International Journal of Non-Linear Mechanics 38 (2003) 659–662 A variational procedure utilizing the assumption of maximum dissipation rate for gradient-dependent elastic–plastic materials S. Baek, A.R. Srinivasa * Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843 3123, USA Received 24 February 2001; received in revised form 20 September 2001 Abstract In this brief note we consider an elastic–plastic material whose strain energy depends not only on the plastic strain but also on spatial gradient. The governing equations of motion are obtained by maximizing the rate of global energy dissipation. The resulting dierential equations are specialized for the case where the strain energy depends only upon the density of geometrically necessary dislocations. ? 2002 Published by Elsevier Science Ltd. 1. Introduction Consider a body B, which at time t occupies a con- guration t (B). The motion of the body, measured from some xed conguration R , and the deforma- tion gradient are given, respectively, by x = X R (X;t ); F:= @X R @X ; (1) where X and x are the positions of a particle x in R and t , respectively. We consider here an elastic–plastic material such as a metal crystal whose internal microstructure (e.g. lattice structure, dislocation networks etc.) changes with deformation. Following Eckart [1] and Rajagopal and Srinivasa [2] we model the internal structure of the material with an evolving natural conguration p(t ) that reects the evolving microstructure. The gradient of the mapping from R to p(t ) is given by G. * Corresponding author. Tel.: +1-979-862-3999; fax: +1-979- 845-3081. E-mail address: asrinivasa@mengr.tamu.edu (A.R. Srinivasa). Our task is to nd the governing equations for the evolution of t and p(t ) or equivalently X R and G as functions of X and t . The basis of our approach here is the fact that plastic ow in crystalline materials is a dissipative process so that, we shall prescribe forms for both the stored energy and the rate of dissipation. 2. The stored energy and the rate of dissipation Let us assume that the material is characterized by a specic stored energy function of the form = ˆ (F; G; ∇G); (2) where ∇≡ @=@X, and that the total specic energy (per unit reference volume) E, i.e. the kinetic energy plus potential energy, is E = 1 2 0 v · v + : (3) As the material deforms, work is supplied or with- drawn from it through the boundary so that at every material point, the specic total energy changes due 0020-7462/03/$ - see front matter ? 2002 Published by Elsevier Science Ltd. PII:S0020-7462(01)00123-8