Chapter 18 Dynamics and Shock Waves in Media with Second Order Phase Transformations Pavel Grinfeld and Michael Grinfeld Abstract We study free oscillations and the weak shock waves in the Ehrenfest liquids, i.e., in the liquids undergoing second order phase transformations. Also, we discuss the second order phase transformations in crystalline solids. Keywords Free oscillations • Thermodynamics • Shock waves • Phase transformations • Hugoniot adiabat 18.1 Introduction Studies of the first order phase transformations “liquid–vapor” last for centuries. Studies of the second order phase transformation have started much later. They have been mostly triggered by the so-called λ-transformations in helium, discovered by Kammerling–Honnes. The first and the simplest model of λ-transformations was suggested by Paul Ehrenfest in the first third of the twentieth century [1–3]. To that end, Ehrenfest introduced the notion of the second order phase transformation. Also, Ehrenfest found the key relationship for the second order phase transformation. The Ehrenfest model of the second order phase transformation has been modified later by Landau and his Soviet school but the key relationship survived this modification [4]. Therefore, it deserves to be coined as the Ehrenfest relationship. The characteristic feature of the second order phase transformation is the absence of the latent heat of the transformation. At the same time, the second order phase transformations are accompanied by considerable jumps of the heat capacity, compressibility, and thermal expansion coefficients. In this paper, we first illustrate some peculiarities in the dynamics of the liquid systems with discontinuous compressibilities. In another context, dynamics of an elastic rod with bi-linear Young modulus has been analyzed in mathematical paper of Maslov and Mosolov [5]. Even with this simplest generalization the behavior of the rod and the types of wave-fronts in it becomes very complex and cumbersome as compared with linear case. However, the Riemann problem for this case still can be addressed analytically. Our interest, however, is different as compared with [5]. We analyze the problem of infinitesimal oscillations of heavy fluid with bi-linear compressibilities, following the paper [6]. In this case, the dispersion equation can be expressed explicitly in transcendental functions. We proceed with reminding the Ehrenfest model of liquids, undergoing second order phase transformations, and then study of shock waves in those liquids. P. Grinfeld Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA e-mail: pg77@drexel.edu M. Grinfeld (*) The U.S. Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD 21005, USA e-mail: michael.greenfield4.civ@mail.mil B. Song et al. (eds.), Dynamic Behavior of Materials, Volume 1: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06995-1_18, #The Society for Experimental Mechanics, Inc. 2015 113
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