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Exact analytical solution of a homogeneous anisotropic piezo-thermoelasic half-space of a hexagonal type under different fields with three theories

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Abstract

In the present paper, the three different theories (coupled theory, Lord–Shulman with one relaxation time and Green-Lindsay with two relaxation times) are applied to study the deformation of a generalized piezo-thermoelastic rotating medium under the influence of gravity and magnetic field. The normal mode analysis is used to obtain the expressions for the displacement components, the electric potential, the temperature, the stress, the strain components and the electric displacements. Comparisons are made with the results predicted by Lord–Shulman (L–S) and Green-Lindsay (G-L) theories in the presence and absence of rotation and the magnetic field as well as gravity.

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Abbreviations

u i :

The mechanical displacement

T:

The absolute temperature

σ ij :

The stress tensor

E i :

The electric field

C ijkl :

The elastic parameters tensor

∊ij :

The dielectric moduli

ρ :

The mass density

K ij :

The heat conduction tensor

C e :

The specific heat at constant strain

\(\varvec{J}\) :

The current density vector

\(\varvec{h}\) :

The induced magnetic field vector

α 1, α 3 :

The coefficients of linear thermal expansion

ɛ 0, μ 0 :

The electric and magnetic permeability respectively

\(v_{p} = \sqrt {{{C_{11} } / \rho }}\) :

The longitudinal wave velocity in the medium

\(\varphi\) :

The electric potential

ɛ ij :

The strain tensor

β ij :

The thermal elastic coupling tensor

D i :

The electric displacement

e ijk :

The piezoelectric moduli

p i :

The pyroelectric moduli

t 0, t 1 :

The thermal relaxation time parameters

T 0 :

The reference temperature

n 1 :

The non-dimensional parameter

\(\varvec{E}\) :

The induced electric field vector

H 0 :

The constant of magnetic field

i :

The imaginary unit

References

  • Abd-alla AN, Alshaikh F (2017) The influence of piezoelectric and electro-magnetic effects on the propagation of Rayleigh surface waves in smart material. J Comput Theor Nanosci 13:8674–8682

    Article  Google Scholar 

  • Akbarzadeh AH, Babaei MH, Chen ZT (2011) Coupled thermopiezoelectric behaviour of a one dimensional functionally graded piezoelectric medium based on C–T theory. J Mech Eng Sci 225:2537–2551

    Article  Google Scholar 

  • Ashida F (1993) Response of a piezothermoelastic plate of crystal class 6 mm subject to axisymmetric heating. Int J Eng Sci 31:373–384

    Article  MATH  Google Scholar 

  • El-Naggar AM, Abd-Alla AM, Ahmed SM (1995) Influence of gravity and initial stress on the propagation of waves in an orthotropic elastic medium. Rev Roum Sci Tech Mec Appl 40:403–408

    MATH  Google Scholar 

  • Grover D, Sharma JN (2011) Transverse vibrations in piezothermoelastic beam resonators. J Intell Mat Syst Struct 23:77–84

    Article  Google Scholar 

  • Hou PF, Luo ZW, Leung AYT (2010) 2D Green’s functions for two-phase orthotropic piezothermoelastic plane. J Intell Mat Sys Struct 21:557–565

    Article  Google Scholar 

  • Marin M (2008) Weak solutions in elasticity of dipolar porous materials. Math Probl Eng 2008:1–8. Art. No. 158908

  • Marin M (2016) An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 51:1127–1133

    Article  MathSciNet  MATH  Google Scholar 

  • Marin M, Baleanu D (2016) On vibrations in thermoelasticity without energy dissipation for micropolar bodies. Bound Value Prob 2016:1–19

    Article  MathSciNet  MATH  Google Scholar 

  • Marin M, Agarwal RP, Mahmoud SR (2013) Modeling a microstretch thermoelastic body with two temperatures. Abstr Appl Anal 2013:1–7. Article ID 583464

  • Mindlin RD (1974) Equations of high frequency vibrations of thermopiezoelectric crystal plates. Int J Solids Struct 10:625–637

    Article  MATH  Google Scholar 

  • Niraula OP, Noda N (2002) Thermal stresses analysis of piezothermoelastic strip with an edge crack. J Thermal Stress 25:389–405

    Article  MATH  Google Scholar 

  • Othman MIA, Ahmed EAA (2015) The effect of rotation on piezothermoelastic medium using different theories. Struct Eng Mech 56:649–665

