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Thermal stresses and thermal expansion coefficients ofn-layered fiber-reinforced composites
Sumce and Technolugy 57 (1997) 249-260 © 1997 ElsevIer SCIence LImIted Pnnted In Northern Ireland. All nghts reserved 0266-3538/97/$17.00
CUlIlpO.lltf.l
P II: SO 266 - 3538 (96) 0 0 13 7 - 6
THERMAL STRESSES AND THERMAL EXPANSION COEFFICIENTS OF n-LAYERED FIBER-REINFORCED COMPOSITES A. Agbossou & J. Pastor Laboratoire MaterlllllX Composites (LaMaCo), ESIGEC, Universite de Savoie, 73376 Le Bourget du Lac, Chambery, Frallce
(Received 22 May 1996: revised 13 September 1996: accepted 1 October 1996)
control the thermal expansion coefficients of the composite. Many studies of thermal stresses and thermal expansion coefficients have been reported in the literature.I-~ Chamis and Sendeckyt have surveyed the theoretical approaches which include the mechanics-of-materials approaches, self-consistent CCA models, variationaL statisticaL and discreteelements methods. The stress field around a coated reinforcement fiber is presented in Refs 7 and 8. The effective thermal expansion coefficients have been predicted by Hashin and Strickman 9 and Maurer. 10 Composites with imperfect interfaces were studied in Ref. 11. All of these thermal behavior studies were carried out with models with two, three 7.H or at most four 4 homogeneous layers despite the facts that they are applied to multi-layered composites 12 and the actual fiber-reinforced composites may have spatial property variations. 13 In this work, we present a thermal self-consistent (TSC) model for n-layered materials. The proposed model is based on Refs 14 and 15 to account for prediction of the composite elastic behavior. In Section 2 of this paper we formulate the problem, and in Section 3 we present two different solutions. Explicit expressions of thermal stresses and thermal expansion coefficients in multi-layered composites are given. In Section 4, we discuss and compare the results with other calculation~ given in the literature such as generalized plane strain (GPS) tensors, In the micromechanical cell model,x and the CCA mode!.17 In Section -1-, we illustrate the application of the n-layered model in the evaluation of thermal stresses and thermal expansion coefficients in materials with spatial property variations.
Abstract In this work we develop a thermal self-consistent (TSe) model for n-layered fiber-reinforced composites. The jitndamental representative volume element is an inclllsion made lip of a fiber coated by (n - 1) layers encapsulated in an infinite equivalent homogeneous medium. On the basis of the energy eqllivalence principle we show that the thermal problem under conditions of no external loading can be treated without having to determine the elastic characteristics of the equivalent medium. Thereby, thermal expansion coefficients and microstress distributions inside the layers have been determined according to the proposed semi-analytical and analytical procedures. Both procedllres, thollgh different in their implementation, give the same results. The model's performance has been characterized throllgh the sllcces4ul reproduction of composite thermal expansion and residual stresses calclliated from similar studies with different micromechanical models. The proposed model cOllld be llseful for (1) analysing the thermal behavior of materials with property gradients, (2) determining thermal expansion coefficients and thermal stresses in n-layer tubing sllch as gas pipes (lnd (3) stud,ving the effects of interfacial imperfections on thermal behavior. CO 1997 Elsevier Science Linwed. All rights reserved
Keywords: thermoelasticity, thermal stresses, thermal expansIOn coefficients, interphase, n-layer fiber composites
1 INTRODUCTION When the bonding between the constituents of a composite is intimate, even under conditions of no external loading, stresses are set up in the composite as a consequence of differences between the thermal expansion coefficients and elastic constants of the constituents, These thermal stresses cause stress concentrations which may initiate yielding or de bonding. Therefore, in design analysis, it is important to understand how these stresses arise and how to
2 FORMULATION OF THE THERMAL SELFCONSISTENT PROBLEM Let us consider a composite consisting of a concentric n-layered, long circular and cylindrical inclusion of 249
250
A. Agbossou. 1. Pastor
(a)
(b)
Fig. 1. Representation of n-layered fiber-reinforced composites. (a) Representative volume element (RYE); 5 r = 51 + 5,,+1: transverse surface area where 51 is the inclusion surface area. (b) Homogeneous medium without inclusion; 5T + 5L = 5: total transverse and lateral surface area where 5L is the lateral surface area.
radius rn embedded in the equivalent homogeneous medium as shown in Fig. lea). In this representative volume element (RYE) we let phase 1 of the fiber inclusion constitute the central core and phase i lie within the shell limited by the cylinder with radii r,_! and rl (i E [l,n]). Each phase is assumed to be homogeneous, elastic and transversely isotropic along the longitudinal direction (z or L) of the inclusion. The outer cylinder with radius rn +! represents the unknown equivalent homogeneous material. In order to deal with the general self-consistent problem, a limit of the outer cylinder is taken as rn+l ~ x. By introducing a further layer, this model becomes an extension of the model of Christensen and Lo.IS The volume fractions of each layer are equal to: (1)
that ( Uo) stored in the equivalent homogeneous medium which contains no inclusion (Fig. 1(b»:
U
=
Uo for rn +! ~
OCJ
(2)
Therefore, the problem for which we present a rigorous solution is that of obtaining the relationship between the configuration shown in Fig. lea) and the equivalent homogeneous medium (Fig. 1(b». With this energy condition (2) we propose the two following hypotheses.
(HI) The stress and the displacement fields can be expressed by using two fixed finite RYEs as shown in Fig. 1 (finite size). The energy of both RYEs are identical when rn+! goes to infinity. (H2) The two fixed finite RYEs: (H2a) are subjected at all points to the same thermal condition ~T with no stress on the lateral surface SL of the RYE, i.e.
n
where i > 1 and
2: VI = 1 1=1
The entire composite (Fig. l(a» is subjected to a uniform and constant temperature change, ~ T. The problem is to determine the thermal stress and displacement fields. The general solution of stress and displacement fields can be obtained by using the techniques presented by Mal and Singh. 19 This solution in the ith layer depends on unknowns which must be determined from the interface and boundary conditions. 7 .!4 The explicit form of these conditions yields fewer equations than basic unknowns. Further equations are obtained by applying the energy condition, namely that the elastic strain energy (U) stored in the infinite area of the composite model (Fig. l(a» is the same as
(H2b) present the following displacement fields III
= zl,(X,y) + EljXj zl, = 0 (i = 1 to 3) on SL
where (J'"n j is antiperiodic on ST and E = a(hom)~T is the unknown thermal deformation tensor. The form of zl is due to the arbitrary length of the RYE and the periodic condition on ST. 20 Writing out the energy condition (2) gives:
f
«(J',A, -
(J'~~E~:) dv = 0
(3)
l'
where V is the total volume of the RYE; (J'lj and E'l are respectively the stress and the strain in the inclusion;
251
Thermal stresses and thermal expansion coefficients and (T~ and EZ are respectively the stress and strain in the equivalent homogeneous medium (Fig. l(b». By using the divergence theorem, and taking into account (T,].] = 0 and (T~.] = 0, eqn (3) can be expressed as:
I
,JV
T,u, ds -
I T~u7
ds
= 0
(4)
iil'
Then (5) where ST and SL are respectively the transverse and lateral surface area of the RVE: T, and u, are the components of respectively stress and displacement vectors on the boundary of the inclusion; and T~ and ll~) represent the same components as above but on the equivalent homogeneous medium which contains no inclusion (Fig. l(b)). Based on the hypotheses HI and H2, we propose to determine stress and displacement fields by extending the effective-modulus model developed by Pagano and Tandon 7 to deal with the thermal self-consistent problem. With regard to the composite thermal behavior, we found (see Section 3) that this problem could be treated without having to determine the elastic characteristics of the homogeneous equivalent medium.
C~.j are the elastic stiffness constants of the ith
layer, e~') = a~)ilT, e~/) = a~)ilT are the expansional strain components along the longitudinal (L or z), and transverse directions (r-8) respectively, and D~'), D~) and D~) are unknown constants of the ith layer.
