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Three-dimensional Green’s functions for transversely isotropic poro-chemo-thermoelastic media

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Abstract

Shale having a porous structure, is sensitive to thermal and chemical stimuli. In order to study the effects of concentrated piont sources on the mechanical behavior of porous materials, we introduce two displacement functions and derive the general solutions of the coupled fields based on the operator theory, superposition principle, and generalized Almansi’s theorem. Two examples are used to introduce the application of general solutions by the liquid-chemical-thermal equilibrium boundary conditions. In the first example, the general solutions are used to solve the problem of semi-infinite transversely isotropic poro-chemo-thermoelastic (PCT) cones subjected to a point fluid source, a point ion source or a point heat source at the vertex. In the other example, the general solutions are used to solve the problem of transversely isotropic PCT media with conical cavities subjected to a point fluid source, a point ion source or a point heat source at the origin. Finally, the contours of the coupled fields of PCT cones and PCT media with a conical cavity are drawn. The numerical results show that the variation of the vertex angle can affect the diffusion trend of the coupled fields.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11902132).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, NO.185 in Qianshan Middle Street, Anshan, 114000, Liaoning, People’s Republic of China

    Zhouwen Shi, Shuaixiang Qi & Di Wu

  2. General ironmaking plant of Bengang Steel Plates Co., Ltd, Benxi, 117000, Liaoning, People’s Republic of China

    Jiadong Han

  3. Sinosteel Anshan Research Institute of Thermo-Energy Co., Ltd, Anshan, 114044, Liaoning, People’s Republic of China

    Longming Fu

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  1. Zhouwen Shi
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Correspondence to Di Wu.

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Appendix A

Appendix A

$$g_{i1} = C_{11} + C_{13} \mu_{i1} s_{i} - \alpha_{1} \mu_{i2} + \chi_{1} \mu_{i3} - \beta_{1} \mu_{i4} ,$$
$$g_{i2} = C_{12} + C_{13} \mu_{i1} s_{i} - \alpha_{1} \mu_{i2} + \chi_{1} \mu_{i3} - \beta_{1} \mu_{i4} ,$$
$$g_{i3} = C_{13} + C_{33} \mu_{i1} s_{i} - \alpha_{3} \mu_{i2} + \chi_{3} \mu_{i3} - \beta_{3} \mu_{i4} ,$$
$$g_{i4} = C_{44} \left( {\mu_{i1} - s_{i} } \right), \, \left( {i = 1,2,3,4,5} \right).$$

Coefficients in Eq. (11)

