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Modelling of a Three Dimensional Thermoelastic Half Space with Three Phase Lags using Memory Dependent Derivative

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Abstract

A mathematical model for a three dimensional isotropic half-space has been formulated to inspect lagging behaviours due to the presence of phase lags in context of memory dependent derivative, as an extension of several existing thermoelastic models like- Green-Naghdi-III, Lord Shulman, and Fourier’s Law etc. The analytical and procedural work has been done in integral transform domain preceded by eigenvalue approach to find the solution from the governing equations. Numerical computations and graphical representation of distribution of non-dimensional stress components, temperature with the effect of three phase lag, kernel function and time-delay has been performed with the help of the efficient mathematical software.

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Abbreviations

x i :

Space variables

\( \lambda ,\mu \) :

Lame’s Constant

u i :

Displacement components

\( \tau \) :

Relaxation Time

\( C_{E} \) :

Specific heat at constant strain

\( \tau_{q} \,,\tau_{v} \,,\tau_{T} \) :

Three-phase-lag

t :

Time variable

\( \rho \) :

Density of the material

\( T_{0} \) :

Reference temperature chosen such that \( \left| {\frac{{T - T_{0} }}{{T_{0} }}} \right| \) ≪ 1

\( \varepsilon = \tfrac{{\gamma^{2} T_{0} }}{{\rho C_{E} (\lambda + 2\mu )}} \) :

Thermal coupling parameter

σ ij :

Stress Components

ω :

Time delay

K (t, ξ):

Kernel function

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Authors and Affiliations

  1. Department of Mathematics, Jadavpur University, Kolkata, 700 032, India

    Debkumar Ghosh & Abhijit Lahiri

  2. Department of Geology, Asutosh College, 92 S. P.Mukherjee Road, Kolkata, 700 026, India

    Ashis Kumar Das

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  1. Debkumar Ghosh
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Appendix

