Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses

doi:10.1016/j.mechrescom.2010.04.006

Mechanics Research Communications

Volume 37, Issue 4, June 2010, Pages 436-440

https://doi.org/10.1016/j.mechrescom.2010.04.006 Get rights and content

Abstract

The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative of order $α$ is used to investigate thermal stresses in an infinite body with a circular cylindrical hole. The solution is obtained applying Laplace and Weber integral transforms. Several examples of problems with Dirichlet and Neumann boundary conditions are presented. Numerical results are illustrated graphically.

Introduction

The standard theory of heat conduction is based on the Fourier law relating the heat flux vector q to the temperature gradient. The classical theory of thermoelasticity investigates stresses caused by the temperature field found from the conventional parabolic heat conduction equation. In non-classical theories the Fourier law as well as the standard heat conduction equation is replaced by more general equations which lead to generalized theories of thermoelasticity. For an extensive bibliography on this subject and further discussion see, for example, Chandrasekharaiah, 1986, Chandrasekharaiah, 1998, Ignaczak (1989), Hetnarski and Ignaczak (1999) and references therein.

The time-nonlocal dependence between the heat flux vector and the temperature gradient with the “long-tale” power kernel can be interpreted in terms of fractional integrals and derivatives (essentials of the fractional calculus are presented in Appendix A): $\begin{matrix} q (t) = - k D_{R L}^{1 - α} grad T (t), & 0 < α < 1, \\ q (t) = - k I^{α - 1} grad T (t), & 1 < α \leq 2, \end{matrix}$ and yields the time-fractional heat conduction equation with Caputo time-fractional derivative of order $0 < α \leq 2$ . In terms of diffusion, such an equation can also be obtained from the continuous time random walk theory (see e.g. Metzler and Klafter, 2000) which extends classical Brownian random walks to variable jump lengths and waiting times between successive jumps. In the jump model, the particle moves instantaneously to a new site. Longer jumps are less probable. The power-law tails make it possible to have very long waiting times, and in the case $0 < α < 1$ , particles, on average, move slower than in ordinary diffusion which corresponds to $α = 1$ . In a superdiffusion regime $1 < α \leq 2$ , particles, on average, move faster than in the ordinary diffusion. From a waiting time perspective, this case corresponds to the velocity model (Zumofen and Klafter, 1993, Metzler and Compte, 1999) which assumes that the particle moves at the constant velocity to the new site. Additional discussion on derivation of fractional transfer equation can also be found in (Nigmatullin, 1984), where it was remarked that the generalized linear transfer equation contains all three classical types of partial equations (elliptic, parabolic, and hyperbolic) turning into each other in a continuous way.

For an infinite medium with a cylindrical cavity, a lot of problems describing interesting phenomena which characterize different theories of thermoelasticity have been solved by many researchers (Abbas and Abd-alla, 2008, Chandrasekharaiah and Keshavan, 1992, Chandrasekharaiah and Srinath, 1997, Furukawa et al., 1990, Youssef, 2009, among others).

A quasi-static uncoupled theory of thermoelasticity based on fractional heat conduction equation was put forward by Povstenko (2005). In this paper, we study axisymmetric thermal stresses in an infinite medium with a circular cylindrical cavity due to various boundary conditions for temperature at the surface of a cavity. The numerical results are given only for the stress tensor components. The interesting reader is referred to the previous paper (Povstenko, 2009), where plots of temperature are presented (see also Povstenko, 2008).

Section snippets

Formulation of the problem

A theory of thermal stresses is governed by the equation of motion in terms of displacements: $μ Δ u + (λ + μ) grad div u - ρ \frac{\partial^{2} u}{\partial t^{2}} = β K grad T,$ the stress–strain–temperature relation: $σ = 2 μ e + (λ tr e - β K T) I,$ and the time-fractional heat conduction equation: $\frac{\partial^{α} T}{\partial t^{α}} + γ_{e} \frac{\partial^{α} tr e}{\partial t^{α}} = a Δ T, 0 < α \leq 2,$ where body forces are neglected, $u$ is the displacement vector, $σ$ the stress tensor, $e$ the linear strain tensor, $T$ the temperature, $λ$ and $μ$ are Lamé constants, $K = λ + 2 μ / 3, β$ is the thermal coefficient of volumetric expansion, $I$ denotes the

The Dirichlet boundary condition

Consider the initial-boundary-value (5), (6) with the instantaneous delta-pulse at the boundary: $r = R : T = U_{0} δ_{+} (t), U_{0} = const.$ We apply the Laplace integral transform with respect to time $t$ and the Weber transform (see Appendix B) with respect to the spatial coordinate $r$ .

