A Trefftz collocation method for multiple interacting spherical nano-inclusions considering the interface stress effect

In this study, a Trefftz collocation method (TCM) is proposed for modeling multiple interacting nano-scale spherical inhomogeneities considering the interface stress effect. The Papkovich–Neuber (P–N) general solutions are used as Trefftz trial functions, which are expressed in terms of spherical harmonics. Non-singular harmonic functions, and singular harmonics from multiple source points are included, facilitating the study of multiple inclusions. Characteristic lengths are used to scale the Trefftz trial functions, to avoid ill-conditioning of the derived system of linear equations. The collocation method is used to enforce boundary conditions. The displacement continuity and the stress jump across the matrix/inclusion interface, which is described by the generalized Young–Laplace equation for solids, are also enforced by the collocation method. Numerical results by the proposed Trefftz method agree well with the available analytical solutions in the literature. The stress distributions of solids containing nano-inhomogeneities show significant size-dependency, in contrast to those for composites without considering the interface stress effect. Interactions of multiple nano-inclusions are also studied, which can be used as benchmark solutions in future studies.