    Article  Google Scholar 

  • Othman MIA, Ahmed EAA (2016) The effect of gravity field on piezo-thermoelastic medium with three theories. J Thermal Stress 39:474–486

    Article  Google Scholar 

  • Othman MIA, Kumar R (2009) Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity. Int Commun Heat Mass Transf 36:513–520

    Article  Google Scholar 

  • Othman MIA, Said SM (2012) The effect of rotation on two-dimensional problem of a fiber-reinforced thermoelastic with one relaxation time. Int J Thermophys 33:160–171

    Article  Google Scholar 

  • Othman MIA, Elmaklizi YD, Said SM (2013) Generalized thermoelastic medium with temperature-dependent properties for different theories under the effect of gravity field. Int J Thermophys 34:512–537

    Google Scholar 

  • Othman MIA, Elmaklizi YD, Ahmed EAA (2017a) Influence of magnetic field on generalized piezo-thermoelastic rotating medium with two relaxation times. Microsyst Technol 23:5599–5612

    Article  Google Scholar 

  • Othman MIA, Elmaklizi YD, Ahmed EAA (2017b) Effect of magnetic field on piezothermo-elastic medium with three theories. Results Phys 7:3361–3368

    Article  Google Scholar 

  • Schoenberg M, Censor D (1973) Elastic waves in rotating media. Q Appl Maths 31:115–125

    Article  MATH  Google Scholar 

  • Sharma MD (2010) Propagation of inhomogeneous waves in anisotropic piezo-thermo-elastic media. Acta Mech 215:307–318

    Article  MATH  Google Scholar 

  • Sharma JN, Kumar M (2000) Plane harmonic waves in piezo-thermoelastic materials. Indian J Eng Mater Sci 7:434–442

    Google Scholar 

  • Sharma JN, Walia V, Gupta SK (2008) Reflection of piezothermoelastic waves from the charge and stress free boundary of a transversely isotropic half space. Int J Eng Sci 46:131–146

    Article  MathSciNet  MATH  Google Scholar 

  • Tianhu H, Xiaogeneg T, Yapeng S (2002) Two-dimensional generalized thermal shock a thick piezoelectric plate of infinite extent. Int J Eng Sci 40:2249–2264

    Article  MATH  Google Scholar 

  • Vel SS, Batra RC (2003) Generalized plane strain thermopiezoelectric analysis of multi-layered plates. J Thermal Stress 26:353–377

    Article  Google Scholar 

  • Walia V, Sharma JN, Sharma PK (2009) Propagation characteristics of thermoelastic waves in piezoelectric (6 mm class) rotating plate. Eur J Mech A Solids 28:569–581

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Mohamed I. A. Othman

  2. Department of Applied Mathematics, Faculty of Science, Cairo University, Cairo, Egypt

    Ethar A. A. Ahmed

Authors
  1. Mohamed I. A. Othman
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  2. Ethar A. A. Ahmed
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Correspondence to Ethar A. A. Ahmed.

