Large Deformations of Inelastic Shells Holm ALTENBACH 1,a and Victor A. EREMEYEV 1,2,b 1 Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, D-39106 Magdeburg, Germany 2 South Scientific Center of RASci & South Federal University, Rostov on Don, Russia a holm.altenbach@iw.uni-halle.de, b eremeyev.victor@gmail.com Keywords: Dislocations; Plasticity; Nonlinear shells. Abstract. In this note we discuss the derivation of two-dimensional governing equations of a non- linear shell made of material containing continuously distributed dislocations. We apply the trough- the-thickness integration procedure to a nonlinear shell-like three-dimensional body with dislocations. This procedure gives the exact equilibrium equations with the stress resultant and couple stress ten- sors. The dual to the surface stress tensors are the surface strain measures which are represented by two surface fields. The first one is the translation vector of the base surface of the shell while the second field is the proper orthogonal tensor expressing rotation of the shell cross-section. Since in the case of solids with dislocations there are no displacement vector field. These fields are interpreted as quantities defined on the nonmaterial base surface of the shell. Introduction Dislocations play an important role in the mechanics of materials because dislocations determine mostly the mechanical properties of solids, see [1, 2, 3]. In particular, creation, annihilation and mo- tion of dislocations are responsible for the plastic behavior of materials. The most achievement in mechanics of dislocated solids are related to three-dimensional (3D) modeling. Only in several pa- pers the problem is considered for two-dimensional (2D) structures, see [4, 5, 6]. The interest to 2D is growing with relation to some applications in mechanics of nano- and microfilms, see for exam- ple [7, 8, 9, 10]. For such structures the motion of dislocations relates with plastic-type behavior of nanotubes, nanofilms, etc. In the case of thin structures the dislocations distribution has specific pecu- liarities due to geometric restrictions which are different from the three-dimensional case, in general. Unlike to [4, 5, 6], where the dislocations are considered on 2D level, that is for 2D equations of plates and shells, here we consider the reduction procedure from 3D equations of solids with disloca- tions to 2D equations of the nonlinear shell theory. For this purposes we use the through-the-thickness integration of the equilibrium equations of the nonlinear elasticity described in [11]. Basic Equations of the Nonlinear Elastic Shell Theory Following [11, 12] we recall the basic equations of the general nonlinear shell theory. We consider a shell as a deformable surface, each point of which has six degrees of freedom, that is three translational degrees and three rotational ones. The interaction between different parts of the shell is described by forces and moments only. In other words, the considered shell model is kinematically equivalent to a two-dimensional Cosserat or micropolar continuum, which can be introduced on the base of the 3D theory presented in [13]. So, this variant of the elastic shell theory is also named micropolar shell theory. The kinematical model of the shell is based on the introduction of a directed material surface ω, which is determined in the actual configuration by r(q 1 ,q 2 ), d k (q 1 ,q 2 ); d k · d m = δ km , where r(q 1 ,q 2 ,t) is the position vector of ω, q 1 ,q 2 ∈ ω are coordinates on ω, d k (q 1 ,q 2 ) are orthonormal vectors called directors, k =1, 2, 3, and δ km is the Kronecker symbol. Key Engineering Materials Vols. 535-536 (2013) pp 76-79 © (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/KEM.535-536.76 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 87.246.223.102-23/10/12,11:51:17)