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Jump relations and discontinuity waves in conductors with memory
Mathematical and Computer Modelling 43 (2006) 138–149 www.elsevier.com/locate/mcm
Jump relations and discontinuity waves in conductors with memory A. Morro ∗ University of Genoa, DIBE, Via Opera Pia 11a, 16145, Italy Received 11 April 2005; accepted 27 April 2005
Abstract This paper investigates a linear functional of the history of a variable which, along with the derivatives, suffers jump discontinuities. The first purpose is to evaluate the jump of the time and space derivatives, of any order, of the functional. Next, the modelling of conductors with memory is considered, in terms of the history of the temperature gradient, and necessary and sufficient thermodynamic restrictions are established. The propagation and evolution of one-dimensional waves is then investigated. The thermodynamic restrictions are shown to guarantee the finiteness of the wave speed and the decay of the discontinuity. c 2005 Elsevier Ltd. All rights reserved.
Keywords: Jump relations; Materials with memory; Discontinuity waves
1. Introduction The model of materials with fading memory is often applied to describe memory properties by letting the response be given by functionals of the whole history of appropriate variables. The dependence on the history results in remarkable consequences on the propagation of discontinuity waves in connection with both the existence of waves and the time evolution of the discontinuities. When dealing with memory functionals, the derivation of jump relations, even for low-order derivatives, is technically quite involved (see [1]). Mathematical difficulties arise because of the nonlinearity and of the dependence on the history. The purpose of this paper is twofold. The first fold is to investigate a linear functional such that g(z, ·) on (−∞, t) is mapped into f (z, t) in the form Z ∞ h(ξ )g(z, t − ξ )dξ (1) f (z, t) = 0
where h is continuously differentiable on R+ , whereas g(z, ·) and its derivatives suffer jump discontinuities. We aim to evaluate the jump of the derivatives ∂tk f, ∂zk f , of any order k. The literature provides low-order derivatives of (stress) nonlinear functionals [2,1]. Here, more compact and definite results for (1) are likely to arise thanks to the linearity of the functional. It is really so, but the calculation is non-trivial, in that g is not continuous and hence an appropriate procedure is in order. The second fold is to apply the results and examine the propagation of one-dimensional waves ∗ Tel.: +39 10 3532786; fax: +39 10 3532777.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2005.04.016
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in conductors with memory. This application is also motivated by the long-time interest in the modelling of heat conductors compatible with discontinuity waves (see, e.g., [3–5]). In this sense it is worth remarking that the wellknown generalized Fourier (or Cattaneo–Maxwell) law amounts to a particular case of the relation (1) where g is the temperature gradient and f is the heat flux [6]. The dependence of h on ξ ∈ R+ , but not on z, reflects the assumption that the material under investigation is homogeneous. This assumption in turn implies that the speed of propagation of discontinuities is constant, which simplifies the expressions of ∂tk f and ∂zk f . Concerning the model of heat conduction, we let the heat flux be a linear functional of the history of the temperature gradient, whereas the internal energy is a function of the temperature. It is then shown that an appropriate condition on the kernel, of the heat flux, is necessary and sufficient for the validity of the second law of thermodynamics. Next, we investigate the propagation condition and the evolution of temperature-rate discontinuity waves. It follows that the thermodynamic restrictions guarantee the finiteness of the wave speeds and the decay of the discontinuity. As a useful connection, the widely applied model involving summed histories is contrasted with the present model based on the histories. This paper shows that summed histories are not necessary to obtain finite wave speeds, and that the two models prove equivalent in linear approximations and are qualitatively different if nonlinear constitutive functionals are considered. 2. Jump relations for a linear functional For later purposes we consider functions g(z, t) on Ω × R, where Ω is an interval in R, Ω ⊆ R (see Fig. 1). The functions g are allowed to suffer a jump discontinuity at the line γ : z = Z (t), where Z (t) is invertible, and to be of class C n outside γ . To fix ideas, let (z 0 , t0 ), z 0 = Z (t0 ) be a point of the line. We consider the functional mapping g to R so that a function f on Ω × R is defined in the form (1) where h ∈ C n−1 on R+ . Denote by [g](z 0 , t0 ) = g(z 0 , t0+ ) − g(z 0 , t0− ) = g(z 0− , t0 ) − g(z 0+ , t0 ) the jump of g at (z 0 , t0 ). If t > t0 then g(z 0 , ·), on (−∞, t), suffers a jump at t0 . If t < t0 then g(z 0 , ·) is continuously differentiable on (−∞, t). We now investigate the differentiability of f . Let z = z 0 , t > t0 . Since g is discontinuous at t0 , the integrand is discontinuous (at ξ = t −t0 ). It is then convenient to write the function f in the form Z t−t0 Z ∞ f (z 0 , t) = h(ξ )g(z 0 , t − ξ )dξ + h(ξ )g(z 0 , t − ξ )dξ. (2) 0
t−t0
Since g(z 0 , ·) is continuously differentiable in (−∞, t0 ) ∪ (t0 , t), then f is differentiable and Z t−t0 h(ξ )∂tg(z 0 , t − ξ )dξ + h(t − t0 )g(z 0 , t0+ ) ∂t f (z 0 , t) = 0 Z ∞ + h(ξ )∂t g(z 0 , t − ξ )dξ − h(t − t0 )g(z 0 , t0− ). t−t0
Hence we can write Z ∂t f (z 0 , t) =
t−t0
h(ξ )∂t g(z 0 , t − ξ )dξ +
0
Z
∞
h(ξ )∂t g(z 0 , t − ξ )dξ + h(t − t0 )[g](z 0 , t0 ), t−t0
where the sum of the two integrals is the integral on R+ . Since h is differentiable and ∂t g(z 0 , ·) is continuously differentiable on (−∞, t0 ) ∪ (t0 , t), then ∂t f (z 0 , t) is differentiable too and it follows that Z ∞ Z t−t0 ∂t2 f (z 0 , t) = h(ξ )∂t2 g(z 0 , t − ξ )dξ + h(ξ )∂t2 g(z 0 , t − ξ )dξ 0
t−t0
+ h(t − t0 )[∂t g](z 0 , t0 ) + h (t − t0 )[g](z 0 , t0 ) 0
where
h0
is the derivative of h. Iteration of the procedure up to the n-th order derivative shows that Z ∞ n−1 X (n−1−k) ∂tn f (z 0 , t) = h(ξ )∂tn g(z 0 , t − ξ )dξ + h (k) (t − t0 )[∂t g](z 0 , t0 ), t > t0 . 0
k=0
(3)
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Fig. 1. At z = z 0 , if t > t0 then g(z, ·) on (−∞, t) is discontinuous at t0 ; if t < t0 then g(z, ·) is continuously differentiable.
If, instead, t < t0 then the integrand in (2) is of class C n and we have Z ∞ ∂tn f (z 0 , t) = h(ξ )∂tn g(z 0 , t − ξ )dξ, t < t0 .
(4)
0
Letting t → t0+ , by (3) we obtain ∂tn f (z 0 , t0+ ) =
∞
Z
h(ξ )∂tn g(z 0 , t0 − ξ )dξ +
0
n−1 X
(n−1−k)
h (k) (0)[∂t
g](z 0 , t0 ).
(5)
k=0
By (4), we have ∂tn f (z 0 , t0− ) =
∞
Z 0
h(ξ )∂tn g(z 0 , t0 − ξ )dξ.