Therefore the problem is to unknown constants. For our problem (Fig. l(a»:
•
determine
(3
x 11)
the perfect interface bonding conditions are U~/) = U~'+l)
at r = r,
(8)
(perfect displacement bonding condition)
(9)
(perfect stress bonding condition)
•
from eqns (6) and (8) the displacement along the fiber direction is verified:
•
from eqn (6) the finite displacement condition at the origin, r = 0, yields
3 METHOD OF SOLUTION
(11)
3.1 Formulation of displacement and stress fields When the composite model in Fig. l(a) is subjected to a uniform temperature change ilT, the displacement and stress fields which verify the equilibrium equation in the ith layer are gIven In cylindrical coordinates (r,8,Z) by:7
(6)
and
Substituting eqn (6) into eqn (10), one obtains: (12) where
The problem is now reduced to determining (211 + I) unknown constants. The relations eqns (8) and (9) yield (211) equations. The final equation is obtained by applying the energy condition (HI). To obtain this equation we set:
E
=
af}'°m)ilT(e/2Je r) + (a~()m)ilT + EO)ee ®ee
=
ef}'°m)(er®er ) + (eLhom) + Eo)ee ®e~
and then let Eo = O. The condition eqn (5) gives •
where: the subscript 1 represents the longitudinal direction z or L,
on the boundary of the RVE (Fig. 1(a) ): on
A. AgbosSOll, J. Pastor
252
[Aj, {Y} and {B} take the form: 1
r1
-rl -
-
•
c(hom)c-
12
LO
_ 2c(hom)_2_ ~
,
r
on
[Aj
•
-
2K~1)
2C~1
, rl
0
- 2K~~1
0
, r3.
2Ki~)
0
0
- r3
0
0
0
-
2CW v I
2C\~)V2
0
2C\~~)V3
1
Finally, we obtain:
r2
L51(eLhom) + Eo){(2C\'d(D\') - e¥l) + C\'/(Eo + e~oml
0
- e~I)/sI - ci~()m)Eo} + 27rL(c\;om)E o 2c~~om)elf'°m)D~n+l) =
2C~~)
0
0
2C~~)
0
-
r=
r2
2K\21 _3
=
on the boundary of the RVE homogeneous medium without inclusion (Fig. l(b)):
1
- r2
0
0
rl
D(n+l)
(13)
r~
1
where 51 is the inclusion surface area, and the symbol ( . Is, means the average over 51' By writing the fixed Eo as Eo = - e~om) we obtain D~n+l) = O. The problem now is to show that Din + l ) = 0 when Eo = O. Due to the linearity of the thermoelastic problem, let the coefficient D
r, 2n~)
0
0
0
0
0
C\~)
-
C(~)
0
cg) -
C(~)
(17)
0
r3
7
0
0
0
r'3
2K~~)
-
C\~)
3
~C(I)V II , ,~l
where B~,;l) and B~:+I) correspond to the value D~n+l) in the thermal problem (i)..T = 1, Eo = 0) and D~n+l1 in elastic problem (LlT = 0, Eo = 1) respectively. According to Ref. 14 the coefficient Bc~~f" vanishes in the elastic problem. Therefore B~,/l) vanishes, yielding to D<;,+I) = O. It should be mentioned that the proof we give of the coefficient D&n+l) vanishing shows that the TSC problem is similar to the Hashin-Rosen thermoelastic problem in which the boundary conditions are on the inclusion of radius rw
Dill DI21 D~21
{Y} =
and
D\4) D~4)
o o {B} =
3.2 Semi-analytical procedure The substitution of Eo = 0 into eqn (13) gives:
D\3) D~')
C\~)er)
+ 2Ki~)ef) -
CWe~ 1_
2KWe lf>
Cg)e~)
+ 2Kg)e~) -
Cgler) -
2Kh~)e!f)
o C\~)et')
The explicit form of the relations eqn (15) gives a linear equation:
[A]{Y} + {B} = 0
(16)
where the components of the vector {Y} are the basic unknowns (D\I),D<;I, ... ,D\n),D~n),elf'°m).eLhom»). In the case of 3-layered fiber-reinforced composites
(18)
+ 2Kg)e~)
~ ( 2C\lde¥) + C\,/e~) )v/
where 2Kilj = C<;i + C<;j and 2CW = c<;d - C<;d. Thus. for a given n-Iayered fiber-reinforced composite the semi-analytical procedure consists of determining the matrix [Aj and the components of {Y} and {B}, and then solving the linear eqn (16) by using the standard Gauss algorithm.
Thermal stresses and thermal expansion coefficients
3.3 Analytical solution In the analytical approach, the problem is to find an analytical expression for the three unknown constants D\'), D~) and D~) of the ith layer. Let X, denote the vector with components Dll) and D~). The perfect bonding conditions at the interface, eqns (8) and (9), yield: T'+I(r,)X'+1 + D¥,+115,+1 + !:lTF,+l _ (n+ll-T,(r,)X,+D, S,+!:lTF,
253
above constants can be expressed as: D\II=
[(2K&'3)a¥'1 -
2K¥~IP\nl
+
C\f~la~'I)b
+
(2Kl{~)M\n'
+ C\21)a]r;, + 2C~4lPjnlb 2Kl{~IM\nl
[( -
-
C\f~I)C
- 2C~4)M¥'la
+ 2K~'3)Q\f;-1 Ib ]r~ + 2C~~)MY')c -
2C~4)Q~;lb
(22)
(19)
D
,
(n+l)
where:
T,(r)=f
[( r 2K~')
F,
= { _
-L,Y1l r
- C\f;I)C
+ 2Kl{~IQ\~-I)b ]r;, + 2C~~IMin)c - 2C~4IQ~;lb (23)
where
Ci'ial'~ 2K~_Ja¥)}
5,
=
n
{C~'i}
a = '2)2C\'iP\' I - C\?al l - 2C\'ia¥»v, n
and D¥,+I) = D~l) (according to eqn (10». The solution of eqn (19) gives: X- l+l -- N(IIX-I
2K~~)M\nl
b = 2: (C\,/ + 2C\'iM\l)v,
with
MI II = 0, p\ II = 0
,~I
IL+ D(n+l + a"TW-1+1 3 1+1
f1
(20)
C
= 2C\~IVI + 2:(2C\'iQ\'I-II)V, [=2
where
Therefore, eqn (12) yields ap:°m l = D~I 1/ J. T. From eqn (8) with i = n. one obtains: D(n) D\n+ll=a\h()ml=Dlnl+~
(24)
r~l
l
The eqns (7), (21). (22), (23) and (24) are the analytical form sought. From these equations, one can determine the thermal stresses and thermal expansion coefficients in an n-layered fiber-reinforced composite.
