Abstract
We present in this work two finite volume methods for the simulation of unidimensional impact problems, both for bars and plane waves, on elastic–plastic solid media within the small strain framework. First, an extension of Lax–Wendroff to elastic–plastic constitutive models with linear and nonlinear hardenings is presented. Second, a high order TVD method based on flux-difference splitting [1] and Superbee flux limiter [2] is coupled with an approximate elastic–plastic Riemann solver for nonlinear hardenings, and follows that of Fogarty [3] for linear ones. Thermomechanical coupling is accounted for through dissipation heating and thermal softening, and adiabatic conditions are assumed. This paper essentially focuses on one-dimensional problems since analytical solutions exist or can easily be developed. Accordingly, these two numerical methods are compared to analytical solutions and to the explicit finite element method on test cases involving discontinuous and continuous solutions. This allows to study in more details their respective performance during the loading, unloading and reloading stages. Particular emphasis is also paid to the accuracy of the computed plastic strains, some differences being found according to the numerical method used. Lax–Wendoff two-dimensional discretization of a one-dimensional problem is also appended at the end to demonstrate the extensibility of such numerical scheme to multidimensional problems.
Highlights
The numerical simulation of hyperbolic initial boundary value problems including extreme loading conditions such as impacts requires the ability to accurately capture and track the wave front of shock waves induced in the medium
The numerical simulation of impacts on dissipative solids has been and is again mainly performed with the classical finite element method coupled with centered differences or Newmark finite difference schemes in time [7, 8], which is implemented in many industrial codes
The finite element method is still popular in the solid mechanics community for, among others, its easy implementation of nonlinearities of partial differential equations, that is for solid-type media it enables to account for history-dependent constitutive equations through appropriate integration algorithms [9] and storage of internal variables at integration points in each element
Summary
The numerical simulation of hyperbolic initial boundary value problems including extreme loading conditions such as impacts requires the ability to accurately capture and track the wave front of shock waves induced in the medium. Since the early work of Wilkins [12] and Trangenstein et al [13], several authors have proposed many ways to simulate impacts on dissipative solid media such as elastic-plastic solids, these can be merely classified into eulerian approaches, generally based on a fractional-step method to treat the plasticity [14, 15, 16, 17, 18] and used for extremely high strain and strain rate problems, and lagrangian approaches [19, 20] that allow to follow the path of material particles and account for refined history-dependent constitutive equations though limited by mesh entanglement, both being coupled with an approximate Riemann or WENO solver. For a plane wave with isotropic and kinematic linear hardenings
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