$$t_{11} = C_{11} \Delta + C_{44} \frac{{\partial^{2} }}{{\partial z^{2} }},\quad t_{12} = - \left( {C_{13} + C_{44} } \right)\frac{\partial }{\partial z},$$
$$t_{13} = \alpha_{1} ,\quad t_{14} = - x_{1} ,\quad t_{15} = \beta_{1}$$
$$t_{21} = - \left( {C_{13} + C_{44} } \right)\Delta \frac{\partial }{\partial z},\quad t_{22} = C_{44} \Delta + C_{33} \frac{{\partial^{2} }}{{\partial z^{2} }}$$
$$t_{23} = - \alpha_{3} \frac{\partial }{\partial z},\quad t_{24} = x_{3} \frac{\partial }{\partial z},\quad t_{25} = - \beta_{3} \frac{\partial }{\partial z}$$
(64)
$$t_{33} = \kappa_{11} \Delta + \kappa_{33} \frac{{\partial^{2} }}{{\partial z^{2} }},\quad t_{44} = \varphi_{11}^{s} \Delta + \varphi_{33}^{s} \frac{{\partial^{2} }}{{\partial z^{2} }},\quad t_{55} = \lambda_{11} \Delta + \lambda_{33} \frac{{\partial^{2} }}{{\partial z^{2} }},$$
$$\begin{gathered} a_{1} = \left( {C_{13} + C_{44} } \right)\left( {\varphi_{33}^{s} \lambda_{33} \alpha_{3} - \kappa_{33} \lambda_{33} x_{3} + \kappa_{33} \varphi_{33}^{s} \beta_{3} } \right) \hfill \\ \, - C_{33} \left( {\varphi_{33}^{s} \lambda_{33} \alpha_{1} - \kappa_{33} \lambda_{33} x_{1} + \kappa_{33} \varphi_{33}^{s} \beta_{1} } \right), \hfill \\ \end{gathered}$$
$$\begin{gathered} b_{1} = \left( {C_{13} + C_{44} } \right)\left[ {\left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right)\alpha_{3} - \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right)\chi_{3} + \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right)\beta_{3} } \right] \hfill \\ \, - C_{44} \left( {\varphi_{33}^{s} \lambda_{33} \alpha_{1} - \kappa_{33} \lambda_{33} \chi_{1} + \kappa_{33} \varphi_{33}^{s} \beta_{1} } \right) \hfill \\ \, - C_{33} \left[ {\left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right)\alpha_{1} - \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right)\chi_{1} + \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right)\beta_{1} } \right], \hfill \\ \end{gathered}$$
$$\begin{gathered} c_{1} = \left( {C_{13} + \, C_{44} } \right)\left( {\varphi_{11}^{s} \lambda_{11} \alpha_{3} - \kappa_{11} \lambda_{11} \chi_{3} + \kappa_{11} \varphi_{11}^{s} \beta_{3} } \right) \, - C_{33} \left( {\varphi_{11}^{s} \lambda_{11} \alpha_{1} - \kappa_{11} \lambda_{11} \chi_{1} + \kappa_{11} \varphi_{11}^{s} \beta_{1} } \right) \hfill \\ \, - C_{44} \left[ {\left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right)\alpha_{1} - \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right)\chi_{1} + \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right)\beta_{1} } \right] \, , \hfill \\ \end{gathered}$$
$$d_{1} = - C_{44 \, } \left( {\varphi_{11}^{s} \lambda_{11} \alpha_{1} - \kappa_{11} \lambda_{11} \chi_{1} + \kappa_{11} \varphi_{11}^{s} \beta_{1} } \right),$$
$$a_{2} = C_{44} \left( {\varphi_{33}^{s} \lambda_{33} \alpha_{3} - \kappa_{33} \lambda_{33} \chi_{3} + \kappa_{33} \varphi_{33}^{s} \beta_{3} } \right),$$
$$\begin{gathered} b_{2} \, = C_{11} \left( {\varphi_{33}^{s} \lambda_{33} \alpha_{3 \, } - \kappa_{33} \lambda_{33} \chi_{3} + \kappa_{33} \varphi_{33}^{s} \beta_{3} } \right) - \left( {C_{13} + C_{44} } \right)\left( {\varphi_{33}^{s} \lambda_{33} \alpha_{1} - \kappa_{33} \lambda_{33} \chi_{1} + \kappa_{33} \varphi_{33}^{s} \beta_{1} } \right) \, \hfill \\ \, + C_{44 \, } \left[ {\left( {\varphi_{33}^{s} \lambda_{11} + \varphi_{11}^{s} \lambda_{33} } \right)\alpha_{3} - \left( {\kappa_{33} \lambda_{11} + \kappa_{11} \lambda_{33} } \right) \, \chi_{3} + \left( {\kappa_{33} \varphi_{11}^{s} + \kappa_{11} \varphi_{33}^{s} } \right) \, \beta_{3} } \right] \, , \hfill \\ \end{gathered}$$
$$\begin{gathered} c_{2} = C_{11} \left[ {\left( {\varphi_{33}^{s} \lambda_{11} + \varphi_{11}^{s} \lambda_{33} } \right)\alpha_{3} \, - \left( {\kappa_{33} \lambda_{11} + \kappa_{11} \lambda_{33} } \right)\chi_{3} + \left( {\kappa_{33} \varphi_{11}^{s} + \kappa_{11} \varphi_{33}^{s} } \right)\beta_{3} } \right] \, \hfill \\ \, - \left( {C_{13} + C_{44} } \right)\left[ {\left( {\varphi_{33}^{s} \lambda_{11} + \varphi_{11}^{s} \lambda_{33} } \right)\alpha_{1} - \left( {\kappa_{33} \lambda_{11} + \kappa_{11} \lambda_{33} } \right)\chi_{1} + \left( {\kappa_{33} \varphi_{11}^{s} + \kappa_{11} \varphi_{33}^{s} } \right)\beta_{1} } \right] \hfill \\ \, + C_{44} \left( {\varphi_{11}^{s} \lambda_{11} \alpha_{3} - \kappa_{11} \lambda_{11} \chi_{3} + \kappa_{11} \varphi_{11}^{s} \beta_{3} } \right), \, \hfill \\ \end{gathered}$$
$$d_{2} \, = C_{11} \left( {\varphi_{11}^{s} \lambda_{11} \alpha_{3} - \kappa_{11} \lambda_{11} \chi_{3} + \kappa_{11} \varphi_{11}^{s} \beta_{3} } \right) - \left( {C_{13} + C_{44} } \right)\left( {\varphi_{11}^{s} \lambda_{11} \alpha_{1} - \kappa_{11} \lambda_{11} \chi_{1} + \kappa_{11} \varphi_{11}^{s} \beta_{1} } \right), \,$$
$$a_{3} = a_{0} \varphi_{33}^{s} \lambda_{33} , \, b_{3} = b_{0} \varphi_{33}^{s} \lambda_{33} + a_{0} \left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right),$$
$$c_{3} = c_{0} \varphi_{33}^{s} \lambda_{33} + b_{0} \left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right) + a_{0} \varphi_{11}^{s} \lambda_{11} ,$$
$$d_{3} = c_{0} \left( {\varphi_{11}^{s} \lambda_{33} + \varphi_{33}^{s} \lambda_{11} } \right) + b_{0} \varphi_{11}^{s} \lambda_{11} , \, e_{3} = c_{0} \varphi_{11}^{s} \lambda_{11} ,$$
$$a_{4} = a_{0} \kappa_{33} \lambda_{33} , \, b_{4} = b_{0} \kappa_{33} \lambda_{33} + a_{0} \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right),$$
$$c_{4} = c_{0} \kappa_{33} \lambda_{33} + b_{0} \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right) + a_{0} \kappa_{11} \lambda_{11} , \,$$
$$d_{4} = c_{0} \left( {\kappa_{11} \lambda_{33} + \kappa_{33} \lambda_{11} } \right) + b_{0} \kappa_{11} \lambda_{11} , \, e_{4} = c_{0} \kappa_{11} \lambda_{11} ,$$
$$a_{5} = a_{0} \kappa_{33} \varphi_{33}^{s} , \, b_{5} = b_{0} \kappa_{33} \varphi_{33}^{s} + a_{0} \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right),$$
$$c_{5} = c_{0} \kappa_{33} \varphi_{33}^{s} + b_{0} \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right) + a_{0} \kappa_{11} \varphi_{11}^{s} , \,$$
$$d_{5} = c_{0} \left( {\kappa_{11} \varphi_{33}^{s} + \kappa_{33} \varphi_{11}^{s} } \right) + b_{0} \kappa_{11} \varphi_{11}^{s} , \, e_{5} = c_{0} \kappa_{11} \varphi_{11}^{s} . \,$$
(65)

