×

Stability of classical shock fronts for compressible hyperelastic materials of Hadamard type. (English) Zbl 1481.74412

Summary: This paper studies the uniform and weak Lopatinskiĭ conditions associated with classical (Lax) shock fronts of arbitrary amplitude for compressible hyperelastic materials of Hadamard type in several space dimensions. Thanks to the seminal works of A. Majda [The existence of multi-dimensional shock fronts. Providence, RI: American Mathematical Society (AMS) (1983; Zbl 0517.76068); The stability of multi-dimensional shock fronts. Providence, RI: American Mathematical Society (AMS) (1983; Zbl 0506.76075)] and G. Métivier [Trans. Am. Math. Soc. 296, 431–479 (1986; Zbl 0619.35075); Commun. Partial Differ. Equations 15, No. 7, 983–1028 (1990; Zbl 0711.35078); Prog. Nonlinear Differ. Equ. Appl. 47, 25–103 (2001; Zbl 1017.35075)], the uniform Lopatinskiĭ condition ensures the local-in-time, multidimensional, nonlinear stability of such fronts. The stability function (also called Lopatinskiĭ determinant) for shocks of arbitrary amplitude in this large class of hyperelastic materials is computed explicitly. This information is used to establish the conditions for uniform and weak shock stability in terms of the parameters of the shock and of the elastic moduli of the material.

MSC:

74J40 Shocks and related discontinuities in solid mechanics
74B20 Nonlinear elasticity
35Q74 PDEs in connection with mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akritas, AG; Akritas, EK; Malaschonok, GI, Various proofs of Sylvester’s (determinant) identity, Math. Comput. Simulation, 42, 4-6, 585-593 (1996) · Zbl 1037.15501
[2] Aron, M., Aizicovici, S.: On a class of deformations of compressible, isotropic, nonlinearly elastic solids. J. Elast.49(2), 175-185, 1997/98 · Zbl 0907.73011
[3] Aubert, G., Necessary and sufficient conditions for isotropic rank-one convex functions in dimension \(2\), J. Elast., 39, 1, 31-46 (1995) · Zbl 0828.73015
[4] Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.63(4), 337-403, 1976/77 · Zbl 0368.73040
[5] Ball, JM; James, RD, Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal., 100, 1, 13-52 (1987) · Zbl 0629.49020
[6] Ball, JM; James, RD, Proposed experimental tests of a theory of fine microstructure and the two-well problem, Phil. Trans. R. Soc. Lond. A, 338, 1650, 389-450 (1992) · Zbl 0758.73009
[7] Benzoni-Gavage, S., Stability of multi-dimensional phase transitions in a van der Waals fluid, Nonlinear Anal. TMA, 31, 1-2, 243-263 (1998) · Zbl 0928.76015
[8] Ball, JM; James, RD, Stability of subsonic planar phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal., 150, 1, 23-55 (1999) · Zbl 0980.76023
[9] Benzoni-Gavage, S.; Freistühler, H., Effects of surface tension on the stability of dynamical liquid-vapor interfaces, Arch. Ration. Mech. Anal., 174, 1, 111-150 (2004) · Zbl 1081.76027
[10] Benzoni-Gavage, S., Serre, D.: Multidimensional hyperbolic partial differential equations: First-order systems and applications. Oxford Mathematical Monographs, The Clarendon Press - Oxford University Press, Oxford, 2007 · Zbl 1113.35001
[11] Bethe, H. A.: On the theory of shock waves for an arbitrary equation of state [Rep. No. 545, Serial No. NDRC-B-237, Office Sci. Res. Develop., U. S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 1942]. In Classic papers in shock compression science, High-press. Shock Compression Condens. Matter, Springer, New York, pp. 421-492, 1998
[12] Bischoff, JE; Arruda, EM; Grosh, K., A new constitutive model for the compressibility of elastomers at finite deformations, Rubber Chem. Technol., 74, 4, 541-559 (2001)
[13] Blatz, PJ; Chompff, A.; Newman, S., On the thermostatic behavior of elastomers, Polymer Networks, 23-45 (1971), New York, NY: Springer Science and Business Media, New York, NY
