Influence of Fractional Parameters in a Functionally Graded Elliptical Plate with Thermally Sensitive Material Features

Document Type : Research Paper

Authors
N. K. Lamba 1
P. P. Bhad 2
V. R. Manthena 2
G. D. Kedar 3
M. I.A. Othman 4
1 Department of Mathematics, Shri Lemdeo Patil Mahavidyalaya, Nagpur, India
2 Department of Mathematics, Priyadarshini J. L. College of Engineering, Nagpur, India
3 Department of Mathematics, RTM Nagpur University, Nagpur, India
4 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt
10.22055/jacm.2025.48181.5037
Abstract
In order to examine the relationship between temperature change and thermoelastic deformation in the assumed functionally graded materials (FGMs), the present work involves fractional-order mathematical modeling of a thermoelastic thick elliptical annular plate, where the material properties of the plate are considered thermo-sensitive. To determine the precise thermal response of the problem, the density, heat capacity, and thermal conductivity are also graded axially, in addition to the thermo-sensitivity. The temperature distribution of the plate is assumed to be zero at the start and end of the thickness variation, with convective heat exchange boundary conditions in the spatial direction. Kirchhoff's variable transformation method is used to eliminate temperature-dependent nonlinearity in the heat transfer equations. The Laplace transform for the initial condition and a perturbation solution are obtained through an asymptotic series expansion. The heat transfer problem is ultimately solved using the modified Mathieu transform, the Taylor series technique, and its inversion, yielding the required temperature distribution function in the Laplace domain. Using nickel as the metal and titanium carbide as the ceramic, a model of a ceramic-metal-based FGM is constructed for numerical computations. All results are graphically presented, showing the distributions of temperature, displacement, and stress curves along the spatial variables for two different fractional-order parameters by varying the plate’s thickness. This study enables a deeper understanding of the thermo-mechanical responses of FGMs through its novel application to FGMs with elliptical geometry and its innovative modeling approach incorporating thermally sensitive material properties. It opens the door to optimized designs in thermal industries. The understanding and improvement of thermal behavior in advanced materials have been significantly enhanced by the discovery of the influence of fractional parameters in an FGM elliptical plate.
Keywords
Functionally graded elliptic plate
Thermo-sensitive
Fractional
Taylor&rsquo
s series
Perturbation
Mathieu transform
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Popovich, V.S., On the solution of heat conduction problems for thermo-sensitive bodies heated by convective heat exchange, Journal of Soviet Mathematics, 63, 1993, 94–97.
[2] Popovich, V.S., Fedai, B.N., The axisymmetric problem of thermoelasticity of a multilayer thermosensitive tube, Journal of Mathematical Sciences, 86, 1997, 2605–2610.
[3] Popovich, V.S., Garmatii, G.Yu., Solution of nonstationary heat-conduction problems for thermosensitive bodies under convective heat exchange, Journal of Mathematical Sciences, 90, 1998, 2037–2041.
[4] Popovich, V.S., Garmatyi, G.Yu., The nonstationary heat-conduction problem fora heat-sensitive space with a spherical cavity, Journal of Mathematical Sciences, 79, 1996, 1478–1482.
[5] Popovich, V.S., Makhorkin, I.M., On the solution of heat-conduction problems for thermo-sensitive bodies, Journal of Mathematical Sciences, 88, 1998, 352–359.
[6] Popovich, V.S., Harmatyi, H.Yu., Vovk, O.M., Thermoelastic state of a thermo-sensitive hollow sphere under the conditions of convective-radiant heat exchange with the environment, Materials Science, 42, 2006, 756–770.
[7] Othman, M.I.A., State space approach to generalized thermoelasticity plane waves with two relaxation times under the dependence of the modulus of elasticity on reference temperature, Canadian Journal of Physics, 81(12), 2003, 1403-1418.
[8] Ezzat, M.A., Othman, M.I.A., El Karamany, A.S., The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity, Journal of Thermal Stresses, 24(12), 2001, 1159-1176.
