Mathematical approaches of thermal and thermoelastic contact

Int. J. Mach. Tools Manufact. Vol. 32. No. 1/2, pp. 129-135, 1992. Printed in Great Britain

0890-6955/9255.00 + .00 Pergamon Press plc

M A T t l E M A T I C A L A P P R O A C H E S OF T H E R M A L A N D T H E R M O E L A S T I C C O N T A C T

C. GARDIN, H. LANCHON-DUCAUQUIS, J. SAINT JEAN PAULIN D. BILLEREY, N. DEBBABI, M. PLATEK Laboratoire d'Energdtique et de M6canique Th6orique et Appliqude 2, avenue de la For~t de Haye, B.P. 160 54504 VANDOEUVRE LES NANCY Cddex, FRANCE SUMMARY Our general objective is to point out, by mathematical modelling, the correlations between the topography of rough surfaces and, some characteristic quantities about their thermal or thermoelastic contact. In a first approach [which was described at the 3rd International Conference on Metrology and Properties of Engineering Surf~Lces (Ref. 1)], we considered the pure thermal conducting contact ; we have then neglected the thermomechanical effects (pressures, stresses, strains), the presence of several competitive small reduced parameters (thickness and proportion of contact, relative conductivity of the interstitial fluid), and also the possibility of a radiative contribution at the level of the interstices. In this paper we describe the first attempts to take into account all these realistic effects. (-d) and reasonably

1) INTRODUCTIOI~! We refer the r,gader to the ref. 1 and 2 for the general description o f the physical situation and we are satisfied here with two diagrams (Fig. 1 and 2) for the u n d e r s t a n d i n g and illustration of the notations. The interstitial fluid

t

temperatures d e s c r i b e d by

(0) fields can the equations.

(i)

(J) (J) - b (J) 0 ( ? ) = O a ijkh Uk,hj j

(2)

(J) C (J) a0(J) - k (J) A 0 (J) = O Po 0t

be

in ~ J for J = I, II; t> a i j k h Uk,h -- X Uk,h 8ij + tt (ui,j + uj,i) expresses the Hook law, t> b = (3Z. + 2~t) ct, where a is the thermal coefficient of dilatation. t> C and k are respectively the coefficients o f specific heat (for constant strain) and thermal conductivity. t> Po is the initial mass per unit of volume.

I

j Th__.~etheoretical interfa__..__ ce ~_~ The defects zone (2d) ~ _ ~ T h e perturbated zone

The state of compressible, or incompressible, fluid, pinched in the interstitial domain f~ f, can i t s e l f be d e s c r i b e d by :

Ftg.1 1.1,) T h e

t, e n e r a l

relations

IT= o (3)

~l p = P 0

constant

number

/

%(f!3k, x) "

t>

....)

V and P0

represent

respectively

the

velocity and the pressure, and Of , again the t e m p e r a t u r e s field. To c o m p l e t e this mathematical model, we have to prescribe: the initial temperature on f2 (union o f ~ I, f2 II and ~ f), the b o u n d a r y conditions on a ~ and also the transmission

~.. f~f Ll

Lkf A0 f = O

conditions on each interface F J K (for {J,K} c

L II

{I,II,f} ). These last two families of relations will be described later.

Fig.2 In fact, C. G A R D I N (Ref. 2) s h o w e d t h a t , if we have t w o homogeneous, isotropic,

(4)

e l a s t i c c y l i n d e r s ~ I a n d t-I l l in i m p e r f e c t mutual contact, submitted to some thermal or mechanical small perturbations, then, t h e i r displacements

0 (x,o) = o

for

x•

f2

1.2) T h e ~ e o m e t r i c a l descrintion We give now some new geometrical descriptions : f2 is also a cylinder, we specify its

129

130

C. GARDINet al.

cross section Z, its length 2L, and an orthogonal coordinate system as it is indicated on the fig.2. We introduce also the functions q~l(Xg_) and 92(X) where X = {Xl,X2} such that :

