Continuously Dislocated Elastic Bodies Subjected to Antiplane Shear D. Sfyris*, N. Charalambakis* and V. K. Kalpakides** *Department of Civil Engineering, Aristotle University of Thessaloniki **Department of Materials Science and Engineering, University of Ioannina

Materially uniform body with inhomogeneities

Infinite cylinder with generators in the axis Anti-plane shear of the form the out-of-plane displacement

Incompressible body Unknown functions: and Elastic energy: 1st P-K: , Equilibrium equations: What expressions are admissible for the field of the defects in order the material to admit states of anti-plane shear?

Screw dislocations Continuous distribution of screw dislocations with Burgers vector and dislocation line parallel in the axes

Differential equations of equilibrium: pressure field: remaining differential equation for : The body admits states of anti-plane shear as if the defects were not present.

Edge dislocations (case A) Continuous distribution of edge dislocations with dislocation line in the axis and Burgers vector in the axis

Differential equations of equilibrium: With and the remaining differential equation takes the form

We choose aluminum

Discretization of the cross section of the cylinder with 764 linear finite elements

Three dimensional plot of the out-of-plane displacement field

Edge dislocations (Case B) Continuous distribution of edge dislocations with dislocation line in the axis and Burgers vector in the axis

Differential equations of equilibrium: Sufficient condition for solution The differential equations that should satisfy

A special solution with additive separated variables Is the following

A schematic representation of the special solution

The rigid body motion is an acceptable solution with the choice

References Knowles, J.K., On finite anti-plane shear for incompressible elastic materials. Journal of the Australian Mathematical Society 19 (Series B), 400-415 (1976). Knowles, J.K., The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids. International Journal of Fracture 13, 611-639 (1977). Knowles, J.K., A note on anti-plane shear for compressible materials in finite elastostatics. Journal of the Australian Mathematical Society 19 (Series B), 1-7 (1977). Noll, W., Materially uniform simple bodies with inhomogeneities. Archives for Rational Mechanics and Analysis 27, 1-32 (1967). Sfyris, D, Charalambakis, N, Kalpakides, V.K., Variational arguments and Noether”s theorem on the nonlinear continuum theory of dislocations. International Journal of Engineering Science 44, 501-512 (2006). Sfyris, D, Charalambakis, N, Kalpakides, V.K., Continuously dislocated elastic bodies subjected to anti-plane shear (submitted). Svendsen, B., A thermodynamic formulation of finite deformation elastoplasticity with hardening based on the concept of material isomorphism. International Journal of Plasticity 14, 473-488 (1998). Valanis, K.C., Panoskaltsis, V.P., Material metric, connectivity and dislocations in continua. Acta Mechanica 175, 77-103 (2005).