[14] Blokhin, A.M.: Uniqueness of the classical solution of a mixed problem for equations of gas dynamics with boundary conditions on a shock wave. Sibirsk. Mat. Zh.23(5), 17-30, 222, 1982 · Zbl 0546.35011
[15] Carroll, MM; Murphy, JG; Rooney, FJ, Plane stress problems for compressible materials, Int. J. Solids Struct., 31, 11, 1597-1607 (1994) · Zbl 0946.74505
[16] Chugainova, AP; Il’ichev, AT; Shargatov, VA, Stability of shock wave structures in nonlinear elastic media, Math. Mech. Solids, 24, 11, 3456-3471 (2019) · Zbl 07273377
[17] Ciarlet, P.G.: Mathematical elasticity. Vol. I: Three-dimensional elasticity, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988. · Zbl 0648.73014
[18] Ciarlet, PG; Geymonat, G., Sur les lois de comportement en élasticité non linéaire compressible, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 295, 4, 423-426 (1982) · Zbl 0497.73017
[19] Coleman, BD; Noll, W., On certain steady flows of general fluids, Arch. Ration. Mech. Anal., 3, 1, 289-303 (1959) · Zbl 0087.19402
[20] Corli, A., Weak shock waves for second-order multi-dimensional systems, Boll. Un. Mat. Ital. B (7), 7, 3, 493-510 (1993) · Zbl 0796.73010
[21] Costanzino, N.; Jenssen, HK; Lyng, G.; Williams, M., Existence and stability of curved multidimensional detonation fronts, Indiana Univ. Math. J., 56, 3, 1405-1461 (2007) · Zbl 1213.76112
[22] Coulombel, J-F; Secchi, P., The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53, 4, 941-1012 (2004) · Zbl 1068.35100
[23] Coulombel, J-F; Secchi, P., Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41, 1, 85-139 (2008) · Zbl 1160.35061
[24] Currie, PK, The attainable region of strain-invariant space for elastic materials, Int. J. Non-Linear Mech., 39, 5, 833-842 (2004) · Zbl 1348.74048
[25] Dacorogna, B., Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension, Discrete Contin. Dyn. Syst. Ser. B, 1, 2, 257-263 (2001) · Zbl 1055.35045
[26] Dafermos, CM, Quasilinear hyperbolic systems with involutions, Arch. Ration. Mech. Anal., 94, 4, 373-389 (1986) · Zbl 0614.35057
[27] Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, fourth ed., 2016 · Zbl 1364.35003
[28] Davies, PJ, A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain, J. Elast., 26, 3, 291-296 (1991) · Zbl 0759.73013
[29] D’ yakov, SP, On the stability of shock waves, Ž. Eksper. Teoret. Fiz., 27, 288-295 (1954) · Zbl 0057.42101
[30] De Tommasi, D.; Puglisi, G.; Zurlo, G., A note on strong ellipticity in two-dimensional isotropic elasticity, J. Elast., 109, 1, 67-74 (2012) · Zbl 1255.74005
[31] Eremeyev, VA; Cloud, MJ; Lebedev, LP, Applications of Tensor Analysis in Continuum Mechanics (2018), Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ · Zbl 1471.74001
[32] Erpenbeck, JJ, Stability of step shocks, Phys. Fluids, 5, 1181-1187 (1962) · Zbl 0111.38403
[33] Fosdick, R., Royer-Carfagni, G.: Multiple natural states for an elastic isotropic material with polyconvex stored energy. J. Elast.60(2000)(3), 223-231, 2001 · Zbl 0992.74012
[34] Freistühler, H., Some results on the stability of non-classical shock waves, J. Partial Differ. Eqs., 11, 1, 25-38 (1998) · Zbl 0903.35006
[35] Freistühler, H.; Plaza, RG, Normal modes and nonlinear stability behaviour of dynamic phase boundaries in elastic materials, Arch. Ration. Mech. Anal., 186, 1, 1-24 (2007) · Zbl 1147.74038
[36] Freistühler, H., Plaza, R.G.: Normal modes analysis of subsonic phase boundaries in elastic materials. In Hyperbolic problems: Theory, Numerics, Applications, Benzoni-Gavage, S., Serre, D. (Eds.), Proceedings of the 11th International Conference on Hyperbolic Problems (HYP2006) held at the École Normale Supérieure, Lyon, July 17-21, 2006, Springer, Berlin, pp. 841-848, 2008 · Zbl 1157.35066
[37] Freistühler, H., Szmolyan, P.: The Lopatinski determinant of small shocks may vanish. Preprint, 2011. arXiv:1102.4279.
