Nonlinear dynamics analysis of cracked rectangular viscoelastic plates

Abstract

The effects of cracks and materials on nonlinear dynamics behaviors of cracked viscoelastic plates were investigated. Based on the nonlinear plate theory and the isotropic viscoelastic constitutive relation, the nonlinear dynamic equations of the viscoelastic thin plates with an all-over part-through surface crack were derived, and the corresponding boundary conditions and the crack continuity conditions were also introduced. In order to satisfy the boundary conditions and the crack continuity conditions, the suitable expressions of stress functions and deflection shape functions were put forward. In the example calculations, the materials of the plate were assumed to be standard linear solid, and the movable simple supports on four edges of the plates were adopted as the boundary conditions, moreover, in the crack continuity conditions, the Rice crack model was applied. Under the action of transversely distributed simple harmonic loads, and with the use of the Galerkin procedure, the numerical results of the bifurcations and chaos of the cracked rectangular viscoelastic plates were obtained and then expressed by the Poincare maps. According to the numerical results and the Poincare maps, the effects of the crack-depth and the crack-location, as well as the viscoelastic parameters, on the bifurcations and chaos of the plates were discussed. And some significant conclusions were obtained: 1) when the crack-depth increases or when the crack-location approaches the center of the plates, the motions of the plates are changed from single period to periodic bifurcations and then to chaos; 2) when the viscoelastic material parameter increases, the motions of the plates are changed contrarily from chaos to periodic bifurcations and then to single period states.