Riemann and Shock Waves in a Porous Liquid-Saturated Geometrically Nonlinear Medium
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 5, September, 2020
RIEMANN AND SHOCK WAVES IN A POROUS LIQUID-SATURATED GEOMETRICALLY NONLINEAR MEDIUM V. I. Erofeev and A. V. Leont′eva
UDC 534.1
Within the framework of the classical Biot theory, the propagation of plane longitudinal waves in a porous liquidsaturated medium is considered with account for the nonlinear connection between deformations and displacements of solid phase. It is shown that the mathematical model accounting for the geometrical nonlinearity of the medium skeleton can be reduced to a system of evolution equations for the displacements of the skeleton of medium and of the liquid in pores. The system of evolution equations, in turn, depending on the presence of viscosity, is reduced to the equation of a simple wave or to the equation externally resembling the Burgers equation. The solution of the Riemann equation is obtained for a bell-shaped initial profile; the characteristic wave breaking is shown. In the second case, the solution is found in the form of a stationary shock wave having the profile of a nonsymmetric kink. The relationship between the amplitude and width of the shock wave front has been established. It is noted that the behavior of nonlinear waves in such media differs from the standard one typical of dissipative nondispersing media, in which the propagation of waves is described by the classical Burgers equation. Keywords: porous medium (Biot medium), geometrical nonlinearity, evolution equation, Riemann wave, generalized Burgers equation, stationary shock wave. The mathematical models of deformed porous materials, both presented in the classical works of M. A. Biot [1, 2], Ya. I. Frenkel [3], L. Ya. Kosachevskii [4] and subsequently in modified forms in [5–30], are widely used in studying the processes proceeding in the geophysics and mechanics of natural and artificial composite materials. The authors of the majority of works in their investigations restrict themselves to the linear theory of pore-toughness despite the fact that, as shown experimentally in [31], the nonlinear effects in liquid-saturated porous media are substantial and are also of definite interest. Moreover, in nature, technique, and technologies porous liquid-saturated materials are frequently encountered that contain cavities filled with a liquid and distributed chaotically. Under certain conditions, the cavities oscillate under the action of an elastic wave and exert a substantial influence on the laws governing the propagation of waves. It is shown in works [32– 34] that along with the geometric nonlinearity (nonlinear connection between deformations and displacements) and physical nonlinearity (nonlinear connection between stresses and deformations) it is important to account for the cavity nonlinearity. Works [35–37] consider the propagation of plane longitudinal waves in a porous liquid-saturated medium with cavities. The behavior of linear and nonlinear waves in cavity-porous media is studied. It is shown that in such media three l
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