Studying the isotherm of the complementary Schaefer Ignaczak  thermodynamical process of the first plane state of elastic strains for the   unbounded micropolar body-Fourier Schaefer-Ignaczak formulas

Kheder Manhal; Mountajab; Monir

doi:https://doi.org/10.54216/PAMDA.030101

Full Length Article

Volume 3 • Issue 1 • PP: 08-20 • 2023

Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas

Kheder Manhal Al-Saleh ^1*

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,

Mountajab Al-Hasan ²

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,

Monir Makhlouf ³

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¹PHD Student. Department of Mathematics, Albaath University, Homs, Syria

²Mountajab Al-Hasan, Prof. Department of Mathematics, Albaath University, Homs, Syria

³Department of Mathematics, Albaath University, Homs, Syria

* Corresponding Author.

DOI https://doi.org/10.54216/PAMDA.030101

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Received: May 2023 Revised: August 2023 Accepted December, 2023

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Abstract

This paper concerns the Ignaczak stress-temperature distribution [2] of the homogenous isotropic 2D micropolar thermodynamical in the first plane state of elastic strain, which discussed by Eringen [9] and Nowacki [8]. In [1] we provide this problem with new analytical method called Schaefer-Ignaczak method. In the paper, we do the following; We prove that the complementary Schaefer-Ignaczak process is an isothermal process for infinite 2D (E-N:5) [6,8], with no stresses and temperature at infinity, and then we find the related Fourier Schaefer-Ignaczak formulas [1] for the classical and complementary behavior of a two-dimensional infinite body (E-N:5), which is a micropolar body.

Keywords

isotherm Schaefer Ignaczak thermodynamical process elastic strains

References

[1] Mountajab Al-Hasan, combining regular solutions of the Schaefer-Ignaczak thermodynamical behaviors relating to the first plane state of elastic strain of the micropolar body subjected to temperature field, Prospects for Applied Mathematics and data Analysis (PAMDA), Vol. 02, No. 01, PP. 27-41, 2023.

[2] Al-Hasan M., Dyszlewicz J. Coupled, Dynamic Micropolar Problems of Thermoelasticity: Stress–Temperature Equations of Motion of Ignaczak Type. In: Hetnarski R.B. (eds) Encyclopedia of Thermal Stresses. Springer, Dordrecht,2014

[3] Hetnarski, R.B., Ignaczak, J., Eslami, M.R., Noda, N., Sumi, N., and Tanigawa, Y. - Theory of Elasticity and Thermal Stresses, Springer Science+Business Media Dordrecht. 2013.

[4] Hetnarski, R.B., and Ignaczak, J., - The Mathematical Theory of Elasticity, Second Edition, CRC Press, Taylor & Francis Group,6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742. 2011

[5] Ignaczak, J., Starzewski, M.O - Thermoelasticity with Finite Wave Speeds, Oxford University Press Inc., New York. 2010

[6] Dyszlewicz, J, Micropolar Theory of Elasticity, in: Series Lectures. Notes in Applied and Computational Mechanics, Vol.15, 356 p, Springer. 2004

[7] Dyszlewicz, J, Selected problems of linear asymmetrical thermoelasticity, Journal of Thermal Stresesses,19,1996

[8] Nowacki, W, Theory of Asymmetric Elasticity, Warsaw, PWN, 1986

[9] Eringen, A. C, Linear theory of micropolar elasticity, J. Math, 1966

[10] Ignaczak, J Tensorial equations of motion for elastic materials with microstructure, in: Trends of Elasticity and Thermoelasticity, Witold Nowacki Ann.Volume, Wolters-Noordhoff Groningen, 1971

[11] Debnath, L& Bhatta, D, Integral Transforms and their Applications, (Second Edition), CRC Press, Boca Raton, Florida, 2007.

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Al-Saleh, Kheder Manhal, Al-Hasan, Mountajab, Makhlouf, Monir. "Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas." Prospects for Applied Mathematics and Data Analysis, vol. Volume 3, no. Issue 1, 2023, pp. 08-20. DOI: https://doi.org/10.54216/PAMDA.030101

Al-Saleh, K., Al-Hasan, M., Makhlouf, M. (2023). Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas. Prospects for Applied Mathematics and Data Analysis, Volume 3(Issue 1), 08-20. DOI: https://doi.org/10.54216/PAMDA.030101

Al-Saleh, Kheder Manhal, Al-Hasan, Mountajab, Makhlouf, Monir. "Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas." Prospects for Applied Mathematics and Data Analysis Volume 3, no. Issue 1 (2023): 08-20. DOI: https://doi.org/10.54216/PAMDA.030101

Al-Saleh, K., Al-Hasan, M., Makhlouf, M. (2023) 'Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas', Prospects for Applied Mathematics and Data Analysis, Volume 3(Issue 1), pp. 08-20. DOI: https://doi.org/10.54216/PAMDA.030101

Al-Saleh K, Al-Hasan M, Makhlouf M. Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas. Prospects for Applied Mathematics and Data Analysis. 2023;Volume 3(Issue 1):08-20. DOI: https://doi.org/10.54216/PAMDA.030101

K. Al-Saleh, M. Al-Hasan, M. Makhlouf, "Studying the isotherm of the complementary Schaefer Ignaczak thermodynamical process of the first plane state of elastic strains for the unbounded micropolar body-Fourier Schaefer-Ignaczak formulas," Prospects for Applied Mathematics and Data Analysis, vol. Volume 3, no. Issue 1, pp. 08-20, 2023. DOI: https://doi.org/10.54216/PAMDA.030101

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