Thermo-mechanical analysis in a simply supported plate using a fractional order approach

Abstract

Keywords: Fractional Order Theory; Heat Source; Rectangular Plate; Thermal Bending Moment; Thermal Stress

Introduction

In the preceding years, the requirements for using fractional calculus to describe complex phenomena, i.e., multiphysics couplings, multiscale couplings, etc., have increased dramatically. Thermoelasticity has made a lot of use of derivatives of fractions recently, and renowned researchers, mathematicians, and scientists are working hard to develop

it further. The domains of practical sciences and engineering are two areas where fractional calculus is used in a variety of applications (included in the books by Podlubny [1], Atanackovi’c et al. [2], Herrmann [3], Magin [4], Mainardi [5], West et al. [6], and Zaslavsky [7]). Povstenko performed a study on thermoelasticity to apply the fractional transfer of heat equation [8], along with a discussion of modified Fourier’s law. The time-based fractional radial distribution

*Corresponding author. *E-mail address: navneetkumarlamba@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey © Author. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

in a sphere was treated by Povstenko [9]. In the context of decoupling thermoelasticity, the heat transmission formula has a fractional component of time for an infinite medium, Povstenko [10] studied thermal loads. Recently, numerous more prominent researchers have made major contributions [11–17] to the advancement of thermoelasticity with fractional techniques. Abouelregal et al. [18] utilized a memory-dependent variant of the heat transmission model and determined thermoelastic rotational nano-beams with changeable thermal characteristics. Using an integral transformation process, Lamba [19] presented the memory-based behaviour in a thick, solid thermoelastic cylindrical cylinder with an internal thermal source. Iyengar and Alwar [20] have given an overall solution for the induced stresses caused by a uniform transfer of temperature flow in a long rectangular plate. Iyengar and Chandrashekhara [21] have established the stress caused by temperature with boundary constraints. Chen [22] uses a direct power series method to tackle the linear issues associated with stress in a rectangular thin plate. Tanigawa and Komatsubara [23] established the thermal variation assessment in the two-dimensional plane stress situation and examined its intensity. Morimoto et al. [24] use the example of thermal buckling of an inhomogeneous plate with rectangular dimensions with heat input. Kulkarni and Deshmukh [25] explored the problem based on the thermal analysis of a quasi-static plate of rectangular dimensions. Also, Deshmukh et al. [26-29] described the effect of thermal bending on the rectangle-shaped plate problem. A discussion of the thermoelastic phenomenon of a thin plate of rectangular dimensions caused by a partially dispersed thermal supply was carried out by Lamba and Khobragade [27]. Further, Lamba and Khobragade [28] explored various aspects related to the three-dimensional inverse modeling of the rectangular plate problem. Manthena et al. [30] analyse in detail the behaviour of heat and mechanical inhomogeneity in the case of a rectangular plate. They also carry out numerical calculations while taking into consideration the fluctuation in inhomogeneous characteristics. A rectangular-shaped plate with non-homogeneous material characteristics that generated heat was studied by Manthena et al. [31] to evaluate the heat distribution, movement, and stresses caused by temperature. Aleck [32] found an approximate explanation for the pressures produced by a uniform temperature change in a thin, camped, rectangular-shaped plate. Sedelnikov et al. [33] developed analytical methods for the study of a homogeneous plate that underwent a shock. Also, the fractional approach is adopted by some more authors in knowing the behaviour of solids as reflected in [34, 35]. Lamba and Deshmukh [36] studied the memory effect in a solid body of infinite length and successfully determined the temperature, displacement, and stress functions. Lamba et al. [37] discussed both space- and time-based fractional ordered behaviour in a finite-length layer under certain boundaries. Demirbas et al. [38] utilized the finite difference method