The solution reads $\bar{T} = - \frac{2 κ^{2}}{π} \int_{0}^{\infty} E_{α, α} (- κ^{2} η^{2}) \frac{J_{0} (ρ η) Y_{0} (η) - Y_{0} (ρ η) J_{0} (η)}{J_{0}^{2} (η) + Y_{0}^{2} (η)} η d η,$ which after using Eq. (7) gives ${\bar{σ}}_{r r} = \frac{2 κ^{2}}{π ρ^{2}} \int_{0}^{\infty} E_{α, α} (- κ^{2} η^{2}) \frac{Y_{0} (η) [ρ J_{1} (ρ η) - J_{1} (η)] - J_{0} (η) [ρ Y_{1} (ρ η) - Y_{1} (η)]}{J_{0}^{2} (η) + Y_{0}^{2} (η)} d η,$ ${\bar{σ}}_{θ θ} = - {\bar{σ}}_{r r} - \bar{T},$ where $E_{α, β} (z)$ is the

The Neumann boundary condition

In the case of instantaneous delta-pulse specification for normal derivative: $r = R : \frac{\partial T}{\partial r} = - W_{0} δ_{+} (t), W_{0} = const,$ the Laplace and Weber transforms technique leads to the following equations: $\bar{T} = - \frac{2 κ^{2}}{π} \int_{0}^{\infty} E_{α, α} (- κ^{2} η^{2}) \frac{J_{0} (ρ η) Y_{1} (η) - Y_{0} (ρ η) J_{1} (η)}{J_{1}^{2} (η) + Y_{1}^{2} (η)} d η,$ ${\bar{σ}}_{r r} = \frac{2 κ^{2}}{π ρ} \int_{0}^{\infty} E_{α, α} (- κ^{2} η^{2}) \frac{J_{1} (ρ η) Y_{1} (η) - Y_{1} (ρ η) J_{1} (η)}{J_{1}^{2} (η) + Y_{1}^{2} (η)} \frac{d η}{η},$ where $\bar{T} = t T / R W_{0}$ , ${\bar{σ}}_{i j} = t σ_{i j} / (2 μ m R W_{0})$ .

The components of the stress tensor are displayed in Fig. 5, Fig. 6.

Discussion and concluding remarks

We have investigated axisymmetric thermal stresses in an infinite medium with a cylindrical cavity in a frame-work of quasi-static uncoupled theory of thermoelasticity based on fractional heat conduction equation with time-fractional derivative of order $0 < α < 2$ subjected to various boundary conditions (Dirichlet and Neumann) for temperature at the surface of a cavity. In the case $0 < α < 1$ , the time-fractional heat conduction equation interpolates the elliptic Helmholtz equation and the parabolic

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Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses

Abstract

Introduction

Section snippets

Formulation of the problem

The Dirichlet boundary condition

The Neumann boundary condition

Discussion and concluding remarks

Phys. Lett. A

Physica A

Phys. Rep.

Mech. Res. Commun.

Effects of thermal relaxations on thermo-elastic interactions in an infinite orthotropic elastic medium with a cylindrical cavity

Arch. Appl. Mech.

Theory of Thermal Stresses

Thermoelasticity with second sound: a review

Appl. Mech. Rev.

Hyperbolic thermoelasticity: a review of recent literature

Appl. Mech. Rev.

Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity

Acta Mech.

Axisymmetric thermoelastic interactions without energy dissipation in an unbounded body with cylindrical cavity

J. Elasticity

Generalized thermoelasticity for an infinite body with a circular cylindrical hole

JSME Int. J.

Integral Transforms and Special Functions in Heat Conduction Problems