Topics in the mathematical theory of nonlinear elasticity.

Abstract

My thesis consists of two interrelated parts. The first part lies in the field of the mathematical theory of nonlinear elasticity, and it concerns the rigorous derivation of theories for elastic shells. The second part concerns modeling and analyzing of shells with residual stresses. The approach for both parts is based on the refined methods in Calculus of Variations (notably the so-called $\Gamma$-convergence) and a combination of the arguments in modern Mathematical Analysis and Riemannian Geometry. More precisely, in chapter 2-3, we derive the von K\'arm\'an theory for variable thickness shells and also the von K\'arm\'an theory for incompressible shells with uniform thickness. In chapter 4-5, we first establish the Kirchhoff theory for non-Euclidean shells and its incompressible counterpart. Then, we also derive the incompatible F\"oppl-von K\'arm\'an theory for prestrained shells with variable thickness, calculate the associated Euler-Lagrange equations and found the convergence of equilibria. Finally, the incompatible F\"oppl-von K\'arm\'an theory for incompressible prestrained shells and the associated Euler-Lagrange equations are investigated.