Direct Approach to a Generalized Fractionalorder Thermoelastic Problem for a Thickcircular Plate with Lord-Shulman and Classical Coupledthermoelastic Model

Abstract

In the present article, we discussed the mathematical solution of generalized fractional order thermoelastic problem for a thick circular plate of finite thickness 2b occupying the space D defined by 0 ≤ r ≤∞, −b ≤ z ≤ b. The problem is discussed within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper and the lower surfaces of the thick plate are traction free and subjected to an axi-symmetric heat supply. The solution is carried out by using integral transform technique and a direct approach. The most general solutions are obtained and represented graphically using MATHCAD. The validity of this intended model is assessed by comparing it with previously published results. According to the authors this model is explicitly useful for the researchers those studying development in theory of hyperbolic thermoelastic diffusion, in material science and designers of new materials.

Introduction: This paper presents an analytical solution to a generalized fractional order thermoelastic problem in a thick circular plate using both the Lord–Shulman (LS) and Classical Coupled Thermoelasticity (CTE) models. The analysis considers a finite plate of thickness 2b, subjected to axisymmetric thermal loading on its traction-free upper and lower surfaces. Employing a direct approach along with integral transforms, we derive general solutions in the Laplace domain without introducing potential functions. Numerical evaluations are carried out using MATHCAD, and results are graphically depicted to highlight variations in temperature, displacement, stress, concentration, and chemical potential. The obtained outcomes show consistency with established results and illustrate the influence of fractional order and time on the thermoelastic behaviour. This model proves valuable for advancing the study of hyperbolic thermoelastic diffusion, material science, and innovative material design.

Objectives: To evaluate numerical solutions for temperature, displacement, stress, concentration, and chemical potential and observe the variations graphically by employing MATHCAD software for two distinct models namely Lord-Shulman & Classical Coupled Thermoelastic model.

Methods: Concerned problem is solved by employing an effective method of direct approach along with integral transforms without introducing potential function.

Results: Evaluated numerical solutions for the field functions such as temperature, displacement, stress, concentration, and chemical potential.

Conclusions: In this paper, we analytically solved the generalized fractional order thermoelastic                                                   problem for a thick circular plate using a direct approach and integral transform techniques. The used approach avoids the use of potential function, offering a simplified and efficient route to obtain exact solution in the Laplace domain. The findings have significant implications for researchers exploring fractional thermoelasticity, particularly in the design of advanced materials and systems subjected to thermal and diffusion effects.