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ISR-affiliated Professor Stuart Antman (Math) has received a one-year, $123K NSF grant for Nonlinear Problems of Solid Mechanics.
The investigator gives careful mathematical treatments of a variety of dynamical and steady-state nonlinear problems for deformable rods, shells, and three-dimensional solid bodies, possibly in contact with moving fluids, variable temperature fields, and electromagnetic fields. The bodies are composed of nonlinearly elastic, plastic, viscoplastic, or magneto-(visco-)elastic materials. In each case, properly invariant, geometrically exact theories encompassing general nonlinear constitutive equations are used. The goals of these studies are to discover new nonlinear effects and new kinds of instabilities, determine thresholds in constitutive equations separating qualitatively different responses, determine general classes of constitutive equations that are both physically and mathematically natural, determine how existence, regularity, and well-posedness depend on material behavior, contribute to the theory of shocks and dissipative mechanisms in solids, and develop new methods of nonlinear analysis and of effective computation for problems of solid mechanics.
For treating the behavior of both new technological materials and old biological materials, such as living tissue, traditional theories, based on such simplifying assumptions as small deformations and linear response, may well fail to capture important physical phenomena and may, in fact, give misleading information. E.g., the study of a specific problem for a traditional theory may indicate that a dangerous instability occurs in a solid body when an externally applied load or frequency exceeds a certain threshold. An analysis of the same problem without the simplifying assumptions may well show that a large class of materials give rise to instabilities at thresholds much lower than those predicted by simplified theories. Since correctly formulated exact nonlinear problems of solid mechanics seldom can be directly subsumed under available mathematical theories for treating the governing equations, one purpose of this project is to develop new mathematical techniques to handle such problems. These techniques can treat whole ranges of materials at one time. The results have consequences for structures under loads (bridges, buildings, biological structures), for control of "smart" structures, and for micro-electro-mechanical systems (MEMs).
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