A Modern Approach to the Complicated Line Integrals in Linear Elastic Dislocation Theory

Abstract

The standard textbook analysis of dislocations is generally limited to the case of infinitely straight screw or edge dislocations, which do not exist. This is due to the complexity of the formulas for arbitrary dislocation loops, i.e., Burger’s equation for the displacement field, the Peach-Köhler equation for the stress field and Blin’s equation for the interaction energy, which involve line integrals along the dislocation loop. The integrands are complex, and integration often involves non-elementary functions. Elaboration of the integrands with symbolic mathematical software produces tensor formulas which can be reused at will. By formulating convenient parametric expressions for the configuration studied and using superposition, mathematical software can be used to perform the integrations for arbitrary Burgers vectors. Often, the resulting expressions for the tensorial fields are very long, but they can be easily incorporated as user-defined formulas for plotting, parametric analysis, and incorporation into routines for energy minimisation or the non-linear equations for force equilibrium. The effectiveness of this approach will be illustrated by the example of short straight dislocations, circular dislocations, the interaction between a pileup and dissociated dislocations in the grain boundary, and the nucleation of dislocations at grain boundaries.