(6)
Hence we obtain the jump of ∂tn f by simply subtracting (6) from (5). We find that [∂tn f ] =
n−1 X
(n−1−k)
h (k) (0)[∂t
g],
n ∈ N,
(7)
k=0 (n−k+1)
where the jumps of ∂tn f and ∂t g are evaluated at the same point (z 0 , t0 ) of γ . If n = 0, the relation (7) holds in the trivial form [ f ] = 0. We now examine partial differentiation with respect to the first argument z. For any value of z ∈ Ω , let t0 = Z −1 (z) so that (z, t0 ) ∈ γ . Also, let dt0 /dz = 1/U so that U is the speed of the point on γ . Of course, we regard t as fixed. First let t > t0 and write Z ∞ Z t−t0 f (z, t) = h(ξ )g(z, t − ξ )dξ + h(ξ )g(z, t − ξ )dξ (8) 0
t−t0
where it is understood that t0 is a function of z. The integrand is continuously differentiable (with respect to z) as ξ ∈ (0, t − t0 (z)) ∪ (t − t0 (z), ∞) and t0 depends on z (see Fig. 1). We find that Z t−t0 1 ∂z f (z, t) = h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )g(z, t0+ ) U 0 Z ∞ 1 + h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )g(z, t0− ) U t−t0
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whence ∂z f (z, t) =
t−t0
Z
h(ξ )∂z g(z, t − ξ )dξ +
0
∞
Z
h(ξ )∂z g(z, t − ξ )dξ − t−t0
1 h(t − t0 )[g](z, t0 ). U
So, differentiation with respect to z of the integral provides −1/U times the jump of the integrand at ξ = t − t0 . Hence further differentiation gives Z ∞ Z t−t0 h(ξ )∂z2 g(z, t − ξ )dξ h(ξ )∂z2 g(z, t − ξ )dξ + ∂z2 f (z, t) = 0
t−t0
1 1 − h(t − t0 )[∂z g](z, t0 ) + ∂z {h(t − t0 )[g](z, t0 )}. U U Upon n times differentiation, we have ∂zn
f (z, t) =
∞
Z 0
h(ξ )∂zn g(z, t − ξ )dξ −
n−1 1 X ∂ k {h(t − t0 )[∂zn−1−k g]}. U k=0 z
(9)
This relation holds for n = 1, 2. If the relation holds for n, then the application of ∂z to ∂zn f produces −
n−1 1 X 1 h(t − t0 )[∂zn g](z, t0 ) − ∂ k+1 {h(t − t0 )[∂zn−1−k g](z, t0 )} U U k=0 z
=−
n 1 1 X h(t − t0 )[∂zn g](z, t0 ) − ∂ k {h(t − t0 )[∂zn−k g](z, t0 )} U U k=1 z
=−
n 1 X ∂ k {h(t − t0 )[∂zn−k g](z, t0 )}. U k=0 z
This shows that the validity of (9) for ∂zn implies the validity for ∂zn+1 . This in turn proves the validity for any n ∈ N. Denote by δ/δt the operator δ = ∂t + U ∂z |t δt where ∂z |t means partial differentiation at constant t. Now it is convenient to observe that, owing to the dependence of t0 on z, ∂z [g](z, t0 ) = ∂z |t [g](z, t0 ) + ∂t [g](z, t0 )
δ[g] 1 = (z, t0 ) U δt
and hence δ[∂ n−1−k g] 1 0 1 h (t − t0 )[∂zn−1−k g](z, t0 ) + h(t − t0 ) z (z, t0 ) U U δt and so on for higher-order differentiations. We obtain k δ k− j [∂zn−1−k g] 1 X k k n−1−k ∂z {h(t − t0 )[∂z g](z, t0 )} = k (−1) j h ( j) (t − t0 ) (z, t0 ). U j=0 j δt k− j ∂z {h(t − t0 )[∂zn−1−k g](z, t0 )} = −
If, instead, t < t0 then g(z, ·) is continuously differentiable and we have Z ∞ h(ξ )∂zn g(z, t − ξ )dξ. ∂zn f (z, t) =
(10)
0
Let t → t0+ in (9) and t → t0− in (10). The difference ∂zn f (z, t0+ ) − ∂zn f (z, t0− ) yields [∂zn f ](z, t) = −
n−1 k δ k− j [∂zn−1−k g] 1 X 1 X j ( j) (−1) h (0) (z, t). U k=0 U k j=0 δt k− j
(11)
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The differentiation operators ∂z and ∂t do not commute. Let t > t0 . Time differentiation of Z ∞ Z t−t0 1 h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )[g](z, t0 ) h(ξ )∂z g(z, t − ξ )dξ + ∂z f (z, t) = U t−t0 0 gives ∂t ∂z f (z, t) =
∞
Z
h(ξ )∂t ∂z g(z, t − ξ )dξ + h(t − t0 )[∂z g](z, t0 ) − 0
1 0 h (t − t0 )[g](z, t0 ). U
In the reverse order, we have Z ∞ 1 h(ξ )∂z ∂t g(z, t − ξ )dξ − h(t − t0 )[∂t g](z, t0 ) + ∂z {h(t − t0 )[g](z, t0 )} ∂z ∂t f (z, t) = U 0 where ∂z {h(t − t0 )[g](z, t0 )} = −
1 δ[g] 1 0 h (t − t0 )[g](z, t0 ) + h(t − t0 ) (z, t0 ). U U δt
Hence, letting t → t0+ and subtracting the limit as t → t0− of the corresponding expressions for t < t0 , we obtain 1 0 h (0)[g](z, t), U 1 1 1 δ[g] [∂z ∂t f ](z, t) = − h(0)[∂t g](z, t) − h 0 (0)[g](z, t) + h(0) (z, t). U U U δt [∂t ∂z f ](z, t) = h(0)[∂z g](z, t) −
(12) (13)
This proves that ∂z and ∂t do not commute. Later on, we show that if [g] = 0 then [∂t g] = −U [∂z g]. As a consequence, if [g] = 0 then ∂z ∂t f = ∂t ∂z f . The results (7) and (11), for any n ∈ N, appear to be new in the literature, where calculations are usually performed by having recourse to low-order partial time derivatives. This section shows that partial space differentiation is decisively more involved. 3. Heat conductor with memory The simplest model of heat conductor is that of a rigid body occupying a region Ω . The pertinent fields are the absolute temperature θ, the internal energy (per unit mass) , and the heat flux vector q. A possible energy supply is described by r . All fields depend on the position x ∈ Ω and the time t ∈ R. The balance of energy is written as ˙ = −∇ · q + r
(14)
where ∇· is the divergence operator. Since the body is undeformable, total time differentiation, denoted by a superposed dot, coincides with partial time differentiation ∂t . The constitutive properties of the conductor are modelled by saying that, at each point x ∈ Ω , the internal energy is a function of the temperature, whereas the heat flux is a linear functional of the history of the temperature gradient. We write Z ∞ = (θ ), q(t) = −κ0 ∇θ (t) − κ(ξ )∇θ t (ξ )dξ (15) 0
where κ is a bounded function on θ t (ξ ) = θ (t − ξ ),
R+ .
The symbol θ t stands for the history of θ up to time t:
ξ ∈ R+ .
According to (15), the dependence of q on ∇θ t is chosen to be linear. The dependence (of , θ, κ, q) on the position x is understood and not written.
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3.1. Thermodynamic restrictions Let η be the entropy density. The second law of thermodynamics is expressed by the inequality Z d (η˙ + ∇ · (q/θ ) − r/θ )dt ≥ 0
(16)
0
to hold at any point x ∈ Ω for every set of periodic functions (η, q, θ, r ) in time with period d. By means of (14), the inequality (16) becomes Z d 1 ˙ η˙ − − 2 q · ∇θ dt ≥ 0. θ θ 0 Because of the periodic character, Z d ˙ η˙ − dt = 0. θ 0 Hence we are left with Z d 1 q · ∇θ dt ≤ 0. θ2 0 This condition holds if and only if Z d q · ∇θdt ≤ 0.
