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Wave Analysis and Representation of Fundamental Solution in Modified Couple Stress Thermoelastic Diffusion with Voids, Nonlocal and Phase Lags

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Abstract

In the present study, we explored a new mathematical formulation involving modified couple stress thermoelastic diffusion (MCTD) with nonlocal, voids and phase lags. The governing equations are expressed in dimensionless form for the further investigating. The desired equations are expressed in terms of elementary functions by assuming time harmonic variation of the field variables (displacement, temperature field, chemical potential and volume fraction field). The fundamental solutions are constructed for the obtained system of equations for steady oscillation and some basic features of the solutions are established. Also, plane wave vibrations has been examined for two dimensional cases. The characteristic equation yields the attributes of waves like phase velocity, attenuation coefficients, specific loss and penetration depth which are computed numerically and presented in form of distinct graphs. Some unique cases are also deduced. The results provide the motivation for the researcher to investigate thermally conducted modified couple stress elastic material under nonlocal, porosity and phase lags impacts as a new class of applicable materials.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Kurukshetra University, 136119, Kurukshetra, Haryana, India

    R. Kumar

  2. School of Chemical Engineering and Physical Sciences, Lovely Professional University, 144411, Phagwara, India

    S. Kaushal &  Pragati

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  2. S. Kaushal
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  3. Pragati
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APPENDIX