[38] Gardner, C.S.: Comment on “Stability of step shocks.” Phys. Fluids6(9), 1366-1367, 1963 · Zbl 0119.20202
[39] Gavrilyuk, S.; Ndanou, S.; Hank, S., An example of a one-parameter family of rank-one convex stored energies for isotropic compressible solids, J. Elast., 124, 1, 133-141 (2016) · Zbl 1337.35146
[40] Giaquinta, M., Hildebrandt, S.: Calculus of variations I. The Lagrangian formalism. vol. 310 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1996 · Zbl 0853.49001
[41] Gorb, Y.; Walton, JR, Dependence of the frequency spectrum of small amplitude vibrations superimposed on finite deformations of a nonlinear, cylindrical elastic body on residual stress, Int. J. Eng. Sci., 48, 11, 1289-1312 (2010) · Zbl 1231.74043
[42] Grabovsky, Y.; Truskinovsky, L., Legendre-Hadamard conditions for two-phase configurations, J. Elast., 123, 2, 225-243 (2016) · Zbl 1334.74069
[43] Grabovsky, Y.; Truskinovsky, L., Explicit relaxation of a two-well Hadamard energy, J. Elast., 135, 1-2, 351-373 (2019) · Zbl 1415.74005
[44] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier/Academic Press, Amsterdam, seventh ed. 2007. Translated from the Russian. Translation edited and with a preface by A. Jeffrey and D. Zwillinger. · Zbl 0918.65002
[45] Hadamard, J., Sur les problèmes aux dérivées partielles et leur signification physique, Princeton Univ. Bull., 13, 49-52 (1902)
[46] Hadamard, J.: Leçons sur la propagation des ondes et les équations de l’hydrodynamique. Librairie Scientifique A. Hermann, Paris, 1903 · JFM 34.0793.06
[47] Hartmann, S.: The class of Simo & Pister-type hyperelasticity relations. Technical Report Fac3-10-02, Technical Report Series, Clausthal University of Technology, 2010
[48] Hayes, M., A remark on Hadamard materials, Quart. J. Mech. Appl. Math., 21, 2, 141-146 (1968) · Zbl 0159.56901
[49] Hersh, R., Mixed problems in several variables, J. Math. Mech., 12, 3, 317-334 (1963) · Zbl 0149.06602
[50] Hill, R., On constitutive inequalities for simple materials - I, J. Mech. Phys. Solids, 16, 4, 229-242 (1968) · Zbl 0162.28702
[51] Hill, R., On constitutive inequalities for simple materials - II, J. Mech. Phys. Solids, 16, 5, 315-322 (1968) · Zbl 0167.54302
[52] Holzapfel, GA, Nonlinear Solid Mechanics (2000), Chichester: Wiley, Chichester · Zbl 0980.74001
[53] Horgan, CO, Remarks on ellipticity for the generalized Blatz-Ko constitutive model for a compressible nonlinearly elastic solid, J. Elast., 42, 2, 165-176 (1996) · Zbl 0852.73019
[54] Horgan, CO; Saccomandi, G., Constitutive models for compressible nonlinearly elastic materials with limiting chain extensibility, J. Elast., 77, 2, 123-138 (2004) · Zbl 1076.74008
[55] Jenssen, H.K., Lyng, G.: Evaluation of the Lopatinski determinant for multi-dimensional Euler equations. Appendix to K. Zumbrun, “Stability of large-amplitude shock waves of compressible Navier-Stokes equations” in The Handbook of Fluid Mechanics, Vol. III, S. Friedlander and D. Serre, eds. North-Holland, Amsterdam, 2004.
