## Weak shock waves and shear bands in compressible, inextensible thermoelastic solids.(English)Zbl 1184.74040

Summary: A study of weak shock waves propagating into a solid, which is compressible but temperature-dependent extensible in a specified direction is presented. The inextensible solid is also considered. The constitutive equations of constrained thermoelastic material are written as the summation of constrained and unconstrained counterparts of the relevant quantities. The equation of motion of weak shock waves, which is recovered by the theory of singular surfaces, reduces to an eigenvalue problem. The solution of this eigenvalue problem yields the speeds of propagation of weak shock waves. In the case of an undeformed solid, the speeds of these waves are explicitly expressed. Additionally, a discussion on the ductility limits of constrained thermoelastic material subjected to the uniaxial and biaxial extensions is presented.

### MSC:

 74J40 Shocks and related discontinuities in solid mechanics 74F05 Thermal effects in solid mechanics
Full Text:

### References:

 [1] Adkins J.E., Rivlin R.S.: Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords. Phil. Trans. R Soc. A 248, 201 (1955) · Zbl 0066.18802 [2] Spencer A.J.M.: Deformations of Fibre-Reinforced Materials. Oxford University Press, London (1972) · Zbl 0238.73001 [3] Spencer A.J.M.: Dynamics of ideal fibre-reinforced rigid-plastic beams. J. Mech. Phys. Solids 22, 147–159 (1974) · Zbl 0278.73030 [4] Horgan C.O., Saccomandi G.: A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids. J. Mech. Phys. Solids 53, 1985–2015 (2005) · Zbl 1176.74026 [5] Weitsman Y.: On wave propagation and energy scattering in materials reinforced by inextensible fibers. Int. J. Solids Struct. 8, 627–650 (1972) · Zbl 0236.73041 [6] Chen P.J., Gurtin M.E.: On wave propagation in inextensible elastic bodies. Int. J. Solids Struct. 10, 275–281 (1974) · Zbl 0269.73034 [7] Chen P.J., Nunziato J.W.: On wave propagation in perfectly heat conducting inextensible elastic bodies. J. Elast. 5, 155–160 (1975) · Zbl 0332.73033 [8] Scott N.H.: Acceleration waves in constrained elastic materials. Arch. Rat. Mech. Anal. 58, 57–75 (1975) · Zbl 0339.73006 [9] Reddy B.D.: The propagation and growth of acceleration waves in constrained thermoelastic materials. J. Elast. 14, 387–402 (1984) · Zbl 0573.73032 [10] Trapp J.A.: Reinforced materials with thermomechanical constraints. Int. J. Eng. Sci. 9, 757–773 (1971) · Zbl 0231.73004 [11] Bleach G.P., Reddy B.D.: The influence of constraints on the properties of acceleration waves in isotropic thermoelastic media. Arch. Rat. Mech. Anal. 98, 31–64 (1987) · Zbl 0617.73008 [12] Scott N.H.: Small vibrations of prestrained constrained elastic materials: the idealized fibre-reinforced material. Int. J. Solids Struct. 27, 1969–1980 (1991) · Zbl 0764.73020 [13] Rogerson G.A., Scott N.H.: Wave propagation in singly-constrained and nearly-constrained elastic materials. Q. J. Mech. Appl. Math. 45, 77–99 (1992) · Zbl 0760.73013 [14] Rogerson G.A., Scott N.H.: Doubly constrained elastic wave propagation. Int. J. Solids Struct. 31, 2769–2792 (1994) · Zbl 0943.74527 [15] Bortoloni L., Pastrone F.: Waves in approximately constrained materials and applications to fiber-reinforced composites. Wave Motion 36, 275–286 (2002) · Zbl 1163.74319 [16] Tonon M.L.: Waves in constrained linear elastic materials. J. Elast. 