Viscoplastic bifurcation buckling of plates

Abstract

A method is proposed according to which the bifurcation buckling load of inelastic rate-sensitive plates subjected to specified boundary conditions can be deduced from the solution of the corresponding perfectly elastic problem under the same boundary conditions. Applications of the method are given for the determination of the buckling stresses of simply supported viscoplastic plates for various values of loading rates and thermal conditions. Both classical and higher-order theories of plates are used in this investigation, and the effect of shear deformation is studied. The numerical results are given for a viscoplastic material that is modeled by the Bodner-Partom unified theory. HE determination of the bucking load of perfectly (i.e., without inelastic effects) elastic structures has received considerable attention (see, for example, Ref. 1 for a recent review). The stability analysis of inelastic plates is much more complicated due to the inherent nonlinearity of the material constitutive law. For total deformation plasticity theory, nu- merous investigations on buckling analysis can be found in the literature (see, for example, Ref. 2). It is well known, however, that incremental plasticity theo- ries are more realistic in the modeling of the behavior of inelastic materials. In the framework of the classical theory of incremental plasticity, Hendelman and Prager3 determined the critical stress of buckling of plates. This work was discussed by Pearson4 who made some comparisons with experimental results. Hutchinson5 studied the effect of initial small imper- fections on the inelastic buckling of plates and shells, while shear deformation effects were investigated by Shrivastava. 6 An extensive survey and bibliography on time-independ ent plastic buckling was given by Bushnell.7 Recently, the influence of material rate sensitivity on the elastic buckling of an eccentrically stiffened panel was investi- gated by perturbation and numerical analysis by Tvergaard.8 For shells of revolution, this effect was investigated by Bodner and Naveh9 using the finite element method, and they con- cluded that viscoplastic material behavior could have an im- portant influence on the buckling of structures in the inelastic range. In the present paper, a method is proposed for the stability analysis of viscoplastic plates. The method is based on the solution of the corresponding perfectly elastic problem that is used in deducing the buckling state. The inelastic plate is initially isotropic, but at later stages of loading plastic defor- mation develops that leads to the appearance of instantaneous anisotropic properties. It is shown that if the buckling solution of the corresponding perfectly elastic anisotropic plate is known, one can readily determine whether or not inelastic buckling occurs at the current instant. The present investigation is concerned with the determina- tion of the critical state that leads to bifurcation of the equi- librium path of a given inelastic plate. An instability criterion is established for the prediction of this bifurcation point. The condition for the occurrence of the bifurcation is the loss of out-of-plane stiffness caused by in-plane loading. This loss of stiffness can be expressed by the fact that the stiffness matrix of structure becomes singluar. In the present paper, the rate-sensitive inelastic behavior of the material is modeled by the unified elastic-viscoplastic the- ory of Bodner and Partom.10 This theory does not require a yield criterion or loading or unloading conditions. In all stages of loading history, the material constitutive law involves both elastic and plastic components, but the latter is negligibly small when the material behavior should be essentially elastic. The proposed method of prediction of the bifurcation buck- ling of viscoplastic structures is illustrated in the cases of plates made of rate-sensitive as well as rate-independent mate- rials. To this end, two types of materials were chosen: com- mercially pure titanium and aluminum alloy. The first one is highly rate sensitive whereas the latter is weakly rate sensitive (at room temperature). Both classical plate theory (CPT) and higher-order shear deformation theory (HSDT) are used in the buckling analysis of the inelastic plates. In the latter type of plate theory, the shear deformation effects are parabolically distributed across the thickness of the plate. Illustrations are given that exhibit the effect of applied loading rate, material rate sensitivity, elevated temperature, and relative thickness on the bifurcation buckling of the in- elastic plates.