(17)
0
For, letting θ be constant at the point under consideration but ∇θ be unrestricted, we obtain (17). Conversely, if (17) holds, then Z d Z d 1 1 q · ∇θ dt ≤ (q · ∇θ )dt ≤ 0 2 θ2 θM 0 0 where θ M is the maximum value of θ in [0, d]. Consequently, the model (15) is compatible with thermodynamics if and only if (17) holds. Denote by κc the half-range cosine transform of κ, namely Z ∞ κc (ω) = κ(ξ ) cos ωξ dξ. 0
The thermodynamic restrictions on the model (15) are stated as follows. Proposition. The model (15) is compatible with thermodynamics if and only if κ0 + κc (ω) ≥ 0,
ω ∈ R.
(18)
Proof. To show that the condition (18) is necessary, we choose the function ∇θ as ∇θ (t) = gˆ cos ωt + g˜ sin ωt where ω ∈ R and gˆ , g˜ are arbitrary vectors. Upon substitution in (15), we find that q(t) · ∇θ (t) = −(κ0 + κc )ˆg2 cos2 ωt − (κ0 + κc )˜g2 sin2 ωt + κs gˆ · g˜ (sin2 ωt − cos2 ωt) − 2(κ0 + κc )ˆg · g˜ cos ωt sin ωt + κs (ˆg2 − g˜ 2 ) cos ωt sin ωt where κs is the half-range sine transform of κ. Hence q · ∇θ is periodic with period d = π/|ω|. Integration over [0, π/|ω|] yields Z π/|ω| π (κ0 + κc )(ˆg2 + g˜ 2 ). 0≥ q(t) · ∇θ (t)dt = − 2|ω| 0 Hence it follows that (18) holds.
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A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
To show that (18) is sufficient, we let ∇θ be periodic with period d and write it as a Fourier series in the form ∇θ (t) =
∞ X
gˆ k cos kωt + g˜ k sin kωt
k=0
where ω = 2π/d. Hence we obtain q(t) · ∇θ (t) = −κ0 −
∞ X
(ˆgk cos kωt + g˜ k sin kωt) · (ˆgh cos hωt + g˜ h sin hωt)
k,h=0 ∞ X
{[ˆgk (κc (kω) cos kωt + κs (kω) sin kωt) + g˜ k (κc (kω) sin kωt − κs (kω) cos kωt)]
h,k=0
· (ˆgh cos hωt + g˜ h sin hωt)}. Integration over [0, 2π/ω] yields Z 2π/ω ∞ πX (κ0 + κc (kω))(ˆg2k + g˜ 2k ). q(t) · ∇θ (t)dt = − ω k=0 0 The condition (18) then implies (17).
As we see in a moment, wave propagation is allowed only if κ0 = 0. That is why we let κ0 = 0, and hence (18) becomes κc (ω) ≥ 0,
ω ∈ R.
(19)
3.2. Restrictions on the initial value of κ and κ 0 We preliminarily establish a relation between κc and κ. Let κ ∈ L 1 (R+ ) and define a function κ on R by letting κ(ξ ), ξ ≥ 0, κ(ξ ) = κ(−ξ ), ξ < 0. Hence the Fourier transform κ F of κ, Z ∞ κ F (ω) = κ(ξ ) exp(−iωξ )dξ −∞
gives κ F (ω) = 2κc (ω). Since κc is even, κc (ω) = κc (−ω), the inverse Fourier transform Z ∞ 1 κ(ξ ) = κ F (ω) exp(iωξ )dω 2π −∞ yields κ(ξ ) =
2 π
Z
∞
κc (ω) cos ωξ dω.
0
The restriction to ξ ≥ 0 gives Z 2 ∞ κ(ξ ) = κc (ω) cos ωξ dω. π 0
(20)
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
It follows at once from (20) and (19) that Z 2 ∞ κc (ω)dω > 0. κ(0) = π 0
145
(21)
Equality corresponds to κ = 0, and hence is excluded. A further relation holds if κ, κ 0 , κ 00 ∈ L 1 (R+ ). An integration by parts gives Z ∞ Z ∞ κ(ξ ) cos ωξ dξ κ 0 (ξ ) sin ωξ dξ = κ(ξ ) sin ωξ |∞ −ω κs0 (ω) = 0 0
0
whence κs0 (ω) = −ωκc (ω). Furthermore, Z ∞ Z ω κ 0 (ξ ) sin ωξ dξ = −κ 0 (ξ ) cos ωξ |∞ + 0 0
∞
κ 00 (ξ ) cos ωξ dξ 0
whence ωκs0 (ω) = κ 0 (0) + κc00 (ω). Upon evaluating the limit as ω → ∞ and applying the Riemann–Lebesgue lemma to κ 00 , we obtain κ 0 (0) = − lim ω2 κc (ω). ω→∞
Since κc (ω) ≥ 0 on R, we obtain κ 0 (0) ≤ 0.
(22)
Finally, we let the specific heat θ = d/dθ be positive, θ > 0.
(23)
In the next section we examine the consequences of (21)–(23) on wave propagation. Remark. A more general model is obtained by letting (t) = ˆ (θ (t), θ t ). In such a case the requirement (16) of the second law results in the inequality Z d ˙ 1 + 2 q · ∇θ dt ≤ 0. θ θ 0
(24)
The exploitation of (24) is generally a hard task. To fix ideas, by way of example we might keep q as in (15) and take Z ∞ p t (t) = ˜ (θ (t)) + ν(ξ )θ (ξ )dξ (25) 0
where p is a positive real. Hence (24) is equivalent to two separate inequalities # Z ∞ p−1 Z ∞ Z d" Z d 1 1 t t ν(ξ )θ (ξ )dξ ν(ξ )θ˙ (ξ )dξ dt ≤ 0, q · ∇θ dt ≤ 0. θ (t) 0 θ2 0 0 0 It is the nonlinear way θ occurs in the first inequality that makes the exploitation difficult. That is why quite often a linear approximation of the inequality (16) is considered [7].
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4. Temperature-rate discontinuity waves For simplicity we restrict attention to one-dimensional waves (see [1,8]). Let Ω ⊆ R be the region occupied by the body and let Z ∈ Ω be a time-dependent point which is the common boundary of two regions Ω + , Ω − . Let φ(z, t) be a function on Ω × R which is continuously differentiable in Ω + and Ω − and let φ approach finite limits φ + , φ − as Z is approached from the interior of Ω + , Ω − . The jump of φ at Z is defined by [φ] = φ − − φ + . Denote by U the speed of displacement of Z , U = Z 0 . Let δ/δt be the time derivative as apparent to an observer moving with Z , so that δφ = ∂t φ + U ∂z φ. δt Hence we have δ[φ] = [∂t φ] + U [∂z φ]. δt As a consequence, ∂φ [φ] = 0 H⇒ = −U [∂z φ]. ∂t
(26)
(27)
If φ(z 0 , t 0 ) suffers a jump discontinuity at (z, t), then [φ(z, t)] = φ(z, t+ ) − φ(z, t− ).
(28)
Consider temperature-rate waves characterized by [θ] = 0 whereas ∂t θ, ∂z θ and higher-order derivatives suffer jump discontinuities across Z . The function (θ ) is then continuous across Z . So does the (z-component of the) heat flux q, which follows from the application of the Kotchine theorem [9]. By (15), the z-component of q, Z ∞ q(z, t) = −κ0 ∂z θ (z, t) + κ(ξ )∂z θ (z, t − ξ )dξ 0
gives [q] = −κ0 [∂z θ]. Hence [q] = 0 is allowed, when [∂z θ] 6= 0, only if κ0 = 0. Consider the equation of energy, θ ∂t θ = −∂z q + r. Letting [r ] = 0, we have θ [∂t θ] = −[∂z q]. Upon the identifications q = f , g = ∂z θ, h = −κ, by (11), with n = 1, we have 1 κ(0)[∂z θ]. U By (27), it follows that [∂z q] =
[∂t θ] = −U [∂z θ].
