Summary
The nonlinear scalar constitutive equations of gases lead to a change in sound speed from point to point as would be found in linear inhomogeneous (and time dependent) media. The nonlinear tensor constitutive equations of solids introduce the additional local effect of solution dependent anisotropy. The speed of a wave passing through a point changes with propagation direction and its rays are inclined to the front. It is an open question wether the widely used operator splitting techniques achieve a dimensional splitting with physically reasonable results for these multi-dimensional problems.
May be this is the main reason why the theoretical and numerical investigations of multi-dimensional wave propagation in nonlinear solids are so far behind gas dynamics. We hope to promote the subject a little by a discussion of some fundamental aspects of the solution of the equations of nonlinear elastodynamics. We use methods of characteristics because they only integrate mathematically exact equations which have a direct physical interpretation.
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References
PAYTON, R.G.: Elastic Wave Propagation in Transversely Isotropic Media. Martinus Nijhoff, The Hague (1983).
JEFFREY, A.; TANIUTI, T.: Non-Linear Wave Propagation with Applications to Physics and Magnetohydrodynamics. Academic Press, New York (1964).
BALLMANN, J.; RAATSCHEN, H.J.; STAAT, M.: High Stress Intensities in Focussing Zones of Waves. In P. Ladevese (Ed.): Local Effects in the Analysis of Structures. Elsevier, Amsterdam (1985).
STAAT, M.: Nichtlineare Wellen in elastischen Scheiben. Dr.-Ing. Thesis, RWTH-Aachen (1987).
KRAWIETZ, A.: Materialtheorie, Springer, Berlin (1986).
STAAT, M.; BALLMANN, J.: Zur Problematik tensorieller Verallgemeinerungen einachsiger nichtlinearer Materialgesetze. To be published in ZAMM.
BRAUN, M.: Wave Propagation in Elastic Membranes. In U. Nigul, J. Engelbrecht (Ed.): Nonlinear Deformation Waves. Springer, Berlin (1983).
PAO, Y.-H.; SACHSE, W.; FUKUOA, H.: Acoustoelasticity and Ultrasonic Measurements of Residual Stresses. Physical Acoustics 17, Academic Press, New York (1984) Chap. 2.
PETROWSKY, I.G.: On the Diffusion of Waves and the Lacunas for Hyperbolic Equations. Rec. Math. (Math. Sbornic) 17 (1945) 289–370.
BURRIDGE, R.: Lacunas in Two-Dimensional Wave Propagation. Proc. Camb. Phil. Soc. 63 (1967) 819–825.
PAYTON, R.G.: Two Dimensional Wave Front Shape Induced in a Homogeneously Strained Elastic Body by a Point Perturbing Body Force. Arch. Rat. Mech. 32 (1969) 311–330 and
PAYTON, R.G.: Two Dimensional Wave Front Shape Induced in a Homogeneously Strained Elastic Body by a Point Perturbing Body Force. Arch. Rat. Mech. 35 (1969) 402–408.
MUSGRAVE, M.J.P.: Crystal Acoustics. Holden-Day, San Francisco (1970).
SAUERWEIN, H.: Anisotropic Waves in Elastoplastic Soils. Int. J. Engng. Sci. 5 (1967) 455–475.
VARLEY, E.; Cumberbatch E.: Non-linear Theory of Wave-front Propagation. J. Inst. Maths Applics, 1 (1965) 101–112.
ERINGEN, A.C.; SUHUBI, E.S., Elastodynamics I, Academic Press, New York (1974).
THOMAS, T.Y.: Plastic Flow and Fracture in Solids. Academic Press, New York (1961).
TRUESDELL, C.; TOUPIN, R.A.: The Classical Field Theories. In S. Flügge (Ed.): Handbuch der Physik III/1. Springer, Berlin (1960).
WANG, C.C.; TRUESDELL, C.: Introduction to Rational Elasticity. Nordhoff, Leyden (1973).
CLIFTON, R.J.: A Difference Method for Plane Problems in Dynamic Elasticity. Quart. of Appl. Math. 25 (1967) 97–116.
BUTLER, D.S.: The Numerical Solution of Hyperbolic Systems of Partial Differential Equations in Three Independent Variables. Proc. Roy. Soc, A 255 (1966) 232–252.
HAINES, D.W.; WILSON, W.D.: Strain-Energy Density Function for Rubberlike Materials, J.Mech. Phys. Solids 27 (1979) 331–343
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© 1989 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Staat, M., Ballmann, J. (1989). Fundamental Aspects of Numerical Methods for the Propagation of Multi-Dimensional Nonlinear Waves in Solids. In: Ballmann, J., Jeltsch, R. (eds) Nonlinear Hyperbolic Equations — Theory, Computation Methods, and Applications. Notes on Numerical Fluid Mechanics, vol 24. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-87869-4_56
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DOI: https://doi.org/10.1007/978-3-322-87869-4_56
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