Linear Shell Theory: Equilibrium and Energy

This lecture covers the expression of the Kirchhoff-Saint Venant energy in a covariant setting, dimensional reduction for shells, equilibrium equations for spherical shells, linear shell theory, strain energy density, and the relation between stress and strain. The instructor explains the concepts of covariant and contravariant components, the trace operation, and the covariant derivative. The lecture also delves into the energy of the Kirchhoff-Saint Venant model, the Second Piola-Kirchhoff stress tensor, and the strain energy density formula. Furthermore, it explores the invariance of physical observables, the gradient of a scalar, and the components of covectors. The presentation concludes with a discussion on the divergence theorem and the application of the divergence theorem in various dimensions.