Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin-Murdoch model of material surfaces

The objective of this work is to present an approach allowing for inclusion of the complete Gurtin-Murdoch material surface equations in methods leading to closed-form formulas defining effective properties of particle-reinforced nanocomposites. Considering that all previous developments of the closed-form formulas for effective properties employ only some parts of the Gurtin-Murdoch model, its complete inclusion constitutes the main focus of this work. To this end, the recently introduced new notion of the energy-equivalent inhomogeneity is generalized to precisely include all terms of the model. The crucial aspect of that generalization is the identification of the energy associated with the last term of the Gurtin-Murdoch equation, i.e., with the surface gradient of displacements. With the help of that definition, the real nanoparticle and its surface possessing its own distinct elastic properties and residual stresses are replaced by an energy-equivalent inhomogeneity with properties incorporating all surface effects. Such equivalent inhomogeneity can then be used in combination with any existing homogenization method. In this work, the method of conditional moments is used to analyze composites with randomly dispersed spherical nanoparticles. Closed-form expressions for effective moduli are derived for both bulk and shear moduli. As numerical examples, nanoporous aluminum is investigated. The normalized bulk and shear moduli of nanoporous aluminum as a function of residual stresses are analyzed and evaluated in the context of other theoretical predictions.