Theory of Fractional Order Generalized Thermoelasticity

2010 ◽  
Vol 132 (6) ◽  
Author(s):  
Hamdy M. Youssef
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
State Space ◽  
Uniqueness Theorem ◽  
Half Space ◽  
Fractional Order ◽  
Elastic Material ◽  
Generalized Thermoelasticity ◽  
New Model ◽  
One Dimensional

In this work, a new model of thermoelasticity theory has been constructed in the context of a new consideration of heat conduction with fractional order, and its uniqueness theorem has been approved also. One-dimensional application for a half-space of elastic material, which is thermally shocked, has been solved by using Laplace transform and state-space techniques. According to the numerical results and its graphs, conclusion about the new theory of thermoelasticity has been constructed.

scholarly journals A Study on Fractional Order Thermoelastic Half Space

2020 ◽  
Vol 25 (4) ◽  
pp. 191-202
Author(s):  
Sourov Roy ◽  
Abhijit Lahiri
Keyword(s):  
Half Space ◽  
Fractional Order ◽  
Equations Of Motion ◽  
Generalized Thermoelasticity ◽  
Dimensional Problem ◽  
Governing Equations ◽  
One Dimensional ◽  
Closed Form Solutions ◽  
Eigen Value ◽  
The Laplace Transform

AbstractIn this paper, we consider a one dimensional problem on a fractional order generalized thermoelasticity in half space subjected to an instantaneous heat source. The Laplace transform as well as eigen value approach techniques are applied to solve the governing equations of motion and heat conduction. Closed form solutions for displacement, temperature and stress are obtained and presented graphically.

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

2018 ◽  
Vol 14 (1) ◽  
pp. 2-15 ◽  
Author(s):  
Jitesh Tripathi ◽  
Shrikant Warbhe ◽  
K.C. Deshmukh ◽  
Jyoti Verma
Keyword(s):  
Mathematical Model ◽  
Heat Source ◽  
Laplace Transform ◽  
Half Space ◽  
Fractional Order ◽  
Order Parameter ◽  
Order Theory ◽  
Numerical Inversion ◽  
Content Type ◽  
Copper Material

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.

scholarly journals Fractional Heat Conduction Models and Thermal Diffusivity Determination

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Monika Žecová ◽  
Ján Terpák
Keyword(s):  
Heat Conduction ◽  
Thermal Diffusivity ◽  
Numerical Methods ◽  
Fractional Order ◽  
Experimental Verification ◽  
Heat Conduction Equation ◽  
Historical Overview ◽  
One Dimensional ◽  
The One ◽  
Fractional Heat Conduction

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper.

scholarly journals Two-Temperature Generalized Thermoviscoelasticity with Fractional Order Strain Subjected to Moving Heat Source: State Space Approach

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Renu Yadav ◽  
Kapil Kumar Kalkal ◽  
Sunita Deswal
Keyword(s):  
Heat Source ◽  
State Space ◽  
Fractional Order ◽  
Mechanical Load ◽  
Generalized Thermoelasticity ◽  
Internal Heat ◽  
Internal Heat Source ◽  
Conductive Temperature ◽  
Special Cases ◽  
Two Temperature

The theory of generalized thermoelasticity with fractional order strain is employed to study the problem of one-dimensional disturbances in a viscoelastic solid in the presence of a moving internal heat source and subjected to a mechanical load. The problem is in the context of Green-Naghdi theory of thermoelasticity with energy dissipation. Laplace transform and state space techniques are used to obtain the general solution for a set of boundary conditions. To tackle the expression of heat source, Fourier transform is also employed. The expressions for different field parameters such as displacement, stress, thermodynamical temperature, and conductive temperature in the physical domain are derived by the application of numerical inversion technique. The effects of fractional order strain, two-temperature parameter, viscosity, and velocity of internal heat source on the field variables are depicted graphically for copper material. Some special cases of interest have also been presented.

scholarly journals The Effect of Magnetic Field and Initial Stress on Fractional Order Generalized Thermoelastic Half-Space

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Sunita Deswal ◽  
Sandeep Singh Sheoran ◽  
Kapil Kumar Kalkal
Keyword(s):  
Magnetic Field ◽  
Initial Stress ◽  
Half Space ◽  
Fractional Order ◽  
Generalized Thermoelasticity ◽  
Order Theory ◽  
Homogeneous Half Space ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Space Formulation

The aim of this paper is to study magneto-thermoelastic interactions in an initially stressed isotropic homogeneous half-space in the context of fractional order theory of generalized thermoelasticity. State space formulation with the Laplace transform technique is used to obtain the general solution, and the resulting formulation is applied to the ramp type increase in thermal load and zero stress. Solutions of the problem in the physical domain are obtained by using a numerical method of the Laplace inverse transform based on the Fourier expansion technique, and the expressions for the displacement, temperature, and stress inside the half-space are obtained. Numerical computations are carried out for a particular material for illustrating the results. Results obtained for the field variables are displayed graphically. Some comparisons have been shown in figures to present the effect of fractional parameter, ramp parameter, magnetic field, and initial stress on the field variables. Some particular cases of special interest have been deduced from the present investigation.

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