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.

在这项研究中，考虑到界面应力效应，开发了计算晶粒（CGs）用于具有纳米级不均匀性的异质材料的微机械建模。每个二维CG（一个虚拟的或数学上定义的多边形的有限大小的几何域）可以包含一个圆形的弹性纳米夹杂物。在本模型中，沿每个CG的外边界假定了CG间兼容的位移场，而独立的Trefftz试验函数被假定为矩阵内部的位移场和每个CG中的包含物。通过特征长度缩放的复数电势用于导出矩阵以及包含物中的Trefftz试验位移场。广义Young-Laplace方程描述了穿过基质/包含物界面的应力跃迁，这是拉格朗日乘法器在新开发的仅边界类型的多场边界变分原理中所弱实施的。引入并行算法可在对包含大量纳米夹杂物的RVE建模时进一步加速计算。给出了单个，多个和大量纳米级不均匀性问题的数值示例，以证明当前开发的用于纳米力学的CG模型的有效性和强大功能。