Rigidity and Buckling of Shells: Toward New Nonlinear Shell Theories

经典线性壳理论对薄结构的变形或临界屈曲载荷的预测往往不能与薄结构的实际性能相匹配。可能导致这种失败的几种现象之一是所谓的“对缺陷的敏感性”，或者能量最小化器的高度振荡性质，即使在非常简单的几何形状（如圆柱体）中也会导致非紧致性。从飞机和汽车部件、建筑、电线杆和其他民用基础设施，到碳纳米管和生物聚合物外壳，对各种薄壳结构的工程师来说，不完美敏感性现象是一个持续的挑战。本课题的目标之一是利用薄结构屈曲的一般理论来量化缺陷敏感现象。另一个目标是解决长期存在的问题，即确定给定壳体的刚度，这被认为是由壳体的主曲率来确定的。虽然确定板的刚度的问题很好理解，确定一个给定的壳的刚度一直是一个长期存在的问题，在连续介质力学。在线弹性条件下，壳薄面上满足Robin边界条件的向量场的最新结果表明，壳的刚度仅由壳中表面的主曲率决定。这个项目的目标之一是ansatz-free下界扩展到非线性几何刚度设置。策略是使用现有的估计诱导人工边界条件通过测试函数以及普遍Korn插值不平等,这减少了线性几何刚度估算Poincar吗?型估计。一旦这样做了，一个新的壳层理论的层次结构将自然而然地与更小的差距，这将包括现有的由伽马收敛推导出来的理论。项目的第二部分是进一步发展薄结构的屈曲理论。该理论有两个组成部分：(i)证明尖锐的Korn不等式，（ii）证明无弯曲平凡分支的存在性。虽然(i)在以前的工作中几乎完全完成了，但（ii）是项目第二部分的目标。研究者和合作者将采用定量隐函数定理的方法和非线性弹性解的线性逼近方法。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。

The predictions made by classical linear shell theories on the deformation or critical buckling load of thin structures often fail to match the actual performance of such physical structures. Among the several phenomena that may be responsible for this failure is so-called "sensitivity to imperfections," or the highly oscillatory nature of energy minimizers resulting in non-compactness even in very simple geometries like cylinders. The imperfection sensitivity phenomenon poses an ongoing challenge to engineers working with a wide range of thin-shelled structures, from aircraft and automotive components, buildings, poles, and other civil infrastructures, to carbon nanotubes and biopolymer shells. One of the goals of this project is to use the general theory of buckling of thin structures to quantify the imperfection sensitivity phenomenon. Another goal is to solve the longstanding problem of determining the rigidity of a given shell, which is believed to be identified by the principal curvatures of the shell. While the problem of determining the rigidity of plates is well understood, determining the rigidity of a given shell has been a longstanding problem in continuum mechanics. Recent results on shells in the linear elasticity setting, with vector fields satisfying Robin boundary conditions on the thin face of the shell, provide evidence that the rigidity of a shell is determined solely by the principal curvatures of the shell's mid-surface. One of the goals of the project is to extend ansatz-free lower bounds to the nonlinear geometric rigidity setting. The strategy is to use existing estimates inducing artificial boundary conditions by test functions as well as a universal Korn interpolation inequality, which reduces the linear geometric rigidity estimate to a Poincar? type estimate. Once this is done, a new hierarchy of shell theories with smaller gaps will naturally follow, which will encompass the existing ones derived by Gamma-convergence. The second part of the project is to further develop the theory of buckling of thin structures. The theory has two ingredients: (i) proving sharp Korn's inequalities, and (ii) proving the existence of bending-free trivial branches. While (i) has been almost entirely done in previous work, (ii) is the goal of the second part of the project. The investigator and collaborators will adopt the approach of quantitative implicit function theorems and quantitative approximation of nonlinear elasticity solutions by the ones in the linear setting.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.