Finite Element Modeling and Computer Design of Anisotropic Elastic Porous Composites with Surface Stresses

Abstract

The chapter presents mathematical modelling and computer design of effective properties of anisotropic porous elastic materials with a nanoscale random structure of porosity. This integrated approach includes the effective moduli method of composite mechanics, the simulation of representative volumes with stochastic porosity and the finite element method. In order to take into account nanoscale sizes of pores, the Gurtin-Murdoch model of surface stresses is used at the borders between material and pores. The general methodology for determination of effective mechanical properties of porous composites is produced for a two-phase bulk (mixture) composite with special conditions for stresses discontinuities at the phase interfaces. The mathematical statements of boundary value problems and the resulting formulas to determine the complete set of effective stiffness moduli of the two-phase composites with arbitrary anisotropy and with surface stresses are described; the generalized problem definitions are formulated and the finite element approximations are given. It is used, that the homogenization procedures for porous composites with surface effects can be considered as special cases of the corresponding procedures for the two-phase composites with interphase stresses if the moduli material of the second phase (nanoinclusions) are negligibly small. These approaches have been implemented in the finite element package ANSYS for a model of nanoporous silicon with cubic crystal system for various values of surface moduli, porosity and number of pores. Model of representative volume was built in the form of a cube, evenly divided into cubic solid finite elements, some of which had been declared as pores. Surface stresses on the boundaries between material and pores were modeled by shell finite elements with the options of membrane stresses. It has been noted that the magnitude of the area of the interphase boundaries has influence on the effective moduli of the porous materials with nanosized stochastic structure.