(29)
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
147
Hence we find that 1 θ − κ(0) 2 [∂t θ] = 0. U Accordingly, temperature-rate waves occur only if the speed U satisfies U2 =
κ(0) . θ
The thermodynamic restriction (21) and the assumption (23) imply that the right-hand side is positive, and hence U exists. 5. Decay of temperature-rate waves Apply (26) to φ = ∂t θ to obtain δ[∂t θ] = [∂t2 θ] + U [∂z ∂t θ]. δt
(30)
An expression of [∂t2 θ] is obtained by differentiating (29) with respect to time and evaluating the jump to get θθ [(∂t θ )2 ] + θ [∂t2 θ] = −[∂t ∂z q]. By (12), we have 1 0 κ (0)[∂z θ]. U Upon substitution, (30) becomes [∂t ∂z q] = −κ(0)[∂z2 θ] +
δ[∂t θ] κ 0 (0) θθ = U ([∂z ∂t θ] + U [∂z2 θ]) − [∂z θ] − [(∂t θ )2 ]. δt U θ θ Owing to (26), [∂z ∂t θ] + U [∂z2 θ] =
δ[∂z θ] , δt
[∂z θ] = −
1 [∂t θ]. U
Hence we obtain δ[∂t θ] κ 0 (0) θθ = [∂t θ] − [(∂t θ )2 ]. δt 2κ(0) 2θ
(31)
Eq. (31) is the evolution equation for the discontinuity [∂t θ]. If ∂t θ = 0 ahead of the wave then [(∂t θ )2 ] = [∂t θ]2 , and (31) becomes the classical equation of Bernoulli type [1]. The nonlinear term is a direct effect of the nonlinearity of (θ). If θθ = 0, then (31) yields [∂t θ](t) = [∂t θ](0) exp(κ 0 (0)t/2κ(0)).
(32)
The restrictions (22) and (21) result in the decay of [∂t θ]. An analogous result holds in linear viscoelasticity (see [1], p. 338) with the kernel κ replaced by the stressrelaxation function G. Remark. If the constitutive equation for is taken in the form (25) then by means of (7) we find that [∂t ] = ˜θ [∂t θ], [∂t2 ] = ˜θ [∂t2 θ] + ˜θθ [(∂t θ )2 ] + γ (t)[∂t θ] where γ (t) = p
∞
Z
ν(ξ )θ (t − ξ )dξ 0
p−1
ν(0).
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As a consequence, the speed U remains unchanged, whereas the evolution equation is given by 0 δ[∂t θ] κ (0) θθ = − γ (t) [∂t θ] − [(∂t θ )2 ]. δt 2κ(0) 2θ
(33)
5.1. Comments on the Cattaneo–Maxwell law It is worth applying the result (32) to the Cattaneo–Maxwell law which, in the one-dimensional case, reads τ ∂t q + q = −k∂z θ, where τ > 0 is viewed as the relaxation time. A straightforward integration with respect to t gives Z k ∞ q(z, t) = − exp(−ξ/τ )∂z θ (z, t − ξ )dξ. τ 0 Hence the results of this paper apply with κ(ξ ) =
k exp(−ξ/τ ). τ
The restriction (21) implies that k > 0. Hence κ 0 (0) = −
κ < 0. τ2
This means that the Cattaneo–Maxwell law, as a particular case of (15), provides temperature-rate waves which strictly decay in time. 6. Models based on summed histories So as to obtain a theory for which thermal disturbances travel with finite speeds, Gurtin and Pipkin [10] established a model of heat conduction in which the constitutive functionals depend on the temperature gradient through the summed history ∇θ t , on R+ , such that Z t t ∇θ (ξ ) = ∇θ (λ)dλ. t−ξ
The model was extended to deformable bodies by Chen and Gurtin [11]. One might think that the dependence on the summed history ∇θ t is the crucial step for the finiteness of the wave speeds. This paper, though, has shown that such is not the case, in that a dependence on the history is compatible with thermodynamics and provides finite wave speeds. Hence the dependences on the history ∇θ t or on the summed history ∇θ t are in fact different ways of modelling the memory features. This in turn suggests that we look for similarities and differences between the two dependences. Restrict attention to linear functionals for the heat flux. The dependence on ∇θ t is expressed in the form Z ∞ Z t 0 q(x, t) = α (ξ ) ∇θ (x, λ)dλ dξ 0
t−ξ
is the derivative of a scalar function α on R+ . An integration by parts gives ∞ Z ∞ Z t q(x, t) = α(ξ ) ∇θ (x, λ)dλ − α(ξ )∇θ (x, t − ξ )dξ.
where
α0
t−ξ
0
0
Assume that lim α(ξ )
ξ →∞
Z
t t−ξ
∇θ (x, λ)dλ = 0.
(34)
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
149
Hence we have ∞
Z
α(ξ )∇θ (x, t − ξ )dξ.
q(x, t) = − 0
Accordingly, with regard to the heat flux, the dependence on the summed history, subject to (34), is equivalent to the dependence of q in (15) once we make the identification κ = −α 0 . In this sense, it is consistent that (33) reduces to the result of Chen [12] if p = 1 and θθ = 0. Of course, nonlinear dependences may prove to be non-equivalent. Differences between the models, relative both to compatibility with thermodynamics and to wave propagation, may arise according to the constitutive assumption for the internal energy. In particular, models with summed histories lead to the heat flux having free energy as a thermodynamic potential [10]. There can be circumstances and technical reasons such that recourse to the summed histories is convenient. Often, though, histories provide more direct and simpler models. References [1] P.J. Chen, Growth and Decay of Waves in Solids, in: C. Truesdell (Ed.), Encyclopedia of Physics, vol. VIa/3, Springer, Berlin, 1973, pp. 303–402. [2] B.D. Coleman, M.E. Gurtin, Waves in materials with memory II. On the growth and decay of one-dimensional acceleration waves, Arch. Ration. Mech. Anal. 19 (1965) 266–298. [3] D.D. Joseph, L. Preziosi, Heat waves, Rev. Modern Phys. 61 (1989) 41–73. [4] A. Morro, T. Ruggeri, Non-equilibrium properties of solids, obtained from second sound measurements, J. Phys. C 21 (1988) 1743–1752. [5] V.A. Cimmelli, Thermodynamics of anisotropic solids near absolute zero, Math. Comput. Modelling 28 (1998) 79–89. [6] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1948) 83–101. [7] M. Fabrizio, B. Lazzari, J.E. Mu˜noz Rivera, Asymptotic behaviour in linear thermoelasticity, J. Math. Anal. Appl. 232 (1999) 138–165. [8] M.F. McCarthy, Singular surfaces and waves, in: A.C. Eringen (Ed.), Continuum Physics II, Academic Press, New York, 1975, pp. 449–521. [9] C. Truesdell, Introduction a` la m´ecanique rationelle des milieux continus, Masson, Paris, 1974, p. 307. [10] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31 (1968) 113–126. [11] P.J. Chen, M.E. Gurtin, On second sound in materials with memory, Z. Angew. Math. Phys. 21 (1970) 232–241. [12] P.J. Chen, On the growth and decay of temperature rate waves of arbitrary form, Z. Angew. Math. Phys. 20 (1969) 448–453.
