Summary
This paper studies the mechanism of shock formation from generalised simple waves in the case of an isotropic incompressible hyperelastic material. Fast and slow waves are identified and it is established that slow simple waves cannot tend to shocks on the wavefront. The time and position of occurrence of shocks when they do form is investigated with the aid of generalised Riemann invariants and a circularly polarised slow wave is identified. The special case of plane shear waves is commented upon.
Zusammenfassung
Diese Arbeit beschäftigt sich mit dem Mechanismus der Ausbildung von Stoßfronten aus verallgemeinerten einfachen Wellen im Fall eines isotropen inkompressiblen hyperelastischen Werkstoffes. Schnelle und langsame Wellen werden identifiziert und es wird festgestellt, daß langsame einfache Wellen nicht zur Ausbildung von Stoßfronten an der Wellenfront führen können. Zeit und Ort des Auftretens von Stoßfronten, sofern sie sich bilden, werden mit Hilfe von verallgemeinerten Riemannschen Invarianten untersucht und eine zirkular-polarisierte langsame Welle wird identifiziert. Der Spezialfall ebener Scherwellen wird diskutiert.
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References
Lax, P. D.: The Initial Value Problem for Nonlinear Hyperbolic Equations in Two Independent Variables. Ann. Math. Studies33, 211–229 (1954).
Courant, R., andD. Hilbert: Methods of Mathematical Physics, Vol. II, pp. 430–433. New York: Wiley, 1962.
Jeffrey, A.: The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations. Arch. Rat. Mech. Anal.14, 27–37 (1963).
Green, A. E.: A Note on Wave Propagation in Initially Deformed Bodies. J. Mech. Phys. Solids11, 119–126 (1963).
Jeffrey, A., andT. Taniuti: Nonlinear Wave Propagation, Section 2.2 and 2.4. New York-London: Academic Press. 1964.
Chu, B. T.: Finite Amplitude Waves in Incompressible Perfectly Elastic Materials. J. Mech. Phys. Solids12, 45–57 (1964).
Bland, D. R.: Dilatational Waves and Shocks in Large Displacement Isentropic Dynamic Elasticity. J. Mech. Phys. Solids12, 245–267 (1964).
Jeffrey, A.: The Formation on Magneto-acoustic Shocks. J. Math. Anal. and Appl.15, 139–150 (1965).
Bland, D. R.: Plane Isentropic Large Displacement Simple Waves in a Compressible Elastic Solid. J. Appl. Maths. and Phys. (ZAMP)16, 752–769 (1965).
Collins, W. D.: One Dimensional Nonlinear Wave Propagation in Incompressible Elastic Materials. Q. J. Mech. and Appl. Maths.19, 259–328 (1966).
Howard, I. C.: Finite Simple Waves in a Compressible Transversely Isotropic Elastic Solid. Q. J. Mech. and Appl. Maths.19, 328–341 (1966).
Collins, W. D.: The Propagation and Interaction of One Dimensional Nonlinear Waves in an Incompressible Isotropic Elastic Half-Space. Q. J. Mech. and Appl. Maths.20, 429–452 (1967).
Jeffrey, A.: The Propagation of Weak Discontinuities on Quasilinear Hyperbolic Systems with Discontinuous Coefficients, Part I—Fundamental Theory. J. Applicable Anal.3, 79–100 (1973).
Jeffrey, A.: The Propagation of Weak Discontinuities in Quasilinear Hyperbolic Systems with Discontinuous Coefficients, Part II—Special Cases and Application. J. Applicable Anal.3 (to appear).
Jeffrey, A., andS. Tin: Waves Over Obstacles on a Shallow Seabed. Proc. R. S. E.A 71, 181–192 (1973).
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Jeffrey, A., Teymur, M. Formation of shock waves in hyperelastic solids. Acta Mechanica 20, 133–149 (1974). https://doi.org/10.1007/BF01374966
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DOI: https://doi.org/10.1007/BF01374966