[9] Srinivas, V.B., Manthena, V.R., Warbhe, S.D., Kedar, G.D., Lamba, N.K., Thermal stresses associated with a thermosensitive multilayered disc analysed due to point heating, International Journal of Applied Mechanics and Engineering, 29(2), 2024, 118-137.
[10] Lamba, N.K., Thermosensitive response of a functionally graded cylinder with fractional order derivative, International Journal of Applied Mechanics and Engineering, 27, 2022, 107–124.
[11] Manthena, V.R., Kedar, G.D., Deshmukh, K.C., Thermal stress analysis of a thermosensitive functionally graded rectangular plate due to thermally induced resultant moments, Multidiscipline Modeling in Materials and Structures, 14, 2018, 857–873.
[12] Lamba, N.K., Manthena, V.R., Bhad, P.P., Srinivas, V.B., Abouelregal, A.E., Thermal characteristics of a multi-layered annular disk with thermosensitive features using a fractional-order heat conduction model, Acta Mechanica, 236, 2025, 937–958.
[13] Manthena, V.R., Uncoupled thermoelastic problem of a functionally graded thermosensitive rectangular plate with convective heating, Archive of Applied Mechanics, 89, 2019, 1627–1639.
[14] Wang, C., Song, Z., Fan, H., Novel evidence theory-based reliability analysis of functionally graded plate considering thermal stress behavior, Aerospace Science and Technology, 146, 2024, 108936.
[15] Kumar, S., Kar, V.R., Nonlinear fully coupled thermoelastic transient analysis of axial functionally graded composite panel, Mechanics Based Design of Structures and Machines, 52, 2024, 3426–3455.
[16] Saadatfar, M., Babazadeh, M.A., Babaelahi, M., Creep analysis in a rotating variable thickness functionally graded disc with convection heat transfer and heat source, Mechanics of Time-Dependent Materials, 28, 2024, 19–41.
[17] Alsaeed, S.S., Abouelregal, A.E., Analysis of thermomechanical responses of functionally graded unbounded materials using an advanced dual-phase delay heat transfer model with higher-order fractional derivatives, ZAMM-Journal of Applied Mathematics and Mechanics, 105(1), 2025, e202400930.
[18] Soleiman, A., Abouelregal, A.E., Fahmy, M.A., Sedighi, H.M., Thermo-mechanical behavior of functionally graded nanoscale beams under fractional heat transfer model with a two-parameter Mittag-Leffler function, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 48, 2024, 1117–1133.
[19] Abouelregal, A.E., Alhassan, Y., Alsaeed, S.S., Elzayady, M.E., Tempered fractional thermal conduction model for magneto elastic solids with spherical holes under time-dependent laser pulse heating, Archive of Applied Mechanics, 95(1), 2024, 27.
[20] Abbas, I.A., Generalized thermoelastic interaction in functional graded material with fractional order three-phase lag heat transfer, Journal of Central South University, 22, 2015, 1606–1613.
[21] Abbas, I.A., A GN model for thermoelastic interaction in a microscale beam subjected to a moving heat source, Acta Mechanica, 226, 2015, 2527–2536.
[22] Alqahtani, Z., Abbas, I.A., Generalized thermoelastic interactions in functional graded materials under laser heat source with a timed pulse, Case Studies in Thermal Engineering, 61, 2024, 105117.
[23] Jabbari, M., Nejad, M.Z., Mechanical and thermal stresses in radially functionally graded hollow cylinders with variable thickness due to symmetric loads, Australian Journal of Mechanical Engineering, 18(sup1), 2018, S108–S121.
[24] Aboudi, J., Pindera, M.J., Arnold, S.M., Higher-order theory for functionally graded materials, Composites Part B: Engineering, 30, 1999, 777–832.
[25] Jin, Z.H., Paulino, G.H., Transient thermal stress analysis of an edge crack in a functionally graded material, International Journal of Fracture, 107, 2001, 73–98.
[26] Kieback, B., Neubrand, A., Riedel, H., Processing techniques for functionally graded materials, Materials Science & Engineering A, 362, 2003, 81–106.