Jump relations and discontinuity waves in conductors with memory A. Morro ∗ University of Genoa, DIBE, Via Opera Pia 11a, 16145, Italy Received 11 April 2005; accepted 27 April 2005
Abstract This paper investigates a linear functional of the history of a variable which, along with the derivatives, suffers jump discontinuities. The first purpose is to evaluate the jump of the time and space derivatives, of any order, of the functional. Next, the modelling of conductors with memory is considered, in terms of the history of the temperature gradient, and necessary and sufficient thermodynamic restrictions are established. The propagation and evolution of one-dimensional waves is then investigated. The thermodynamic restrictions are shown to guarantee the finiteness of the wave speed and the decay of the discontinuity. c 2005 Elsevier Ltd. All rights reserved.
Keywords: Jump relations; Materials with memory; Discontinuity waves
1. Introduction The model of materials with fading memory is often applied to describe memory properties by letting the response be given by functionals of the whole history of appropriate variables. The dependence on the history results in remarkable consequences on the propagation of discontinuity waves in connection with both the existence of waves and the time evolution of the discontinuities. When dealing with memory functionals, the derivation of jump relations, even for low-order derivatives, is technically quite involved (see [1]). Mathematical difficulties arise because of the nonlinearity and of the dependence on the history. The purpose of this paper is twofold. The first fold is to investigate a linear functional such that g(z, ·) on (−∞, t) is mapped into f (z, t) in the form Z ∞ h(ξ )g(z, t − ξ )dξ (1) f (z, t) = 0
where h is continuously differentiable on R+ , whereas g(z, ·) and its derivatives suffer jump discontinuities. We aim to evaluate the jump of the derivatives ∂tk f, ∂zk f , of any order k. The literature provides low-order derivatives of (stress) nonlinear functionals [2,1]. Here, more compact and definite results for (1) are likely to arise thanks to the linearity of the functional. It is really so, but the calculation is non-trivial, in that g is not continuous and hence an appropriate procedure is in order. The second fold is to apply the results and examine the propagation of one-dimensional waves ∗ Tel.: +39 10 3532786; fax: +39 10 3532777.
E-mail address: [email protected]. c 2005 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2005.04.016
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
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in conductors with memory. This application is also motivated by the long-time interest in the modelling of heat conductors compatible with discontinuity waves (see, e.g., [3–5]). In this sense it is worth remarking that the wellknown generalized Fourier (or Cattaneo–Maxwell) law amounts to a particular case of the relation (1) where g is the temperature gradient and f is the heat flux [6]. The dependence of h on ξ ∈ R+ , but not on z, reflects the assumption that the material under investigation is homogeneous. This assumption in turn implies that the speed of propagation of discontinuities is constant, which simplifies the expressions of ∂tk f and ∂zk f . Concerning the model of heat conduction, we let the heat flux be a linear functional of the history of the temperature gradient, whereas the internal energy is a function of the temperature. It is then shown that an appropriate condition on the kernel, of the heat flux, is necessary and sufficient for the validity of the second law of thermodynamics. Next, we investigate the propagation condition and the evolution of temperature-rate discontinuity waves. It follows that the thermodynamic restrictions guarantee the finiteness of the wave speeds and the decay of the discontinuity. As a useful connection, the widely applied model involving summed histories is contrasted with the present model based on the histories. This paper shows that summed histories are not necessary to obtain finite wave speeds, and that the two models prove equivalent in linear approximations and are qualitatively different if nonlinear constitutive functionals are considered. 2. Jump relations for a linear functional For later purposes we consider functions g(z, t) on Ω × R, where Ω is an interval in R, Ω ⊆ R (see Fig. 1). The functions g are allowed to suffer a jump discontinuity at the line γ : z = Z (t), where Z (t) is invertible, and to be of class C n outside γ . To fix ideas, let (z 0 , t0 ), z 0 = Z (t0 ) be a point of the line. We consider the functional mapping g to R so that a function f on Ω × R is defined in the form (1) where h ∈ C n−1 on R+ . Denote by [g](z 0 , t0 ) = g(z 0 , t0+ ) − g(z 0 , t0− ) = g(z 0− , t0 ) − g(z 0+ , t0 ) the jump of g at (z 0 , t0 ). If t > t0 then g(z 0 , ·), on (−∞, t), suffers a jump at t0 . If t < t0 then g(z 0 , ·) is continuously differentiable on (−∞, t). We now investigate the differentiability of f . Let z = z 0 , t > t0 . Since g is discontinuous at t0 , the integrand is discontinuous (at ξ = t −t0 ). It is then convenient to write the function f in the form Z t−t0 Z ∞ f (z 0 , t) = h(ξ )g(z 0 , t − ξ )dξ + h(ξ )g(z 0 , t − ξ )dξ. (2) 0
t−t0
Since g(z 0 , ·) is continuously differentiable in (−∞, t0 ) ∪ (t0 , t), then f is differentiable and Z t−t0 h(ξ )∂tg(z 0 , t − ξ )dξ + h(t − t0 )g(z 0 , t0+ ) ∂t f (z 0 , t) = 0 Z ∞ + h(ξ )∂t g(z 0 , t − ξ )dξ − h(t − t0 )g(z 0 , t0− ). t−t0
Hence we can write Z ∂t f (z 0 , t) =
t−t0
h(ξ )∂t g(z 0 , t − ξ )dξ +
0
Z
∞
h(ξ )∂t g(z 0 , t − ξ )dξ + h(t − t0 )[g](z 0 , t0 ), t−t0
where the sum of the two integrals is the integral on R+ . Since h is differentiable and ∂t g(z 0 , ·) is continuously differentiable on (−∞, t0 ) ∪ (t0 , t), then ∂t f (z 0 , t) is differentiable too and it follows that Z ∞ Z t−t0 ∂t2 f (z 0 , t) = h(ξ )∂t2 g(z 0 , t − ξ )dξ + h(ξ )∂t2 g(z 0 , t − ξ )dξ 0
t−t0
+ h(t − t0 )[∂t g](z 0 , t0 ) + h (t − t0 )[g](z 0 , t0 ) 0
where
h0
is the derivative of h. Iteration of the procedure up to the n-th order derivative shows that Z ∞ n−1 X (n−1−k) ∂tn f (z 0 , t) = h(ξ )∂tn g(z 0 , t − ξ )dξ + h (k) (t − t0 )[∂t g](z 0 , t0 ), t > t0 . 0
k=0
(3)
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Fig. 1. At z = z 0 , if t > t0 then g(z, ·) on (−∞, t) is discontinuous at t0 ; if t < t0 then g(z, ·) is continuously differentiable.
If, instead, t < t0 then the integrand in (2) is of class C n and we have Z ∞ ∂tn f (z 0 , t) = h(ξ )∂tn g(z 0 , t − ξ )dξ, t < t0 .
(4)
0
Letting t → t0+ , by (3) we obtain ∂tn f (z 0 , t0+ ) =
∞
Z
h(ξ )∂tn g(z 0 , t0 − ξ )dξ +
0
n−1 X
(n−1−k)
h (k) (0)[∂t
g](z 0 , t0 ).
(5)
k=0
By (4), we have ∂tn f (z 0 , t0− ) =
∞
Z 0
h(ξ )∂tn g(z 0 , t0 − ξ )dξ.
