Wave Propagation in Two-Temperature Porothermoelasticity

Abstract

In the present paper, the governing equations for two-temperature generalized porothermoelasticity are formulated in accordance with Green and Naghdi theory of thermoelasticity without energy dissipation. Two-dimensional plane wave solution of these governing equations indicates the existence of one shear vertical and four coupled longitudinal waves in porothermoelastic medium. A problem on reflection of longitudinal and shear waves is considered at a thermally insulated and stress-free surface of a generalized porothermoelastic solid half-space. The appropriate potentials for incident and reflected waves satisfy the required boundary conditions at free surface of the half-space and a non-homogeneous system of five equations in reflection coefficients is obtained. The expressions for energy ratios of reflected waves are obtained for incidence of both longitudinal and shear waves. The numerical results are obtained for values of porosity lying between 0.01 and 0.5, which are suitable for most of rocks present in the earth’s crust. The experimental data of kerosene-saturated sandstone are selected for numerical computations to observe the effects of two-temperature parameters and porosity on the energy ratios of reflected waves.

Appendices

Appendices

1.1 Appendix 1

The expressions for A, B, C, D and E have been obtained as

$$\begin{aligned} A &= {} ({\rho }_{11} {\rho }_{22} - {\rho }^2_{12})(F_{11} F_{22} - F_{12} F_{21}),\\ B &= {} (F_{12} F_{21} - F_{11} F_{22}) [\rho _{22}(\lambda + 2 \mu ) + \rho _{11} R - 2 \rho _{12} Q] \\&+ (K^{*s} F_{22} + K^{*f} F_{11} ) (\rho _{12}^2 - \rho _{11} \rho _{22})\\&+ T_0 [F_{11} (2 \rho _{12} R_{12} R_{22} - \rho _{11} R^2_{22} - \rho _{22} R^2_{12}) + F_{22} (2 \rho _{12} R_{11} R_{21} - \rho _{22} R^2_{11} - \rho _{11} R^2_{21}) \\&+ (F_{12} + F_{21})(\rho _{22} R_{11} R_{22} + \rho _{11} R_{21} R_{22} - \rho _{12} R_{12} R_{21} - \rho _{12} R_{11} R_{22})],\\ C &= {} (F_{11} F_{22} - F_{12} F_{21}) [(\lambda + 2 \mu ) R - Q^2] (K^{*s} F_{22} + K^{*f} F_{11})[(\lambda + 2 \mu ) \rho _{22} \\&+ \rho _{11} R - 2 \rho _{12} Q] \\&+ K^{*s} T_0 [\rho _{11} R^2_{22} + \rho _{22} R^2_{12} - 2 \rho _{12} R_{12} R_{22}] + K^{*f} T_0 [\rho _{11} R^2_{21} + \rho _{22} R^2_{11} - 2 \rho _{12} R_{11} R_{21}] \\&+ K^{*s} K^{*f} ({\rho }_{11} {\rho }_{22} - {{\rho }_{12}}^2) + T^2_0 (R^2_{11} R^2_{22} + R^2_{12} R^2_{21} - 2 R_{11} R_{21} R_{12} R_{22}) \\&+ T_0 [(\lambda + 2 \mu ) \{R^2_{21} F_{22} + R^2_{22} F_{11} - R_{21} R_{22} (F_{12} + F_{21})\} \\&+ Q \{(F_{12} + F_{21}) (R_{11} R_{22} + R_{12} R_{21}) - 2 R_{11} R_{21} F_{22} - 2 R_{12} R_{22} F_{11}\} \\&+ R \{ R^2_{11} F_{22} + R^2_{12} F_{11} - R_{11} R_{12} (F_{12} + F_{21})\}], \\ D &= {} -K^{*s} K^{*f} [\rho _{22}(\lambda + 2 \mu ) + \rho _{11} R - 2 \rho _{12} Q] + K^{*s} [Q^2 F_{22} + 2Q R_{12} R_{22} - R R^2_{12} \\&- (\lambda + 2 \mu ) (T_0 R^2_{22} + R F_{22})] + K^{*f} [Q^2 F_{11} + 2Q R_{11} R_{21} - R R^2_{11} \\&- (\lambda + 2 \mu ) (T_0 R^2_{21} + R F_{11})],\\ E& = {} K^{*s} K^{*f}[R(\lambda + 2 \mu ) - Q^2], \end{aligned}$$

where $K^{*s} = \frac{K^s}{1+a^{*s}k^2}$ and $K^{*s} = \frac{K^f}{1+a^{*f}k^2}$.

1.2 Appendix 2

The expressions for ${\eta }_i, {\zeta }_i$ and ${\xi }_i$ (i = 1, 2,…, 4) are derived as

$$\begin{aligned} {\eta }_i= & {} -\bigg [\displaystyle \frac{\Delta T_0 R_{11} {V_{i}}^2 +(K^{*s}-{V_{i}}^2F_{11})(R_{22}W_1-R_{12}W_2)+F_{12}{V_i}^2(R_{21}W_1-R_{11}W_2)}{\Delta T_0 R_{21} {V_{i}}^2 +(K^{*s}-{V_{i}}^2 F_{11})(R_{22}W_2-R_{12}W_3)+F_{12}{V_i}^2(R_{21}W_2-R_{11}W_3)}\bigg ], \end{aligned}$$

(34)

$$\begin{aligned} {\zeta }_i= & {} \frac{k^2_i}{(1+a^{*s}{k^2_i})}\bigg [\displaystyle \frac{R_{22}W_1-R_{12} W_2+ \eta _i (R_{22} W_2-R_{12}W_3)}{\Delta }\bigg ], \end{aligned}$$

(35)

$$\begin{aligned} {\xi }_i= & {} -\frac{k^2_i}{(1+a^{*f}{k^2_i})}\bigg [\displaystyle \frac{R_{21}W_1-R_{11} W_2+ \eta _i (R_{21} W_2-R_{11}W_3)}{\Delta }\bigg ], \end{aligned}$$

(36)

where

$$\begin{aligned} \Delta = R_{12} R_{21} - R_{11} R_{22}, ~W_1 = (\lambda + 2\mu ) - {\rho }_{11} {V_i}^2,~W_2 = Q - {\rho }_{12} {V_i}^2,~W_3 = R - {\rho }_{22} {V_i}^2. \end{aligned}$$