Higher-Order Hypoelastic-Plastic Beam Elements in Nonlinear Explicit Dynamics

In this paper, a mass lumped, rate-type C 0 continuous finite element model of arbitrarily higher-order is developed using well-known hypoelastic-plastic type constitutive relations for a shear deformable (Timoshenko) beam. Higher-order elements provide true curvature, improved performance in bending and buckling, and avoid hourglass modes and numerical locking without ad hoc treatments. Historically, however, first-order elements have been predominant in explicit finite element codes because of difficulties achieving robust and reliable mass-lumping. Although known to exhibit some deficiencies in certain regimes, hypoelastic type constitutive relations possess sufficient theoretical rigor and have been successfully and extensively used in production finite element codes for the nonlinear applications of interest herein. First, the governing equations of motion for a Timoshenko beam element capable of finite deformation are developed. We spatially discretize using a standard weak-form Galerkin finite element approximation and optimal C 0 Lagrange shape functions of arbitrarily higher order and integrate throughout time using a central-difference method and a lumped mass matrix. Then, the thermodynamical consistency of the hypoelastic-plastic rate type formulation is proved starting with a Gibbs potential and its viability for the considered elasto-plasticity applications is discussed. We present several numerical examples of nonlinear finite deformation demonstrating the accuracy and utility of the lumped-mass beam elements developed herein and evaluate the performance of increased order compared to standard linear elements.