(6)
Hence we obtain the jump of ∂tn f by simply subtracting (6) from (5). We find that [∂tn f ] =
n−1 X
(n−1−k)
h (k) (0)[∂t
g],
n ∈ N,
(7)
k=0 (n−k+1)
where the jumps of ∂tn f and ∂t g are evaluated at the same point (z 0 , t0 ) of γ . If n = 0, the relation (7) holds in the trivial form [ f ] = 0. We now examine partial differentiation with respect to the first argument z. For any value of z ∈ Ω , let t0 = Z −1 (z) so that (z, t0 ) ∈ γ . Also, let dt0 /dz = 1/U so that U is the speed of the point on γ . Of course, we regard t as fixed. First let t > t0 and write Z ∞ Z t−t0 f (z, t) = h(ξ )g(z, t − ξ )dξ + h(ξ )g(z, t − ξ )dξ (8) 0
t−t0
where it is understood that t0 is a function of z. The integrand is continuously differentiable (with respect to z) as ξ ∈ (0, t − t0 (z)) ∪ (t − t0 (z), ∞) and t0 depends on z (see Fig. 1). We find that Z t−t0 1 ∂z f (z, t) = h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )g(z, t0+ ) U 0 Z ∞ 1 + h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )g(z, t0− ) U t−t0
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whence ∂z f (z, t) =
t−t0
Z
h(ξ )∂z g(z, t − ξ )dξ +
0
∞
Z
h(ξ )∂z g(z, t − ξ )dξ − t−t0
1 h(t − t0 )[g](z, t0 ). U
So, differentiation with respect to z of the integral provides −1/U times the jump of the integrand at ξ = t − t0 . Hence further differentiation gives Z ∞ Z t−t0 h(ξ )∂z2 g(z, t − ξ )dξ h(ξ )∂z2 g(z, t − ξ )dξ + ∂z2 f (z, t) = 0
t−t0
1 1 − h(t − t0 )[∂z g](z, t0 ) + ∂z {h(t − t0 )[g](z, t0 )}. U U Upon n times differentiation, we have ∂zn
f (z, t) =
∞
Z 0
h(ξ )∂zn g(z, t − ξ )dξ −
n−1 1 X ∂ k {h(t − t0 )[∂zn−1−k g]}. U k=0 z
(9)
This relation holds for n = 1, 2. If the relation holds for n, then the application of ∂z to ∂zn f produces −
n−1 1 X 1 h(t − t0 )[∂zn g](z, t0 ) − ∂ k+1 {h(t − t0 )[∂zn−1−k g](z, t0 )} U U k=0 z
=−
n 1 1 X h(t − t0 )[∂zn g](z, t0 ) − ∂ k {h(t − t0 )[∂zn−k g](z, t0 )} U U k=1 z
=−
n 1 X ∂ k {h(t − t0 )[∂zn−k g](z, t0 )}. U k=0 z
This shows that the validity of (9) for ∂zn implies the validity for ∂zn+1 . This in turn proves the validity for any n ∈ N. Denote by δ/δt the operator δ = ∂t + U ∂z |t δt where ∂z |t means partial differentiation at constant t. Now it is convenient to observe that, owing to the dependence of t0 on z, ∂z [g](z, t0 ) = ∂z |t [g](z, t0 ) + ∂t [g](z, t0 )
δ[g] 1 = (z, t0 ) U δt
and hence δ[∂ n−1−k g] 1 0 1 h (t − t0 )[∂zn−1−k g](z, t0 ) + h(t − t0 ) z (z, t0 ) U U δt and so on for higher-order differentiations. We obtain k δ k− j [∂zn−1−k g] 1 X k k n−1−k ∂z {h(t − t0 )[∂z g](z, t0 )} = k (−1) j h ( j) (t − t0 ) (z, t0 ). U j=0 j δt k− j ∂z {h(t − t0 )[∂zn−1−k g](z, t0 )} = −
If, instead, t < t0 then g(z, ·) is continuously differentiable and we have Z ∞ h(ξ )∂zn g(z, t − ξ )dξ. ∂zn f (z, t) =
(10)
0
Let t → t0+ in (9) and t → t0− in (10). The difference ∂zn f (z, t0+ ) − ∂zn f (z, t0− ) yields [∂zn f ](z, t) = −
n−1 k δ k− j [∂zn−1−k g] 1 X 1 X j ( j) (−1) h (0) (z, t). U k=0 U k j=0 δt k− j
(11)
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The differentiation operators ∂z and ∂t do not commute. Let t > t0 . Time differentiation of Z ∞ Z t−t0 1 h(ξ )∂z g(z, t − ξ )dξ − h(t − t0 )[g](z, t0 ) h(ξ )∂z g(z, t − ξ )dξ + ∂z f (z, t) = U t−t0 0 gives ∂t ∂z f (z, t) =
∞
Z
h(ξ )∂t ∂z g(z, t − ξ )dξ + h(t − t0 )[∂z g](z, t0 ) − 0
1 0 h (t − t0 )[g](z, t0 ). U
In the reverse order, we have Z ∞ 1 h(ξ )∂z ∂t g(z, t − ξ )dξ − h(t − t0 )[∂t g](z, t0 ) + ∂z {h(t − t0 )[g](z, t0 )} ∂z ∂t f (z, t) = U 0 where ∂z {h(t − t0 )[g](z, t0 )} = −
1 δ[g] 1 0 h (t − t0 )[g](z, t0 ) + h(t − t0 ) (z, t0 ). U U δt
Hence, letting t → t0+ and subtracting the limit as t → t0− of the corresponding expressions for t < t0 , we obtain 1 0 h (0)[g](z, t), U 1 1 1 δ[g] [∂z ∂t f ](z, t) = − h(0)[∂t g](z, t) − h 0 (0)[g](z, t) + h(0) (z, t). U U U δt [∂t ∂z f ](z, t) = h(0)[∂z g](z, t) −
(12) (13)
This proves that ∂z and ∂t do not commute. Later on, we show that if [g] = 0 then [∂t g] = −U [∂z g]. As a consequence, if [g] = 0 then ∂z ∂t f = ∂t ∂z f . The results (7) and (11), for any n ∈ N, appear to be new in the literature, where calculations are usually performed by having recourse to low-order partial time derivatives. This section shows that partial space differentiation is decisively more involved. 3. Heat conductor with memory The simplest model of heat conductor is that of a rigid body occupying a region Ω . The pertinent fields are the absolute temperature θ, the internal energy (per unit mass) , and the heat flux vector q. A possible energy supply is described by r . All fields depend on the position x ∈ Ω and the time t ∈ R. The balance of energy is written as ˙ = −∇ · q + r
(14)
where ∇· is the divergence operator. Since the body is undeformable, total time differentiation, denoted by a superposed dot, coincides with partial time differentiation ∂t . The constitutive properties of the conductor are modelled by saying that, at each point x ∈ Ω , the internal energy is a function of the temperature, whereas the heat flux is a linear functional of the history of the temperature gradient. We write Z ∞ = (θ ), q(t) = −κ0 ∇θ (t) − κ(ξ )∇θ t (ξ )dξ (15) 0
where κ is a bounded function on θ t (ξ ) = θ (t − ξ ),
R+ .
The symbol θ t stands for the history of θ up to time t:
ξ ∈ R+ .
According to (15), the dependence of q on ∇θ t is chosen to be linear. The dependence (of , θ, κ, q) on the position x is understood and not written.
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3.1. Thermodynamic restrictions Let η be the entropy density. The second law of thermodynamics is expressed by the inequality Z d (η˙ + ∇ · (q/θ ) − r/θ )dt ≥ 0
(16)
0
to hold at any point x ∈ Ω for every set of periodic functions (η, q, θ, r ) in time with period d. By means of (14), the inequality (16) becomes Z d 1 ˙ η˙ − − 2 q · ∇θ dt ≥ 0. θ θ 0 Because of the periodic character, Z d ˙ η˙ − dt = 0. θ 0 Hence we are left with Z d 1 q · ∇θ dt ≤ 0. θ2 0 This condition holds if and only if Z d q · ∇θdt ≤ 0.
