RIGOROUS DERIVATION OF MODERATE AND HIGH-CONTRAST NONLINEAR COMPOSITE PLATE THEORIES

最近有几个结果是关于从全三维弹性方程推导二维模型（即板，壳）的。相关处理通常从引入一个代表板厚的小参数开始，并在参数趋于零时寻求通过极限的程序。在线性化弹性的背景下，相关的二维极限理论，有时被称为“Kirchhoff-Love理论”，自20世纪70年代以来得到了广泛的研究。然而，严格的非线性、框架无关的板理论最近才在膜极限背景下以及在弯曲和von Karman制度中得到解决。上述非线性板理论的严格方法可以被称为经典，因为它们是在假设底层存储能量函数具有均匀矫顽力的情况下工作的。然而，尝试理解当制造薄体的材料允许某些部分比其他部分承受更大的应变时，会产生什么额外效应，这在实际中是很重要的。限制这些最新结果适用性的另一个特点是，它们只适用于均匀板，而目前工业上使用的绝大多数材料是“复合材料”，其数学三维理论已经相当发达，至少在经典的周期背景下。本项目将解决现有理论的这两个缺陷。在更广泛的数学背景下，这将服务于三个主要目标：1)通过提供更广泛的模型（包括一些更现实的模型）的严格推导，使板块的数学理论更加完整；2)弥合均质化理论与现有非线性板块理论之间的差距；3)对现实板的整体响应的面内非定域性等“非经典”效应进行了研究。在应用方面，它将产生复合材料设计的新发展，以及最近一类难以通过试验和错误获得的超材料。

There are several results obtained quite recently that are concerned with the derivation of 2D models (i.e. plates, shells) from the equations of the full 3D elasticity. The related treatment usually starts by introducing a small parameter which represents the plate thickness, and seeks a procedure for passing to the limit as the parameter tends to zero. In the setting of linearised elasticity the related 2D limit theory, which is sometimes referred to as the ``Kirchhoff-Love theory'', has been extensively studied starting since 1970's. However, the rigorous nonlinear, frame-indifferent, plate theories have only been addressed recently in the membrane-limit context and also in the flexural and von Karman regimes.The above rigorous approaches to nonlinear plate theories can be termed classical in the sense that they work under the assumption of uniform coercivity of the underlying stored-energy function. It would be practically important, however, to try to understand what additional effects emerge when the material of which a thin body is made allows for significantly larger strains in some parts of it than in others. Another feature that limits the applicability of these recent results is that they only apply to homogeneous plates, while the vast majority of the materials used nowadays in industry are ``composites'', whose mathematical 3D theory has been fairly well developed, at least in the classical periodic context. The present project will address these two shortcomings of the existing theories. In the wider mathematical context this will serve three major objectives: 1) To make the mathematical theory of plates more complete by providing a rigorous derivation of a wider class of models, including some that are more realistic; 2) To bridge the gap between the homogenisation theory and the existing nonlinear plate theories; 3) To make an advance on ``non-classical'' effects, such as the in-plane non-locality in the overall response, for realistic plates. In the applied context, it will generate new developments in the design of composites, and a more recent class of metamaterials , which are difficult to obtain experimentally by trial and error.