Journal ArticleDOI

# Effect of fractional parameter on plane waves in a rotating elastic medium under fractional order generalized thermoelasticity

26 Nov 2012-International Journal of Applied Mechanics (Imperial College Press)-Vol. 04, Iss: 03, pp 1250030

AbstractIn ["Theory of fractional order generalized thermoelasticity," Journal of Heat Transfer132, 2010] Youssef has proposed a model in generalized thermoelasticity based on the fractional order time derivatives. The current manuscript is concerned with a two-dimensional generalized thermoelastic coupled problem for a homogeneous isotropic and thermally conducting thermoelastic rotating medium in the context of the above fractional order generalized thermoelasticity with two relaxation time parameters. The normal mode analysis technique is used to solve the resulting non-dimensional coupled governing equations of the problem. The resulting solution is then applied to two concrete problems. The effect of the fractional parameter and the time instant on the variations of different field quantities inside the elastic medium are analyzed graphically in the presence of rotation.

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Abstract: The widespread interest on the hereditary behavior of biological and bioinspired materials motivates deeper studies on their macroscopic minimal" state. The resulting integral equations for the detected relaxation and creep power-laws, of exponent $\beta$, are characterized by fractional operators. Here strains in $SBV_{loc}$ are considered to account for time-like jumps. Consistently, starting from stresses in $L_{loc}^{r}$, $r\in [1,\beta^{-1}], \, \, \beta\in(0,1)$ we reconstruct the corresponding strain by extending a result in [42]. The minimal" state is explored by showing that different histories delivering the same response are such that the fractional derivative of their difference is zero for all times. This equation is solved through a one-parameter family of strains whose related stresses converge to the response characterizing the original problem. This provides an approximation formula for the state variable, namely the residual stress associated to the difference of the histories above. Very little is known about the microstructural origins of the detected power-laws. Recent rheological models, based on a top-plate adhering and moving on functionally graded microstructures, allow for showing that the resultant of the underlying microstresses" matches the action recorded at the top-plate of such models, yielding a relationship between the macroscopic state and the microstresses".

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Journal ArticleDOI
, Dong Liu1
Abstract: In this paper, a new formulation for the generalized thermoelasticity in an isotropic elastic medium with temperature-dependent material properties is established. The governing equations for the generalized axisymmetric plane strain problem are derived. The asymptotic solutions for an infinite cylinder with the boundary subjected to the thermal shock are obtained under the linear assumption. Numerical results for the propagation of the thermal and elastic waves and the distributions of the displacement, temperature and stresses are given and illustrated graphically. Using these solutions, some phenomenon involved in the generalized thermoelastic problem are obtained, and the jumps at the wavefronts are observed clearly. The comparison is made with results obtained in the temperature-independent case and the influence of the temperature dependency of material properties on the propagation of thermal and elastic waves are also discussed.

16 citations

Journal ArticleDOI
Abstract: We investigate the unsteady flow of a viscous fluid near a vertical heated plate. The momentum and energy equations are considered as fractional differential equations with respect to the time t. Solutions of the initial-boundary values problem are determined by means of the Laplace transform technique and are represented by means of the Wright functions. The fundamental solution for the temperature field is obtained. This allows obtaining the temperature field for different conditions on the wall temperature. A numerical case is analyzed in order to obtain information regarding the influence of the fractional parameters on the temperature and velocity fields. Some physical aspects of the fluid behavior are presented by graphical illustrations.

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Journal ArticleDOI
Xiong Chunbao, Guo Ying1
Abstract: A one-dimensional generalized magnetothermoelastic problem of a thermoelastic rod with finite length is investigated in the context of the fractional order thermoelasticity. The rod with variable properties, which are temperature-dependent, is fixed at both ends and placed in an initial magnetic field, and the rod is subjected to a moving heat source along the axial direction. The governing equations of the problem in the fractional order thermoelasticity are formulated and solved by means of Laplace transform in tandem with its numerical inversion. The distributions of the nondimensional temperature, displacement, and stress in the rod are obtained and illustrated graphically. The effects of the temperature-dependent properties, the velocity of the moving heat source, the fractional order parameter, and so forth on the considered variables are concerned and discussed in detail, and the results show that they significantly influence the variations of the considered variables.

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### Cites methods from "Effect of fractional parameter on p..."

• ...By employing this theory, Sarkar and Lahiri [25] concerned a two-dimensional generalized thermoelastic problem for a rotating elastic medium....

[...]

• ...[25] N. Sarkar and A. Lahiri, “Effect of fractional parameter on plane waves in a rotating elastic medium under fractional order generalized thermoelasticity,” International Journal of Applied Mechanics, vol. 4, no. 3, Article ID 1250030, 2012....

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##### References
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Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.

2,861 citations

Journal ArticleDOI
Abstract: A unified treatment is presented of thermoelasticity by application and further developments of the methods of irreversible thermodynamics. The concept of generalized free energy introduced in a previous publication plays the role of a thermoelastic potential'' and is used along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement. The general laws of thermoelasticity are formulated in a variational form along with a minimum entropy production principle. This leads to equations of the Lagrangian type, and the concept of thermal force is introduced by means of a virtual work definition. Heat conduction problems can then be formulated by the methods of matrix algebra and mechanics. This also leads to the very general property that the entropy density obeys a diffusion‐type law. General solutions of the equations of thermoelasticity are also given using the Papkovitch‐Boussinesq potentials. Examples are presented and it is shown how the generalized coordinate method may be used to calculate the thermoelastic internal damping of elastic bodies.

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Journal ArticleDOI
01 Apr 1971

642 citations

Journal ArticleDOI
Michele Caputo
TL;DR: The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials.
Abstract: The model of dissipation based on memory introduced by Caputo is generalized and checked with experimental dissipation curves of various materials

623 citations

Journal ArticleDOI
Abstract: A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.

416 citations