(5)

f

~l=

{X ~ t l , - L I < x 3 < 9 1 (X__)}

i~II= [

{Xe

(ll)

~,92(X_..)
f l f = {X e 11,91 (X__) < x 3 < 9 2 (X__) }

The middle plane

x 3 = O being defined by :

S (92 (X_.)- 91 (.X.)) dx 1 dx 2 = O Z two functions 9ct are supposed

The

to

contain all the information about the roughness of the two solid surfaces when 0= O. 1.3) I n v e n t o r y

the

Three data

of

the

small

aarameters

small parameters are related to 9 a (a = 1,2). In fact it is easy to

guess that each 9 a

is strongly fluctuating ; we

can introduce, either : (6) d = 2~ (suP92(X__)-inftPl(X__)) with

2L=LI+

; X~Z

L I I or 1

(7)

is very much smaller than 1. The 4 th obvious small p a r a m e t e r is the relative conductivity of the interstitial fluid that may be defined by

x ~ x

with 9 (-X-)= 9 2 X(X(X(X(-~ 91 (X) and define the thickness

of

with k s = max (kI,k II)

1.4) S u m m a r y of the two first stens In the two methods exposed in 1986 (Ref. 1 and 3) we have just described the situation of a pure thermal conducting phenomenon, r e t a i - n i n g only the small p a r a m e t e r e. In the first case, we admitted that t h e distribution of roughness was a periodical on e and, with the k n o w l e d g e of the representative cell (i.e the bidimensional period) of the contact area , we s h o w e d how to compute the contact r e s i s t a n c e by the "homogenization techniques". In the second case, we have produced only l o w e r and upper bounds for the c o n t a c t r e s i s t a n c e ; the necessary data for that were the statistical information (expectation, variance and two point correlation function) relative to the distribution of conductivities in the contact area, which is made of three different media. 2) THE T H E R M O E L A S T I C CONTACT (Ref. 2)

sup 9(X)

n = ~

k f-kf ks ,

contact by 2d or

2.1) The c o n t e x t : by a dimensional analysis we obtained the relatively simplified system (1) (2) (3) that we completed by: i) The t r a n s m i s s i o n conditions at the interfaces F J K • With the notation I[g] ~K' to r

2n.

If we assume, which is realistic, that the double d i s t r i b u t i o n of r o u g h n e s s along the two surfaces is s t a t i s t i c a l l y h o m o g e n e o u s , we may define a characteristic sample of roughness and two directions OXl, ox 2 along which there

denote the jump of a quantity g across F JK, the continuty of, the temperature, the heat flux, the displacement and the normal stresses at each of the interfaces F JK implies.

exists two characteristic sizes h 1, h 2, small by

(12)

[0]

(13)

3O [ k ~]--K dN F"

comparison

with

L,

such

that

e a = ha/L

could he considered as a kind of reduced half period of r o u g h n e s s in the x (x direction (8)

(a=l,2)

a small

FI II= {X e f~ such as 9(X__) = O}

is a surface of non zero measure ,,'1. (F I I I ) ; the specialists of engineering surfaces ensure that if A ( Z ) is the measure of Y. ( cross section of t~), then the proportion of contact p defined by A. ( r I II) (lO)

p -

=o

(14)

[7]

FIII = o

parameter.

We may also assume that the set (9)

JK = 0

; then:

e = max (el, e2),

is also

F

,,,Tt(z)

(15)

(16)

[aijkh Uk,h - b O 8ij ] i.l.iI N j = o .(a(J) • (J) ) ~ ijkh Uk,h" b(J) 0(J) 5.lj Nj = - Po Ni o n F Jf

for J=I or II

N being the unit normal vector to the interface F JK, oriented for instance in the direction of increasing x 3 .

