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The incompressible limit method and Rayleigh waves in incompressible layered nonlocal orthotropic elastic media

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Abstract

In this paper, we introduce the incompressible limit method to nonlocal orthotropic elastic solids under plane strain. Then, applying it we investigate Rayleigh waves propagating in a nonlocal orthotropic elastic half-space coated with a nonlocal orthotropic elastic layer in which the layer and the half-space may be compressible or incompressible, and they are in welded contact with each other. Our main aim is to derive explicit secular equations of Rayleigh waves. These secular equations are derived using the nonlocal stress boundary conditions, not the local stress ones as previously used. First, the secular equation for the compressible case (both the half-space and the layer are compressible) is derived by using the effective boundary condition technique. Then, the secular equations for the incompressible cases (at least one of the half-space and the layer is incompressible) are obtained by taking the incompressible limit of the obtained compressible secular equation. The simple and immediate derivation of the incompressible secular equations proves the convenience and powerfulness of the incompressible limit technique. Some numerical examples are carried out to show the strong effect of the incompressibility and the nonlocality on the Rayleigh wave velocity. Remarkably, while the nonlocality of half-spaces makes the Rayleigh wave velocity decreasing, the nonlocality of layers can increase the Rayleigh wave velocity.

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Acknowledgements

Vu Thi Ngoc Anh was funded by Vingroup JSC and supported by the Postdoctoral Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, Code VINIF.2021.STS.02.

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  1. Faculty of Mathematics, Mechanics and Informatics, University of Science, Vietnam National University, Hanoi, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    V. T. N. Anh & P. C. Vinh

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Appendices

Appendix A: The coefficients of the secular equation (48)

$$\begin{aligned} \bar{A}_0= & {} 2\bar{\beta }_1^*\bar{\beta }_2^*\bar{\gamma }_1^*\bar{\gamma }_2^*[\alpha ^*] -[\bar{\alpha }^*;\bar{\beta }^*\bar{\gamma }^*]^+[\gamma ^*;\beta ^*]-\bar{\gamma }_1^*\bar{\gamma }_2^*[\bar{\alpha }^*;\bar{\beta }^*]^+[\beta ^*]-\bar{\beta }_1^*\bar{\beta }_2^*[\bar{\gamma }^*]^+[\alpha ^*;\gamma ^*],\nonumber \\ \bar{B}_0= & {} [\bar{\beta }^{*2};\bar{\gamma }^{*2}]^+[\alpha ^*]-[\bar{\alpha }^*\bar{\beta }^*;\bar{\gamma }^*]^+[\gamma ^*;\beta ^*]-[\bar{\alpha }^*\bar{\beta }^*;\bar{\gamma }^{*2}]^+[\beta ^*]-[\bar{\beta }^{*2};\bar{\gamma }^*]^+[\alpha ^*;\gamma ^*],\nonumber \\ \bar{C}_0= & {} \bar{\beta }_2^*\bar{\gamma }_1^*[\bar{\gamma }^*][\alpha ^*;\beta ^*]-\bar{\beta }_1^*\bar{\gamma }_2^* [\bar{\alpha }^*;\bar{\beta }^*][\gamma ^*],\, \bar{D}_0=-\bar{\beta }_1^*\bar{\gamma }_2^*[\bar{\gamma }^*][\alpha ^*;\beta ^*]+\bar{\beta }_2^*\bar{\gamma }_1^* [\bar{\alpha }^*;\bar{\beta }^*][\gamma ^*],\nonumber \\ \bar{E}_0= & {} -\bar{A}_0+[\bar{\gamma }^*][\bar{\alpha }^*;\bar{\beta }^*][\gamma ^*;\beta ^*], \end{aligned}$$
(68)

where:

