Mathematical Modelling of Nonlinear Waves in Layered Elastic Waveguides with Inhomogeneities

The proposed multi-disciplinary project lies at the intersection of three major areas: Mathematical Theory of Nonlinear Waves, Solid Mechanics and Non-destructive Testing of Materials and Structures. The aim of the project is to develop a mathematical theory of nonlinear waves propagating in layered elastic waveguides with extended inhomogeneities representing damage/delamination. This will be achieved using mathematical models of different complexity and the application of a wide range of analytical methods and numerical simulations. Theoretical predictions will be tested experimentally to verify the mathematical theory. Traditionally, nonlinear waves in fluids and solids are studied by different communities of researchers. However, it becomes increasingly clear that there are many analogies in the mathematical approaches to seemingly different problems. One of the best known problems in the area of nonlinear waves in inhomogeneous media is the classical fluid mechanics problem of the dynamics of surface gravity waves propagating over smooth bottom topography. This problem has been studied in various formulations, including monochromatic waves, shallow-water solitons and wave packets. The present project aims at the theoretical and experimental study of nonlinear wave processes in imperfectly bonded layered elastic waveguides with extended interfacial inhomogeneities (modelling poorly bonded areas), which can be considered, in a sense, as an analogue of the classical problem described above. Indeed, we can assume that coupling between the layers varies slowly along a layered waveguide (modelling the areas where the cohesive forces between the layers are weakened by the presence of defects). We can then use the variational formulation of the problem and methods developed for surface gravity waves propagating over variable topography to describe the slow evolution of a nonlinear wave (such as a strain soliton, monochromatic wave or a wave packet). In particular, given the initial velocity of the strain soliton, we expect to be able to find the characteristics of the damaged region sufficient to cause the propagation failure of the soliton. This last effect can be potentially used for the non-destructive testing of layered structures, using nonlinear waves.Note that although similar mathematical methods have already been developed for some classical fluid and solid mechanics problems, their application to the described problem of nonlinear elasticity will be completely new and highly nontrivial.