Mathematical Sciences: Nonlinear Elasticity and the Calculusof Variations

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With this award the principal investigator will study models of nonlinear elasticity that describe the behavior of elastic materials at moderate to large deformations. In contrast to linear models nonlinear models usually require for their solution more sophisticated types of mathematical analysis. On the theoretical side the principal investigator will study the weak continuity of the Jacobian determinant and the choice of a space of admissible deformations. On the applied side he will study the stability of elastic composite materials such as foam rubber, and he will analyze solid-solid phase transitions in elastic crystals. A goodly number of everyday phenomena involving elastic materials can be explained using what is called linear elasticity theory. This linear theory concerns the behavior of elastic materials in response to small deformations. However when the deformations are moderate to large in size the linear theory must be replaced with a more complicated nonlinear theory. A nonlinear theory is also required if the material is a composite, such as foam rubber or fibre-reinforced rubber. With this award the principal investigator will use mathematical analysis to study models of nonlinear elastic materials.