Piezothermoelasticity in an infinite slab within the dual-phase-lag model

Abstract

The system of equations of the generalized piezothermoelasticity in anisotropic medium with dual-phase-lag model is established. The exact expressions for displacement components, the temperature, stress components, electric potential and electric displacements are given in the physical domain and illustrated graphically. These expressions are calculated numerically for the problem. Comparisons are made with the results predicted by Lord–Shulman theory and dual-phase-lag model. It is shown that the results from both theories are close to each other for thin slabs, while they differ considerably for thick ones. The present results are of interest in studying the thermomechanical response of piezoelectric sheets under different working thermal, mechanical and electrical conditions.

Appendices

Appendix 1

Some important coefficients appearing in the text:

$$\begin{aligned} \delta _{1}&=\frac{C_{11}}{\rho v_{p}^{2}}, \delta _{2} =\frac{C_{23}}{ \rho v_{p}^{2}}, \delta _{3} =\frac{(C_{13}+C_{23})}{\rho v_{p}^{2}}, \\ \delta _{4}&=\frac{(e_{31}+e_{15})}{e_{33}}, \delta _{5} =\frac{C_{33}}{ \rho v_{p}^{2}}, \\ \delta _{6}&=\frac{e_{15}}{e_{33}}, \delta _{7} =-\frac{\beta _{3}}{ \beta _{1}}, \delta _{8} =\frac{(e_{15}+e_{31})}{\rho v_{p}^{2}},\\ \delta _{9}&=\frac{e_{15}}{\rho v_{p}^{2}}, \delta _{10} =\frac{e_{33}}{ \rho v_{p}^{2}}, \\ \delta _{11}&=-\frac{\epsilon _{11}}{e_{33}}, \delta _{12} =-\frac{ \epsilon _{33}}{e_{33}}, \delta _{13} =\frac{P_{3}}{\beta _{1}}, \\ \delta _{14}&=\frac{K_{1}\omega ^{*}}{\rho C_{T}v_{p}^{2}}, \delta _{15}= \frac{K_{3}\omega ^{*}}{\rho C_{T}v_{p}^{2}}, \\ \delta _{16}&=\frac{\beta _{1}^{2}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}}, \delta _{17} =\frac{\beta _{1}\beta _{3}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}}, \delta _{18} =-\frac{P_{3}\beta _{1}T_{0}}{\rho C_{T}e_{33}}. \end{aligned}$$

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

Appendix 2

Some important coefficients appearing in the text for $n=1,2,3,4$:

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

Appendix 3

$$\begin{aligned}&\sum _{n=1}^{4}H_{5n}M_{n}\sum _{n=1}^{4}G_{5n}R_{n}=-f_{1}^{*}, \\&\sum _{n=1}^{4}H_{6n}M_{n}+\sum _{n=1}^{4}G_{6n}R_{n}=0, \\&\sum _{n=1}^{4}H_{3n}M_{n}+\sum _{n=1}^{4}G_{3n}R_{n}=f_{2}^{*}, \\&\sum _{n=1}^{4}k_{n}H_{2n}M_{n}\sum _{n=1}^{4}G_{2n}R_{n}=0 \\&\sum _{n=1}^{4}M_{n}{\mathrm{e}}^{-k_{n}l}+\sum _{n=1}^{4}R_{n}{\mathrm{e}}^{k_{n}l}\overset{}{=}0,\\&\sum _{n=1}^{4}H_{1n}M_{n}{\mathrm{e}}^{-k_{n}l}+\sum _{n=1}^{4}G_{1n}R_{n}{\mathrm{e}}^{k_{n}l}\overset{}{=}0, \\&\sum _{n=1}^{4}H_{2n}M_{n}{\mathrm{e}}^{-k_{n}l}+\sum _{n=1}^{4}G_{2n}R_{n}{\mathrm{e}}^{k_{n}l}\overset{}{=}0, \\&\sum _{n=1}^{4}H_{3n}M_{n}{\mathrm{e}}^{-k_{n}l}+\sum _{n=1}^{4}G_{3n}R_{n}{\mathrm{e}}^{k_{n}l}\overset{}{=}f_{3}^{*}. \end{aligned}$$

