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Kinks versus Shocks

  • Conference paper
Shock Induced Transitions and Phase Structures in General Media

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 52))

Abstract

Localized phase transitions as well as shock waves can often be modeled by material discontinuities satisfying Rankine-Hugoniot (RH) jump conditions. The use of Maxwell, Gibbs-Thompson, Hertz-Knudsen, and similar (supplementary to RH) relations in the theory of dynamic phase changes suggest that the classical system of jump conditions is at least incomplete in the case of phase transitions. While the propagation of a shock wave is completely determined by the conservations laws, the boundary conditions of the problem and the condition that the entropy increases in the process, the same is not true for the propagation of phase boundaries. Additional condition must be added to the RH conditions in order to provide sufficient data for the unique determination of the transformation process. The necessity was tacitly assumed by those who attacked the calculation of the phase boundary velocity without even trying to determine this parameter from the conservation laws and boundary conditions alone.

In order to be able to point out the contrast between shock waves and phase boundaries we treat both of them on the basis of the same assumption that the process takes place over a zone of finite width and consider a rather general model of the internal structure of the interface with special emphasis on the interplay between dispersion and dissipation effects. Our extended model of the continuum, capable of describing a “thick” interface, incorporates a weak form of nonlocality together with a number of dissipative mechanisms. Analysis of a model-type solution of the structure problem clarifies the distinction between supersonic (shock) and subsonic(kink) discontinuities and provides explicit examples of additional jump relations in the case of kinks (which simulate subsonic phase boundaries).

It is emphasized, that an extended system of jump relations for kinks may depend on the ratios of internal scales of length introduced by a more detailed description. An original theory, which provides discontinuous solutions, must therefore be complemented by these nondimensional parameters, even though internal scales by themselves are considered to be zero in this theory.

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  1. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, 55455, MN, USA

    L. Truskinovsky

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  1. Numerical Mathematics Division # 1534, Sandia National Labs, P.O. Box 5800, 87184, Albuquerque, NM, USA

    J. E. Dunn

  2. Department of Aerospace Engineering and Mechanics, University of Minnesota, 55455, Minneapolis, MN, USA

    Roger Fosdick

  3. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    Marshall Slemrod

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Truskinovsky, L. (1993). Kinks versus Shocks. In: Dunn, J.E., Fosdick, R., Slemrod, M. (eds) Shock Induced Transitions and Phase Structures in General Media. The IMA Volumes in Mathematics and its Applications, vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8348-2_11

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