(17)
0
For, letting θ be constant at the point under consideration but ∇θ be unrestricted, we obtain (17). Conversely, if (17) holds, then Z d Z d 1 1 q · ∇θ dt ≤ (q · ∇θ )dt ≤ 0 2 θ2 θM 0 0 where θ M is the maximum value of θ in [0, d]. Consequently, the model (15) is compatible with thermodynamics if and only if (17) holds. Denote by κc the half-range cosine transform of κ, namely Z ∞ κc (ω) = κ(ξ ) cos ωξ dξ. 0
The thermodynamic restrictions on the model (15) are stated as follows. Proposition. The model (15) is compatible with thermodynamics if and only if κ0 + κc (ω) ≥ 0,
ω ∈ R.
(18)
Proof. To show that the condition (18) is necessary, we choose the function ∇θ as ∇θ (t) = gˆ cos ωt + g˜ sin ωt where ω ∈ R and gˆ , g˜ are arbitrary vectors. Upon substitution in (15), we find that q(t) · ∇θ (t) = −(κ0 + κc )ˆg2 cos2 ωt − (κ0 + κc )˜g2 sin2 ωt + κs gˆ · g˜ (sin2 ωt − cos2 ωt) − 2(κ0 + κc )ˆg · g˜ cos ωt sin ωt + κs (ˆg2 − g˜ 2 ) cos ωt sin ωt where κs is the half-range sine transform of κ. Hence q · ∇θ is periodic with period d = π/|ω|. Integration over [0, π/|ω|] yields Z π/|ω| π (κ0 + κc )(ˆg2 + g˜ 2 ). 0≥ q(t) · ∇θ (t)dt = − 2|ω| 0 Hence it follows that (18) holds.
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To show that (18) is sufficient, we let ∇θ be periodic with period d and write it as a Fourier series in the form ∇θ (t) =
∞ X
gˆ k cos kωt + g˜ k sin kωt
k=0
where ω = 2π/d. Hence we obtain q(t) · ∇θ (t) = −κ0 −
∞ X
(ˆgk cos kωt + g˜ k sin kωt) · (ˆgh cos hωt + g˜ h sin hωt)
k,h=0 ∞ X
{[ˆgk (κc (kω) cos kωt + κs (kω) sin kωt) + g˜ k (κc (kω) sin kωt − κs (kω) cos kωt)]
h,k=0
· (ˆgh cos hωt + g˜ h sin hωt)}. Integration over [0, 2π/ω] yields Z 2π/ω ∞ πX (κ0 + κc (kω))(ˆg2k + g˜ 2k ). q(t) · ∇θ (t)dt = − ω k=0 0 The condition (18) then implies (17).
As we see in a moment, wave propagation is allowed only if κ0 = 0. That is why we let κ0 = 0, and hence (18) becomes κc (ω) ≥ 0,
ω ∈ R.
(19)
3.2. Restrictions on the initial value of κ and κ 0 We preliminarily establish a relation between κc and κ. Let κ ∈ L 1 (R+ ) and define a function κ on R by letting κ(ξ ), ξ ≥ 0, κ(ξ ) = κ(−ξ ), ξ < 0. Hence the Fourier transform κ F of κ, Z ∞ κ F (ω) = κ(ξ ) exp(−iωξ )dξ −∞
gives κ F (ω) = 2κc (ω). Since κc is even, κc (ω) = κc (−ω), the inverse Fourier transform Z ∞ 1 κ(ξ ) = κ F (ω) exp(iωξ )dω 2π −∞ yields κ(ξ ) =
2 π
Z
∞
κc (ω) cos ωξ dω.
0
The restriction to ξ ≥ 0 gives Z 2 ∞ κ(ξ ) = κc (ω) cos ωξ dω. π 0
(20)
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
It follows at once from (20) and (19) that Z 2 ∞ κc (ω)dω > 0. κ(0) = π 0
145
(21)
Equality corresponds to κ = 0, and hence is excluded. A further relation holds if κ, κ 0 , κ 00 ∈ L 1 (R+ ). An integration by parts gives Z ∞ Z ∞ κ(ξ ) cos ωξ dξ κ 0 (ξ ) sin ωξ dξ = κ(ξ ) sin ωξ |∞ −ω κs0 (ω) = 0 0
0
whence κs0 (ω) = −ωκc (ω). Furthermore, Z ∞ Z ω κ 0 (ξ ) sin ωξ dξ = −κ 0 (ξ ) cos ωξ |∞ + 0 0
∞
κ 00 (ξ ) cos ωξ dξ 0
whence ωκs0 (ω) = κ 0 (0) + κc00 (ω). Upon evaluating the limit as ω → ∞ and applying the Riemann–Lebesgue lemma to κ 00 , we obtain κ 0 (0) = − lim ω2 κc (ω). ω→∞
Since κc (ω) ≥ 0 on R, we obtain κ 0 (0) ≤ 0.
(22)
Finally, we let the specific heat θ = d/dθ be positive, θ > 0.
(23)
In the next section we examine the consequences of (21)–(23) on wave propagation. Remark. A more general model is obtained by letting (t) = ˆ (θ (t), θ t ). In such a case the requirement (16) of the second law results in the inequality Z d ˙ 1 + 2 q · ∇θ dt ≤ 0. θ θ 0
(24)
The exploitation of (24) is generally a hard task. To fix ideas, by way of example we might keep q as in (15) and take Z ∞ p t (t) = ˜ (θ (t)) + ν(ξ )θ (ξ )dξ (25) 0
where p is a positive real. Hence (24) is equivalent to two separate inequalities # Z ∞ p−1 Z ∞ Z d" Z d 1 1 t t ν(ξ )θ (ξ )dξ ν(ξ )θ˙ (ξ )dξ dt ≤ 0, q · ∇θ dt ≤ 0. θ (t) 0 θ2 0 0 0 It is the nonlinear way θ occurs in the first inequality that makes the exploitation difficult. That is why quite often a linear approximation of the inequality (16) is considered [7].
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4. Temperature-rate discontinuity waves For simplicity we restrict attention to one-dimensional waves (see [1,8]). Let Ω ⊆ R be the region occupied by the body and let Z ∈ Ω be a time-dependent point which is the common boundary of two regions Ω + , Ω − . Let φ(z, t) be a function on Ω × R which is continuously differentiable in Ω + and Ω − and let φ approach finite limits φ + , φ − as Z is approached from the interior of Ω + , Ω − . The jump of φ at Z is defined by [φ] = φ − − φ + . Denote by U the speed of displacement of Z , U = Z 0 . Let δ/δt be the time derivative as apparent to an observer moving with Z , so that δφ = ∂t φ + U ∂z φ. δt Hence we have δ[φ] = [∂t φ] + U [∂z φ]. δt As a consequence, ∂φ [φ] = 0 H⇒ = −U [∂z φ]. ∂t
(26)
(27)
If φ(z 0 , t 0 ) suffers a jump discontinuity at (z, t), then [φ(z, t)] = φ(z, t+ ) − φ(z, t− ).
(28)
Consider temperature-rate waves characterized by [θ] = 0 whereas ∂t θ, ∂z θ and higher-order derivatives suffer jump discontinuities across Z . The function (θ ) is then continuous across Z . So does the (z-component of the) heat flux q, which follows from the application of the Kotchine theorem [9]. By (15), the z-component of q, Z ∞ q(z, t) = −κ0 ∂z θ (z, t) + κ(ξ )∂z θ (z, t − ξ )dξ 0
gives [q] = −κ0 [∂z θ]. Hence [q] = 0 is allowed, when [∂z θ] 6= 0, only if κ0 = 0. Consider the equation of energy, θ ∂t θ = −∂z q + r. Letting [r ] = 0, we have θ [∂t θ] = −[∂z q]. Upon the identifications q = f , g = ∂z θ, h = −κ, by (11), with n = 1, we have 1 κ(0)[∂z θ]. U By (27), it follows that [∂z q] =
[∂t θ] = −U [∂z θ].
