Finite strain plasticity, the stress condition and a complete shell model

Abstract

The null stress (s33 = 0) and incompressibility (J = 1) conditions in finite strain elasto-plastic shell analysis are studied in closed-form and implemented with a variant of the combined control by Ritto-Correa and Camotim. Coupling between constitutive laws and shell kinematics results from the satisfaction of either of the conditions; nonlocality results from the coupling. We prove that the conditions are, in general, incompatible. A new thickness-deformable is studied in terms of kinematics and strong-ellipticity. The affected continuum laws are derived and, in the discrete form, it is shown that thickness degrees-of-freedom and enhanced strains are avoided: a mixed displacement-shear strain shell element is used. Both hyperelastic and elasto-plastic constitutive laws are tested. Elasto-plasticity follows Lee’s decomposition and direct smoothing of the complementarity condition. A smooth root finder is employed to solve the resulting algebraic problem. Besides closed-form examples, numerical examples consisting of classical and newly proposed benchmarks are solved.