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Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation
International Journal of Engineering Science 43 (2005) 977–991 www.elsevier.com/locate/ijengsci
Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation Y.Z. Povstenko Institute of Mathematics and Computer Science, Jan Długosz University of Czeßstochowa, al. Armii Krajowej 13/15, 42-200 Czeßstochowa, Poland Received 14 April 2004; received in revised form 5 March 2005; accepted 5 March 2005
Abstract The theory of diffusive stresses based on the diffusion-wave equation with time-fractional derivative of fractional order a is formulated. The non-parabolic diffusion equation is a mathematical model of a wide range of important physical phenomena and can be obtained as a consequence of the non-local constitutive equation for the matter flux vector with the long-tale power time-non-local kernel. Because the considered equation in the case 1 6 a 6 2 interpolates the parabolic equation (a = 1) and the wave equation (a = 2), the proposed theory interpolates a classical theory of diffusive stresses and that without energy dissipation introduced by Green and Naghdi. The stresses caused by a source of diffusion in an unbounded solid are found in one-dimensional and axially symmetric cases (for plane deformation). Numerical results for the concentration and stress distributions are given and illustrated graphically. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The theory of diffusive strain is governed by a system of partial differential equations consisting of the equations of elasticity with gradient of diffusion taken into account and the balance equation of diffusion. The problem can be treated as an uncoupled in the case when the influence of the time change of strain on a flux of diffusion is neglected. The first theoretical investigation of E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.03.004
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diffusive strain dates back to Pidstryhach (spelled also Pidstrygach and Podstrigach) [1–4]. For additional references, further generalizations and discussion see [5–9]. The conventional theory of diffusive strain is based on the principles of the classical theory of diffusion, specifically on the classical Fick law J ¼ j grad c;
ð1:1Þ
where J is the matter flux, c is the concentration and j is the diffusion conductivity. In combination with the balance equation for mass the Fick law leads to the classical diffusion equation oc ¼ j Dc þ qQ; ð1:2Þ ot with q being the mass density, t the time, Q the mass source. In the past two decades, a considerable research efforts has been expended to study anomalous diffusion which is characterized by the time-fractional equation q
oa c ¼ j Dc þ qQ; ð1:3Þ ota where the parameter 0 < a < 2. Two types of anomalous transport can be distinguished. The subdiffusion regime is exemplified by the value 0 < a < 1, whereas the superdiffusion regime is characterized by the value 1 < a < 2. In subdiffusion, particles move slower, in superdiffusion particles move faster than in the ordinary diffusion. The limiting case a = 2 is known as ballistic diffusion [10]. Anomalous diffusive transport appears to be a universal experimental phenomenon. A number of works has been published dealing with anomalous transport in amorphous [11], colloid [12], glassy [13,14] and porous [10,15] materials, in fractals [16–18] and percolation clusters [15], comb structures [19], dielectrics and semiconductors [20,21], polymers [22,23], random [24] and disordered [25] media. Various mathematical aspects concerning the existence, uniqueness, well-posedness, Sommerfeld condition for non-parabolic diffusion equation are discussed in [26–30]. The purpose of this paper is to formulate the theory of diffusive strain of elastic solid based on the time-fractional diffusion-wave equation and to study the stresses in an unbounded solid exerted by a source of diffusion in one-dimensional and axially symmetric cases (for plane deformation). q
2. Preliminaries In this section we recall the main ideas of fractional calculus [31–34]. The notion of the Riemann–Liouville fractional integral is introduced as a natural generalization of the wellknown n-fold repeated integral Inf(t) written in a convolution type form Z t 1 a1 a I f ðtÞ ¼ ðt sÞ f ðsÞ ds; a > 0; ð2:1Þ CðaÞ 0 where C(a) is the gamma function.
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The Riemann–Liouville derivative of the fractional order a is defined by the following formula: Z t dn 1 na1 a ðt sÞ f ðsÞ ds ; n 1 < a < n. ð2:2Þ DRL f ðtÞ ¼ n dt Cðn aÞ 0 There are other possibilities to introduce fractional derivatives [35–38]. One of the alternative definitions of the fractional derivative was proposed by Caputo [37,38] Z t 1 dn f ðsÞ a ðt sÞna1 ds; n 1 < a < n. ð2:3Þ DC f ðtÞ ¼ Cðn aÞ 0 dsn The rules for the Laplace transforms of the Riemann–Liouville fractional integral and fractional derivative read LfI a f ðtÞg ¼
1 Lff ðtÞg; sa
ð2:4Þ
n1 X L DaRL f ðtÞ ¼ sa Lff ðtÞg Dk I na f ð0þ Þsn1k ;
n 1 < a < n;
ð2:5Þ
k¼0
where s is the transform variable. The Laplace transform rule for the Caputo derivative has the following form: n1 X L DaC f ðtÞ ¼ sa Lff ðtÞg f k ð0þ Þsa1k ;
n 1 < a < n.
ð2:6Þ
k¼0
The Caputo fractional derivative is a regularization in the time origin for the Riemann–Liouville fractional derivative by incorporating the relevant initial conditions [39]. In this paper we shall use the Caputo fractional derivative omitting the index C. The major utility of this type fractional derivative is caused by the treatment of differential equations of fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer (not fractional) order, even if the governing equation is of fractional order [34,40]. It is common knowledge that from mathematical viewpoint the Fick law in the theory of diffusion and the Fourier law in the theory of heat conduction are identical. During the last four decades, non-classical thermoelasticity theories in which the Fourier law and the heat equation are replaced by more general equations, have been formulated. For an extensive bibliography on this subject see [41–46]. Here, we mention the time-non-local constitutive equation for heat flux with a general non-local kernel [47–50] and with ‘‘short-tale’’ exponential kernel [43,51,52]. The time-fractional diffusion equation (1.3) corresponds to non-local constitutive equation for the matter flux J with the power time-non-local kernel describing ‘‘long-tale’’ memory.
