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Nonlocal and micropolar effects in a transversely isotropic functionally graded thermoelastic solid under an inclined load

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Abstract

The objective of this study is to analyze the thermo-mechanical interactions occurring in a nonlocal transversely isotropic functionally graded (nonhomogeneous) micropolar thermoelastic half-space when subjected to an inclined load, based on the Lord and Shulman (LS) theory. The material properties are assumed to be graded exponentially along the \(z\)-direction. Utilizing the normal mode technique, the exact expressions for physical fields such as normal displacement, normal stress, shear stress, temperature field, and couple stress are derived. Numerical computation of the derived results is performed for a material resembling a magnesium crystal, and graphical representations are presented to illustrate the impacts of nonhomogeneity parameter, material’s anisotropy, time, nonlocal parameter, microinertia, and the inclination angle of the applied load on the variations of different physical fields. Some specific cases of interest have been deduced from the present investigation.

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References

  • Abbas, I.A., Kumar, R.: Interaction due to a mechanical source in transversely isotropic micropolar media. J. Vib. Control 20, 1663–1670 (2014)

    Article  MathSciNet  Google Scholar 

  • Aboudi, J., Pindera, M.J., Arnold, S.M.: Thermoelastic theory for the response of materials functionally graded in two directions. Int. J. Solids Struct. 33, 931–966 (1996)

    Article  Google Scholar 

  • Abouelregal, A.E., Moaaz, O., Khalil, K.M., Abouhawwash, M., Nasr, M.E.: Micropolar thermoelastic plane waves in microscopic materials caused by Hall-current effects in a two-temperature heat conduction model with higher-order time derivatives. Arch. Appl. Mech. (2023). https://doi.org/10.1007/s00419-023-02362-y

    Article  Google Scholar 

  • Barak, M.S., Dhankhar, P.: Effect of inclined load on a functionally graded fiber-reinforced thermoelastic medium with temperature-dependent properties. Acta Mech. 233, 3645–3662 (2022)

    Article  MathSciNet  Google Scholar 

  • Barak, M.S., Dhankhar, P.: Thermo-mechanical interactions in a rotating nonlocal functionally graded transversely isotropic elastic half-space. Z. Angew. Math. Mech. 103, e202200319 (2023)

    Article  MathSciNet  Google Scholar 

  • Barak, M.S., Poonia, R., Devi, S., Dhankhar, P.: Nonlocal and dual-phase-lag effects in a transversely isotropic exponentially graded thermoelastic medium with voids. Z. Angew. Math. Mech. e202300579 (2024). https://doi.org/10.1002/zamm.202300579

  • Biot, M.: Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27, 240–253 (1956)

    Article  MathSciNet  Google Scholar 

  • Challamel, N., Grazide, C., Picandet, V., Perrot, A., Zhang, Y.: A nonlocal Fourier’s law and its application to the heat conduction of one-dimensional and two-dimensional thermal lattices. C. R. Mecanique 344, 388–401 (2016)

    Article  Google Scholar 

  • Edelen, D.G.B., Laws, N.: On the thermodynamics of systems with nonlocality. Arch. Ration. Mech. Anal. 43, 24–35 (1971)

    Article  MathSciNet  Google Scholar 

  • Edelen, D.G.B., Green, A.E., Laws, N.: Nonlocal continuum mechanics. Arch. Ration. Mech. Anal. 43, 36–44 (1971)

    Article  MathSciNet  Google Scholar 

  • Eringen, A.C.: Linear theory of micropolar elasticity. J. Math. Mech. 15, 909–923 (1966)

    MathSciNet  Google Scholar 

  • Eringen, A.C.: Foundation of Micropolar Thermoelasticity. Courses and Lectures, vol. 23. Springer, Wien (1970)

    Book  Google Scholar 

  • Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)

