Rigidity and Buckling of Shells: Toward New Nonlinear Shell Theories

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

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.