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Thermoelasticity at cryogenic temperatures
cQ2cb7462;92 s5.w c .oo
IN. 1. Non-Linear Mechanics. Vol. 27, No. 2 pp. 251-263. 1992 Printed in Great Britain.
THERMOELASTICITY
Pergamon Reu
AT CRYOGENIC
plc
TEMPERATURES
G. CAMGLIA Dipartimento di Matematica, Universita di Geneva, 16132 Geneva, Italy
A. MORRO DIBE, Universita di Genova, 16145 Genova, Italy and B. STRAUGHAN Department of Mathematics, University of Glasgow, Glasgow G12 SQW, Scotland, U.K. (Received 21 September 1990; accepted 2 January 1991)
Abstract-A theory of thermoelasticity is developed which is suitable for application at cryogenic temperatures. The thermodynamic functions are so chosen as to give correct predictions with experimental findings of second-sound wave speeds in NaF, in the temperature range 8-20 K and in Bi, in the temperature range l-4 K. After the non-linear theory is presented, a linearized theory is developed. A uniqueness theorem is provided for the linearized theory on an unbounded domain. The question of a half-space heated on its boundary is addressed and, in particular, the question of transverse elastic wave propagation is studied. Finally, a simplified theory in which only unidirectional solutions are allowed is examined.
1. INTRODUCTION
The phenomenon of second-sound (or thermal pulse propagation) has been experimentally observed under accurate conditions in solid He’ and He4 (see, e.g. [l] and the references therein), in sodium fluoride C2-43, in bismuth [S] and in sodium iodide and lithium fluoride [3]. Thermal wave propagation is a low-temperature phenomenon; for example, the above experiments are in the range l-20 K. It is found that the thermal wave speed decreases appreciably as the temperature increases and tends to an asymptotic value as absolute zero is approached, see [3]. In a recent paper Coleman and Newman [6] study heat propagation in a rigid heat conductor and curve-fit thermal wave speeds to mean temperature for NaF and Bi. They find an excellent fit with the empirical formula 1
uz=-
A + Be
(1.1)
where U (cm/s) is the thermal wave speed, and for NaF in the temperature (0) range 10.0 I, 8 I; 18.5 K, they select A = 9.09 x 10-12, B = 2.22 x 10-i’, n = 3.10, while for Bi in the range 1.4 I 0 I 4.0 K, they select A = 9.07 x lo-“, B = 7.58 x lo-i3, n = 3.75. A thermodynamically consistent theory for thermal pulse propagation was recently developed by Morro and Ruggeri [7]. Their paper develops a non-linear theory for which the constitutive variables depend on the temperature only. Basically their equations are
v-q = 0 T(e)4+ de + (1+ rf$q = 0 c,(e)e +
(l-2) (1.3)
where q is the heat flux, K is a thermal conductivity, a superposed dot denotes (partial) time differentiation, and I- = - K[5Ke-4 + (5 - @e+] (1.4) T = K(A’e-3 + w-3)
(1.5)
where A = A/E, B’ = B/E, E being a constant; for NaF, E = 23 erg Ke4 gives a good fit, [6]. The function co is the equilibrium specific heat and Coleman and Newman [6] obtain an Contributed by K. R. Rajagopal. 251
G. CAVIGLIA er al
252
excellent data fit with
co(e) = &P.
(1.6)
While equation (1.3) bears some resemblance to the classical Cattaneo theory [S], a fundamental difference is that T is not constant. Certainly, T is bounded below, and so the commonly used assertion that the limit r + 0 in Cattaneo’s theory (T is Cattaneo’s relaxation time) yields the classical theory of heat conduction does not apply here. However, the thermal
wave speed of [7] satisfies
and this is matched to the empirical fit ofequation (l.l), [6], Thus the Morro-Ruggeri (MR) theory is in agreement with experimentally observed wave speeds, but the parameter Tcannor physically go to zero (although for mathematical purposes it may be convenient to sometimes allow this). Shock behaviour in the MR theory is analysed in [9]. While theoretical work on second-sound has concentrated on explaining the behaviour of the thermal wave, the experiments of McNelly er al. [3] and Jackson et al. [2] demonstrate the existence of three distinct waves. They used a very pure crystal of NaF and evaporated manganin heaters and lead detectors onto opposing faces of the crystal and hence transmitted heat pulses through the sample. Their observations clearly indicate a longitudinal elastic wave which travels fastest, a transverse elastic wave and a temperature wave. For temperatures below 8 K the transverse wave and the thermal wave appear to travel with the same speed, or at least the experiment was unable to detect any appreciable difference. However, at temperatures above 8 K the temperature wave slows down and three distinct waves are observed, the longitudinal one being fastest, transverse next, and the thermal one slowest. In order to theoretically examine the experiments outlined above it is necessary to construct an appropriate theory of thermoelasticity. It is to this goal that our paper is addressed. We derive a theory of thermoelasticity by analogy with the MR rigid heat conductor theory and investigate the consequences. Other second-sound thermoelastic theories (see e.g. [lo]), allow the Cattaneo parameter T to go to zero and hence identify the thermal wave as the fast one: this is clearly incompatible with the experimental findings. According to the MR model, the limit T ---t0 is not allowed and so we have a chance of obtaining good agreement with experiments. The full problem of explaining three-dimensional thermoelastic pulse propagation is very difficult. Previous studies of the subject testify to this. Early work of Lockett and Sneddon [l l] (see also [123), shows how the problem of a half-space heated along its boundary may be treated by Fourier transform methods, but no attempt is made to explicitly calculate the three-dimensional inverse transforms. Likewise, later work has been unable to solve the three-dimensional problem and has concentrated on one-dimensional solutions (see [13] for the classical theory and [lo, 143 for various second-sound theories). We are here unable to fully resolve the three-dimensional problem, but we produce several useful results which help to provide a global picture. An outline of the paper is now given. Section 2 develops a non-linear theory, consistent with that of [7] for a rigid solid, employing an internal-variable theory. This theory is linearized in Section 3; even though the equations given there are linear, the coefficients depend on the equilibrium temperature and so the feature of the correct behaviour of the speed of a thermal wave is retained. In Section 4 we examine the well-posedness of the linear theory in an unbounded domain. Section 5 treats the question of transverse waves in a half-space heated along the wall; the uniqueness theorem of Section 4 is thus relevant. Finally, in Section 6, we examine the half-space problem when dependence on only one space v&able is allowed. It is proved that this simplification problem to a truly one-dimensional one. For completeness, one-dimensional problem is given. 2. THERMODYNAMIC
reduces the three-dimensional an approximate solution to the
SCHEME
Since we have in mind a thermoelastic solid we let the Cauchy stress tensor T, the heat flux q and the free energy $ be functions of the deformation gradient F, the temperature 0, the spatial temperature gradient g, and an internal vector variable 5. By the material
253
Thcrmoelasticity at cryogenic temperatures
objectivity we conclude that T = F?;(C, 8, G, E)FT q = Fij(C,f?, G, 5) $ = $(C, 8, G, Z)
where T, Q, $ are arbitrary
on Sym x Rx Vx V and C = FTF, G = FTg,
functions
1 _ FTC. _Let Y = JF- l T(F- l)T be the second Piola-Kirchhoff
stress tensor, Q = JF- ‘q the heat and p,, the reference mass density. The entropy
flux vector in the reference configuration inequality can be given the form
- po($+@)+fY*C-;Q*G>O for the constitutive functions 6, 4, P, Q of C, 8, G, Z. Accordingly -po(~c.~+~~e+~~.~+~~.~+rlB)+~Y.~-$Q.G~O and this holds if and only if Jc = 0 tl= -$0,
(2.1)
Y = 2Po6c
pot&4
+Q*GsO
(2.2)
where the subscripts C, 0, G, E denote partial derivatives. Then any model is admjssible provided that II/ = $(C, 8, E) and q, Y, Q satisfy equations (2.1) and (2.2). Of course the solution to equations (2.1) and (2.2) is non-unique. We look for quite a general non-linear model which allows for wave propagation and generalizes Fourier’s law of heat conduction. For definiteness, letting Q be independent of G, we start from the evolution equation for Z in the form & z.=mG-nZ (2.3) where m, n are functions of 8, C and n > 0. Substitution in equation (2.2) leads to Q=p,@&,
&.Z>O.