Thermal and Thermoelastic Contact

Remark : It is important to point out that the condition (13) on the interfaces F J f (J = I, II) means that we n e g l e c t e d the radiating flux which is produced in the zone where the contact is broken.

ii) Some conditions; thai

particular is

imposed temperatme +LII: (17)

~I u =o

(lg)

01 = o o n

to

boundary

say,

clamping

at each extremity

-L

I

on x3 = + L II

x 3 = -L 1 , 0 II = T > o on x3 =

neither m e c h a n i c a l loading the lateral boundary OflL

nor heat

contact between the two solids begins in X = 0 (eventually, simultaneously along the whole interface, in the peculiar case where ATo = o). BTR/ATR = _R 2 : then the

contact between the two solids simultaneously along the c i r c u m f e r e n c e

2 2~2 Xl+X2=K .

transfer at

It is easy to show that T O exists

8iet

, for J = I or I I ,

]

and

is

W e m a y w r i t e e x p l i c i t c o n d i t i o n s on parameters to have at least one T R < T o then we net = o

are in the situation showed on the fig. 3 ; if these conditions are not satisfied, then we are in the situation showed on the fig. 4.

et=1,2

~ (20)

begins

unique; the uniqueness o f T R is not obvious.

(J) (J) _ b (J) 0 (J) aietk h 0 k , h on 0D J

= ~T

for two kinds of critical temperatures. E i t h e r T O such that BT o = 0 : then, the

,

+L II.

(19)

So because the parabolo~'dic shape o f the surface defined by x 3 (X), we have to look

O r T R such that

x3 = _LI, ~u l I = o

on

and,

131

k (J) eqO(J) net = o on ~ (LJ)

1

l

V/!

for J = I or II

a/T < TR < TO

2.2) The

results

in absence

of roughness

:

d>o,o=o

Peculiar h y p o t h e s i s (we supose in the following that, Cpl and q~2 have been reduced by L as well as the spatial coordinates): b/T = TR < TO CPl(X_.)=-d,

cp2(2D=+d;

d>o. Fig. 3

i) We obtain the : x p l i c i t temperature field and c o n t a c t r e s i s t a n c e . The two only small parameters are d and ikf

which intervene only by

their ratio 2 d ~ f ; 1:he thermal behaviour is clear whatever their relative order o f magnitude. ii) In the case o f a circular cross section, we obtain the explicit strained shapes o f the two cylinders; and tha~:, as far as the imposed temperature is smaller than a critical one, Tcr, for which the two cylinders begin to be in contact. We obtain in particular

I )) a/T
R

! | b/T = TO < TR

f IPT (-~) = (P2T (X..) - (PIT (X.) (21)

Fig. 4 In fact the critical temperature is

i f a n 6 o n l y i f ~T ( X ) ~ o

Tcr = rain (T 0 , TR)

Here A T and ]3T are only depending on the

ii) As fas as T < Tcr : we can for each T compute

g e o m e t r i c a l and thermal c h a r a c t e r i s t i c s o f the three m e d i a (in fact the elastic effects do not appear as far as the contact is not established).

the s e c o n d a p p r o x i m a t i o n o f the t e m p e r a t u r e field 0 for the new geometric configuration, and then, deduce a new contact resistance which will depend on T in a decreasing way.

132

C. GARDIN et al.

iii) For T-> Tcr, the mathematical model has to be modified, alternative,

because we have to write some unilateral, new transmission

conditions at the solid/solid inter-face F I II

2.3) Tile result for i n t e r f a c e s with roughness Peculiar h y p o t h e s i s : we approach the real interface by a periodical double distribution of roughness, and we assume that the only small parameter is e. The m u l t i c o n n e c t e d domain occupied by the union of the two solids, depends obviously on e; we call it now f~e; that introduces some technical new difficulties to compute the approximation of the displacement field ~e. u To obtain the homogenized solution ~ e in three main steps.

i) We

replace

0 e in

we proceed

(1)

by

the

homogenized solution 0 ° of (2), (3), (12), (13) computed in MAKAYA (Ref. 1, 3) in the pure thermal

problem.

