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Plane wave propagation in a piezo-thermoelastic rotating medium within the dual-phase-lag model


We investigate the effect of rotation on plane wave propagation in a half-space of a piezo-thermoelastic material within the frame of dual-phase-lag model. Normal mode technique is used to obtain analytic expressions for the displacement components, temperature and stress components. Numerical results for the quantities of practical interest are given in the physical domain and illustrated graphically. Comparison is carried out between the results predicted by the dual-phase-lag model and Lord–Shulman theory, in the presence or absence of rotation. It is believed that the present results may be useful in the design and construction of different pyro/piezoelectric devices, such as gyroscopes and sensors.

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\(u_{i}\) :

The mechanical displacement

T :

Absolute temperature

\(\sigma _{ij}\) :

Stress tensor

\(E_{i}\) :

Electric field

\(C_{ijkl}\) :

Elastic stiffness tensor

\(\in _{ij}\) :

The dielectric moduli

\(\tau _{\theta }\) :

Phase lag of temperature gradient

\(K_{ij}\) :

Heat conduction tensor

\(C_{T}\) :

Specific heat at constant strain

\(\alpha _{1} ,\alpha _{3}\) :

Coefficients of linear thermal expansion

\(v_{p}=\sqrt{\frac{1}{\rho }C_{11}}\) :

Longitudinal wave velocity

\(\varphi\) :

Electric potential

\(\varepsilon _{ij}\) :

Strain tensor

\(\beta _{ij}\) :

Thermoelastic tensor

\(D_{i}\) :

Electric displacement

\(e_{ijk}\) :

Piezoelectric tensor

\(p_{i}\) :

Pyroelectric moduli

\(\tau _{q}\) :

Phase lag of the heat flux

\(T_{0}\) :

Reference temperature

\(\rho\) :

Mass density


  1. Abbas IA, Zenkour AM (2014) Dual-phase-lag model on thermoelastic interactions in a semi-infinite medium subjected to a ramp-type heating. J Comput Theor Nanosci 11(3):642–645

  2. Abd-Alla AN, Alsheikh FA (2009) Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses. Arch Appl Mech 79(9):843–857

  3. Abou-Dina MS, Dhaba EL, Ghaleb AF, Rawy EK (2017) A model of nonlinear thermo-electroelasticity in extended thermo-electroelasticity in extended thermoelasticity. Int J Eng Sci 119:29–39

  4. Ahmed EAA, Abou-Dina MS, El Dhaba AR (2019) Effect of gravity on piezo-thermoelasticity within the dual-phase-lag model. Microsyst Technol 25:1–10

  5. Alshaikh FA (2012) The mathematical modelling for studying the influence of the initial stresses and relaxation times on reflection and refraction waves in piezothermoelastic half-space. Appl Math 3:819–832

  6. Ciesielski M (2017) Analytical solution of the dual phase lag equation describing the laser heating of thin metal film. J Appl Math Comput Mech 16(1):33–40

  7. Hou PF, Leung AYT (2009) Three-dimensional Green\(^{\prime }\)s functions for two-phase transversely isotropic piezothermoelastic media. J Intell Mater Syst Struct 16(5):1915–1923

  8. Kumar R, Sharma N, Lata P, Marin M (2018) Reflection of plane waves at micropolar piezothermoelastic half-space. CMST 24(1):113–124

  9. Li L, Wei PJ (2014) The piezoelectric and piezomagnetic effect on the surface wave velocity of magneto-electroelastic solids. J Sound Vib 333(8):2312–2326

  10. Mahmoud SR (2016) An analytical solution for the effect of initial stress, rotation, magnetic field and a periodic loading in a thermoviscoelastic medium with a spherical cavity. Mech Adv Mater Struct 23(1):1–7

  11. Mindlin RD (1974) Equations of high frequency vibrations of thermo-piezo-electric plate. Int J Solids Struct 10:625–637

  12. Othman MIA (2004) Effect of rotation on plane waves in generalized thermoelasticity with two relaxation times. Int J Solids Struct 41:2939–2956

  13. Othman MIA, Ahmed EAA (2015) The effect of rotation on piezo-thermoelastic medium using different theories. Struct Eng Mech 56(4):649–665

  14. Othman MIA, Ahmed EAA (2016) Influence of the gravitational field on piezo-thermoelastic rotating medium with G–L theory. Eur Phys J Plus 131:1–12

  15. Othman MIA, Elmaklizi YD, Ahmed EAA (2017) Influence of magnetic field on generalized piezo-thermoelastic rotating medium with two relaxation times. Microsyst Technol 23(12):5599–5612

  16. Othman MIA, Hasona WM, Eraki EEM (2013) Influence of gravity field and rotation on a generalized thermoelastic medium using a dual-phase-lag model. J Thermoelast 1(4):12–22

  17. Othman MIA, Hasona WM, Eraki EEM (2014) Effect of rotation on micro-polar generalized thermoelasticity with two-temperatures using a dual-phase-lag model. Can J Phys 92(2):149–158

  18. Quintanilla R, Racke R (2006) A note on stability of dual-phase-lag heat conduction. Int J Heat Mass Transf 49:1209–1213

  19. Roy Choudhuri SK (2007) One-dimensional thermoelastic waves in elastic half-space with dual phase lag effects. J Mech Mater Struct 2(3):489–503

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

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

  22. Singh B, Kumari S, Singh J (2017) Propagation of Rayleigh wave in two temperature dual phase lag thermoelasticity. Mech Mech Eng 21(1):105–116

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Correspondence to Ethar A. A. Ahmed.