Appendix

$$ A = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ {M_{51} } & {M_{52} } & {M_{53} } & {M_{54} } & {M_{55} } & {M_{56} } & {M_{57} } & {M_{58} } \\ {M_{61} } & {M_{62} } & {M_{63} } & {M_{64} } & {M_{65} } & {M_{66} } & {M_{67} } & {M_{68} } \\ {M_{71} } & {M_{72} } & {M_{73} } & {M_{74} } & {M_{75} } & {M_{76} } & {M_{77} } & {M_{78} } \\ {M_{81} } & {M_{82} } & {M_{83} } & {M_{84} } & {M_{85} } & {M_{86} } & {M_{87} } & {M_{88} } \\ \end{array} } \right] $$
$$ \underrightarrow v = \left[ {U_{1} \,U_{2} \,U_{3} \,\,\varTheta \,\,U_{1}^{\prime } \,U_{2}^{\prime } \,U_{3}^{\prime } \,\,\varTheta^{\prime}} \right]^{T} $$
$$ \begin{aligned} &M_{51} = A_{11} ,M_{52} = 0,M_{53} = 0,M_{54} = 0,M_{55} = 0, \hfill \\ &M_{56} = A_{12} ,M_{57} = A_{13} ,M_{58} = A_{14} , \hfill \\ &M_{61} = 0,\,\,M_{62} = B_{11} ,\,M_{63} = B_{12} ,\,M_{64} = B_{13} \,,M_{65} = B_{14} ,M_{66} = 0,\,\,\,M_{67} = 0\,,\,\,\,\,\,M_{68} = 0 \hfill \\ &M_{71} = 0,\,\,M_{72} = C_{11} ,\,M_{73} = C_{12} ,\,M_{74} = C_{13} ,\,M_{75} = C_{14} ,M_{76} = 0,\,\,M_{77} = 0\,,\,\,\,M_{78} = 0 \hfill \\ &M_{81} = 0,\,\,M_{82} = D_{11} ,\,M_{83} = D_{12} ,\,M_{84} = D_{13} ,\,M_{85} = D_{14} ,M_{86} = 0,\,M_{87} = 0\,,\,\,M_{88} = 0 \hfill \\ \end{aligned} $$
$$ \begin{aligned} A_{11} = s^{2} + \varepsilon_{3} \left( {\psi_{2}^{2} + \psi_{3}^{2} } \right), \hfill \\ A_{12} = - i\,\varepsilon_{1} \,\psi_{2} , \hfill \\ A_{13} = - i\,\varepsilon_{1} \,\psi_{3} , \hfill \\ A_{14} = 1 + \frac{{\varepsilon_{3} \,\nu \,G}}{\omega }, \hfill \\ \end{aligned} $$
$$ \begin{aligned} B_{11} = \frac{{s^{2} + \psi_{2}^{2} }}{{\varepsilon_{3} }}, \hfill \\ B_{12} = \frac{{\varepsilon_{1} \,\psi_{2\,} \psi_{3} }}{{\varepsilon_{3} }} \hfill \\ B_{13} = \frac{{i\,\psi_{2} }}{{\varepsilon_{3} }}\left( {1 + \frac{{\varepsilon_{3} \,\nu \,G}}{\omega }} \right), \hfill \\ B_{14} = - \frac{{i\,\varepsilon_{1} \,\psi_{2} }}{{\varepsilon_{3} }}, \hfill \\ \end{aligned} $$
$$ \begin{aligned} C_{11} = \frac{{\varepsilon_{1} \,\psi_{2\,} \psi_{3} }}{{\varepsilon_{3} }}, \hfill \\ C_{12} = \frac{{s^{2} + \psi_{3}^{2} + \varepsilon_{3} \psi_{2}^{2} }}{{\varepsilon_{3} }}, \hfill \\ C_{13} = \frac{{i\,\psi_{3} }}{{\varepsilon_{3} }}\left( {1 + \frac{{\varepsilon_{3} \,\nu \,G}}{\omega }} \right), \hfill \\ C_{14} = - \frac{{i\,\varepsilon_{1} \,\psi_{3} }}{{\varepsilon_{3} }}, \hfill \\ \end{aligned} $$