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Appendices

Appendix A

$$\begin{aligned} \delta_{1} = \frac{{C_{11} + \mu_{0} H_{0}^{2} }}{{\rho v_{p}^{2} }},\;\delta_{2} = \frac{{C_{44} }}{{\rho v_{p}^{2} }},\;\delta_{3} = \frac{{C_{13} + C_{44} + \mu_{0} H_{0}^{2} }}{{\rho v_{p}^{2} }},\;\delta_{4} = \frac{{(e_{31} + e_{15} )}}{{e_{33} }},\;\delta_{5} = \left( {1 + \frac{{\varepsilon_{0} \mu_{0}^{2} H_{0}^{2} }}{\rho }} \right), \hfill \\ \delta_{6} = \frac{{C_{33} + \mu_{0} H_{0}^{2} }}{{\rho v_{p}^{2} }},\;\delta_{7} = \frac{{e_{15} }}{{e_{33} }},\;\delta_{8} = - \frac{{\beta_{3} }}{{\beta_{1} }},\;\delta_{9} = \frac{{(e_{15} + e_{31} )}}{{\rho v_{p}^{2} }},\;\delta_{10} = \frac{{e_{15} }}{{\rho v_{p}^{2} }},\;\delta_{11} = \frac{{e_{33} }}{{\rho v_{p}^{2} }}, \hfill \\ \delta_{12} = - \frac{{ \in_{11} }}{{e_{33} }},\;\delta_{13} = - \frac{{ \in_{33} }}{{e_{33} }},\;\delta_{14} = \frac{{p_{3} }}{{\beta_{1} }},\;\delta_{15} = \frac{{K_{1} \omega^{*} }}{{\rho C_{e} {\kern 1pt} v_{p}^{2} }},\;\delta_{16} = \frac{{K_{3} \omega^{*} }}{{\rho C_{e} {\kern 1pt} v_{p}^{2} }},\;\delta_{17} = \frac{{\beta_{1}^{2} {\kern 1pt} T_{0} }}{{\rho^{2} C_{e} {\kern 1pt} v_{p}^{2} }}, \hfill \\ \delta_{18} = \frac{{\beta_{1} \beta_{3} T_{0} }}{{\rho^{2} C_{e} v_{p}^{2} }},\;\delta_{19} = - \frac{{p_{3} \beta_{1} T_{0} }}{{\rho C_{e} e_{33} }}. \hfill \\ \end{aligned}$$
(52)
$$\begin{aligned} A_{1} = - \frac{{(a^{2} \delta_{1} - a^{2} c^{2} \delta_{5} - \Omega^{2} )}}{{\delta_{2} }},\;A_{2} = \frac{{ia\delta_{3} }}{{\delta_{2} }},\;A_{3} = \frac{{iag + 2ia{\kern 1pt} c{\kern 1pt} \Omega }}{{\delta_{2} }},\;A_{4} = \frac{{ia\delta_{4} }}{{\delta_{2} }}, \hfill \\ A_{5} = - \frac{{ia(1 - i{\kern 1pt} a{\kern 1pt} c{\kern 1pt} t_{1} )}}{{\delta_{2} }},\;A_{6} = \frac{{ia\delta_{3} }}{{\delta_{6} }},\;A_{7} = - \frac{{iag + 2ia{\kern 1pt} c{\kern 1pt} \Omega }}{{\delta_{6} }},\;A_{8} = - \frac{{(a^{2} \delta_{2} - a^{2} c^{2} \delta_{5} - \Omega^{2} )}}{{\delta_{6} }}, \hfill \\ A_{9} = \frac{1}{{\delta_{6} }},\;A_{10} = - \frac{{a^{2} \delta_{7} }}{{\delta_{6} }},\;A_{11} = \frac{{\delta_{8} (1 - ia{\kern 1pt} c{\kern 1pt} t_{1} )}}{{\delta_{6} }},\;A_{12} = \frac{{ia\delta_{9} }}{{\delta_{11} }},\;A_{13} = - \frac{{a^{2} \delta_{10} }}{{\delta_{11} }},\;A_{14} = \frac{{\delta_{13} }}{{\delta_{11} }}, \hfill \\ A_{15} = - \frac{{a^{2} \delta_{12} }}{{\delta_{11} }},\;A_{16} = \frac{{\delta_{14} (1 - ia{\kern 1pt} c{\kern 1pt} t_{1} )}}{{\delta_{11} }},\;A_{17} = \frac{{i{\kern 1pt} a{\kern 1pt} \delta_{17} }}{{\delta_{16} }}(ia{\kern 1pt} {\kern 1pt} c + a^{2} c^{2} n_{1} {\kern 1pt} t_{0} ),\;A_{18} = \frac{{\delta_{18} }}{{\delta_{16} }}(ia{\kern 1pt} c + a^{2} c^{2} n_{1} t_{0} ), \hfill \\ A_{19} = \frac{{\delta_{19} }}{{\delta_{16} }}(ia{\kern 1pt} c + a^{2} c^{2} n_{1} t_{0} ),\;A_{20} = - \frac{{(a^{2} \delta_{15} - ia{\kern 1pt} c - a^{2} c^{2} t_{0} )}}{{\delta_{16} }}. \hfill \\ \end{aligned}$$
