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

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Abstract

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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Abbreviations

\(u_{i}\) :

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

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

    Ethar A. A. Ahmed, M. S. Abou-Dina & A. F. Ghaleb

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  1. Ethar A. A. Ahmed
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  2. M. S. Abou-Dina
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  3. A. F. Ghaleb
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Correspondence to Ethar A. A. Ahmed.

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Appendices

Appendices

1.1 Appendix A

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

and

$$\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}$$

1.2 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). https://doi.org/10.1007/s00542-019-04567-0

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