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Vibrational analysis of Love waves in a viscoelastic composite multilayered structure

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Abstract

A unified procedure is presented to investigate the dispersion and damping behavior of Love waves in a double-layered structure. The double-layered structure is comprised of a viscoelastic fiber-reinforced medium (Medium I) embedded between a viscoelastic sandy layer (Medium II) and a viscoelastic porous semi-infinite medium (Medium III). The displacement vectors for the respective medium have been obtained using analytical techniques. Using the separation of variables method and admissible boundary conditions at the free surface and the interfaces, the complex frequency equation is obtained in closed form. The substantial impact of all material parameters involved in the considered model on the phase velocity as well as damped velocity has been observed numerically and shown by graphical illustrations. The outcomes from this analysis may serve as a dynamic tool in various fields including geophysics, geotechnical, and earthquake engineering.

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Acknowledgements

The authors convey their sincere thanks to DST-SERB for funding by the research project no. DST-SERB/2018-2019/599/AM entitled “Analytical modelling of surface waves in fiber-reinforced and micro-polar media”.

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  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Dharmendra Kumar, Santimoy Kundu, Raju Kumhar & Shishir Gupta

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Appendix

Appendix

$$\begin{aligned} \alpha _1= & {} (\mu _t +i \omega \mu _t^{\prime }) = \mu _t(1 + i \epsilon _2),\\ \alpha _2= & {} (\mu _l +i \omega \mu _l^{\prime }-\mu _t -i \omega \mu _t^{\prime })\\= & {} \mu _t\left( \frac{\mu _l}{\mu _t} + i \epsilon _1 -1 - i \epsilon _2\right) ,\\ D_1= & {} (2 i k a_1 + P_1 a_3) =X_1 + i X_2,\\ D_2= & {} ({P_1}^{2} + 4 {\phi _2}^{2}) = X_3 + i X_4,\\ D_3= & {} (P_1 \alpha _1 + a_3 \alpha _2 D_1 -2 \phi _3 {\overline{L}}) = Y_5 + i Y_6,\\ D_4= & {} (a_3 \alpha _2 (2 i k P_1 a_1 + a_3 D_2) = Y_7 + i Y_8,\\ D_5= & {} (a_3 \alpha _2 (a_3 \alpha _2 {D_1}^{2} + 4 {a_3}^{2} {\phi _2}^{2}) = Y_9 + i Y_{10}, \end{aligned}$$
