Dynamic Response of Strain Rate Dependent Glass/Epoxy Composite Beams Using Finite Difference Method

Abstract

This paper deals with a numerical analysis of the transient response of composite beams with strain rate dependent mechanical properties by use of a finite difference method. The equations of motion based on Timoshenko beam theory are derived. The geometric nonlinearity effects are taken into account with von Karman large deflection theory. The finite difference method in conjunction with Newmark average acceleration method is applied to solve the differential equations. A modified progressive damage model which accounts for strain rate effects is developed based on the material property degradation rules and modified Hashin-type failure criteria and added to the finite difference model. The components of the model are implemented into a computer code in Mathematica 6. Glass/epoxy laminated composite beams with constant and strain rate dependent mechanical properties under dynamic load are analyzed. Effects of strain rate on dynamic response of the beam for various stacking sequences, load and boundary conditions are investigated. orthotropic beam which incorporates the effects of transverse shear and normal deformations. Obst and Rakesh (2) studied the nonlinear static and transient response of laminated beams using a one-dimensional finite element formulation based on higher-order displacement model which accounts for geometric nonlinearities and a parabolic shear strain distribution through the thickness. Marur and Kant (3) used higher-order shear-deformable refined theories, based on isoparametric elements, for transient dynamic analysis of symmetric and un-symmetric sandwich and composite beams. Khdeir (4) developed an analytical solution of the classical, first- and third-order laminate beam to study the transient response of antisymmetric cross-ply laminated beams with generalized boundary conditions and for arbitrary loadings. Kant, et al. (5) reported an analytical solution to the natural frequency analysis of composites and sandwich beams based on a higher order refined theory. Gong and Lam (6) investigated the transient response of layered composite beams subjected to underwater shock involving the effects of structural damping and stiffness. None of these literatures considers the effects of strain rate. In this paper, a macro-mechanical approach using finite difference method and progressive damage modeling algorithm which accounts for geometric nonlinearity effects, transverse shear strain effects and the effects of strain rate is presented. Coupled equations of motion for a laminated composite beam based on Timoshenko beam theory are reduced to ordinary differential equations in time domain using finite difference approximations for displacements. Newmark time integration scheme in association with Newton-Raphson iteration method are applied to solve the system of equations. The variations of mechanical properties due to strain rate and failure are taken into account using empirical relations and sudden material property degradation rules, respectively. A computer code in Mathematica 6 is developed to implement the numerical procedures. The effects of strain rate on transient response of composite beams with various stacking sequences, loadings and boundary conditions are presented and discussed.