zbMATH — the first resource for mathematics

Weak shock waves and shear bands in thermoelastic solids. (English) Zbl 1171.74028
Summary: The linear weak shock wave (acoustic wave) propagation and the existence of shear bands are examined in finitely deformed thermoelastic solids within the framework of the theory of singular surfaces. The jumps of certain field variables across the shock wave front are obtained by using Taylor series expansions, and the propagation condition is obtained by using a strain-energy function corresponding to Duhamel-Neumann expression. We determine the propagation speeds of weak shock waves for a particular state of deformation and for general dilation. The formation of shear bands and the magnitudes of critical stretches are obtained for uniaxial and biaxial extensions and for uniform dilation.
Reviewer: Reviewer (Berlin)

74J40 Shocks and related discontinuities in solid mechanics
74F05 Thermal effects in solid mechanics
74B20 Nonlinear elasticity
Full Text: DOI
[1] Rice J.R.: The localization of plastic deformation. In: Koiter, W.T.(eds) Theoretical and Applied Mechanics, pp. 207–220. North-Holland, Amsterdam (1976)
[2] Mengi Y., McNiven H.D., Erdem A.Ü.: A theory for the formation of Lüders bands in a plate subjected to uniaxial tension. Int. J. Solids Struct. 11, 813–825 (1975) · Zbl 0325.73062 · doi:10.1016/0020-7683(75)90004-9
[3] Needleman A.: Dynamic shear band development in plane strain. J. Appl. Mech. 56, 1–9 (1989) · doi:10.1115/1.3176046
[4] Reddy B.D.: The occurrence of surface instabilities and shear bands in plane strain deformation of an elastic half space. Q. J. Mech. Appl. Math. 36, 337–350 (1983) · Zbl 0531.73029 · doi:10.1093/qjmam/36.3.337
[5] Gültop T.: Existence of shear bands in hyperelastic solids. Mech. Res. Commun. 29, 431–436 (2002) · Zbl 1094.74551 · doi:10.1016/S0093-6413(02)00248-3
[6] Gültop, T., Alyavuz, B.: Existence of shear bands in thermoelastic solids. In: Proceedings of the 6th European Solid Mechanics Conference ESMC 2006, Budapest, Hungary (2006) · Zbl 1171.74028
[7] Gültop, T., Alyavuz, B.: Weak shock wave propagation and shear band formation in constrained thermoelastic solids. In: Aköz, A.Y., Engin, H., Gülcat, Ü., Hacınlıyan, A. (eds.) Proceedings of the 15th National Mechanics Conference, Isparta, Turkey, pp. 465–475 (2007)
[8] Ogden R.W.: Large deformation isotropic elasticity–on the correlation of theory and experiment for incompressible rubberlike solids. Proc. R. Soc. Lond. A 326, 565–584 (1972) · Zbl 0257.73034 · doi:10.1098/rspa.1972.0026
[9] Ogden R.W.: Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. R. Soc. Lond. A 328, 567–583 (1972) · Zbl 0245.73032 · doi:10.1098/rspa.1972.0096
[10] Holzapfel G.A.: Nonlinear Solid Mechanics, A Continuum Approach For Engineering. Wiley, Chichester (2000) · Zbl 0980.74001
[11] Ciarlet P.G.: Mathematical Elasticity, Vol. 1: Three Dimensional Elasticity. Amsterdam, North-Holland (1988) · Zbl 0648.73014
[12] Lubarda V.A.: On thermodynamic potentials in linear thermoelasticity. Int. J. Solids Struct. 41, 7377–7398 (2004) · Zbl 1076.74003 · doi:10.1016/j.ijsolstr.2004.05.070
[13] Hadamard, J.: Leçons sur la propagation des ondes et les equations de l’hydrodynamique, Paris (1903) · JFM 34.0793.06
[14] Hill R.: Acceleration waves in solids. J. Mech. Phys. Solids 10, 1–16 (1962) · Zbl 0111.37701 · doi:10.1016/0022-5096(62)90024-8
[15] Chadwick P., Powdrill B.: Singular surfaces in linear thermoelasticity. Int. J. Eng. Sci. 3, 561–595 (1965) · doi:10.1016/0020-7225(65)90009-1
[16] Matsumoto E.: Behaviour of the stationary singular points in one-dimensional materials with internal state variables. Acta Mech. 39, 241–249 (1981) · Zbl 0479.73005 · doi:10.1007/BF01170345
[17] Osinov, V.A., Wu, W.: Wave speeds, shear bands and the second-order work for incrementally nonlinear constitutive models. Acta Mech. doi: 10.1007/s00707-008-0008-8 · Zbl 1254.74060
[18] Achenbach J.D.: The influence of heat conduction on propagating stress jumps. J. Mech. Phys. Solids 16, 273–282 (1968) · doi:10.1016/0022-5096(68)90035-5
[19] Inan E.: Decay of weak shock waves in hyperelastic solids. Acta Mech. 23, 103–111 (1975) · Zbl 0322.73017 · doi:10.1007/BF01177672
[20] Ukeje E.: Weak shock waves in heat conducting thermoelastic materials. Int. J. Eng. Sci. 20, 1275–1290 (1982) · Zbl 0495.73010 · doi:10.1016/0020-7225(82)90054-4
[21] Ukeje E.: Weak shock waves in non-heat conducting thermoelastic materials–variation of amplitude of the weak shocks. Int. J. Eng. Sci. 19, 1187–1201 (1981) · Zbl 0472.73004 · doi:10.1016/0020-7225(81)90140-3
[22] Eringen A.C., Şuhubi E.S.: Elastodynamics Vol. I. Academic Press, New York (1975)
[23] Cohen H., Tallin A.G.: Waves in thermo-viscoelastic rods. Acta Mech. 42, 85–97 (1982) · Zbl 0491.73029 · doi:10.1007/BF01176515
[24] Currò C., Sugiyama M., Suzumura H., Valenti G.: Weak shock waves in isotropic solids at finite temperatures up to melting point. Continuum Mech. Thermodyn. 18, 395–409 (2007) · Zbl 1160.74370 · doi:10.1007/s00161-006-0033-6
[25] Achenbach J.D.: The propagation of stress discontinuities according to the coupled equations of thermoelasticity. Acta Mech. 3, 342–351 (1967) · doi:10.1007/BF01181493
[26] Gültop T.: Weak shock waves in constrained thermoelastic solids. Arch. Appl. Mech. 72, 511–521 (2002) · Zbl 1084.74525 · doi:10.1007/s00419-002-0234-9
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.