Thermoelastic wave propagation in a piezoelectric layered half-space within the dual-phase-lag model

Abstract

We investigate linear, thermoelastic wave propagation in a layered piezoelectric material composed of a slab bonded to a half-space substrate of a dissimilar material, within dual-phase-lag model and under thermomechanical loads. One of the aims of the present work is to formulate a set of boundary conditions that is compatible with the field equations. Normal mode technique is used to obtain a solution to the considered problem. The model allows for a jump in temperature at the interface, and this can be used to evaluate a material constant of the slab. It turns out, particularly, that a normal electric field is generated outside the structure. Numerical results for all quantities of practical interest are obtained and discussed. A relation between the jump in temperature and the slab thickness is given. These results may be useful in determining the values of the material constants of the slab, in particular the thermal relaxation times.

Appendix A

Coefficients appearing in Eqs. (14a)–(14d)

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

Coefficients appearing in Eqs. (16a)–(16d)

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

$\displaystyle A=\frac{-1}{A_{12}-A_{7}}( A_{13}-A_{8}-A_{1}A_{7}+A_{3}A_{5}+A_{1}A_{12}-A_{3}A_{10}+A_{6}A_{12}-A_{7}A_{11}$ $+A_{7}A_{18} +A_{9}A_{17}+A_{12}A_{18}-A_{14}A_{17}-A_{2}A_{5}A_{12}+A_{2}A_{7}A_{10}$ $+A_{7}A_{14}A_{16}-A_{9}A_{12}A_{16}),$

$\displaystyle B=\frac{1}{A_{12}-A_{7}} (-A_{1}A_{8}+A_{1}A_{13}+A_{6}A_{13}-A_{8}A_{11}-A_{8}A_{18}+A_{13}A_{18}+A_{1}A_{6}A_{12} -A_{1}A_{7}A_{11}+A_{3}A_{5}A_{18}-A_{4}A_{5}A_{17}+A_{4}A_{7}A_{15}+A_{1}A_{9}A_{17}-A_{3}A_{9}A_{15}\,+$ $A_{1}A_{12}A_{18}-A_{3}A_{10}A_{18}+A_{4}A_{10}A_{17}-A_{4}A_{12}A_{15}-A_{1}A_{14}A_{17}+A_{3}A_{14}A_{15}+A_{6}A_{12}A_{18}-A_{7}A_{11}A_{18}-A_{6}A_{14}A_{17}+A_{9}A_{11}A_{17}+A_{8}A_{14}A_{16}-A_{9}A_{13}A_{16}-A_{2}A_{5}A_{12}A_{18}$ $+\,A_{2}A_{7}A_{10}A_{18}+A_{4}A_{5}A_{12}A_{16}-A_{4}A_{7}A_{10}A_{16}+A_{1}A_{7}A_{14}A_{16}-A_{1}A_{9}A_{12}A_{16}+A_{2}A_{5}A_{14}A_{17}-A_{2}A_{7}A_{14}A_{15}-A_{2}A_{9}A_{10}A_{17} +A_{2}A_{9}A_{12}A_{15}-A_{3}A_{5}A_{14}A_{16}+A_{3}A_{9}A_{10}A_{16}),$

$\displaystyle C=\frac{-1}{A_{12}-A_{7}} (A_{1}A_{6}A_{13}-A_{1}A_{8}A_{11}-A_{1}A_{8}A_{18}+A_{4}A_{8}A_{15}+A_{1}A_{13}A_{18}-A_{4}A_{13}A_{15}+A_{6}A_{13}A_{18}-A_{8}A_{11}A_{18}+A_{1}A_{6}A_{12}A_{18}-A_{1}A_{7}A_{11}A_{18}+A_{3}A_{5}A_{11}A_{18}-A_{3}A_{6}A_{10}A_{18}-$ $A_{4}A_{5}A_{11}A_{17}+A_{4}A_{6}A_{10}A_{17}-A_{4}A_{6}A_{12}A_{15}+A_{4}A_{7}A_{11}A_{15}-A_{1}A_{6}A_{14}A_{17}+A_{1}A_{9}A_{11}A_{17}-A_{2}A_{5}A_{13}A_{18}+A_{2}A_{8}A_{10}A_{18}+A_{3}A_{6}A_{14}A_{15} - A_{3}A_{9}A_{11}A_{15} +A_{4}A_{5}A_{13}A_{16} - A_{4}A_{8}A_{10}A_{16}$ $+A_{1}A_{8}A_{14}A_{16}-A_{1}A_{9}A_{13}A_{16} - A_{2}A_{8}A_{14}A_{15}+A_{2}A_{9}A_{13}A_{15}),$

$\displaystyle E=\frac{1}{A_{12}-A_{7}} (A_{1}A_{6}A_{13}A_{18}-A_{1}A_{8}A_{11}A_{18}-A_{4}A_{6}A_{13}A_{15}+A_{4}A_{8}A_{11}A_{15}).$

