric theory of defect mechanics that will enable one to solve several open problems in the nonlinear mechanics of solids with distributed line and point defects. There is very little known about defects in the setting of nonlinear elasticity. Several new exact solutions will be obtained for the stress fields of nonlinear solids with distributed dislocations, disclinations, point defects, and their combinations. Not much is known about the interaction of defects in the presence of large strains and our preliminary results show that the geometric theory will be very useful in making progress in this direction. Dynamics and stability studies of nonlinear defects are exciting and intellectually challenging problems that will be investigated during the course of this project.