The

temperature

0 ° can be

considered in fact as a first approximation of 0 e. ii) We have now to solve an elastic

problem on f~e; we c o m p u t e then the homogenized corresponding p r o b l e m by usual techniques, and we use the ideas of CIORANESCU D., SAINT JEAN PAULIN J. (Ref. 4) to pass through the difficulties of the dependence on e of f~. The result obtained here is the following one : ----> the best approximation uO of u e is the solution of a new equation of Elasticity type, formulated on ~. (22)

[

qijkh

]

o Uk,h , j -- fij

+ gij

Po

]

,j

The new elastic coefficients are no longer fluctuating in the contact zone x 3 e ] - d , + d [ ; they can be obtained, as well as the functions fii and gij' by simple computations on a reference cell ; we use for that a finite element method. The second member can be interpreted as some new body forces due to the thermal stresses and to the pressure of the interstitial fluid. The boundary conditions for u ° are '~o u (-L I) = u O (+L I I ) - - 0 as imposed for -u) e

(23) and, on

(24)

the lateral boundary 3 ~ L ,

(qictkhUk,h

+fict

0° + gict Po )

iii) The solution u o of (22) (23) (24) can also be c o m p u t e d by, a finite e l e m e n t method. --3'0 Remark: From u we can compute the new shapes ~PIT(X) and cp2T (X) of the interfaces cPl(X), cP2(X__) ( w i c h

are no longer strongly fluctuating)

and then, c o m p u t e the c o r r e s p o n d i n g perature field which can be considered better approximation of 0 e .

temas a

3) STUDY A C C O R D I N G TO THE ORDERS OF THE DIFFERENT SMALL PARAMETERS 3.1) M o t i v a t i o n s Usually the interstitial fluid which is a gas is a bad heat conductor, while the two solids in contact are very good ones; then the most part of the flux lines passes t h r o u g h the rough excressences in contact; it is observed (see paticularly fig.l) that the perturbated zone is bigger than the strict region of roughness (defects zone). This phenomenon called " c o n s t r i c t i o n " is in fact more e s p e c i a l l y emphasized as the contrast of conductivities between solids and fluid is bigger and the thickness of contact is smaller. We are in presence of a boundary layer phenomenon ; this one is not put in the limelight in our precedent theories, by the fact that we considered only the small parameter e. It is interesting for instance to give the following examples : if the interstitial fluid is the air at the ordinary temperature then l~f = 5.2

10-4

if the solids are in steel

l~f = 1.2 10 -4 if the solids are in aluminium we may also guess that the very small proportion of contact p, has something to do with the "constriction" phenomenon. In the following, we give first a systematic inventory of all the differentiated situations ; after that we shall describe briefly the "method of m a t c h e d asymptotic e x p a n s i o n s " which, will be particularly adapted to described the boundary layer phenomenon.

3.2) S y s t e m a t i c inventory of the different s i t u a t i o n s (in the case of a pure conducting thermal contact) By some mathematical studies of convergences (Ref. 5), we have obtained the following informations: i) If I~fis the smallest parameter, the best approximation of 0 is the temperature of the same problem when the void replaces the interstitial fluid (problem of heat conduction accross a body with small c a v i t i e s in the neighbourhood of the interface x 3 = 0). Let us call Po this new problem.

nct

=0 on

If now, ~ is the s e c o n d smallest para-meter, we can approach the temperature 0 by the solution of the preceeding problem where the cavities have been flattened in "fissures" on their middle surfaces.