to determine the thermo-mechanical behaviour of the plate for different compositional gradient exponents. Eslami [39] presented the development of the theory of thermoelasticity by considering solid plates. Gunasekar et al. [40, 41] presented the transform approach for the solution of integro-differential equations. Using a basic fractional calculus method, Raghavendran et al. [42] investigated approaches to various families of fractional integro-differential equations The present manuscript is prepared on the basis of the literature given in the book title “Thermal Stresses” by Noda et al. [43] and we modified this basic idea and prepared the mathematical modelling. No recent literature was studied by any researcher except the work done by one of the authors, Deshmukh et al. [29], by considering problem of circular plate with bending moments. The above work is based on only the classical heat conduction of a rectangular plate with a bending moment. Whereas in the present manuscript, we modified the classical heat conduction of the problem by introducing the fractional derivative in the heat transfer equation, which interpolates the classical heat conduction equation, and the impact of the retarded response is discussed successfully. In this article, we have studied the heat transfer equations using fractional order theory and discussed the effects of thermal stresses with an additional heat source and thermal bending moments using fractional order theory. Till date, only studies have shown the thermal bending moment for classical heat conduction equations, whereas this paper, considered heat conduction containing a fractional order approach, which predicts the retarded response, and it also interpolates classical heat conduction equations. This kind of work has not been cited so far, which contributes to the development of thermoelasticity based on fractional analysis and is useful for various material structure designs under the influence of additional heat generation. This is our new and novel contribution to the field of thermoelasticity.

Modelling Of Fractional Thermoelastic Problem

Let’s assume a rectangular plate with dimension {𝐷: 0 < 𝑥 < 𝑎, 0 < 𝑦 < 𝑏, 0 < 𝑧 < 𝑐}, which is maintained at a temperature 𝐹(𝑥, 𝑦, 𝑧) initially. All of the boundary constraints are maintained at absolute zero. In such cases, it is crucial to calculate the thermoelasticity in conjunction with a rectangular plate that is simply supported. The shape of a plate that is rectangular with a simple support is shown in Figure 1. To better fit this problem to physical conditions, fractional derivatives of time order are used for the fluctuation of temperature, and an additional thermal source is also used in setting up the heat transfer equation. The rectangular plate’s bending stiffness or thermal produced resultant moment are both factors in the fundamental equation of deflection. This type of modeling can play an important role

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

Figure 1. Geometrical design of rectangular-shaped plate with source of heat under fraction order theory. in classifying materials based on fracture parameters and is capable of describing the actual behaviour of materials. Governing Heat Transfer Equation The fractional derivative-based modeling of heat conduction for homogeneous and isotropic rectangular-shaped solids with fractional order differential and heat generation is given by

(4) where 𝑇denotes the function of temperature flow, 𝐴 and 𝑘 stands for the thermal conductivity and thermal diffusivity of the material of the plate respectively and 𝛼is the fractional order parameter. Also 𝑔(𝑥, 𝑦, 𝑧, 𝑡) denotes the instantaneous point source of heat situated at any point inside the rectangular solid plate. Since the thickness of the assumed plate is believed to be insignificant, one observes a rectangular plate with simple support that has had a thermal load applied a × b dimensions. Bending Moments, Thermal Deflection and Thermal Stresses Following [26, 29], the basic formula and related boundary circumstances in the Cartesian system of coordinates are

(7) where 𝑤 and 𝜈 are notations denote the deflection and Poisson’s ratio respectively. The plate’s bending stiffness and the consequent moment caused by the heat, respectively, are denoted by the letters 𝑅 and 𝑀𝑇 are defined as (8)

and (9) where 𝑎𝑡 and 𝐸, respectively, represent the linear heat expansion coefficient and Young’s modulus. One focuses on the equilibrium condition in the 𝑥 and 𝑦 in-plate dimensions. Consequently, the in-plate resulting forces becomes (10)

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

The definition of the resultant moments per unit length of the plate reads as follows: (11)

(20d) The problem is mathematically formulated in Equations (1) through (20). Analytical Approach to Solve a Mathematical Modeled Problem The integral transformation method—which is covered in the subsection below—is used to get the solution to the heat transfer equation for rectangular-shaped solids with fractional order differential and heat generation.