$$\begin{aligned} \epsilon _1= & {} \omega \frac{\mu _l' }{{\mu _t }},\quad \epsilon _2 =\omega \frac{\mu _t' }{{\mu _t }},\\ \epsilon _3= & {} \omega \frac{{N}^{*}}{{N }}, \quad \epsilon _4 = \omega \frac{{L}^{*}}{{L }},\\ X_1= & {} (Y_1 a_3 - 2 k_1 a_1 \delta ), X_2 = (2 k_1 a_1 + a_3 y_2),\\ X_3= & {} ({Y_1}^2 - {Y_2}^2 + 4 {x_2}^2 - 4 {y_2}^2), X_4 = (2 Y_1 Y_2 + 8 x_2 y_2),\\ X_5= & {} (1+(\frac{\mu _l}{\mu _t}-1){a_1}^2)(1-{\delta }^2)-2 \delta (\epsilon _2+(\epsilon _1-\epsilon _2){a_1}^2),\\ X_6= & {} (\epsilon _2 + (\epsilon _1-\epsilon _2){a_1}^2)(1-{\delta }^2)+2 \delta \left( 1+\left( \frac{\mu _l}{\mu _t}-1\right) {a_1}^2\right) ,\\ X_7= & {} (Y_1 a_3 - 2 k_1 a_1 \delta ), X_8 = (2 k_1 a_1 + a_3 Y_2),\\ Y_1= & {} \frac{a_1 a_3 k_1 \left( (1 - \frac{\mu _l}{\mu _t}) \delta - (\epsilon _1-\epsilon _2)-{(\frac{\mu _l}{\mu _t} - 1)}^2 {a_3}^2 \delta +\left( \frac{\mu _l}{\mu _t}-1\right) \epsilon _2- \epsilon _2 \delta ( \epsilon _1 -\epsilon _2)-{( \epsilon _1 - \epsilon _2)}^2 {a_3}^2\delta \right) }{{\left( 1+(\frac{\mu _l}{\mu _t}-1){a_3}^2\right) }^2 + {\left( \epsilon _2 + (\epsilon _1 - \epsilon _2){a_3}^2\right) }^2},\\ Y_2= & {} \frac{a_1 a_3 k_1 \left( ( \frac{\mu _l}{\mu _t} - 1) - \delta (\epsilon _1 - \epsilon _2) +{( \frac{\mu _l}{\mu _t} - 1)}^2 {a_3}^2 + \epsilon _2 (\frac{\mu _l}{\mu _t} - 1) \delta + \epsilon _2 (\epsilon _1 - \epsilon _2) + {(\epsilon _1 - \epsilon _2)}^2 {a_3}^2\right) }{{{\left( 1 + (\frac{\mu _l}{\mu _t} - 1){a_3}^2\right) }^2 + {\left( \epsilon _2 + (\epsilon _1 - \epsilon _2){a_3}^2\right) }^2}},\\ Y_3= & {} \frac{\rho _2 {\omega }^2(1+(\frac{\mu _l}{\mu _t}-1){a_3}^2)-\mu _t {k_1}^2\left( (1+(\frac{\mu _l}{\mu _t}-1){a_3}^2) X_5 + (\epsilon _2+(\epsilon _1-\epsilon _2){a_3}^2) X_6 \right) }{\mu }_t{\left( 1+(\frac{\mu _l}{\mu _t}-1){a_3}^2\right) }^2\\&+{\left( \epsilon _2+(\epsilon _1-\epsilon _2){a_3}^2\right) }^2,\\ Y_4= & {} \frac{\left( \mu _t {k_1}^2 \left( X_5 \right) (\epsilon _2 +(\epsilon _1 - \epsilon _2){a_3}^2) - (1 + (\frac{\mu _l}{\mu _t}-1){a_3}^2)\left( X_6 \right) \right) -\rho _2 {\omega }^2(\epsilon _2+(\epsilon _1-\epsilon _2){a_3}^2)}{\mu _t{\left( 1+(\frac{\mu _l}{\mu _t}-1){a_3}^2\right) }^2 + {\left( \epsilon _2+(\epsilon _1-\epsilon _2){a_3}^2\right) }^2}, \end{aligned}$$
$$\begin{aligned} x_1= & {} \sqrt{r_1}\cos \frac{\theta _1}{2},\quad y_1 = \sqrt{r_1}\sin \frac{\theta _1}{2},\\ r_1 \cos \theta _1= & {} \frac{\omega ^2}{{\beta _1}^2 \eta _1\left( \frac{1}{{\eta _1}^2}+{\varpi }^2\right) }-(1-{\delta }^2){k_1}^2,\\ r_1 \sin \theta _1= & {} - \frac{{\omega }^2 \varpi }{{\beta _1}^2 \left( \frac{1}{{\eta _1}^2}+{\varpi }^2 \right) }- 2\delta {k_1}^2,\\ x_2= & {} \sqrt{r_2}\cos \frac{\theta _2}{2}, \quad y_2 = \sqrt{r_2}\sin \frac{\theta _2}{2},\\ r_2 \cos \theta _2= & {} \frac{4 Y_3 - {Y_1}^2 + {Y_2}^2}{4},\quad r_2 \sin \theta _2 = \frac{2 Y_4 - Y_1 Y_2}{2},\\ x_3= & {} \sqrt{r_3}\cos \frac{\theta _3}{2}, \quad y_3 = \sqrt{r_3}\sin \frac{\theta _3}{2},\\ r_3 \cos \theta _3= & {} \frac{N {k_1}^2 (1-{\delta }^2)+ \epsilon _3 \epsilon _4 N {k_1}^2 (1-{\delta }^2)-{\omega }^2 d_1-2 \epsilon _3 N {k_1}^2 \delta + 2 \epsilon _4 N {k_1}^2 \delta }{L(1+{\epsilon _4}^2)},\\ r_3 \sin \theta _3= & {} \frac{\epsilon _3 N {k_1}^2 (1-{\delta }^2)-\epsilon _4 N {k_1}^2 (1-{\delta }^2)+\epsilon _4 {\omega }^2 d_1 + 2 N {k_1}^2 \delta + 2 \epsilon _3 \epsilon _4 N {k_1}^2 \delta }{L(1+{\epsilon _4}^2)},\\ Y_5= & {} \left( \mu _t\left( (Y_1 - Y_2 \epsilon _2)+ a_3((\frac{\mu _l}{\mu _t}-1)(Y_1 a_3 -2 k_1 a_1 \delta )- (\epsilon _1-\epsilon _2)(2 k_1 a_1 + a_3 Y_2))\right) -2 L (x_3 - y_3 \epsilon _4)\right) ,\\ Y_6= & {} \left( \mu _t \left( (Y_2 + Y_1 \epsilon _2)+ a_3 ((\epsilon _1-\epsilon _2)(Y_1 a_3 - 2 k_1 a_1 \delta )+ (\frac{\mu _l}{\mu _t}-1)(2 k_1 a_1 + a_3 Y_2))\right) -2 L (y_3 + x_3 \epsilon _4)\right) ,\\ Y_7= & {} a_3 \mu _t \left( (\frac{\mu _l}{\mu _t}-1)\left( a_3 ({Y_1}^2 - {Y_2}^2 + 4 {x_2}^2 - 4 {y_2}^2)- 2 a_1 k_1 (Y_1 \delta + Y_2)\right) \right. \\&\left. - (\epsilon _1-\epsilon _2)\left( a_3 (2 Y_1 Y_2 + 8 x_2 y_2)+ 2 a_1 k_1 (Y_1 - Y_2 \delta )\right) \right) ,\\ Y_8= & {} a_3 \mu _t\left( (\frac{\mu _l}{\mu _t}-1)\left( a_3 (2 Y_1 Y_2 + 8 x_2 y_2)+ 2 a_1 k_1 (Y_1 - Y_2 \delta )\right) \right. \\&\left. + (\epsilon _1 - \epsilon _2)\left( a_3 ({Y_1}^2 - {Y_2}^2 + 4 {x_2}^2 - 4 {y_2}^2)-2 a_1 k_1 (Y_1 \delta + Y_2)\right) \right) ,\\ Y_9= & {} {a_3}^2 \mu _t (\frac{\mu _l}{\mu _t}-1)\left( \mu _t \left( (\frac{\mu _l}{\mu _t}-1)((X_7)^2 -(X_8)^2)- 2 (\epsilon _1 - \epsilon _2)( X_7)(X_8)\right) + 4 a_3 ({x_2}^2 - {y_2}^2)\right) \\&- {a_3}^2 \mu _t (\epsilon _1 - \epsilon _2) \left( \mu _t \left( 2 (\frac{\mu _l}{\mu _t}-1)(X_7)(X_8) + (\epsilon _1 - \epsilon _2)((X_7)^2 - (X_8)^2)\right) + 8 a_3 x_2 y_2\right) ,\\ Y_{10}= & {} (\epsilon _1 - \epsilon _2) {a_3}^2 \mu _t \left( \mu _t \left( (\frac{\mu _l}{\mu _t}-1)((X_7)^2 - (X_8)^2) - 2 (\epsilon _1 - \epsilon _2)(X_7)(X_8)\right) + 4 a_3 ({x_2}^2 - {y_2}^2)\right) \\&+ {a_3}^2 \mu _t (\frac{\mu _l}{\mu _t}-1)\left( \mu _t \left( 2 (\frac{\mu _l}{\mu _t}-1)(X_7)(X_8) + (\epsilon _1 - \epsilon _2)((X_7)^2 - (X_8)^2)\right) + 8 a_3 x_2 y_2 \right) ,\\ \end{aligned}$$
