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The initial stress effect on a thermoelastic micro-elongated solid under the dual-phase-lag model

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Abstract

This paper's main objective is to study the two-dimensional deformation of the thermoelastic micro-elongated solid with the effect of initial stress on the model of dual-phase-lag and the theory of Lord and Shulman (L-S). The interface of the elastic half-space and the thermoelastic micro-elongated half-space is utilized to apply mechanical force. The solution has been made using the normal mode analysis method. For aluminum epoxy, numerical simulations are done and graphically presented to demonstrate the presence and complete absence of initial stress on the various physical quantities.

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Abbreviations

\(P\) :

Initial pressure

εij :

The strain tensor where \(\varepsilon_{ij} = \frac{1}{2}(u_{i,j} + u_{j,i} )\)

j0 :

Microinertia

a0, λ0, λ1 :

Micro-elongational constants

\(T\) :

Absolute temperature

\(T_{0}\) :

Reference temperature

\(\tau_{\theta }\) :

Temperature gradient parameter

\(\tau_{q}\) :

Heat flux parameter

\({\varvec{u}}^{{\text{e}}}\) :

Displacement vector in an elastic medium

\(\rho^{{\text{e}}}\) :

Density in an elastic medium

\(\lambda^{{\text{e}}} ,\mu^{{\text{e}}}\) :

Lame's constants in an elastic medium

\(k^{e}\) :

Thermal conductivity in an elastic medium

\(\rho\) :

Density in micro-elongated medium

\({\varvec{u}}\) :

Displacement vector in micro-elongated medium

\(\sigma_{ij}\) :

Component of stress tensor for micro-elongated medium

\(\varphi\) :

Micro-elongational scalar

\(k\) :

Thermal conductivity in micro-elongated medium

\(c_{{\text{e}}}\) :

Specific heat at the constant strain in micro-elongated medium

\(\lambda ,\mu\) :

Lame's constants in micro-elongated medium

\(c_{{\text{e}}}^{{\text{e}}}\) :

Specific heat at the constant strain in an elastic medium

\(\alpha_{{t_{1} }} \,,\,\alpha_{{t_{2} }}\) :

Coefficient of linear thermal expansion where \(\beta_{0} = (\,3\lambda + 2\,\mu \,)\,\alpha_{{t_{1} }}\)\(\beta_{1} = (\,3\lambda + 2\,\mu \,)\,\alpha_{{t_{2} }}\)