3. Problem formulation The stressed–strained state of a solid is governed by the equilibrium equation in terms of displacements lDu þ ðk þ lÞ grad div u ¼ bc Kgrad c;
ð3:1Þ
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the stress–strain-concentration relation r ¼ 2le þ ðk tr e bc KcÞI
ð3:2Þ
and the time-fractional diffusion equation oa c ¼ aDc þ Qðx; tÞ; ð3:3Þ ota where u is the displacement vector, r the stress tensor, e the linear strain tensor, c the concentration, a the diffusivity coefficient, k and l are Lame´ constants, K = k + 2l/3, bc is the diffusion coefficient of volumetric expansion, I denotes the unit tensor. It is well known that a solution to the classical diffusion equation oc ¼ aDc þ Qðx; tÞ; ot with initial condition t¼0:
ð3:4Þ
c ¼ f ðxÞ;
ð3:5Þ
is expressed by the Poisson formula Z Z tZ cðx; tÞ ¼ Eðjx nj; tÞf ðnÞ dn þ Eðjx nj; t sÞQðn; sÞ dn ds; Rn
ð3:6Þ
Rn
0
where n is the dimension of a space ! 1 jxj2 Eðjxj; tÞ ¼ pffiffiffiffiffiffiffi n exp ; 4at ð2 patÞ
ð3:7Þ
is the fundamental solution, the same for both the Cauchy problem oc ¼ aDc; ot t ¼ 0 : c ¼ dðxÞ
ð3:8Þ ð3:9Þ
and the source problem oc ¼ aDc þ dðxÞdðtÞ. ot
ð3:10Þ
Here d(x) is DiracÕs delta. In the case of anomalous diffusion we have the following formula: Z tZ Z E1 ðjx nj; tÞf ðnÞ dn þ E2 ðjx nj; t sÞQðn; sÞ dn ds; cðx; tÞ ¼ Rn
0
Rn
instead of (3.6) and hence two different fundamental solutions E1 ðjxj; tÞ
and E2 ðjxj; tÞ;
corresponding to the Cauchy problem and to the source problem.
ð3:11Þ
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The fundamental solution to the Cauchy problem for anomalous diffusion equation and the stresses corresponding to it are discussed in the accompanying paper [53]. Here we restrict our consideration to the fundamental solution for the source problem. As in the case of classical thermoelasticity or the theory of diffusive strain we use the representation of stresses in terms of the displacement potential U [54]
2 o dij D U; ð3:12Þ rij ¼ 2l oxi oxj where dij is the Kronecker delta. The displacement potential is determined from the equation 1 þ m bc ; 1m 3 with m being the Poisson ratio. DU ¼ mc;
ð3:13Þ
m¼
4. Solution in one-dimensional case In one-dimensional case, Eq. (3.3) with Q(x, t) = qd(x)d(t) reads oa c o2 c ¼ a 2 þ qdðxÞdðtÞ; ð4:1Þ ota ox where q is the strength of mass source. Using the Laplace transform with respect to time (denoted by the asterisk) we obtain the ordinary differential equation for the concentration d2 c
sa c ¼ qdðxÞ; dx2 having the solution pffiffi a=2 q c ¼ pffiffiffi a=2 eðjxj= aÞs . 2 as
ð4:2Þ
a
ð4:3Þ
The inversion of the Laplace transform in (4.3) leads to the following expression: a q c ¼ pffiffiffi 1a=2 H z; ; 2 at 2
ð4:4Þ
where H(z; b) is the Hanyga auxiliary function of the similarity variable jxj z ¼ pffiffiffi b ; at
b ¼ a=2;
0 < b < 1;
ð4:5Þ
having the following series representation: H ðz; bÞ ¼
1 X n¼0
ð1Þn zn ; n!Cðbn þ bÞ
0 < b < 1.
ð4:6Þ
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The function H(z; b) is a particular case of the Wright function [55] 1 X zn W ðz; k; lÞ ¼ ; k > 1; l > 0; n!Cðkn þ lÞ n¼0
ð4:7Þ
corresponding to k = b, l = b and negative values of the first independent variable H ðz; bÞ ¼ W ðz; b; bÞ. It is obvious that in the case b = 1/2 the Gaussian auxiliary function is obtained
2 1 1 z H z; ¼ pffiffiffi exp . 2 4 p Other known particular cases correspond to the values b = 1/3 and b = 2/3 [28]
1 H z; ¼ 31=3 Ai0 ð31=3 zÞ; 3
2 2 3 2=3 4=3 2 z Þ exp z ; H z; ¼ 3 Aið3 3 27
ð4:8Þ
ð4:9Þ
ð4:10Þ ð4:11Þ
where Ai(x) is the Airy function, the prime denotes the derivative. We also present expressions for cases b = 1/4 (a = 1/2) and b = 3/4 (a = 3/2). On the one hand, all these expressions allow us to test the numerical algorithms for computing series representation of the auxiliary function, especially for z > 1. On the other hand, these expressions can be used to obtain the corresponding solutions in more complicated axially symmetric case. For this purpose we use the Fourier transform with respect to the spatial coordinate x (denoted by a bar) with n being the transform variable. For a = 1/2 we obtain q 1 c ¼ pffiffiffiffiffiffi pffiffi . 2p s þ an2 Inverting the Laplace transform yields pffi 1 1 2 2 1 2 4 p ffiffiffiffi ffi an G1=2 ðn; tÞ ¼ L tÞ; expða n tÞ erfc ðan ¼ pffiffi pt s þ an2
ð4:12Þ
ð4:13Þ
where 2 erfc x ¼ pffiffiffi p
Z
1
2
ev dv;
ð4:14Þ
x
is the complementary error function. Inverting the Fourier transform leads us to the integral representation of the solution Z q 1 c¼ G1=2 ðn; tÞ cosðxnÞ dn p 0 Z pffi q 1 1 2 2 2 4 pffiffiffiffiffi an expða n tÞ erfc ðan tÞ cosðxnÞ dn. ð4:15Þ ¼ p 0 pt
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pffi Inserting (4.14) into (4.15), using substitution v ¼ u2 þ an2 t and changing integration with respect to n and u we obtain
2 2 Z 1
q z 2 u2 zu 1=u4 1 1e exp du; ð4:16Þ c ¼ pffiffiffi pffiffiffi 3=4 4 8 2 2p at 0 where the following formula [56]:
2 pffiffiffi
Z 1 x2 x p 2 bn2 ne cosðxnÞ dn ¼ 3=2 1 exp ; 2b 4b 4b 0 has been used. In the case a = 3/2 we have 1 1 G3=2 ðn; tÞ ¼ L s3=2 þ an2 h 2 i 1 z1 2pi=3 z22 2pi=3 z23 ¼ e erfc z þ e e erfc z þ e e erfc z . 1 2 3 3a1=3 n2=3 Here
pffi z1 ¼ a1=3 tn2=3 ;
pffi z2 ¼ epi=3 a1=3 tn2=3 ;
pffi z3 ¼ epi=3 a1=3 tn2=3 .