[56] Jiang, Q.; Knowles, JK, A class of compressible elastic materials capable of sustaining finite anti-plane shear, J. Elast., 25, 3, 193-201 (1991) · Zbl 0766.73021
[57] John, F., Plane elastic waves of finite amplitude. Hadamard materials and harmonic materials, Comm. Pure Appl. Math., 19, 309-341 (1966) · Zbl 0139.43401
[58] John, F., Finite amplitude waves in a homogeneous isotropic elastic solid, Commun. Pure Appl. Math., 30, 4, 421-446 (1977) · Zbl 0404.73023
[59] Kirkinis, E.; Ogden, RW; Haughton, DM, Some solutions for a compressible isotropic elastic material, Z. Angew. Math. Phys., 55, 1, 136-158 (2004) · Zbl 1064.74024
[60] Knowles, JK, A note on anti-plane shear for compressible materials in finite elastostatics, J. Austral. Math. Soc. Ser. B, 20, 1, 1-7 (1977) · Zbl 0363.73044
[61] Knowles, J.K., Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Ration. Mech. Anal.63(1976)(4), 321-336, 1977 · Zbl 0351.73061
[62] Kreiss, H-O, Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., 23, 277-298 (1970) · Zbl 0188.41102
[63] Kubo, R., Large elastic deformation of rubber, J. Phys. Soc. Japan, 3, 312-317 (1948)
[64] Kulikovskiĭ, AG; Chugaĭnova, AP, On the stability of quasi-transverse shock waves in anisotropic elastic media, Prikl. Mat. Mekh., 64, 6, 1020-1026 (2000) · Zbl 0984.74041
[65] Lax, PD, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10, 537-566 (1957) · Zbl 0081.08803
[66] Le Tallec, P.: Numerical methods for nonlinear three-dimensional elasticity. In: Handbook of Numerical Analysis, Numerical Methods for Solids (Part 1). Ciarlet, P.G., Lions, J.L. (Eds.), vol. 3 of Handbook of Numerical Analysis, Elsevier Science B.V., Amsterdam, pp. 465-622, 1994 · Zbl 0875.73234
[67] Levinson, M.; Burgess, IW, A comparison of some simple constitutive relations for slightly compressible rubber-like materials, Int. J. Mech. Sci., 13, 6, 563-572 (1971)
[68] Lopatinskiĭ, J.B.: The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type. Dopovīdī Akad. Nauk Ukraïn. RSR Ser. A1970, 592-594, 668, 1970 · Zbl 0201.12903
[69] Mac Donald, BJ, Practical Stress Analysis with Finite Elements (2011), Dublin: Glasnevin Publishing, Dublin
[70] Majda, A., The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 43, 281, v + 93 (1983) · Zbl 0517.76068
[71] Majda, A., The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc., 41, 275, iv + 95 (1983) · Zbl 0506.76075
[72] Majda, A.: Compressible fluid flow and systems of conservation laws in several space variables, vol. 53 of Applied Mathematical Sciences, Springer-Verlag, New York, 1984 · Zbl 0537.76001
[73] Meille, S.; Garboczi, EJ, Linear elastic properties of 2D and 3D models of porous materials made from elongated objects, Modelling Simul. Mater. Sci. Eng., 9, 5, 371-390 (2001)
[74] Métivier, G., Interaction de deux choc pour un systéme de deux lois de conservation, en dimension deux d’espace, Trans. Amer. Math. Soc., 296, 431-479 (1986) · Zbl 0619.35075
[75] Métivier, G., Stability of multidimensional weak shocks, Commun. Partial Differ. Eqs., 15, 7, 983-1028 (1990) · Zbl 0711.35078
[76] Métivier, G.: Stability of multidimensional shocks. In: Advances in the Theory of Shock Waves, Freistühler, H., Szepessy, A. (Eds.), vol. 47 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, pp. 25-103, 2001 · Zbl 1017.35075
[77] Mills, N.: Polymer Foams Handbook, Engineering and Biomechanics Applications and Design Guide, Butterworth-Heinemann, Amsterdam, 2007
[78] Nemat-Nasser, S.; Shatoff, HD, A consistent numerical method for the solution of nonlinear elasticity problems at finite strains, SIAM J. Appl. Math., 20, 462-481 (1971) · Zbl 0234.73011
[79] Ogden, RW, Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids, Proc. R. Soc. Lond. A Math. Phys. Sci., 328, 1575, 567-583 (1972) · Zbl 0245.73032
[80] Ogden, RW, Non-linear elastic deformations (1984), Chichester and John Wiley: Ellis Horwood, Chichester and John Wiley · Zbl 0541.73044