References

  1. I.S. Sokolnikoff, Mathematical Theory of Elasticity, 2nd edn. (Mcgraw-Hill Book Company, New York, 1956)

    MATH  Google Scholar 

  2. J.-M. Duhamel, J. de l’École Polytechnique. 15, 1–57 (1837)

    Google Scholar 

  3. K. Neumann, Vorlesungen Uber Die Theorie Der Elasticitat (Meyer, Breslau, 1885)

    MATH  Google Scholar 

  4. M.A. Biot, J. Appl. Phys. 27, 240–253 (1956)

    Article  ADS  MathSciNet  Google Scholar 

  5. H.W. Lord, Y. Shulman, J. Mech. Phys. Solids 15, 299–309 (1967)

    Article  ADS  Google Scholar 

  6. M.I.A. Othman, J. Therm. Stresses 25, 1027–1045 (2002)

    Article  Google Scholar 

  7. A.E. Green, K.A. Lindsay, J. Elast. 2, 1–7 (1972)

    Article  Google Scholar 

  8. D.Y. Tzou, J. Heat Transfer 117, 8–16 (1995)

    Article  Google Scholar 

  9. D.Y. Tzou, Int. J. Heat Mass Transf. 38, 3231–3231 (1995)

    Article  Google Scholar 

  10. M.A. Ezzat, M.I.A. Othman, Int. J. Eng. Sci. 38, 107–120 (2000)

    Article  Google Scholar 

  11. M. Marin, S. Vlase, M. Paun, AIP Adv. 5(3), 037113 (2015)

    Article  ADS  Google Scholar 

  12. A.A. Khan, R. Bukhari, M. Marin, R. Ellahi, Heat Transfer Res. 50(11), 1061–1080 (2019)

    Article  Google Scholar 

  13. T. Saeed, I. Abbas, M. Marin, Symmetry. 12(3), 488 (2020)

    Article  Google Scholar 

  14. M.I.A. Othman, M. Marin, Results Phys. 7, 3863–3872 (2017)

    Article  ADS  Google Scholar 

  15. E.M. Abd-Elaziz, M. Marin, M.I.A. Othman, Symmetry. 11(3), 413–430 (2019)

    Article  Google Scholar 

  16. M.A. Ezzat, M.I.A. Othman, A.S. El Karamany, J. Therm. Stresses 24(5), 411–432 (2001)

    Article  Google Scholar 

  17. M.I.A. Othman, Y.D. Elmaklizi, S.M. Said, Int. J. Thermophys. 34(3), 521–537 (2013)

    Article  ADS  Google Scholar 

  18. S. Shaw, B. Mukhopadhyay, Appl. Math. Comput. 218, 6304–6313 (2012)

    MathSciNet  Google Scholar 

  19. S. Shaw, B. Mukhopadhyay, J. Eng. Phys. Thermophys. 86, 716–722 (2013)

    Article  Google Scholar 

  20. P. Ailawalia, S.K. Sachdeva, D.S. Pathania, Int. J. Appl. Mech. Eng. 20, 717–731 (2015)

    Article  Google Scholar 

  21. P. Ailawalia, S.K. Sachdeva, D.S. Pathania, J. Theor. Appl. Mech. 46, 65–82 (2016)

    Article  Google Scholar 

  22. P. Ailawalia, A. Singla, Mech. Mech. Eng. 23, 233–240 (2019)

    Article  Google Scholar 

  23. M.I.A. Othman, E.E.M. Eraki, S.Y. Atwa, M.F. Ismail, J. Appl. Math. Mech. 101, e202100109 (2021). https://doi.org/10.1002/zamm.202100109

    Article  Google Scholar 

  24. M.I.A. Othman, R.S. Tantawi, E.E.M. Eraki, Microsyst. Technol. 23, 5587–5598 (2017)

    Article  Google Scholar 

  25. M.I.A. Othman, M.E.M. Zidan, I.E.A. Mohamed, Steel Compos. Struct. An Int’l J. 38, 355–363 (2021)

    Google Scholar 

  26. W.M. Ewing, W.S. Jardetzky, F. Press, A. Beiser, Phys. Today 10, 27 (1957)

    Article  ADS  Google Scholar 

  27. K.E. Bullen, An Introduction to Theory of Seismology (Cambridge University Press, Cambridge, 1963)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Mohamed I. A. Othman & Ebtesam E. M. Eraki

  2. Department of Engineering Mathematics and Physics, Higher Institute of Engineering, Shorouk Academy, P.O. Box 11837, Al- Shorouk City, Egypt

    Sarhan Y. Atwa & Mohamed F. Ismail

Authors
  1. Mohamed I. A. Othman
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  2. Sarhan Y. Atwa
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  3. Ebtesam E. M. Eraki
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  4. Mohamed F. Ismail
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Correspondence to Mohamed F. Ismail.