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Appendix A

$$\begin{aligned} \delta _{1}&=\frac{C_{11}}{\rho v_{p}^{2}}, \delta _{2} =\frac{C_{44}}{ \rho v_{p}^{2}}, \\ \delta _{3}&=\frac{(C_{13}+C_{44})}{\rho v_{p}^{2}}, \delta _{4} =\frac{(e_{31}+e_{15})}{e_{33}}, \delta _{5} =\frac{C_{33}}{ \rho v_{p}^{2}}, \\ \displaystyle \delta _{6}&=\frac{e_{15}}{e_{33}},\displaystyle \delta _{7} =-\frac{\beta _{3}}{\beta _{1}},\\ \displaystyle \delta _{8}&=\frac{ (e_{15}+e_{31})}{\rho v_{p}^{2}},\displaystyle \delta _{9} =\frac{e_{15}}{ \rho v_{p}^{2}},\\ \displaystyle \delta _{10}&=\frac{e_{33}}{\rho v_{p}^{2}},\\ \displaystyle \delta _{11}&=-\frac{\epsilon _{11}}{e_{33}},\displaystyle \delta _{12} =-\frac{\epsilon _{33}}{e_{33}},\\ \displaystyle \delta _{13}&= \frac{P_{3}}{\beta _{1}},\displaystyle \delta _{14} =\frac{K_{1}\omega ^{*}}{\rho C_{T}v_{p}^{2}},\\ \displaystyle \delta _{15}&=\frac{ K_{3}\omega ^{*}}{\rho C_{T}v_{p}^{2}}, \\ \displaystyle \delta _{16}&=\frac{\beta _{1}^{2}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}},\displaystyle \delta _{17} =\frac{\beta _{1}\beta _{3}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}},\\ \displaystyle \delta _{18}&=-\frac{ P_{3}\beta _{1}T_{0}}{\rho C_{T}e_{33}}. \end{aligned}$$


$$\begin{aligned} \displaystyle A_{1}= & \frac{a^{2}c^{2}-a^{2}\delta _{1}+\Omega ^{2}}{\delta _{2}}, \\ \displaystyle A_{2}= & \frac{ia\delta _{3}}{\delta _{2}},\\ \displaystyle A_{3}= & \frac{2iac\Omega }{\delta _{2}}, \\ \displaystyle A_{4}= & \frac{ia\delta _{4}}{\delta _{2}},\\ \displaystyle A_{5}= & \frac{-ia}{\delta _{2}}, \\ \displaystyle A_{6}= & \frac{ ia\delta _{3} }{\delta _{5}}, \\ \displaystyle A_{7}= & \frac{-2iac\Omega }{\delta _{5}},\\ \displaystyle A_{8}= & \frac{a^{2}c^{2}-a^{2}\delta _{2}+\Omega ^{2}}{\delta _{5}},\\ \displaystyle A_{9}= & \frac{1}{\delta _{5}}, \\ \displaystyle A_{10}= & -\frac{a^{2}\delta _{6}}{\delta _{5}},\\ \displaystyle A_{11}= & \frac{\delta _{7}}{\delta _{5}},\\ \displaystyle A_{12}= & \frac{ia\delta _{8}}{\delta _{10}}, \\ \displaystyle A_{13}= & -\frac{a^{2}\delta _{9}}{\delta _{10}},\\ \displaystyle A_{14}= & \frac{\delta _{12}}{\delta _{10}},\\ \displaystyle A_{15}= & -\frac{a^{2}\delta _{11}}{\delta _{10}}, \\ \displaystyle A_{16}= & \frac{\delta _{13}}{\delta _{10}},\\ \displaystyle A_{17}= & -\frac{a^{2}c\delta _{16}(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })}, \\ \displaystyle A_{18}= & \frac{iac\delta _{17}(1-iac\tau _{q}) }{\delta _{15}(1-iac\tau _{\theta })}, \\ \displaystyle A_{19}= & \frac{iac\delta _{18}(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })}, \\ \displaystyle A_{20}= & \frac{-a^{2}\delta _{14}(1-iac\tau _{\theta })+iac(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })}, \\ \displaystyle A= & \frac{-1}{A_{14}-A_{9}}\left( A_{14}A_{20}+A_{15}-A_{16}A_{19}+A_{8}A_{14}\right. \\&\left. -A_{9}A_{20}-A_{9}A_{13}A_{9}A_{16}A_{18}-A_{10}+A_{11}A_{19}\right. \\&-\left. A_{11}A_{14}A_{18}+A_{1}A_{14}-A_{1}A_{19}A_{2}A_{6}A_{14}\right. \\&\left. +A_{2}A_{9}A_{12}+A_{4}A_{6}-A_{4}A_{12}\right) ,\\ \displaystyle B= & \frac{1}{(A_{14}-A_{9})} (A_{5}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}+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}-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}-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}+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}-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}-A_{5}A_{9}A_{12}A_{18}+A_{5}A_{9}A_{17}),\\ \displaystyle C= & \frac{-1}{(A_{14}-A_{9})} (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}\\&-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}\\&-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}+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}-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}),\\ \displaystyle E= & \frac{1}{(A_{14}-A_{9})} (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}).\\ \end{aligned}$$