To determine E we require that, under stationary the form
conditions,
(2.4) Fourier’s law is recovered in
Q=-KG
(2.5)
K being a function of 8, C whose values are positive-definite tensors. Now, as e = 0, by equation (2.3) it follows that
G=
-;z.
This suggests that we set Q=;KI.
(2.6)
Then the first relation in equation (2.4) becomes
whence lj/=lJ(e,C)+~=*Kz 2p, em2-
The inequality in equation (2.4) holds identically. The internal energy e = $ + 8q is given by e=$-eJg++~.
(mnK -*d-2!&
de poem
2.
>
254
G. CAWGLIA
et al.
It is convenient, and not severely restrictive, to let e be independent of Z:, and then of Q. This occurs if and only if nK = M(C). a
In addition we take R to be positive definite. In conclusion, the constitutive equations are given by $==$(e,C)+;Z.Rr
(2.7)
Y=2p&++R,P If= -&-)Z.K~
(2.8)
along with equation (2.3). The positive definiteness of ft ensures’q is maximal at equilibrium. Further, we can write the balance equations as poii = Vx.S + pOb
- p,e&d
(2.9)
+ I+&~.~) = - Vx~(p&mZ)
(2.10)
where the relations (2.1) have been taken into account. 3. LINEARIZED
VERSION
OF
THE DYNAMIC
PROBLEM
Also with a view to standardizing experimental settings it is worth examining the dynamics of a thermoelastic solid in the linearized version. Consider an equilibrium configuration 9 and denote by W* the present configuration. The particle occupying the position X in the reference configuration gc, is taken to be at x in 9I and at x* in W*. In W we let the temperature 8 be uniform and then G = 0, Z = 0. The corresponding values in 9* are denoted by 0*, G*, E*. Accordingly, denote by C*, C the values of the Cauthy-Green tensor in 9*, W. Let
u = x* - x,
3 = e* - 8,
A=E*.
Linearization consists in writing the model equations to the first order in u, 9, A. Then we proceed by keeping only the terms up to the first-order. By direct substitution we have C* - C = ZF’(sym Vu)F. By equations (2.7) and (2.8) Y* = 2p,[&
+ &c*
- C) + $&].
Letting Y be the value in 9, namely the Piola-Kirchhoff
pre-stress tensor, we have
Y* = Y + A(F’VuF) - B9 where A = 2p0 $cc, B = - 2p,J0c. On substituting in p,,%* = Vx-S* + peb
rearranging and taking into account the equilibrium equation Vx.FY +pOb=O we obtain in component notation, POiii = [(SijJFNh’FKI1 Gl + ~iHFjM~HK.HN)Uj.N - FiHBHKQl.K, where T is the Cauchy pre-stress tensor. In the present case F = I and thus we obtain POci = [(6ijL +
AiljkJUj.t
-
Bil$l.I*
The balance of energy, Poe*[&8(e*,c*)f9*+ tJOc(e*,c*)mc*~= v~q~~m(e*)Wc*)n*l
(3.1)
Thermoclasticity
in the linear approximation
at cryogenic
255
temperatures
becomes
&A + BBS(sym Vu) = - p,m@R*(VA)
(3.2)
where c = - P,,$~,+ Similarly, the evolution equation for A takes the form A + mV3 + nA = 0.
(3.3)
Of course the quantities c, m, n are evaluated at the configuration 9 where the temperature 8 is uniform and C = 1. Since F = 1 and A = Z, the identification, [equation (2.6)]
allows us to rewrite equation (3.3) as ratj + V3 + aq = 0
(3.4)
where we have set T=n-‘,
8=&R-‘.
(3.5)
Also, it is convenient to rewrite equation (3.2) as c$ + B*sym(Vir) = -iv-q.
(3.6)
In the isotropic case Aijrl
I where
K
=
iJij6,,
1;:j
=
rdij
Bij
=
BSij
Qij =
+
/l(6i,6js
+
6&6jr)
(3.7)
h*-qij
is a constant. 4.
UNIQUENESS
AND
CONTINUOUS
FOR
AN UNBOUNDED
DEPENDENCE
ON THE
DATA
DOMAIN
Since we envisage the problem of heating the boundary of a half-space we include in this section a method for establishing uniqueness and continuous dependence on the data for a general unbounded domain. We treat the general linear, anisotropic case for a domain R*, exterior to a bounded domain C!, in W3. The boundary of R, r, is assumed sufficiently regular to allow applications of the divergence theorem. The method is not restricted to an exterior domain, however; appropriate details for a half-space follow from Section 5, mutaris mutandis.
Denote by BR the ball, radius R, and suppose the coordinate system is selected so that 0 E R. Then define RR to be RR=BR-Rwhen R is large enough that BR IJ Q. We suppose that equations (3.1), (3.4) and (3.6) hold in the region R x (0, T) for some prescribed time T ( > 0). For boundary conditions we suppose that u = a(x, t),
xeT*,
r~(0, 7-I
rm= 8(x,t),
XErZ,
tE(O,Tl
3 = 0(x, t),
XEI-,
TV@,
(4.1)
Tl
where r, u r2 = r, with rl # 0 (other types of mixed boundary conditions can be considered: such a case is encountered in Section 5). At infinity we require that u,
vu,
vu,
q-n,
3
(4.2)
grow no faster than an arbitrary exponential, linear in r ( E 1x1)as r + co. The initial data
G. CAVICLIA t-r al.
256
considered are u(x, 0) = G(x),
li(x, 0) = lP(x),
9(x, 0) = P(x), XER.
q(x, 0) = qO(xX
(4.3)
Since equations (3.1), (3.4) and (3.6) are linear and we are studying uniqueness and continuous dependence on the data we consider two solutions x = (v, 4). x+ = (v*, +*) which satisfy equations (3.1), (3.4) and (3.6) together with equations (4.1), (4.2) and (4.3). For uniqueness and continuous dependence on the initial data it is sufficient to consider x, x* satisfying equation (4.1) for the same data functions, with different u”, lie, go in the initial data study. At this point we concentrate on uniqueness and hence take x, x* to satisfy equation (4.3) for the same functions u”, 9’. Define u = v* - v, 0 = +* - C$and CTto be the total stress associated with u. The boundary-initial value for (u, 3) is then pOii=
[(dijrkl
+
Ailjk)Uj.k
-
(4.4)
Bil91.1
0
(4.5)
‘UijQj + 9.i + Uijqj = 0
(4.6)
CA + Bijtii. j -t iqi,
i =
equations (4.4H4.6) holding on R x (0, T]; bn = 0 on r2 x(O,T],
u=Oonr,x(O,T], u = 0,
6 = 0,
9 = 0,
q = 0,
$=Oonrx(O,T]
(4.7)
when t = 0.
(4.8)
We define Sij &.l + Ailjk = ciljk and suppose that ciljk 4jk
(4.9)
70
for some constant y. ( > 0), and (4.10) for some constant 7. We are now in a position to show the only solution to equations (4.4x4.8) is the zero solution. Let ( *)R denote integration over RR, e.g. (f)R
=
1
fdf’. n.
