Rayleigh-Type Surface Waves in a Swelling Porous Half-Space

Abstract

The present work deals with the propagation of Rayleigh-type surface waves in a swelling porous elastic half-space consisting of three phases, namely solid matrix, liquid (viscous) and gas (inviscid). Using Eringen’s theory of swelling porous media, the governing equations are first solved by potential method. Frequency equation of Rayleigh-type waves has been derived, which is found to be irrational due to the presence of radicals in it. This irrational equation has been rationalized into a polynomial, which is then solved numerically for a specific porous model consisting of sandstone, water (viscous) and carbon dioxide as solid, liquid and gas phases, respectively. The nature of Rayleigh-type surface waves in the considered swelling porous medium is found to be inhomogeneous. Two modes of Rayleigh-type surface waves are noticed: One of them is the counterpart of the classical Rayleigh wave, while the second mode of Rayleigh-type surface waves arises due to the presence of either liquid or gas phases of the swelling porous medium. The variation of phase speeds and the corresponding attenuations of Rayleigh-type surface waves are depicted graphically against frequency parameter for the selected model. In the considered model, the swelling parameter has negligible effect on the propagation speeds of Rayleigh-type surface modes. It is also observed that in the absence of swelling, there still exist two modes of Rayleigh-type waves. The effect of the viscosity of the liquid constituent present in the pores is also examined on the phase speeds and attenuations. The results of Gales (Eur J Mech A Solids, 23:345–357 2004) for the cases of fluid saturation alone and gas saturation alone have also been deduced analytically as special cases from the present formulation.

Appendices

Appendix 1

The explicit expressions of the various coefficients are as:

$$\begin{aligned}&A=a_{12}A_{1}+a_{13}A_{2}+a_{11}A_{3},B=-\chi _{f}A_{1}-\chi _{g}A_{2}+\varOmega _{s}A_{3}+a_{12}B_{1}+a_{13}B_{2}+a_{11}B_{3},\\&C=-\chi _{f}B_{1}-\chi _{g}B_{2}+\varOmega _{s}B_{3}+a_{12}C_{1}+a_{13}C_{2}+a_{11}C_{3},D=-\chi _{f}C_{1}-\chi _{g}C_{2}+\varOmega _{s}C_{3},\\&{\bar{B}}=\bar{a}_{11}{\bar{B}}_3,\qquad {\bar{C}}=-\chi _{g}{\bar{B}}_{2}+\varOmega _{s}{\bar{B}}_{3}+{\bar{a}}_{11}C_{3},\\&A_{1}=a_{13}a_{23}-a_{12}a_{33},B_{1}=(a_{13}\chi _{fg}+a_{33}\chi _{f})-(a_{23}\chi _{g}+a_{12}\varOmega _{g}),\\&C_{1}=\varOmega _{g}\chi _{f}-\chi _{g}\chi _{fg},\\&A_{2}=a_{12}a_{23}-a_{13}a_{22},B_{2}=(a_{12}\chi _{fg}+a_{22}\chi _{g})-(a_{23}\chi _{f}+a_{13}\varOmega _{f}),\\&C_{2}=\varOmega _{f}\chi _{g}-\chi _{f}\chi _{fg},\\&A_{3}=a_{22}a_{33}-a^2_{23},\quad B_{3}=a_{33}\varOmega _{f}+a_{22}\varOmega _{g}-2a_{23}\chi _{fg},\quad C_{3}=\varOmega _{f}\varOmega _{g}-\chi ^2_{fg},\\&{\bar{B}}_{2}={\bar{a}}_{22}\chi _{g},{\bar{B}}_{3}={\bar{a}}_{22}\varOmega _{g},\\&a_{11}=\lambda +2\mu ,\quad a_{12}=-\sigma ^{f},\quad a_{13}=-\sigma ^{g},\quad a_{22}=-[\sigma ^{ff}+\iota \omega (\lambda _{\nu }+2\mu _{\nu })],\\&a_{23}=-\sigma ^{fg},\\&a_{33}=-\sigma ^{gg},{\bar{a}}_{11}=\mu ,{\bar{a}}_{22}=-\iota \omega \mu _{\nu },\chi _{f}=\chi _{ff}+\chi _{fg},\chi _{g}=\chi _{gg}+\chi _{fg},\\&\chi _{ff}=\frac{\iota }{\omega }\xi ^{ff},\\&\chi _{gg}=\frac{\iota }{\omega }\xi ^{gg},\chi _{fg}=\frac{\iota }{\omega }\xi ^{fg},\varOmega _{s}=\rho ^{s}_{0}+\chi _{f}+\chi _{g},\varOmega _{f}=\rho ^{f}_{0}+\chi _{ff},\varOmega _{g}=\rho ^{g}_{0}+\chi _{gg}. \end{aligned}$$

