This project will involve research on a number of nonlinear problems in solid mechanics. These problems include: (1) dynamical problems of viscoelasticity and dissipation and shocks (including one-dimensional problems for nonlinearly viscoelastic bodies and self-sustained oscillations, Hopf bifurcations, and fluid-solid interactions); (2) asymptotics of light flexible bodies; (3) bifurcation problems of nonlinear elastoplasticity; (4) contact problems; (5) coexistence of phases in anisotropic media; and (6) other problems such as inverse bifurcation problems, nonlinear magnetoelasticity, and control problems. Each problem will be given an exact formulation (without ad hoc geometric simplifications) and each problem employs general constitutive equations satisfying the standard invariance conditions and mild physical restrictions. The use of such constitutive equations illuminates the mathematical structure of the governing equations and permits the detection of new effects associated with material response. The goals of the research are to detect new phenomena, especially those associated with instabilities and the occurrence of thresholds, to treat concrete problems that illuminate the general theories, and to develop analytic tools for the treatment of both general and specific problems. A variety of dynamic and steady-state nonlinear problems for rods, shells, and three-dimensional solid bodies will be studied. The bodies are composed of nonlinearly elastic, viscoelastic, plastic, or magnetoelastic materials. In each case, fully invariant, geometrically exact theories encompassing general nonlinear constitutive equations are to be used. The goals of these studies are to discover new nonlinear effects, determine thresholds in constitutive equations separating qualitatively different responses, examine important kinds of instabilities, contribute to the theory of shocks and dissipative mechanisms in solids, and develop new methods of nonlinear analysis for problems of solid mechanics.