Thermal and Thermoelastic Contact

If it is e w h i c h is t h e s e c o n d s m a l l e s t p a r a m e t e r , it is possible to show (following ref. 1 or 2 and 4) that we can approach the temperature 0 by the solution of a n e w homogenized v e r s i o n of Po, in which the zone of cavities is replazed by a smooth equivalent interface; we have then to compute a s p e c i f i c contact resistance, b i g g e r than the one corresponding to k f ~ o . ii) If k f = rl = o The approached thermal behaviour, when e is small by comparison with p, will have to be deducted from the works of SANCHEZ-PALANCLk E. and TELEGA. iii) I f d is the s m a l l e s t p a r a m e t e r The best approximation of 0 is the unperturbated temperature field corresponding to the solution of the p r o b l e m w i t h p e r f e c t contact. iv) If 11 is the s m a l l e s t p a r a m e t e r The best approximation of 0, is the solution of the problem, wheTe the two solids are in perfect contac't along a common fluctuating i n t e r f a c e (corresponding to the middle surface of the two rough interfaces). So because e is still a small parameter, a n h o m o g e n i z a t i o n is still n e c e s s a r y , to replace the fluctuating interface by a flat one, and to evaluate if the i n t r o d u c t i o n of a c o n t a c t resistance is still necessary. R e m a r k : when d o1" 11 are equal to 0, the orders of 1~f and p are without object, because there is no more interstitial fluid, and p > 1. v) If e is t h e s m a l l e s t p a r a m e t e r , the best a p p r o x i m a t i o n of 0 is the one obtained by MAKAYA ; see paragraph 1.4. above and ref.(1 and 3). If d is t h e second s m a l l e s t p a r a m e t e r , the best approach,~d temperature is the one corresponding to the perfect contact (this result is also contained in ref. 3). R e m ~ t r k : To conclude this inventory it is necessary to say that : - Some abstract mathematical studies could help to obtain good approximation according to the order of p (proportion of contact). - A lot of situation.,, are still without conclusions;

133

than 1 ; for this reason, it is natural to try a dilatation of this zone in order to study better the details of the perturbation (in particular for instance, the phenomenon of constriction). The method of matched asymptotic expansions, that we describe below, is quite adapted to this kind of situation ; we shall have to combine here this method with the one of m u l t i p l e seale e x p a n s i o n s which is necessary in the technics of "homogenization". Combination of the two asvmototic methods for the study of a thermal contact It is n e c e s s a r y to recall that, when we considered e as the unique small parameter (Ref. 1 and 3), we looked for an asymptotic expansion of the solution 0 e of the type. 0e~)= 0o(.X_, Y_) + e 01(X, Y_.)+ c2 0 2 ( ~ , Y) + o(e2) KS

with

Y = { Y l , Y 2 } , yet= - £- ,

ct= 1 or2

The change of scale, from xec to Ycx , was i n t r o d u c e d to dilate the reference cell of roughness ; the simultaneous presence of X and 21(_ in the terms Ok , was to point out the common importance of and minorscale particle in the approximation

the macroscale represented by X represented by Y (position of the reference cell). The homogenized 0 o was defined by

0 o (_~, Y_Y_.)= lim 0 e ~ ) when e ~ o , with Y fixed. .It was proved, in fact, that 0 o does not depend on

Y.

If we w a n t now to point out, that rl is also a s m a l l p a r a m e t e r , we h a v e first to define t h e r e l a t i v e o r d e r of m a g n i t u d e of e and 11 by some precisions of the type : f

(25)

l i m 11( e ) = o w h e n e --* o and something~ else between the two situations lim

--- o w h e n

e ~

o

V p > o

or

lira

--= EP

o when

e --* o

V p >0

We have then to introduce the dilatation x3 (26) z = --

for instance when l~f and d, or l~f and e, or still d and e are s i m u l t a n e o u s l y the two smallest parameters. - In some of the situations mentioned above; it would be good I:o improve the results of convergence.

in order to point out the importance of the proximity or distance from the "contact" and to build an "inner approximation" of the solution Oerl defined by

3.3) U s e f u l n e s s Gf the m a t c h e d e x n a n s i o n technics. Preliminary remark. It is obvious th~Lt the temperature field is perturbated by the roughness, essentially in the neighbourhood of the contact zone, when the reduced thickness rl (or d) is much smaller

with the rule retained in (25), and, Y_.. and z fixed, this "inner approximation" should be a priori valid "near" the "contact". W e m a y also build an "outer approximation" of oe~ defined by _'

MTM 32-1/2--J

( 27 )

0 (_X.,Y.Y_,z) = lira 0 erl x(.x)when rl ---)o and e ---)o,

X = {Xl,X2}.

