Effect of gravity on piezo-thermoelasticity within the dual-phase-lag model

Abstract

The equations of generalized piezo-thermoelasticity within the frame of dual-phase-lag model are used to study the effect of gravitational force on the behavior of a half-space. Analytic expressions for the displacement components, temperature, stress and strain tensors components are obtained using normal mode analysis. Numerical results for the field quantities of practical interest are given in the physical domain and illustrated graphically. Comparison is made between the results predicted by Lord–Shulman theory and dual-phase-lag (DPL) model. The results obtained by applying both of the L–S theory and DPL model are shown to be very close to each other except in determining one of the components of the electric displacement where the results differ and in general the effect of the presence of gravity is to weaken the absolute values of the physical quantities except in the case of the same component of the electric displacement.

Appendix A

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

and

$$\begin{aligned}&A_{1}=\frac{a^{2}c^{2}-a^{2}\delta _{1}}{\delta _{2}},&A_{2}=\frac{ia\delta _{3}}{\delta _{2}},&A_{3}= \frac{iag}{\delta _{2}},&A_{4}=\frac{ia\delta _{4}}{\delta _{2} }, \\&A_{5}=\frac{-ia}{\delta _{2}},&A_{6}=\frac{ ia\delta _{3}}{\delta _{5}},&A_{7}=\frac{-iag}{\delta _{5}},&A_{8}=\frac{a^{2}c^{2}-a^{2}\delta _{2}}{\delta _{5}}, \\&A_{9}=\frac{1}{\delta _{5}},&A_{10}=-\frac{ a^{2}\delta _{6}}{\delta _{5}},&A_{11}=\frac{\delta _{7}}{ \delta _{5}},&A_{12}=\frac{ia\delta _{8}}{\delta _{10}}, \\&A_{13}=-\frac{a^{2}\delta _{9}}{\delta _{10}},&A_{14}=\frac{\delta _{12}}{\delta _{10}},&A_{15}=-\frac{ a^{2}\delta _{11}}{\delta _{10}},&A_{16}=\frac{\delta _{13}}{ \delta _{10}}, \\&A_{17}=-\frac{a^{2}c\delta _{16}(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })},&A_{18}=\frac{iac\delta _{17}(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })},&A_{19}=\frac{iac\delta _{18}(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })},&\\ \end{aligned}$$

$$\begin{aligned} A_{20}=\frac{-a^{2}\delta _{14}(1-iac\tau _{\theta })+iac(1-iac\tau _{q})}{\delta _{15}(1-iac\tau _{\theta })} \end{aligned}$$

$$\begin{aligned}A&=\frac{-1}{(A_{14}-A_{9})}\left(A_{14}A_{20}+A_{15}-A_{16}A_{19}+A_{8}A_{14}-A_{9}A_{20}\right.\\ &\left.-A_{9}A_{13} +A_{9}A_{16}A_{18}-A_{10}+A_{11}A_{19}\right.\\ &\left.-A_{11}A_{14}A_{18}+A_{1}A_{14}-A_{1}A_{9} -A_{2}A_{6}A_{14}+A_{2}A_{9}A_{12}+A_{4}A_{6}-A_{4}A_{12}\right) ,\end{aligned}$$

$$\begin{aligned} B&=\frac{1}{(A_{14}-A_{9})}(A_{5}A_{20}+A_{8}A_{14}A_{20}+A_{8}A_{15}-A_{8}A_{16}A_{19}-A_{9}A_{13}A_{20}-A_{10}A_{13}\\&\quad+A_{10}A_{20}+A_{10}A_{16}A_{18}+A_{11}A_{13}A_{19}-A_{11}A_{15}A_{18}+A_{1}A_{14}A_{20}+A_{1}A_{15}\\&\quad-A_{1}A_{16}A_{19}+A_{1}A_{8}A_{14}\\&\quad-A_{1}A_{9}A_{13}-A_{1}A_{9}A_{20}+A_{1}A_{9}A_{16}A_{18}-A_{1}A_{10}+A_{1}A_{11}A_{19}-A_{1}A_{11}A_{14}A_{18}\\&\quad-A_{2}A_{6}A_{14}A_{20}\\&\quad-A_{2}A_{6}A_{15}+A_{2}A_{6}A_{16}A_{19}+A_{2}A_{9}A_{12}A_{20}-A_{2}A_{9}A_{16}A_{17}+A_{2}A_{10}A_{12}\\&\quad-A_{2}A_{11}A_{12}A_{19}+A_{2}A_{11}A_{14}A_{17}-A_{3}A_{7}A_{14}\\&\quad+A_{4}A_{6}A_{13}+A_{4}A_{6}A_{20}-A_{4}A_{6}A_{16}A_{18}-A_{4}A_{12}A_{20}\\&\quad+A_{4}A_{16}A_{17}-A_{4}A_{8}A_{12}+A_{4}A_{11}A_{12}A_{18}-A_{4}A_{11}A_{17}\\&\quad+A_{5}A_{6}A_{19}+A_{5}A_{6}A_{14}A_{18}+A_{5}A_{12}A_{19}\\&\quad-A_{5}A_{14}A_{17}-A_{5}A_{9}A_{12}A_{18}+A_{5}A_{9}A_{17}), \end{aligned}$$

