### Coupled Non-Stationary Thermoelastic Fields In A Rigidly Fixed Round Plate

Document Type : Research Paper

Author

Samara State Technical University, Office 206, 244 Molodogvardeyskaya str., Samara, 443100, Russia

Abstract

The mathematical formulation of thermoelasticity problems includes coupled non-self-adjoint differential equations of motion and heat conduction. The problem of integrating them and constructing a general solution leads, as a rule, to the study of only the heat conduction equation or to the analysis of thermoelasticity problems in an unconnected formulation. However, for a better assessment of thermomechanical fields, it becomes necessary to construct coupled analytical solutions in a three-dimensional formulation. Therefore, the development of effective analytical methods and algorithms for calculating elastic systems is currently one of the urgent problems of modern science. In this problem, a mathematical calculation model is developed and a closed solution of the coupled axisymmetric non-stationary problem of the theory of thermoelasticity for a rigidly fixed isotropic plate is constructed. Design ratios are obtained by the method of finite biorthogonal transformations and are valid for an external temperature effect arbitrary in time (boundary conditions for thermal conductivity of the 1st kind). Software that allows to analyze the effect of coupled thermoelastic fields on the temperature field and the stress-strain state of the structure has been developed. Numerical analysis of the results shows that for a given external temperature effect, the rigidity of an elastic system (physical and mechanical characteristics and geometric dimensions) has a significant effect on its thermoelastic field. The developed calculation algorithm finds its application in the design of enclosing structures in the form of single-layer and multi-layer plates.