This system is cast in the following matricial form:

$$\begin{aligned} {\mathbf {M}}=\left( \begin{array}{cccccccc} M_{1},&M_{2},&M_{3},&M_{4},&R_{1},&R_{2},&R_{3},&R_{4} \end{array} \right) ^\mathrm{T}, \end{aligned}$$

where the superscript T is used here after to denote the transpose of the superscripted matrix and A is the matrix of coefficients given as

$$\begin{aligned} {\mathbf {A}}=\left( \begin{array}{cccccccc} H_{51} &{} H_{61} &{} H_{31} &{} -k_{1}H_{21} &{} {\mathrm{e}}^{-k_{1}l} &{} {H}_{11} {e}^{-k_{1}l} &{} k_{1}H_{21}{\mathrm{e}}^{-k_{1}l} &{} H_{31}{\mathrm{e}}^{-k_{1}l} \\ H_{52} &{} H_{62} &{} H_{32} &{} -k_{2}H_{22} &{} {\mathrm{e}}^{-k_{2}l} &{} H_{12}{\mathrm{e}}^{-k_{2}l} &{} -k_{2}H_{22}{\mathrm{e}}^{-k_{2}l} &{} H_{32}{\mathrm{e}}^{-k_{2}l} \\ H_{53} &{} H_{63} &{} H_{33} &{} -k_{3}H_{23} &{} {\mathrm{e}}^{-k_{3}l} &{} H_{13}{\mathrm{e}}^{-k_{3}l} &{} -k_{3}H_{23}{\mathrm{e}}^{-k_{3}l} &{} H_{33}{\mathrm{e}}^{-k_{3}l} \\ H_{54} &{} H_{64} &{} H_{34} &{} -k_{4}H_{24} &{} {\mathrm{e}}^{-k_{4}l} &{} H_{14}{\mathrm{e}}^{-k_{4}l} &{} -k_{4}H_{24}{\mathrm{e}}^{-k_{4}l} &{} H_{34}{\mathrm{e}}^{-k_{4}l} \\ G_{51} &{} G_{61} &{} G_{31} &{} k_{1}G_{21} &{} {\mathrm{e}}^{k_{1}l} &{} G_{11}{\mathrm{e}}^{k_{1}l} &{} k_{1}H_{21}{\mathrm{e}}^{k_{1}l} &{} G_{31}{\mathrm{e}}^{k_{1}l} \\ G_{52} &{} G_{62} &{} G_{32} &{} k_{2}G_{22} &{} {\mathrm{e}}^{k_{2}l} &{} G_{12}{\mathrm{e}}^{k_{2}l} &{} k_{2}H_{22}{\mathrm{e}}^{k_{2}l} &{} G_{32}{\mathrm{e}}^{k_{2}l} \\ G_{53} &{} G_{63} &{} G_{33} &{} k_{3}G_{23} &{} {\mathrm{e}}^{k_{3}l} &{} G_{13}{\mathrm{e}}^{k_{3}l} &{} k_{3}H_{23}{\mathrm{e}}^{k_{3}l} &{} G_{33}{\mathrm{e}}^{k_{3}l} \\ G_{54} &{} G_{64} &{} G_{34} &{} k_{4}G_{24} &{} {\mathrm{e}}^{k_{4}l} &{} G_{14}{\mathrm{e}}^{k_{4}l} &{} k_{4}H_{24}{\mathrm{e}}^{k_{4}l} &{} G_{34}{\mathrm{e}}^{k_{4}l} \end{array} \right) ^\mathrm{T} \end{aligned}$$

and F is the following vector of constants

$$\begin{aligned} {\mathbf {F}}=\left( \begin{array}{cccccccc} -f_{1}^{*},&0,&f_{2}^{*},&0,&0,&0,&0,&f_{2}^{*} \end{array} \right) ^\mathrm{T}, \end{aligned}$$

The solution of this system of equations takes the form:

$$\begin{aligned} {\mathbf {M}}={\mathbf{A}}^{-1}{\mathbf {F}} \end{aligned}$$