$$\begin{aligned} T_1= & {} 4 \mu _t \left( x_2 (1 + {a_3}^2 (\frac{\mu _l}{\mu _t} - 1)) - y_2 (\epsilon _2 + {a_3}^2 (\epsilon _1 - \epsilon _2))\right) ,\\ T_2= & {} 4 \mu _t \left( y_2 (1 + {a_3}^2 (\frac{\mu _l}{\mu _t} - 1)) + x_2 (\epsilon _2 + {a_3}^2 (\epsilon _1 - \epsilon _2))\right) , \\ T_3= & {} \mu _1 \left( (\frac{x_1 S_3 - y_1 S_4}{\eta _1}) - \varpi (y_1 S_3 + x_1 S_4 )\right) - L (x_3 - y_3 \epsilon _4),\\ T_4= & {} \varpi (x_1 S_3 - y_1 S_4) + (\frac{y_1 S_3 + x_1 S_4}{\eta _1}) - L (y_3 + x_3 \epsilon _4),\\ T_5= & {} - 2 \mu _1 \left( (\frac{x_1}{\eta _1} - y_1 \varpi )(S_3 Y_5 - S_4 Y_6) - (\frac{y_1}{\eta _1}+ x_1 \varpi )(S_4 Y_5 + S_3 Y_6)\right) , \\ T_6= & {} - 2 \mu _1 \left( (\frac{y_1}{\eta _1} + x_1 \varpi )(S_3 Y_5 - S_4 Y_6) + (\frac{x_1}{\eta _1}- y_1 \varpi )(S_4 Y_5 + S_3 Y_6)\right) ,\\ T_7= & {} {\mu _t}^2 \left( (1 - {\epsilon _2}^2) X_3 - 2 \epsilon _2 X_4 \right) + \left( 2 \mu _t (Y_7 - \epsilon _2 Y_8) + Y_9 \right) ,\\ T_8= & {} {\mu _t}^2 \left( 2 \epsilon _2 X_3 + (1 - {\epsilon _2}^2) X_4\right) + \left( 2 \mu _t (\epsilon _2 Y_7 + Y_8) + Y_{10}\right) ,\\ T_9= & {} - 2 L \left( (X_1 x_3 - X_2 y_3) - \epsilon _4 (X_2 x_3 + X_1 y_3)\right) - L \left( (Y_1 x_3 - Y_2 y_3) - \epsilon _4 (Y_2 x_3 + Y_1 y_3)\right) ,\\ T_{10}= & {} - 2 L \left( (X_2 x_3 + X_1 y_3) + \epsilon _4 (X_1 x_3 - X_2 y_3)\right) - L \left( (Y_2 x_3 + Y_1 y_3) + \epsilon _4 (Y_1 x_3 - Y_2 y_3)\right) ,\\ T_{11}= & {} \left( (\frac{x_1}{\eta _1} - y_1 \varpi )( S_3 (x_3 - y_3 \epsilon _4) - S_4 (y_3 + x_3 \epsilon _4)) - (\frac{y_1}{\eta _1} + x_1 \varpi )( S_4 (x_3 - y_3 \epsilon _4) + S_3 (y_3 + x_3 \epsilon _4))\right) ,\\ T_{12}= & {} \left( (\frac{y_1}{\eta _1} + x_1 \varpi )( S_3 (x_3 - y_3 \epsilon _4) - S_4 (y_3 + x_3 \epsilon _4)) + (\frac{x_1}{\eta _1} - y_1 \varpi )( S_4 (x_3 - y_3 \epsilon _4) + S_3 (y_3 + x_3 \epsilon _4))\right) ,\\ T_{13}= & {} \mu _2 \left( \sqrt{\frac{c^2}{{\beta _2}^2}-1}\right) \left( \mu _1 \sqrt{\frac{c^2}{{\beta _1}^2}-1} \; \tan \left( \sqrt{\frac{c^2}{{\beta _1}^2}-1}\right) H_2 - \mu _3 k_1 \left( \sqrt{1 - \frac{c^2}{{\beta _4}^2}}\right) \right) ,\\ T_{14}= & {} \mu _1 \mu _3 \left( \sqrt{\frac{c^2}{{\beta _1}^2}-1}\; \tan \left( \sqrt{\frac{c^2}{{\beta _1}^2}-1}\right) H_2 \sqrt{1-\frac{c^2}{{\beta _4}^2}}\right) + {\mu _2}^2 k_1 \left( \frac{c^2}{{\beta _2}^2}-1\right) ,\\ T_{15}= & {} a_1 k_1 \mu _3 \sqrt{1 - \frac{c^2}{{\beta _4}^2}},\\ S_3= & {} \frac{\tan (x_1 H_2)(1 - \tanh ^2 (y_1 H_2))}{1 + \tan ^2 (x_1 H_2) \tanh ^2 (y_1 H_2)},\\ S_4= & {} \frac{\tanh (y_1 H_2)(1 + \tan ^2 (x_1 H_2))}{1 + \tan ^2 (x_1 H_2) \tanh ^2 (y_1 H_2)}. \end{aligned}$$

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Kumar, D., Kundu, S., Kumhar, R. et al. Vibrational analysis of Love waves in a viscoelastic composite multilayered structure. Acta Mech 231, 4199–4215 (2020). https://doi.org/10.1007/s00707-020-02767-8

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