Elasticity: on the interface between statics and dynamics

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Elasticity is the branch of continuum mechanics modelling the behaviour of elastic solids, that is bodies that deform under loading but return to their original configuration once the loads are removed. These include metal alloys, polymers, and biological materials such as steels, shape-memory alloys, rubbers, liquid crystals, soft tissues, and cell membranes, with numerous important theoretical and engineering applications in the physical and life sciences.The mathematical treatment of elasticity, as all models in continuum mechanics, lies within the area of partial differential equations (PDEs), particularly nonlinear PDEs, with a meaningful distinction between static and dynamic problems, i.e. problems with and without time-dependence, describing respectively bodies in and out of equilibrium. Static problems rely on the theory of elliptic PDEs and the calculus of variations, popular mathematical areas with remarkable applications in modern materials science. On the other hand, dynamic problems in elasticity require tools from the theory of hyperbolic PDEs and conservation laws which are typical in the study of fluids, e.g. the Euler equations. Yet the interaction between these different areas of mathematics is limited and the tools from hyperbolic PDEs are not sharp enough for applications in elasticity. Indeed, many of these tools rely on the convexity of an energy associated to each model, an assumption which is too strong for the theory of elasticity due to physical considerations. Instead, the much weaker notion of quasiconvexity needs to be assumed for energies in elasticity, a notion which has been studied extensively within the calculus of variations since its discovery in the 1950s. Still, quasiconvexity remains poorly understood and introduces significant challenges even for static problems in elasticity. In dynamics, the role of quasiconvexity is far less understood and mathematical results exploiting the quasiconvexity property in the context of dynamics are very limited. The same is true in the study of numerical approximations for problems in elasticity, static and dynamic. Once viscosity effects are also accounted for in dynamic models for elasticity, additional modelling and mathematical complications arise due to the so-called axiom of frame-indifference, an assumed physical invariance of the constitutive laws in continuum mechanics under a change of observers. Evidently, the current mathematical theory on the role of quasiconvexity in dynamics and numerical approximations for elasticity, as well as the incorporation of viscosity effects is unsatisfactory. These problems lie on the interface between the calculus of variations and hyperbolic PDEs, two largely disconnected areas, and tackling them requires a combination of tools from both fields. It is the aim of the project to address such problems in elasticity and initiate an important and timely study in the wide, yet under-explored, interface between these fields.