The equations for moments’ equilibrium about the x and y axes are

Solution of Heat Transfer Equation To obtain the desired temperature function, one constructs the “Triple Analytic Transformation” and its inverted formula.

(15) where the shearing forces are denoted by 𝑄𝑥, 𝑄𝑦. The thermal stress elements are stated as [26] in terms of the resulting forces and resulting moments as

(22) where the kernels are 𝑆(𝜙𝑚, 𝑥), 𝑆(𝜃𝑛, 𝑦) and 𝑆(𝜓𝑝, 𝑧) are obtained and are defined as

when the equations (1) to (4) are transformed using the triple integral transform described in (21), one obtains

By employing the theorem known as Green’s to solve the problem (25), one can derive an expression such as (26)

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

(27) Solving the equation (26), one obtains (28) where 𝐸𝛼,𝛼 denotes for the Mittag Leffler function. To obtain a mathematical expression of the temperature function, the triple integral transformation is inverted repeatedly by applying the inversion formula (22) to equation (27)

(35) Using equations (10),(29),(30),(31),(33),(34), and (35) in equations (16), (17), and (18), we get the thermal stress functions mathematically as

Determination of Bending Moments, Thermal Deflection and Thermal Stresses Using equation (29) in equations (8) and (19), one obtains the expression of the resultant moment and resultant forces as below: (30)

(38) Convergence Analysis Numerical computations were made on different axes (𝑥 and 𝑦) to assess the impact of steady heat supply on the plate’s extreme ends. Consider

(31) Equations (30) and (31) represent the resultant moment and resultant forces respectively. Utilizing equation (30) in (5) the expression of thermal deflection obtained as (32) Using equations (30) and (32) in (11), (12) and (13), one obtains the resultant moments as,

The terms sine and cosine are also constrained. The required condition for convergence has been met, so it is simple to confirm that all of the aforementioned series are convergent by using D’Alembert’s ratio test. Additionally, for large values of m, n, and p, the term in the calculation for temperature, deflection, and thermal stresses is minimal and converges to zero at infinity. Numerical Computation To make computations as simple as possible, we set the functions in dimensionless form as Dimensionless form:

Material Properties The computational analysis has been done for a plate with a rectangular shape made entirely of copper-based material properties [26, 29].

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

Thermal conductivity, 𝐴 = 386𝑊𝑚−1𝑘−1 Thermal diffusivity, 𝑘 = 112.34 × 10−6𝑚2𝑠−1 Density, 𝜌 = 8954𝑘𝑔𝑚−3 Specific heat, 𝑐𝑝 = 383𝐽𝑘𝑔−1𝑘−1 Poisson ratio, 𝜈 = 0.35 Coefficient of linear thermal expansion, 𝑎𝑡 = 16.5 × 10−6𝑘−1 Lames constant, 𝜇 = 26.67

Dimensions The following dimensions are set for numerical computations: Length of a rectangular plate, 𝑎 = 2𝑚 Breadth of a rectangular plate, 𝑏 = 1𝑚 Height of rectangular plate, 𝑐 = 0.5𝑚

Graphical Illustration And Discussion

Arbitrary establishing heat supply 𝑇 = 𝑇0 and instantaneous point heating source of strength 𝑔0 located at location (𝑥0, 𝑦0, 𝑧0) inside the material spontaneously releases heat at the time 𝑇 = 𝜏. 𝑔(𝑥, 𝑦, 𝑧, 𝑡) = 𝑔0𝛿(𝑥 − 𝑥0)𝛿(𝑦 − 𝑦0)𝛿(𝑧 − 𝑧0)𝛿(𝑡 − 𝑡0)