Coefficients in Eq. (20)

$$\begin{gathered} \eta_{i\varsigma } = a_{\varsigma } s_{i}^{6} - b_{\varsigma } s_{i}^{4} + c_{\varsigma } s_{i}^{2} - d_{\varsigma } , \, \left( {\varsigma = 1,2; \, \tau = 3,4,5} \right), \hfill \\ \eta_{3\tau } = \left( {a_{0} s_{\tau }^{4} - b_{0} s_{\tau }^{2} + c_{0} } \right)\left( {\varphi_{33}^{s} s_{\tau }^{2} - \varphi_{11}^{s} } \right)\left( {\lambda_{33} s_{\tau }^{2} - \lambda_{11} } \right), \hfill \\ \eta_{4\tau } = \left( {a_{0} s_{\tau }^{4} - b_{0} s_{\tau }^{2} + c_{0} } \right)\left( {\kappa_{33} s_{\tau }^{2} - \kappa_{11} } \right)\left( {\lambda_{33} s_{\tau }^{2} - \lambda_{11} } \right), \hfill \\ \eta_{5\tau } = \left( {a_{0} s_{\tau }^{4} - b_{0} s_{\tau }^{2} + c_{0} } \right)\left( {\kappa_{33} s_{\tau }^{2} - \kappa_{11} } \right)\left( {\varphi_{33}^{s} s_{\tau }^{2} - \varphi_{11}^{s} } \right),\quad \left( {i \, = 1,2,3,4,5} \right). \, \hfill \\ \end{gathered}$$
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Shi, Z., Qi, S., Han, J. et al. Three-dimensional Green’s functions for transversely isotropic poro-chemo-thermoelastic media. Arch Appl Mech 93, 3427–3460 (2023). https://doi.org/10.1007/s00419-023-02448-7

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  • DOI: https://doi.org/10.1007/s00419-023-02448-7

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