$$ \begin{aligned} D_{11} = \frac{{\varepsilon_{12} }}{{\varepsilon_{9} }},\,\, \hfill \\ D_{12} = \frac{{\varepsilon_{13} \,}}{{\varepsilon_{9} }},\,\,\, \hfill \\ D_{13} = \frac{{\varepsilon_{10} \,}}{{\varepsilon_{9} }},\,\, \hfill \\ D_{14} = \frac{{\varepsilon_{14} \,}}{{\varepsilon_{9} }}\,\, \hfill \\ \end{aligned} $$
$$ \begin{aligned} \beta_{1} = i\,\varepsilon_{4} \,\psi_{2} \hfill \\ \beta_{2} = i\,\varepsilon_{4} \,\psi_{3} \hfill \\ \beta_{3} = - \left( {1 + \frac{{\varepsilon_{3} \,\nu \,G}}{\omega }} \right), \hfill \\ \beta_{4} = i\,\psi_{2} , \hfill \\ \beta_{5} = i\,\psi_{3} , \hfill \\ \beta_{6} = i\,\varepsilon_{2} \,\psi_{2} , \hfill \\ \beta_{7} = i\,\varepsilon_{2} \,\psi_{3} , \hfill \\ \end{aligned} $$
$$ \begin{aligned} \varepsilon_{1} = \frac{\lambda + \mu }{\lambda + 2\mu };\varepsilon_{2} = \frac{\mu }{\lambda + 2\mu };\varepsilon_{3} = \frac{1}{{c_{0} \eta^{2} }};\,\,\varepsilon_{4} = \frac{\lambda }{\lambda + 2\mu }; \hfill \\ \varepsilon_{5} = \frac{{T_{0} \,\gamma^{2} }}{{k\,\,\eta \,\,\left( {\lambda + 2\mu } \right)}};\,\,\varepsilon_{6} = \frac{{k\,c_{0}^{4} \,\eta^{2} \,\tau_{T} }}{{k\,^{*} }};\,\,\,\varepsilon_{7} = \frac{{c_{0} \,\eta \,\,\tau_{v}^{*} }}{{k\,^{*} }};\,\,\, \hfill \\ \varepsilon_{8} = \frac{k}{{k\,^{*} }};\,\,\varepsilon_{9} = \left( {1 + \frac{{\tau_{\theta } \,\,G}}{\omega }} \right)\left( {1 + \varepsilon_{6} \,s^{2} + + \varepsilon_{7} \,s} \right);\,\,\, \hfill \\ \end{aligned} $$
$$ \begin{aligned} \varepsilon_{10} = \varepsilon_{8} \left( {s + \frac{{\tau_{q} \,s\,G}}{\omega } + \frac{{\tau_{q}^{2} \,s^{2} \,G}}{2\,\omega }} \right) + \varepsilon_{9} \left( {\psi_{2}^{2} + \psi_{3}^{2} } \right);\,\, \hfill \\ \varepsilon_{11} = \varepsilon_{5} \,\varepsilon_{8} \left( {s + \frac{{\tau_{q} \,s\,G}}{\omega } + \frac{{\tau_{q}^{2} \,s^{2} \,G}}{2\,\omega }} \right); \hfill \\ \end{aligned} $$
$$ \begin{aligned} \varepsilon_{12} = i\,\psi_{2\,} \varepsilon_{5} \,\varepsilon_{8} \left( {s + \frac{{\tau_{q} \,s\,G}}{\omega } + \frac{{\tau_{q}^{2} \,s^{2} \,G}}{2\,\omega }} \right);\,\,\, \hfill \\ \varepsilon_{13} = i\,\psi_{3\,} \varepsilon_{5} \,\varepsilon_{8} \left( {s + \frac{{\tau_{q} \,s\,G}}{\omega } + \frac{{\tau_{q}^{2} \,s^{2} \,G}}{2\,\omega }} \right); \hfill \\ \end{aligned} $$

