Rigidity and Buckling of Shells: Toward New Nonlinear Shell Theories

经典的线性壳理论对薄壁结构的变形或临界屈曲载荷的预测往往与此类物理结构的实际性能不符。在可能导致这种失败的几个现象中，有所谓的“对缺陷的敏感性”，也就是能量最小器的高度振荡性质，即使在非常简单的几何形状(如圆柱体)中也会导致不紧凑。缺陷敏感性现象给从事各种薄壳结构工作的工程师带来了持续的挑战，从飞机和汽车部件、建筑物、电杆和其他民用基础设施，到碳纳米管和生物聚合物壳。本项目的目标之一是利用薄壁结构屈曲的一般理论来量化缺陷敏感现象。另一个目标是解决一个长期存在的问题，即确定一个给定的壳的刚性，这被认为是通过壳的主曲率来确定的。虽然确定板的刚度的问题是众所周知的，但确定给定壳体的刚度一直是连续介质力学中的一个长期问题。在壳体薄面满足Robin边界条件的情况下，关于线弹性壳体的最新结果表明，壳体的刚度仅由壳体中表面的主曲率决定。该项目的目标之一是将没有ansatz的下界扩展到非线性几何刚性设置。其策略是利用现有的由检验函数引入人工边界条件的估计，以及一个普遍的Korn插值不等式，从而将线性几何刚性估计简化为Poincar？键入Estimate。一旦做到这一点，一个新的壳理论层级将自然地出现，它将包括由伽玛收敛派生的现有壳理论。项目的第二部分是进一步发展薄壁结构的屈曲理论。该理论有两个组成部分：(I)证明了尖锐的Korn不等式；(Ii)证明了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)是该项目第二部分的目标。研究人员和合作者将采用定量隐函数定理和线性环境下的非线性弹性解的定量近似的方法。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。

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.