Eq. (4.18) can be rewritten in terms of the function w(z) h
pffi 1 2=3 4=3 1=3 2=3 G3=2 ðn; tÞ ¼ t exp a n t erfc a n 3a1=3 n2=3 pffi pffi i þe2pi=3 w epi=6 a1=3 n2=3 t þ e2pi=3 w epi=6 a1=3 n2=3 t ; where w(z) is defined as follows [57]:
Z z 2i 2 2 2 wðzÞ ¼ ez erfc ðizÞ ¼ ez 1 þ pffiffiffi eu du . p 0 Rearranging Eq. (4.20) gives ! pffiffiffi pffi ( p ffiffi ffi 3 p t c G3=2 ðn; tÞ ¼ pffiffiffi ec erfc c þ 2ec=2 cos 2 3 3 c ! ) pffiffiffi pffiffiffi Z 1 4 c 3 2 cð1 v2 Þ dv ; þ pffiffiffi ecð1v Þ=2 cos 2 p 0 where c = a2/3n4/3t, and Z q 1 G3=2 ðn; tÞ cosðxnÞ dn. c¼ p 0
ð4:17Þ
ð4:18Þ
ð4:19Þ
ð4:20Þ
ð4:21Þ
ð4:22Þ
ð4:23Þ
It follows from Eqs. (3.12) and (3.13) that the non-zero components of the stress tensor are: ryy ¼ rzz ¼ 2lmc.
ð4:24Þ
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It should be noted that the function ez erfc z has the asymptotic expansion [57]
1 1 1 3 2 3þ 5 ; ez erfc z pffiffiffi 4z p z 2z
ð4:25Þ
valid for z ! 1, jarg zj < 3p/4. The latter condition is fulfilled by jarg zij, i = 1, 2, 3, where zi are introduced by (4.19). As 1 1 1 þ e2pi=3 þ e2pi=3 ¼ 0; z1 z2 z3
1 1 1 þ e2pi=3 3 þ e2pi=3 3 ¼ 0. 3 z1 z2 z3
It follows from (4.18) and (4.25) that: 3 G3=2 ðn; tÞ pffiffiffi 5=2 4 4 pa2 t n
for
n ! 1.
ð4:26Þ
Hence, the integral (4.23) is convergent for all x including x = 0 and the concentration c and the components of the stress tensor ryy and rzz have no singularity at x = 0. We also present the solution for the limiting case a = 2. From Eq. (4.1) we obtain q 1 c ¼ pffiffiffiffiffiffi 2 2p s þ an2 and after inverting the Laplace and Fourier transforms [58], pffiffiffi q c ¼ pffiffiffi H ð at jxjÞ; 2 a
ð4:27Þ
ð4:28Þ
where H(x) is the Heaviside function. Distributions of non-dimensional concentration and stress are shown in Fig. 1 for various values of a.
Fig. 1. Dependence of concentration (or stress) on the similarity variable.
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5. Solution in axially symmetric case q In an axially symmetric case of two variables, Eq. (3.3) with Qðx; tÞ ¼ 2pr dðrÞdðtÞ has the form
2 a oc o c 1 oc q ¼a þ þ dðrÞdðtÞ. ð5:1Þ a 2 ot or r or 2pr
The Laplace transform with respect to time t and the Hankel transform with respect to polar coordinate r will be employed. The non-zero components of the stress tensor expressed in terms of displacement potential U read rzz ¼ rrr þ rhh ¼ 2lDU;
2 o U 1 oU ; rrr rhh ¼ 2l or2 r or
ð5:2Þ
or taking into account Eq. (3.13) ð0Þ
ð0Þ hh ¼ 2lmcð0Þ ; ð0Þ r zz ¼ r rr þ r ð2Þ
hh ¼ 2lmcð0Þ ; ð2Þ r rr r
ð5:3Þ
where a bar marks the Hankel transform of the order indicated by a number in parentheses. We consider two particular cases of Eq. (5.1) corresponding to a = 1/2 and a = 3/2. Similarly to a method used for a = 1/2 in one-dimensional case, we obtain Z 1 q G1=2 ðn; tÞJ 0 ðrnÞn dn c¼ 2p 0 Z 1 pffi q 1 2 2 2 4 pffiffiffiffiffi an expða n tÞ erfc ðan tÞ J 0 ðrnÞn dn; ð5:4Þ ¼ 2p 0 pt where Jn(r) is the Bessel function of the first kind of the order n. In the subsequent text we shall use the following formulae [56]:
2 Z 1 1 r2 r 3 bn2 e J 0 ðrnÞn dn ¼ 2 1 exp ; ð5:5Þ 4b 4b 2b 0
2 Z 1 r2 r 3 bn2 e J 2 ðrnÞn dn ¼ 3 exp . ð5:6Þ 4b 8b 0 After some rearranging expression (5.4) taking into account (5.5) we arrive at
2 Z 1 q q2 u qu 1=u2 ð1 e Þ 1 c ¼ 3=2 exp du; 8p at 0 8 8
ð5:7Þ
where the similarity variable q = a1/2t1/4r has been chosen. Having the solution (5.7) for the concentration c and making use of (5.6) we obtain from (5.3) that
2 Z 1 q qu 1=u2 rrr ¼ lm ð1 e Þ exp du; 8atp3=2 0 8
2 ð5:8Þ Z 1 q q2 u qu 1=u2 ð1 e Þ 1 rhh ¼ lm exp du. 8atp3=2 0 4 8
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Fig. 2. Dependence of concentration on the similarity variable.