1 C(l+1 12 { 1 } 2C(l+11 _ 2
- C(')12 -
,+1 -
r
22
1
It;+1 =
4 NUMERICAL RESULTS AND DISCUSSION
(C\'tl1al+ 11 + 2K~tlla¥+I) - C\'ial ' - 2K~ia¥l) { 1 }
4.1 Comparison of TSC model and other calculations Section 3 provided semi-analytical and analytical procedures for determining the thermal stresses and thermal expansion coefficients of an n-layered fiber-reinforced composite subjected to a uniform temperature change. Although the two procedures lead to different expressions. they yield the same results. However. the analytical expressions seem to be more useful when the composite properties vary continuously. Owing to the lack of published results on thermal stresses in n-layered (n > 3) composites, we compare the present results with other theoretical predictions in three-layer composites (fiber / coating/ matrix). Consider first the special case of a continuous fiber embedded in an infinite matrix with uniform eigenstrain in the inclusion (Eshelby's problem). The corresponding stresses determined by G PS tensors 16 and the proposed method are given in Table 1. The stresses in the fiber domain (first layer) and in the matrix domain (second and third layers) coincide with
-r;
2C~2+1)
By using the iterative approach analogous to the one developed by Herve and Zaoui,14 eqn (20) can be expressed as:
Xa l = Q(l)X I + D¥,+l I M,+l + !::.TP,+l
(21)
where
K
P,+I = It;+1 + The
determination
of
,
~(I]N{j»)lt; the
three
unknowns
(D\l),D~I,D~'I) is now reduced to finding the two
constants Dlli and D~n+ll. By using the energy condition and eqn (12), the
A. Agbossolt, f. Pastor
254
Table 1. Comparison of thermal stress values using GPS tensor and TSC model: I1T=
Data for model composites
EL
ET
(GPa)
(GPa)
84 84 84 84 4 4
84 84 84 84 4 4
VLT
0·22 0·22 0·22 0·22 0·40 0·40
VTT
0·22 0·22 0·22 0·22 0·40 0·40
Thermal expansion coefficients
aL aT (10- 6 °e l ) (10- 60 e
4·9 0 0 4·9 60 60
-ec
4·9 0 0 4·9 30 30
l
)
TSC (hom) aL (1O- 00 e
GPS IT~l) (kPa)
(hom)
)
aT (1O- 00 e
l
1·470
1·470
10·740
34·287
those obtained by Eshelby's method and GPS tensors. It can be noted that Eshelby's method to compute the stress fields in the matrix is quite elaborate. 4 Although the GPS tensor concept is different from the model presented, the same coincidences are observed in composites which are not isotropic thermal expansions (Table 1). Table 2 compares the results of the three concentric cylinders model (EM model, effective modulus strain approach), the micromechanical cell (MCM) model,S and the present (TSC) model. This comparison was done on coated Nicalon-fiber/BMAS composites subjected to a uniform temperature change. The composite constituents are elastic and isotropic. Data for the model composites appear in the tables. The result of the present model is consistent with the results of Pagano and Tandon (EM strain approach),7 Gardner et al.,8 and Benveniste et al. 21 We observe that the TSC model gives slightly higher values than the MCM model and that the difference in the results falls within the range of experimental uncertainty. Tn order to give numerical results for the n-Iayered TSC model we have selected a composite with four layers (fiber, first coating, second coating, matrix). The material properties are representative of Nicalon fibers, and barium-magnesium alumino-silicate (BMAS). This composite has a ratio of coating thickness to radius of matrix outer cylinder equal to 0·05, which corresponds to a coating volume fraction of ~0·08. Table 3 gives the thermal expansion and thermal stresses for the above composites. Figure 2 shows the radial stress distributions predicted by the present model as a function of the normalized radial distance from the fiber center. The stresses have been normalized with respect to the product of the Young's modulus, the thermal expansion coefficient of the matrix and the temperature change (E 4 cx 4 !J.T). One can observe that the thermal homogeneity of the coating significantly affects the thermal stresses and the thermal expansion coefficient. In order to prove the validity of the TSC model in
=0·30;
V2
+ V3 =0·70;
'3 =1;
and
Thermal stresses at fiber/coating interface
TSC l
Vi
GPS IT;!) (kPa)
TSC IT~I) (kPa)
TSC IT;!) (kPa)
)
-369,384 -184'692 -369'384
528·273
85·645
-184'692
528·273
85·645
the heterogeneous fiber-reinforced composite, we compare the TSC results with the direct solution using the CCA model. The analysis is carried out under the conditions previously studied by Jayaraman et al.17 The composite (T300/Ni/ Al-6061) with nickel-coated fibers has been considered in this study. Table 4 gives the properties of the composite constituents. The interphase elastic properties are modelled as follows. (a) Constant variation: E2 = ENl
(ENl = 207 GPa)
(25)
(b) Power variation: E2 = Pr Q
(P = 2·218 X lOW, Q = - 4·544) (26)
(c) Reciprocal variation:
E 2 =P/(r-Q)
(P=1·773xlO lO ,
Q = 0·522)
(27)
(d) Cubic variation:
E2
=
Pr 3 + Qr 2 + Rr + S
(28)
(P = - 4·665 X lOB, Q = 1·046 X 10
R = - 7·824 X
1013,
S = 1·958 X
14
,
10 13 )
Figure 3 shows the curves corresponding to the above interphase Young's modulus. The composite fiber volume fraction is VI = 0·36, and the coating volume fraction is set at 0·248 which corresponds to (r2 - rd/2rl = 0·15. This composite is subjected to a uniform temperature change of -1°C. In order to determine the composite thermal stresses we finely discretize the interphase Young's modulus and then apply the analytical relationships. Figure 4 compares thermal stresses in the radial direction as determined by the TSC model with the direct solution by using the CCA model. 17 We note that the CCA analysis using a continuous variation of E2(r) yields the same results as the TSC model using a discretized approximation for E2(r).
0·08
()·no
()'32
U,
2()() 34·5 106
(GPa)
(GPa)
200 34·5 106
EI
EL
()'298 ()·31 0·232
VIT
0·298 0·31 ()·232
"IT
O'L
;./ 1·8 2·7
(10-noc I) 3·2 1·8 2·7
"I (10-noC 1)
)
32·7
(kPa)
(T~'
- 57·7
(kPa)
(~)
U c
- 46·9
(kPa)
(T~3J
NumerIC (MCM)
33·5
(kPa)
atl)
- 60·3
(kPa)
l2)
U c
- 47·1
(kPa)
(3)
U c
EffectIve modulus (EM)
33·39
(kPa)
(T~J )
- 59·33
(kPa)
(T;2J
- 47·77
(kPa)
(T~Jl
Thermal self-consistent (TSC)
0·04
()·no
Ca,e (c)
0'04
0·04
0·04
1'.1
()'32
0·32
0·32
V4
200 34·5 34·5 106 200 34·5 34·5 106 200 34·5 34·5 106
EL (GPa)
200 34'5 34·5 106 200 34·5 34'5 106 200 34'5 34·5 106
EI (GPa)
Ca~e
(c) coatIng thermally
ani~otropic,
(aLh' Lo.!lmg) =
ac
nJ
0'~2nd COdling)
*'
COdtlllgl)
Lo.ttm~))
0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232
V T1
Q'L
3·2 1·8 1·8 2·7 3·2 0·18 1·8 2·7 3·2 1·8 1·8 2·7
(lO-ooC 1)
= uV-nu L:O,tllng) # (aL2 nJ. L(l.Jlmg) = a.(/nd UhttJn g »). (a\ht LOdt1I1gl = a\;nct ClMtlll g ) .
0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232
VLI'
compm,Ile~
Case (a): coatIng thermally IsotropIC, (T~J<,t co,ltllHU = a 11ht cnJllIlg) = Case (b): coating thermally Inhomogeneous, (O'~ hi lOdlll1g) = ay~t
0·04
0·60
Case (h)
0·04
0·60
V2
Case (a)
VI
Data for model
3·2 1'8 1·8 2·7 3·2 0·18 1·8 2'7 3·2 0·18 0·18 2'7
Cl:'r
(l0- n oc- 1 )
3·069
3·056
3·071
(10 noc-I)
U'L
(hom)
(hom)
2·828
- 34·277
78·037
139·526
- 36·833
(kPa)
2·869
U~2)
59·330
(kPa)
- 33·391
U~I)
(kPa)
78·037
58·526
59·330
lT~3)
Axial stresses at interface
2·936
(l0- hO C- 1 )
aT
Thermal expansion coeffiCients
(kPa)
44·760
44·307
47·777
U(4) c
Table 3. Thermal expansion coefficients and axial stresses in 4-layered fiber-reinforced composite using TSC model in coated Nicalon-fiber/BMAS composites as function of thermal coating properties (ratio of coating thickness to radius of matrix outer cylinder equal to 0'05, !1T = rC)
v,
VI
Data for model composites
Table 2. Comparison of axial microstresses determined by the micromechanical cell model (MCM),8 effective modulus (EM) modef and TSC model, in Nicalon-fiber/BMAS composite (ratio of coating thickness to radius of matrix outer cylinder equal to 0·05, !1T = - rc)
N
U1 U1
Cj
~
~ §.
a
~
;::s
'"o·
;::s
-Gi:::l'"
!2..
