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This paper considers the propagation of plane longitudinal waves in a liquidsaturated
porous medium with allowance for the nonlinear relationship between deformations
and displacements of the solid phase. This porous liquid-saturated medium is examined herein
within the framework of the classical Biot's theory. It is shown that a mathematical model
allowing for a geometric nonlinearity may be reduced to a system of evolutionary equations
with respect to displacements of the medium skeleton and liquid in pores. The system of
evolutionary equations, in its turn, depending on the availability of viscosity, is reduced to a
simple wave equation or the generalized Burgers equation. The solution of the Riemann
equation is obtained for a bell-shaped initial profile. The solution for the generalized Burgers
equation has been found in the form of a stationary shock wave. The relationship between the
amplitude and width of the shock wave front is established.
Keywords: porous medium (Biot's medium); geometrical nonlinearity; evolutionary equation; the Riemann wave; stationary shock wave. |
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