Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T07:49:48.920Z Has data issue: false hasContentIssue false
  • English
  • Français

1D Exact Elastic-Perfectly Plastic Solid Riemann Solver and Its Multi-Material Application

Part of: Plastic materials, materials of stress-rate and internal-variable type Special kinds of problems Equations of mathematical physics and other areas of application Coupling of solid mechanics with other effects Hyperbolic equations and systems

Published online by Cambridge University Press:  17 January 2017

Si Gao  and
Tiegang Liu
Si Gao*
Affiliation:
LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
Tiegang Liu*
Affiliation:
LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
*
*Corresponding author. Email:[email protected] (S. Gao), [email protected] (T. G. Liu)
*Corresponding author. Email:[email protected] (S. Gao), [email protected] (T. G. Liu)
Get access

Abstract

The equation of state (EOS) plays a crucial role in hyperbolic conservation laws for the compressible fluid. Whereas, the solid constitutive model with elastic-plastic phase transition makes the analysis of the solid Riemann problem more difficult. In this paper, one-dimensional elastic-perfectly plastic solid Riemann problem is investigated and its exact Riemann solver is proposed. Different from previous works treating the elastic and plastic phases integrally, we resolve the elastic wave and plastic wave separately to understand the complicate nonlinear waves within the solid and then assemble them together to construct the exact Riemann solver for the elastic-perfectly plastic solid. After that, the exact solid Riemann solver is associated with the fluid Riemann solver to decouple the fluid-solid multi-material interaction. Numerical tests, including gas-solid, water-solid high-speed impact problems are simulated by utilizing the modified ghost fluid method (MGFM).

Keywords

Riemann problemelastic-plastic solidghost fluid methodmulti-material flows

MSC classification

Secondary: 35L50: Initial-boundary value problems for first-order hyperbolic systems 35Q74: PDEs in connection with mechanics of deformable solids 74C05: Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials) 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74M20: Impact
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 3 , June 2017 , pp. 621 - 650
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Godunov, S. K., A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), pp. 357–393.Google Scholar
[2] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3th ed., Springer-Verlag, 2009.CrossRefGoogle Scholar
[3] Einfeldt, B., On Godunov-Type methods for gas dynamics, SIAM J. Numer. Anal., 25 (1988), pp. 294–318.CrossRefGoogle Scholar
[4] Einfeldt, B., Munz, C. D., Roe, P. L. and Sjögreen, B., On Godunov-Type methods near low densities, J. Comput. Phys., 92 (1991), pp. 273–295.CrossRefGoogle Scholar
[5] Haten, A., Lax, P. D. and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (2002), pp. 35–61.CrossRefGoogle Scholar
[6] Toro, E. F., Spurse, M. and Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock Waves, 4 (1994), pp. 25–34.Google Scholar
[7] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.Google Scholar
[8] Engquist, B. and Osher, S., One sided difference approximations for nonlinear conservation laws, Math. Comput., 36 (1981), pp. 321–351.Google Scholar
[9] Osher, S. and Solomon, F., Upwind difference schemes for hyperbolic conservation laws, Math. Comput., 38 (1982), pp. 339–374.Google Scholar
[10] Liu, T. G., Khoo, B. C. and Yeo, K. S., Ghost fluid method for strong shock impacting on material interface, J. Comput. Phys., 190 (2003), pp. 651–681.CrossRefGoogle Scholar
[11] Liu, T. G., Xie, W. F. and Khoo, B. C., The modified ghost fluid method for coupling of fluid and structure constituted with Hydro-Elasto-Plastic equation of state, SIAM J. Sci. Comput., 33 (2008), pp. 1105–1130.Google Scholar
[12] Liu, T. G., Chowdhury, A. W. and Khoo, B. C., The modified ghost fluid method applied to fluid-elastic structure interaction, Adv. Appl. Math. Mech., 3 (2011), pp. 611–632.CrossRefGoogle Scholar
[13] Kaboudian, A. and Khoo, B. C., The ghost solid method for the elastic solid-solid interface, J. Comput. Phys., 257 (2014), pp. 102–125.CrossRefGoogle Scholar
[14] LeFloch, P. G. and Olsson, F., A second-order Godunov method for the conservation laws of nonlinear elastidynamics, Impact Comput. Sci. Eng., 2 (1990), pp. 318–354.CrossRefGoogle Scholar
[15] Garaizar, X., Solution of a Riemann problem for elasticity, J. Elasticity., 26 (1991), pp. 43–63.Google Scholar
[16] Miller, G. H., An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics, J. Comput. Phys., 193 (2003), pp. 198–225.Google Scholar
[17] Barton, P. T., Drikakis, D., Romenski, E. and Titarev, V. A., Exact and approximate solutions of Riemann problems in non-linear elasticity, J. Comput. Phys., 228 (2009), pp. 7046–7068.CrossRefGoogle Scholar
[18] López Ortega, A., Lombardini, M., Pullin, D. I. and Meiron, D. I., Numerical simulation of elastic-plastic solid mechanics using an Eulerian stretch tensor approach and HLLD Riemann solver, J. Comput. Phys., 257 (2014), pp. 414–441.CrossRefGoogle Scholar
[19] Trangenstein, J. A. and Pember, R. B., The Riemann problem for longitudinal motion in an elastic-plastic bar, SIAM J. Sci. State. Comput., 12 (1991), pp. 180–207.Google Scholar
[20] Lin, X., Numerical Computation of Stress Waves in Solids, Akademie Verlag, 1996.Google Scholar
[21] Wang, F., Glimm, J. G., Grove, J. W., Plohr, B. J. and Sharp, D. H., A conservative Eulerian numerical scheme for elasto-plasticity and application to plate impact problems, Impact Comput. Sci. Eng., 5 (1993), pp. 285–308.CrossRefGoogle Scholar
[22] Plohr, B. and Sharp, B., A conservative formulation for plasticity, Adv. Appl. Math., 13 (1992), pp. 462–493.CrossRefGoogle Scholar
[23] Miller, G. H. and Colella, P., A high-order Eulerian Godunov Method for elastic-plastic flow in solids, J. Comput. Phys., 167 (2001), pp. 131–176.Google Scholar
[24] Trangenstein, J. A. and Colella, P., A higher-order Godunov method for modeling finite deformation in elastic-plastics solids, Commun. Pure Appl. Math., 44 (1991), pp. 41–100.CrossRefGoogle Scholar
[25] Wilkins, M. L., Calculation of elastic-plastic flow, Meth. Comput. Phys., 3 (1964), pp. 211–263.Google Scholar
[26] Tang, H. S. and Sotiropoulos, F., A second-order Godunov method for wave problems in coupled solid-water-gas systems, J. Comput. Phys., 151 (1999), pp. 790–815.Google Scholar
[27] Fedkiw, R. P., Marquina, A. and Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. Comput. Phys., 148 (1999), pp. 545–578.CrossRefGoogle Scholar
[28] Smooler, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, 1983.CrossRefGoogle Scholar
[29] Fedkiw, R. P., Aslam, T., Merriman, B. and Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the Ghost Fluid Method), J. Comput. Phys., 152 (1999), pp. 457–492.CrossRefGoogle Scholar
[30] Guo, Y. H., Li, R. and Yao, C. B., A numerical method on Eulerian grids for two-phase compressible flow, Adv. Appl. Math. Mech., 8 (2016), pp. 187–212.CrossRefGoogle Scholar
[31] Xu, L. and Liu, T. G., Modified ghost fluid method as applied to fluid-plate interaction, Adv. Appl. Math. Mech., 6 (2014), pp. 24–48.Google Scholar