Analysis of Deformation, Buckling, and Fracture of Materials: From Composite Materials to Thin Domains

制造的设备和结构通常是包含细长部件的复合材料。为了避免结构失效并预测其力学行为，需要了解材料在载荷下的力学行为，即结构的刚度、柔性、屈曲和断裂，并推导出复合材料的有效行为，广义地说，这相当于建立复合材料有效性能的严格界限。 现有的工程薄结构理论主要依赖于形式的渐近展开和近似，而一般来说，他们预测薄材料的变形和断裂相当准确，他们不能这样做，在某些情况下的原因不太清楚，因为往往没有一个数学上严格和全面的机制，验证制度的有效性，即使当结构的几何形状是简单的。另一方面，现有的数学薄结构理论仍然包含几个未解决的或未经验证的制度有关的几何形状的结构和能量和负载的大小。该项目旨在解决薄结构的变形，屈曲和断裂建模中的这些缺点，通过建立新的工具来获得数学上严格的薄结构理论，这取决于结构的几何参数，并研究复合材料和印刷材料，以获得其有效性能的新界限。 该项目将为研究生提供研究培训机会。该项目的第一部分涉及连续介质和断裂力学。虽然可展壳的变形和刚度的数学理论很好理解，但常号高斯曲率壳和薄域的数学理论就不那么清楚了。与此同时，薄结构屈曲的研究是众所周知的复杂和欠发达。众所周知，铰接薄壳的线性几何刚度取决于壳体的高斯曲率，本项目旨在证明在非线性情况下也是如此，并改进常号高斯曲率薄域的理论。该项目还计划通过一种新的细长结构屈曲理论来处理薄结构屈曲的建模问题，并在断裂力学方面，推导出数学上严格的壳体断裂格里菲斯理论，适用于高斯曲率为零或常数的壳体。该项目的第二部分旨在通过所谓的极端准凸二次型来推导复合材料和印刷材料（称为超材料）的有效特性的新界限。该项目将使用应用分析和真实的代数和凸几何的技术和工具。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Manufactured devices and structures are often composites containing slender parts. To avoid structural failures and predict their mechanical behavior, one needs to understand material mechanical behavior under loading, namely the rigidity, flexibility, buckling, and fracture of structures, and to derive the effective behavior of composites, which broadly speaking amounts to establishing tight bounds on a composite's effective properties. Existing engineering thin structure theories mostly rely on formal asymptotic expansions and approximations, and while generally they predict the deformation and fracture of thin materials quite accurately, they fail to do so in some cases for reasons not well understood, since often there is not a mathematically rigorous and comprehensive mechanism for verification of the regime of validity, even when the structural geometry is simple. On the other hand, existing mathematical thin structure theories still contain several unsolved or unverified regimes concerning the geometry of the structure and the energy and loading magnitudes. This project aims to tackle these shortcomings in the modeling of deformation, buckling, and fracture of thin structures, by building new tools to obtain mathematically rigorous thin structure theories depending on geometric parameters of the structure, and to study composites and printed materials to derive new bounds on their effective properties. The project will provide research training opportunities for graduate students.The first part of the project deals with continuum and fracture mechanics. While the mathematical theory of deformation and rigidity for developable shells is well understood, it is less so for constant-sign Gaussian curvature shells and thin domains. At the same time the study of buckling of thin structures is known to be complex and underdeveloped. It is known that the linear geometric rigidity of shells with pinned thin faces depends on the Gaussian curvature of the shell, and this project aims at showing that this is indeed the case in the nonlinear setting too and at improving the theories for constant-sign Gauss curvature thin domains. The project plans also to tackle the modeling of buckling of thin structures by means of a new slender structure buckling theory, and, on the side of fracture mechanics, to derive a mathematically rigorous shell fracture Griffith theory for shells with vanishing or constant-sign Gaussian curvature. The second part of the project intends to derive new bounds on the effective properties of composites and printed materials, known as metamaterials, by means of the so-called extremal quasiconvex quadratic forms. The project will use techniques and tools from applied analysis and real algebraic and convex geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.