The effective stiffness of a nanoporous rod

Abstract

279 Many nanomaterials have abnormal physical prop� erties, which differ considerably from the properties of bulk materials. One of the explanations for these dif� ferences consists in the presence of surface effects, the role of which can be extremely large for nanodimen� sional structures in comparison with those in classical mechanics [1]. The purpose of this work is analysis of the influence of surface effects on the elastic characteristics of nan� oporous materials. Two models are considered. The first one is based on taking into account the surface stresses [1–4]. The surface stresses τ are the generali� zation of the surface tension known in the theory of capillarity for the case of solids. As is shown in [1, 5], taking into account surface stresses results in increas� ing stiffness of nanoporous materials. This phenome� non is similar to increasing flexural stiffness of nano� plates in comparison with the plates of macroscopic sizes [6, 7]. The second model uses the approach of the theory of composite materials [8–10]. In this approach, the surface effects are taken into account due to the surface layer of finite thickness with elastic moduli differing from those of the basic material (the matrix). Here the increase or decrease in the rod stiff� ness depends on the relation between the elastic mod� uli of the surface layer and the matrix. The effective stiffness can both decrease and increase with decreas� ing pore radius. On the basis of these two approaches, we proposed a complex model combining both the presence of surface stresses and the surface layer with the properties that differ from those of the matrix. PROBLEM FORMULATION We consider the problem on the tension–compres� sion of a linear elastic rectilinear rod. Let the rod have a circular cross section of radius R. We consider that n cylindrical pores with identical radii r (Fig. 1) are located in parallel to the rod axis. We designate the area occupied with pores in the rod cross section as S = πnr 2 . We assume also that the rod cross section is symmetric so that it is not subjected to bending under tension. A regularly distributed load, which is stati� cally equivalent to forces P, acts on the rod end faces. We designate the Young’s modulus of the rod mate� rial as E. For a large number (n � 1) of pores, the rod can be considered as a homogeneous cylinder made of transversally isotropic material. We designate the cor� responding effective longitudinal Young’s modulus