Fractional Order Generalized Electro-Magneto-Thermo-Elasticity with Magnetic Monopoles and Geometrical Nonlinearity

2013 ◽  
Vol 273 ◽  
pp. 162-166 ◽  
Author(s):  
Ya Jun Yu ◽  
Xiao Geng Tian
Keyword(s):  
Electromagnetic Field ◽  
Numerical Methods ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Inverse Method ◽  
Geometrical Nonlinearity ◽  
Generalized Thermoelasticity ◽  
Magnetic Monopoles ◽  
Rate Dependent ◽  
Temperature Rate

Recently, Youssef developed the fractional order generalized thermoelasticity (FOGTE) in the context of extended thermoelasticity (ETE). In this work, we extended the concept of fractional calculus into the temperature rate dependent thermoelasticity (TRDTE) and introduced the unified form of the two cases. Upon introducing the electromagnetic field with magnetic monopoles and considering the geometrical nonlinearity, we proposed a fractional order generalized electro-magneto- thermo-elasticity (FOGEMm-poleTEg-non) with magnetic monopoles (m-pole) and geometrical nonlinearity (g-non). To deal with multi-physics problems using numerical methods, we obtained a generalized variational theorem by using the semi-inverse method.

scholarly journals Stability in constrained temperature-rate-dependent thermoelasticity

2017 ◽  
Vol 22 (8) ◽  
pp. 1738-1763
Author(s):  
Amnah M Alharbi ◽  
Nigel H Scott
Keyword(s):  
Heat Conduction Equation ◽  
Deformation Gradient ◽  
Generalized Thermoelasticity ◽  
Rate Of Change ◽  
Harmonic Waves ◽  
Rate Dependent ◽  
Thermoelastic Material ◽  
Anisotropic Temperature ◽  
Temperature Rate ◽  
Anisotropic And Isotropic Materials

In an anisotropic temperature-rate-dependent thermoelastic material four plane harmonic waves may propagate in any direction, all dispersive and attenuated, and all stable in the sense that their amplitudes remain bounded in the direction of travel. In this paper, the material is additionally assumed to suffer an internal constraint of the deformation-temperature type, i.e. the temperature is a prescribed function of the deformation gradient. In this constrained thermoelastic material four waves continue to propagate but instabilities are now found. Constrained temperature-rate-dependent thermoelasticity is then combined with generalized thermoelasticity in which the rate of change of heat flux also appears in the heat conduction equation. Four waves again propagate but instabilities are found as before. Anisotropic and isotropic materials are both considered.

scholarly journals Modified Green–Lindsay thermoelasticity wave propagation in elastic materials under thermal shocks

2020 ◽  
Author(s):  
Farshad Shakeriaski ◽  
Maryam Ghodrat ◽  
Juan Escobedo-Diaz ◽  
Masud Behnia
Keyword(s):  
Wave Propagation ◽  
Thermal Shock ◽  
Generalized Thermoelasticity ◽  
Stress Wave Propagation ◽  
Rate Dependent ◽  
Classic Theory ◽  
Finite Strain Theory ◽  
Dependent Model ◽  
Temperature Rate ◽  
Acceptable Agreement

Abstract In this study, a nonlinear numerical method is presented to solve the governing equations of generalized thermoelasticity in a large deformation domain of an elastic medium subjected to thermal shock. The main focus of the study is on the modified Green–Lindsay thermoelasticity theory, solving strain and temperature rate-dependent model using finite strain theory. To warrant the continuity of the finding responses at the boundary after the applied shock, higher order elements are adopted. An analytical solution is provided to validate the numerical findings and an acceptable agreement between the two presented solutions is obtained. The findings revealed that stress and thermal waves have distinct interactions and a harmonic temperature variation may lead to a systematic uniform stress distribution. Besides, a notable difference in the results predicted by the modified Green–Lindsay model and classic theory is observed. It is also found that the modified Green–Lindsay theory is more efficient in determining the wave propagation phenomenon. Furthermore, the findings established that thermal shock induces tensile stresses in the structure immediately after the shock, and the perceived phenomenon mainly depends on the defined boundary conditions. The results show that the strain rate can have a significant influence on the displacement and stress wave propagation in a structure subjected to thermal shock and these impacts may be more considerable with mechanical loading.

scholarly journals Good (and Not So Good) Practices in Computational Methods for Fractional Calculus

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 324 ◽  
Author(s):  
Kai Diethelm ◽  
Roberto Garrappa ◽  
Martin Stynes
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Computational Approach ◽  
Numerical Treatment ◽  
Good Practices ◽  
Order Case ◽  
Fractional Differential

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.

scholarly journals Effect of Rotation in an Orthotropic Elastic Slab

2017 ◽  
Vol 22 (1) ◽  
pp. 163-174
Author(s):  
S. Santra ◽  
A. Lahiri ◽  
N.C. Das
Keyword(s):  
Normal Mode Analysis ◽  
Equations Of Motion ◽  
Generalized Thermoelasticity ◽  
Mode Analysis ◽  
Rate Dependent ◽  
Temperature Rate ◽  
Elastic Slab ◽  
Basic Equations ◽  
Matrix Differential ◽  
Vector Matrix

Abstract The fundamental equations of the two dimensional generalized thermoelasticity (L-S model) with one relaxation time parameter in orthotropic elastic slab has been considered under effect of rotation. The normal mode analysis is used to the basic equations of motion and heat conduction equation. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by the eigenvalue approach. The field variables in the space time domain are obtained numerically. The results corresponding to the cases of conventional thermoelasticity CTE), extended thermoelasticity (ETE) and temperature rate dependent thermoelasticity (TRDTE) are compared by means of graphs.

TEMPERATURE RATE DEPENDENT GENERALIZED THERMOELASTICITY WITH MODIFIED OHM'S LAW

2012 ◽  
Vol 01 (04) ◽  
pp. 1250031 ◽  
Author(s):  
NANTU SARKAR ◽  
ABHIJIT LAHIRI
Keyword(s):  
Temperature Gradient ◽  
Half Space ◽  
Generalized Thermoelasticity ◽  
Mode Analysis ◽  
Plane Boundary ◽  
Ohm’S Law ◽  
Rate Dependent ◽  
Ohm's Law ◽  
Temperature Rate ◽  
Physical Quantities

The present work is concerned with an electro-magneto-thermoelastic coupled problem for a homogeneous, isotropic, thermally and electrically conducting semi-infinite solid medium in two-dimensional space where an initial magnetic field with constant intensity acts parallel to the plane boundary of the half-space. The surface of the half-space taking as traction free which is subjected to a thermal shock. The modified Ohm's law including the temperature gradient and charge density effect to the governing equations of the generalized thermoelasticity under the temperature rate dependent thermoelasticity (TRDTE) proposed by Green and Lindsay (GL model) has been introduced. Normal mode analysis together with eigenvalue approach technique is used to obtain the general solutions for the physical quantities. Numerical results for the physical quantities are illustrated graphically and analyzed. The graphical results indicate that the effect of the coefficient connecting the temperature gradient and the electric current density of the modified Ohm's law on the plane waves is very pronounced. Comparison are made with the results obtained in the absence of the above coefficient.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

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