Modelling the viscoelastic-plastic deformation of flexible reinforced plates with account of weak resistance to transverse shear

Abstract

A mathematical model of viscoelastic-plastic deformation of flexible plates cross-reinforced in planes parallel to the middle plane is developed on the basis of the algorithm of time steps. The deformations of the components of the composition of the plates are assumed to be small and they are decomposed into elastic and plastic components. The viscoelastic behavior of the composition materials is subject to the relations of the Maxwell–Boltzmann body. Inelastic deformation is defined by the equations of the theory of plastic flow with isotropic hardening. The normal stresses in the transverse direction are linearly approximated over the thickness of the plates. Therefore, linear deformations in the transverse direction and their velocities are excluded from the governing equations for the composition components. The weakened resistance of fibrous plates to transverse shear is taken into account in the framework of the non-classical Reddy bending theory. The geometric nonlinearity of the problem is considered in the Karman approximation. The formulated initial-boundary value problems are solved numerically using an explicit scheme of the “cross” type. To obtain a stable numerical scheme, an artificial technique is used: the stresses in the Maxwell – Boltzmann viscoelastic relations at the current discrete time are expressed in terms of the stress velocity by the trapezoid formula with a step back. The elastic-plastic and viscoelastic-plastic bending dynamic behavior is investigated for the relatively thick orthogonally reinforced fiberglass rectangular plates under the action of explosive loads. It is shown that a change in the structure of reinforcement leads to a change in the value of the residual deflection of the structure. It was found that the amplitude of oscillations of the reinforced plate in the vicinity of the initial time instant significantly exceeds the value of the residual deflection.