~
'"~
;::s :;::,..,
i:::l
~
~
~
§.
:;!
~ ~
256
A. AgbosSOll, 1. Pastor 0.00
It
000
n t
1 -0.01
r
-001
a .. s •.
-0.02
rJ:J
-;
:t
:t
'IJ
~ .......
it
t
e ,
-0.02
:a
-0.03
~
=:
"C
...-; ~
N
-0.03
-0.04
E C
-0.05 075
Z
0.80
085
095
0.90
100
-0.04
Fiber
Matrix
'----=::::C:====:::;:::==::"'"
-0.05 L_--L.._ _.l-_~_ _- L - _ - - - ' -_ _--L-_ _ 0.0 0.2 0.6 0.4
0.8
1.0
Normalized Radial Distance from Fiber Center Fig. 2. Thermal radial stress in 4-layered fiber-reinforced composite: ( - - ) coating thermally isotropic, case (a) in Table 3; ( ... ) coating thermally inhomogeneous, case (b) in Table 3; and (- - -) coatmg thermally anisotropic, case (c) in Table 3. 1: Fiber domain; 2: first coating domain; 3: second coating domain: 4: matrix domain.
When the variation laws are not simple, the direct solution from the CCA model becomes complex. The advantage of the proposed n-Iayered model is that it is capable of treating any problem with continuous property variations by fine discretization.
4.2 Influencing of coating and fiber anisotropy in controlling thermal performance In this section a parametric study is done to examine the influence of coating properties on the thermal stresses and the coefficients of thermal expansion of composites. We focus our analysis on the effects of coating and fiber anisotropy on the thermal expansion coefficient for fixed values of fiber, coating and matrix volume fraction (VI = 0·500, V2 = 0·005 and VJ = 0·459 respectively). Two cases were analysed by using an isotropic matrix: (a) the coating and fiber properties are isotropic (i.e. fiber-glass/epoxy, see Table 4). The coating Young's modulus and thermal expansion coefficients vary over a wide range of values: and
(b) the coating properties are isotropic, while the fiber thermal and elastic properties are anisotropic (i.e. T300/epoxy, see Table 4). The coating Young's modulus and thermal expansion coefficients vary over the same range of values as in case (a). The composites are subjected to a uniform temperature change of - 1°C. Figure 5 represents, in three-dimensional (3D) diagrams, the normalized transverse expansion coefficients (CI'!p0m)/CI'J) of the above composites as a function of normalized coating Young's modulus (E~/ E,) and thermal expansion coefficient (CI'~/ 0'3)' It is observed that the fiber anisotropy significantly changes the transverse expansion coefficients. In case (a), the maximum thermal expansion coefficient occurs when 0")0'., is large and E~/EJ = 1: whereas in case (b) the maximum is reached when C1'~/CI'3 and E2/ E J are both high, which corresponds to perfect interface conditions. 19 However, in both case (a) and case (b) when E~/ EJ < 1 and 0'2/0'3 < 1, an increase in (E 2 /E 3 , C1'~/CI'3) yields a decrease in C1'!p0m)/CI'3' This implies that: (1) for lower coating properties, the fiber
Table 4. Properties of the composite constituents used in the parametric studies
Material Graphite T300 Ni AI 6061 Glass fiber Epoxy matrix
EL
(GPa)
ET
(GPa)
226·0 207·0 68·6
22·0 207·0 68·6
2·9
2·9
no
no
(hum)
(hom)
VLT
VTl
aL (10" eel)
(10-" eel)
0·30 0·31 0·345 0·22 0·35
0·42 0·31 0·345 0·22 0·35
-]·5 13·3 23·5 4·9 60·0
27-0 13·3 23·5 4·9 60·0
aT
257
Thermal stresses and thermal expansion coefficients 24'\0\\
r----r---,---,----,-------,r------,
4.3 Effect of interfacial imperfections on the micromechanical stress and thermal expansion coefficient Let us consider the proposed relationships 22 of properties in the interfacial region of polymer-matrix composite materials. The interphase follows an exponential law: E 2 (r) = E3
06
063
a 75
072
069
066
v 2 (r) =
078
VJ
+ (yE)
- E3)R(r)
(29)
+ (f3v I
-
v,)R(r)
(30)
where:
Normalized Radial Distance
Fig. 3. Interphase modulus variation for T300/N)/ AI-6061 according to Jayaraman et a/.17 (r2 - r\)/2r\ = 0'15, i.e. v2~0'248: - - (a, constant variation): - - - - (b, power variation): - - - (c, reciprocal variation); - - - (d, cubic variation).
_)
R (r
1 - rexp(l- r) 1 - fi exp( 1 - fi)
=-------"----''-------
and K
anisotropy has no effect on the thermal expansion coefficient; and (2) E2/ E J = 1 may give the lower bound of the thermal expansion coefficient. Figure 6 presents, in 3D diagrams, the normalized radial stress determined at the fiber/coating interface (17 r (r = r)/E3aij.T) as a function of a normalized coating Young's modulus (E2/ E 3 ) and thermal expansion coefficient (a2/ (3)' It is observed that the fiber anisotropy has no effect on axial thermal stresses. However, as expected, an increase in the ratios E2/ E3 and a2/ a 3 yields an increase in interfacial stress. When E2/ E3 = 0, which corresponds to debonding at the interface, the axial stress at the fiber/coating interface goes to zero.
K
y=-+A f3
= 1- (E3/ E ) 1-(vdv3)
A = (E 3 /E I )
-
(VI/V3)
l-(v)/vJ)
In the light of the present state of technology and research on spatial property variations in the interfacial zone of polymer composites, eqns (29) and (30) are reasonable and suitable for parametric studies. One can see clearly the relationship between the material inhomogeneity, the fibre/matrix adhesion efficiency ( y) and the volume fractions of the constituents (v), V2)' The model of Kakavas et al. 22 depends directly on the parameters y and f3, and accounts for an abrupt variation of elastic properties at the fiber/interphase boundary (y = f3 = 1 corresponds to continuous variation of the elastic properties at the interface of fiber and interphase). 011 010
0)0
- (a) -- (b) ---- (e) -
009 008 007
..
006
;a
003
00
002
~
001
til til
(d)
<1/
005
ijj ';
=.::
o
<1/
005 004
.!::
0
..8
-002
'; -001 <:>
:z
-003 -004
-005
(d)
----------.... . •••. (e)
-005
(b)
-006 -007
o
02
04
06
08
Normalized Radial Distance
10
-008 0
(a) 0.2
04
06
08
10
Normalized Radial Distance (I)
(II)
Fig. 4. Comparison of the radial stress distribution along the r aXIs under a temperature change of -IT for T300/Ni/AI-6061 (v\ = 0·36: V2 ~ 0·248, (r2 - r\)/2r\ = 0·15): (1) n-Iayered thermal self-consistent model (/1 = 10): (II) direct solution using CCA model. 17
A. Agbossou, J. Pastor
258
atom) a3
6
atom) 20
ii3
4 10
2 100
E2 E3
100
aZ a3
E2 E3
0.01
0·2 1:1
3
0.01
(a)
(b)
Fig. 5. Normalized transverse expansion coefficients 0'~hom)/0'3 as a function of normalized coating Young's modulus E2/E3 and thermal expansion coefficient 0'2/0'3: (a) isotropic fiber composite; (b) anisotropic fiber composite (EU E~ ~ 10). VI = 0·50, V:, = 0·005 and v, = 0·495.
Figure 7 shows the vanatlOn of the Young's modulus and Poisson's ratio within the interphase of the analysed composites. The constituent volume fractions of these composites are VI = 0·60, V2 = 0·08 and V3 = 0·32. The study is conducted for a uniform temperature change of - 1°C.