Appendix 2

The explicit expressions of the various coefficients used in the determinantal equation of the Rayleigh wave are as:

$$\begin{aligned} r_{1i}= & {} 2({\bar{a}}_{11}+f_i{\bar{a}}_{22}),r_{1J}={\bar{a}}_{11}+{\bar{f}}_j{\bar{a}}_{22}\,,r_{2i}=L_i+M_i+N_i~,~~r_{3i}=1-f_i\,,\\ r_{3J}= & {} 1-{\bar{f}}_j\,,r_{5i}=1-g_i\,,r_{5J}=1-{\bar{g}}_j\,,\text{ where }\,i=1,2,3;\,j=1,2;\,J = j+3. \end{aligned}$$

The explicit expressions of the various coefficients used in the secular equation of the Rayleigh wave are as:

$$\begin{aligned} d_0= & {} c_0^2 - c_1^2 b_{\alpha 3}^2 b_{\beta 1}^2 - c_2^2 b_{\alpha 3}^2 b_{\beta 2}^2 + c_3^2 b_{\beta 1}^2 b_{\beta 2}^2\,, d_1 = 2 c_0 c_3 - 2 c_1 c_2 b_{\alpha 3}^2\\ c_0= & {} b_0^2 - b_1^2 b_{\alpha 2}^2 b_{\alpha 3}^2 - b_2^2 b_{\alpha 2}^2 b_{\beta 1}^2 + b_3^2 b_{\alpha 3}^2 b_{\beta 1}^2 - b_4^2 b_{\alpha 2}^2 b_{\beta 2}^2 + b_5^2 b_{\alpha 3}^2 b_{\beta 2}^2\\&+\,b_6^2 b_{\beta 1}^2 b_{\beta 2}^2 - b_7^2 b_{\alpha 2}^2 b_{\alpha 3}^2 b_{\beta 1}^2 b_{\beta 2}^2\,,\\ c_1= & {} 2 \left( b_0 b_3 - b_1 b_2 b_{\alpha 2}^2 + b_5 b_6 b_{\beta 2}^2 - b_4 b_7 b_{\alpha 2}^2 b_{\beta 2}^2\right) \,,\\ c_2= & {} 2\left( b_0 b_5 - b_1 b_4 b_{\alpha 2}^2 + b_3 b_6 b_{\beta 1}^2 - b_2 b_7 b_{\alpha 2}^2 b_{\beta 1}^2\right) \,,\\ c_3= & {} 2\left( b_0 b_6 - b_2 b_4 b_{\alpha 2}^2 + b_3 b_5 b_{\alpha 3}^2 - b_1 b_7 b_{\alpha 2}^2 b_{\alpha 3}^2\right) \,,\\ b_0= & {} -a_1^2 b_{\alpha 1}^2 + a_2^2 b_{\alpha 2}^2 + a_3^2 b_{\alpha 3}^2 - a_4^2 b_{\alpha 1}^2 b_{\alpha 2}^2 b_{\beta 1}^2 - a_5^2 b_{\alpha 1}^2 b_{\alpha 3}^2 b_{\beta 1}^2 + a_6^2 b_{\alpha 2}^2 b_{\alpha 3}^2 b_{\beta 1}^2 \\&- a_7^2 b_{\alpha 1}^2 b_{\alpha 2}^2 b_{\beta 2}^2 - a_8^2 b_{\alpha 1}^2 b_{\alpha 3}^2 b_{\beta 2}^2 + a_9^2 b_{\alpha 2}^2 b_{\alpha 3}^2 b_{\beta 2}^2 - a_{10}^2 b_{\alpha 1}^2 b_{\alpha 2}^2 b_{\alpha 3}^2 b_{\beta 1}^2 b_{\beta 2}^2\,,\\ b_1= & {} 2\left( a_2 a_3 - a_4 a_5 b_{\alpha 1}^2 b_{\beta 1}^2 - a_7 a_8 b_{\alpha 1}^2 b_{\beta 2}^2\right) \,,\\ b_2= & {} -2\left( a_1 a_4 b_{\alpha 1}^2 - a_3 a_6 b_{\alpha 3}^2 + a_{10} a_8 b_{\alpha 1}^2 b_{\alpha 3}^2 b_{\beta 2}^2\right) \,,\\ b_3= & {} -2\left( a_1 a_5 b_{\alpha 1}^2 - a_2 a_6 b_{\alpha 2}^2 + a_{10} a_7 b_{\alpha 1}^2 b_{\alpha 2}^2 b_{\beta 2}^2\right) \,,\\ b_4= & {} -2\left( a_1 a_7 b_{\alpha 1}^2 - a_3 a_9 b_{\alpha 3}^2 + a_{10} a_5 b_{\alpha 1}^2 b_{\alpha 3}^2 b_{\beta 1}^2\right) \,, \\ \end{aligned}$$