69, 15–39 (2002) · Zbl 1171.74365 [17] Gültop T.: Weak shock waves in constrained thermoelastic solids. Arch. Appl. Mech. 72, 511–521 (2002) · Zbl 1084.74525 [18] Fleck N.A.: Compressive failure of fiber composites. Adv. Appl. Mech. 33, 43–117 (1997) · Zbl 0958.74054 [19] Merodio J., Pence T.J.: Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation: I Mechanical equilibrium. J. Elast. 62, 119–144 (2001) · Zbl 1007.74022 [20] Merodio J., Ogden R.W.: Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Struct. 40, 4707–4727 (2003) · Zbl 1054.74721 [21] Ciarlet P.G.: Mathematical Elasticity: Three Dimensional Elasticity, vol.$$\sim$$1. North-Holland, Amsterdam (1988) · Zbl 0648.73014 [22] Casey J.: Treatment of internally constrained materials. J. Appl. Mech. T. ASME 62, 542–544 (1995) · Zbl 0845.73015 [23] Casey J., Krishnaswamy S.: A characterization of internally constrained thermoelastic materials. Math. Mech. Solids 3, 71–89 (1998) · Zbl 1001.74552 [24] Lubarda V.A.: On thermodynamic potentials in linear thermoelasticity. Int. J. Solids Struct. 41, 7377–7398 (2004) · Zbl 1076.74003 [25] Eringen A.C., Şuhubi E.S.: Elastodynamics vol. I. Academic Press, New York (1975) [26] Truesdell C., Toupin R.A.: Classical field theories of mechanics. In: Flügge, S. (eds) Handbuch der physik III/1, Springer, Berlin (1960) · Zbl 0119.19201 [27] Eringen A.C.: Mechanics of Continua, 2nd edn. Krieger Publishing Company, New York (1980) · Zbl 0436.76006 [28] 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 [29] Gültop T.: Existence of shear bands in hyperelastic solids. Mech. Res. Commun. 29, 431–436 (2002) · Zbl 1094.74551 [30] Abeyaratne R., Triantafyllidis N.: The emergence of shear bands in plane strain. Int. J. Solids Struct. 12, 1113–1134 (1981) · Zbl 0476.73082 [31] Rice J.R.: The localization of plastic deformation. In: Koiter, W.T. (eds) Theoretical and applied mechanics, pp. 207–220. North-Holland, Amsterdam (1976) [32] 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 [33] Needleman A.: Dynamic shear band development in plane strain. J. Appl. Mech. 56, 1–9 (1989) [34] Zhang Y.Q., Lu Y., Qiang H.F.: Influence of damage on properties of strain localization in geomaterials at plane stress and plane strain. Arch. Appl. Mech. 74, 102–117 (2004) · Zbl 1158.74467 [35] Alyavuz, B., Gültop, T.: Weak shock waves and shear bands in thermoelastic materials. Acta. Mech. (2008) doi: 10.1007/s00707-008-0117-4 · Zbl 1171.74028 [36] Bardet J.P.: A comprehensive review of strain localization in elastoplastic soils. Comput. Geo. 10, 163–188 (1990) [37] Triantafyllidis N., Abeyaratne R.: Instabilities of a finitely deformed fiber-reinforced elastic material. J. Appl. Mech. T ASME 50, 149–156 (1983) · Zbl 0511.73036 [38] Kurashige M.: On elastostatic shocks in an ideal fiber-reinforced composite (Case of plane deformation). Technol. Rep. Tohoku Univ. 49, 115–128 (1984) [39] Hadamard, J.: Leçons sur la propagation des ondes et les equations de l’hydrodynamique. Paris (1903) · JFM 34.0793.06 [40] Hill R.: Acceleration waves in solids. J. Mech. Phys. Solids 10, 1–16 (1962) · Zbl 0111.37701 [41] Mandel J.: Conditions de stabilite et postulat de drucker. In: Kravtencko, J., Sirieys, P.M. (eds) Rheology and soil mechanics, Springer, Berlin (1966) [42] Gültop, T., Alyavuz, B.: Existence of shear bands in thermoelastic solids. In: Proceedings of the 6th European Solid Mechanics Conference. Budapest, Hungary, 28 August–1 September (2006) · Zbl 1171.74028
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.