1.1 Appendix B

Coefficients appearing in Eqs. (20a)–(20e)

$$\begin{aligned} \displaystyle H_{1n}= & {} -\frac{s_{1n}}{s_{2n}} ,\\ \displaystyle H_{2n}= & {} -\frac{q_{1n}+q_{2n}H_{1n}}{q_{3n}}, \\ \displaystyle H_{3n}= & {} -\frac{ (k_{n}^{2}+A_{1})+(-A_{2}k_{n}+A_{3})H_{1n}-A_{4}k_{n}H_{2n}}{A_{4}}, \\ \displaystyle H_{4n}= & {} r_{1}-l_{1}k_{n}H_{1n}-l_{2}k_{n}H_{2n}-H_{3n},\\ \displaystyle H_{5n}= & {} r_{2}-\delta _{5}k_{n}H_{1n}-k_{n}H_{2n}+\delta _{7}H_{3n},\\ \displaystyle H_{6n}= & {} -\delta _{2}k_{n}+r_{3}H_{1n}+r_{4}H_{2n}],\\ \displaystyle H_{7n}= & {} -l_{3}k_{n}+r_{5}H_{1n}+r_{6}H_{2n}, \\ H_{8n}= & {} r_{7}-l_{6}k_{n}H_{1n}-l_{7}k_{n}H_{2n}+l_{8}H_{3n}, \\ n= & {} 1,2,3,4.\\ \displaystyle G_{1n}= & {} -\frac{y_{1n}}{y_{2n}}, \\ \displaystyle G_{2n}= & {} -\frac{z_{4n}+z_{5n}G_{1n}}{z_{6n}}, \\ \displaystyle G_{3n}= & {} -\frac{ (k_{n}^{2}+A_{1})+A_{2}k_{n}G_{1n}+A_{3}k_{n}G_{2n}}{A_{4}},\\ \displaystyle G_{4n}= & {} r_{1}+l_{1}k_{n}G_{1n}+l_{2}k_{n}G_{2n}+G_{3n}, \\ \displaystyle G_{5n}= & {} r_{2}+\delta _{5}k_{n}G_{1n}+k_{n}G_{2n}+\delta _{7}G_{3n},\\ \displaystyle G_{6n}= & {} [\delta _{2}k_{n}+r_{3}G_{1n}+r_{4}G_{2n}],\\ \displaystyle G_{7n}= & {} l_{3}k_{n}+r_{5}G_{1n}+r_{6}G_{2n},\\ \end{aligned}$$

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

1.2 Appendix C

Matrices appearing in Eq. (22)

$$\begin{aligned} \Lambda _{1}= & {} \left( \begin{array}{cccc} e^{-k_{1s}d} &{} e^{-k_{2s}d} &{} e^{-k_{3s}d} &{} e^{-k_{4s}d} \\ e^{-k_{1s}d}H_{11s} &{} e^{-k_{2s}d}H_{12s} &{} e^{-k_{3s}d}H_{13s} &{} e^{-k_{4s}d}H_{14s} \\ e^{-k_{1s}d}H_{61s} &{} e^{-k_{2s}d}H_{62s} &{} e^{-k_{3s}d}H_{63s} &{} e^{-k_{4s}d}H_{64s} \\ e^{-k_{1s}d}H_{51s} &{} e^{-k_{2s}d}H_{52s} &{} e^{-k_{3s}d}H_{53s} &{} e^{-k_{4s}d}H_{54s} \end{array} \right) , \\ \Lambda _{2}= & {} \left( \begin{array}{cccc} e^{k_{1s}d} &{} e^{k_{2s}d} &{} e^{k_{3s}d} &{} e^{k_{4s}d} \\ e^{k_{1}d}G_{11s} &{} e^{k_{2}d}G_{12s} &{} e^{k_{3}d}G_{13s} &{} e^{k_{4}d}G_{14s} \\ e^{k_{1}d}G_{61s} &{} e^{k_{2}d}G_{62s} &{} e^{k_{3}d}G_{63s} &{} e^{k_{4}d}G_{64s} \\ e^{k_{1}d}G_{51s} &{} e^{k_{2}d}G_{52s} &{} e^{k_{3}d}G_{53s} &{} e^{k_{4}d}G_{54s} \end{array} \right) , \\ \Lambda _{3}= & {} \left( \begin{array}{cccc} -e^{-k_{1h}d} &{} -e^{-k_{2h}d} &{} -e^{-k_{3h}d} &{} -e^{-k_{4h}d} \\ -e^{-k_{1h}d}H_{11h} &{} -e^{-k_{2h}d}H_{12h} &{} -e^{-k_{3h}d}H_{13h} &{} -e^{-k_{4h}d}H_{14h} \\ -e^{-k_{1h}d}H_{61h} &{} -e^{-k_{2h}d}H_{62h} &{} -e^{-k_{3h}d}H_{63h} &{} -e^{-k_{4h}d}H_{64h} \\ -e^{-k_{1h}d}H_{51h} &{} -e^{-k_{2h}d}H_{52h} &{} -e^{-k_{3h}d}H_{53h} &{} -e^{-k_{4h}d}H_{54h} \end{array} \right) ,\\ \Lambda _{4}= & {} \left( \begin{array}{cccc} K_{3s}e^{-k_{1s}d}H_{31s} &{} K_{3s} e^{-k_{2s}d}H_{32s} &{} K_{3s}e^{-k_{3s}d}H_{33s} &{} K_{3s}k_{2s}e^{-k_{3s}d}H_{34s} \\ -K_{3s}k_{1s}e^{-k_{1s}d}H_{31s} &{} -K_{3s}k_{2s} e^{-k_{2s}d}H_{32s} &{} -K_{3s}k_{3s}e^{-k_{3s}d}H_{33s} &{} -K_{3s}k_{4s}e^{-k_{3s}d}H_{34s} \end{array} \right) ,\\ \end{aligned}$$