Fig. 3. Dependence of rrr on the similarity variable.
Y.Z. Povstenko / International Journal of Engineering Science 43 (2005) 977–991
For a = 3/2 we have Z 1 q G3=2 ðn; tÞJ 0 ðrnÞn dn; c¼ 2p 0 Z q 1 rrr þ rhh ¼ lm G3=2 ðn; tÞJ 0 ðrnÞn dn; p 0 Z q 1 G3=2 ðn; tÞJ 2 ðrnÞn dn; rrr rhh ¼ lm p 0 where G3/2(n, t) is the same as in one-dimensional case and is defined by (4.22). For completeness we also present the classical solution [54] for a = 1
2 q q exp ; c¼ 4pat 4
Fig. 4. Dependence of rhh on the similarity variable.
987
ð5:9Þ
ð5:10Þ
ð5:11Þ
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2 q q rrr ¼ lm 1 exp ; 2 patq 4
2 q q2 q 1þ exp 1 ; rhh ¼ lm patq2 2 4
ð5:12Þ
where q = a1/2t1/2r, and the solution for the limiting case a = 2 c¼
q H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pat 1 q2
ð5:13Þ
q H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; pat 1 q2 " # q 2 q2 H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; rrr rhh ¼ lm H ðq 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pat q2 1 q2 ð1 þ 1 q2 Þ rrr þ rhh ¼ lm
ð5:14Þ
where q = a1/2t1r. In the frame-work of a thermoelasticity without energy dissipation proposed by Green and Naghdi [59] and corresponding to the wave equation for the temperature (a = 2) the similar problems were considered by Chandrasekharaiah and Srinath [60,61]. Distributions of concentration and stresses exerted by a source of diffusion are shown in Figs. 2–4 for various values of a.
6. Concluding remarks 1. The theory of diffusive strain based on the time-fractional diffusion-wave equation describing the anomalous mass transfer has been proposed. The cases 0 < a < 1, a = 1, 1 < a < 2 and a = 2 correspond to subdiffusion (weak diffusion), normal diffusion, superdiffusion (strong diffusion) and ballistic diffusion, respectively. 2. The stresses corresponding to the fundamental solutions to the source problem for the timefractional diffusion-wave equation have been found in one-dimensional and axially symmetric cases (for plane deformation). 3. According to (4.26) the integrals in (5.9) and (5.10) are convergent for all r P 0. In axially symmetric case the fundamental solution to the source problem, and therefore the components of the stress tensor, have no singularity at the origin r = 0. It can be calculated from (5.7), (5.9), (5.11) and (5.13) that the non-dimensional concentration c0 = atc/q has the following values at the origin r = 0 a ¼ 1=2 : a ¼ 3=2 :
1 ; 8p 3 c0 ð0Þ ¼ ; 8p
c0 ð0Þ ¼
a¼1: a¼2:
2 ; 8p 4 c0 ð0Þ ¼ . 8p c0 ð0Þ ¼
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The situation differs radically from that in the case of the Cauchy problem for the considered time-fractional equation [53], where the concentration and stresses for a = 1/2 and a = 3/2 are divergent at r = 0 and have the logarithmic singularity.
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[25] S. Havlin, D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 36 (1987) 695–798. [26] H. Berens, U. Westphal, A Cauchy problem for a generalized wave equation, Acta Sci. Math. (Szeged) 29 (1968) 93–106. [27] A.N. Kochubei, Fractional order diffusion, Differen. Equat. 26 (1990) 485–492. [28] A. Hanyga, Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. A 458 (2002) 933–957. [29] A. Hanyga, Multidimensional solutions of space-fractional diffusion equations, Proc. R. Soc. Lond. A 457 (2001) 2993–3005. [30] A. Hanyga, Multidimensional solutions of space-time-fractional diffusion equations, Proc. R. Soc. Lond. A 458 (2002) 429–450. [31] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. [32] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993. [33] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. [34] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. [35] M.M. Dzrbashyan, A.B. Nersesyan, On the use of some integrodifferential operators, Dokl. Acad. Sci. USSR 121 (1958) 210–213 (in Russian). [36] M.M. Dzrbashyan, A.B. Nersesyan, Fractional derivatives and the Cauchy problem for differential equation of fractional order, Izv. Acad. Sci. Armenian SSR 3 (1968) 3–29 (in Russian). [37] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Part II, Geophys. J. Roy. Astron. Soc. 13 (1967) 529–539. [38] M. Caputo, Elasticita` e Dissipazione, Zanichelli, Bologna, 1969 (in Italian). [39] R. Gorenflo, F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech. 50 (1998) 377–388. [40] F. Mainardi, R. Gorenflo, On Mittag–Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. 118 (2000) 283–299. [41] Ya.S. Podstrigach, Yu.M. Kolyano, Generalized Thermomechanics, Naukova Dumka, Kiev, 1976 (in Russian). [42] N. Petrov, G. Brankov, Modern Problems of Thermodynamics, Mir, Moscow, 1986 (in Russian). [43] D.S. Chandrasekharaiah, Thermoelasticity with second sound: a review, Appl. Mech. Rev. 39 (1986) 355–376. [44] D.S. Chandrasekharaiah, Hiperbolic thermoelasticity: a review of recent literature, Appl. Mech. Rev. 51 (1998) 705–729. [45] K.K. Tamma, X. Zhou, Macroscale and microscale thermal transport and thermomechanical interaction: some noteworthy perspectives, J. Therm. Stresses 21 (1998) 405–449. [46] R.B. Hetnarski, J. Ignaczak, Generalized thermoelasticity, J. Therm. Stresses 22 (1999) 451–476. [47] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968) 113–126. [48] J.W. Nunziato, On heat conduction in materials with memory, Q. Appl. Math. 29 (1971) 187–204. [49] F.R. Norwood, Transient thermal waves in the general theory of heat conduction with finite wave speeds, J. Appl. Mech. 39 (1972) 673–676. [50] T.B. Moodi, R.J. Tait, On thermal transients with finite wave speeds, Acta Mech. 50 (1983) 97–104. [51] C. Cattaneo, Sur une forme de lÕe´quation de la chaleur e´liminant le paradoxe dÕune propagation instantane´e, C.R. Acad. Sci. 247 (1958) 431–433. [52] P. Vernotte, Les paradoxes de la the´orie continue de lÕe´quation de la chaleur, C.R. Acad. Sci. 246 (1958) 3154– 3155. [53] Y.Z. Povstenko, Fractional heat conduction equation and associated thermal stresses, J. Therm. Stresses 28 (2005) 83–102. [54] W. Nowacki, Thermoelasticity, Polish Scientific Publishers, Warszawa, 1986. [55] A. Erde´lyi, W. Magnus, F. Oberhettinger, F. TricomiHigher Transcendental Functions, vol. 3, McGraw-Hill, New York, 1955. [56] I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980. [57] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972.
