New frontiers in the mathematics of solids

固体力学是研究固体在外力或位移、温度变化和其他因素的作用下如何变形的学科。中心模型是弹性理论，其中应力（作用在材料内表面上的每单位面积的力）是应变的规定函数，而对该理论的各种修改使得能够描述其他效应，例如塑性流动和损伤。固体力学的控制方程是高度非线性的偏微分方程组，其数学性质，例如，当溶液存在时，以及它们如何依赖于重要的参数，人们知之甚少。该建议的目的是在固体力学的数学和溶液的计算方面进行一项基础广泛的研究计划，集中在三个重要的应用领域。第一个领域涉及合金中微观结构图案的形成，由相变引起的，其中下面的晶格在临界温度下经历形状的变化（例如，从立方到四方）。这些图案对于确定材料的日常性质非常重要。我们的研究将第一次尝试用数学的方法来描述这些图案是如何形成的。第二个领域是断裂力学，它研究材料何时以及如何破裂和断裂。这是一个具有重大技术意义的大科学领域。我们的研究将集中在潜在的令人兴奋的新的数学模型的骨折，不作出猜测的位置和形式的新的骨折表面，并允许有效的数值计算这些表面。第三个领域涉及固体力学在医学上的应用。力学对于理解人体的许多部分变得越来越重要。我们将研究骨骼和组织如何生长的模型，并将其应用于肿瘤，特别是结肠癌，以及如何通过观察乳房弹性特性的变化来帮助检测乳房肿瘤。该计划将涉及与实验学家，显微镜学家和医学研究人员的密切合作，但同时它将借鉴并试图加深我们对基本方程的数学理解，这对于所有应用是共同的。它将涉及结合建模技能（例如，模型如何与原子或细胞相互作用相关），方程的数学分析，设计有效的计算算法，以及与那些想要使用结果的人（工程师，材料科学家和医生）互动。

Solid mechanics is the study of how solids deform under the action of applied forces or displacements, changes of temperature and other factors. The central model is that of elasticity theory, in which the stress (the force per unit area acting across internal surfaces in the material) is a prescribed function of the strain, while various modifications to this theory enable other effects, such as plastic flow and damage, to be described.The governing equations of solid mechanics are highly nonlinear systems of partial differential equations, the mathematical properties of which, such as when solutions exist and how they depend on important parameters, are poorly understood.The aim of the proposal is to conduct a broadly based programme of research on the mathematics of solid mechanics and computation of solutions, concentrating on three important areas of applications.The first area concerns the formation of patterns of microstructure in alloys, arising from phase transformations in which the underlying crystal lattice undergoes a change of shape (for example, from cubic to tetragonal) at a critical temperature. These patterns are of importance for determining the everyday properties of the material. Our research will attempt for the first time to describe mathematically how these patterns form.The second area is fracture mechanics, which is the study of when and how materials crack and break. This is a large scientific field of great technological importance. Our research will focus on potentially exciting new mathematical models of fracture, which do not make guesses as to the position and form of new fracture surfaces, and which allow effective numerical computation of these surfaces.The third area concerns applications of solid mechanics to medicine. Mechanics is becoming increasingly important for the understanding of many parts of the human body. We will study models of how bone and tissue grow, with applications to tumours and in particular colon cancer, and how the detection of breast tumours can be aided by observing changes in the elastic properties of the breast.This programme will involve close collaboration with experimentalists, microscopists and medical researchers, but at the same time it will draw on and attempt to deepen our mathematical understanding of the underlying equations, which are common to all the applications. It will involve combining skills in modelling (for example, how the models are related to atomic or cellular interactions), in mathematical analysis of the equations, in devising effective computational algorithms, and in interacting with those wanting to use the results (engineers, materials scientists, and doctors).