    Article  MathSciNet  Google Scholar 

  • Eringen, A.C.: Theory of nonlocal thermoelasticity. Int. J. Eng. Sci. 12, 1063–1077 (1974)

    Article  Google Scholar 

  • Eringen, A.C.: Plane waves in nonlocal micropolar elasticity. Int. J. Eng. Sci. 22, 1113–1121 (1984)

    Article  Google Scholar 

  • Eringen, A.C.: Theory of micropolar elasticity. In: Microcontinuum Field Theories I: Foundations and Solids, pp. 101–248. Springer, New York (1999)

    Chapter  Google Scholar 

  • Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    Google Scholar 

  • Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)

    Article  MathSciNet  Google Scholar 

  • Eringen, A.C., Suhubi, E.S.: Nonlinear theory of micro-elastic solids I. Int. J. Eng. Sci. 2, 189–203 (1964)

    Article  MathSciNet  Google Scholar 

  • Gunghas, A., Kumar, R., Deswal, S., Kalkal, K.K.: Influence of rotation and magnetic fields on a functionally graded thermoelastic solid subjected to a mechanical load. J. Math. 19, 1–16 (2019)

    Article  MathSciNet  Google Scholar 

  • Kalkal, K.K., Gunghas, A., Deswal, S.: Two-dimensional magneto-thermoelastic interactions in a micropolar functionally graded solid. Mech. Based Des. Struct. Mach. 48, 348–369 (2020)

    Article  Google Scholar 

  • Kalkal, K.K., Deswal, S., Poonia, R.: Reflection of plane waves in a rotating non-local fiber-reinforced transversely isotropic thermoelastic medium. J. Therm. Stresses 46, 276–292 (2023)

    Article  Google Scholar 

  • Khurana, A., Tomar, S.K.: Reflection of plane longitudinal waves from the stress-free boundary of a nonlocal micropolar solid half-space. J. Mech. Mater. Struct. 8, 95–107 (2013)

    Article  Google Scholar 

  • Kumar, R., Gupta, R.R.: Propagation of waves in transversely isotropic micropolar generalized thermoelastic half space. Int. Commun. Heat Mass Transf. 37, 1452–1458 (2010)

    Article  Google Scholar 

  • Lord, H.W., Shulman, Y.A.: Generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967)

    Article  Google Scholar 

  • Lotfy, K., Tantawi, R.S.: Photo-thermal-elastic interaction in a functionally graded material (FGM) and magnetic field. Silicon 12, 295–303 (2020)

    Article  Google Scholar 

  • Nowacki, W.: Couple stresses in the theory of thermoelasticity I. Bull. Acad. Pol. Sci., Sér. Sci. Tech. 14, 129–138 (1966a)

    Google Scholar 

  • Nowacki, W.: Couple stresses in the theory of thermoelasticity II. Bull. Acad. Pol. Sci., Sér. Sci. Tech. 14, 263–272 (1966b)

    Google Scholar 

  • Nowacki, W.: Couple stresses in the theory of thermoelasticity III. Bull. Acad. Pol. Sci., Sér. Sci. Tech. 14, 801–809 (1966c)

    Google Scholar 

  • Poonia, R., Deswal, S., Kalkal, K.K.: Propagation of plane waves in a nonlocal transversely isotropic thermoelastic medium with voids and rotation. Z. Angew. Math. Mech. 103, e202200493 (2023)

    Article  MathSciNet  Google Scholar 

  • Sarkar, N., Tomar, S.K.: Plane waves in nonlocal thermoelastic solid with voids. J. Therm. Stresses 42, 580–606 (2019)

    Article  Google Scholar 

  • Sheoran, S.S., Chaudhary, S., Deswal, S.: Thermo-mechanical interactions in a nonlocal transversely isotropic material with rotation under Lord–Shulman model. Waves Random Complex Media (2021). https://doi.org/10.1080/17455030.2021.1986648

