Antman receives NSF grant to study nonlinear problems of solid mechanics

ISR-affiliated Professor Stuart Antman (Math) has been awarded a two-year, $288,362 National Science Foundation grant for Nonlinear Problems of Solid Mechanics.

Abstract
The researchers will treat a variety of dynamical and steady-state nonlinear problems for deformable rods, shells, and three-dimensional solid bodies that can suffer large and rapid deformations. The bodies are composed of nonlinearly elastic, viscoelastic, plastic, viscoplastic, or magnetoelastic materials. In each case, properly invariant, geometrically exact theories encompassing general nonlinear constitutive equations are used. The goals of these studies are to (i) discover new nonlinear effects, (ii) determine thresholds in constitutive equations separating qualitatively different responses, (iii) determine general classes of constitutive equations that are both physically and mathematically natural, (iv) examine and predict important kinds of instabilities, (v) determine how existence, regularity, and well-posedness of solutions of the governing quasilinear systems of partial differential equations depend on material behavior, (vi) contribute to the theory of shocks and dissipative mechanisms in solids, and (vii) develop new methods of nonlinear analysis and of effective computation for problems of solid mechanics. Antman is continuing to make carefully formulated physical theories accessible to mathematicians, and to make modern techniques of applicable analysis accessible to scientists and engineers, with special attention to graduate students.

The researchers hope to develop powerful new mathematical methods for analyzing the large and rapid deformations of solid bodies. This work has applications both to new technological materials including "smart" materials and to old biological materials, such as living tissue. (Although it has been known for 50 years how to derive the equations governing such deformations cleanly from first principles without ad hoc approximations, the mathematical tools for treating these equations are only now being developed and refined.) The specific problems under study include (1) the deformation of thin shell-like bodies in a fluid flow, (2) the loss of stability of such a shell subjected to pulsating pressures, (3) the swimming of eels, (4) the rigorous justification of useful approximations for the motions of the body when the loads it bears are applied slowly or when the loads are very large or when the body is of light weight, (5) the buckling of shells, (6) strange effects for spinning bodies, (7) the development and the preclusion of shocks (which are large and sudden discontinuities in velocity and deformation), and (8) the construction of effective methods of computation. Antman is continuing to make carefully formulated physical theories accessible to mathematicians, and to make modern techniques of applicable mathematics accessible to scientists and engineers, with special attention to graduate students.

Published June 19, 2007