Define the function g by g(x) = exp( - rr). (4.11) Multiply eCptiOn (4.4) by gUi, integrate over RR, and then integrate by parts and use equations (4.7) and (4.2) to obtain $
[~
+
!!(gCiljkUi.Iuj,k)R]
=
-
(Bilg.lGi)R
+
-
QgCiljk
Uj.
kci
(4.12)
dS
where rR is the boundary of BR and dS is the surface element. Thanks to equation (4.7) the boundary integral over I- is zero. We let R + cc and due to equation (4.2) the integral over rR likewise vanishes, provided we pick CY large enough. Denoting by ( * ) integration over R, we find from equation (4.12) $~[
+
(gciljkui.I~i,k)l
=
-
+
1
~gciljkuj.k~i
>
.
t4.13)
Now, muhiply equation (4.5) by g9 and integrate over RR. After an integration by parts, and allowing R -+ co, we find
=Bij(gtii9.j) - i(qi,iSg) - ZBij
.
(4.14)
Finally, multiply equation (4.6) by gqi/8, integrate over RR, integrate by parts and allow
257
Thermoelasticity at cryogenic temperatures
R + cc, to obtain (4.15) Equations (4.13H4.15) are added together to find
We now use the arithmetic-geometric mean inequality on the right-hand side of equation (4.16), and employ the bounds (4.9) and (4.10). and the positive definiteness of aii to show that there exists a constant k such that, defining
F(t) =
1
+
(gCiljk”i,luj,k)
C(9’) +
+
i(Uijqiqj)
[
1
we derive from equation (4.16) Recalling the initial conditions (4.3), this inequality may be integrated to show vt E [O, T].
F(r) 5 0, Hence, we find Ui -
0,
9 3 0,
qi S 0
on R
X
[O, T].
This completes the proof of the uniqueness result. To establish continuous dependence on the initial data one argues similarly, although z has to be chosen carefully. The details of this calculation are easily worked out by combining the above analysis with the method of Galdi and Rionero [16] (see also [17]).
5. A HALF-SPACE
HEATED
ALONG
THE
BOUNDARY
We now have a general uniqueness theorem for the dynamical linear problem and turn to examine a mathematical counterpart to the experimental studies reported in the Introduction. Let x > 0 be the half-space occupied by the thennoelastic material. We restrict attention now to a pre-stressed isotropic body, and so equations (3.1), (3.4) and (3.6) hold with the constitutive equation (3.7), namely p,,ii = (z + p)Au + (/J + I.)V(V*u) - /YVS
(5.1)
cs+j?v.i++=o
(5.2)
Ttj+KVQ+q=o.
(5.3)
On the boundary x = 0 we suppose the temperature is prescribed, namely, S(O, Y, z, t) = &(Y, For our first result concerning the propagation u = constant,
z, 0.
(5.4)
of transverse waves we assume also that
QYX = u,, = 0,
on x = 0.
(5.5)
Here a is the total stress, namely a=aVu+ZpsymVu+IV.ul-jI91
(5.6)
and II = (u, u, w). Conditions (5.5) are evidently reasonable for a boundary for which 9, is constant in equation (5.4). By taking the curl of equation (5.1), we have for 4 = V x u POT= (a + PM
hr.&Y ‘7:2-1
(5.7)
G. CAVIGLIA ec al.
258
The components of C are i?r ?u --5;-.--cz zz
c
dw dc ---
ax'ax
(5.8)
The boundary conditions (5.5) are, on x = 0, u = constant,
(5.9) (5.10) (5.11)
Since u is constant on x = 0, it follows that Zu du -_E__= ~z - 0, ZL’
on x = 0
and so from equations (5.10) and (5.11) cc -z Z.u
0,
q z=O,
onx=O
and hence, <, = CZE 0
on x = 0.
(5.12)
Moreover, differentiating equations (5.10) and (5.1 l),
from which it follows that ai, -=0 L?.Y
onx=O.
(5.13)
Thus we have to solve equation (5.7) in (x > 0}, subject to equations (5.12) and (5.13). Consider an origin 0 on x = 0 and let RR be the hemisphere radius R centred on 0, for x > 0. We repeat the argument of Section 4 for a hemisphere to show V x u 3 0. Define rR to be the surface of the hemisphere and denote again by ( *)R integration over RR. For g:= exp( - Jr), 6 > 0 being a constant to be chosen and rz = x *x, we consider F(r) = (9PoZ% The growth conditions imposed are that t, Vc grow no faster than an arbitrary exponential in t. Then by a calculation similar to that of Section 4 we find k’ = 2(,~ + r)(j’.A& = - (2 +Y&IVC
where &IR is the whole boundary of RR. Observe that thanks to equations (5.12) and (5.13), the boundary integral over that part of dRR lying on x = 0 is zero. Next, let R + co in equation (5.14) and select 6 large enough to dominate any exponential growth. By using the arithmetic-geometric mean inequality in equation (5.14) we may easily determine a constant k, such that for G = F + (2 + cl)(gVC.VC)R it follows from equation (5.14) that d 5 k,G. Hence, if the half-space is initially at rest then C(x, 0) E i’(x, 0) 3 0,
x > 0,
259
Thermoelasticity at cryogenic temperatures
and so G(r) < 0 which leads to l E 0 and hence 4 = 0 in x > 0. Therefore, for the boundary conditions (5.5), and for a half-space initially at rest we necessarily have V x u z 0. This means that transverse waves do not exist. It is important to realize that the condition u E constant on x = 0 while seemingly reasonable for 3, E constant on x = 0 is not to be expected for inhomogeneous 3,. Hence, non-constant 3i may still lead to transverse waves. In an experiment, it will be difficult to achieve 3i constant conditions. We finally consider the problem of solving equations (5.1H5.4) for a half-space initially at rest, i.e. u=ri~O inx>O. It is possible to write down a formal solution to this problem in terms of inverse Laplace and Fourier transforms. However, we have not been able to fully determine the inversions. We finish by including a heuristic solution for the equivalent problem for a rigid heat conductor. Despite the simplicity of the model, the difficulties encountered here are non-trivial. For a rigid heat conductor we consider the problem: (5.15)
r3 + 3 = KAQ, in (x > 0, (y, Z)E R*} with 3 - 9 c 0,
at t = 0
(5.16)
and (5.17)
3(0, y, f, f) = 3, (Y, z, t).
We take the Laplace transform ofequation (5.15) in I and the Fourier transform of the result in y and in z. We next identify the result with the corresponding one-space dimensional problem in ([18], p. 202). Using the method there and using the Fourier inversion theorem, see e.g. [19], we find
where cr = ;Jl
- 4Kr(4* + s2)
and Ii is the standard Bessel function. Of course, care must be taken with the inverse Fourier transforms since the expression inside the square root in CJchanges sign: certainly a technique such as that described on p. 122 of [19] will be required. In view of the difficulty in the above example, the corresponding thermoelastic problem presents a challenge: its solution would, however, be very valuable in connection with the experiments mentioned in the Introduction. 6. THE
THREE-DIMENSIONAL WITH
UNI-DIRECTIONAL
HALF-SPACE
PROBLEM
DEPENDENCE
In we consider the half-space problem of Section 5, but assume u = u(x, t), 3 = 3(x, t), q = q(x, t), then equations (3.1), (3.4), (3.6) and (3.7) reduce to
pOii = (2 + 2~ + i.)u” - B3’
(6.1)
poi;’ = (2 + jJ)u”
(6.2)
poz = (2 + /J) WI
(6.3)
G. CAVKXIAer 01.
260
tcj + h-3’ + q =
0,
Trjv + 4” = 0
(6.5)
where u = (tc, c, w), q = qx, qy denote qu and q._, and a prime “ ’ ” denotes differentiation with respect to x. The above equations hold on {X > Oj x R* x (0, oz). For an initially quiescent body we have the initial conditions uEilr3eqE0,
I = 0.
