Mathematical Modelling of Nonlinear Waves in Layered Elastic Waveguides with Inhomogeneities

拟建的多学科项目涉及三个主要领域：非线性波数学理论、固体力学和材料与结构无损检测。该项目的目的是发展非线性波在具有扩展不均匀性的层状弹性波导中传播的数学理论，表示损伤/分层。这将通过使用不同复杂性的数学模型和广泛的分析方法和数值模拟的应用来实现。理论预测将通过实验来验证数学理论。传统上，流体和固体中的非线性波是由不同的研究团体研究的。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。非均匀介质中非线性波领域中最著名的问题之一是表面重力波在光滑底部地形上传播的经典流体力学问题。这个问题已经在各种公式中进行了研究，包括单色波、浅水孤子和波包。本项目旨在对具有扩展界面不均匀性的非完美键合层状弹性波导（模拟键合不良区域）中的非线性波过程进行理论和实验研究，从某种意义上说，这可以视为上述经典问题的模拟。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（模拟层之间的凝聚力因缺陷的存在而减弱的区域）。然后，我们可以使用问题的变分公式和在可变地形上传播的表面重力波的方法来描述非线性波（如应变孤子、单色波或波包）的缓慢演化。特别是，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坏区域的特征。最后一种效应可以潜在地用于层状结构的无损检测，使用非线性波。请注意，虽然类似的数学方法已经用于一些经典的流体和固体力学问题，但它们在所描述的非线性弹性问题上的应用将是全新的和非常重要的。

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.