Appendix B

$$\begin{aligned} \displaystyle H_{1n}&= -\frac{s_{1n}}{s_{2n}}, \\ \displaystyle H_{2n}&= -\frac{q_{1n}+q_{2n}H_{1n}}{q_{3n}},\\ \displaystyle H_{3n}&= -\frac{(k_{n}^{2}+A_{1})+(-A_{2}k_{n}+A_{3})H_{1n}-A_{4}k_{n}H_{2n}}{A_{5}},\\ \displaystyle H_{4n}&= [r_{1}-l_{1}k_{n}H_{1n}-l_{2}k_{n}H_{2n}-H_{3n}], \\ \displaystyle H_{5n}&= [r_{2}-\delta _{5}k_{n}H_{1n}-k_{n}H_{2n}+\delta _{7}H_{3n}],\\ \displaystyle H_{6n}&= [-\delta _{2}k_{n}+r_{3}H_{1n}+r_{4}H_{2n}],\\ \displaystyle H_{7n}&= [-l_{3}k_{n}+r_{5}H_{1n}+r_{6}H_{2n}],\\ \displaystyle H_{8n}&= [r_{7}-l_{6}k_{n}H_{1n}-l_{7}k_{n}H_{2n}+l_{8}H_{3n}]. \\&\quad n=1,2,3,4.\\ \displaystyle q_{1n}&= A_{11}k_{n}^{3}+(A_{1}A_{11}-A_{5}A_{6})k_{n}+A_{5}A_{7},\\ \displaystyle q_{2n}&= (-A_{2}A_{11}+A_{5})k_{n}^{2}+A_{3}A_{11}k_{n}+A_{5}A_{8},\\ \displaystyle q_{3n}&= (-A_{4}A_{11}+A_{5}A_{9})k_{n}^{2}+A_{5}A_{10}, \\ \displaystyle q_{4n}&= A_{16}k_{n}^{3}+(A_{1}A_{16}-A_{5}A_{12})k_{n},\\ \displaystyle q_{5n}&= (A_{5}-A_{2}A_{16})k_{n}^{2}+A_{3}A_{16}k_{n}+A_{5}A_{13}, \\ \displaystyle q_{6n}&= (A_{5}A_{14}-A_{4}A_{16})k_{n}^{2}+A_{5}A_{15}.\\ \displaystyle s_{1n}&= q_{1n}q_{6n}-q_{3n}q_{4n}, \\ \displaystyle s_{2n}&= q_{2n}q_{6n}-q_{3n}q_{5n}.\\ \displaystyle l_{1}&= \frac{C_{13}}{\rho v_{p}^{2}}, \\ \displaystyle l_{2}&= \frac{e_{31}}{e_{33}}, \\ \displaystyle l_{3}&= \frac{e_{15}\beta _{1}T_{0}}{e\rho v_{p}^{2}},\\ \displaystyle l_{4}&= -\frac{\epsilon _{11}\beta _{1}T_{0}}{ee_{33}},\\ \displaystyle l_{5}&= \frac{e_{31}\beta _{1}T_{0}}{e\rho v_{p}^{2}},\\ \displaystyle l_{6}&= \frac{e_{33}\beta _{1}T_{0}}{e\rho v_{p}^{2}}, \ \ \ \ \ \ \ \\ \displaystyle l_{7}&= -\frac{\epsilon _{33}\beta _{1}T_{0}}{ee_{33}}, \ \ \ \\ \displaystyle l_{8}&= -\frac{P_{3}T_{0}}{e}. \\&\displaystyle \{r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7}\}=ia\{\delta _{1},l_{1},\delta _{2},\delta _{6},l_{3},l_{4},l_{5}\} \end{aligned}$$

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Ahmed, E.A.A., Abou-Dina, M.S. & Ghaleb, A.F. Plane wave propagation in a piezo-thermoelastic rotating medium within the dual-phase-lag model. Microsyst Technol 26, 969–979 (2020).

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