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High order curvilinear finite elements for elastic-plastic Lagrangian dynamics. (English) Zbl 1351.76057
Summary: This paper presents a high-order finite element method for calculating elastic-plastic flow on moving curvilinear meshes and is an extension of our general high-order curvilinear finite element approach for solving the Euler equations of gas dynamics in a Lagrangian frame [1,2]. In order to handle transition to plastic flow, we formulate the stress-strain relation in rate (or incremental) form and augment our semi-discrete equations for Lagrangian hydrodynamics with an additional evolution equation for the deviatoric stress which is valid for arbitrary order spatial discretizations of the kinematic and thermodynamic variables. The semi-discrete equation for the deviatoric stress rate is developed for 2D planar, 2D axisymmetric and full 3D geometries. For each case, the strain rate is approximated via a collocation method at zone quadrature points while the deviatoric stress is approximated using an $$L2$$ projection onto the thermodynamic basis. We apply high order, energy conserving, explicit time stepping methods to the semi-discrete equations to develop the fully discrete method. We conclude with numerical results from an extensive series of verification tests that demonstrate several practical advantages of using high-order finite elements for elastic-plastic flow.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 74S05 Finite element methods applied to problems in solid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 76A10 Viscoelastic fluids
TENSOR; MFEM
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