(53)
$$\begin{aligned} A = - \left( {\frac{1}{{A_{14} - A_{9} }}} \right)(A_{14} A_{20} + A_{15} - A_{16} A_{19} + A_{8} A_{14} - A_{9} A_{13} - A_{9} A_{20} + A_{9} A_{16} A_{18} - A_{10} \hfill \\ + A_{11} A_{19} - A_{11} A_{14} A_{18} + A_{1} A_{14} - A_{1} A_{9} - A_{2} A_{6} A_{14} + A_{2} A_{9} A_{12} + A_{4} A_{6} - A_{4} A_{12} ) \hfill \\ \end{aligned}$$
(54)
$$\begin{aligned} B = \left( {\frac{1}{{A_{14} - A_{9} }}} \right)(A_{15} A_{20} + A_{8} A_{14} A_{20} + A_{8} A_{15} - A_{8} A_{16} A_{19} - A_{9} A_{13} A_{20} - A_{10} A_{13} - A_{10} A_{20} \hfill \\ + A_{10} A_{16} A_{18} + A_{11} A_{13} A_{19} - A_{11} A_{15} A_{18} + A_{1} A_{14} A_{20} + A_{1} A_{15} - A_{1} A_{16} A_{19} + A_{1} A_{8} A_{14} \hfill \\ - A_{1} A_{9} A_{13} - A_{1} A_{9} A_{20} + A_{1} A_{9} A_{16} A_{18} - A_{1} A_{10} + A_{1} A_{11} A_{19} - A_{1} A_{11} A_{14} A_{18} - A_{2} A_{6} A_{14} A_{20} \hfill \\ - A_{2} A_{6} A_{15} + A_{2} A_{6} A_{16} A_{19} + A_{2} A_{9} A_{12} A_{20} - A_{2} A_{9} A_{16} A_{17} + A_{2} A_{10} A_{12} - A_{2} A_{11} A_{12} A_{19} \hfill \\ + A_{2} A_{11} A_{14} A_{17} - A_{3} A_{7} A_{14} + A_{4} A_{6} A_{13} + A_{4} A_{6} A_{20} - A_{4} A_{6} A_{16} A_{18} - A_{4} A_{12} A_{20} + A_{4} A_{16} A_{17} \hfill \\ - A_{4} A_{8} A_{12} + A_{4} A_{11} A_{12} A_{18} - A_{4} A_{11} A_{17} - A_{5} A_{6} A_{19} + A_{5} A_{6} A_{14} A_{18} + A_{5} A_{12} A_{19} - A_{5} A_{14} A_{17} \hfill \\ - A_{5} A_{9} A_{12} A_{18} + A_{5} A_{9} A_{17} ) \hfill \\ \end{aligned}$$
(55)
$$\begin{aligned} C = - \left( {\frac{1}{{A_{14} - A_{9} }}} \right)(A_{8} A_{15} A_{20} - A_{10} A_{13} A_{20} + A_{1} A_{15} A_{20} + A_{1} A_{8} A_{14} A_{20} + A_{1} A_{8} A_{15} - A_{1} A_{8} A_{16} A_{19} \hfill \\ - A_{1} A_{9} A_{13} A_{20} - A_{1} A_{10} A_{13} - A_{1} A_{10} A_{20} + A_{1} A_{10} A_{16} A_{18} + A_{1} A_{11} A_{13} A_{19} - A_{1} A_{11} A_{15} A_{18} \hfill \\ - A_{2} A_{6} A_{15} A_{20} + A_{2} A_{10} A_{12} A_{20} - A_{2} A_{10} A_{16} A_{17} + A_{2} A_{11} A_{15} A_{17} - A_{3} A_{7} A_{14} A_{20} - A_{3} A_{7} A_{15} + A_{3} A_{7} A_{16} A_{19} \hfill \\ + A_{4} A_{6} A_{13} A_{20} - A_{4} A_{8} A_{12} A_{20} + A_{4} A_{8} A_{16} A_{17} - A_{4} A_{11} A_{13} A_{17} - A_{5} A_{6} A_{13} A_{19} + A_{5} A_{6} A_{15} A_{18} \hfill \\ - A_{5} A_{15} A_{17} + A_{5} A_{8} A_{12} A_{19} - A_{5} A_{8} A_{14} A_{17} + A_{5} A_{9} A_{13} A_{17} - A_{5} A_{10} A_{12} A_{18} + A_{5} A_{10} A_{17} ) \hfill \\ \end{aligned}$$
(56)
$$E = \left( {\frac{1}{{A_{14} - A_{9} }}} \right)(A_{1} A_{8} A_{15} A_{20} - A_{1} A_{10} A_{13} A_{20} - A_{3} A_{7} A_{15} A_{20} - A_{5} A_{8} A_{15} A_{17} + A_{5} A_{10} A_{13} A_{17} )$$
(57)