Figure 8 presents the thermal stresses as functions of interphase inhomogeneity and adhesion efficiency. We note a strong effect of the adhesion parameter y on the stresses. A high stress variation was found to exist within the interphase. In the fiber, the radial and axial stresses decrease when adhesion between the
600 .r. .r.
.r. '" Z
2
400
2
Z
OJ 400
~
-'3
":;J
~
~
":;J
":;J
'"
'"
e)
'-' .~
::::
OJ 200
-,; E
:i
z
E
200
':
0
0
001
001
(aJ
(bJ
Fig. 6. Normalized radial stress at the fiber/coating interface as a function of normalized coating Young's modulus E2/E3 and thermal expansion coefficient a2/0'3: (a) isotropic fiber composite; (b) amsotropic fiber composite (EU E~ ~ 10). VI = 0·500, V:, = 0·005 and V, = 0·495.
259
Thermal stresses and thermal expansion coefficients 0.4 60
0.3
40 0.2
20
0.1
o
0.775
r1
13
0.785
0.805
0.795
0.815
Normalized Interphase Radius
0.78
r
13
0.82
Normalized Interphase Radius
Fig. 7. Variation of the elastic properties within the interphase according to Ref. 22.
fiber and matrix is poor. One can also see that adhesion efficiency has no effect on the stresses in the matrix domain. Table 5 shows the thermal expansion coefficient as a function of the adhesion parameters. An increase in adhesion efficiency causes an increase in thermal expansion coefficients. The higher values of (XL when the adhesion efficiency increases favor the deformation of composite structures subjected to temperature changes.
5 CONCLUSION A thermal self-consistent model for n-layered fiber-reinforced composites subjected to a uniform temperature change has been investigated in this paper. On the basis of the energy balance criterion we have shown that, by using the thermal self-consistent model. the problem with no external load can be treated without having to determine the elastic
1.0
0.81
0.8
0.79
,.,===========,--.----,
= 0·60 and
VI
Ve
r
l':i
= o·om~.
characteristics of the homogeneous medium. Therefore we propose semi-analytical and analytical procedures for determining the thermal behavior. Both procedures, different in their implementation, give the same results. However, the analytical expressions seem to be more useful when the composite properties vary continuously. Owing to the lack of published results on thermal Table 5. Thermal expansion coefficients of the glassfiber/epoxy composite as a function of adhesion between fiber and matrix. The interphase properties follow the exponential law of variation presented in Ref. 22 (hnm)
(h(lm)
Adhesion efficiency (y)
O'L
O'T
(10 "oC I)
(lO-n0C- I )
6·671 7·677
28·94 29·01 29·34
0·1 0·5 1
9·079
10r------------------------..-----,
y =0.1
0.8
Y=0.5
y=l
o
=0.1 Y=0.5
y
0.6
-10
y=l
0.4 -20
0.2
-30~~--~~--~--~~--~~U-~~
0.2
0.4
0.6
Nonnahzed Radial Distance
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Nonnahzed RadIal Distance
Fig. 8. Radial stress and axial stress along the r axis under a temperature change of -1 DC for different values of the adhesion parameter in the composite with inhomogeneous interphase; V\ = 0·60, Ve = 0·008 and v, = 0·32. The interpha~e properties follow the exponential law of variation presented in Fig. 7.
260
A. Agbossou. J. Pastor
stresses in n-Iayered (n > 3) composites, we have compared the present results with other theoretical predictions in three-phase composites. The present model successfully reproduces the composite residual stresses calculated by similar studies using different micromechanical models. We have illustrated the application of the n-Iayered model in the evaluation of thermal stresses and thermal expansion coefficients in materials with spatial property variation. It is shown, based on the CCA model. that the thermal self-consistent (TSC) model can be used to treat any problem with continuous property variations by fine discretization. In the light of the present state of technology and research on spatial property variations in the interfacial domain of polymer composites, we have examined the influence of different coating properties on the thermal stresses and coefficients of thermal expansion of the composites. We found that fiber anisotropy and interphase imperfections significantly change the thermal expansion coefficients and the thermal microstresses. The thermal stresses in the fiber decrease when adhesion between fiber and matrix is poor. We also note a strong increase of the composite expansion coefficients with an increase of the fiber/matrix adhesion efficiency. showing that an increase in adhesion efficiency would lead to significant deformations of composite structures subjected to temperature changes. The proposed model could be useful for: 1. analysing the thermal behavior of materials with property gradients: 2. determining thermal expansion coefficients and thermal stresses in n-Iayered media such as gas pipes or tubing: 3. studying the effect of interfacial imperfections on thermal behavior; and 4. providing material guidance for controlling the micromechanical failure modes.
ACKNOWLEDGEMENT The authors acknowledge Mr Mike Rueck for his help in preparing this manuscript. REFERENCES 1. Asamoah, N. K. & Wood, G., Thermal self-straining of fiber-reinforced materials. f. Strain Ana!., 5 (1970) 88-97. 2. Williams, J. H. & Kousiounelos, P. N., Thermoplastic fibre coatings enhance composite strength and toughness. Fibre SCI. Technol., 11 (1978) 83-88.
3. Iesan, D., Thermal stresses in composite cylinders. f. Thermal Stress, 3 (1980) 495-508. 4. Mikata, Y. & Taya, M., Stress field in a coated continuous fiber composite subjected to thermomechanical loadings. 1. Compos. Mater.. 19 (1985) 554-579. 5. Arsenault, R. J. & Taya, M., Thermal residual stress in metal matrix composite. Acta Metall., 35 (1987) 651-659. 6. Chamis, C C & Sendeckyj, G. P., Critique on the theories predicting properties of fibrous composites. f. Compos. Mater., 23 (1968) 332-358. 7. Pagano, N. J. & Tandon, G. P., Elastic of multidirectional coated-fiber composites. Compos. Sci. Techno!., 31 (1988) 273-293. 8. Gardner, S. D., Charles, n, Pittman, J. R. & Hackett, R. M., Residual thermal stresses in filamentary polymer-matrix composites containing an elastomeric interphase. f. Compos. Mater., 278 (1993) 830-860. 9. Hashin, Z. & Strickman, S., A variational approach to the theory of the elastic behaviour of multiphase materials. f. Mech. Phys. Solids, 10 (1963) 127-140. 10. Maurer, F. H. J., in Polymer Composites, ed. B. Sedlacek, Elsevier, New York, 1986, p. 399. 11. Hashin, Z., Thermoelastic properties of particulate composites with imperfect interface. f. Mech. Phys. Solids, 39 (1991) 745-762. 12. Theocaris, P. S., The Mesophase Concept in Composites. Springer-Verlag, Berlin, 1987. 13. Sottos, N. R., McCullough, R. L. & Guceri, Thermal stresses due to property gradients at the fiber/matrix interface. In Proc. 3rd foint ASCE/ASME Mechanics Con/., La Jolla, CA, 1989, pp. 11-20. 14. Herve, E. & Zaoui, A, Elastic behaviour of mUltiply coated fibre-reinforced composites. Int. f. Eng. Sci., 3310 (1995) 1419-1433. 15. Pastor, J. & Nguyen, V., Une resolution du probleme autocoherente a n-phases. In 12eme Congres Franr:ais de Mecamque, Vol. 2, AUM, Paris, 1995, pp. 129-132. 16. Xie, Z. Y. & Chandra, N., Application of GPS tensors to fiber reinforced composites. 1. Compos. Mater., 2914 (1993) 1885-1907. 17. Jayaraman, K. & Reifsnider, K. L., Residual stresses in a composite with continuously varying Young's modulus in fiber/matrix interphase. f. Compos. Mater., 266 (1992) 770-791. 18. Christensen, R. M. & Lo, R. H., Solutions for effective shear properties in three phase sphere and cylinder models. f. Mecl!. Phys. Solids, 27 (1979) 315-330. 19. Mal, A. K. & Singh, S. J., Deformation of Elastic Solids. Prentice Hall, Englewood Cliffs, NJ, 1991. 20. Francescato, P. & Pastor, 1., Limit analysis and homogenization: predicting limit loads of periodic heterogeneous materials. Ellr. 1. Mech. A-Solids (in press). 21. Benveniste, Y., Dvorak, G. J. & Chen, T., Mech. Mater., 7 (1989) 305-317. 22. Kakavas, P. A, Anifantis, N. K., Baxevanakis, K., Katsareas, D. E. & Papnicolaou, G. C, The effect of interfacial imperfections on the micromechanical stress and strain distribution in fiber reinforced composites. 1. Mater. Sci., 30 (1995) 4541-4548.
