2D PROBLEM FOR A SPHERE IN THE FRACTIONAL ORDER THEORY THERMOELASTICITY TO AXISYMMETRIC TEMPERATURE DISTRIBUTION

2022 ◽  
Vol 11 (1) ◽  
pp. 1-15
Author(s):  
S.G. Khavale ◽  
K.R. Gaikwad
Keyword(s):  
Temperature Distribution ◽  
Fractional Order ◽  
Direct Method ◽  
Order Theory ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Copper Material ◽  
Mathcad Software ◽  
Axisymmetric Temperature ◽  
A Direct Method

In the present article, we implement the fractional thermoelasticity theory to a 2D issue for a sphere whose surface is free from traction, subject to a provided axisymmetric temperature distribution of heat. The medium is supposed to be quiescent initially. A direct method is used to get a solution and the Laplace transform technique is used. Mathematical models for copper material are designed as a particular instance. Numerical results are computed with help of Mathcad software and graphically represented and the fractional-order parameter effect has been explained.

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.

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 Application of Fractional Order Theory of Thermoelasticity in an Elliptical Disk and Associated Thermal Stresses

2020 ◽  
Vol 25 (3) ◽  
pp. 169-180
Author(s):  
S. Thakare ◽  
Y. Panke ◽  
K. Hadke
Keyword(s):  
Temperature Distribution ◽  
Fractional Order ◽  
Heat Supply ◽  
Numerical Evaluation ◽  
Thermal Stresses ◽  
Order Theory ◽  
Curved Surface ◽  
Mathieu Function ◽  
Zero Temperature ◽  
Thermal Deflection

AbstractIn this article, a time fractional-order theory of thermoelasticity is applied to an isotropic homogeneous elliptical disk. The lower and upper surfaces of the disk are maintained at zero temperature, whereas the sectional heat supply is applied on the outer curved surface. Thermal deflection and associated thermal stresses are obtained in terms of Mathieu function of the first kind of order 2n. Numerical evaluation is carried out for the temperature distribution, Thermal deflection and thermal stresses and results of the resulting quantities are depicted graphically.

scholarly journals A direct method of moving spheres on fractional order equations

2017 ◽  
Vol 272 (10) ◽  
pp. 4131-4157 ◽  
Author(s):  
Wenxiong Chen ◽  
Yan Li ◽  
Ruobing Zhang
Keyword(s):  
Fractional Order ◽  
Direct Method ◽  
Moving Spheres ◽  
A Direct Method

scholarly journals Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2084
Author(s):  
Oscar Martínez-Fuentes ◽  
Fidel Meléndez-Vázquez ◽  
Guillermo Fernández-Anaya ◽  
José Francisco Gómez-Aguilar
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Lyapunov Functions ◽  
Fractional Derivatives ◽  
Direct Method ◽  
The Laplace Transform ◽  
Uniformly Convergent ◽  
The Stability ◽  
Stability Results ◽  
Stability Concept

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

Thermoelastic fractional derivative model for exciting viscoelastic microbeam resting on Winkler foundation

2020 ◽  
pp. 107754632095652 ◽  
Author(s):  
Ahmed E Abouelregal
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Governing Equation ◽  
Axial Load ◽  
Generalized Thermoelasticity ◽  
Beam Theory ◽  
Order Theory ◽  
Winkler Foundation ◽  
Laplace Transform Technique ◽  
Viscoelastic Microbeam

In the current investigation, the thermoelastic vibration of a viscoelastic microbeam resting on the Winkler foundation is studied using the fractional-order theory. To describe the damping of the viscoelastic material according to experimental results, the Kelvin–Voigt model is replaced by a new form with a fractional-order derivative. The generalized thermoelasticity model and Euler–Bernoulli beam theory are used to construct the governing equation. The microbeam is subjected to axial load, ultrafast laser heating, and varying sinusoidal heat. The governing equation is then solved using the Laplace transform technique to determine the deflection and thermoelastic interaction responses of microbeams. The effects of many parameters such as the coefficient of viscosity, axial load, fractional derivative order, laser pulse duration, and foundation parameter on the microbeam response are explained and discussed in detail.

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