W5)
Furthermore, we assume that on the half-space boundary x = 0 we have the temperature prescribed and stress-free conditions, namely 30 t) = 3,(f)
(6.7)
6,, = (z + 2p + i.)u’ - p3 = 0
(6-S)
We show that the boundary-initial value problem (6.1)-(6.10) reduces strictly to a one space-dimension problem, i.e. we demonstrate c E w 5 qy 3 q_ = 0. Before proving this result, we remark that in practice this will not preclude transverse waves. Certainly the boundary heating is never uniform, and in a finite region expansion effects induce c, w components which in turn will enhance the propagation of a transverse wave. Nevertheless, from the mathematical point of view our result is precise: it shows the need for a three-dimensional solution. Of course, the one-dimensional problem is of interest in its own right and so we include an approximate analysis for a one-dimensional heat pulse. From equations (6.5) we may immediately deduce that qy(x, t) = q&x, O)exp( - rt) = 0.
The show u is zero (the case for w follows in the same manner, mutatis murandis) observe that c satisfies C-s2r”=O, ti’ = 0 at x = 0; in (.x > Oj x (t > 0); r:=tY!=O
atr=O
(6. I 1)
where we have set s2 = (x -t &/pO. In addition we require Ic’I, 161have at most exponential growth in X, as .Y-+ 3~.
(6.12)
The idea is to use a weighted energy technique (cf. [16]) as in Sections 4 and 5. BY introducing g(x) = exp( - 4.x) (b > 0, it can be shown that i zzz+s2$E where
* W) = t
gti2 dx -t- s2 jOx U
got2
d-x).
0
Hence, E(t)lO*u=O since c’= 0 when t = 0. The problem (6.1)-(6.10), therefore, reduces to solving poii = (x + :” + i.)u” - B3’ c3 + /?a’ + ;q’ = 0
(6.13)
I 54 -t h.3’ + q = 0 in {x > Oj x (r
7
0}, together with u=ti=$=q=O
whenr=O
(6.14)
Thermoelasticity
at cryogenic
261
temperatures
and (z + 2,~ + j.)u’ = 83
3(0, f) = 3,(r);
at x = 0.
(6.15)
A formal solution is easily found by using Laplace transforms (cf. [lo], [ 133). The idea is to introduce the Laplace transforms of U, 3, 4. e.g. 0 W(x,p) = exp( - pt)u(x, r)dt I0 and then transform equations (6.13) and (6.15). We specifically consider a boundary condition of a heat pulse, (6.16)
3,(t) = 3, [H(t) - H(t - E)]
where 3, is a constant. By transforming (6.13) and eliminating variables, Is, $ are seen to satisfy (6.17) where h = U or 9, and where n = (x + 2~ + &‘po, b = x/(1 + rp)O, and the boundary conditions become
where we have put f(P)
=
1 - exp( - cp)
p
The roots of equation (6.17) satisfy the fourth-order abm’ -
.
equation
PP’ f cp’ = 0 acp + bp2 + PO ) (
and neglecting growing solutions (in x) at infinity we find (!Rm, 2 0) U = dr exp( - mix) -I- d2exp( - m2x),
3 = dJexp( - mix) + &exp( - m2x).
From the boundary conditions and compatibility with equation (6.13), d,, dz, dJ, d4 are given by &Bmrf hh2f dr = d 2=poa(mS - mf)’
p04d
d4
=
_
-
d)
b2 - ami)3tf. a(m: - mf)
An asymptotic solution for p 4 1 (which may be interpreted as yielding the short time, t + 1, behaviour) may be found as in [lo]. By carrying out expansions for m, and for the appropriate terms in dr , . . . , d4, and performing the inverse Laplace transform it may be shown that u(u’ , t)
2:
Qexp( Pea
-h,x)~-~[(r-~)H(r-~)-(r-~-E)H(I-~-~)]
+~(+,h,)[(r-~)2H(l-~)-(i--~-L)2H(I-~-C)]} 03 - =exp(
- h2x) { t - v“f(i-$(I-;)-(+)ff(+)]
Xi?
G.
3(x, t)
”
CAVIGLIAet al.
+{(,sI- l)[H(I-~)-H(I-;-E)]exp(-hlr) L)[H(I-6)-H(r-;-E)]exp(-h2.v,
-(at2-
+a(T1 -
+[l
-(I-;--E)H(I-~-E)]eXP(-h,.y~
in which the various constants have the forms v
1,2
2ah-
-
)
( MfJZ
1.2-
h I.2 =
1
N _t J=
23!2\/CC
N=ac+c
i\/I = PO’
St =
(N/M - 2ah.c)
(A4 + &P--zG)“2 UC +
s
UK
P’ -
T f
“‘342 - 4dlCcs’
=
2
UK(MN
-
=
2ati
2UKc)
JXK-ZZ
M&_/z-zE 5 1.2
K
PO >
(
N,/= ’
CL.2
=
+ (NM 2alCJZ
?KaC)
.
From the mathematical viewpoint it is worth observing formally the above solution admits the possibility that T + 0 * V2-+ x, a situation which Sharma [lo] interprets as being consistent with T + 0 meaning the classical theory with infinite speed of thermal waves is approached. What our theory predicts is thus mathematically consistent with previous work. However, it must be stressed that since c = c(0), 5 = s(O), we cannot allow r -+ 0 in V,. The function r is limited below as shown in Section 1. Thus, we have a means to yield a fast elastic wave and a thermal wave which behaves in accordance with the experimental findings, namely that the speed of the thermal wave decreases as the mean temperature is increased. Of course, it has to be noted that the thermal wave had been observed at cryogenic temperatures. Acknowledgement-This research was performed supported by the Italian CNR. Partial support
while B. S. was a Visiting Professor, at the University of Genova. to G.C. and A.M. from CNR is gratefully acknowledged.
REFERENCES 1. C. C. Ackerman and W. C. Overton, Second sound in Helium-3. Phys. Rev. Lett. 22, 764-766 (1969). 2. H. E. Jackson, C. T. Walker and T. F. .McNelly. Second sound in NaF. Phys. Rev. Letr. 25. 26-29 (1970). 3. T. F. McNelly, S. J. Rogers, D. J. Channin, R. J. Rollefson, W. M. Goubau, G. E. Schmidt, J. A. Krumhansl and R. 0. Pohl, Heat pulses in NaF: onset of second sound. Phys. Rev. Left. 24. 100-102 (1970). 4. R. J. Hardy and S. S. Jaswal, Veloc;ty of second sound in NaF. Phys. Rea. B3. 4385-4387 (1971). 5. V. Narayanamurti and R. C. Dynes, Observation of second sound in Bismuth. Phys. Rec. Left. 28. 1401-1405 (1972). 6. B. D. Coleman and D. C. Newman, Implications of a nonlinearity in the theory of second sound in solids. Php. Rev. B37, 1492-1498 (1988). 7. A. Morro and T. Ruggeri, Non-equilibrium properties of solids obtained from second-sound measurements. J. Phys. C: Solid State Phys. 21, 1743-1752 (1988).