[27] Nemat-Alla, M., Reduction of thermal stresses by developing two-dimensional functionally graded materials, International Journal of Solids and Structures, 40, 2003, 7339–7356.
[28] Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A., Ford, R.G., Functionally graded materials: Design, Processing and Applications, Materials Technology Series, Springer Science & Business Media, 5, 2013.
[29] Ebrahimi, F., Salari, E., Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment, Acta Astronautica 113, 2015, 29–50.
[30] Gupta, A., Talha, M., Recent development in modeling and analysis of functionally graded materials and structures, Progress in Aerospace Sciences, 79, 2015,1–14.
[31] Ebrahimi, F., Reza Barati, M., Vibration analysis of nonlocal beams made of functionally graded material in thermal environment, European Physical Journal - Plus, 131, 2016, 279.
[32] Burlayenko, V.N., Altenbach, H., Sadowski, T., Dimitrova, S.D., Bhaskar, A., Modelling functionally graded materials in heat transfer and thermal stress analysis by means of graded finite elements, Applied Mathematical Modelling, 45, 2017, 422–438.
[33] Dehrouyeh-Semnani, A.M., On the thermally induced non-linear response of functionally graded beams, International Journal of Engineering Science, 125, 2018, 53–74.
[34] Manthena, V.R., Kedar, G.D., Mathematical modeling of thermoelastic state of a functionally graded thermally sensitive thick hollow cylinder with internal heat generation, International Journal of Thermodynamics, 21, 2018, 202–212.
[35] Trabelsi, S., Frikha, A., Zghal, S., Dammak, F., Thermal post-buckling analysis of functionally graded material structures using a modified FSDT, International Journal of Mechanical Sciences, 144, 2018, 74–89.
[36] Saleh, B., Jiang, J., Fathi, R., Al-hababi, T., Xu, Q., Wang, L., Song, D., Ma, A., 30 Years of functionally graded materials: An overview of manufacturing methods, Applications and Future Challenges, Composites Part B: Engineering, 201, 2020, 108376.
[37] Zhang, C., Chen, F., Huang, Z., Jia, M., Chen, G., Ye, Y., Lin, Y., Liu, W., Chen, B., Shen, Q., Zhang, L., Lavernia, E.J., Additive manufacturing of functionally graded materials: A review, Materials Science & Engineering A, 764, 2019, 138209.
[38] El-Galy, I.M., Saleh, B.I., Ahmed, M.H., Functionally graded materials classifications and development trends from industrial point of view, SN Applied Sciences, 1, 2019, 1378.
[39] Dhital, S., Rokaya, A., Kaizer, M.R., Zhang, Y., Kim, J., Accurate and efficient thermal stress analyses of functionally graded solids using incompatible graded finite elements, Composite Structures, 222, 2019, 110909.
[40] Povstenko, Y.Z., Fractional Cattaneo-type equations and generalized thermo-elasticity, Journal of Thermal Stresses, 34, 2011, 97–114.
[41] Povstenko, Y., Fractional heat conduction and related theories of thermo-elasticity, Springer International, Publishing, 2024.
[42] Povstenko, Y., Theories of thermal stresses based on space–time-fractional telegraph equations, Computers & Mathematics with Applications, 64, 2012, 3321–3328.
[43] Povstenko, Y., Signaling problem for time-fractional diffusion-wave equation in a half-space in the case of angular symmetry, Nonlinear Dynamics, 59, 2010, 593–605.
[44] Povstenko, Y., Time-fractional radial heat conduction in a cylinder and associated thermal stresses, Archive of Applied Mechanics, 82, 2012, 345–362.
[45] Povstenko, Y., Fractional Thermoelasticity, Springer, 2015.
[46] Povstenko, Y., Kyrylych, T., Fractional thermoelasticity problem for a plane with a line crack under heat flux loading, Journal of Thermal Stresses, 41, 2018, 1313–1328.
[47] Povstenko, Y., Ostoja-Starzewski,M., Fractional telegraph equation under moving time-harmonic impact, International Journal of Heat and Mass Transfer, 182, 2022, 121958.