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Appendices

Appendix 1

$$\begin{gathered} a_{1} = \frac{\mu }{{\rho c_{1}^{2} }},\;a_{2} = \frac{\lambda + \mu }{{\rho \,c_{1}^{2} }},\;a_{3} = \frac{{\beta_{1} \lambda_{0} c_{1}^{2} }}{{a_{0} \beta_{0} \,w^{ * 2} }},\;a_{4} = \frac{{\lambda_{1} c_{1}^{2} }}{{a_{0} \,w^{ * 2} }},\;a_{5} = \frac{{\lambda_{0}^{2} }}{{a_{0} \rho \,w^{ * 2} }},\;a_{6} = \frac{{\rho j_{0} c_{1}^{2} }}{{2a_{0} }}, \hfill \\ \,a_{7} = \frac{{\rho c_{e} c_{1}^{2} }}{{kw^{ * } }},\;a_{8} = \frac{{\beta_{0}^{2} T_{0} }}{{k\rho w^{ * } }},\;a_{9} = \frac{{\beta_{0} \beta_{1} T_{0} c_{1}^{2} }}{{k\lambda_{0} w^{ * } }},\;a_{10} = \frac{\lambda }{{\rho c_{1}^{2} }},\;a_{11} = \frac{\lambda \, + \,2\mu }{{\beta_{0} \,T_{0} }},\;a_{12} = \frac{{\mu + \tfrac{1}{2}(\lambda + 2\mu )P}}{{\rho c_{1}^{2} }}, \hfill \\ a_{13} = \frac{{\mu - \tfrac{1}{2}(\lambda + 2\mu )P}}{{\rho \,c_{1}^{2} }},\;\delta_{1} = a_{1} {\kern 1pt} + {\kern 1pt} {\kern 1pt} \frac{P}{2},\;\delta_{2} = a_{1} b^{2} + a_{2} b^{2} + \omega^{2} ,\;\delta_{3} = iba_{2} - \frac{P}{2}ib,\;\delta_{4} = a_{1} {\kern 1pt} + {\kern 1pt} a_{2} , \hfill \\ \delta_{5} = a_{1} b^{2} + \frac{P}{2}b^{2} + \omega^{2} ,\;\delta_{6} = iba_{5} ,\;\delta_{7} = b^{2} \, + \,a_{4} \, + \,a_{6} \omega^{2} ,\;\delta_{8} = 1 + \tau_{\theta } \omega ,\;\delta_{9} = 1 + \tau_{q} \omega , \hfill \\ \delta_{10} = i{\kern 1pt} b{\kern 1pt} a_{8} \omega {\kern 1pt} \delta_{9} ,\;\delta_{11} = a_{8} \omega \,\delta_{9} ,\;\delta_{12} = b^{2} \delta_{8} + a_{7} \omega \,\delta_{9} ,\;\delta_{13} = a_{9} \omega , \hfill \\ \end{gathered}$$
$$A = \frac{ - 1}{{\delta_{1} \delta_{4} \delta_{8} }}\left[ {a_{5} \delta_{1} \delta_{8} - \delta_{1} \delta_{5} \delta_{8} - \delta_{2} \delta_{4} \delta_{8} - \delta_{3}^{2} \delta_{8} - \delta_{1} \delta_{4} \delta_{12} - \delta_{1} \delta_{4} \delta_{7} \delta_{8} } \right]$$
$$B = \frac{1}{{\delta_{1} \delta_{4} \delta_{8} }}\left[ \begin{gathered} \delta_{2} \delta_{11} + \delta_{3} \delta_{10} - a_{3} \delta_{1} \delta_{11} - a_{5} \delta_{2} \delta_{8} - a_{5} \delta_{1} \delta_{12} - a_{5} \delta_{1} \delta_{13} + ib\delta_{3} \delta_{11} - ib\delta_{4} \delta_{10} + \delta_{2} \delta_{5} \delta_{8} \hfill \\ - \delta_{3} \delta_{6} \delta_{8} + \delta_{1} \delta_{5} \delta_{12} + \delta_{2} \delta_{4} \delta_{12} + \delta_{3}^{2} \delta_{12} + \delta_{1} \delta_{7} \delta_{11} + ia_{5} b\delta_{3} \delta_{8} + a_{3} \delta_{1} \delta_{4} \delta_{13} + ib\delta_{4} \delta_{6} \delta_{8} + \delta_{1} \delta_{5} \delta_{7} \delta_{8} \hfill \\ + \delta_{2} \delta_{4} \delta_{7} \delta_{8} + \delta_{3}^{2} \delta_{7} \delta_{8} + \delta_{1} \delta_{4} \delta_{7} \delta_{12} \hfill \\ \end{gathered} \right]$$
$$\begin{gathered} C = \frac{ - 1}{{\delta_{1} \delta_{4} \delta_{8} }}[a_{3} \delta_{2} \delta_{11} + a_{3} \delta_{3} \delta_{10} + a_{5} \delta_{2} \delta_{12} + a_{5} \delta_{2} \delta_{13} + ib\delta_{5} \delta_{10} - \delta_{2} \delta_{5} \delta_{12} - \delta_{2} \delta_{7} \delta_{11} - \delta_{3} \delta_{7} \delta_{10} + \delta_{3} \delta_{6} \delta_{12} \hfill \\ + \delta_{3} \delta_{6} \delta_{13} + iba_{3} \delta_{3} \delta_{11} - iba_{3} \delta_{4} \delta_{10} + iba_{5} \delta_{3} \delta_{12} + iba_{5} \delta_{4} \delta_{13} - a_{3} \delta_{1} \delta_{5} \delta_{13} - a_{3} \delta_{2} \delta_{4} \delta_{13} - a_{3} \delta_{3}^{2} \delta_{13} \hfill \\ - ib\delta_{5} \delta_{6} \delta_{8} - ib\delta_{3} \delta_{7} \delta_{11} + ib\delta_{4} \delta_{7} \delta_{10} - ib\delta_{4} \delta_{6} \delta_{12} - ib\delta_{4} \delta_{6} \delta_{13} - \delta_{2} \delta_{5} \delta_{7} \delta_{8} - \delta_{1} \delta_{5} \delta_{7} \delta_{12} \hfill \\ - \delta_{2} \delta_{4} \delta_{7} \delta_{12} - \delta_{3}^{2} \delta_{7} \delta_{12} ] \hfill \\ E = \frac{1}{{\delta_{1} \delta_{4} \delta_{8} }}[iba_{3} \delta_{5} \delta_{10} + a_{3} \delta_{2} \delta_{5} \delta_{13} - ib\delta_{5} \delta_{7} \delta_{10} + ib\delta_{5} \delta_{6} \delta_{12} + ib\delta_{5} \delta_{6} \delta_{13} + \delta_{2} \delta_{5} \delta_{7} \delta_{12} ] \hfill \\ \end{gathered}$$
$$\begin{gathered} H_{1n} = \frac{{k_{n} \delta_{2} + ibk_{n} \delta_{3} - \delta_{1} k_{n}^{3} }}{{(ib\delta_{4} - \delta_{3} )k_{n}^{2} - ib\delta_{5} }},\;H_{2n} = \frac{{k_{n}^{3} \delta_{11} H_{1n} + a_{5} \delta_{11} k_{n} H_{1n} + \delta_{7} \delta_{11} H_{1n} k_{n} - k_{n}^{2} \delta_{10} - \delta_{6} \delta_{13} + \delta_{7} \delta_{10} }}{{\delta_{7} \delta_{8} k_{n}^{2} + \delta_{12} k_{n}^{2} - \delta_{8} k_{n}^{4} - \delta_{7} \delta_{12} - a_{3} \delta_{13} }}, \hfill \\ H_{3n} = \frac{{a_{5} k_{n} H_{1n} + a_{3} H_{2n} - \delta_{6} }}{{\delta_{7} - k_{n}^{2} }},\;H_{4n} = ib - a_{10} k_{n} H_{1n} - H_{2n} + H_{3n} ,\;H_{5n} = iba_{10} - k_{n} H_{1n} - H_{2n} + H_{3n} , \hfill \\ H_{6n} = iba_{13} H_{1n} - a_{12} k_{n} . \hfill \\ \end{gathered}$$