    Article  Google Scholar 

  • Suhubi, E.S., Eringen, A.C.: Nonlinear theory of micro-elastic solids II. Int. J. Eng. Sci. 2, 389–404 (1964)

    Article  MathSciNet  Google Scholar 

  • Wang, X., Ren, X., Zhou, H., Yu, J., Li, K.: Dynamics of thermoelastic Lamb waves in functionally graded nanoplates based on the modified nonlocal theory. Appl. Math. Model. 117, 142–161 (2023)

    Article  MathSciNet  Google Scholar 

  • Yu, J., Wang, X., Zhang, X., Li, Z., Li, F.: An analytical integration Legendre polynomial series approach for Lamb waves in fractional order thermoelastic multilayered plates. Math. Methods Appl. Sci. 45, 7631–7651 (2022)

    Article  MathSciNet  Google Scholar 

  • Zheng, M., Ma, H., Lyu, Y., Lu, C., He, C.: Derivation of circumferential guided waves equations for a multilayered laminate composite hollow cylinder by state-vector and Legendre polynomial hybrid formalism Compos. Struct. 255, 112950 (2021)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, Indira Gandhi University, Meerpur (Rewari), 122502, Haryana, India

    Priti Dhankhar & M. S. Barak

  2. Department of Mathematics, Government College, Nalwa (Hisar), 125037, Haryana, India

    Ravinder Poonia

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  1. Priti Dhankhar
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  2. M. S. Barak
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  3. Ravinder Poonia
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Contributions

[Priti Dhankhar] contributed to the study conception and design. Material preparation, data collection and analysis were performed by [Priti Dhankhar], [M.S. Barak] and [Ravinder Poonia]. The first draft of the manuscript was written by [Priti Dhankhar] and all the authors commented on previous versions of the manuscript. Also, all the authors read and approved the final manuscript.

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Correspondence to Ravinder Poonia.