Thermoelasticity at cryogenic temperatures
263
8. C. Cattaneo. Sulla conduzione del calore. Arri St-m. &fat. Fis. Lenin.‘tiodena 3. 83101 (1948). 9. T. Ruggeri, A. Muracchini and L. Seccia, Shock waves and second sound in a rigid heat conduction. A critical temperature for NaF and Bi. Phys. Rev. L.etr. 64, 2640-2643. 10. J. N. Sharrna, Some considerations on generalized thermoelasticity. Int. J. engng Sci. 25. 1387-1392 (1987). Il. F. J. Locket1 and I. N. Sneddon. Propagation of thermal stresses in an infinite medium. Proc. Edinburgh Math. Sot. 11, 237-244 (1959). 12. 1. N. Sneddon, The Linear Theory 01 Thermoelasricity, CIME Vol. 119. Springer, Wien (1974). 13. C. V. Massalas. E. Anagnostaki and V. K. Kalpakidis. Some considerations on the coupled thermoelastic problems. Inr. J. engng Sci. 23, 677-683 (1985). 14. D. S. Chandrasekharaiah, Tberrnoelasticity with second sound: a review. Appl. Mech. Rec. 39, 355-376 (1986). 15. D. Iesan, Prestressed Bodies. Pitman Res. Notes in Math.. Longman. Harlow (1989). 16. G. P. Galdi and S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity. Springer, Berlin (1985). 17. B. Straughan, Nonexistence, Insrability and Weighted Energy Methods in Fluid Dynamics ond Related Theories. Pitman, London (1982). 18. H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Morhemocics. Dover, New York (1963). 19. I N. Sneddon, Fourier Transforms. McGraw-Hill, New York (1951).
IN. 1. Non-Linear Mechanics. Vol. 27, No. 2 pp. 251-263. 1992 Printed in Great Britain.
THERMOELASTICITY
Pergamon Reu
AT CRYOGENIC
plc
TEMPERATURES
G. CAMGLIA Dipartimento di Matematica, Universita di Geneva, 16132 Geneva, Italy
A. MORRO DIBE, Universita di Genova, 16145 Genova, Italy and B. STRAUGHAN Department of Mathematics, University of Glasgow, Glasgow G12 SQW, Scotland, U.K. (Received 21 September 1990; accepted 2 January 1991)
Abstract-A theory of thermoelasticity is developed which is suitable for application at cryogenic temperatures. The thermodynamic functions are so chosen as to give correct predictions with experimental findings of second-sound wave speeds in NaF, in the temperature range 8-20 K and in Bi, in the temperature range l-4 K. After the non-linear theory is presented, a linearized theory is developed. A uniqueness theorem is provided for the linearized theory on an unbounded domain. The question of a half-space heated on its boundary is addressed and, in particular, the question of transverse elastic wave propagation is studied. Finally, a simplified theory in which only unidirectional solutions are allowed is examined.
1. INTRODUCTION
The phenomenon of second-sound (or thermal pulse propagation) has been experimentally observed under accurate conditions in solid He’ and He4 (see, e.g. [l] and the references therein), in sodium fluoride C2-43, in bismuth [S] and in sodium iodide and lithium fluoride [3]. Thermal wave propagation is a low-temperature phenomenon; for example, the above experiments are in the range l-20 K. It is found that the thermal wave speed decreases appreciably as the temperature increases and tends to an asymptotic value as absolute zero is approached, see [3]. In a recent paper Coleman and Newman [6] study heat propagation in a rigid heat conductor and curve-fit thermal wave speeds to mean temperature for NaF and Bi. They find an excellent fit with the empirical formula 1
uz=-
A + Be
(1.1)
where U (cm/s) is the thermal wave speed, and for NaF in the temperature (0) range 10.0 I, 8 I; 18.5 K, they select A = 9.09 x 10-12, B = 2.22 x 10-i’, n = 3.10, while for Bi in the range 1.4 I 0 I 4.0 K, they select A = 9.07 x lo-“, B = 7.58 x lo-i3, n = 3.75. A thermodynamically consistent theory for thermal pulse propagation was recently developed by Morro and Ruggeri [7]. Their paper develops a non-linear theory for which the constitutive variables depend on the temperature only. Basically their equations are
v-q = 0 T(e)4+ de + (1+ rf$q = 0 c,(e)e +
(l-2) (1.3)
where q is the heat flux, K is a thermal conductivity, a superposed dot denotes (partial) time differentiation, and I- = - K[5Ke-4 + (5 - @e+] (1.4) T = K(A’e-3 + w-3)
(1.5)
where A = A/E, B’ = B/E, E being a constant; for NaF, E = 23 erg Ke4 gives a good fit, [6]. The function co is the equilibrium specific heat and Coleman and Newman [6] obtain an Contributed by K. R. Rajagopal. 251
G. CAVIGLIA er al
252
excellent data fit with
co(e) = &P.
(1.6)
While equation (1.3) bears some resemblance to the classical Cattaneo theory [S], a fundamental difference is that T is not constant. Certainly, T is bounded below, and so the commonly used assertion that the limit r + 0 in Cattaneo’s theory (T is Cattaneo’s relaxation time) yields the classical theory of heat conduction does not apply here. However, the thermal
wave speed of [7] satisfies
and this is matched to the empirical fit ofequation (l.l), [6], Thus the Morro-Ruggeri (MR) theory is in agreement with experimentally observed wave speeds, but the parameter Tcannor physically go to zero (although for mathematical purposes it may be convenient to sometimes allow this). Shock behaviour in the MR theory is analysed in [9]. While theoretical work on second-sound has concentrated on explaining the behaviour of the thermal wave, the experiments of McNelly er al. [3] and Jackson et al. [2] demonstrate the existence of three distinct waves. They used a very pure crystal of NaF and evaporated manganin heaters and lead detectors onto opposing faces of the crystal and hence transmitted heat pulses through the sample. Their observations clearly indicate a longitudinal elastic wave which travels fastest, a transverse elastic wave and a temperature wave. For temperatures below 8 K the transverse wave and the thermal wave appear to travel with the same speed, or at least the experiment was unable to detect any appreciable difference. However, at temperatures above 8 K the temperature wave slows down and three distinct waves are observed, the longitudinal one being fastest, transverse next, and the thermal one slowest. In order to theoretically examine the experiments outlined above it is necessary to construct an appropriate theory of thermoelasticity. It is to this goal that our paper is addressed. We derive a theory of thermoelasticity by analogy with the MR rigid heat conductor theory and investigate the consequences. Other second-sound thermoelastic theories (see e.g. [lo]), allow the Cattaneo parameter T to go to zero and hence identify the thermal wave as the fast one: this is clearly incompatible with the experimental findings. According to the MR model, the limit T ---t0 is not allowed and so we have a chance of obtaining good agreement with experiments. The full problem of explaining three-dimensional thermoelastic pulse propagation is very difficult. Previous studies of the subject testify to this. Early work of Lockett and Sneddon [l l] (see also [123), shows how the problem of a half-space heated along its boundary may be treated by Fourier transform methods, but no attempt is made to explicitly calculate the three-dimensional inverse transforms. Likewise, later work has been unable to solve the three-dimensional problem and has concentrated on one-dimensional solutions (see [13] for the classical theory and [lo, 143 for various second-sound theories). We are here unable to fully resolve the three-dimensional problem, but we produce several useful results which help to provide a global picture. An outline of the paper is now given. Section 2 develops a non-linear theory, consistent with that of [7] for a rigid solid, employing an internal-variable theory. This theory is linearized in Section 3; even though the equations given there are linear, the coefficients depend on the equilibrium temperature and so the feature of the correct behaviour of the speed of a thermal wave is retained. In Section 4 we examine the well-posedness of the linear theory in an unbounded domain. Section 5 treats the question of transverse waves in a half-space heated along the wall; the uniqueness theorem of Section 4 is thus relevant. Finally, in Section 6, we examine the half-space problem when dependence on only one space v&able is allowed. It is proved that this simplification problem to a truly one-dimensional one. For completeness, one-dimensional problem is given. 2. THERMODYNAMIC
reduces the three-dimensional an approximate solution to the
SCHEME
Since we have in mind a thermoelastic solid we let the Cauchy stress tensor T, the heat flux q and the free energy $ be functions of the deformation gradient F, the temperature 0, the spatial temperature gradient g, and an internal vector variable 5. By the material
253
Thcrmoelasticity at cryogenic temperatures
objectivity we conclude that T = F?;(C, 8, G, E)FT q = Fij(C,f?, G, 5) $ = $(C, 8, G, Z)
where T, Q, $ are arbitrary
on Sym x Rx Vx V and C = FTF, G = FTg,
functions
1 _ FTC. _Let Y = JF- l T(F- l)T be the second Piola-Kirchhoff
stress tensor, Q = JF- ‘q the heat and p,, the reference mass density. The entropy
flux vector in the reference configuration inequality can be given the form
- po($+@)+fY*C-;Q*G>O for the constitutive functions 6, 4, P, Q of C, 8, G, Z. Accordingly -po(~c.~+~~e+~~.~+~~.~+rlB)+~Y.~-$Q.G~O and this holds if and only if Jc = 0 tl= -$0,
(2.1)
Y = 2Po6c
pot&4
+Q*GsO
(2.2)
where the subscripts C, 0, G, E denote partial derivatives. Then any model is admjssible provided that II/ = $(C, 8, E) and q, Y, Q satisfy equations (2.1) and (2.2). Of course the solution to equations (2.1) and (2.2) is non-unique. We look for quite a general non-linear model which allows for wave propagation and generalizes Fourier’s law of heat conduction. For definiteness, letting Q be independent of G, we start from the evolution equation for Z in the form & z.=mG-nZ (2.3) where m, n are functions of 8, C and n > 0. Substitution in equation (2.2) leads to Q=p,@&,
&.Z>O.
