×

zbMATH — the first resource for mathematics

An Eulerian finite volume scheme for large elastoplastic deformations in solids. (English) Zbl 1183.74331
Summary: Conservative formulations of the governing laws of elastoplastic solid media have distinct advantages when solved using high-order shock capturing methods for simulating processes involving large deformations and shock waves. In this paper one such model is considered where inelastic deformations are accounted for via conservation laws for elastic strain with relaxation source terms. Plastic deformations are governed by the relaxation time of tangential stresses. Compared with alternative Eulerian conservative models, the governing system consists of fewer equations overall. A numerical scheme for the inhomogeneous system is proposed based upon the temporal splitting. In this way the reduced system of non-linear elasticity is solved explicitly, with convective fluxes evaluated using high-order approximations of Riemann problems locally throughout the computational mesh. Numerical stiffness of the relaxation terms at high strain rates is avoided by utilizing certain properties of the governing model and performing an implicit update. The methods are demonstrated using test cases involving large deformations and high strain rates in one-, two-, and three-dimensions.

MSC:
74S10 Finite volume methods applied to problems in solid mechanics
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Benson, Computational methods in Lagrangian and Eulerian hydrocodes, Computer Methods in Applied Mechanics and Engineering 99 pp 235– (1992) · Zbl 0763.73052
[2] Godunov, Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates, Journal of Applied Mechanics and Technical Physics 13 pp 868– (1972)
[3] Kondaurov, Equations of elastoviscoplastic medium with finite deformations, Journal of Applied Mechanics and Technical Physics 23 pp 584– (1982)
[4] Plohr, A conservative formulation for plasticity, Advances in Applied Mathematics 13 pp 462– (1992) · Zbl 0771.73022
[5] Godunov, Elements of Continuum Mechanics and Conservation Laws (2003) · Zbl 1031.74004
[6] LeFloch, A second-order Godunov method for the conservation laws of nonlinear elastodynamics, Impact Computer Science and Engineering 2 pp 318– (1990)
[7] Titarev, MUSTA-type upwind fluxes for non-linear elasticity, International Journal for Numerical Methods in Engineering 73 pp 897– (2008) · Zbl 1159.74046
[8] Vorobiev, Application of schemes on moving grids for numerical simulation of hypervelocity impact problems, International Journal of Impact Engineering 17 pp 891– (1995)
[9] Wang, A conservative Eulerian numerical scheme for elastoplasticity and application to plate impact problems, Impact of Computing in Science and Engineering 5 pp 285– (1993) · Zbl 0787.73076
[10] Walter J, Yu D, Plohr BJ, Grove J, Glimm J. An algorithm for Eulerian front tracking for solid deformation. Stony Brook AMS Preprint 2000; SUNYSB-AMS-00-24.
[11] Miller, A high-order Eulerian Godunov method for elastic-plastic flow in solids, Journal of Computational Physics 167 pp 131– (2001) · Zbl 0997.74078
[12] Kluth, Perfect plasticity and hyperelastic models for isotropic materials, Continuum Mechanics and Thermodynamics 20 pp 173– (2008) · Zbl 1172.74008
[13] Barton, Exact and approximate solutions of Riemann problems in non-linear elasticity, Journal of Computational Physics (2009) · Zbl 1172.74032
[14] Romenski, Godunov Methods: Theory and Applications (2001)
[15] Wilkins ML. Calculation of elastic-plastic flow. UCRL Technical Report, UCRL-7322, 1963.
[16] Merzhievsky, Relation of dislocation kinetics with dynamic characteristics in modelling mechanical behaviour of materials, Journal de Physique C3 49 pp 457– (1988)
[17] Merzhievskii, Construction of the time dependence of the relaxation of tangential stresses on the state parameters of a medium, Journal of Applied Mechanics and Technical Physics 21 pp 716– (1980)
[18] Resnyanky, DYNA-modelling of the high-velocity impact problems with a split-element algorithm, International Journal of Impact Engineering 27 pp 709– (2002)
[19] Godunov, Interpolation formulas for maxwell viscosity of certain metals as a function of shear-strain intensity and temperature, Journal of Applied Mechanics and Technical Physics 15 pp 526– (1974)
[20] Eberle, Computational Fluid Dynamics: VKI Lecture Series (1987)
[21] Jiang, Efficient implementation of weighted ENO schemes, Journal of Computational Physics 126 pp 202– (1996) · Zbl 0877.65065
[22] Balsara, Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order accuracy, Journal of Computational Physics 160 pp 405– (2000) · Zbl 0961.65078
[23] Jin, Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation, Journal of Computational Physics 122 pp 51– (1995) · Zbl 0840.65098
[24] Chen, Hyperbolic conservation laws with stiff relaxation terms and entropy, Communications on Pure and Applied Mathematics 47 pp 787– (1994) · Zbl 0806.35112
[25] Pareschi, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, Journal of Scientific Computing 25 pp 129– (2005) · Zbl 1203.65111
[26] Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics 77 pp 439– (1988) · Zbl 0653.65072
[27] Liu, Ghost fluid method for strong shock impacting on material interface, Journal of Computational Physics 190 pp 651– (2003) · Zbl 1076.76592
[28] Miller, A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing, Journal of Computational Physics 183 pp 26– (2002) · Zbl 1057.76558
[29] Nourgaliev, High-fidelity interface tracking in compressible flows: unlimited anchored adaptive level set, Journal of Computational Physics 224 pp 836– (2007) · Zbl 1124.76043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.