On the theoretical foundations of thin solid and liquid shells

This paper gives a concise summary of the general theoretical framework suitable to describe shells with solid-like and liquid-like behavior. Thin-shell kinematics are considered and used to derive the equilibrium equations from linear- and angular-momentum balance. Based on the mechanical power balance and the mechanical dissipation inequality, the constitutive equations for the hyperelastic material behavior of constrained shells are derived and their material stability is examined. Various constitutive examples are considered and assessed for their stability. The governing weak form of the formulation is derived and decomposed into in-plane and out-of-plane components. The presented work provides a very general framework for a unified description of solid and liquid shells and illustrates what leads to their loss of material stability. This framework serves as a basis for developing computational shell formulations based on rotation-free shell discretizations. Therefore the full linearization of the formulation is also presented here.