To determine E we require that, under stationary the form
conditions,
(2.4) Fourier’s law is recovered in
Q=-KG
(2.5)
K being a function of 8, C whose values are positive-definite tensors. Now, as e = 0, by equation (2.3) it follows that
G=
-;z.
This suggests that we set Q=;KI.
(2.6)
Then the first relation in equation (2.4) becomes
whence lj/=lJ(e,C)+~=*Kz 2p, em2-
The inequality in equation (2.4) holds identically. The internal energy e = $ + 8q is given by e=$-eJg++~.
(mnK -*d-2!&
de poem
2.
>
254
G. CAWGLIA
et al.
It is convenient, and not severely restrictive, to let e be independent of Z:, and then of Q. This occurs if and only if nK = M(C). a
In addition we take R to be positive definite. In conclusion, the constitutive equations are given by $==$(e,C)+;Z.Rr
(2.7)
Y=2p&++R,P If= -&-)Z.K~
(2.8)
along with equation (2.3). The positive definiteness of ft ensures’q is maximal at equilibrium. Further, we can write the balance equations as poii = Vx.S + pOb
- p,e&d
(2.9)
+ I+&~.~) = - Vx~(p&mZ)
(2.10)
where the relations (2.1) have been taken into account. 3. LINEARIZED
VERSION
OF
THE DYNAMIC
PROBLEM
Also with a view to standardizing experimental settings it is worth examining the dynamics of a thermoelastic solid in the linearized version. Consider an equilibrium configuration 9 and denote by W* the present configuration. The particle occupying the position X in the reference configuration gc, is taken to be at x in 9I and at x* in W*. In W we let the temperature 8 be uniform and then G = 0, Z = 0. The corresponding values in 9* are denoted by 0*, G*, E*. Accordingly, denote by C*, C the values of the Cauthy-Green tensor in 9*, W. Let
u = x* - x,
3 = e* - 8,
A=E*.
Linearization consists in writing the model equations to the first order in u, 9, A. Then we proceed by keeping only the terms up to the first-order. By direct substitution we have C* - C = ZF’(sym Vu)F. By equations (2.7) and (2.8) Y* = 2p,[&
+ &c*
- C) + $&].
Letting Y be the value in 9, namely the Piola-Kirchhoff
pre-stress tensor, we have
Y* = Y + A(F’VuF) - B9 where A = 2p0 $cc, B = - 2p,J0c. On substituting in p,,%* = Vx-S* + peb
rearranging and taking into account the equilibrium equation Vx.FY +pOb=O we obtain in component notation, POiii = [(SijJFNh’FKI1 Gl + ~iHFjM~HK.HN)Uj.N - FiHBHKQl.K, where T is the Cauchy pre-stress tensor. In the present case F = I and thus we obtain POci = [(6ijL +
AiljkJUj.t
-
Bil$l.I*
The balance of energy, Poe*[&8(e*,c*)f9*+ tJOc(e*,c*)mc*~= v~q~~m(e*)Wc*)n*l
(3.1)
Thermoclasticity
in the linear approximation
at cryogenic
255
temperatures
becomes
&A + BBS(sym Vu) = - p,m@R*(VA)
(3.2)
where c = - P,,$~,+ Similarly, the evolution equation for A takes the form A + mV3 + nA = 0.
(3.3)
Of course the quantities c, m, n are evaluated at the configuration 9 where the temperature 8 is uniform and C = 1. Since F = 1 and A = Z, the identification, [equation (2.6)]
allows us to rewrite equation (3.3) as ratj + V3 + aq = 0
(3.4)
where we have set T=n-‘,
8=&R-‘.
(3.5)
Also, it is convenient to rewrite equation (3.2) as c$ + B*sym(Vir) = -iv-q.
(3.6)
In the isotropic case Aijrl
I where
K
=
iJij6,,
1;:j
=
rdij
Bij
=
BSij
Qij =
+
/l(6i,6js
+
6&6jr)
(3.7)
h*-qij
is a constant. 4.
UNIQUENESS
AND
CONTINUOUS
FOR
AN UNBOUNDED
DEPENDENCE
ON THE
DATA
DOMAIN
Since we envisage the problem of heating the boundary of a half-space we include in this section a method for establishing uniqueness and continuous dependence on the data for a general unbounded domain. We treat the general linear, anisotropic case for a domain R*, exterior to a bounded domain C!, in W3. The boundary of R, r, is assumed sufficiently regular to allow applications of the divergence theorem. The method is not restricted to an exterior domain, however; appropriate details for a half-space follow from Section 5, mutaris mutandis.
