1. 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

2. Materially uniform body with inhomogeneities

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

4. 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?

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

6. 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.

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

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

9. We choose aluminum

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

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

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

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

14. A special solution with additive separated variables
Is the following

15. A schematic representation of the special solution

16. The rigid body motion is an acceptable solution
with the choice

17. References
Knowles, J.K., On finite anti-plane shear for incompressible elastic materials. Journal of the Australian Mathematical Society 19 (Series B), (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, (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, (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, (1998).
Valanis, K.C., Panoskaltsis, V.P., Material metric, connectivity and dislocations in continua. Acta Mechanica 175, (2005).