Keywords

Main Subjects

[1]          W. Nowacki, 1975, Dynamic problems of thermoelasticity, Springer Science & Business Media,
[2]          Y. S. Podstrigach, V. Lomakin, Y. M. Kolyano, Thermoelasticity of Bodies of Nonuniform Structure, Moscow, 1984.
[3]          M. Mohammadi, A. Farajpour, M. Goodarzi, H. Mohammadi, Temperature Effect on Vibration Analysis of Annular Graphene Sheet Embedded on Visco-Pasternak Foundati, Journal of Solid Mechanics, Vol. 5, No. 3, pp. 305-323, 2013.
[4]          M. Mohammadi, M. Goodarzi, M. Ghayour, A. Farajpour, Influence of in-plane pre-load on the vibration frequency of circular graphene sheet via nonlocal continuum theory, Composites Part B: Engineering, Vol. 51, pp. 121-129, 2013.
[5]          M. Goodarzi, M. Mohammadi, A. Farajpour, M. Khooran, Investigation of the effect of pre-stressed on vibration frequency of rectangular nanoplate based on a visco-Pasternak foundation, 2014.
[6]          S. Asemi, A. Farajpour, M. Mohammadi, Nonlinear vibration analysis of piezoelectric nanoelectromechanical resonators based on nonlocal elasticity theory, Composite Structures, Vol. 116, pp. 703-712, 2014.
[7]          S. R. Asemi, M. Mohammadi, A. Farajpour, A study on the nonlinear stability of orthotropic single-layered graphene sheet based on nonlocal elasticity theory, Latin American Journal of Solids and Structures, Vol. 11, No. 9, pp. 1515-1540, 2014.
[8]          M. R. Farajpour, A. Rastgoo, A. Farajpour, M. Mohammadi, Vibration of piezoelectric nanofilm-based electromechanical sensors via higher-order non-local strain gradient theory, Micro & Nano Letters, Vol. 11, No. 6, pp. 302-307, 2016.
[9]          M. Hosseini, M. Shishesaz, A. Hadi, Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness, Thin-Walled Structures, Vol. 134, pp. 508-523, 2019.
[10]        M. Mohammadi, A. Farajpour, M. Goodarzi, H. Shehni nezhad pour, Numerical study of the effect of shear in-plane load on the vibration analysis of graphene sheet embedded in an elastic medium, Computational Materials Science, Vol. 82, pp. 510-520, 2014/02/01/, 2014.
[11]        M. Mohammadi, M. Ghayour, A. Farajpour, Free transverse vibration analysis of circular and annular graphene sheets with various boundary conditions using the nonlocal continuum plate model, Composites Part B: Engineering, Vol. 45, No. 1, pp. 32-42, 2013.
[12]        M. Mohammadi, M. Ghayour, A. Farajpour, Analysis of Free Vibration Sector Plate Based on Elastic Medium by using New Version of Differential Quadrature Method, Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering, Vol. 3, No. 2, pp. 47-56, 2010.
[13]        M. Mohammadi, A. Rastgoo, Primary and secondary resonance analysis of FG/lipid nanoplate with considering porosity distribution based on a nonlinear elastic medium, Mechanics of Advanced Materials and Structures, Vol. 27, No. 20, pp. 1709-1730, 2020/10/15, 2020.
[14]        H. Moosavi, M. Mohammadi, A. Farajpour, S. H. Shahidi, Vibration analysis of nanorings using nonlocal continuum mechanics and shear deformable ring theory, Physica E: Low-dimensional Systems and Nanostructures, Vol. 44, No. 1, pp. 135-140, 2011/10/01/, 2011.
[15]        M. Mohammadi, A. Farajpour, A. Moradi, M. Hosseini, Vibration analysis of the rotating multilayer piezoelectric Timoshenko nanobeam, Engineering Analysis with Boundary Elements, Vol. 145, pp. 117-131, 2022/12/01/, 2022.
[16]        S. Asemi, A. Farajpour, H. Asemi, M. Mohammadi, Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM, Physica E: Low-dimensional Systems and Nanostructures, Vol. 63, pp. 169-179, 2014.
[17]        H. Harmatij, M. Król, V. Popovycz, Quasi-static problem of thermoelasticity for thermosensitive infinite circular cylinder of complex heat exchange, Advances in Pure Mathematics, Vol. 3, No. 4, pp. 430-437, 2013.
[18]        D. A. Shlyakhin, Z. M. Kusaeva, The associated non-stationary thermal elasticity problem for a two-layer plate, IOP Conference Series: Materials Science and Engineering, Vol. 1015, No. 1, pp. 012009, 2021/01/01, 2021.
[19]        I. Makarova, Solution of an unrelated thermoelasticity problem with boundary conditions of the first kind, Bulletin of the Samara state technical university. Ser. Phys.-Math. Sci, Vol. 28, No. 3, pp. 191-5, 2012.
[20]        S. Sargsyan, Mathematical model of micropolar thermo-elasticity of thin shells, Journal of Thermal Stresses, Vol. 36, No. 11, pp. 1200-1216, 2013.
[21]        K. Verma, Thermoelastic waves in anisotropic plates using normal mode expansion method with thermal relaxation time, World Academy of Science, Engineering and Technology, Vol. 37, pp. 573-580, 2008.
[22]        A. Zhornik, V. Zhornik, P. Savochka, On a thermoelasticity problem for a solid cylinder News of the Southern Federal University, Techn. Sci, Vol. 9, No. 1, pp. 63-9, 2012.
[23]        S. Lychev, Y. Senitskii, Nonsymmetric finite integral transformations and their application to visco-elasticity problems, Vestnik Samarskogo Gosudarstvennogo Universitetea. Estestvennonauchnaya Seriya, Vol. 2002, 01/01, 2002.
[24]        S. Lychev, The related dynamical problem of thermoelasticity for a finite cylinder, Bulletin of the Samara State University, Vol. 4, No. 30, pp. 112-24, 2003.
[25]        M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016.
[26]        Y. E. Senitskii, Solution of coupled dynamic thermoelasticity problem for an infinite cylinder and sphere, Soviet Applied Mechanics, Vol. 18, No. 6, pp. 514-520, 1982.
[27]        S. Lychev, A. Manzhirov, S. V. Joubert, Closed solutions of boundary-value problems of coupled thermoelasticity, Mechanics of solids, Vol. 45, No. 4, pp. 610-623, 2010.