CUlIlpO.lltf.l
P II: SO 266 - 3538 (96) 0 0 13 7 - 6
THERMAL STRESSES AND THERMAL EXPANSION COEFFICIENTS OF n-LAYERED FIBER-REINFORCED COMPOSITES A. Agbossou & J. Pastor Laboratoire MaterlllllX Composites (LaMaCo), ESIGEC, Universite de Savoie, 73376 Le Bourget du Lac, Chambery, Frallce
(Received 22 May 1996: revised 13 September 1996: accepted 1 October 1996)
control the thermal expansion coefficients of the composite. Many studies of thermal stresses and thermal expansion coefficients have been reported in the literature.I-~ Chamis and Sendeckyt have surveyed the theoretical approaches which include the mechanics-of-materials approaches, self-consistent CCA models, variationaL statisticaL and discreteelements methods. The stress field around a coated reinforcement fiber is presented in Refs 7 and 8. The effective thermal expansion coefficients have been predicted by Hashin and Strickman 9 and Maurer. 10 Composites with imperfect interfaces were studied in Ref. 11. All of these thermal behavior studies were carried out with models with two, three 7.H or at most four 4 homogeneous layers despite the facts that they are applied to multi-layered composites 12 and the actual fiber-reinforced composites may have spatial property variations. 13 In this work, we present a thermal self-consistent (TSC) model for n-layered materials. The proposed model is based on Refs 14 and 15 to account for prediction of the composite elastic behavior. In Section 2 of this paper we formulate the problem, and in Section 3 we present two different solutions. Explicit expressions of thermal stresses and thermal expansion coefficients in multi-layered composites are given. In Section 4, we discuss and compare the results with other calculation~ given in the literature such as generalized plane strain (GPS) tensors, In the micromechanical cell model,x and the CCA mode!.17 In Section -1-, we illustrate the application of the n-layered model in the evaluation of thermal stresses and thermal expansion coefficients in materials with spatial property variations.
Abstract In this work we develop a thermal self-consistent (TSe) model for n-layered fiber-reinforced composites. The jitndamental representative volume element is an inclllsion made lip of a fiber coated by (n - 1) layers encapsulated in an infinite equivalent homogeneous medium. On the basis of the energy eqllivalence principle we show that the thermal problem under conditions of no external loading can be treated without having to determine the elastic characteristics of the equivalent medium. Thereby, thermal expansion coefficients and microstress distributions inside the layers have been determined according to the proposed semi-analytical and analytical procedures. Both procedllres, thollgh different in their implementation, give the same results. The model's performance has been characterized throllgh the sllcces4ul reproduction of composite thermal expansion and residual stresses calclliated from similar studies with different micromechanical models. The proposed model cOllld be llseful for (1) analysing the thermal behavior of materials with property gradients, (2) determining thermal expansion coefficients and thermal stresses in n-layer tubing sllch as gas pipes (lnd (3) stud,ving the effects of interfacial imperfections on thermal behavior. CO 1997 Elsevier Science Linwed. All rights reserved
Keywords: thermoelasticity, thermal stresses, thermal expansIOn coefficients, interphase, n-layer fiber composites
1 INTRODUCTION When the bonding between the constituents of a composite is intimate, even under conditions of no external loading, stresses are set up in the composite as a consequence of differences between the thermal expansion coefficients and elastic constants of the constituents, These thermal stresses cause stress concentrations which may initiate yielding or de bonding. Therefore, in design analysis, it is important to understand how these stresses arise and how to
2 FORMULATION OF THE THERMAL SELFCONSISTENT PROBLEM Let us consider a composite consisting of a concentric n-layered, long circular and cylindrical inclusion of 249
250
A. Agbossou. 1. Pastor
(a)
(b)
Fig. 1. Representation of n-layered fiber-reinforced composites. (a) Representative volume element (RYE); 5 r = 51 + 5,,+1: transverse surface area where 51 is the inclusion surface area. (b) Homogeneous medium without inclusion; 5T + 5L = 5: total transverse and lateral surface area where 5L is the lateral surface area.
radius rn embedded in the equivalent homogeneous medium as shown in Fig. lea). In this representative volume element (RYE) we let phase 1 of the fiber inclusion constitute the central core and phase i lie within the shell limited by the cylinder with radii r,_! and rl (i E [l,n]). Each phase is assumed to be homogeneous, elastic and transversely isotropic along the longitudinal direction (z or L) of the inclusion. The outer cylinder with radius rn +! represents the unknown equivalent homogeneous material. In order to deal with the general self-consistent problem, a limit of the outer cylinder is taken as rn+l ~ x. By introducing a further layer, this model becomes an extension of the model of Christensen and Lo.IS The volume fractions of each layer are equal to: (1)
that ( Uo) stored in the equivalent homogeneous medium which contains no inclusion (Fig. 1(b»:
U
=
Uo for rn +! ~
OCJ
(2)
Therefore, the problem for which we present a rigorous solution is that of obtaining the relationship between the configuration shown in Fig. lea) and the equivalent homogeneous medium (Fig. 1(b». With this energy condition (2) we propose the two following hypotheses.
(HI) The stress and the displacement fields can be expressed by using two fixed finite RYEs as shown in Fig. 1 (finite size). The energy of both RYEs are identical when rn+! goes to infinity. (H2) The two fixed finite RYEs: (H2a) are subjected at all points to the same thermal condition ~T with no stress on the lateral surface SL of the RYE, i.e.
n
where i > 1 and
2: VI = 1 1=1
The entire composite (Fig. l(a» is subjected to a uniform and constant temperature change, ~ T. The problem is to determine the thermal stress and displacement fields. The general solution of stress and displacement fields can be obtained by using the techniques presented by Mal and Singh. 19 This solution in the ith layer depends on unknowns which must be determined from the interface and boundary conditions. 7 .!4 The explicit form of these conditions yields fewer equations than basic unknowns. Further equations are obtained by applying the energy condition, namely that the elastic strain energy (U) stored in the infinite area of the composite model (Fig. l(a» is the same as
(H2b) present the following displacement fields III
= zl,(X,y) + EljXj zl, = 0 (i = 1 to 3) on SL
where (J'"n j is antiperiodic on ST and E = a(hom)~T is the unknown thermal deformation tensor. The form of zl is due to the arbitrary length of the RYE and the periodic condition on ST. 20 Writing out the energy condition (2) gives:
f
«(J',A, -
(J'~~E~:) dv = 0
(3)
l'
where V is the total volume of the RYE; (J'lj and E'l are respectively the stress and the strain in the inclusion;
251
Thermal stresses and thermal expansion coefficients and (T~ and EZ are respectively the stress and strain in the equivalent homogeneous medium (Fig. l(b». By using the divergence theorem, and taking into account (T,].] = 0 and (T~.] = 0, eqn (3) can be expressed as:
I
,JV
T,u, ds -
I T~u7
ds
= 0
(4)
iil'
Then (5) where ST and SL are respectively the transverse and lateral surface area of the RVE: T, and u, are the components of respectively stress and displacement vectors on the boundary of the inclusion; and T~ and ll~) represent the same components as above but on the equivalent homogeneous medium which contains no inclusion (Fig. l(b)). Based on the hypotheses HI and H2, we propose to determine stress and displacement fields by extending the effective-modulus model developed by Pagano and Tandon 7 to deal with the thermal self-consistent problem. With regard to the composite thermal behavior, we found (see Section 3) that this problem could be treated without having to determine the elastic characteristics of the homogeneous equivalent medium.
