A systematic approach to derive dynamic equations for homogeneous and functionally graded micropolar plates

Abstract

This work considers a systematic derivation process to obtain hierarchies of dynamical equations for micropolar plates being either homogeneous or with a functionally graded (FG) material variation over the thickness. Based on the three dimensional micropolar continuum theory, a power series expansion technique of the displacement and micro-rotation fields in the thickness coordinate of the plate is adopted. The construction of the sets of plate equations is systematized by the introduction of recursion relations which relates higher order powers of displacement and micro-rotation terms with the lower order terms. This results in variationally consistent partial differential plate equations of motion and pertinent boundary conditions. Such plate equations can be constructed in a systematic fashion to any desired truncation order, where each equation order is hyperbolic and asymptotically correct. The resulting lowest order flexural plate equation is seen to be of a generalized Mindlin type. The numerical results illustrate that the present approach may render accurate solutions of benchmark type for both homogeneous and functionally graded micropolar plates provided higher order truncations are used. Moreover, low order truncations render new sets of plate equations that can act as engineering plate equations, e.g. of a generalized Mindlin type.