Nonlinear size-dependent finite element analysis of functionally graded elastic tiny-bodies

• Nonlinear size-dependent FE analysis of FG tiny-bodies considering surface energy effects.
• Study effects of surface energy and the residual stress in the bulk induced by the pre-surface tension.
• A new finite element is developed to allow insertion of surface energy to the total surface energy.
• Study effects of surface energy and the 2nd order displacement gradient on the behavior of FG bodies.
• An algorithm is provided for computing the equivalent Young’s modulus for FG triangular elements.

In this paper, a nonlinear size-dependent finite element model incorporating surface energy effects is developed to study the mechanical behavior of tiny elastic functionally graded (FG) bodies. Here the classical elasticity theory is modified to incorporate the surface energy effects. Most of previous studies assumed that the surface energy depends only on the 2D symmetric infinitesimal surface strains and neglects the second-order products of surface strains/displacement gradients. These descriptions assume a small strain deformation of the surface and neglect the pre-strain that is, already, developed on the surface – before loading – due to the pre-tension stress σ0. Here in this paper, the pre-strain is considered which forces the surface to a state of large strain after loading instead of small strain. In this sense, in the presence of initial surface tension, the theory of surface elasticity is a hybrid formulation characterized by linearized bulk elastic material and second order finite deformation of the surface. In the proposed finite element model, the surface energy effect is taken into account in the derivation of the element stiffness matrix for the material elements located very close to the boundary surface. The proposed model is then used to study the effects of surface energy, including the 2nd order displacement gradient, on the mechanical behavior of plane-strain functionally graded elastic body.