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[58] P.K. Kythe, Fundamental Solutions for Differential Operators and Applications, Birkha¨user, Boston, 1996. [59] A.E. Green, P.M. Naghdi, Thermoelasticity without energy dissipation, J. Elast. 31 (1993) 189–208. [60] D.S. Chandrasekharaiah, K.S. Srinath, Thermoelastic interactions without energy dissipation due to a line heat source, Acta Mech. 128 (1998) 243–251. [61] D.S. Chandrasekharaiah, K.S. Srinath, Thermoelastic interactions without energy dissipation due to a point heat source, J. Elast. 50 (1998) 97–108.
Stresses exerted by a source of diffusion in a case of a non-parabolic diffusion equation Y.Z. Povstenko Institute of Mathematics and Computer Science, Jan Długosz University of Czeßstochowa, al. Armii Krajowej 13/15, 42-200 Czeßstochowa, Poland Received 14 April 2004; received in revised form 5 March 2005; accepted 5 March 2005
Abstract The theory of diffusive stresses based on the diffusion-wave equation with time-fractional derivative of fractional order a is formulated. The non-parabolic diffusion equation is a mathematical model of a wide range of important physical phenomena and can be obtained as a consequence of the non-local constitutive equation for the matter flux vector with the long-tale power time-non-local kernel. Because the considered equation in the case 1 6 a 6 2 interpolates the parabolic equation (a = 1) and the wave equation (a = 2), the proposed theory interpolates a classical theory of diffusive stresses and that without energy dissipation introduced by Green and Naghdi. The stresses caused by a source of diffusion in an unbounded solid are found in one-dimensional and axially symmetric cases (for plane deformation). Numerical results for the concentration and stress distributions are given and illustrated graphically. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction The theory of diffusive strain is governed by a system of partial differential equations consisting of the equations of elasticity with gradient of diffusion taken into account and the balance equation of diffusion. The problem can be treated as an uncoupled in the case when the influence of the time change of strain on a flux of diffusion is neglected. The first theoretical investigation of E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2005.03.004
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diffusive strain dates back to Pidstryhach (spelled also Pidstrygach and Podstrigach) [1–4]. For additional references, further generalizations and discussion see [5–9]. The conventional theory of diffusive strain is based on the principles of the classical theory of diffusion, specifically on the classical Fick law J ¼ j grad c;
ð1:1Þ
where J is the matter flux, c is the concentration and j is the diffusion conductivity. In combination with the balance equation for mass the Fick law leads to the classical diffusion equation oc ¼ j Dc þ qQ; ð1:2Þ ot with q being the mass density, t the time, Q the mass source. In the past two decades, a considerable research efforts has been expended to study anomalous diffusion which is characterized by the time-fractional equation q
oa c ¼ j Dc þ qQ; ð1:3Þ ota where the parameter 0 < a < 2. Two types of anomalous transport can be distinguished. The subdiffusion regime is exemplified by the value 0 < a < 1, whereas the superdiffusion regime is characterized by the value 1 < a < 2. In subdiffusion, particles move slower, in superdiffusion particles move faster than in the ordinary diffusion. The limiting case a = 2 is known as ballistic diffusion [10]. Anomalous diffusive transport appears to be a universal experimental phenomenon. A number of works has been published dealing with anomalous transport in amorphous [11], colloid [12], glassy [13,14] and porous [10,15] materials, in fractals [16–18] and percolation clusters [15], comb structures [19], dielectrics and semiconductors [20,21], polymers [22,23], random [24] and disordered [25] media. Various mathematical aspects concerning the existence, uniqueness, well-posedness, Sommerfeld condition for non-parabolic diffusion equation are discussed in [26–30]. The purpose of this paper is to formulate the theory of diffusive strain of elastic solid based on the time-fractional diffusion-wave equation and to study the stresses in an unbounded solid exerted by a source of diffusion in one-dimensional and axially symmetric cases (for plane deformation). q
2. Preliminaries In this section we recall the main ideas of fractional calculus [31–34]. The notion of the Riemann–Liouville fractional integral is introduced as a natural generalization of the wellknown n-fold repeated integral Inf(t) written in a convolution type form Z t 1 a1 a I f ðtÞ ¼ ðt sÞ f ðsÞ ds; a > 0; ð2:1Þ CðaÞ 0 where C(a) is the gamma function.