C~.j are the elastic stiffness constants of the ith
layer, e~') = a~)ilT, e~/) = a~)ilT are the expansional strain components along the longitudinal (L or z), and transverse directions (r-8) respectively, and D~'), D~) and D~) are unknown constants of the ith layer.
Therefore the problem is to unknown constants. For our problem (Fig. l(a»:
•
determine
(3
x 11)
the perfect interface bonding conditions are U~/) = U~'+l)
at r = r,
(8)
(perfect displacement bonding condition)
(9)
(perfect stress bonding condition)
•
from eqns (6) and (8) the displacement along the fiber direction is verified:
•
from eqn (6) the finite displacement condition at the origin, r = 0, yields
3 METHOD OF SOLUTION
(11)
3.1 Formulation of displacement and stress fields When the composite model in Fig. l(a) is subjected to a uniform temperature change ilT, the displacement and stress fields which verify the equilibrium equation in the ith layer are gIven In cylindrical coordinates (r,8,Z) by:7
(6)
and
Substituting eqn (6) into eqn (10), one obtains: (12) where
The problem is now reduced to determining (211 + I) unknown constants. The relations eqns (8) and (9) yield (211) equations. The final equation is obtained by applying the energy condition (HI). To obtain this equation we set:
E
=
af}'°m)ilT(e/2Je r) + (a~()m)ilT + EO)ee ®ee
=
ef}'°m)(er®er ) + (eLhom) + Eo)ee ®e~
and then let Eo = O. The condition eqn (5) gives •
where: the subscript 1 represents the longitudinal direction z or L,
on the boundary of the RVE (Fig. 1(a) ): on
A. AgbosSOll, J. Pastor
252
[Aj, {Y} and {B} take the form: 1
r1
-rl -
-
•
c(hom)c-
12
LO
_ 2c(hom)_2_ ~
,
r
on
[Aj
•
-
2K~1)
2C~1
, rl
0
- 2K~~1
0
, r3.
2Ki~)
0
0
- r3
0
0
0
-
2CW v I
2C\~)V2
0
2C\~~)V3
1
Finally, we obtain:
r2
L51(eLhom) + Eo){(2C\'d(D\') - e¥l) + C\'/(Eo + e~oml
0
- e~I)/sI - ci~()m)Eo} + 27rL(c\;om)E o 2c~~om)elf'°m)D~n+l) =
2C~~)
0
0
2C~~)
0
-
r=
r2
2K\21 _3
=
on the boundary of the RVE homogeneous medium without inclusion (Fig. l(b)):
1
- r2
0
0
rl
D(n+l)
(13)
r~
1
where 51 is the inclusion surface area, and the symbol ( . Is, means the average over 51' By writing the fixed Eo as Eo = - e~om) we obtain D~n+l) = O. The problem now is to show that Din + l ) = 0 when Eo = O. Due to the linearity of the thermoelastic problem, let the coefficient D
r, 2n~)
0
0
0
0
0
C\~)
-
C(~)
0
cg) -
C(~)
(17)
0
r3
7
0
0
0
r'3
2K~~)
-
C\~)
3
~C(I)V II , ,~l
where B~,;l) and B~:+I) correspond to the value D~n+l) in the thermal problem (i)..T = 1, Eo = 0) and D~n+l1 in elastic problem (LlT = 0, Eo = 1) respectively. According to Ref. 14 the coefficient Bc~~f" vanishes in the elastic problem. Therefore B~,/l) vanishes, yielding to D<;,+I) = O. It should be mentioned that the proof we give of the coefficient D&n+l) vanishing shows that the TSC problem is similar to the Hashin-Rosen thermoelastic problem in which the boundary conditions are on the inclusion of radius rw
Dill DI21 D~21
{Y} =
and
D\4) D~4)
o o {B} =
3.2 Semi-analytical procedure The substitution of Eo = 0 into eqn (13) gives:
D\3) D~')
C\~)er)
+ 2Ki~)ef) -
CWe~ 1_
2KWe lf>
Cg)e~)
+ 2Kg)e~) -
Cgler) -
2Kh~)e!f)
o C\~)et')
The explicit form of the relations eqn (15) gives a linear equation:
[A]{Y} + {B} = 0
(16)
where the components of the vector {Y} are the basic unknowns (D\I),D<;I, ... ,D\n),D~n),elf'°m).eLhom»). In the case of 3-layered fiber-reinforced composites
(18)
+ 2Kg)e~)
~ ( 2C\lde¥) + C\,/e~) )v/
where 2Kilj = C<;i + C<;j and 2CW = c<;d - C<;d. Thus. for a given n-Iayered fiber-reinforced composite the semi-analytical procedure consists of determining the matrix [Aj and the components of {Y} and {B}, and then solving the linear eqn (16) by using the standard Gauss algorithm.
Thermal stresses and thermal expansion coefficients
3.3 Analytical solution In the analytical approach, the problem is to find an analytical expression for the three unknown constants D\'), D~) and D~) of the ith layer. Let X, denote the vector with components Dll) and D~). The perfect bonding conditions at the interface, eqns (8) and (9), yield: T'+I(r,)X'+1 + D¥,+115,+1 + !:lTF,+l _ (n+ll-T,(r,)X,+D, S,+!:lTF,
253
above constants can be expressed as: D\II=
[(2K&'3)a¥'1 -
2K¥~IP\nl
+
C\f~la~'I)b
+
(2Kl{~)M\n'
+ C\21)a]r;, + 2C~4lPjnlb 2Kl{~IM\nl
[( -
-
C\f~I)C
- 2C~4)M¥'la
+ 2K~'3)Q\f;-1 Ib ]r~ + 2C~~)MY')c -
2C~4)Q~;lb
(22)
(19)
D
,
(n+l)
where:
T,(r)=f
[( r 2K~')
F,
= { _
-L,Y1l r
- C\f;I)C
+ 2Kl{~IQ\~-I)b ]r;, + 2C~~IMin)c - 2C~4IQ~;lb (23)
where
Ci'ial'~ 2K~_Ja¥)}
5,
=
n
{C~'i}
a = '2)2C\'iP\' I - C\?al l - 2C\'ia¥»v, n
and D¥,+I) = D~l) (according to eqn (10». The solution of eqn (19) gives: X- l+l -- N(IIX-I
2K~~)M\nl
b = 2: (C\,/ + 2C\'iM\l)v,
with
MI II = 0, p\ II = 0
,~I
IL+ D(n+l + a"TW-1+1 3 1+1
f1
(20)
C
= 2C\~IVI + 2:(2C\'iQ\'I-II)V, [=2
where
Therefore, eqn (12) yields ap:°m l = D~I 1/ J. T. From eqn (8) with i = n. one obtains: D(n) D\n+ll=a\h()ml=Dlnl+~
(24)
r~l
l
The eqns (7), (21). (22), (23) and (24) are the analytical form sought. From these equations, one can determine the thermal stresses and thermal expansion coefficients in an n-layered fiber-reinforced composite.