Appendix 2

$$\begin{gathered} l_{1} = \frac{{\lambda^{e} + 2\mu^{e} }}{{\rho^{e} c_{1}^{e2} }},l_{2} = \frac{{\lambda^{e} + \mu^{e} }}{{\rho^{2} c_{1}^{e2} }},\;l_{3} = \frac{{\mu^{e} }}{{\rho^{e} c_{1}^{e2} }},\;l_{4} = \frac{{\lambda^{e} }}{{\rho^{e} c_{1}^{e2} }},\;\ell_{1} = b^{2} l_{1} + \omega^{2} ,\;\ell_{2} = l_{3} b^{2} + \omega^{2} , \hfill \\ G = \frac{{b^{2} l_{2}^{2} - \delta_{1} l_{1} - \delta_{2} l_{3} }}{{l_{1} l_{3} }},\;N = \frac{{\delta_{1} \delta_{2} }}{{l_{1} l_{3} }},\;L_{1n} = \frac{{l_{3} r_{n}^{2} - \delta_{1} }}{{ibl_{2} r_{n} }},\;L_{1(n + 2)} = \frac{{l_{3} r_{n}^{2} - \delta_{1} }}{{ - \;ibl_{2} r_{n} }},\;L_{2n} = ibl_{1} - r_{n} l_{4} L_{1n} , \hfill \\ L_{2(n + 2)} = ibl_{1} + r_{n} l_{4} L_{1(n + 2)} ,\;L_{3n} = ibl_{4} - l_{1} r_{n} L_{1n} ,\;L_{3(n + 2)} = ibl_{4} - l_{1} r_{n} L_{1(n + 2)} , \hfill \\ L_{4n} = ibl_{3} L_{1n} - l_{3} r_{n} ,\;L_{4(n + 2)} = i{\kern 1pt} b{\kern 1pt} l_{3} L_{1(n + 2)} + l_{3} r_{n} . \hfill \\ \end{gathered}$$

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Othman, M.I.A., Atwa, S.Y., Eraki, E.E.M. et al. The initial stress effect on a thermoelastic micro-elongated solid under the dual-phase-lag model. Appl. Phys. A 127, 697 (2021). https://doi.org/10.1007/s00339-021-04809-x

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