[81] Pence, TJ; Gou, K., On compressible versions of the incompressible neo-Hookean material, Math. Mech. Solids, 20, 2, 157-182 (2015) · Zbl 07278984
[82] Plaza, RG, Multidimensional stability of martensite twins under regular kinetics, J. Mech. Phys. Solids, 56, 4, 1989-2018 (2008) · Zbl 1162.74422
[83] Prasolov, V.V.: Problems and theorems in linear algebra, vol. 134 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by D. A. Leĭtes. · Zbl 0803.15001
[84] Rivlin, RS, Large elastic deformations of isotropic materials. I. Fundamental concepts, Philos. Trans. Roy. Soc. London. Ser. A., 240, 459-490 (1948) · Zbl 0029.32605
[85] Rivlin, RS; Ericksen, JL, Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal., 4, 2, 323-425 (1955) · Zbl 0064.42004
[86] Roberts, A.E.: Stability of a steady plane shock. Los Alamos Scientific Laboratory, Report No. LA-299, 1945
[87] Serre, D.: Systems of Conservation Laws 1. Hyperbolicity, entropies, shock waves. Cambridge University Press, Cambridge 1999. Translated from the 1996 French original by I. N. Sneddon. · Zbl 0930.35001
[88] Serre, D.: Systems of Conservation Laws 2. Geometric structures, oscillations and initial-boundary value problems. Cambridge University Press, Cambridge 2000. Translated from the 1996 French original by I. N. Sneddon. · Zbl 0936.35001
[89] Serre, D., La transition vers l’instabilité pour les ondes de choc multi-dimensionnelles, Trans. Amer. Math. Soc., 353, 12, 5071-5093 (2001) · Zbl 1078.35521
[90] Sfyris, D., The strong ellipticity condition under changes in the current and reference configuration, J. Elast., 103, 2, 281-287 (2011) · Zbl 1273.74026
[91] Šilhavý, M.: The mechanics and thermodynamics of continuous media, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 · Zbl 0870.73004
[92] Simo, JC; Miehe, C., Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation, Comput. Meth. Appl. Mech. Eng., 98, 1, 41-104 (1992) · Zbl 0764.73088
[93] Simo, JC; Pister, KS, Remarks on rate constitutive equations for finite deformation problems: computational implications, Comput. Meth. Appl. Mech. Eng., 46, 2, 201-215 (1984) · Zbl 0525.73042
[94] Simo, JC; Taylor, RL, Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms, Comput. Methods Appl. Mech. Engrg., 85, 3, 273-310 (1991) · Zbl 0764.73104
[95] Simpson, HC; Spector, SJ, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Ration. Mech. Anal., 84, 1, 55-68 (1983) · Zbl 0526.73026
[96] Thorpe, MF; Jasiuk, I., New results in the theory of elasticity for two-dimensional composites, Proc. R. Soc. Lond. A Math. Phys. Sci., 438, 1904, 531-544 (1992) · Zbl 0806.73042
[97] Trabelsi, K., Nonlinear thin plate models for a family of Ogden materials, C. R. Math. Acad. Sci. Paris, 337, 12, 819-824 (2003) · Zbl 1032.74035
[98] Truesdell, C., General and exact theory of waves in finite elastic strain, Arch. Ration. Mech. Anal., 8, 263-296 (1961) · Zbl 0111.37703
[99] Truesdell, C.; Noll, W., The non-linear field theories of mechanics (2004), Berlin: Springer-Verlag, Berlin · Zbl 1068.74002
[100] Truesdell, C., Toupin, R.: The classical field theories. In Handbuch der Physik, Bd. III/1, S. Flügge, ed., Springer, Berlin, pp. 226-793, 1960. appendix, pp. 794-858. With an appendix on tensor fields by J. L. Ericksen.
[101] Wang, Y.; Aron, M., A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elast., 44, 1, 89-96 (1996) · Zbl 0876.73030
[102] Zumbrun, K.: Multidimensional stability of planar viscous shock waves. In: Advances in the Theory of Shock Waves, Freistühler, H., Szepessy, A. (Eds.), vol. 47 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, pp. 307-516, 2001 · Zbl 0989.35089
[103] Zumbrun, K.; Friedlander, S.; Serre, D., Stability of large-amplitude shock waves of compressible Navier-Stokes equations, Handbook of mathematical fluid dynamics, 311-533 (2004), Amsterdam: North-Holland, Amsterdam · Zbl 1222.35156
[104] Zumbrun, K.; Serre, D., Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J., 48, 3, 937-992 (1999) · Zbl 0944.76027
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.