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The Riemann–Liouville derivative of the fractional order a is defined by the following formula: Z t dn 1 na1 a ðt sÞ f ðsÞ ds ; n 1 < a < n. ð2:2Þ DRL f ðtÞ ¼ n dt Cðn aÞ 0 There are other possibilities to introduce fractional derivatives [35–38]. One of the alternative definitions of the fractional derivative was proposed by Caputo [37,38] Z t 1 dn f ðsÞ a ðt sÞna1 ds; n 1 < a < n. ð2:3Þ DC f ðtÞ ¼ Cðn aÞ 0 dsn The rules for the Laplace transforms of the Riemann–Liouville fractional integral and fractional derivative read LfI a f ðtÞg ¼
1 Lff ðtÞg; sa
ð2:4Þ
n1 X L DaRL f ðtÞ ¼ sa Lff ðtÞg Dk I na f ð0þ Þsn1k ;
n 1 < a < n;
ð2:5Þ
k¼0
where s is the transform variable. The Laplace transform rule for the Caputo derivative has the following form: n1 X L DaC f ðtÞ ¼ sa Lff ðtÞg f k ð0þ Þsa1k ;
n 1 < a < n.
ð2:6Þ
k¼0
The Caputo fractional derivative is a regularization in the time origin for the Riemann–Liouville fractional derivative by incorporating the relevant initial conditions [39]. In this paper we shall use the Caputo fractional derivative omitting the index C. The major utility of this type fractional derivative is caused by the treatment of differential equations of fractional order for physical applications, where the initial conditions are usually expressed in terms of a given function and its derivatives of integer (not fractional) order, even if the governing equation is of fractional order [34,40]. It is common knowledge that from mathematical viewpoint the Fick law in the theory of diffusion and the Fourier law in the theory of heat conduction are identical. During the last four decades, non-classical thermoelasticity theories in which the Fourier law and the heat equation are replaced by more general equations, have been formulated. For an extensive bibliography on this subject see [41–46]. Here, we mention the time-non-local constitutive equation for heat flux with a general non-local kernel [47–50] and with ‘‘short-tale’’ exponential kernel [43,51,52]. The time-fractional diffusion equation (1.3) corresponds to non-local constitutive equation for the matter flux J with the power time-non-local kernel describing ‘‘long-tale’’ memory.
3. Problem formulation The stressed–strained state of a solid is governed by the equilibrium equation in terms of displacements lDu þ ðk þ lÞ grad div u ¼ bc Kgrad c;
ð3:1Þ
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the stress–strain-concentration relation r ¼ 2le þ ðk tr e bc KcÞI
ð3:2Þ
and the time-fractional diffusion equation oa c ¼ aDc þ Qðx; tÞ; ð3:3Þ ota where u is the displacement vector, r the stress tensor, e the linear strain tensor, c the concentration, a the diffusivity coefficient, k and l are Lame´ constants, K = k + 2l/3, bc is the diffusion coefficient of volumetric expansion, I denotes the unit tensor. It is well known that a solution to the classical diffusion equation oc ¼ aDc þ Qðx; tÞ; ot with initial condition t¼0:
ð3:4Þ
c ¼ f ðxÞ;
ð3:5Þ
is expressed by the Poisson formula Z Z tZ cðx; tÞ ¼ Eðjx nj; tÞf ðnÞ dn þ Eðjx nj; t sÞQðn; sÞ dn ds; Rn
ð3:6Þ
Rn
0
where n is the dimension of a space ! 1 jxj2 Eðjxj; tÞ ¼ pffiffiffiffiffiffiffi n exp ; 4at ð2 patÞ
ð3:7Þ
is the fundamental solution, the same for both the Cauchy problem oc ¼ aDc; ot t ¼ 0 : c ¼ dðxÞ
ð3:8Þ ð3:9Þ
and the source problem oc ¼ aDc þ dðxÞdðtÞ. ot
ð3:10Þ
Here d(x) is DiracÕs delta. In the case of anomalous diffusion we have the following formula: Z tZ Z E1 ðjx nj; tÞf ðnÞ dn þ E2 ðjx nj; t sÞQðn; sÞ dn ds; cðx; tÞ ¼ Rn
0
Rn
instead of (3.6) and hence two different fundamental solutions E1 ðjxj; tÞ
and E2 ðjxj; tÞ;
corresponding to the Cauchy problem and to the source problem.
ð3:11Þ
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The fundamental solution to the Cauchy problem for anomalous diffusion equation and the stresses corresponding to it are discussed in the accompanying paper [53]. Here we restrict our consideration to the fundamental solution for the source problem. As in the case of classical thermoelasticity or the theory of diffusive strain we use the representation of stresses in terms of the displacement potential U [54]
2 o dij D U; ð3:12Þ rij ¼ 2l oxi oxj where dij is the Kronecker delta. The displacement potential is determined from the equation 1 þ m bc ; 1m 3 with m being the Poisson ratio. DU ¼ mc;
ð3:13Þ
m¼
4. Solution in one-dimensional case In one-dimensional case, Eq. (3.3) with Q(x, t) = qd(x)d(t) reads oa c o2 c ¼ a 2 þ qdðxÞdðtÞ; ð4:1Þ ota ox where q is the strength of mass source. Using the Laplace transform with respect to time (denoted by the asterisk) we obtain the ordinary differential equation for the concentration d2 c
sa c ¼ qdðxÞ; dx2 having the solution pffiffi a=2 q c ¼ pffiffiffi a=2 eðjxj= aÞs . 2 as
ð4:2Þ
a
ð4:3Þ
The inversion of the Laplace transform in (4.3) leads to the following expression: a q c ¼ pffiffiffi 1a=2 H z; ; 2 at 2
ð4:4Þ
where H(z; b) is the Hanyga auxiliary function of the similarity variable jxj z ¼ pffiffiffi b ; at
b ¼ a=2;
0 < b < 1;
ð4:5Þ
having the following series representation: H ðz; bÞ ¼
1 X n¼0
ð1Þn zn ; n!Cðbn þ bÞ
0 < b < 1.
