Magnetic field effect on piezo-thermoelastic wave propagation in a half-space within dual-phase-lag


The current work is concerned with the study of wave propagation in a half-space of a piezo-thermoelastic material under a bias tangential magnetic field within dual-phase-lag (DPL). This is relevant to the design and performance of piezoelectric devices working under a bias magnetic field, for example, the DC magnetic field piezoelectric sensors widely used in various areas of technology. The characteristic time for the problem under consideration is in the range of picoseconds, which is comparable to the thermal relaxation times in many metals. Exact analytic expressions for the mechanical displacement, temperature, stress, electric potential, electric displacement, electric current and induced magnetic field are obtained using normal mode analysis. Plots are provided and discussed for rather large intensities of the magnetic field. Comparison is carried out between the results predicted by DPL and Lord–Shulman theory (LS) in the presence or absence of magnetic field. The presented results clearly indicate that the effect of magnetic field is more pronounced for LS than for DPL. It was found that for magnetic field intensities of \(10^7\) Tesla, both theories give almost identical results. Beyond this threshold value, sensible deviations between the theories start to appear and grow with the intensity of the magnetic field. In connection with experimental measurements, the present investigation may help in calculating numerical values of different material parameters and in assessing the efficiency of the two considered theories.

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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

\({\mathbf {J}}\) :

Current density vector

\({\mathbf {h}}\) :

Induced magnetic field vector

\(\varphi \) :

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

\({\mathbf {E}}\) :

Induced electric field vector

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

Coefficients of linear thermal expansion

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

Longitudinal wave velocity


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Appendix 1

$$\begin{aligned}&\delta _{1} =\frac{C_{11}+\mu _{0}H_{0}^{2}}{\rho v_{p}^{2}}, \delta _{2} =\frac{C_{44}}{\rho v_{p}^{2}}, \\&\delta _{3} =\frac{ (C_{13}+C_{44}+\mu _{0}H_{0}^{2})}{\rho v_{p}^{2}}, \delta _{4} =\frac{(e_{31}+e_{15})}{e_{33}}, \\&\delta _{5} =(1+\frac{\varepsilon _{0}\mu _{0}^{2}H_{0}^{2}}{\rho }), \delta _{6} =\frac{C_{33}+\mu _{0}H_{0}^{2}}{\rho v_{p}^{2}},\\&\delta _{7} =\frac{e_{15}}{e_{33}}, \delta _{8} =-\frac{\beta _{3}}{\beta _{1}}, \\&\delta _{9} =\frac{ (e_{15}+e_{31})}{\rho v_{p}^{2}}, \delta _{10} =\frac{e_{15} }{\rho v_{p}^{2}}, \\&\delta _{11} =\frac{e_{33}}{\rho v_{p}^{2}}, \delta _{12} =-\frac{\epsilon _{11}}{e_{33}}, \\&\delta _{13} =-\frac{\epsilon _{33}}{e_{33}}, \delta _{14} =\frac{P_{3}}{ \beta _{1}}, \\&\delta _{15} =\frac{K_{1}\omega ^{*}}{\rho C_{T}v_{p}^{2}}, \delta _{16} =\frac{K_{3}\omega ^{*}}{\rho C_{T}v_{p}^{2}}, \\&\delta _{17} =\frac{\beta _{1}^{2}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}}, \delta _{18} =\frac{\beta _{1}\beta _{3}T_{0}}{\rho ^{2}C_{T}v_{p}^{2}}, \\&\delta _{19} =-\frac{P_{3}\beta _{1}T_{0}}{\rho C_{T}e_{33}} \end{aligned}$$


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

Appendix 2

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

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Ahmed, E.A.A., Abou-Dina, M.S. & Ghaleb, A.F. Magnetic field effect on piezo-thermoelastic wave propagation in a half-space within dual-phase-lag. Indian J Phys (2020).

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  • Piezoelectricity
  • Magnetic field
  • Dual-phase-lag model
  • Normal modes
  • Generalized thermoelasticity