Theorems of Linear Elasticity extended to Gradient Elasticity and their Applications

经典的弹性理论是工程师日常生活中不可或缺的一部分。它在19世纪初到20世纪中叶之间建立在坚实的理论基础上。它的发展可以被认为是完整的。遗憾的是，它的范围是有限的：它对尺寸不敏感，当边界数据中出现不连续时，它包含应力和位移的奇异性，并且不能包括边界和表面能量。因此，它仅限于典型的工程应用。对于微米和纳米范围内的微观成分或现象的描述，它只是部分适用。应变梯度弹性是经典弹性的自然扩展，在应变梯度弹性中，位移场出现了更高的导数。许多论文表明，经典弹性理论的局限性可以通过梯度展开来克服，而不会模糊结构和材料特性之间的通常分离，就像其他非局部理论一样。不幸的是，还不可能像经典弹性理论那样为梯度弹性理论建立一个完整的坚实的基础。这不是一个纯粹的学术问题。部件的日益小型化和微结构材料的有针对性的发展要求我们超越经典的弹性理论。此外，通过去除经典弹性力学的奇异性，我们能够应用许多准则(例如，断裂准则和流动准则)，这些准则通常是在柯西应力中表达的，也可以在边界不连续处应用。这大大增加了弹性理论的适用性。在拟议的项目中，经典弹性理论的成熟理论基础将扩展到应变梯度弹性。为此，人们提出了一种推广的公理理论，其中约2/3已转移到梯度理论中。我们试图完成这一移交，这是德国项目合作伙伴工作的核心。俄罗斯项目合作伙伴关心的是具体的应用。例如，具有纯位移或纯应力边界条件的边值问题的唯一性定理被应用于齐次化。例如，有了它们，无限矩阵中椭圆包含的EShelby基本解就可以推广。另一个应用是横观各向同性纤维增强复合材料，它的尺度转换和刚度张量的特性都需要研究。最后，将在梁实验中研究梯度弹性的圣维南原理。

The classical theory of elasticity is an integral part of the daily routine of engineers. It was placed on a firm theoretical foundation between the beginning of the 19th century and the mid-20th century. Its development can be considered complete.Unfortunately, its scope is limited: It is size insensitive, it contains singularities in the stresses and displacements when discontinuities appear in the boundary data, and can not include boundary and surface energies. Thus, it is limited to typical engineering applications. For the description of micro-components or phenomena in the micron- and nanometer range it is only partially suitable.A natural extension of classical elasticity is the strain gradient elasticity, in which higher derivatives of the displacement field appear. It has been shown in numerous papers that the limitations of classical elasticity theory can be overcome with gradient expansion without blurring the usual separation between structure- and material properties, as is the case with alternative nonlocal theories. Unfortunately, it has not yet been possible to develop a complete solid foundation for gradient elasticity as it exists for classical elasticity theory.This is not a purely academic matter. The increasing miniaturization of components and the targeted development of micro-structured materials require us to go beyond the classical theory of elasticity. Furthermore, by removing the singularities of classical elasticity, we are able to apply a number of criteria (e.g. fracture and flow criteria), which are usually formulated in the Cauchy stresses, also in the vicinity of boundary discontinuities. This significantly increases the applicability of the elasticity theory.In the proposed project, the well-established theoretical foundations of classical elasticity are to be expanded to strain gradient elasticity. For this purpose, a generalizing axiomatic theory has been worked out, about 2/3 of which have already been transferred to the gradient theory. We try to complete this transfer, which is the core of the work of the German project partner. The Russian project partner is concerned with specific applications. For example, uniqueness theorems for boundary value problems with pure displacement or pure stress boundary conditions are applied in homogenization. With them, for example, the Eshelby fundamental solution of an elliptical inclusion in an infinite matrix can be extended. Another application are transversely isotropic fiber-reinforced composites, for which both a scale transition and the specific properties of the stiffness tensor are to be investigated. Finally, the de Saint-Venant principle for gradient elasticity will be investigated in beam experiments.