Fractional order generalized thermoelastic response in a half space due to a periodically varying heat source

Abstract

Purpose The present work is concerned with the solution of a fractional-order thermoelastic problem of a two-dimensional infinite half space under axisymmetric distributions in which lower surface is traction free and subjected to a periodically varying heat source. The thermoelastic displacement, stresses and temperature are determined within the context of fractional-order thermoelastic theory. To observe the variations of displacement, temperature and stress inside the half space, the authors compute the numerical values of the field variables for copper material by utilizing Gaver-Stehfast algorithm for numerical inversion of Laplace transform. The effects of fractional-order parameter on the variations of field variables inside the medium are analyzed graphically. The paper aims to discuss these issues. Design/methodology/approach Integral transform technique and Gaver-Stehfast algorithm are applied to prepare the mathematical model by considering the periodically varying heat source in cylindrical co-ordinates. Findings This paper studies a problem on thermoelastic interactions in an isotropic and homogeneous elastic medium under fractional-order theory of thermoelasticity proposed by Sherief (Ezzat and El-Karamany, 2011b). The analytic solutions are found in Laplace transform domain. Gaver-Stehfast algorithm (Ezzat and El-Karamany, 2011d; Ezzat, 2012; Ezzat, El Karamany, Ezzat, 2012) is used for numerical inversion of the Laplace transform. All the integrals were evaluated using Romberg’s integration technique (El-Karamany et al., 2011) with variable step size. A mathematical model is prepared for copper material and the results are presented graphically with the discussion on the effects of fractional-order parameter. Research limitations/implications Constructed purely on theoretical mathematical model by considering different parameters and the functions. Practical implications The system of equations in this paper may prove to be useful in studying the thermal characteristics of various bodies in real-life engineering problems by considering the time fractional derivative in the field equations. Originality/value In this problem, the authors have used the time fractional-order theory of thermoelasticity to solve the problem for a half space with a periodically varying heat source to control the speed of wave propagation in terms of heat and elastic waves for different conductivity like weak conductivity, moderate conductivity and super conductivity which is a new and novel contribution.

Fractional heat conduction in a rectangular plate with bending moments

“…Tripathi et al [15] solved a fractional order thermoelastic deflection in a thin circular plate with constant temperature distribution within the context of the quasi-static theory. Tripathi et al [16] solved a problem on fractional order generalized thermoelastic response in a half space due to a periodically varying heat source.…”

Analysis of a Thin Layer Formation of Third-Grade Fluid

“…Some readings about research work in coating efficaciously indulge the non-linear work and their comparisons in leading TGF [8][9][10][11][12][13][14][15][16][17][18]. Sullivan et al [12] premeditated the coating depth in surface coating size/width of TGF by implementing lubrication estimates with numerical and investigational outcomes for NNF. The influence of elasto-plastic material of blade surface protector with weaker viscoelastic fluids has been investigated as well in [13].…”

“…( ) at different values of β and varying ranges of . (a) with the s range of [0-2]; (b) with the s range of [0-3]; (c) with the s range of [0-5]; (d) with the s range of [0-10].Now using from Equation(12) …”