Mathematical Modelling of Nonlinear Waves in Layered Elastic Waveguides with Inhomogeneities

拟议的多学科项目位于三个主要领域的交叉点：非线性波的数学理论，固体力学和材料和结构的无损检测。该项目的目的是发展一个数学理论的非线性波传播的层状弹性波导与扩展的不均匀性代表损伤/分层。这将通过使用不同复杂程度的数学模型和应用广泛的分析方法和数值模拟来实现。理论预测将通过实验进行测试，以验证数学理论。传统上，流体和固体中的非线性波由不同的研究团体研究。然而，越来越清楚的是，在看似不同的问题的数学方法中有许多类比。在非均匀介质中的非线性波领域中，最著名的问题之一是在光滑海底地形上传播的表面重力波动力学的经典流体力学问题。这个问题已经研究了各种配方，包括单色波，浅水孤子和波包。本项目的目的是理论和实验研究的非线性波过程中不完全结合的层状弹性波导与扩展的界面不均匀性（建模不良结合的地区），这可以被认为是，在某种意义上，作为一个类似的经典问题上述。实际上，我们可以假设层之间的耦合沿着分层波导缓慢变化（对层之间的内聚力由于缺陷的存在而减弱的区域进行建模）。然后，我们可以使用变分制定的问题和方法开发的表面重力波传播的变化的地形来描述一个非线性波（如应变孤立子，单色波或波包）的缓慢演变。特别地，给定应变孤子的初始速度，我们期望能够找到足以导致孤子传播失败的损伤区域的特征。这最后的效果可以潜在地用于非破坏性测试的层状结构，使用nonlinear waves.Note，虽然类似的数学方法已经开发了一些经典的流体和固体力学问题，其应用所描述的问题的非线性弹性将是全新的和高度非平凡的。

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.