Denote by BR the ball, radius R, and suppose the coordinate system is selected so that 0 E R. Then define RR to be RR=BR-Rwhen R is large enough that BR IJ Q. We suppose that equations (3.1), (3.4) and (3.6) hold in the region R x (0, T) for some prescribed time T ( > 0). For boundary conditions we suppose that u = a(x, t),
xeT*,
r~(0, 7-I
rm= 8(x,t),
XErZ,
tE(O,Tl
3 = 0(x, t),
XEI-,
TV@,
(4.1)
Tl
where r, u r2 = r, with rl # 0 (other types of mixed boundary conditions can be considered: such a case is encountered in Section 5). At infinity we require that u,
vu,
vu,
q-n,
3
(4.2)
grow no faster than an arbitrary exponential, linear in r ( E 1x1)as r + co. The initial data
G. CAVICLIA t-r al.
256
considered are u(x, 0) = G(x),
li(x, 0) = lP(x),
9(x, 0) = P(x), XER.
q(x, 0) = qO(xX
(4.3)
Since equations (3.1), (3.4) and (3.6) are linear and we are studying uniqueness and continuous dependence on the data we consider two solutions x = (v, 4). x+ = (v*, +*) which satisfy equations (3.1), (3.4) and (3.6) together with equations (4.1), (4.2) and (4.3). For uniqueness and continuous dependence on the initial data it is sufficient to consider x, x* satisfying equation (4.1) for the same data functions, with different u”, lie, go in the initial data study. At this point we concentrate on uniqueness and hence take x, x* to satisfy equation (4.3) for the same functions u”, 9’. Define u = v* - v, 0 = +* - C$and CTto be the total stress associated with u. The boundary-initial value for (u, 3) is then pOii=
[(dijrkl
+
Ailjk)Uj.k
-
(4.4)
Bil91.1
0
(4.5)
‘UijQj + 9.i + Uijqj = 0
(4.6)
CA + Bijtii. j -t iqi,
i =
equations (4.4H4.6) holding on R x (0, T]; bn = 0 on r2 x(O,T],
u=Oonr,x(O,T], u = 0,
6 = 0,
9 = 0,
q = 0,
$=Oonrx(O,T]
(4.7)
when t = 0.
(4.8)
We define Sij &.l + Ailjk = ciljk and suppose that ciljk 4jk
(4.9)
70
for some constant y. ( > 0), and (4.10) for some constant 7. We are now in a position to show the only solution to equations (4.4x4.8) is the zero solution. Let ( *)R denote integration over RR, e.g. (f)R
=
1
fdf’. n.
Define the function g by g(x) = exp( - rr). (4.11) Multiply eCptiOn (4.4) by gUi, integrate over RR, and then integrate by parts and use equations (4.7) and (4.2) to obtain $
[~
+
!!(gCiljkUi.Iuj,k)R]
=
-
(Bilg.lGi)R
+
-
QgCiljk
Uj.
kci
(4.12)
dS
where rR is the boundary of BR and dS is the surface element. Thanks to equation (4.7) the boundary integral over I- is zero. We let R + cc and due to equation (4.2) the integral over rR likewise vanishes, provided we pick CY large enough. Denoting by ( * ) integration over R, we find from equation (4.12) $~[
+
(gciljkui.I~i,k)l
=
-
+
1
~gciljkuj.k~i
>
.
t4.13)
Now, muhiply equation (4.5) by g9 and integrate over RR. After an integration by parts, and allowing R -+ co, we find
=Bij(gtii9.j) - i(qi,iSg) - ZBij
.
(4.14)
Finally, multiply equation (4.6) by gqi/8, integrate over RR, integrate by parts and allow
257
Thermoelasticity at cryogenic temperatures
R + cc, to obtain (4.15) Equations (4.13H4.15) are added together to find
We now use the arithmetic-geometric mean inequality on the right-hand side of equation (4.16), and employ the bounds (4.9) and (4.10). and the positive definiteness of aii to show that there exists a constant k such that, defining
F(t) =
1
+
(gCiljk”i,luj,k)
C(9’) +
+
i(Uijqiqj)
[
1
we derive from equation (4.16) Recalling the initial conditions (4.3), this inequality may be integrated to show vt E [O, T].
F(r) 5 0, Hence, we find Ui -
0,
9 3 0,
qi S 0
on R
X
[O, T].
This completes the proof of the uniqueness result. To establish continuous dependence on the initial data one argues similarly, although z has to be chosen carefully. The details of this calculation are easily worked out by combining the above analysis with the method of Galdi and Rionero [16] (see also [17]).
5. A HALF-SPACE
HEATED
ALONG
THE
BOUNDARY
We now have a general uniqueness theorem for the dynamical linear problem and turn to examine a mathematical counterpart to the experimental studies reported in the Introduction. Let x > 0 be the half-space occupied by the thennoelastic material. We restrict attention now to a pre-stressed isotropic body, and so equations (3.1), (3.4) and (3.6) hold with the constitutive equation (3.7), namely p,,ii = (z + p)Au + (/J + I.)V(V*u) - /YVS
(5.1)
cs+j?v.i++=o
(5.2)
Ttj+KVQ+q=o.
(5.3)
On the boundary x = 0 we suppose the temperature is prescribed, namely, S(O, Y, z, t) = &(Y, For our first result concerning the propagation u = constant,
z, 0.
(5.4)
of transverse waves we assume also that
QYX = u,, = 0,
on x = 0.
(5.5)
Here a is the total stress, namely a=aVu+ZpsymVu+IV.ul-jI91
(5.6)
and II = (u, u, w). Conditions (5.5) are evidently reasonable for a boundary for which 9, is constant in equation (5.4). By taking the curl of equation (5.1), we have for 4 = V x u POT= (a + PM
hr.&Y ‘7:2-1
(5.7)
G. CAVIGLIA ec al.
258
The components of C are i?r ?u --5;-.--cz zz
c
dw dc ---
ax'ax
(5.8)
The boundary conditions (5.5) are, on x = 0, u = constant,
(5.9) (5.10) (5.11)
Since u is constant on x = 0, it follows that Zu du -_E__= ~z - 0, ZL’
on x = 0
and so from equations (5.10) and (5.11) cc -z Z.u
0,
q z=O,
onx=O
and hence, <, = CZE 0
on x = 0.
(5.12)
Moreover, differentiating equations (5.10) and (5.1 l),
from which it follows that ai, -=0 L?.Y
onx=O.
(5.13)
Thus we have to solve equation (5.7) in (x > 0}, subject to equations (5.12) and (5.13). Consider an origin 0 on x = 0 and let RR be the hemisphere radius R centred on 0, for x > 0. We repeat the argument of Section 4 for a hemisphere to show V x u 3 0. Define rR to be the surface of the hemisphere and denote again by ( *)R integration over RR. For g:= exp( - Jr), 6 > 0 being a constant to be chosen and rz = x *x, we consider F(r) = (9PoZ% The growth conditions imposed are that t, Vc grow no faster than an arbitrary exponential in t. Then by a calculation similar to that of Section 4 we find k’ = 2(,~ + r)(j’.A& = - (2 +Y&IVC
where &IR is the whole boundary of RR. Observe that thanks to equations (5.12) and (5.13), the boundary integral over that part of dRR lying on x = 0 is zero. Next, let R + co in equation (5.14) and select 6 large enough to dominate any exponential growth. By using the arithmetic-geometric mean inequality in equation (5.14) we may easily determine a constant k, such that for G = F + (2 + cl)(gVC.VC)R it follows from equation (5.14) that d 5 k,G. Hence, if the half-space is initially at rest then C(x, 0) E i’(x, 0) 3 0,
x > 0,
259
Thermoelasticity at cryogenic temperatures
and so G(r) < 0 which leads to l E 0 and hence 4 = 0 in x > 0. Therefore, for the boundary conditions (5.5), and for a half-space initially at rest we necessarily have V x u z 0. This means that transverse waves do not exist. It is important to realize that the condition u E constant on x = 0 while seemingly reasonable for 3, E constant on x = 0 is not to be expected for inhomogeneous 3,. Hence, non-constant 3i may still lead to transverse waves. In an experiment, it will be difficult to achieve 3i constant conditions. We finally consider the problem of solving equations (5.1H5.4) for a half-space initially at rest, i.e. u=ri~O inx>O. It is possible to write down a formal solution to this problem in terms of inverse Laplace and Fourier transforms. However, we have not been able to fully determine the inversions. We finish by including a heuristic solution for the equivalent problem for a rigid heat conductor. Despite the simplicity of the model, the difficulties encountered here are non-trivial. For a rigid heat conductor we consider the problem: (5.15)
r3 + 3 = KAQ, in (x > 0, (y, Z)E R*} with 3 - 9 c 0,
at t = 0
(5.16)
and (5.17)
3(0, y, f, f) = 3, (Y, z, t).