(29)
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Hence we find that 1 θ − κ(0) 2 [∂t θ] = 0. U Accordingly, temperature-rate waves occur only if the speed U satisfies U2 =
κ(0) . θ
The thermodynamic restriction (21) and the assumption (23) imply that the right-hand side is positive, and hence U exists. 5. Decay of temperature-rate waves Apply (26) to φ = ∂t θ to obtain δ[∂t θ] = [∂t2 θ] + U [∂z ∂t θ]. δt
(30)
An expression of [∂t2 θ] is obtained by differentiating (29) with respect to time and evaluating the jump to get θθ [(∂t θ )2 ] + θ [∂t2 θ] = −[∂t ∂z q]. By (12), we have 1 0 κ (0)[∂z θ]. U Upon substitution, (30) becomes [∂t ∂z q] = −κ(0)[∂z2 θ] +
δ[∂t θ] κ 0 (0) θθ = U ([∂z ∂t θ] + U [∂z2 θ]) − [∂z θ] − [(∂t θ )2 ]. δt U θ θ Owing to (26), [∂z ∂t θ] + U [∂z2 θ] =
δ[∂z θ] , δt
[∂z θ] = −
1 [∂t θ]. U
Hence we obtain δ[∂t θ] κ 0 (0) θθ = [∂t θ] − [(∂t θ )2 ]. δt 2κ(0) 2θ
(31)
Eq. (31) is the evolution equation for the discontinuity [∂t θ]. If ∂t θ = 0 ahead of the wave then [(∂t θ )2 ] = [∂t θ]2 , and (31) becomes the classical equation of Bernoulli type [1]. The nonlinear term is a direct effect of the nonlinearity of (θ). If θθ = 0, then (31) yields [∂t θ](t) = [∂t θ](0) exp(κ 0 (0)t/2κ(0)).
(32)
The restrictions (22) and (21) result in the decay of [∂t θ]. An analogous result holds in linear viscoelasticity (see [1], p. 338) with the kernel κ replaced by the stressrelaxation function G. Remark. If the constitutive equation for is taken in the form (25) then by means of (7) we find that [∂t ] = ˜θ [∂t θ], [∂t2 ] = ˜θ [∂t2 θ] + ˜θθ [(∂t θ )2 ] + γ (t)[∂t θ] where γ (t) = p
∞
Z
ν(ξ )θ (t − ξ )dξ 0
p−1
ν(0).
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A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
As a consequence, the speed U remains unchanged, whereas the evolution equation is given by 0 δ[∂t θ] κ (0) θθ = − γ (t) [∂t θ] − [(∂t θ )2 ]. δt 2κ(0) 2θ
(33)
5.1. Comments on the Cattaneo–Maxwell law It is worth applying the result (32) to the Cattaneo–Maxwell law which, in the one-dimensional case, reads τ ∂t q + q = −k∂z θ, where τ > 0 is viewed as the relaxation time. A straightforward integration with respect to t gives Z k ∞ q(z, t) = − exp(−ξ/τ )∂z θ (z, t − ξ )dξ. τ 0 Hence the results of this paper apply with κ(ξ ) =
k exp(−ξ/τ ). τ
The restriction (21) implies that k > 0. Hence κ 0 (0) = −
κ < 0. τ2
This means that the Cattaneo–Maxwell law, as a particular case of (15), provides temperature-rate waves which strictly decay in time. 6. Models based on summed histories So as to obtain a theory for which thermal disturbances travel with finite speeds, Gurtin and Pipkin [10] established a model of heat conduction in which the constitutive functionals depend on the temperature gradient through the summed history ∇θ t , on R+ , such that Z t t ∇θ (ξ ) = ∇θ (λ)dλ. t−ξ
The model was extended to deformable bodies by Chen and Gurtin [11]. One might think that the dependence on the summed history ∇θ t is the crucial step for the finiteness of the wave speeds. This paper, though, has shown that such is not the case, in that a dependence on the history is compatible with thermodynamics and provides finite wave speeds. Hence the dependences on the history ∇θ t or on the summed history ∇θ t are in fact different ways of modelling the memory features. This in turn suggests that we look for similarities and differences between the two dependences. Restrict attention to linear functionals for the heat flux. The dependence on ∇θ t is expressed in the form Z ∞ Z t 0 q(x, t) = α (ξ ) ∇θ (x, λ)dλ dξ 0
t−ξ
is the derivative of a scalar function α on R+ . An integration by parts gives ∞ Z ∞ Z t q(x, t) = α(ξ ) ∇θ (x, λ)dλ − α(ξ )∇θ (x, t − ξ )dξ.
where
α0
t−ξ
0
0
Assume that lim α(ξ )
ξ →∞
Z
t t−ξ
∇θ (x, λ)dλ = 0.
(34)
A. Morro / Mathematical and Computer Modelling 43 (2006) 138–149
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Hence we have ∞
Z
α(ξ )∇θ (x, t − ξ )dξ.
q(x, t) = − 0
Accordingly, with regard to the heat flux, the dependence on the summed history, subject to (34), is equivalent to the dependence of q in (15) once we make the identification κ = −α 0 . In this sense, it is consistent that (33) reduces to the result of Chen [12] if p = 1 and θθ = 0. Of course, nonlinear dependences may prove to be non-equivalent. Differences between the models, relative both to compatibility with thermodynamics and to wave propagation, may arise according to the constitutive assumption for the internal energy. In particular, models with summed histories lead to the heat flux having free energy as a thermodynamic potential [10]. There can be circumstances and technical reasons such that recourse to the summed histories is convenient. Often, though, histories provide more direct and simpler models. References [1] P.J. Chen, Growth and Decay of Waves in Solids, in: C. Truesdell (Ed.), Encyclopedia of Physics, vol. VIa/3, Springer, Berlin, 1973, pp. 303–402. [2] B.D. Coleman, M.E. Gurtin, Waves in materials with memory II. On the growth and decay of one-dimensional acceleration waves, Arch. Ration. Mech. Anal. 19 (1965) 266–298. [3] D.D. Joseph, L. Preziosi, Heat waves, Rev. Modern Phys. 61 (1989) 41–73. [4] A. Morro, T. Ruggeri, Non-equilibrium properties of solids, obtained from second sound measurements, J. Phys. C 21 (1988) 1743–1752. [5] V.A. Cimmelli, Thermodynamics of anisotropic solids near absolute zero, Math. Comput. Modelling 28 (1998) 79–89. [6] C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1948) 83–101. [7] M. Fabrizio, B. Lazzari, J.E. Mu˜noz Rivera, Asymptotic behaviour in linear thermoelasticity, J. Math. Anal. Appl. 232 (1999) 138–165. [8] M.F. McCarthy, Singular surfaces and waves, in: A.C. Eringen (Ed.), Continuum Physics II, Academic Press, New York, 1975, pp. 449–521. [9] C. Truesdell, Introduction a` la m´ecanique rationelle des milieux continus, Masson, Paris, 1974, p. 307. [10] M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31 (1968) 113–126. [11] P.J. Chen, M.E. Gurtin, On second sound in materials with memory, Z. Angew. Math. Phys. 21 (1970) 232–241. [12] P.J. Chen, On the growth and decay of temperature rate waves of arbitrary form, Z. Angew. Math. Phys. 20 (1969) 448–453.