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Green’s functions for infinite orthotropic, hygro-electro-magneto-thermoelastic materials

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Abstract

The current study is attempted to explore the general solution for the plane problem of an orthotropic hygro-electro-magneto-thermoelastic material (HEMTM). A complete general solution is accomplished by the aid of operator theory and superposition principle, which satisfies twelfth order partial differential equation. Moreover, the obtained general solution is simplified in terms of six harmonic functions and is helpful for boundary value problem of HEMTM. As an application, the fundamental solution for a steady point moisture source combined with heat source in an infinite HEMTM is derived using the general solution. Consequently, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions. Along with this, numerical simulations are conducted using the general solution and graphically illustrated.

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Abbreviations

\({u_{i}}\) :

Elastic displacements (m)

\(\Phi \) :

Electric potential (N m/C)

\(\psi \) :

Magnetic potential (I)

\(\theta \) :

Changes of temperature (K)

m :

Moisture concentration (kg/m\(^3\))

\(D_{i}\) :

Electric displacements (C/m\(^2\))

\(g_{ij}\) :

Magneto-electric coefficients (Ns/VC)

\(p_{3}\) :

Pyroelectric coefficients (C/m\(^2\)K)

\(\chi _{3}\) :

Hygroelectric coefficients (Cm/kg)

\(c_{ij}\) :

Elastic coefficients (N/m\(^2\))

\(q_{i}\) :

Moisture flux (kg/m\(^2\))

\(h_{i}\) :

Heat flux (W/m\(^2\))

\(\sigma _{ij}\) :

Component of stresses (N/m\(^2\))

\(e_{ij}\) :

Piezoelectric coefficients (C/m\(^2\))

\(d_{ij}\) :

Piezomagnetic coefficients (N/Am)

\(\alpha _{i}\) :

Thermal expansion coefficients (1/K)

\(\beta _{i}\) :

Moisture expansion coefficients (m\(^3\)/kg)

\(\uplambda _{ij}\) :

Conduction coefficients of heat (W/Km)

\(B_{i}\) :

Magnetic induction (M/IT\(^2\))

\(m_{3}\) :

Pyromagnetic coefficients (C/m\(^2\) K)

\(\nu _{3}\) :

Hygromagnetic coefficients (N m\(^2\)/A kg)

\(\mu _{ij}\) :

Magnetic constants (Ns\(^2\)/C\(^2\))

\(\epsilon _{ij}\) :

Dielectric constant (C\(^2\)/N m\(^2\))

\(k_{ij}\) :

Conduction coefficients of moisture (kg/m\(^2\) s Pa)

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 52071298) and ZhongYuan Science and Technology Innovation Leadership Program (No. 214200510010).

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Authors and Affiliations

  1. Henan Academy of Big Data/School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Muzammal Hameed Tariq & Jingli Ren

  2. School of Mechanics and Engineering Science, Zhengzhou University, Zhengzhou, 450001, China

    Huayang Dang

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Appendix

Appendix

$$\begin{aligned} t_{11}&=\left( c_{11}\frac{\partial ^2}{\partial x^2}+c_{44}\frac{\partial ^2}{\partial z^2}\right) ,~~~~t_{12}=(c_{13}+c_{44})\frac{\partial ^2}{\partial x\partial z},~~~t_{13}=(e_{31}+e_{15})\frac{\partial ^2}{\partial x\partial z},\\ t_{14}&=(d_{15}+d_{31})\frac{\partial ^2}{\partial x\partial z},~~~t_{15}=-\beta _{1}\frac{\partial }{\partial x},\,\,\,t_{16}=-\alpha _{1}\frac{\partial }{\partial x},\\ t_{21}&=(c_{13}+c_{44})\frac{\partial ^2}{\partial x\partial z},~~~t_{22}=c_{44}\frac{\partial ^2}{\partial x^2}+c_{33}\frac{\partial ^2}{\partial z^2},~~~t_{23}=e_{15}\frac{\partial ^2}{\partial x^2}+e_{33}\frac{\partial ^2}{\partial z^2},\\ t_{24}&=d_{15}\frac{\partial ^2}{\partial x^2}+d_{33}\frac{\partial ^2}{\partial z^2},~~~t_{25}=-\beta _{3}\frac{\partial }{\partial z},\,\,\,t_{26}=-\alpha _{3}\frac{\partial }{\partial z},\\ t_{31}&=(e_{15}+e_{31})\frac{\partial ^2}{\partial x\partial z},~~~~~t_{32}=e_{15}\frac{\partial ^2}{\partial x^2}+e_{33}\frac{\partial ^2}{\partial z^2},~~~~~t_{33}=-\left( \epsilon _{11}\frac{\partial ^2}{\partial x^2}+\epsilon _{33}\frac{\partial ^2}{\partial z^2}\right) ,\\ t_{34}&=-\left( g_{11}\frac{\partial ^2}{\partial x^2}+g_{33}\frac{\partial ^2}{\partial z^2}\right) ,~~~~~t_{35}=\chi _{3}\frac{\partial }{\partial z},\,\,\,t_{36}=p_{3}\frac{\partial }{\partial z},\\ t_{41}&=(d_{15}+d_{31})\frac{\partial ^2}{\partial x\partial z},~~~~t_{42}=d_{15}\frac{\partial ^2}{\partial x^2}+d_{33}\frac{\partial ^2}{\partial z^2},~~~~t_{43}=-\left( g_{11}\frac{\partial ^2}{\partial x^2}+g_{33}\frac{\partial ^2}{\partial z^2}\right) ,\\ t_{44}&=-\left( \mu _{11}\frac{\partial ^2}{\partial x^2}+\mu _{33}\frac{\partial ^2}{\partial z^2}\right) ,~~~t_{45}=\nu _{3}\frac{\partial }{\partial z},\,\,\,t_{46}=m_{3}\frac{\partial }{\partial z},\\ t_{51}&=0,~~t_{52}=0,~~t_{53}=0,~~t_{54}=0,\,\,\,t_{55}=k_{11}\frac{\partial ^2}{\partial x^2}+k_{33}\frac{\partial ^2}{\partial z^2} ,~~t_{56}=0,\\ t_{61}&=0,~~t_{62}=0,~~t_{63}=0,~~t_{64}=0,~~t_{65}=0 ,~~t_{66}=\uplambda _{11}\frac{\partial ^2}{\partial x^2}+\uplambda _{33}\frac{\partial ^2}{\partial z^2}. \end{aligned}$$

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Tariq, M.H., Dang, H. & Ren, J. Green’s functions for infinite orthotropic, hygro-electro-magneto-thermoelastic materials. Arch Appl Mech 92, 3325–3342 (2022). https://doi.org/10.1007/s00419-022-02239-6

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