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Appendix

Appendix

$$\begin{aligned} & A_{1}=\frac{L_{12}}{L_{11}},\;\;A_{2}=\frac{L_{13}}{L_{11}},\;\;A_{3}= \frac{L_{14}}{L_{11}},\;\;A_{4}=\frac{L_{15}}{L_{11}},\;\;A_{5}= \frac{L_{16}}{L_{11}},\;\;A_{6}=\frac{L_{17}}{L_{11}}, \\ & A_{7}=\frac{L_{18}}{L_{11}},\;\;A_{8}=\frac{L_{19}}{L_{11}},\;\;L_{11}=I_{11}I_{22}I_{43}, \\ &L_{12}=I_{11}I_{22}I_{44}+I_{11}I_{23}I_{43}+I_{12}I_{22}I_{43}+ I_{11}I_{22}I_{33}I_{43}, \\ &L_{13}=T_{1}+T_{2}+T_{3}, L_{14}=T_{4}+T_{5}+T_{6}+T_{7}+T_{8}, \\ &L_{15}=T_{9}+T_{10}+T_{11}+T_{12}+T_{13}+T_{14}+T_{15}+T_{16}+T_{17}, \\ &L_{16}=T_{18}+T_{19}+T_{20}+T_{21}+T_{22}+T_{23}+T_{24}+T_{25}+T_{26}+T_{27}, \\ &L_{17}=T_{28}+T_{29}+T_{30}+T_{31}+T_{32}+T_{33}+T_{34}+T_{35}+T_{36}, \\ &L_{18}=T_{37}+T_{38}+T_{39}+T_{40}+T_{41}, L_{19}=+T_{42}+T_{43},\\ &T_{1}=I_{11}I_{22}I_{45} - I_{14}^{2}I_{43} + I_{11}I_{23}I_{44} + I_{11}I_{24}I_{43}+ I_{12}I_{22}I_{44} + I_{12}I_{23}I_{43}, \\ &T_{2}=I_{13}I_{22}I_{43}- I_{17}I_{22}I_{41}+ I_{11}I_{22}I_{33}I_{44} + I_{11}I_{22}I_{34}I_{43} + I_{11}I_{23}I_{33}I_{43}, \\ &T_{3}=I_{12}I_{22}I_{33}I_{43}- I_{11}I_{25}I_{32}I_{43}, \\ &T_{4}=I_{11}I_{23}I_{45} - I_{14}I_{15}I_{43} - I_{14}I_{21}I_{43} - I_{14}^{2}I_{44}+ I_{11}I_{24}I_{44}+ I_{12}I_{22}I_{45}, \\ &T_{5}=+ I_{12}I_{23}I_{44} + I_{12}I_{24}I_{43}+ I_{13}I_{22}I_{44} + I_{13}I_{23}I_{43} - I_{17}I_{23}I_{41} - I_{18}I_{22}I_{41}, \\ &T_{6}=- I_{14}^{2}I_{33}I_{43} + I_{11}I_{22}I_{33}I_{45} + I_{11}I_{22}I_{34}I_{44}+ I_{11}I_{23}I_{33}I_{44}+ I_{11}I_{23}I_{34}I_{43}, \\ &T_{7}=I_{11}I_{24}I_{33}I_{43}+ I_{12}I_{22}I_{33}I_{44} + I_{12}I_{22}I_{34}I_{43} + I_{12}I_{23}I_{33}I_{43}+ I_{13}I_{22}I_{33}I_{43}, \\ &T_{8}=- I_{11}I_{25}I_{32}I_{44} - I_{11}I_{26}I_{32}I_{43}-I_{12}I_{25}I_{32}I_{43} - I_{17}I_{22}I_{33}I_{41}, \\ &T_{9}=I_{14}I_{17}I_{42} - I_{14}I_{15}I_{44} - I_{14}^{2}I_{45} - I_{14}I_{21}I_{44}- I_{15}I_{21}I_{43} + I_{11}I_{24}I_{45}, \\ &T_{10}=- I_{11}I_{27}I_{42} + I_{12}I_{23}I_{45}+ I_{12}I_{24}I_{44} + I_{13}I_{22}I_{45} + I_{13}I_{23}I_{44} + I_{13}I_{24}I_{43}, \\ &T_{11}= I_{14}I_{27}I_{41} - I_{17}I_{24}I_{41} - I_{18}I_{23}I_{41} - I_{14}^{2}I_{33}I_{44}- I_{14}^{2}I_{34}I_{43} - I_{14}I_{15}I_{33}I_{43}, \\ &T_{12}= I_{14}I_{16}I_{32}I_{43}- I_{14}I_{21}I_{33}I_{43} + I_{11}I_{22}I_{34}I_{45} + I_{11}I_{23}I_{33}I_{45}+ I_{11}I_{23}I_{34}I_{44}, \\ &T_{13}= I_{11}I_{24}I_{33}I_{44} + I_{11}I_{24}I_{34}I_{43}+ I_{12}I_{22}I_{33}I_{45} + I_{12}I_{22}I_{34}I_{44} + I_{12}I_{23}I_{33}I_{44}, \\ &T_{14}= I_{12}I_{23}I_{34}I_{43} + I_{12}I_{24}I_{33}I_{43} + I_{13}I_{22}I_{33}I_{44}+ I_{13}I_{22}I_{34}I_{43}+ I_{13}I_{23}I_{33}I_{43}, \\ &T_{15}= - I_{16}I_{22}I_{31}I_{43}- I_{11}I_{25}I_{32}I_{45}- I_{11}I_{26}I_{32}I_{44} - I_{12}I_{25}I_{32}I_{44}- I_{12}I_{26}I_{32}I_{43}, \\ &T_{16}= - I_{13}I_{25}I_{32}I_{43} + I_{14}I_{25}I_{31}I_{43}- I_{17}I_{22}I_{34}I_{41} - I_{17}I_{23}I_{33}I_{41}- I_{18}I_{22}I_{33}I_{41}, \\ &T_{17}= I_{17}I_{25}I_{32}I_{41}, \\ &T_{18}= I_{14}I_{18}I_{42}-I_{14}I_{15}I_{45}-I_{14}I_{21}I_{45}-I_{15}I_{21}I_{44} +I_{17}I_{21}I_{42}+I_{12}I_{24}I_{45}, \\ &T_{19}= -I_{12}I_{27}I_{42}+I_{13}I_{23}I_{45} +I_{13}I_{24}I_{44}+I_{15}I_{27}I_{41}-I_{18}I_{24}I_{41}-I_{14}^{2}I_{33}I_{45}, \\ &T_{20}= -I_{14}^{2}I_{34}I_{44}-I_{14}I_{15}I_{33}I_{44}-I_{14}I_{15}I_{34}I_{43} +I_{14}I_{16}I_{32}I_{44}+I_{14}I_{17}I_{33}I_{42}, \\ &T_{21}=-I_{14}I_{21}I_{33}I_{44} -I_{14}I_{21}I_{34}I_{43}-I_{15}I_{21}I_{33}I_{43}+I_{16}I_{21}I_{32}I_{43} +I_{11}I_{23}I_{34}I_{45}, \\ &T_{22}=I_{11}I_{24}I_{33}I_{45} +I_{11}I_{24}I_{34}I_{44}-I_{11}I_{27}I_{33}I_{42} +I_{12}I_{22}I_{34}I_{45}+I_{12}I_{23}I_{33}I_{45}, \\ &T_{23}=I_{12}I_{23}I_{34}I_{44}+I_{12}I_{24}I_{33}I_{44} +I_{12}I_{24}I_{34}I_{43}+I_{13}I_{22}I_{33}I_{45} +I_{13}I_{22}I_{34}I_{44}, \\ &T_{24}=I_{13}I_{23}I_{33}I_{44} +I_{13}I_{23}I_{34}I_{43}+I_{13}I_{24}I_{33}I_{43}-I_{16}I_{22}I_{31}I_{44} -I_{16}I_{23}I_{31}I_{43}, \\ &T_{25}=-I_{11}I_{26}I_{32}I_{45}-I_{12}I_{25}I_{32}I_{45} -I_{12}I_{26}I_{32}I_{44}-I_{13}I_{25}I_{32}I_{44}-I_{13}I_{26}I_{32}I_{43}, \\ &T_{26}=I_{14}I_{25}I_{31}I_{44}+I_{14}I_{26}I_{31}I_{43} +I_{15}I_{25}I_{31}I_{43} +I_{14}I_{27}I_{33}I_{41}-I_{17}I_{23}I_{34}I_{41}, \\ &T_{27}=-I_{17}I_{24}I_{33}I_{41} -I_{18}I_{22}I_{34}I_{41}-I_{18}I_{23}I_{33}I_{41}+I_{17}I_{26}I_{32}I_{41} +I_{18}I_{25}I_{32}I_{41}, \\ &T_{28}=I_{18}I_{21}I_{42}-I_{15}I_{21}I_{45}+I_{13}I_{24}I_{45}-I_{13}I_{27}I_{42} -I_{14}^{2}I_{34}I_{45}-I_{14}I_{15}I_{33}I_{45}, \\ &T_{29}=-I_{14}I_{15}I_{34}I_{44} +I_{14}I_{16}I_{32}I_{45}+I_{14}I_{17}I_{34}I_{42}+I_{14}I_{18}I_{33}I_{42} -I_{14}I_{21}I_{33}I_{45}, \\ &T_{30}=-I_{14}I_{21}I_{34}I_{44}-I_{15}I_{21}I_{33}I_{44} -I_{15}I_{21}I_{34}I_{43} +I_{16}I_{21}I_{32}I_{44}+I_{17}I_{21}I_{33}I_{42}, \\ &T_{31}=I_{11}I_{24}I_{34}I_{45}-I_{11}I_{27}I_{34}I_{42}+I_{12}I_{23}I_{34}I_{45} +I_{12}I_{24}I_{33}I_{45}+I_{12}I_{24}I_{34}I_{44}, \\ &T_{32}=-I_{12}I_{27}I_{33}I_{42} +I_{13}I_{22}I_{34}I_{45}+I_{13}I_{23}I_{33}I_{45} +I_{13}I_{23}I_{34}I_{44} +I_{13}I_{24}I_{33}I_{44}, \\ &T_{33}=I_{13}I_{24}I_{34}I_{43}-I_{16}I_{22}I_{31}I_{45} -I_{16}I_{23}I_{31}I_{44}-I_{16}I_{24}I_{31}I_{43}-I_{12}I_{26}I_{32}I_{45}, \\ &T_{34}=-I_{13}I_{25}I_{32}I_{45}-I_{13}I_{26}I_{32}I_{44}+I_{14}I_{25}I_{31}I_{45} +I_{14}I_{26}I_{31}I_{44}+I_{15}I_{25}I_{31}I_{44}, \\ &T_{35}=I_{15}I_{26}I_{31}I_{43} -I_{17}I_{25}I_{31}I_{42} +I_{14}I_{27}I_{34}I_{41}+I_{15}I_{27}I_{33}I_{41} -I_{16}I_{27}I_{32}I_{41}, \\ &T_{36}=-I_{17}I_{24}I_{34}I_{41}-I_{18}I_{23}I_{34}I_{41} -I_{18}I_{24}I_{33}I_{41}+I_{18}I_{26}I_{32}I_{41}, \\ &T_{37}=I_{14}I_{18}I_{34}I_{42}-I_{14}I_{15}I_{34}I_{45}-I_{14}I_{21}I_{34}I_{45} -I_{15}I_{21}I_{33}I_{45}-I_{15}I_{21}I_{34}I_{44}, \\ &T_{38}=I_{16}I_{21}I_{32}I_{45} +I_{17}I_{21}I_{34}I_{42}+I_{18}I_{21}I_{33}I_{42} +I_{12}I_{24}I_{34}I_{45} -I_{12}I_{27}I_{34}I_{42}, \\ &T_{39}=I_{13}I_{23}I_{34}I_{45} +I_{13}I_{24}I_{33}I_{45} +I_{13}I_{24}I_{34}I_{44}-I_{13}I_{27}I_{33}I_{42}-I_{16}I_{23}I_{31}I_{45}, \\ &T_{40}=-I_{16}I_{24}I_{31}I_{44}-I_{13}I_{26}I_{32}I_{45}+I_{14}I_{26}I_{31}I_{45} +I_{15}I_{25}I_{31}I_{45}+I_{15}I_{26}I_{31}I_{44}, \\ &T_{41}=-I_{17}I_{26}I_{31}I_{42} -I_{18}I_{25}I_{31}I_{42}+I_{15}I_{27}I_{34}I_{41}-I_{18}I_{24}I_{34}I_{41}, \\ &T_{42}=-I_{15}I_{21}I_{34}I_{45}+I_{18}I_{21}I_{34}I_{42} +I_{13}I_{24}I_{34}I_{45} -I_{13}I_{27}I_{34}I_{42}-I_{16}I_{24}I_{31}I_{45}, \\ &T_{43}=I_{16}I_{27}I_{31}I_{42} +I_{15}I_{26}I_{31}I_{45}-I_{18}I_{26}I_{31}I_{42}. \end{aligned}$$

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Dhankhar, P., Barak, M.S. & Poonia, R. Nonlocal and micropolar effects in a transversely isotropic functionally graded thermoelastic solid under an inclined load. Mech Time-Depend Mater (2024). https://doi.org/10.1007/s11043-024-09687-3

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