Below Figures 2 to 5 represent the graphical plotting temperature distribution, deflection variation, and stress dispersion along axial directions for different fractional parameters. The value of 𝑦 is fixed at 0.5𝑚 respectively for plotting along 𝑥 axis, similarly the value of 𝑥 is fixed at 1𝑚 for plotting along 𝑦axis and for the plotting along 𝑥 and 𝑦 axis, the value of 𝑧 and time 𝑡are fixed at 0.25𝑚 and 0.5 respectively. Figure 2(a) and 2(b) display the display of dimensionless temperature 𝑇∗ and dimensionless defection distribution 𝜔∗

Figure 2(a). Influence of fractional parameters on temperature behaviour along the 𝑥∗ axis.

Figure 2(b). Influence of fractional parameters on deflection behaviour along the 𝑥∗ axis.

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

along dimensionless 𝑥∗ axis for various values of fractional parameters 𝛼 = 0.5, 1, 1.5, 2. It can be seen in both plots that temperature and defect rise in the 𝑥∗ direction from the initial edge to about the midway and decrease towards the extreme edge. Additionally, it is evident that the largest temperature distribution and deflection are found in the middle, it could be caused by an effect of the source point of heat. The speed of thermal signal transmission is discovered to be directly related to the variation of the fraction-order parameter 𝛼, and temperature and deflection adopt a uniform pattern for various values of fractional parameters 𝛼. Furthermore, deflection 𝜔∗at points 𝑥∗ = 0 and 𝑥∗ = 2 is zero, this agrees with the necessary boundary criteria stated in equation (6).

For various values of the fractions parameters 𝛼 = 0.5, 1, 1.5, 2, Figure 3(a) and 3(b) demonstrate the fluctuation of without dimension temperature 𝑇∗and defection distribution 𝜔∗ along dimensionless 𝑦∗ axis. The rate of thermal signal propagation is shown to be directly related to the parameters denoting the fraction-order differentiation in both plots. Additionally, when one moves into the 𝑦∗ direction from the initial edge towards the middle, the temperature and deflection rise and then fall as one approaches the extreme edge. Further, slight discrimination in both temperature and deflection distribution is noted as compared to distribution in 𝑥∗direction. Also, deflection 𝜔∗ is zero at 𝑦∗ = 0 and 𝑦∗ = 1 which matches with the prescribed boundary condition defined in equation (7).

Figure 3(a). Influence of fractional parameters on temperature behaviour along the 𝑦∗ axis.

Figure 3(b). Influence of fractional parameters on deflection behaviour along the 𝑦∗ axis.

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

Figure 4(a), 4(b) and 4(c) demonstrate the graphical view of dimensionless stress distributions 𝜎𝑥𝑥∗, 𝜎𝑦𝑦∗, 𝜎𝑥𝑦∗ ,

Figure 4(a). Influence of fractional parameters on stress 𝜎𝑥𝑥∗ behaviour along the 𝑥∗ axis.

Figure 4(b). Influence of fractional parameters on stress 𝜎𝑦𝑦∗ behaviour along the 𝑥∗ axis.

Figure 4(c). Influence of fractional parameters on stress 𝜎𝑥𝑦∗ behaviour along the 𝑥∗ axis.

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

parameters 𝛼 = 0.5, 1, 1.5,2. The following observations are noted from graphical plotting: (a) Stress function 𝜎𝑥𝑥∗ grows in the vicinity 0 ≤ 𝑥∗ ≤ 0.74,

