Formulation of Eshelby’s inclusion problem by BIEM (boundary integral equation method) and PDD (parametric dislocation dynamics)

• Eshelby’s inclusion problem is implemented by surface and line discretization forms.
• Equilibrium displacement at the inclusion boundary is computed by Green’s function.
• Eigen strain for PDD is assumed by Voltera and Somigliana dislocation loops.
• Strain energy of inclusion by BIE is derived from traction and eigen displacement.
• Strain energy of inclusion by PDD is computed from double line integral formula.

Eshelby’s inclusion problem is implemented by boundary integral equation method (BIEM) and parametric dislocation dynamics (PDD), which are surface- and line-discretization approaches based on Green’s function method. In BIEM calculations, unknown constraint displacements at the inclusion boundary are deduced from surface integration associated with eigenstrain, and then unknown tractions are solved by a linear system of equations. Field quantities and strain energy are readily calculated by using the boundary tractions and displacements. In the meantime, inclusion stress by PDD is directly calculated by line integration over the dislocation elements, where the burgers vector satisfies geometrical argument of eigenstrain of inclusion. In particular, double line integral formula associated with dislocation interaction is utilized for elastic energy calculation. Computation efficiency of BIEM and PDD is mutually compared as a function of the number of Gauss integration points. Accuracy of field quantities and strain-energy of inclusion are verified by comparison of numerical results to analytical solutions.