Mathematical computational grains for direct numerical simulations of nanocomposites with a large number of nano-inclusions, using parallel computations

In this study, computational grains (CGs) are developed for micromechanical modelling of heterogeneous materials with nanoscale inhomogeneities, considering the interface stress effect. Each two-dimensional CG, which is a virtual or mathematically defined finite-sized geometrical domain of a polygonal shape, can include a circular elastic nano inclusion. In the present model, along the outer-boundary of each CG an inter-CG compatible displacement field is assumed, while independent Trefftz trial functions are assumed as displacement fields inside the matrix and the inclusion within each CG. Complex potentials scaled by characteristic lengths are used to derive the Trefftz trial displacement fields in the matrix as well as the inclusion. The stress jump across the matrix/inclusion interface is described by the generalized Young–Laplace equation, which is enforced in a weak sense by Lagrange multipliers in a newly-developed boundary-only-type multi-field boundary variational principle. A parallel algorithm is introduced to further accelerate the computation when modelling an RVE containing a large number of nano-inclusions. Numerical examples for problems of a single, multiple, and a large number of nanoscale inhomogeneities are given to demonstrate the validity and the power of the currently developed CG model for nanomechanics.