134

(28)

C. GARDIN et al.

0 (x,Y)=lim

0 erl(3.) w h e n l ] ~ o a n d

c ---~o, x = {Xl , x 2 , x 3 } with the rule (25), and, Y, x 3 fixed. We have proved that this last approximation, valid "for" from the contact, is nothing else than the solution 0 o already obtained in (Ref.1, 3). It will be necessary yet to connect the two approximations 0 and 0 by a classical "matched processus" which allows to obtain an approximation valid everywhere. This new type of study is not yet finished. 4) F I R S T APPROACH OF A PURE C O N D U C T I V E AND R A D I A T I V E T H E R M A L CONTACT 4.1) M o t i v a t i o n s The "I.R.S.I.D." which is the French Research Institute for Iron and Steel Industry askes us to examine the part that the radiative transfer could take in a thermal contact, particularly between two metals. K n o w i n g that the e x c h a n g e d r a d i a t i v e energy between two particules P1 and P2 is proportional to the difference of the 4 th power of the c o r r e s p o n d i n g t e m p e r a t u r e s , it was natural to look for an e s t i m a t e of the temperature along respectively the surfaces x 3 = tP1 (A) and x3=~P2(x ). Before any computation it is natural to expect that, the less conductive the contact is the more the difference of temperatures between f~I and ~ I I will be important ; we may then anticipate that if I~ f and (or) p are very small parameters, then the radiative transfer will be important (that has already been showed by experimentation ; see ref. 6). In the following we relate two steps in the study of the radiative contribution. We first show several estimates of the respective parts of conduction and radiation according to different values of T, k f and d. After being convinced that the radiative transfer is a priori, not negligible we begin a new mathematical model by the choice of some realistic hypothesis and the research of the place where the r a d i a t i o n will i n t e r v e n e in the different relations. 4.2) First e s t i m a t e s of the relative Dart 0[ c o n d u c t i v e and r a d i a t i v e t r a n s f e r For the sakes of simplicity we considere the "pseudo contact" without roughness already introduced above in 2.2 ; the purely conductive temperature field is then explicitly obtained with the following numerical data corresponding to steel for ~qI and aluminium for ~ I I : k I = 40 kcal/m h °c, k II = 175 kcal/m h °c, We

LI= 0,2 m L II = 0,3 m show below three tables, where

the

notations are clear ; they correspond respectively to the variations of T, k f and d. The two last columns indicate the respective percentages of the conductive and radiative flux c o r r e s p o n d i n g to the surface temperatures in -d and +d showed in the two first columns.

T (x3=LIl)]

0 (-d)

200 400 600 800 I000

9,67 19,35 29,25 39 48,75

°C °C °C °C °C

% @cond[ % ~rad 96,76 78,35 52,55 31,84 19,3

196,67 393,35 590,01 786,68 983,35

3,2 21,65 47,45 68,16 80,7

Table 1: V a r i a t i o n of i m p o s e d t e m p e r a t u r e : T (For : k f = 0,0208 kcal/m h °C ; d = 10 -3 m)

kf kcal/m h °C

0,1 0,05 0,03 0,02 0,01

IO(-d)I O(d) 112,99 64,58 41,1 29,25 14,59

% [ % ~ rad cond. 81,72 18,28

I

561,58 578 586,02 590,01 595,04

Table

34,75

65,24

2 :

V a r i a t i o n of the fluid c o n d u c t i v i t y : k f (For:T=600°C;d= 10 -3 m)