[28]        Y. È. Senitskii, A multicomponent generalized finite integral transformation and its application to nonstationary problems in mechanics, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, No. 4, pp. 57-63, 1991.
[29]        Y. È. Senitskii, A biorthogonal multicomponent finite integral transformation and its application to boundary value problems in mechanics, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, No. 8, pp. 71-81, 1996.
[30]        V. Kovalev, N. Radaev Yu, R. Revinsky, Passage of a generalized GHIII-thermoelastic wave through a waveguide with a heat-permeable wall, Bulletin of the Saratov University, New. ser. Ser. Math. Mechan. Inform, Vol. 11, No. 1, pp. 59-70, 2011.
[31]        V. A. Kovalev, Y. N. Radayev, D. Semenov, Coupled dynamic problems of hyperbolic thermoelasticity, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Vol. 9, No. 5, pp. 94-127, 2009.
[32]        N. GHAYOUR, A. SEDAGHAT, M. MOHAMMADI, WAVE PROPAGATION APPROACH TO FLUID FILLED SUBMERGED VISCO-ELASTIC FINITE CYLINDRICAL SHELLS, JOURNAL OF AEROSPACE SCIENCE AND TECHNOLOGY (JAST), Vol. 8, No. 1, pp. -, 2011.
[33]        A. Farajpour, M. Danesh, M. Mohammadi, Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics, Physica E: Low-dimensional Systems and Nanostructures, Vol. 44, No. 3, pp. 719-727, 2011.
[34]        A. Farajpour, M. Mohammadi, A. Shahidi, M. Mahzoon, Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E: Low-dimensional Systems and Nanostructures, Vol. 43, No. 10, pp. 1820-1825, 2011.
[35]        M. Danesh, A. Farajpour, M. Mohammadi, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications, Vol. 39, No. 1, pp. 23-27, 2012.
[36]        A. Farajpour, A. Shahidi, M. Mohammadi, M. Mahzoon, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures, Vol. 94, No. 5, pp. 1605-1615, 2012.
[37]        M. Mohammadi, M. Goodarzi, M. Ghayour, S. Alivand, Small scale effect on the vibration of orthotropic plates embedded in an elastic medium and under biaxial in-plane pre-load via nonlocal elasticity theory, 2012.
[38]        M. Mohammadi, A. Farajpour, M. Goodarzi, R. Heydarshenas, Levy Type Solution for Nonlocal Thermo-Mechanical Vibration of Orthotropic Mono-Layer Graphene Sheet Embedded in an Elastic Medium, Journal of Solid Mechanics, Vol. 5, No. 2, pp. 116-132, 2013.
[39]        A. Farajpour, A. Rastgoo, M. Mohammadi, Surface effects on the mechanical characteristics of microtubule networks in living cells, Mechanics Research Communications, Vol. 57, pp. 18-26, 2014/04/01/, 2014.
[40]        M. Mohammadi, A. Farajpour, A. Moradi, M. Ghayour, Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment, Composites Part B: Engineering, Vol. 56, pp. 629-637, 2014.
[41]        M. Mohammadi, A. Farajpour, M. Goodarzi, F. Dinari, Thermo-mechanical vibration analysis of annular and circular graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures, Vol. 11, pp. 659-682, 2014.
[42]        M. Mohammadi, A. Moradi, M. Ghayour, A. Farajpour, Exact solution for thermo-mechanical vibration of orthotropic mono-layer graphene sheet embedded in an elastic medium, Latin American Journal of Solids and Structures, Vol. 11, No. 3, pp. 437-458, 2014.
[43]        M. Safarabadi, M. Mohammadi, A. Farajpour, M. Goodarzi, Effect of surface energy on the vibration analysis of rotating nanobeam, 2015.
[44]        H. Asemi, S. Asemi, A. Farajpour, M. Mohammadi, Nanoscale mass detection based on vibrating piezoelectric ultrathin films under thermo-electro-mechanical loads, Physica E: Low-dimensional Systems and Nanostructures, Vol. 68, pp. 112-122, 2015.
[45]        M. Goodarzi, M. Mohammadi, M. Khooran, F. Saadi, Thermo-Mechanical Vibration Analysis of FG Circular and Annular Nanoplate Based on the Visco-Pasternak Foundation, Journal of Solid Mechanics, Vol. 8, No. 4, pp. 788-805, 2016.
[46]        M. Baghani, M. Mohammadi, A. Farajpour, Dynamic and Stability Analysis of the Rotating Nanobeam in a Nonuniform Magnetic Field Considering the Surface Energy, International Journal of Applied Mechanics, Vol. 08, No. 04, pp. 1650048, 2016.
[47]        A. Farajpour, M. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, Vol. 227, No. 7, pp. 1849-1867, 2016.
[48]        A. Farajpour, M. H. Yazdi, A. Rastgoo, M. Loghmani, M. Mohammadi, Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates, Composite Structures, Vol. 140, pp. 323-336, 2016.
[49]        M. Mohammadi, M. Safarabadi, A. Rastgoo, A. Farajpour, Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, Vol. 227, No. 8, pp. 2207-2232, 2016.
[50]        A. Farajpour, A. Rastgoo, M. Mohammadi, Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment, Physica B: Condensed Matter, Vol. 509, pp. 100-114, 2017.
[51]        M. Mohammadi, M. Hosseini, M. Shishesaz, A. Hadi, A. Rastgoo, Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads, European Journal of Mechanics - A/Solids, Vol. 77, pp. 103793, 2019/09/01/, 2019.
[52]        M. Mohammadi, A. Rastgoo, Nonlinear vibration analysis of the viscoelastic composite nanoplate with three directionally imperfect porous FG core, Structural Engineering and Mechanics, An Int'l Journal, Vol. 69, No. 2, pp. 131-143, 2019.
[53]        D. Shlyakhin, Z. M. Dauletmuratova, Nonstationary axisymmetric problem of thermo-elasticity for a rigidly fixed circular plate, Eng. J.: Science and Innovation, Vol. 5, No. 77, 2018.
[54]        A. Butkovskii, Characteristics of systems with distributed parameters, A textbook Nauka Moscow, 1979.
###### Volume 53, Issue 3September 2022Pages 348-355
• Receive Date: 03 May 2022
• Revise Date: 28 June 2022
• Accept Date: 29 June 2022
• First Publish Date: 29 June 2022