Effects of nano-voids and nano-cracks on the elastic properties of a host medium: XFEM modeling with the level-set function and free surface energy

This work deals with the influences of nano-heterogeneities in the form of voids/cavities or cracks on the elastic properties of a host medium. With a relatively large ratio of apparent surface to volume and particularly strong physical interactions with the surrounding medium at nano-scale, nano-heterogeneities can potentially affect the elastic properties of the parent medium (matrix) containing them in a significant manner. This has been reported by various theoretical and experimental studies, some of which are discussed in the present paper. To describe the positive (reinforcement) or negative (degradation) effect of the nano-heterogeneities from the modeling perspective, it is necessary to take into account the energy of interfaces/surfaces between nano-heterogeneities and the matrix, which, because of the relatively large extent of their apparent surface and their strong physical interaction with their neighborhood, can no longer be neglected compared to those of the volume energy. Thus, to account for the effects of interfaces/surfaces in a nanostructured heterogeneous medium, the coherent interface model is considered in the present investigation within a periodic homogenization procedure. In this interface/surface model, the displacement vector is assumed to be continuous across the interface while the stress vector is considered to be discontinuous and satisfying the Laplace–Young equations. To solve these equations coupled to the classical mechanical equilibrium problem, a numerical simulation tool is developed in a two-dimensional (2D) context using the extended finite element method and the level-set functions. The developed numerical tool is then used to carry out a detailed analysis about the effect of nano-heterogeneities on the overall mechanical properties of a medium. The nano-heterogeneities are present in the medium initially as cylindrical cavities (circular in 2D) before being reduced to plane cracks (line in 2D) by successive flattenings.