From the MATLAB program we can validate the above expressions (55)–(57).

Appendix B

$$\begin{aligned} H_{1n} = - \frac{{s_{1n} }}{{s_{2n} }},\;H_{2n} = - \frac{{(q_{1n} + q_{2n} H_{1n} )}}{{q_{3n} }},\;H_{3n} = - \frac{1}{{A_{5} }}[(k_{n}^{2} + A_{1} ) + ( - A_{2} k_{n} + A_{3} )H_{1n} - A_{4} k_{n} H_{2n} ], \hfill \\ H_{4n} = r_{1} - l_{2} k_{n} H_{1n} - l_{3} k_{n} H_{2n} + r_{2} H_{3n} ,H_{5n} = r_{3} - l_{4} k_{n} H_{1n} - k_{n} H_{2n} + r_{4} H_{3n} , \hfill \\ H_{6n} = - \delta_{2} k_{n} + r_{5} H_{1n} + r_{6} H_{2n} ,H_{7n} = - l_{5} k_{n} + r_{7} H_{1n} + r_{8} H_{2n} , \hfill \\ H_{8n} = r_{9} - l_{8} k_{n} H_{1n} - l_{9} k_{n} H_{2n} + r_{10} H_{3n} ,{\text{n}} = 1,2,3,4. \hfill \\ \end{aligned}$$
(58)
$$\begin{aligned} q_{1n} = A_{11} k_{n}^{3} + (A_{1} A_{11} - A_{5} A_{6} )k_{n} + A_{5} A_{7} ,\;q_{2n} = ( - A_{2} A_{11} + A_{5} )k_{n}^{2} + A_{3} A_{11} k_{n} + A_{5} A_{8} , \hfill \\ q_{3n} = ( - A_{4} A_{11} + A_{5} A_{9} )k_{n}^{2} + A_{5} A_{10} ,\;q_{4n} = A_{16} k_{n}^{3} + (A_{1} A_{16} - A_{5} A_{12} )k_{n} , \hfill \\ q_{5n} = (A_{5} - A_{2} A_{16} )k_{n}^{2} + A_{3} A_{16} k_{n} + A_{5} A_{13} ,\;q_{6n} = (A_{5} A_{14} - A_{4} A_{16} )k_{n}^{2} + A_{5} A_{15} , \hfill \\ \end{aligned}$$
(59)
$$\text{s}_{1n} = q_{1n} q_{6n} - q_{3n} q_{4n} ,\;\text{s}_{2n} = q_{2n} q_{6n} - q_{3n} q_{5n} ,$$
(60)
$$\begin{aligned} l_{1} = \frac{{C_{11} }}{{\rho v_{p}^{2} }},\;l_{2} = \frac{{C_{13} }}{{\rho v_{p}^{2} }},\;l_{3} = \frac{{e_{31} }}{{e_{33} }},\;l_{4} = \frac{{C_{33} }}{{\rho v_{p}^{2} }},\;l_{5} = \frac{{e_{15} \beta_{1} T_{0} }}{{e\rho \,v_{p}^{2} }},\;l_{6} = - \frac{{ \in_{11} \beta_{1} T_{0} }}{{e\,e_{33} }}, \hfill \\ l_{7} = \frac{{e_{31} \beta_{1} T_{0} }}{{e\rho \,v_{p}^{2} }},\;l_{8} = \frac{{e_{33} \beta_{1} T_{0} }}{{e\rho \,v_{p}^{2} }},\;l_{9} = - \frac{{ \in_{33} \beta_{1} T_{0} }}{{e\,e_{33} }},\;l_{10} = \frac{{p_{3} T_{0} }}{e}, \hfill \\ \end{aligned}$$
(61)
$$\begin{aligned} \{ r_{1} ,r_{3} ,r_{5} ,r_{6} ,r_{7} ,r_{8} ,r_{9} \} = ia\{ l_{1} ,l_{2} ,\delta_{2} ,\delta_{7} ,l_{5} ,l_{6} ,l_{7} \} ,\;r_{2} = - (1 - i{\kern 1pt} a{\kern 1pt} c{\kern 1pt} t_{1} ),\;r_{4} = \delta_{8} (1 - i{\kern 1pt} a{\kern 1pt} c{\kern 1pt} t_{1} ), \hfill \\ r_{10} = l_{10} (1 - i{\kern 1pt} a{\kern 1pt} c{\kern 1pt} t_{1} ) \hfill \\ \end{aligned}$$
(62)

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Othman, M.I.A., Ahmed, E.A.A. Exact analytical solution of a homogeneous anisotropic piezo-thermoelasic half-space of a hexagonal type under different fields with three theories. Microsyst Technol 25, 1423–1435 (2019). https://doi.org/10.1007/s00542-018-4089-6

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