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Fractional diffusion equation and diffusive stresses

Yuriy Povstenko1,∗

1 Institute of Mathematics and Computer Science, Jan Długosz University of Czestochowa, Armii Krajowej 13/15,42-200 Czestochowa, Poland.

A quasi-static uncoupled theory of diffusive stresses based on the time-fractional diffusion equation is considered. The Caputofractional derivative of order α is used. In particular, the proposed theory interpolates the classical theory of diffusive stressesand that without energy dissipation. The fundamental solution to the second Cauchy problem for the fractional diffusionequation in a plane as well as the associated diffusive stresses are obtained in the case α = 3/2.

1 Introduction

The theory of diffusive stresses is an integration of the theory of diffusion and the theory of elasticity. In this paper we restrictthe discussion to the effect of diffusion on the deformation of a linear isotropic elastic solid. The inverse effect of deformationon diffusion will not be considered.

The conventional theory of diffusive stresses is formulated on the principles of the classical theory of diffusion, specificallyon the classical Fick law, which relates the matter flux vector to the concentration gradient. It is well known that frommathematical wievpoint the Fick law and the Fourier law as well as and the diffusion equation and the heat conductionequation are indentical. During the last three decades, nonclassical theories in which the Fick law and the Fourier law and,consequently, the diffusion and the heat conduction equations are replaced by more general equations, have been formulated.For an extensive bibliography on this subject see, for example, [1-5]. The time-nonlocal dependence between the flux vectorsand corresponding gradients with “long-tale” power kernel can be interpreted in terms of fractional integrals and derivativesand yields the fractional diffusion (or heat conduction) equation with time derivative of fractional order α.

A quasi-static uncoupled theory of diffusive (or thermal) stresses based on this equation was proposed in [6, 7]. Becausethe time-fractional diffusion (or heat conduction) equation in the case 1 ≤ α ≤ 2 interpolates the standard heat conductionequation (α = 1) and the wave equation (α = 2), the proposed theory interpolates the classical thermoelasticity and thethermoelasticity without energy dissipation introduced by Green and Naghdi [8].

2 Statement of the problem

The stressed-strained state of a solid is governed by the equilibrium equation in terms of displacements

µ∆u + (λ + µ)grad divu = βKgrad c, (1)

the stress-strain-concentration relation

σ = 2µe + (λ tr e− βKc)I, (2)

the time-fractional diffusion equation

∂αc

∂tα= a∆c + Q(x, t), (3)

where u is the displacement vector, σ the stress tensor, e the linear strain tensor, c the concentration, a the diffusivity co-efficient, Q(x, t) the mass source, λ and µ are Lame constants, K = λ + 2µ/3, β is the diffusive coefficient of volumetricexpansion, I denotes the unit tensor, ∂α

∂tαis the Caputo fractional derivative [9,10]

dαf

dtα=

1

Γ(n − α)

∫ t

0

(t − τ)n−α−1 dnf(τ)

dτndτ, n − 1 < α < n, (4)

with the following Laplace transform rule

L

{dαf(t)

dtα

}= sαL {f(t)} −

n−1∑k=0

f (k)(0+)sα−1−k, n − 1 < α < n. (5)

∗ Corresponding author E-mail: [email protected]

PAMM · Proc. Appl. Math. Mech. 7, 2040007–2040008 (2007) / DOI 10.1002/pamm.200700134

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Equation (3) should be subject to initial conditions:

t = 0 : c = U(x) for 0 < α ≤ 2 (6)

and

t = 0 :∂c

∂t= W (x) for 1 < α ≤ 2. (7)

3 The fundamental solution to the second Cauchy problem

In two-dimensional case the diffusive stresses exerted by a source of diffusion Q(r, t) = q2πr δ+(r) δ+(t) were considered in

[6], the stresses associated with the solution to the first Cauchy problem with U(r) = u2πr δ+(r) were obtained in [7]. In this

paper, we consider the second Cauchy problem with W (r) = w2πr δ+(r). The solution is obtained using the Laplace integral

transform with respect to time t and the Hankel transform with respect to the radial coordinate r and has the following form

c∗ =w

2π

sα−2

sα + aξ2, (8)

where asterisk denotes the transforms, s is the Laplace transform variable, ξ is the Hankel transform variable.For α = 3/2 we obtain

c =w

2π

∫∞

0

H3/2(ξ, t)J0(rξ) ξ dξ. (9)

As in a case of classical thermoelasticity or diffusional elasticity we use the representation of stresses in terms of displace-ment potential and get

σrr + σθθ = −µmw

π

∫∞

0

H3/2(ξ, t)J0(rξ) ξ dξ, (10)

σrr − σθθ = −µmw

π

∫∞

0

H3/2(ξ, t)J2(rξ) ξ dξ, (11)

where

H3/2(ξ, t) = L−1

{1√

s(s3/2 + aξ2)

}=

t

3γ

{eγerfc

√γ − 2e−γ/2 cos

(√3

2γ +

π

3

)

+4√

γ√π

∫ 1

0

exp

[−1

2γ(1 − v2)

]cos

[√3

2γ(1 − v2) − π

3

]dv

}, γ = a2/3ξ4/3t, (12)

Jn(r) is the Bessel function of the first kind of the order n.

References

[1] D. S. Chandrasekharaiah, Appl. Mech. Rev. 39, 355 (1986).[2] D. S. Chandrasekharaiah, Appl. Mech. Rev. 51, 705 (1998).[3] K. K. Tamma and X. Zhou, J. Thermal Stresses 21, 405 (1998).[4] R. B. Hetnarski and J. Ignaczak, J. Thermal Stresses 22, 451 (1999).[5] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000).[6] Y. Z. Povstenko, Int. J. Engng Sci. 43, 977 (2005).[7] Y. Z. Povstenko, J. Thermal Stresses 28, 83 (2005).[8] A. E. Green and P. M. Naghdi, J. Elast. 31, 189 (1993).[9] R. Gorenflo and F. Mainardi, in: Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri and F. Mainardi

(Springer, New York, 1997), p. 223.[10] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam,

2006), p. 90.


ICIAM07 Contributed Papers 2040008