A Classical, Nonlinear Thermodynamic Theory of Elastic Shells Based on a Single Constitutive Assumption

Abstract

By integrating along a thickness-like coordinate, exact two-dimensional equations of motion and a Second Law of Thermodynamics (a Clausius-Duhem Inequality) are derived, without approximation, for a shell-like body. The theory derived is called classical because the only stress measures that appear are resultants and couples. Construction of a Virtual Power Identity automatically produces associated extensional and bending strains that are nonlinear in the deformed position \(\bar{\mathbf {y}}\) of a reference surface and a rotation tensor \(\hbox{\mathversion {bold}$\mathsf{Q}$}\). The only approximations come when the Virtual Power Identity is augmented by heating and an internal energy and the resulting expression taken as the First Law of Thermodynamics (Conservation of Energy) for the shell. A Legendre-Fenchel Transformation is introduced to remove possible ill-conditioning when certain approximations are introduced into the strain-energy density.