Results And Discussion

Based on numerical computation, the following significant outcome summaries were obtained: · The deflection on the boundary at 𝑥 = 𝑎 and 𝑦 = 𝑏 is seen to be insignificant since the moments and shearing forces in the x and y directions are negligible, which fulfilled the uniqueness and existence of the problem. · The thermal stresses 𝜎𝑥𝑥 = 𝜎𝑦𝑦 = 0 at 𝑥 = 𝑎, 𝑦 = 𝑏, which shows that they are traction-free. · Numerical representation shows that higher thermal response is observed for a higher value of fractional order parameter. · Due to the initial constant heat input from internal heat generation, significant stresses and deflections develop in a rectangular region. · Both the resulting components of stress and the shear stress components vary greatly from the initial edge to the outermost edge of the rectangular plate and conform to the specified mathematical boundary conditions. · The variation in the temperature, deflection, and stress functions is confined to a restricted range, and no

variation is observed excluding this range. This phenomenon reflects the fact that waves propagate with finite speed. The creation of novel structural materials for physical processing is dependent on the wave propagation rate since changing values of the fractional parameters have an impact on it. The result can be seen schematically. It can be observed that the resultant moments caused by heat and forces are strongly influenced by the response of the weak, medium, and superconductivity, which have a direct influence on the characteristics of the thermal variation in temperature, deflection, and stress distribution. The effects of temperature, deflection, and stress functions for different fraction order parameters show the weak, medium, and superconductivity. Also, for fractional parameters 𝛼 = 1 and 𝛼 = 2 heat transfer interpolates the diffusion and wave equation and then effects on bending moments and stresses are shown in the figures, which represent the limiting case.

Conclusion

An analytical approach is used to effectively analyse the thermoelastic behaviour for a simply supported thin rectangular plate that has a thermal bending moment and a heat source. Here, the concept of resultant forces and moments per unit length of the plate is used to know the exact thermal response of the rectangular plate under the fractional theory, additionally, by excluding the in-plane resultant forces and accounting for the thermal bending moments, the stress components are assessed. A rectangular copper plate was numerically calculated, and the results of thermal change, including temperature and deflection along different axes based on different fractional parameters, are effectively presented and discussed.

Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025

Finally, the authors study fractional order theory to investigate the thermal effect of the time fractional derivative on a rectangular plate, which deals with the memory effect and interpolates the classical heat conduction equation for a rectangular plate. Also due to fractional order theory effectiveness of heat and elastic waves inputting 𝛼 = 0, 𝛼 = 1, 𝛼 = 2 it predicts the classical heat conduction equations i.e. Helmholtz, diffusion, and wave equation. Also noting for 𝛼 = 1, it is the limiting case of the study by Deshmukh et al. [29].

Future Scope Of The Results

It may be helpful to build various thermo-mechanical structures with bending moments and subjected to additional heat sources by using the researched inhomogeneous boundary value issue of a rectangular plate under fractional thermoelasticity, which is essential for capturing the real behaviour of materials. Also, this work might potentially be extended to investigate the thermal bending moment by introducing the concept of memory-dependent derivatives to investigate the behaviour of materials in practical circumstances.

Acknowledgment

The authors are very thankful to respected Prof. Dr. Ahmet Selim Dalkılıç, Editor-in-Chief, for their valuable guidance to improve the manuscript, as well as to respected reviewers for their valuable comments for improving the quality of research work.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