APPENDIX

$${{M}_{1}} = - {{a}_{6}}{{a}_{{25}}}{{a}_{{28}}}{{a}_{{31}}},\quad {{M}_{2}} = \left[ {{{a}_{{28}}}{{a}_{{31}}}\left( {{{a}_{{25}}}{{a}_{{26}}} - {{a}_{6}}{{\omega }^{2}} + {{a}_{4}}{{a}_{8}}} \right) - {{a}_{6}}{{a}_{{25}}}\left( {{{a}_{{29}}}{{a}_{{31}}} + {{a}_{{28}}}{{a}_{{32}}}} \right) - {{a}_{6}}{{a}_{{27}}}{{a}_{{31}}}} \right],$$
$${{M}_{3}} = \left[ {{{a}_{{25}}}{{a}_{{26}}}\left( {{{a}_{{29}}}{{a}_{{31}}} + {{a}_{{25}}}{{a}_{{29}}}} \right) - {{a}_{6}}{{a}_{{25}}}\left( {{{a}_{{29}}}{{a}_{{32}}} - {{a}_{{16}}}{{a}_{{20}}}{{\omega }^{4}}} \right) - {{\omega }^{2}}\left( {{{a}_{9}}{{a}_{{15}}}{{a}_{{31}}}{{a}_{{25}}}} \right.} \right.$$
$$\left. { + \;{{a}_{{11}}}{{a}_{{25}}}{{a}_{{21}}}{{a}_{{28}}} - {{a}_{{23}}}{{a}_{{25}}}{{a}_{{28}}} + {{a}_{6}}{{a}_{{29}}}{{a}_{{31}}} + {{a}_{6}}{{a}_{{28}}}{{a}_{{32}}} - {{a}_{8}}{{a}_{{15}}}{{a}_{{31}}} - {{a}_{6}}{{a}_{{16}}}{{a}_{{30}}} + {{a}_{5}}{{a}_{8}}{{a}_{{21}}}{{a}_{{28}}}} \right)$$
$$\left. { - \;{{\omega }^{2}}\left( {{{a}_{5}}{{a}_{6}}{{a}_{{20}}}{{a}_{{27}}}} \right) + {{a}_{4}}{{a}_{8}}\left( {{{a}_{{29}}}{{a}_{{31}}} + {{a}_{{28}}}{{a}_{{32}}}} \right) + {{a}_{{27}}}{{a}_{{31}}}\left( {{{a}_{4}}{{a}_{9}} + {{a}_{{26}}}} \right) + {{a}_{{30}}}\left( {{{a}_{4}}{{a}_{{11}}}{{a}_{{28}}} + {{a}_{5}}{{a}_{6}}{{a}_{{29}}}} \right)} \right],$$
$${{M}_{4}} = \left[ {{{a}_{{25}}}{{a}_{{26}}}\left( {{{a}_{{29}}}{{a}_{{32}}} - {{a}_{{16}}}{{a}_{{20}}}{{\omega }^{4}}} \right) - {{a}_{6}}{{a}_{{25}}}\left( {{{a}_{{29}}}{{a}_{{32}}} - {{a}_{{16}}}{{a}_{{20}}}{{\omega }^{4}}} \right) - {{\omega }^{2}}\left( {{{a}_{9}}{{a}_{{15}}}{{a}_{{32}}}{{a}_{{25}}} + {{a}_{{11}}}{{a}_{{25}}}{{a}_{{21}}}{{a}_{{29}}}} \right.} \right.$$
$$\left. { - \;{{a}_{{26}}}{{a}_{{29}}}{{a}_{{31}}} - {{a}_{{26}}}{{a}_{{28}}}{{a}_{{32}}} + {{a}_{6}}{{a}_{{29}}}{{a}_{{32}}} + {{a}_{4}}{{a}_{9}}{{a}_{{16}}}{{a}_{{30}}} + {{a}_{4}}{{a}_{{11}}}{{a}_{{20}}}{{a}_{{27}}} - {{a}_{8}}{{a}_{{15}}}{{a}_{{32}}}} \right) - {{\omega }^{2}}\left( {{{a}_{{26}}}{{a}_{{16}}}{{a}_{{30}}}} \right.$$
$$\left. { + \;{{a}_{{11}}}{{a}_{{21}}}{{a}_{{27}}}} \right) + {{\omega }^{2}}\left( {{{a}_{{11}}}{{a}_{{15}}}{{a}_{{30}}} - {{a}_{5}}{{a}_{8}}{{a}_{{21}}}{{a}_{{29}}} + {{a}_{5}}{{a}_{{26}}}{{a}_{{20}}}{{a}_{{27}}} - {{a}_{5}}{{a}_{9}}{{a}_{{27}}}{{a}_{{21}}} + {{a}_{5}}{{a}_{9}}{{a}_{{15}}}{{a}_{{30}}}} \right)$$
$$ + \;{{\omega }^{4}}\left( {{{a}_{{25}}}{{a}_{9}}{{a}_{{16}}}{{a}_{{21}}} + {{a}_{{11}}}{{a}_{{15}}}{{a}_{{20}}}{{a}_{{25}}} - {{a}_{9}}{{a}_{{15}}}{{a}_{{31}}} - {{a}_{{11}}}{{a}_{{21}}}{{a}_{{28}}}} \right) + {{a}_{{16}}}{{\omega }^{4}}\left( {{{a}_{4}}{{a}_{{20}}} - {{a}_{8}}{{a}_{{21}}}} \right)$$
$$\left. { + \;{{a}_{5}}{{a}_{8}}{{a}_{{15}}}{{a}_{{20}}}{{\omega }^{4}} + {{a}_{6}}{{a}_{{16}}}{{a}_{{20}}}{{\omega }^{6}} + {{a}_{4}}{{a}_{{32}}}\left( {{{a}_{8}}{{a}_{{29}}} + {{a}_{9}}{{a}_{{27}}}} \right) + {{a}_{{26}}}{{a}_{{27}}}{{a}_{{32}}} + {{a}_{{29}}}{{a}_{{30}}}\left( {{{a}_{4}}{{a}_{{11}}} - {{a}_{5}}{{a}_{{26}}}} \right)} \right],$$
$${{M}_{5}} = \left[ {\left( {{{a}_{{26}}}{{a}_{{29}}}{{a}_{{32}}}{{\omega }^{2}} - {{a}_{{16}}}{{a}_{{20}}}{{a}_{{26}}}{{\omega }^{6}} - {{a}_{9}}{{a}_{{15}}}{{a}_{{32}}}{{\omega }^{4}}} \right) + {{\omega }^{6}}\left( {{{a}_{9}}{{a}_{{16}}}{{a}_{{21}}} + {{a}_{{11}}}{{a}_{{15}}}{{a}_{{20}}}} \right) - {{a}_{{11}}}{{a}_{{21}}}{{a}_{{29}}}{{\omega }^{4}}} \right],$$

where

$${{a}_{{25}}} = {{\omega }^{2}}\xi _{1}^{2} - ({{a}_{1}} + {{a}_{2}}),\quad {{a}_{{26}}} = - {{a}_{7}} + {{a}_{{10}}}i\omega + {{\omega }^{2}},\quad {{a}_{{27}}} = - {{a}_{{15}}}{{\omega }^{2}},\quad {{a}_{{28}}} = - \left( {{{a}_{{13}}}\tau _{{v}}^{1} + \tau _{T}^{1}} \right),$$
$${{a}_{{29}}} = \tau _{q}^{1}{{a}_{{14}}}{{\omega }^{2}},\quad {{a}_{{30}}} = - {{a}_{{21}}}{{\omega }^{2}},\quad {{a}_{{31}}} = - {{a}_{{19}}}\tau _{R}^{1} - \tau _{P}^{1},\quad {{a}_{{32}}} = {{\omega }^{2}}\tau _{\eta }^{1}.$$

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Kumar, R., Kaushal, S. & Pragati Wave Analysis and Representation of Fundamental Solution in Modified Couple Stress Thermoelastic Diffusion with Voids, Nonlocal and Phase Lags. Russ Math. 68, 31–51 (2024). https://doi.org/10.3103/S1066369X24700099

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