Theory and analysis of shells undergoing finite elastic-plastic strains and rotations

Summary In this paper theory and analysis of shells undergoing finite elastic and finite plastic strains and rotations are presented. The shell kinematics are based on a relaxed normality hypothesis allowing transverse normal material fibers to be stretched and bended, whereas shear deformations are neglected. Lagrangean logarithmic membrane and logarithmic bending strain measures are introduced, and it is shown that they can be additively decomposed into purely elastic and purely plastic parts for superposed moderately large strains and unrestricted rotations. The logarithmic strain measures are used to formulate thermodynamic-based constitutive equations for isotropic elastic and plastic material behavior with isotropic and kinematic hardening induced by continuous plastic flow. To analyse path-dependent elastic-plastic shell deformations by iterative procedures the application of logarithmic strain measures allows to realize load steps with corresponding moderate strains and unrestricted rotations. The moderate strain restriction for superposed deformations can be assured by an appropriate update procedure. Formulae are given to determine exactly the rotational change of the reference configuration during the update. Finally, the principle of virtual work with corresponding elastic-plastic material tensor is formulated and it is shown that the weak form of the virtual work leads to the Lagrangean equilibrium equations and boundary conditions well-known from the nonlinear theory of elastic shells.

Notation

3-D and 2-D continuum F deformation gradient - U right Cauchy-Green stretch tensor - R rotation tensor - E Green strain tensor - e Almansi strain tensor - H Lagrangean logarithmic strain tensor - Kirchhoff stress tensor in the 3-D continuum - Cauchy stress tensor - () ^e reference to an elastic deformation - () ^p reference to a plastic deformation - (^–), (⁺) reference to a first and a second superposed deformation, respectively - (^*) reference to an alternative decomposition of the superposed deformation - AB composition of the two tensorsA andB - A ^T transposed ofA - A ² square ofA - A ^–1 inverse ofA Shell theory u displacement field of the shell space - v displacement field of the middle surface - h, shell thickness in the initial and the current state - thickness coordinate, - g _i ,g ⁱ base vectors in the undeformed shell space,i{1, 2, 3} - base vectors in the deformed shell space - a ,a surface base vectors on the undeformed middle surface =0, {1, 2} - , surface base vectors on the deformed reference surface =0 - n, unit normal vector on the surface =0 in the initial and the deformed configuration, respectively - b, curvature tensor of the surface =0 in the initial and current state - Green membrane strain tensor - Green bending strain tensor - Green second order strain tensor - g logarithmic membrane strain tensor - k logarithmic bending strain tensor - p logarithmic second order strain tensor - T plane Kirchhoff stress tensor - N stress resultant tensor - stress couple tensor - second order stress resultant tensor