References

1. Podlubny I. Fractional Differential Equations. ry-dependent derivative of heat conduction model. Academic Press, San Diego, 1999. Arch Appl Mech 2023;93:197–220. [CrossRef] Sigma J Eng Nat Sci, Vol. 43, No. 4, pp. 1140−1151, August, 2025 1151
2. Lamba N. Impact of Memory-dependent Response of a Thermoelastic Thick Solid Cylinder. J Appl clamped along an edge. ASME J Appl Mech Comput Mech 2023;9:1135–1143. 2021;16:118–122. [CrossRef]
3. Iyengar KT, Sundara RS, Alwar RS. Thermal stresses in a long rectangular plate constrained at one of AS. Investigation of the stress-strain state of a rectan- the shorter edges. Int J Mech Sci 1962;4:485–490. gular plate after a temperature shock. Mathematics [CrossRef] 2023;11:638. [CrossRef]
4. Iyengar KT, Sundara RS, Chandrashekhara K. Thermal stresses in rectangular plates. Appl Sci Res Efficiency and Effectiveness of Rectangular Fin with 1996;15:141–160. [CrossRef] Fractional Order Energy Balance Equation. Int J
5. Chen PYP. Transient thermal stresses in a rectan- Thermodyn 2021;24:216–225. [CrossRef] gular plate due to non-uniform heat transfer coeffi- [35] Gaikwad K, Khavale S. Fractional Order Transient cients. J Therm Stresses 1998;11:115–125. [CrossRef] Thermoelastic Stress Analysis of a Thin Circular
6. Tanigawa Y, Komatsubara Y. Thermal stress analysis Sector Disk. Int J Thermodyn 2022;25:1–8. [CrossRef] of a rectangular plate and its thermal stress intensity [36] Lamba NK, Deshmukh KC. Memory dependent factor for compressive stress field. J Therm Stresses response in an infinitely long thermoelastic solid 1997;20:517–542. [CrossRef] circular cylinder. PNRPU Mech Bull 2024;1:5–12.
7. Morimoto T, Tanigawa Y, Kawamura R. Thermal [CrossRef] buckling analysis of inhomogeneous rectangular [37] Lamba N, Verma J, Deshmukh KC (R2028). A Brief plate due to uniform heat supply. J Therm Stresses Note on Space Time Fractional Order Thermoelastic 2003;26:1151–1170. [CrossRef] Response in a Layer. Appl Appl Math: An Int J
8. Kulkarni VS, Deshmukh KC. A brief note on qua- (AAM) 2023;18:1. si-static thermal stresses in a rectangular plate. Far
9. Demirbas MD, Apalak MK, Ekici R. Determination East J Appl Math 2007;26:349–360. of three-dimensional thermo-mechanical behaviour
10. Deshmukh KC, Khandait MV, Warbhe SD, Kulkarni of bidirectional functionally graded rectangular VS. Thermal stresses in a simply supported plate plates with the theory of elasticity. Mech Adv Mater with thermal bending moments. Int J Appl Math Struct 2024;31:1534–1562. [CrossRef] Mech 2010;6:1–12.
11. Lamba NK, Khobragade NW. Thermoelastic prob- lem of a thin rectangular plate due to partially Stresses in Plates and Shells. Cham: Springer Nature distributed heat supply. Int J Appl Math Mech Switzerland; 2024:43–143. [CrossRef] 2012;8:1–11. [40] Gunasekar T, Raghavendran P, Santra SS, Majumder
12. Lamba NK, Khobragade NW. Integral transform D, Baleanu D, Balasundaram H. Application of methods for inverse problem of heat conduction with Laplace transform to solve fractional integro-dif- known boundary of a thin rectangular object and its ferential equations. J Math Comput Sci-JM stresses. J Therm Sci 2012;21:459–465. [CrossRef] 2024;33:225–237. [CrossRef]
13. Deshmukh KC, Khandait MV, Kumar R. Thermal stresses in a simply supported plate with thermal Transform Approach to Fractional Integro- bending moments with heat sources. Mater Phys Differential Equations. J Comput Anal Appl Mech 2014;21:135–146. 2024;33:358–371. [CrossRef]
14. Manthena VR, Lamba NK, Kedar GD. Transient thermoelastic problem of a nonhomogeneous rect- Santra SS, Majumder D, Baleanu D. Solving frac- angular plate. J Therm Stresses 2017;40:627–640. tional integro-differential equations by Aboodh [CrossRef] transform. J Math Comput Sci-JM 2024;32:229–240.
15. Manthena VR, Lamba N, Kedar GD. Thermoelastic [CrossRef] analysis of a rectangular plate with nonhomoge- [43] Noda N, Hetnarski RB, Tanigawa Y. Thermal neous material properties and internal heat source. J Stresses, 2nd Edition. Taylor Francis, New York, Solid Mech 2018;10:200–215. 2003.

Share and Cite