New frontiers in the mathematics of solids

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).