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，人们需要了解材料在载荷下的力学行为，即结构的刚度、柔韧性、屈曲和断裂，并推导复合材料的有效行为，广义上讲，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大多依赖于形式渐近展开和近似，虽然它们通常相当准确地预测薄材料的变形和断裂，但由于一些不太清楚的原因，它们在某些情况下无法做到这一点，因为通常没有数学上严格和全面的机制来验证有效性制度，即使结构几何很简单。另一方面，现有的数学薄结构理论仍然包含一些关于结构几何、能量和载荷量级的未解决或未经验证的机制。本项目旨在通过建立新的工具来获得基于结构几何参数的数学上严格的薄结构理论，并通过研究复合材料和印刷材料来推导其有效性能的新界限，从而解决薄结构变形、屈曲和断裂建模中的这些缺陷。该项目将为研究生提供研究培训机会。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已经很好地理解了，但对于常符号高斯曲率壳和薄域来说，这还不是很清楚。同时，薄结构屈曲的研究是复杂和不发达的。众所周知,与面容钉薄壳的线性几何刚度取决于壳的高斯曲率,和这个项目旨在表明,这确实是非线性的情况下设置,在提高理论constant-sign高斯曲率薄域。该项目还计划通过一种新的细长结构屈曲理论来解决薄结构屈曲的建模问题，并在断裂力学方面，为具有消失或常符号高斯曲率的壳导出数学上严格的壳断裂Griffith理论。该项目的第二部分旨在通过所谓的极值拟凸二次型，推导复合材料和印刷材料（称为超材料）有效性能的新界限。该项目将使用应用分析和真正的代数和凸几何的技术和工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.