In this proposal the principal investigator plans to continue his studies of the formation and propagation of shock waves in stiff polymeric materials and to develop a mathematical theory of finite deformations of membranes. The latter theory is useful in providing better estimates of the errors incurred when employing stress analysis to wrinkled membranes. The principal investigator will also study the mathematical properties of certain idealized materials that have applications to the twinning of crystals and the behavior of shape-memory alloys. Materials scientists very often study the properties of materials by testing their responses to applied stresses; one could call this the experimental approach. Mathematicians interested in the behavior of materials usually take a different approach which involves the theoretical analysis of mathematical models of materials. In this proposal the principal investigator will use advanced mathematics to try to understand twinning in crystals, necking in polymers and the behavior of shape-memory alloys.