We take the Laplace transform ofequation (5.15) in I and the Fourier transform of the result in y and in z. We next identify the result with the corresponding one-space dimensional problem in ([18], p. 202). Using the method there and using the Fourier inversion theorem, see e.g. [19], we find
where cr = ;Jl
- 4Kr(4* + s2)
and Ii is the standard Bessel function. Of course, care must be taken with the inverse Fourier transforms since the expression inside the square root in CJchanges sign: certainly a technique such as that described on p. 122 of [19] will be required. In view of the difficulty in the above example, the corresponding thermoelastic problem presents a challenge: its solution would, however, be very valuable in connection with the experiments mentioned in the Introduction. 6. THE
THREE-DIMENSIONAL WITH
UNI-DIRECTIONAL
HALF-SPACE
PROBLEM
DEPENDENCE
In we consider the half-space problem of Section 5, but assume u = u(x, t), 3 = 3(x, t), q = q(x, t), then equations (3.1), (3.4), (3.6) and (3.7) reduce to
pOii = (2 + 2~ + i.)u” - B3’
(6.1)
poi;’ = (2 + jJ)u”
(6.2)
poz = (2 + /J) WI
(6.3)
G. CAVKXIAer 01.
260
tcj + h-3’ + q =
0,
Trjv + 4” = 0
(6.5)
where u = (tc, c, w), q = qx, qy denote qu and q._, and a prime “ ’ ” denotes differentiation with respect to x. The above equations hold on {X > Oj x R* x (0, oz). For an initially quiescent body we have the initial conditions uEilr3eqE0,
I = 0.
W5)
Furthermore, we assume that on the half-space boundary x = 0 we have the temperature prescribed and stress-free conditions, namely 30 t) = 3,(f)
(6.7)
6,, = (z + 2p + i.)u’ - p3 = 0
(6-S)
We show that the boundary-initial value problem (6.1)-(6.10) reduces strictly to a one space-dimension problem, i.e. we demonstrate c E w 5 qy 3 q_ = 0. Before proving this result, we remark that in practice this will not preclude transverse waves. Certainly the boundary heating is never uniform, and in a finite region expansion effects induce c, w components which in turn will enhance the propagation of a transverse wave. Nevertheless, from the mathematical point of view our result is precise: it shows the need for a three-dimensional solution. Of course, the one-dimensional problem is of interest in its own right and so we include an approximate analysis for a one-dimensional heat pulse. From equations (6.5) we may immediately deduce that qy(x, t) = q&x, O)exp( - rt) = 0.
The show u is zero (the case for w follows in the same manner, mutatis murandis) observe that c satisfies C-s2r”=O, ti’ = 0 at x = 0; in (.x > Oj x (t > 0); r:=tY!=O
atr=O
(6. I 1)
where we have set s2 = (x -t &/pO. In addition we require Ic’I, 161have at most exponential growth in X, as .Y-+ 3~.
(6.12)
The idea is to use a weighted energy technique (cf. [16]) as in Sections 4 and 5. BY introducing g(x) = exp( - 4.x) (b > 0, it can be shown that i zzz+s2$E where
* W) = t
gti2 dx -t- s2 jOx U
got2
d-x).
0
Hence, E(t)lO*u=O since c’= 0 when t = 0. The problem (6.1)-(6.10), therefore, reduces to solving poii = (x + :” + i.)u” - B3’ c3 + /?a’ + ;q’ = 0
(6.13)
I 54 -t h.3’ + q = 0 in {x > Oj x (r
7
0}, together with u=ti=$=q=O
whenr=O
(6.14)
Thermoelasticity
at cryogenic
261
temperatures
and (z + 2,~ + j.)u’ = 83
3(0, f) = 3,(r);
at x = 0.
(6.15)
A formal solution is easily found by using Laplace transforms (cf. [lo], [ 133). The idea is to introduce the Laplace transforms of U, 3, 4. e.g. 0 W(x,p) = exp( - pt)u(x, r)dt I0 and then transform equations (6.13) and (6.15). We specifically consider a boundary condition of a heat pulse, (6.16)
3,(t) = 3, [H(t) - H(t - E)]
where 3, is a constant. By transforming (6.13) and eliminating variables, Is, $ are seen to satisfy (6.17) where h = U or 9, and where n = (x + 2~ + &‘po, b = x/(1 + rp)O, and the boundary conditions become
where we have put f(P)
=
1 - exp( - cp)
p
The roots of equation (6.17) satisfy the fourth-order abm’ -
.
equation
PP’ f cp’ = 0 acp + bp2 + PO ) (
and neglecting growing solutions (in x) at infinity we find (!Rm, 2 0) U = dr exp( - mix) -I- d2exp( - m2x),
3 = dJexp( - mix) + &exp( - m2x).
From the boundary conditions and compatibility with equation (6.13), d,, dz, dJ, d4 are given by &Bmrf hh2f dr = d 2=poa(mS - mf)’
p04d
d4
=
_
-
d)
b2 - ami)3tf. a(m: - mf)
An asymptotic solution for p 4 1 (which may be interpreted as yielding the short time, t + 1, behaviour) may be found as in [lo]. By carrying out expansions for m, and for the appropriate terms in dr , . . . , d4, and performing the inverse Laplace transform it may be shown that u(u’ , t)
2:
Qexp( Pea
-h,x)~-~[(r-~)H(r-~)-(r-~-E)H(I-~-~)]
+~(+,h,)[(r-~)2H(l-~)-(i--~-L)2H(I-~-C)]} 03 - =exp(
- h2x) { t - v“f(i-$(I-;)-(+)ff(+)]
Xi?
G.
3(x, t)
”
CAVIGLIAet al.
+{(,sI- l)[H(I-~)-H(I-;-E)]exp(-hlr) L)[H(I-6)-H(r-;-E)]exp(-h2.v,
-(at2-
+a(T1 -
+[l
-(I-;--E)H(I-~-E)]eXP(-h,.y~
in which the various constants have the forms v
1,2
2ah-
-
)
( MfJZ
1.2-
h I.2 =
1
N _t J=
23!2\/CC
N=ac+c
i\/I = PO’
St =
(N/M - 2ah.c)
(A4 + &P--zG)“2 UC +
s
UK
P’ -
T f
“‘342 - 4dlCcs’
=
2
UK(MN
-
=
2ati
2UKc)
JXK-ZZ
M&_/z-zE 5 1.2
K
PO >
(
N,/= ’
CL.2
=
+ (NM 2alCJZ
?KaC)
.
From the mathematical viewpoint it is worth observing formally the above solution admits the possibility that T + 0 * V2-+ x, a situation which Sharma [lo] interprets as being consistent with T + 0 meaning the classical theory with infinite speed of thermal waves is approached. What our theory predicts is thus mathematically consistent with previous work. However, it must be stressed that since c = c(0), 5 = s(O), we cannot allow r -+ 0 in V,. The function r is limited below as shown in Section 1. Thus, we have a means to yield a fast elastic wave and a thermal wave which behaves in accordance with the experimental findings, namely that the speed of the thermal wave decreases as the mean temperature is increased. Of course, it has to be noted that the thermal wave had been observed at cryogenic temperatures. Acknowledgement-This research was performed supported by the Italian CNR. Partial support
while B. S. was a Visiting Professor, at the University of Genova. to G.C. and A.M. from CNR is gratefully acknowledged.
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Thermoelasticity at cryogenic temperatures
263
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