Analysis of Three-Dimensional Non-Homogeneous Fractional order Thermoelastic Problem of Thick Rectangular Plate with Internal Heat Generation

Abstract

This paper investigates the thermal behaviour of a three-dimensional thick rectangular plate governed by a fractional-order derivative. The plate, occupying the region                                    , is initially at an arbitrary temperature distribution . For , the plate experiences internal heat generation represented by  Btu/hr ft, while all boundary surfaces are maintained at zero temperature. The governing model is formulated using the Caputo fractional derivative, and analytical solutions for temperature, displacement, and thermal stresses are derived through the integral transform technique. Numerical simulations are carried out using PTC Mathcad software, and the results are illustrated graphically. The study highlights the influence of the fractional-order parameter on heat conduction, displacement, and stress distribution, demonstrating the effectiveness of fractional-order thermoelastic models in capturing nonlocal thermal effects.

Introduction: The study of thermoelastic behaviour in solid materials has gained significant importance due to its wide applications in aerospace, mechanical, and civil engineering. Classical thermoelastic theories often assume instantaneous heat propagation, which contradicts physical reality. To overcome this limitation, fractional-order thermoelasticity has been introduced as an effective approach to model heat conduction with memory and nonlocal effects. In this paper, a three-dimensional non-homogeneous thick rectangular plate with internal heat generation is analysed using the Caputo fractional derivative. The objective is to investigate how the fractional-order parameter (α) influences temperature, displacement, and stress distributions. The proposed model provides a more generalized framework that includes both the classical and wave-type heat conduction as special cases, thus offering a realistic representation of transient thermal behaviour in thick plates.

Objectives:

1. To investigate the thermal behaviour of a three-dimensional thick rectangular plate subjected to internal heat generation.


2. To analyse the effect of the fractional-order parameter (α) on temperature, displacement, and thermal stresses.


3. To demonstrate the usefulness of fractional-order thermoelastic models in capturing nonlocal and memory effects in heat conduction.

Methods: The governing equations are derived using the Caputo fractional derivative for time-fractional heat conduction. Integral transform techniques are employed to obtain analytical solutions for temperature, displacement, and stress fields. Mittag-Leffler functions are used in the solutions to describe the fractional behaviour. Numerical simulations are performed in PTC Mathcad Prime for a thick rectangular plate using physical constants.

Results: The temperature distribution decreases with increasing fractional order α and is maximum at the centre of the plate. Displacement components show symmetrical behaviour, increasing from the edges and vanishing at the centre. Stress components are compressive, peaking at the middle of the plate and zero at the edges. The fractional order parameter α significantly affects heat transfer. Graphs demonstrate clear variation between classical (α = 1) and fractional-order (α ≠ 1) thermoelastic behaviour.

Conclusions: For α = 1, the model reduces to the classical heat diffusion equation, while α = 2 corresponds to the wave equation. The fractional order 0 < α < 1, α = 1, and 1 < α < 2 represent weak, normal, and strong conductivity, respectively. Stress distributions are tensile/compressive and follow normal curves in all directions, directly proportional to α. The fractional order α governs nonlocal heat transfer, influencing both response time and temperature overshoot. The model effectively describes transient thermoelastic behavior in thick plates and can be applied to other nonhomogeneous materials.