[48] Kaur, I., Singh, K., Fractional order strain analysis in thick circular plate subjected to hyperbolic two temperature, Partial Differential Equation in Applied Mathematics, 4, 2021, 100130.
[49] Lamba, N.K., Impact of memory-dependent response of a thermoelastic thick solid cylinder, Journal of Applied and Computational Mechanics, 9, 2023, 1135–1143.
[50] Lamba, N., Verma, J., Deshmukh, K., A brief note on space time fractional order thermoelastic response in a layer, Applications and Applied Mathematics: An International Journal, 18(1), 2023, 18.
[51] Elhagary, M.A., Thermoelastic diffusion in a half space due to fractional order theory, Waves in Random and Complex Media, 2024, 1–23.
[52] Othman, M.I.A., Atef, H.M., Conformable fractional order theory in thermo-elasticity, Mechanics of Solids, 59, 2024, 1180–1193.
[53] Othman, M.I.A., Sur, A., Mondal, S., Memory-dependent derivative and magnetic field for a rotating thermoelastic medium with voids under thermal loading due to the laser pulse, Mechanics of Solids, 59, 2024, 2059–2076.
[54] Othman, M.I.A., Sarkar, N., Atwa, S.Y., Effect of fractional parameter on plane waves of generalized magneto–thermoelastic diffusion with reference temperature-dependent elastic medium, Computers & Mathematics with Applications, 65, 2013, 1103–1118.
[55] Jiji, L.M., Heat Conduction, Springer-Verlag Berlin and Heidelberg GmbH & Co. K, Berlin Heidelberg, 2009.
[56] Bhad, P.P., Khalsa, L.H., Varghese, V., Transient thermoelastic problem in a confocal elliptical disc with internal heat sources, Advances in Mathematical Sciences and Applications, 25, 2016, 43–61.
[57] Sugano, Y., Kondoh, Y., An analytical solution for a plane thermal stress problem expressed in elliptical coordinates (3rd Report. Steady-state thermal stresses in a confocal hollow elliptical plate), Transactions of the Japan Society of Mechanical Engineers, Ser. A, 55, 1989.
[58] Noda, N., Thermal stresses in materials with temperature-dependent properties, Applied Mechanics Reviews, 44, 1991, 383–397.
[59] Bhad, P.P., Khalsa, L.H., Varghese, V., Stress analysis in thermosensitive elliptical plate with simply supported edge and impulsive thermal load, Journal of Solid Mechanics, 10(2), 2018, 326-337.
[60] Yıldız, T., Thermomechanical Vibration Response of Solid and Foam FGM Nano Actuator/Sensor Plates., Journal of Vibration Engineering & Technologies, 12, 2024, 1281–1297.
[61] Yıldız, T., Esen, I., On the effect of the Casimir, van der Waals and electrostatic forces on the thermomechanical buckling of sandwich smart piezo magnetic nanosensor/switch plates, Microsystem Technologies, 31, 2025, 1525–1545.
[62] Yildiz, T., Kabave Kilinçarslan, S., Esen, I., Vibration simulation of sandwich nano-smart plate with an auxetic core and piezo-electro-magnetic face layers based on sinusoidal higher-order theory, Arabian Journal for Science and Engineering, 2024, https://doi.org/10.1007/s13369-024-09746-4.
[63] Raut, G.N., Varghese, V., Khobragade, N.W., On the plane strain and plane stress solutions of uniformly heated functionally graded solid cylinder or disk problems, Advances in Mathematical Sciences and Applications, 19(2), 2009, 403.
[64] Manthena, V.R., Bhad, P.P., Lamba, N.K., Kedar, G.D., Abbas, I.A., Effect of a Laser Moving Heat Source on a Functionally Graded Elliptical Plate's Thermal Behavior Using a Fractional‐Order Mechanism, Heat Transfer, 2025, https://doi.org/10.1002/htj.23308.
Journal of Applied and Computational Mechanics
Volume 12, Issue 1
Winter 2026
Pages 176-188