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Study of wave propagation in an infinite solid due to a line heat source under Moore–Gibson–Thompson thermoelasticity


In the present work, we investigate the effect on thermal and elastic behaviour of a homogenous isotropic unbounded elastic medium caused by a continuous line heat source. The formulation of the problem is carried out for a recent generalized thermoelasticity theory based on Quintanilla–Moore–Gibson–Thompson (QMGT) heat conduction model that considers the thermal displacement and temperature rate term in the constitutive relation of heat transport. To study this problem, the potential function approach accompanied with the Laplace transform technique is applied to derive the analytical solution in the Laplace transform domain. The inversion of the Laplace transform is performed analytically by using the short-time approximation method. A detailed analysis of the analytical results is estimated for the QMGT model and compared with other generalized thermoelasticity theories described previously. The analytical solution for the field variables is further validated by graphical representation of the numerical solution of the present problem by considering the material properties of copper material. Some important observations related to this new model arising out of the present investigation are highlighted.

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Correspondence to Robin Vikram Singh.

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Different notations have been used for the solutions of field variables given by Eqs. (38)–(41) which are given as follows:

$$\begin{aligned}&c_{11}=\tau _{q},\quad c_{12}=1-\tau _{q}a_{0}, \\&c_{31}=\frac{1}{\sqrt{a_{41}}},\quad c_{32}=\frac{-a_{42}}{2\left( a_{41}\right) ^{\frac{3}{2}}}, \\&c_{61}=\frac{1}{\sqrt{b_{41}}},\quad c_{62}=\frac{-b_{42}}{2\left( b_{41}\right) ^{\frac{3}{2}}}, \\&c_{81}=\left( a_{41}\right) ^{2}-\left( b_{41}\right) ^{2},\quad c_{82}=2\left( a_{41}a_{42}+b_{41}b_{42}\right) , \\&c_{91}=\frac{1}{c_{81}},\quad c_{92}=\frac{-c_{82}}{c_{81}^{2}}, \\&d_{11}=c_{11}c_{91},d_{12}=c_{12}c_{91}+c_{11}c_{92}, \\&d_{21}=\left( a_{41}\right) ^{2}-1,\quad d_{22}=2a_{41}a_{42}, \\&d_{31}=\left( b_{41}\right) ^{2}-1,\quad d_{32}=2b_{41}b_{42}, \\&d_{41}=c_{31}d_{21},\quad d_{42}=c_{32}d_{21}+c_{31}d_{22}, \\&d_{51}=c_{61}d_{31},\quad d_{52}=c_{62}d_{31}+c_{61}d_{32}, \\&d_{61}=d_{11}d_{51},\quad d_{62}=d_{12}d_{51}+d_{11}d_{52}, \\&d_{71}=d_{11}d_{41},\quad d_{72}=d_{12}d_{41}+d_{11}d_{42}, \\&d_{81}=d_{71},\quad d_{82}=d_{72},\quad d_{83}=-d_{61},\quad d_{84}=-d_{62}, \\&d_{91}=a_{41}c_{31},\quad d_{92}=a_{42}c_{31}+a_{41}c_{32}, \\&f_{11}=b_{41}c_{61},\quad f_{12}=b_{42}c_{61}+b_{41}c_{62}, \\&f_{21}=d_{11}d_{91},\quad f_{22}=d_{12}d_{91}+d_{11}d_{92}, \\&f_{31}=d_{11}f_{11},\quad f_{32}=d_{12}f_{11}+d_{11}f_{12}, \\&f_{41}=a_{1}f_{21},\quad f_{42}=a_{1}f_{22},\quad f_{43}=-a_{1}f_{31},\quad f_{44}=-a_{1}f_{32}, \\&g_{11}=-\left( \frac{\lambda _{1}-1}{r}\right) a_{41},\quad g_{12}=-\left( \frac{\lambda _{1}-1}{r}\right) a_{42}, \\&g_{21}=-\left( \frac{\lambda _{1}-1}{r}\right) b_{41},\quad g_{22}=-\left( \frac{\lambda _{1}-1}{r}\right) b_{42}, \\&g_{31}=c_{31},\quad g_{32}=c_{32}+c_{31}g_{11}, \\&g_{41}=c_{61},\quad g_{42}=c_{62}+c_{61}g_{21}, \\&g_{51}=g_{31}d_{11},\quad g_{52}=g_{32}d_{11}+g_{31}d_{12}, \\&g_{61}=g_{41}d_{11},\quad g_{62}=g_{42}d_{11}+g_{41}d_{12}, \\&g_{71}=-g_{61},\quad g_{72}=-g_{62}, \\&g_{81}=\left( \lambda _{1}-1\right) \left( a_{41}\right) ^{2}+1,\quad g_{82}=\left( \lambda _{1}-1\right) \left( 2a_{41}a_{42}\right) +\frac{a_{41}}{r}, \\&g_{91}=\left( \lambda _{1}-1\right) \left( b_{41}\right) ^{2}+1,\quad g_{92}=\left( \lambda _{1}-1\right) \left( 2b_{41}b_{42}\right) +\frac{b_{41}}{r}, \\&h_{11}=c_{31}g_{81},\quad h_{12}=c_{31}g_{82}+c_{32}g_{81}, \\&h_{21}=c_{61}g_{91},\quad h_{22}=c_{61}g_{92}+c_{62}g_{91}, \\&h_{31}=a_{1}d_{11}h_{11},\quad h_{32}=a_{1}\left( d_{12}h_{11}+d_{11}h_{12}\right) , \\&h_{41}=a_{1}d_{11}h_{21},\quad h_{42}=a_{1}\left( d_{12}h_{21}+d_{11}h_{22}\right) , \\&h_{51}=-h_{41},\quad h_{52}=-h_{42}. \end{aligned}$$

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Singh, R.V., Mukhopadhyay, S. Study of wave propagation in an infinite solid due to a line heat source under Moore–Gibson–Thompson thermoelasticity. Acta Mech 232, 4747–4760 (2021).

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