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含有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作用下的力学行为，即结构的刚性、柔性、屈曲和断裂，并推导出复合材料的有效行为，广义地讲，这相当于对复合材料的有效性能建立严格的界限。现有的工程薄结构理论大多依赖于形式的渐近展开和近似，虽然它们通常非常准确地预测薄材料的变形和断裂，但在某些情况下，由于不太清楚的原因，它们无法做到这一点，因为往往没有一个严格和全面的数学机制来验证有效性，即使结构几何很简单。另一方面，现有的数学薄壁结构理论仍然包含一些关于结构几何、能量和载荷大小的未解或未经验证的区域。该项目旨在解决薄结构变形、屈曲和断裂建模方面的这些缺陷，通过建立新的工具来根据结构的几何参数获得数学上严格的薄结构理论，并研究复合材料和印刷材料，以得出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。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，但对于常号高斯曲率壳和薄域，这一理论就不那么清楚了。同时，薄壁结构屈曲的研究也比较复杂和不发达。众所周知，具有固定薄面的壳体的线性几何刚度依赖于壳体的高斯曲率，本课题的目的是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解决薄结构的屈曲建模问题，并在断裂力学方面，为具有消失或常号高斯曲率的壳体推导出严格的壳体断裂格里菲斯理论。该项目的第二部分旨在通过所谓的极值拟凸二次形式，推导出复合材料和印刷材料(称为超材料)的有效性质的新界限。该项目将使用应用分析、真实代数和凸几何的技术和工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.