$$\begin{aligned} C&=\frac{-1}{(A_{14}-A_{9})}(A_{8}A_{15}A_{20}-A_{10}A_{13}A_{20}+A_{1}A_{15}A_{20}+A_{1}A_{8}A_{14}A_{20}+A_{1}A_{8}A_{15}\\&\quad-A_{1}A_{8}A_{16}A_{19}-A_{1}A_{9}A_{13}A_{20}-A_{1}A_{10}A_{13}-A_{1}A_{10}A_{20}+A_{1}A_{10}A_{16}A_{18}\\&\quad+A_{1}A_{11}A_{13}A_{19}-A_{1}A_{11}A_{15}A_{18}-A_{2}A_{6}A_{15}A_{20}\\&\quad+A_{2}A_{10}A_{12}A_{20}-A_{2}A_{10}A_{16}A_{17}+A_{2}A_{11}A_{15}A_{17}-A_{3}A_{7}A_{14}A_{20}\\&\quad-A_{3}A_{7}A_{15}+A_{3}A_{7}A_{16}A_{19}+A_{4}A_{6}A_{13}A_{20}-A_{4}A_{8}A_{12}A_{20}\\&\quad+A_{4}A_{8}A_{16}A_{17}-A_{4}A_{11}A_{13}A_{17}-A_{5}A_{6}A_{13}A_{19}+A_{5}A_{6}A_{15}A_{18}\\&\quad-A_{5}A_{15}A_{17}+A_{5}A_{8}A_{12}A_{19}-A_{5}A_{8}A_{14}A_{17}+A_{5}A_{9}A_{13}A_{17}\\&\quad-A_{5}A_{10}A_{12}A_{18}+A_{5}A_{10}A_{17}), \end{aligned}$$

$$\begin{aligned} E&=\frac{1}{(A_{14}-A_{9})}(A_{1}A_{8}A_{15}A_{20}-A_{1}A_{10}A_{13}A_{20}-A_{3}A_{7}A_{15}A_{20}\\&\quad-A_{5}A_{8}A_{15}A_{17}+A_{5}A_{10}A_{13}A_{17}). \end{aligned}$$

1.1 Appendix B

$$\begin{aligned}&H_{1n}=-\frac{s_{1n}}{s_{2n}}, H_{2n}=-\frac{q_{1n}+q_{2n}H_{1n}}{q_{3n}},&H_{3n}=-\frac{(k_{n}^{2}+A_{1})+(-A_{2}k_{n}+A_{3})H_{1n}-A_{4}k_{n}H_{2n}}{ A_{5}},\\&H_{4n}=[r_{1}-l_{1}k_{n}H_{1n}-l_{2}k_{n}H_{2n}-H_{3n}],&H_{5n}=[r_{2}-\delta _{5}k_{n}H_{1n}-k_{n}H_{2n}+\delta _{7}H_{3n}],\\&H_{6n}=[-\delta _{2}k_{n}+r_{3}H_{1n}+r_{4}H_{2n}],&H_{7n}=[-l_{3}k_{n}+r_{5}H_{1n}+r_{6}H_{2n}],\\&H_{8n}=[r_{7}-l_{6}k_{n}H_{1n}-l_{7}k_{n}H_{2n}+l_{8}H_{3n}].&n=1,2,3,4.\\&q_{1n}=A_{11}k_{n}^{3}+(A_{1}A_{11}-A_{5}A_{6})k_{n}+A_{5}A_{7},&\\&q_{2n}=(-A_{2}A_{11}+A_{5})k_{n}^{2}+A_{3}A_{11}k_{n}+A_{5}A_{8},&\\&q_{3n}=(-A_{4}A_{11}+A_{5}A_{9})k_{n}^{2}+A_{5}A_{10},&q_{4n}=A_{16}k_{n}^{3}+(A_{1}A_{16}-A_{5}A_{12})k_{n},\\&q_{5n}=(A_{5}-A_{2}A_{16})k_{n}^{2}+A_{3}A_{16}k_{n}+A_{5}A_{13},&q_{6n}=(A_{5}A_{14}-A_{4}A_{16})k_{n}^{2}+A_{5}A_{15}.\\&s_{1n}=q_{1n}q_{6n}-q_{3n}q_{4n},&s_{2n}=q_{2n}q_{6n}-q_{3n}q_{5n}. \end{aligned}$$

$$\begin{aligned} l_{1}=\frac{C_{13}}{\rho v_{p}^{2}},\quad l_{2}=\frac{e_{31}}{e_{33}},\quad l_{3}=\frac{e_{15}\beta _{1}T_{0}}{e\rho v_{p}^{2}},\quad l_{4}=-\frac{\epsilon _{11}\beta _{1}T_{0}}{ee_{33}}, \end{aligned}$$

$$\begin{aligned} l_{5}=\frac{e_{31}\beta _{1}T_{0}}{e\rho v_{p}^{2}},\quad l_{6}=\frac{e_{33}\beta _{1}T_{0}}{e\rho v_{p}^{2}},\quad l_{7}=-\frac{\epsilon _{33}\beta _{1}T_{0}}{ee_{33}},\quad l_{8}=\frac{P_{3}T_{0}}{e}. \end{aligned}$$

$$\begin{aligned} \{r_{1},r_{2},r_{3},r_{4},r_{5},r_{6},r_{7}\}=ia\{\delta _{1},l_{1},\delta _{2},\delta _{6},l_{3},l_{4},l_{5}\} \end{aligned}$$