ð4:6Þ
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The function H(z; b) is a particular case of the Wright function [55] 1 X zn W ðz; k; lÞ ¼ ; k > 1; l > 0; n!Cðkn þ lÞ n¼0
ð4:7Þ
corresponding to k = b, l = b and negative values of the first independent variable H ðz; bÞ ¼ W ðz; b; bÞ. It is obvious that in the case b = 1/2 the Gaussian auxiliary function is obtained
2 1 1 z H z; ¼ pffiffiffi exp . 2 4 p Other known particular cases correspond to the values b = 1/3 and b = 2/3 [28]
1 H z; ¼ 31=3 Ai0 ð31=3 zÞ; 3
2 2 3 2=3 4=3 2 z Þ exp z ; H z; ¼ 3 Aið3 3 27
ð4:8Þ
ð4:9Þ
ð4:10Þ ð4:11Þ
where Ai(x) is the Airy function, the prime denotes the derivative. We also present expressions for cases b = 1/4 (a = 1/2) and b = 3/4 (a = 3/2). On the one hand, all these expressions allow us to test the numerical algorithms for computing series representation of the auxiliary function, especially for z > 1. On the other hand, these expressions can be used to obtain the corresponding solutions in more complicated axially symmetric case. For this purpose we use the Fourier transform with respect to the spatial coordinate x (denoted by a bar) with n being the transform variable. For a = 1/2 we obtain q 1 c ¼ pffiffiffiffiffiffi pffiffi . 2p s þ an2 Inverting the Laplace transform yields pffi 1 1 2 2 1 2 4 p ffiffiffiffi ffi an G1=2 ðn; tÞ ¼ L tÞ; expða n tÞ erfc ðan ¼ pffiffi pt s þ an2
ð4:12Þ
ð4:13Þ
where 2 erfc x ¼ pffiffiffi p
Z
1
2
ev dv;
ð4:14Þ
x
is the complementary error function. Inverting the Fourier transform leads us to the integral representation of the solution Z q 1 c¼ G1=2 ðn; tÞ cosðxnÞ dn p 0 Z pffi q 1 1 2 2 2 4 pffiffiffiffiffi an expða n tÞ erfc ðan tÞ cosðxnÞ dn. ð4:15Þ ¼ p 0 pt
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983
pffi Inserting (4.14) into (4.15), using substitution v ¼ u2 þ an2 t and changing integration with respect to n and u we obtain
2 2 Z 1
q z 2 u2 zu 1=u4 1 1e exp du; ð4:16Þ c ¼ pffiffiffi pffiffiffi 3=4 4 8 2 2p at 0 where the following formula [56]:
2 pffiffiffi
Z 1 x2 x p 2 bn2 ne cosðxnÞ dn ¼ 3=2 1 exp ; 2b 4b 4b 0 has been used. In the case a = 3/2 we have 1 1 G3=2 ðn; tÞ ¼ L s3=2 þ an2 h 2 i 1 z1 2pi=3 z22 2pi=3 z23 ¼ e erfc z þ e e erfc z þ e e erfc z . 1 2 3 3a1=3 n2=3 Here
pffi z1 ¼ a1=3 tn2=3 ;
pffi z2 ¼ epi=3 a1=3 tn2=3 ;
pffi z3 ¼ epi=3 a1=3 tn2=3 .
Eq. (4.18) can be rewritten in terms of the function w(z) h
pffi 1 2=3 4=3 1=3 2=3 G3=2 ðn; tÞ ¼ t exp a n t erfc a n 3a1=3 n2=3 pffi pffi i þe2pi=3 w epi=6 a1=3 n2=3 t þ e2pi=3 w epi=6 a1=3 n2=3 t ; where w(z) is defined as follows [57]:
Z z 2i 2 2 2 wðzÞ ¼ ez erfc ðizÞ ¼ ez 1 þ pffiffiffi eu du . p 0 Rearranging Eq. (4.20) gives ! pffiffiffi pffi ( p ffiffi ffi 3 p t c G3=2 ðn; tÞ ¼ pffiffiffi ec erfc c þ 2ec=2 cos 2 3 3 c ! ) pffiffiffi pffiffiffi Z 1 4 c 3 2 cð1 v2 Þ dv ; þ pffiffiffi ecð1v Þ=2 cos 2 p 0 where c = a2/3n4/3t, and Z q 1 G3=2 ðn; tÞ cosðxnÞ dn. c¼ p 0
ð4:17Þ
ð4:18Þ
ð4:19Þ
ð4:20Þ
ð4:21Þ
ð4:22Þ
ð4:23Þ
It follows from Eqs. (3.12) and (3.13) that the non-zero components of the stress tensor are: ryy ¼ rzz ¼ 2lmc.
ð4:24Þ
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It should be noted that the function ez erfc z has the asymptotic expansion [57]
1 1 1 3 2 3þ 5 ; ez erfc z pffiffiffi 4z p z 2z
ð4:25Þ
valid for z ! 1, jarg zj < 3p/4. The latter condition is fulfilled by jarg zij, i = 1, 2, 3, where zi are introduced by (4.19). As 1 1 1 þ e2pi=3 þ e2pi=3 ¼ 0; z1 z2 z3
1 1 1 þ e2pi=3 3 þ e2pi=3 3 ¼ 0. 3 z1 z2 z3
It follows from (4.18) and (4.25) that: 3 G3=2 ðn; tÞ pffiffiffi 5=2 4 4 pa2 t n
for
n ! 1.
ð4:26Þ
Hence, the integral (4.23) is convergent for all x including x = 0 and the concentration c and the components of the stress tensor ryy and rzz have no singularity at x = 0. We also present the solution for the limiting case a = 2. From Eq. (4.1) we obtain q 1 c ¼ pffiffiffiffiffiffi 2 2p s þ an2 and after inverting the Laplace and Fourier transforms [58], pffiffiffi q c ¼ pffiffiffi H ð at jxjÞ; 2 a
ð4:27Þ
ð4:28Þ
where H(x) is the Heaviside function. Distributions of non-dimensional concentration and stress are shown in Fig. 1 for various values of a.