$$\begin{aligned} \bar{\alpha }_j^*&=\,\dfrac{(\bar{e}_3+1)\bar{s}_j}{1-\bar{e}r_v^2x-(\bar{e}_2-\bar{e}r_v^2x)\bar{s}_j^2},\,\bar{\beta }_j^*=\dfrac{r_\mu (\bar{s}_j-\bar{\alpha }^*_j)}{1+\bar{e}(1-\bar{s}_j^2)},\,\bar{\gamma }_j^*=r_\mu \dfrac{\bar{e}_3+\bar{e}_2\bar{s}_j\bar{\alpha }^*_j}{1+\bar{e}(1-\bar{s}_j^2)},\nonumber \\ [\alpha ^*]&=\,\theta ^*\Big [x-e_1+ex-(1-ex)\sqrt{P} \Big ],\nonumber \\ [\beta ^*]&=\,-{(1+e)\big [x-e_1+ex+(e_3+xe)\sqrt{P} \big ]- e \big [(e_1-ex-x)(S+\sqrt{P})+(e_3+xe)P \big ] } ,\nonumber \\ [\gamma ^*]&=\,\Big \{(1+e)e_2(1-ex)+e\big [e_3(e_3+1)-e_2(e_1-ex-x) \big ] \Big \} \sqrt{P}\sqrt{S+2\sqrt{P}},\nonumber \\ [\alpha ^*;\beta ^*]&=\,(e_1-ex-x)\Big [1+\dfrac{e}{ e_2-ex}(e_2+e_3+x) \Big ]\sqrt{S+2\sqrt{P}},\nonumber \\ [\alpha ^*;\gamma ^*]&=\, (1+e)\Big [\dfrac{e_1-ex-x}{e_2-ex}(ee_3x+e_2)-e_3(1-ex)\sqrt{P} \Big ]\nonumber \\&+e\sqrt{P}\Big [\dfrac{e_1-ex-x}{e_2-ex}(e_3xe+e_2)+e_3(1-ex)(S+\sqrt{P}) \Big ],\nonumber \\ [\gamma ^*;\beta ^*]&=\, \big [e_3^2-e_2(e_1-ex-x)+ee_3x\big ]\sqrt{P}+ x(e_1-ex-x) \dfrac{e_2(1+e)+ee_3}{e_2-ex} \end{aligned}$$
(69)

in which:

$$\begin{aligned} \begin{aligned} S=&\,\dfrac{[ e_1-x(1+e)](e_2-ex)+[1-x(1+e)](1-ex)-(e_3+1)^2}{(1-ex)(e_2-ex)},\\ P=&\,\dfrac{[e_1-x(1+e)][1-x(1+e)]}{(1-ex)(e_2-ex)}, \end{aligned} \end{aligned}$$
(70)

and \(\bar{s}_j\,(j=1,2)\) are determined by (18) with \(\bar{S}\) and \(\bar{P}\) being calculated by:

$$\begin{aligned} \begin{aligned} \bar{S}=&\,\dfrac{[ \bar{e}_1-r_v^2x(1+\bar{e})](\bar{e}_2-\bar{e}r_v^2x)+[1-r_v^2x(1+\bar{e})](1-\bar{e}r_v^2x)-(\bar{e}_3+1)^2}{(1-\bar{e}r_v^2x)(\bar{e}_2-\bar{e}r_v^2x)},\\ \bar{P}=&\,\dfrac{[\bar{e}_1-r_v^2x(1+\bar{e})][1-r_v^2x(1+\bar{e})]}{(1-\bar{e}r_v^2x)(\bar{e}_2-\bar{e}r_v^2x)}, \end{aligned} \end{aligned}$$
(71)

Appendix B: The coefficients of the secular equation (62)

$$\begin{aligned} \hat{A}_0&=2\bar{\beta }_1'\bar{\beta }_2'\bar{\gamma }_1'\bar{\gamma }_2'[\alpha ^*] +[\bar{\alpha }';\bar{\beta }'\bar{\gamma }']^+[\gamma ^*;\beta ^*]+\bar{\gamma }_1'\bar{\gamma }_2'[\bar{\beta }']^+[\beta ^*]-\bar{\beta }_1'\bar{\beta }_2'[\bar{\alpha }';\bar{\gamma }']^+[\alpha ^*;\gamma ^*],\nonumber \\ \hat{B}_0&=[\bar{\beta }'^2;\bar{\gamma }'^2]^+[\alpha ^*]+[\bar{\alpha }'\bar{\gamma }';\bar{\beta }']^+[\gamma ^*;\beta ^*]+[\bar{\beta }';\bar{\gamma }'^2]^+[\beta ^*]-[\bar{\alpha }'\bar{\gamma }';\bar{\beta }'^2]^+[\alpha ^*;\gamma ^*], \nonumber \\ \hat{C}_0&=-\bar{\beta }_2'\bar{\gamma }_1'[\bar{\alpha }';\bar{\gamma }'][\alpha ^*;\beta ^*]-\bar{\beta }_1'\bar{\gamma }_2' [\bar{\beta }'][\gamma ^*],\, \hat{D}_0=\bar{\beta }_1'\bar{\gamma }_2'[\bar{\alpha }';\bar{\gamma }'][\alpha ^*;\beta ^*]+\bar{\beta }_2'\bar{\gamma }_1' [\bar{\beta }'][\gamma ^*],\nonumber \\ \hat{E}_0&=-\hat{A}_0-[\bar{\alpha }';\bar{\gamma }'][\bar{\beta }'][\gamma ^*;\beta ^*], \end{aligned}$$
(72)

here \(\bar{\alpha }'_j=\bar{s}_j\), \(\bar{\beta }_j',\,\bar{\gamma }_j'\) are determined by (55), and: \([\alpha ^*],\,[\beta ^*],\,[\gamma ^*],\,[\alpha ^*,\beta ^*],\,[\alpha ^*;\gamma ^*],\,[\gamma ^*;\beta ^*]\) are given by (69).

