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Solution of a Riemann problem for elasticity. (English) Zbl 0755.73024
Summary: This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.

74B20 Nonlinear elasticity
35L60 First-order nonlinear hyperbolic equations
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[1] D.R.Bland, Plane isentropic displacement simple waves in a compressible elastic solid, Z. Angew. Math. Phys. 16 (1965) 752-769.
[2] I-L.Chern, J.Glimm, O.McBryan, B.Plohr, and S.Yaniv, Front tracking for gas dynamics, J. Comp. Phys. 62 (1986) 83-110. · Zbl 0577.76068
[3] A.Chorin, Random choice solutions of hyperbolic systems. J. Comp. Phys. 22 (1976) 517-533. · Zbl 0354.65047
[4] P.Daripa, J.Glimm, B.Lindquist, M.Maesumi, and O.McBryan, On the simulation of heterogeneous petroleum reservoirs, in Numerical Simulation in Oil Recovery, M.F.Wheeler (ed.), pp. 89-103, Springer-Verlag, New York (1988).
[5] P.Daripa, J.Glimm, B.Lindquist, and O.McBryan, Polymer floods: a case study of nonlinear wave analysis and of instability control in tertiary oil recovery, SIAM J. Appl. Math. 48 (1988) 353-373. · Zbl 0641.76093
[6] X.Garaizar, The small anisotropy formulation of elastic deformations, Acta Applicandae Mathematicae 14 (1989) 259-268. · Zbl 0683.73018
[7] C.L.Gardner, Supersonic interface instabilities of accelerated surfaces and jets, Phys. of Fluids 29 (1986) 690-695. · Zbl 0593.76071
[8] J.Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure and Appl. Math. 18 (1965) 695-715. · Zbl 0141.28902
[9] J.Glimm, E.Isaacson, D.Marchesin, and O.McBryan, Front tracking for hyperbolic systems, Adv. in Appl. Math. 2 (1981) 91-119. · Zbl 0459.76069
[10] J.Glimm, B.Lindquist, O.McBryan, and L.Padmanabhan, A front tracking reservoir simulator, five-spot validation studies and the water coning problem, in Frontiers in Applied Mathematics, vol. 1, SIAM, Philadelphia (1983). · Zbl 0546.76115
[11] J.Glimm, C.Klingenberg, O.McBryan, B.Plohr, D.Sharp, and S.Yaniv, Front tracking and two dimensional Riemann problems, Adv. in Appl. Math. 6 (1985) 259-290. · Zbl 0631.76068
[12] J. Glimm, J. Grove, B. Lindquist, O.A. McBryan, and G. Tryggvason, The bifurcation of tracked scalar waves, SIAM Jour. Sci. Stat. Comp. 9, Jan (1988). · Zbl 0636.65132
[13] S.K.Godunov, Finite-difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sbornik 47 (1959) 271-306. · Zbl 0171.46204
[14] J.W. Grove, Front tracking and shock-contact interactions, Advances in Computer Methods for Partial Differential Equations, vol. VI, IMACS (1987).
[15] M.E.Gurtin, Topics in finite elasticity, CBMS Reg. Conf. Series, vol. 35, SIAM, New York (1981).
[16] M.E.Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York (1981). · Zbl 0559.73001
[17] A.Hanyga, Mathematical Theory of Non-linear Elasticity, Ellis Horwood Limited, Chichester, England (1985). · Zbl 0561.73011
[18] B.VanLeer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comp. Phys. 32 (1977) 101-136.
[19] J.E.Marsden and T.J.R.Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, N.J. (1983). · Zbl 0545.73031
[20] C.Moler and J.Smoller, Elementary interactions in quasilinear hyperbolic systems, Arch. Rat. Mech. Anal. 37 (1970) 309-322. · Zbl 0192.44901
[21] B.Plohr, J.Glimm, and O.McBryan, Applications of front tracking to two-dimensional gas dynamics calculations, in Lecture Notes in Engineering, J.Chandra and J.Flaherty (ed.), vol. 3, pp. 180-191, Springer Verlag, New York (1983).
[22] B.Plohr, Shockless acceleration of thin plates modeled by a tracked random choice method, AIAA J. 26 (1988) 470-478. · Zbl 0665.76094
[23] Bradley J.Plohr and D.Sharp, A conservative Eulerian formulation of the equations for elastic flow, Adv. Appl. Math. 9 (1988) 481-499. · Zbl 0663.73012
[24] D.J.Steinberg, S.G.Cochran, and M.W.Guinan, A constitutive model for metals applicable at high-strain rate, J. Appl. Phys. 51 (1980) 1498-1504.
[25] Erdogan S. Suhubi, Thermoelastic solids, in Continuum Physics, A.C. Eringen (ed.), vol. II, pp. 173-265 (1975).
[26] Z.Tang and T.C.T.Ting, Wave curves for the Riemann problem of plane waves in simple isotropic elastic solids, Int. J. Engng. Sci. 25 (1987) 1343-1381. · Zbl 0637.73020
[27] Paul R.Woodward and PhillipColella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comp Phys. 54 (1984) 115-173. · Zbl 0573.76057
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