Abstract
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.
Similar content being viewed by others
References
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)
Spencer A.J.M.: Deformations of Fibre-Reinforced Materials. Oxford University Press, London (1972)
Spencer A.J.M.: Dynamics of ideal fibre-reinforced rigid-plastic beams. J. Mech. Phys. Solids 22, 147–159 (1974)
Horgan C.O., Saccomandi G.: A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids. J. Mech. Phys. Solids 53, 1985–2015 (2005)
Weitsman Y.: On wave propagation and energy scattering in materials reinforced by inextensible fibers. Int. J. Solids Struct. 8, 627–650 (1972)
Chen P.J., Gurtin M.E.: On wave propagation in inextensible elastic bodies. Int. J. Solids Struct. 10, 275–281 (1974)
Chen P.J., Nunziato J.W.: On wave propagation in perfectly heat conducting inextensible elastic bodies. J. Elast. 5, 155–160 (1975)
Scott N.H.: Acceleration waves in constrained elastic materials. Arch. Rat. Mech. Anal. 58, 57–75 (1975)
Reddy B.D.: The propagation and growth of acceleration waves in constrained thermoelastic materials. J. Elast. 14, 387–402 (1984)
Trapp J.A.: Reinforced materials with thermomechanical constraints. Int. J. Eng. Sci. 9, 757–773 (1971)
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)
Scott N.H.: Small vibrations of prestrained constrained elastic materials: the idealized fibre-reinforced material. Int. J. Solids Struct. 27, 1969–1980 (1991)
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)
Rogerson G.A., Scott N.H.: Doubly constrained elastic wave propagation. Int. J. Solids Struct. 31, 2769–2792 (1994)
Bortoloni L., Pastrone F.: Waves in approximately constrained materials and applications to fiber-reinforced composites. Wave Motion 36, 275–286 (2002)
Tonon M.L.: Waves in constrained linear elastic materials. J. Elast. 69, 15–39 (2002)
Gültop T.: Weak shock waves in constrained thermoelastic solids. Arch. Appl. Mech. 72, 511–521 (2002)
Fleck N.A.: Compressive failure of fiber composites. Adv. Appl. Mech. 33, 43–117 (1997)
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)
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)
Ciarlet P.G.: Mathematical Elasticity: Three Dimensional Elasticity, vol.~1. North-Holland, Amsterdam (1988)
Casey J.: Treatment of internally constrained materials. J. Appl. Mech. T. ASME 62, 542–544 (1995)
Casey J., Krishnaswamy S.: A characterization of internally constrained thermoelastic materials. Math. Mech. Solids 3, 71–89 (1998)
Lubarda V.A.: On thermodynamic potentials in linear thermoelasticity. Int. J. Solids Struct. 41, 7377–7398 (2004)
Eringen A.C., Şuhubi E.S.: Elastodynamics vol. I. Academic Press, New York (1975)
Truesdell C., Toupin R.A.: Classical field theories of mechanics. In: Flügge, S. (eds) Handbuch der physik III/1, Springer, Berlin (1960)
Eringen A.C.: Mechanics of Continua, 2nd edn. Krieger Publishing Company, New York (1980)
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)
Gültop T.: Existence of shear bands in hyperelastic solids. Mech. Res. Commun. 29, 431–436 (2002)
Abeyaratne R., Triantafyllidis N.: The emergence of shear bands in plane strain. Int. J. Solids Struct. 12, 1113–1134 (1981)
Rice J.R.: The localization of plastic deformation. In: Koiter, W.T. (eds) Theoretical and applied mechanics, pp. 207–220. North-Holland, Amsterdam (1976)
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)
Needleman A.: Dynamic shear band development in plane strain. J. Appl. Mech. 56, 1–9 (1989)
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)
Alyavuz, B., Gültop, T.: Weak shock waves and shear bands in thermoelastic materials. Acta. Mech. (2008) doi:10.1007/s00707-008-0117-4
Bardet J.P.: A comprehensive review of strain localization in elastoplastic soils. Comput. Geo. 10, 163–188 (1990)
Triantafyllidis N., Abeyaratne R.: Instabilities of a finitely deformed fiber-reinforced elastic material. J. Appl. Mech. T ASME 50, 149–156 (1983)
Kurashige M.: On elastostatic shocks in an ideal fiber-reinforced composite (Case of plane deformation). Technol. Rep. Tohoku Univ. 49, 115–128 (1984)
Hadamard, J.: Leçons sur la propagation des ondes et les equations de l’hydrodynamique. Paris (1903)
Hill R.: Acceleration waves in solids. J. Mech. Phys. Solids 10, 1–16 (1962)
Mandel J.: Conditions de stabilite et postulat de drucker. In: Kravtencko, J., Sirieys, P.M. (eds) Rheology and soil mechanics, Springer, Berlin (1966)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Alyavuz, B., Gültop, T. Weak shock waves and shear bands in compressible, inextensible thermoelastic solids. Arch Appl Mech 79, 1145–1161 (2009). https://doi.org/10.1007/s00419-009-0298-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-009-0298-x