$$\begin{aligned} \Lambda _{5}= & {} \left( \begin{array}{cccc} K_{3s}e^{k_{1s}d}G_{31s} &{} K_{3s} e^{k_{2s}d}G_{32s} &{} K_{3s}e^{k_{3s}d}H_{33s} &{} K_{3s}k_{2s}e^{k_{3s}d}G_{34s} \\ -K_{3s}k_{1s}e^{k_{1s}d}G_{31s} &{} -K_{3s}k_{2s} e^{k_{2s}d}G_{32s} &{} -K_{3s}k_{3s}k_{4s}e^{k_{3s}d}G_{33s} &{} -K_{3s}k_{4s}k_{1s}e^{k_{3s}d}G_{34s} \end{array} \right) ,\\ \Lambda _{6}= & {} \left( \begin{array}{cccc} -K_{3s}e^{-k_{1s}d}H_{31s} &{} -K_{3s} e^{-k_{2s}d}H_{32s} &{} -K_{3s}e^{-k_{3s}d}H_{33s} &{} -K_{3s}k_{2s}e^{-k_{3s}d}H_{34s} \\ K_{3s}k_{1s}e^{-k_{1s}d}H_{31s} &{} K_{3s}k_{2s} e^{-k_{2s}d}H_{32s} &{} K_{3s}k_{3s}e^{-k_{3s}d}H_{33s} &{} K_{3s}k_{4s}e^{-k_{3s}d}H_{34s} \end{array} \right) ,\\ \Lambda _{7}= & {} \left( \begin{array}{cccc} e^{-k_{1s}d}H_{21s} &{} e^{-k_{2s}d}H_{22s} &{} e^{-k_{3s}d}H_{23s} &{} e^{-k_{4s}d}H_{24s} \\ e^{-k_{1s}d}H_{81s} &{} e^{-k_{2s}d}H_{82s} &{} e^{-k_{3s}d}H_{83s} &{} e^{-k_{4s}d}H_{84s} \\ H_{61s} &{} H_{62s} &{} H_{63s} &{} H_{64s} \\ H_{51s} &{} H_{52s} &{} H_{53s} &{} H_{54s} \end{array} \right) , \\ \Lambda _{8}= & {} \left( \begin{array}{cccc} e^{k_{1s}d}G_{21} &{} e^{k_{2}d}G_{22s} &{} e^{k_{3}d}G_{23s} &{} e^{k_{4}d}G_{24s} \\ e^{k_{1s}d}G_{81} &{} e^{k_{2}d}G_{82s} &{} e^{k_{3}d}G_{83s} &{} e^{k_{4}d}G_{84s} \\ G_{61s} &{} G_{62s} &{} G_{63s} &{} G_{64s} \\ G_{51s} &{} G_{52s} &{} G_{53s} &{} G_{54s} \end{array} \right) , \\ \Lambda _{9}= & {} \left( \begin{array}{cccc} -e^{-k_{1h}d}H_{21h} &{} -e^{-k_{2h}d}H_{22h} &{} -e^{-k_{3h}d}H_{23h} &{} -e^{-k_{4h}d}H_{24h} \\ -e^{-k_{1h}d}H_{81h} &{} -e^{-k_{2h}d}H_{82h} &{} -e^{-k_{3h}d}H_{83h} &{} -e^{-k_{4h}d}H_{84h} \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \end{array} \right) , \\ \Lambda _{10}= & {} \left( \begin{array}{cccc} H_{31s} &{} H_{32s} &{} H_{33s} &{} H_{34s} \\ H_{21s} &{} H_{22s} &{} H_{23s} &{} H_{24s} \end{array} \right) , \quad \Lambda _{11}=\left( \begin{array}{cccc} G_{31s} &{} G_{32s} &{} G_{33s} &{} G_{34s} \\ G_{21s} &{} G_{22s} &{} G_{23s} &{} G_{24s} \end{array} \right) , \\ \Lambda _{12}= & {} \left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right) . \end{aligned}$$