Nonlinear vibration of cracked rectangular Mindlin plate with in-plane preload

Abstract

In this paper, the nonlinear vibration of a Mindlin plate with a side crack is investigated considering an in-plane preload. Special corner functions are incorporated into the admissible functions of displacements to describe the singularity in stress at the crack tip as well the discontinuities in displacements and rotations across the crack. Based on the Mindlin plate theory and von Kármán's nonlinear plate theory, the potential and kinetic energies of the cracked loaded plated are constructed. The Ritz method with the special admissible functions is employed to determine the initial in-plane stress resultants. The natural frequencies and modes of the plate with or without in-plane preload are also determined through the Ritz method. The nonlinear dynamical equations of the plate are derived using the Hamilton's principle, and are discretized into second-order ordinary differential equations through the Galerkin's method with the three lowest order modes obtained by the Ritz method in free vibration. These equations are transformed into of first-order ordinary differential equations and are solved via the Runge-Kutta algorithm. The nonlinear dynamical response of the plate with various parameters of crack and in-plane preload under transverse harmonic excitation is presented through time history of responses, phase portraits and bifurcation diagrams in conjunction with the maximum Lyapunov exponent. It is demonstrated that the nonlinear dynamics of the plate is complicated due to the crack and in-plane compressive preload.