Peridynamics with Strain Gradient Elasticity to Account for Microstructural Size Effect

Abstract

This study proposes the development of a peridynamic (PD) model with strain gradient elasticity (SGE) for size effect on scaling of structural strength. Peridynamic theory introduces damage into the constitutive relations in a natural way. It will enable the investigation of the combined effect of PD and SGE length scale parameters on the stiffness and strength of the material. The primary challenges of general gradient elasticity are the vast number of constitutive parameters. Also, it requires two additional non-classical boundary conditions arising from the presence of fourth order spatial derivatives in the equation of motion. Considering a simplified SGE model with commonly accepted length parameters, the PD form of the equilibrium equations are established for one- and two-dimensional analysis. The PD with SGE (PDSG) equation of motion is without any spatial derivatives and allows for the imposition of displacement constraints and non-zero tractions in the form of a body force density. The PD equations are derived in their bond-based and state-based forms. This derivation presents a novel approach to write the bond-based and ordinary state-based force density vectors for PD and PDSG in terms of the PD functions provided by the Peridynamic differential operator (PDDO). The resulting equations present two length parameters: the horizon of a material point in PD and the characteristic length in SGE theory. The PDSG is first applied to study the deformation response of a single-walled carbon nanotube (SWCNT) subjected to an axial load, and subsequently its longitudinal vibration. To verify the two-dimensional PDSG formulation a thin film is modeled to mimic the one-dimensional SWCNT problem and subsequently compared with the one-dimensional analytical solution. Another benchmark problem of a thin film subjected to tangential displacement is compared to its analytical solution. The dynamic response is compared to a point-wise computational solution with nonclassical boundary conditions. Finally, a plate with and without a crack is modeled to showcase the capability of the PDSG model.