Dynamic finite element analysis of a gradient elastic bar with micro-inertia

摘要

We study the dynamic behavior of a first strain gradient elastic bar with micro-inertia by means of the finite element method. The partial differential equation describing the motion of the bar expressed in terms of displacement is of fourth order with respect to the spatial variable, therefore, for the standard Galerkin formulation, Hermite elements are required. Consistent mass matrices are employed. The results are validated by comparison with some special cases where the exact solutions are derivable. Dispersion relations for the longitudinal waves are also derived. The effect of micro-inertia in the dynamic response of the bar is analyzed and comparisons are made with the classical elastic case. It is found that the micro-inertia parameter considerably affects the dynamic response of the bar and the dispersion characteristics of longitudinal waves. Keywords Micro-mechanics - Gradient elasticity - Micro-inertia - Dispersion - Modal analysis - Eigenfrequency - Finite elements