Mixed Lagrangian formulation for size‐dependent couple stress elastodynamic and natural frequency analyses

Abstract

SummaryCouple stress formulations have been given much attention lately because of the possibility to explain cases, when the classical theory of elasticity fails to describe adequately the mechanical behavior. Such cases may include size‐dependent stiffness, high stress gradients, and the response of materials with microstructure. Here, a new mixed Lagrangian formulation is developed for elastodynamic response within consistent size‐dependent skew‐symmetric couple stress theory. With a specific choice of mixed variables, the formulation can be written with only C0 continuity requirements, without the need to introduce a Lagrange multiplier or penalty method. Furthermore, this formulation permits, for the first time, the determination of natural frequencies within the consistent couple stress theory. Details for the strong form of the equilibrium equations, constitutive model relations, boundary conditions, and the corresponding weak form are provided. In addition, the discrete forms are also discussed with two approaches for reducing variables. Several simple two‐dimensional computational example problems are then examined, along with a brief investigation of the effect of couple stress on natural frequencies, which exhibit size‐dependence for most, but not all, modes. Copyright © 2016 John Wiley & Sons, Ltd.