A Periodic, Non-Singular Dislocation Energy Kernel Based on Gradient Elasticity and Ewald Summation

Plastic deformation of crystalline solids is primarily caused by the movement of line-like defects, the dislocations. In order to develop a coarse grained continuum description of dislocations (continuum dislocation dynamics, CDD) [1] we seek to derive a local density approximation as suggested in [2] directly from discrete dislocation data on periodic domains. This comprises obtaining the (average) energy from discrete dislocation configurations on periodic domains, while simultaneously avoiding the singularity at the dislocation core, as it results from classical linear elasticity. With regard to the latter, we employ a Helmholtz-type gradient elasticity theory which has been worked out for infinite domains in [3]. In the current work, we present the incorporation of periodicity into isotropic gradient elasticity of Helmholtz type to compute dislocation interaction energies and self-energies on periodic domains. Periodicity is introduced using the Ewald summation (or Ewald splitting) method, which is well known in numerical simulations of molecular dynamics and electrostatics. This approach combines the numerical advantages of periodic formulations with the physical consistency of non-singular elasticity, making it ideal for large-scale simulations of dislocation networks and other crystalline defects. We discuss the choice of the so-called splitting parameter in dependence of the domain discretization. Furthermore, we provide comparisons of energies obtained directly from DDD data with those obtained based on CDD density fields derived from the former.

References

1. T. Hochrainer, S. Sandfeld, M. Zaiser, and P. Gumbsch, Continuum dislocation dynamics: towards a physical theory of crystal plasticity, Journal of the Mechanics and Physics of Solids, 63:167–178, 2014.
2. M. Zaiser, Local density approximation for the energy functional of three-dimensional dislocation systems, Physical Review B, 92(17):174120, 2015.
3. M. Lazar, The fundamentals of non-singular dislocations in the theory of gradient elasticity: dislocation loops and straight dislocations, International Journal of Solids and Structures, 50(2):352–362, 2013.