Regularized variational formulation for nonlinear dynamics of viscoplastic plates

Abstract

In this work we present novel approach to Reissner–Mindlin plates, targeting the nonlinear dynamics applications and in particular robust scheme for improved performance in vibration and wave propagation computations, where the standard finite element isoparametric interpolations no longer provide the best approximation property. The first key ingredient of the present development is regularized variational formulation, recast in terms of stress resultants (bending moments and shear forces), which allows any convenient interpolations. The discrete approximation is here constructed by using the lowest order Raviart–Thomas vector space employed to discretize bending moment and shear force fields, providing the continuity of the stress resultants projections across the boundary between adjacent plate elements. The latter is of particular interest for dynamic analysis for it can capture smooth approximation for wave propagation phenomena. This particular space discrete approximation is further combined with the most appropriate time-integration schemes capable of enforcing either energy conservation or decay of high frequency modes, which delivers the schemes with higher stability and robustness in long-term computations. The developments are carried out for viscoplastic constitutive behavior of plate material, where the corresponding evolution equations are obtained by appealing to the penalty-like form of the principle of maximum plastic dissipation. The choice of viscoplasticity (rather than plasticity) is crucial in that allows development of second-order scheme by integrating simultaneously the momentum balance and viscoplastic strains evolution equations. Several numerical simulations are given to illustrate a very satisfying performance of the proposed hybrid-stress discrete approximation for thick plate element and of the energy conserving/decaying schemes.