Asymptotic analysis of boundary value problems for strongly inhomogeneous multi-layered elastic plates

民用、航空航天和汽车工程如今面临着令人兴奋的工业发展，激发了新的多功能和可持续材料在结构部件中实施。经历重大技术变革的一个关键领域是多层结构的设计，这种结构的特点往往是各层在机械和几何特性上的高度对比。各种各样的复合材料在许多应用中出现，包括在汽车工程中广泛使用的轻质结构部件，目前主要关注绿色汽车生产。此外，现代航空航天工业得益于使用多层结构，其中包含新的轻质多功能材料，如气凝胶，为飞机提供隔热和隔音。尽管工业需求迅速增长，但对于具有强垂直不均匀性的薄弹性层压板的建模，尚未开发出一致的数学方法。造成这种情况的主要原因是机械和几何特性的对比，导致许多额外的问题参数。另一个基本问题是，高对比度层压板的最实际重要的低频振动表现出新的特征，这些特征以前没有在均匀或弱非均匀结构中观察到。复杂的低频响应显着复杂化了旨在结构优化和非破坏性评估的数值和实验数据的解释。该项目将为高对比度多层薄板提供一个相当通用的近似模型，涵盖广泛的问题参数，并涉及沿边缘的低频运动和边界条件的数学一致方程。边界条件的推导历来是薄弹性结构严格理论的主要挑战，目前只在均质框架下进行过尝试。

Civil, aerospace and automotive engineering are nowadays facing exciting industrial developments, inspiring new multifunctional and sustainable materials to be implemented within structural components. A key area, experiencing major technological changes, is concerned with design of multi-layered structures which are often characterised by a high contrast in mechanical and geometric properties of the layers. A diverse variety of such composites arises in numerous applications, including lightweight structural components widely exploited in automotive engineering which is currently strongly focused on green car production. In addition, the modern aerospace industry benefits from making use of multi-layered structures incorporating new lightweight multifunctional materials, such as aerogels, providing both thermal and acoustical insulation in aircrafts. In spite of rapidly growing industrial demands, a consistent mathematical approach for modelling of thin elastic laminates with a strong vertical inhomogeneity has not yet been developed. The main reason for this is presence of the contrast in mechanical and geometrical characteristics, resulting in numerous extra problem parameters. Another fundamental problem is that the most practically important low-frequency vibrations of high-contrast laminates manifest novel features which has not been previously observed for homogeneous or weakly inhomogeneous structures. A sophisticated low-frequency response significantly complicates interpretation of numerical and experimental data aimed at structural optimisation and non-destructive evaluation.This project will provide a fairly universal approximate model for high-contrast multi-layered thin plates covering a broad range of problem parameters and involving the mathematically consistent equations of the low-frequency motion and the boundary conditions along the edges. The derivation of the boundary conditions is traditionally the main challenge in the rigorous theories for thin elastic structures and have been attempted only within the homogeneous framework.