Appendix C: The coefficients of the secular equation (64)

$$\begin{aligned} \hat{A}_0&=2\bar{\beta }_1'\bar{\beta }_2'\bar{\gamma }_1'\bar{\gamma }_2'[\alpha ']+[\bar{\alpha }';\bar{\beta }'\bar{\gamma }']^+[\gamma ';\beta ']+\bar{\gamma }_1'\bar{\gamma }_2'[\bar{\beta }']^+[\beta ']-\bar{\beta }_1'\bar{\beta }_2'[\bar{\alpha }';\bar{\gamma }']^+[\alpha ';\gamma '],\nonumber \\ \hat{B}_0&=[\bar{\beta }'^2;\bar{\gamma }'^2]^+[\alpha ']+[\bar{\alpha }'\bar{\gamma }';\bar{\beta }']^+[\gamma ';\beta ']+[\bar{\beta }';\bar{\gamma }'^2]^+[\beta ']-[\bar{\alpha }'\bar{\gamma }';\bar{\beta }'^2]^+[\alpha ';\gamma '],\nonumber \\ \hat{C}_0&=-\bar{\beta }_2'\bar{\gamma }_1'[\bar{\alpha }';\bar{\gamma }'][\alpha ';\beta ']-\bar{\beta }_1'\bar{\gamma }_2' [\bar{\beta }'][\gamma '],\,\,\hat{D}_0=\bar{\beta }_1'\bar{\gamma }_2'[\bar{\alpha }';\bar{\gamma }'][\alpha ';\beta ']+\bar{\beta }_2'\bar{\gamma }_1' [\bar{\beta }'][\gamma '],\nonumber \\ \hat{E}_0&=-\hat{A}_0-[\bar{\alpha }';\bar{\gamma }'][\bar{\beta }'][\gamma ';\beta '], \end{aligned}$$
(73)

where \(\bar{\alpha }'_j=\bar{s}_j\), \(\bar{\beta }_j',\,\bar{\gamma }_j'\) are determined by (55), and \([\alpha '],\,[\beta '],\,[\gamma '],\,[\alpha ',\beta '],\,[\alpha ';\gamma '],\,[\gamma ';\beta ']\) are given by (61).

Appendix D: The coefficients of the secular equation (66)

$$\begin{aligned} \hat{A}_0&=2\bar{\beta }_1^*\bar{\beta }_2^*\bar{\gamma }_1^*\bar{\gamma }_2^*[\alpha '] -[\bar{\alpha }^*;\bar{\beta }^*\bar{\gamma }^*]^+[\gamma ';\beta ']-\bar{\gamma }_1^*\bar{\gamma }_2^*[\bar{\alpha }^*;\bar{\beta }^*]^+[\beta ']-\bar{\beta }_1^*\bar{\beta }_2^*[\bar{\gamma }^*]^+[\alpha ';\gamma '], \nonumber \\ \hat{B}_0&=[\bar{\beta }^{*2};\bar{\gamma }^{*2}]^+[\alpha ']-[\bar{\alpha }^*\bar{\beta }^*;\bar{\gamma }^*]^+[\gamma ';\beta ']-[\bar{\alpha }^*\bar{\beta }^*;\bar{\gamma }^{*2}]^+[\beta ']-[\bar{\beta }^{*2};\bar{\gamma }^*]^+[\alpha ';\gamma '], \nonumber \\ \hat{C}_0&=\bar{\beta }_2^*\bar{\gamma }_1^*[\bar{\gamma }^*][\alpha ';\beta ']-\bar{\beta }_1^*\bar{\gamma }_2^* [\bar{\alpha }^*;\bar{\beta }^*][\gamma '],\, \hat{D}_0=-\bar{\beta }_1^*\bar{\gamma }_2^*[\bar{\gamma }^*][\alpha ';\beta ']+\bar{\beta }_2^*\bar{\gamma }_1^* [\bar{\alpha }^*;\bar{\beta }^*][\gamma '], \nonumber \\ \hat{E}_0&=-\hat{A}_0+[\bar{\gamma }^*][\bar{\alpha }^*;\bar{\beta }^*][\gamma ';\beta '], \end{aligned}$$
(74)

where \(\bar{\alpha }_j^*,\,\bar{\beta }_j^*,\,\bar{\gamma }_j^*\) are determined by (69), and: \([\alpha '],\,[\beta '],\,[\gamma '],\,[\alpha ',\beta '],\,[\alpha ';\gamma '],\,[\gamma ';\beta ']\) are given by (61).

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Anh, V.T.N., Vinh, P.C. The incompressible limit method and Rayleigh waves in incompressible layered nonlocal orthotropic elastic media. Acta Mech 234, 403–421 (2023). https://doi.org/10.1007/s00707-022-03319-y

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