Elasticity: on the interface between statics and dynamics

弹性力学是连续介质力学的分支，模拟弹性固体的行为，即物体在载荷下变形，但一旦载荷被移除就恢复到其原始构型。这些材料包括金属合金、聚合物和生物材料，如钢、形状记忆合金、橡胶、液晶、软组织和细胞膜，在物理和生命科学中具有许多重要的理论和工程应用。弹性的数学处理，如同连续介质力学中的所有模型一样，属于偏微分方程（PDE）领域，特别是非线性偏微分方程，在静态和动态问题之间有意义的区别，即具有和不具有时间依赖性的问题，分别描述处于和不处于平衡状态的物体。静态问题依赖于椭圆偏微分方程理论和变分法，这是在现代材料科学中具有显着应用的流行数学领域。另一方面，弹性力学中的动力学问题需要双曲偏微分方程和守恒律理论的工具，这在流体研究中是典型的，例如欧拉方程。然而，这些不同的数学领域之间的相互作用是有限的，从双曲偏微分方程的工具是不够尖锐的弹性应用。事实上，这些工具中的许多依赖于与每个模型相关联的能量的凸性，由于物理考虑，这一假设对于弹性理论来说太强了。相反，弹性能量需要假设弱得多的拟凸性概念，自20世纪50年代发现以来，这一概念在变分法中得到了广泛的研究。尽管如此，拟凸性仍然知之甚少，甚至在弹性力学的静态问题中也引入了重大的挑战。在动力学中，拟凸性的作用很少被理解，并且在动力学背景下利用拟凸性性质的数学结果非常有限。在弹性力学、静力学和动力学问题的数值近似研究中也是如此。一旦粘性效应也占在动态模型的弹性，额外的建模和数学的复杂性出现，由于所谓的公理框架无差异，假设物理不变的本构关系在连续介质力学下的观察者的变化。显然，目前关于拟凸性在动力学和弹性数值近似中的作用以及粘性效应的结合的数学理论是不令人满意的。这些问题位于变分法和双曲偏微分方程之间的界面上，这是两个基本上不相关的领域，解决它们需要结合两个领域的工具。该项目的目的是解决这些弹性问题，并在这些领域之间广泛但未充分探索的界面中开展重要而及时的研究。

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.