\( \begin{aligned} & d_{1} = (h_{24} h_{13} - h_{14} h_{23} )(h_{22} h_{33} - h_{32} h_{23} ) - (h_{34} h_{23} - h_{24} h_{33} )(h_{12} h_{23} - h_{22} h_{13} ), \\ & d_{2} = (h_{34} h_{23} - h_{24} h_{33} )(h_{11} h_{23} - h_{21} h_{13} ) - (h_{24} h_{13} - h_{14} h_{23} )(h_{21} h_{33} - h_{31} h_{23} ), \\ & d_{3} = (h_{12} h_{21} - h_{11} h_{22} )(h_{21} h_{34} - h_{31} h_{24} ) - (h_{22} h_{31} - h_{21} h_{32} )(h_{11} h_{24} - h_{14} h_{21} ), \\ & d_{4} = (h_{11} h_{23} - h_{21} h_{13} )(h_{22} h_{33} - h_{32} h_{23} ) - (h_{12} h_{23} - h_{22} h_{13} )(h_{21} h_{33} - h_{31} h_{23} ), \\ \end{aligned} \),

$$ \begin{aligned} h_{11} = M_{51} - \lambda^{2} \,\,,h_{12} = \lambda M_{56} \,\,,h_{13} = \lambda M_{57} \,\,,h_{14} = \lambda M_{58} \hfill \\ h_{21} = M_{65} \,\,,h_{22} = M_{62} \, - \lambda^{2} \,,h_{23} = M_{63} \,\,,h_{24} = M_{64} \hfill \\ h_{31} = \lambda M_{75} \,,h_{32} = M_{72} \,\,,h_{33} = M_{73} \, - \lambda^{2} \,,h_{34} = \lambda M_{74} \hfill \\ h_{41} = \lambda M_{85} \,\,,h_{42} = M_{82} \,\,,h_{43} = M_{83} \,\,,h_{44} = M_{84} - \lambda^{2} \hfill \\ \end{aligned} $$
$$ R_{1i} = \left( { - \lambda_{i} \,d_{1i} + \sum\limits_{k = 1}^{3} {\beta_{k} \,d_{(k + 1)i} } } \right)e^{{ - \lambda_{i} x_{1} }} ,,i = 1(1)4 $$
$$ R_{2i} = \left( { - \lambda_{i} \,\,\varepsilon_{4} \,d_{1i} + \beta_{4} \,d_{2i} + \beta_{2} \,d_{3i} + \beta_{3} \,d_{4i} } \right)e^{{ - \lambda_{i} x_{1} }} ,i = 1(1)4 $$
$$ R_{3i} = \left( { - \lambda_{1} \,\varepsilon_{4} \,d_{1i} + \beta_{1} \,d_{2i} + \beta_{5} \,d_{3i} + \beta_{3} \,d_{4i} } \right)e^{{ - \lambda_{i} x_{1} }} ;,i = 1(1)4 $$
$$ R_{4i} = \left( {\beta_{6} \,d_{1i} - \varepsilon_{2} \,\lambda_{i} \,d_{2i} } \right)e^{{ - \lambda_{i} x_{1} }} ,i = 1(1)4 $$
$$ R_{5i} = \left( {\beta_{7} \,d_{1i} - \varepsilon_{2} \,\lambda_{i} \,d_{3i} } \right)e^{{ - \lambda_{i} x_{1} }} ,i = 1(1)4 $$
$$ R_{6i} = \left( {\beta_{7} \,d_{2i} + \beta_{6} \,d_{3i} } \right)e^{{ - \lambda_{i} x_{1} }} ,i = 1(1)4 $$
$$ R_{7i} = d_{4i} \,e^{{ - \lambda_{i} x_{1} }} i = 1(1)4 $$
$$ \begin{aligned} l_{1j} = R_{1j} (0),\,\,\,\,l_{4j} = R_{4j} (0),\,\,\,\,l_{5j} = R_{5j} ,\,\, \hfill \\ \,l\,_{7j} = R_{7j} (0),\,\,\,\,where\,j = 1,2,3,4 \hfill \\ \end{aligned} $$
$$ and\,\,A_{i} = \frac{{D_{i} }}{\Delta }\,,i = 1,2,3,4 $$
$$ \begin{aligned} D_{1} = \left[ {\begin{array}{*{20}c} 0 & {l_{12} } & {l_{13} } & {l_{14} } \\ 0 & {l_{42} } & {l_{43} } & {l_{44} } \\ 0 & {l_{52} } & {l_{53} } & {l_{54} } \\ {r^{*} } & {l_{72} } & {l_{73} } & {l_{74} } \\ \end{array} } \right],D_{2} = \left[ {\begin{array}{*{20}c} {l_{11} } & 0 & {l_{13} } & {l_{14} } \\ {l_{41} } & 0 & {l_{43} } & {l_{44} } \\ {l_{51} } & 0 & {l_{53} } & {l_{54} } \\ {l_{71} } & {r^{*} } & {l_{73} } & {l_{74} } \\ \end{array} } \right], \hfill \\ D_{3} = \left[ {\begin{array}{*{20}c} {l_{11} } & {l_{12} } & 0 & {l_{14} } \\ {l_{41} } & {l_{42} } & 0 & {l_{44} } \\ {l_{51} } & {l_{52} } & 0 & {l_{54} } \\ {l_{71} } & {l_{72} } & {r^{*} } & {l_{74} } \\ \end{array} } \right],D_{4} = \left[ {\begin{array}{*{20}c} {l_{11} } & {l_{12} } & {l_{13} } & 0 \\ {l_{41} } & {l_{42} } & {l_{43} } & 0 \\ {l_{51} } & {l_{52} } & {l_{53} } & 0 \\ {l_{71} } & {l_{72} } & {l_{73} } & {r^{*} } \\ \end{array} } \right],\Delta = \left[ {\begin{array}{*{20}c} {l_{11} } & {l_{12} } & {l_{13} } & {l_{14} } \\ {l_{41} } & {l_{42} } & {l_{43} } & {l_{44} } \\ {l_{51} } & {l_{52} } & {l_{53} } & {l_{54} } \\ {l_{71} } & {l_{72} } & {l_{73} } & {l_{74} } \\ \end{array} } \right] \hfill \\ \end{aligned} $$

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Ghosh, D., Das, A.K. & Lahiri, A. Modelling of a Three Dimensional Thermoelastic Half Space with Three Phase Lags using Memory Dependent Derivative. Int. J. Appl. Comput. Math 5, 154 (2019). https://doi.org/10.1007/s40819-019-0731-y

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