$$\begin{aligned} b_5= & {} -2\left( a_1 a_8 b_{\alpha 1}^2 - a_2 a_9 b_{\alpha 2}^2 + a_{10} a_4 b_{\alpha 1}^2 b_{\alpha 2}^2 b_{\beta 1}^2\right) \,,\\ b_6= & {} -2\left( a_4 a_7 b_{\alpha 1}^2 b_{\alpha 2}^2 + a_5 a_8 b_{\alpha 1}^2 b_{\alpha 3}^2 - a_6 a_9 b_{\alpha 2}^2 b_{\alpha 3}^2\right) \,,\\ b_7= & {} -2(a_1 a_{10} + a_5 a_7 + a_4 a_8) b_{\alpha 1}^2\,,\\ a_1= & {} (f_{23} f_{32} - f_{22} f_{33}) \left( f_{15} f_{44} f_{51} + f_{11}^2 f_{54} - f_{15} f_{41} f_{54} + f_{14} f_{41} f_{55}\right. \\&\left. -f_{11} (f_{14} f_{51} + f_{44} f_{55})\right) ,\\ a_2= & {} (f_{23} f_{31} - f_{21} f_{33}) \left( -f_{15} f_{44} + f_{15} f_{42} f_{54} + f_{11} (f_{14} - f_{12} f_{54})\right. \\&\left. - f_{14} f_{42}f_{55}+ f_{12} f_{44} f_{55}\right) ,\\ a_3= & {} (f_{22} f_{31} - f_{21} f_{32}) \left( -f_{11} f_{14} + f_{15} f_{44} + f_{11} f_{13} f_{54} - f_{15} f_{43} f_{54} \right. \\&\left. + f_{14} f_{43}f_{55} - f_{13} f_{44} f_{55}\right) ,\\ a_4= & {} (f_{24} f_{33} - f_{23} f_{34}) \left( f_{11}^2 - f_{15} f_{41} + f_{15} f_{42} f_{51} + f_{12} f_{41} f_{55}\right. \\&\left. - f_{11} (f_{12} f_{51}+ f_{42} f_{55})\right) ,\\ a_5= & {} -(f_{24} f_{32} - f_{22} f_{34}) \left( f_{11}^2 - f_{15} f_{41} + f_{15} f_{43} f_{51} + f_{13} f_{41} f_{55}\right. \\&\left. - f_{11} (f_{13} f_{51} + f_{43} f_{55})\right) ,\\ a_6= & {} (f_{24} f_{31} - f_{21} f_{34}) \left( f_{11} (f_{12} - f_{13}) - f_{15} f_{42} + f_{15} f_{43} + f_{13} f_{42} f_{55} - f_{12} f_{43} f_{55}\right) ,\\ a_7= & {} (f_{23} - f_{25} f_{33}) \left( -f_{14} f_{41} + f_{11} f_{44} + f_{14} f_{42} f_{51} - f_{12} f_{44} f_{51} \right. \\&\left. + f_{12} f_{41} f_{54} - f_{11} f_{42} f_{54}\right) ,\\ a_8= & {} (f_{22} - f_{25} f_{32}) \left( -f_{11} f_{44} + f_{13} f_{44} f_{51} + f_{14} (f_{41} - f_{43} f_{51}) - f_{13} f_{41} f_{54} + f_{11} f_{43} f_{54}\right) ,\\ a_9= & {} -(f_{21} - f_{25} f_{31}) \left( f_{14} (f_{42} - f_{43}) - f_{12} f_{44} + f_{13} f_{44} - f_{13} f_{42} f_{54} + f_{12} f_{43} f_{54}\right) ,\\ a_{10}= & {} (f_{24} - f_{25} f_{34}) \left( -f_{12} f_{41} + f_{13} f_{41} + f_{11} f_{42} - f_{11} f_{43} - f_{13} f_{42} f_{51} + f_{12} f_{43} f_{51}\right) , \end{aligned}$$

where

$$\begin{aligned}&f_{11}=k_r,f_{12} = \frac{r_{12}}{r_{11}r_{52}}k_r,f_{13}= \frac{r_{13}}{r_{11}r_{53}}k_r,f_{14} = \frac{r_{14}}{r_{11}}\left( 2k_r^2-k_{\beta 1}^2\right) ,\\&f_{15}=\frac{r_{15}}{r_{11}r_{35}}\left( 2k_r^2-k_{\beta 2}^2\right) ,\\&f_{21}=k_r^2-\frac{r_{21}}{r_{11}}k_{\alpha 1}^2,f_{22} = \frac{1}{r_{11}r_{52}}\left( r_{12}k_r^2-r_{22}k_{\alpha 2}^2\right) ,f_{23} = \frac{1}{r_{11}r_{53}}\left( r_{13}k_r^2-r_{23}k_{\alpha 3}^2\right) ,\\&f_{24} = \frac{2r_{14}}{r_{11}}k_r,\\&f_{25}=\frac{2r_{15}}{r_{11}r_{35}}k_r,f_{31} = r_{31}k_r,f_{32} = \frac{r_{32}}{r_{52}}k_r,f_{33} = \frac{r_{33}}{r_{53}}k_r,f_{34} = r_{34},f_{41}=r_{31},\\&f_{42}=\frac{r_{32}}{r_{52}},f_{43} = \frac{r_{33}}{r_{53}},f_{44} = r_{34}k_r,f_{51} = r_{51},f_{54} = r_{54}k_r,f_{55} = \frac{r_{55}}{r_{35}}k_r. \end{aligned}$$