0 (-d) d (en m) (en o C) 2.10-3 1,5.10-3 10-3 5.10-4 2,5.10-4

15,23 20,03 29,25 54,9 97,7

0(d)

%~cond

%Orad

(en o C) (approxi (approxi 594,83 593,52 590,01 581,25 566,57

mation) 52,56

mation) 47,44

52,15

47,85

Table 3 : V a r i a t i o n of t h i c k n e s s of c o n t a c t 2d (For : T = 600 °C ; k f = 0,0208 kcal / m h °C) In ¢onclusion of this very rough estimate we can make three remarks. The radiative transfer has to be taken in account and specially when we can expect that one of the solid body is sensibly hotter than the other one. We have the confirmation that the smaller ~, f is the more p r e p o n d e r a n t the part of radiative transfer will be. The influence of d seems to be without effect ; but we can suppose that this fact is due to the too big simplification ("pseudo contact without roughness") of the problem.

Thermal and Thermoelastic Contact

4.3) S n e c i f i c h v n o t h e s i s , a n d olace of the r a d i a t i v e e o n t r i b u l i o n in t h e m a t h e m a t i c model a) Hvoothesis relative to the radiative oroDerties Of the three media We make the following realistic assumptions: . T h e two solids a r e " g r a y , o p a q u e bodies" which means : on the one hand, that the emissivity does not depend on the wavelenght of the radiation, on the other hand, that the absorption of radiation takes place only in a very small depth from the boundary. - The fluid is " p e r f e c t l y t r a n s p a r e n t " which means that it does not take part to the radiatif exchange. b) Intervention of th~ r~li~iQn in the model With the hypothesis mentioned above, the new thermal model consists in the relations (2) (4) and (29)

p f c f -30f ~t

kf A Of = O

in ~ f ;

The transmission conditions (12) and (13) are still valid (the secc,nd one only for J = I and K = II). For the interface between solids and fluids (JK = If or ill() the relation (13) is modified as follows.

1

135

H. LANCHON, B. MAKAYA, A. MIRGAUX, J. SAINT JEAN PAUL1N, E. KRONER "A mathematical study to obtain quantitative effects of roughness in technical problems" Wear, 109 (1986) 99-111 C. GARDIN "Contribution h 1'6rude du contact thermo61astique" Th~se de rInstitut National Polytechnique de Lorraine, Nancy, D6cembre 1988 B. MAKAYA " I n f l u e n c e de I ' ~ t a t de r u g o s i t ~ h u n interface solide-solide sur la t r a n s m i s s i o n de la c h a l e u r d ' u n milieu /a un a u t r e " Th~se de 3 ° cycle, Institut National Polytechnique de Lorraine, Nancy, Juin 1983 D. CIORANESCU, J. SAINT JEAN PAULIN "Homogenisation in open sets with holes" J. Math. Anal. Appl. 71 (1979) 590-607

5) H. LANCHON-DUCAUQUIS, k I 301 = k f ~0f FIf 3N ~ - + Ot~ II+ (1- e I) ¢I1~I on

J. SAINT JEAN PAULIN (not yet published)

6) M. LAURENT k f 30 f = k II 30 II

~-

~

+ ¢:II"-)I+ (i-II) ~I~II on FfII

....>

where N is the unit normal to the interfaces that we choose here to orientate in the direction of increasing x3; ~ I ~ J l is the radiative density of flux furnished by the hottest solid body ~ I to the other one f~II and vice versa for ~II -->I;~I and e II are the ~II.

emissivities

of the

solid

bodies ~ I and

Conclusion The next crucial step in the exploration of the pure conductive, reactive thermal contact is to compute, in a way as realistic as possible, the term ~II ~ I in any case of figure. 4.4)

" C o n t r i b u t i o n ~ I ' e t u d e des ~changes de c h a l e u r au c o n t a c t de deux m a t e r i a u x " Th~se de 3 ° cycle, Lyon , Mars 1969