Abstract
For infinite perfectly elastic Mooney materials, nonlinear plane waves are examined in both two and three dimensions. In two dimensions, longitudinal and shear plane waves are examined, while in three dimensions, longitudinal and torsional plane waves are considered. These exact dynamic deformations, applying to the incompressible perfectly elastic Mooney material, can be viewed as extensions of the corresponding static deformations first derived by Adkins [1] and Klingbeil and Shield [2]. Furthermore, the Mooney strain-energy function is the most general material admitting nontrivial dynamic deformations of this type. For two dimensions the determination of plane wave solutions reduces to elementary mathematical analysis, while in three dimensions an integral of the governing system of highly nonlinear ordinary differential equations is determined. In the latter case, solutions corresponding to particular parameter values are shown graphically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.E. Adkins, A reciprocal property of the finite plane strain equations. J. Mech. Phys. Solids 6 (1985) 267–275.
W.W. Klingbeil and R.T. Shield, Large-deformation analyses of bonded elastic mounts. Z. angew. Math. Phys. 17 (1966) 281–305.
J.M. Hill and T. Myers, The combined compression and shear of a rectangular rubber block. Z. angew. Math. Phys. 43 (1992) 911–923.
J.M. Hill and A.I. Lee, Combined compression and torsion of circular cylindrical pads of rubber. J. Mech. Phys. Solids 37 (1989) 175–190.
C. Truesdell and R.A. Toupin, The Classical Field Theories, ed. S. Flugge, Encyclopedia of Physics, Vol. III/1. Springer, Berlin (1960).
C. Truesdell and W. Noll, The Non-linear Field Theories of Mechanics, ed. S. Flugge, Encyclopedia of Physics, Vol. III/3. Springer, Berlin (1965).
J.A. Knowles, Large amplitude oscillations of a tube of incompressible material. Quart. Appl. Math. 18 (1960) 71–77.
J.A. Knowles, On a class of oscillations in the finite-deformation theory of elasticity. J. Appl. Mech. 29 (1962) 283–286.
C.-C. Wang and C. Truesdell, Introduction to Rational Elasticity, ed. W.O. Williams, Mechanics of Continua. Noordoff Internat. Publishing, Leyden (1973).
H.-H. Dai and Q. Bi, Exact solutions for the large axially symmetric deformations of a neo-Hookean rod subjected to static loads. Quart. J. Mech. Appl. Math. 54 (2001) 39–56.
H.-H. Dai, Nonlinear dispersive waves in a circular rod composed of a Mooney-Rivlin material. In: Y. Fu and R.W. Ogden (eds), Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series. Cambridge Univ. Press, Cambridge (2001) pp. 392–432.
H.-H. Dai and X.-H. Zhao, Nonlinear travelling waves in a rod composed of a modified Mooney-Rivlin material. I Bifurcation of critical points and the non-singular case. Proc. Roy. Soc. London A 455 (1999) 3845–3874.
H.-H. Dai and Y. Huo, Solitary shock waves and other travelling waves in a general compressible hyperelastic rod. Proc. Roy. Soc. London A 456 (2000) 331–363.
J.M. Hill, Partial solutions of finite elasticity-plane deformations. Z. angew. Math. Phys. 24 (1973) 401–408.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hill, J.M., Dai, HH. Nonlinear Plane Waves in Finite Deformable Infinite Mooney Elastic Materials. Journal of Elasticity 67, 71–80 (2002). https://doi.org/10.1023/A:1023902332681
Issue Date:
DOI: https://doi.org/10.1023/A:1023902332681