Fig. 1. Dependence of concentration (or stress) on the similarity variable.
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5. Solution in axially symmetric case q In an axially symmetric case of two variables, Eq. (3.3) with Qðx; tÞ ¼ 2pr dðrÞdðtÞ has the form
2 a oc o c 1 oc q ¼a þ þ dðrÞdðtÞ. ð5:1Þ a 2 ot or r or 2pr
The Laplace transform with respect to time t and the Hankel transform with respect to polar coordinate r will be employed. The non-zero components of the stress tensor expressed in terms of displacement potential U read rzz ¼ rrr þ rhh ¼ 2lDU;
2 o U 1 oU ; rrr rhh ¼ 2l or2 r or
ð5:2Þ
or taking into account Eq. (3.13) ð0Þ
ð0Þ hh ¼ 2lmcð0Þ ; ð0Þ r zz ¼ r rr þ r ð2Þ
hh ¼ 2lmcð0Þ ; ð2Þ r rr r
ð5:3Þ
where a bar marks the Hankel transform of the order indicated by a number in parentheses. We consider two particular cases of Eq. (5.1) corresponding to a = 1/2 and a = 3/2. Similarly to a method used for a = 1/2 in one-dimensional case, we obtain Z 1 q G1=2 ðn; tÞJ 0 ðrnÞn dn c¼ 2p 0 Z 1 pffi q 1 2 2 2 4 pffiffiffiffiffi an expða n tÞ erfc ðan tÞ J 0 ðrnÞn dn; ð5:4Þ ¼ 2p 0 pt where Jn(r) is the Bessel function of the first kind of the order n. In the subsequent text we shall use the following formulae [56]:
2 Z 1 1 r2 r 3 bn2 e J 0 ðrnÞn dn ¼ 2 1 exp ; ð5:5Þ 4b 4b 2b 0
2 Z 1 r2 r 3 bn2 e J 2 ðrnÞn dn ¼ 3 exp . ð5:6Þ 4b 8b 0 After some rearranging expression (5.4) taking into account (5.5) we arrive at
2 Z 1 q q2 u qu 1=u2 ð1 e Þ 1 c ¼ 3=2 exp du; 8p at 0 8 8
ð5:7Þ
where the similarity variable q = a1/2t1/4r has been chosen. Having the solution (5.7) for the concentration c and making use of (5.6) we obtain from (5.3) that
2 Z 1 q qu 1=u2 rrr ¼ lm ð1 e Þ exp du; 8atp3=2 0 8
2 ð5:8Þ Z 1 q q2 u qu 1=u2 ð1 e Þ 1 rhh ¼ lm exp du. 8atp3=2 0 4 8
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Fig. 2. Dependence of concentration on the similarity variable.
Fig. 3. Dependence of rrr on the similarity variable.
Y.Z. Povstenko / International Journal of Engineering Science 43 (2005) 977–991
For a = 3/2 we have Z 1 q G3=2 ðn; tÞJ 0 ðrnÞn dn; c¼ 2p 0 Z q 1 rrr þ rhh ¼ lm G3=2 ðn; tÞJ 0 ðrnÞn dn; p 0 Z q 1 G3=2 ðn; tÞJ 2 ðrnÞn dn; rrr rhh ¼ lm p 0 where G3/2(n, t) is the same as in one-dimensional case and is defined by (4.22). For completeness we also present the classical solution [54] for a = 1
2 q q exp ; c¼ 4pat 4
Fig. 4. Dependence of rhh on the similarity variable.
987
ð5:9Þ
ð5:10Þ
ð5:11Þ
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2 q q rrr ¼ lm 1 exp ; 2 patq 4
2 q q2 q 1þ exp 1 ; rhh ¼ lm patq2 2 4
ð5:12Þ
where q = a1/2t1/2r, and the solution for the limiting case a = 2 c¼
q H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2pat 1 q2
ð5:13Þ
q H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; pat 1 q2 " # q 2 q2 H ð1 qÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; rrr rhh ¼ lm H ðq 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pat q2 1 q2 ð1 þ 1 q2 Þ rrr þ rhh ¼ lm
ð5:14Þ
where q = a1/2t1r. In the frame-work of a thermoelasticity without energy dissipation proposed by Green and Naghdi [59] and corresponding to the wave equation for the temperature (a = 2) the similar problems were considered by Chandrasekharaiah and Srinath [60,61]. Distributions of concentration and stresses exerted by a source of diffusion are shown in Figs. 2–4 for various values of a.
6. Concluding remarks 1. The theory of diffusive strain based on the time-fractional diffusion-wave equation describing the anomalous mass transfer has been proposed. The cases 0 < a < 1, a = 1, 1 < a < 2 and a = 2 correspond to subdiffusion (weak diffusion), normal diffusion, superdiffusion (strong diffusion) and ballistic diffusion, respectively. 2. The stresses corresponding to the fundamental solutions to the source problem for the timefractional diffusion-wave equation have been found in one-dimensional and axially symmetric cases (for plane deformation). 3. According to (4.26) the integrals in (5.9) and (5.10) are convergent for all r P 0. In axially symmetric case the fundamental solution to the source problem, and therefore the components of the stress tensor, have no singularity at the origin r = 0. It can be calculated from (5.7), (5.9), (5.11) and (5.13) that the non-dimensional concentration c0 = atc/q has the following values at the origin r = 0 a ¼ 1=2 : a ¼ 3=2 :
1 ; 8p 3 c0 ð0Þ ¼ ; 8p
c0 ð0Þ ¼
a¼1: a¼2:
2 ; 8p 4 c0 ð0Þ ¼ . 8p c0 ð0Þ ¼
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989
The situation differs radically from that in the case of the Cauchy problem for the considered time-fractional equation [53], where the concentration and stresses for a = 1/2 and a = 3/2 are divergent at r = 0 and have the logarithmic singularity.
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