1 C(l+1 12 { 1 } 2C(l+11 _ 2
- C(')12 -
,+1 -
r
22
1
It;+1 =
4 NUMERICAL RESULTS AND DISCUSSION
(C\'tl1al+ 11 + 2K~tlla¥+I) - C\'ial ' - 2K~ia¥l) { 1 }
4.1 Comparison of TSC model and other calculations Section 3 provided semi-analytical and analytical procedures for determining the thermal stresses and thermal expansion coefficients of an n-layered fiber-reinforced composite subjected to a uniform temperature change. Although the two procedures lead to different expressions. they yield the same results. However. the analytical expressions seem to be more useful when the composite properties vary continuously. Owing to the lack of published results on thermal stresses in n-layered (n > 3) composites, we compare the present results with other theoretical predictions in three-layer composites (fiber / coating/ matrix). Consider first the special case of a continuous fiber embedded in an infinite matrix with uniform eigenstrain in the inclusion (Eshelby's problem). The corresponding stresses determined by G PS tensors 16 and the proposed method are given in Table 1. The stresses in the fiber domain (first layer) and in the matrix domain (second and third layers) coincide with
-r;
2C~2+1)
By using the iterative approach analogous to the one developed by Herve and Zaoui,14 eqn (20) can be expressed as:
Xa l = Q(l)X I + D¥,+l I M,+l + !::.TP,+l
(21)
where
K
P,+I = It;+1 + The
determination
of
,
~(I]N{j»)lt; the
three
unknowns
(D\l),D~I,D~'I) is now reduced to finding the two
constants Dlli and D~n+ll. By using the energy condition and eqn (12), the
A. Agbossolt, f. Pastor
254
Table 1. Comparison of thermal stress values using GPS tensor and TSC model: I1T=
Data for model composites
EL
ET
(GPa)
(GPa)
84 84 84 84 4 4
84 84 84 84 4 4
VLT
0·22 0·22 0·22 0·22 0·40 0·40
VTT
0·22 0·22 0·22 0·22 0·40 0·40
Thermal expansion coefficients
aL aT (10- 6 °e l ) (10- 60 e
4·9 0 0 4·9 60 60
-ec
4·9 0 0 4·9 30 30
l
)
TSC (hom) aL (1O- 00 e
GPS IT~l) (kPa)
(hom)
)
aT (1O- 00 e
l
1·470
1·470
10·740
34·287
those obtained by Eshelby's method and GPS tensors. It can be noted that Eshelby's method to compute the stress fields in the matrix is quite elaborate. 4 Although the GPS tensor concept is different from the model presented, the same coincidences are observed in composites which are not isotropic thermal expansions (Table 1). Table 2 compares the results of the three concentric cylinders model (EM model, effective modulus strain approach), the micromechanical cell (MCM) model,S and the present (TSC) model. This comparison was done on coated Nicalon-fiber/BMAS composites subjected to a uniform temperature change. The composite constituents are elastic and isotropic. Data for the model composites appear in the tables. The result of the present model is consistent with the results of Pagano and Tandon (EM strain approach),7 Gardner et al.,8 and Benveniste et al. 21 We observe that the TSC model gives slightly higher values than the MCM model and that the difference in the results falls within the range of experimental uncertainty. Tn order to give numerical results for the n-Iayered TSC model we have selected a composite with four layers (fiber, first coating, second coating, matrix). The material properties are representative of Nicalon fibers, and barium-magnesium alumino-silicate (BMAS). This composite has a ratio of coating thickness to radius of matrix outer cylinder equal to 0·05, which corresponds to a coating volume fraction of ~0·08. Table 3 gives the thermal expansion and thermal stresses for the above composites. Figure 2 shows the radial stress distributions predicted by the present model as a function of the normalized radial distance from the fiber center. The stresses have been normalized with respect to the product of the Young's modulus, the thermal expansion coefficient of the matrix and the temperature change (E 4 cx 4 !J.T). One can observe that the thermal homogeneity of the coating significantly affects the thermal stresses and the thermal expansion coefficient. In order to prove the validity of the TSC model in
=0·30;
V2
+ V3 =0·70;
'3 =1;
and
Thermal stresses at fiber/coating interface
TSC l
Vi
GPS IT;!) (kPa)
TSC IT~I) (kPa)
TSC IT;!) (kPa)
)
-369,384 -184'692 -369'384
528·273
85·645
-184'692
528·273
85·645
the heterogeneous fiber-reinforced composite, we compare the TSC results with the direct solution using the CCA model. The analysis is carried out under the conditions previously studied by Jayaraman et al.17 The composite (T300/Ni/ Al-6061) with nickel-coated fibers has been considered in this study. Table 4 gives the properties of the composite constituents. The interphase elastic properties are modelled as follows. (a) Constant variation: E2 = ENl
(ENl = 207 GPa)
(25)
(b) Power variation: E2 = Pr Q
(P = 2·218 X lOW, Q = - 4·544) (26)
(c) Reciprocal variation:
E 2 =P/(r-Q)
(P=1·773xlO lO ,
Q = 0·522)
(27)
(d) Cubic variation:
E2
=
Pr 3 + Qr 2 + Rr + S
(28)
(P = - 4·665 X lOB, Q = 1·046 X 10
R = - 7·824 X
1013,
S = 1·958 X
14
,
10 13 )
Figure 3 shows the curves corresponding to the above interphase Young's modulus. The composite fiber volume fraction is VI = 0·36, and the coating volume fraction is set at 0·248 which corresponds to (r2 - rd/2rl = 0·15. This composite is subjected to a uniform temperature change of -1°C. In order to determine the composite thermal stresses we finely discretize the interphase Young's modulus and then apply the analytical relationships. Figure 4 compares thermal stresses in the radial direction as determined by the TSC model with the direct solution by using the CCA model. 17 We note that the CCA analysis using a continuous variation of E2(r) yields the same results as the TSC model using a discretized approximation for E2(r).
0·08
()·no
()'32
U,
2()() 34·5 106
(GPa)
(GPa)
200 34·5 106
EI
EL
()'298 ()·31 0·232
VIT
0·298 0·31 ()·232
"IT
O'L
;./ 1·8 2·7
(10-noc I) 3·2 1·8 2·7
"I (10-noC 1)
)
32·7
(kPa)
(T~'
- 57·7
(kPa)
(~)
U c
- 46·9
(kPa)
(T~3J
NumerIC (MCM)
33·5
(kPa)
atl)
- 60·3
(kPa)
l2)
U c
- 47·1
(kPa)
(3)
U c
EffectIve modulus (EM)
33·39
(kPa)
(T~J )
- 59·33
(kPa)
(T;2J
- 47·77
(kPa)
(T~Jl
Thermal self-consistent (TSC)
0·04
()·no
Ca,e (c)
0'04
0·04
0·04
1'.1
()'32
0·32
0·32
V4
200 34·5 34·5 106 200 34·5 34·5 106 200 34·5 34·5 106
EL (GPa)
200 34'5 34·5 106 200 34·5 34'5 106 200 34'5 34·5 106
EI (GPa)
Ca~e
(c) coatIng thermally
ani~otropic,
(aLh' Lo.!lmg) =
ac
nJ
0'~2nd COdling)
*'
COdtlllgl)
Lo.ttm~))
0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232
V T1
Q'L
3·2 1·8 1·8 2·7 3·2 0·18 1·8 2·7 3·2 1·8 1·8 2·7
(lO-ooC 1)
= uV-nu L:O,tllng) # (aL2 nJ. L(l.Jlmg) = a.(/nd UhttJn g »). (a\ht LOdt1I1gl = a\;nct ClMtlll g ) .
0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232 0·298 0·31 0·31 0·232
VLI'
compm,Ile~
Case (a): coatIng thermally IsotropIC, (T~J<,t co,ltllHU = a 11ht cnJllIlg) = Case (b): coating thermally Inhomogeneous, (O'~ hi lOdlll1g) = ay~t
0·04
0·60
Case (h)
0·04
0·60
V2
Case (a)
VI
Data for model
3·2 1'8 1·8 2·7 3·2 0·18 1·8 2'7 3·2 0·18 0·18 2'7
Cl:'r
(l0- n oc- 1 )
3·069
3·056
3·071
(10 noc-I)
U'L
(hom)
(hom)
2·828
- 34·277
78·037
139·526
- 36·833
(kPa)
2·869
U~2)
59·330
(kPa)
- 33·391
U~I)
(kPa)
78·037
58·526
59·330
lT~3)
Axial stresses at interface
2·936
(l0- hO C- 1 )
aT
Thermal expansion coeffiCients
(kPa)
44·760
44·307
47·777
U(4) c
Table 3. Thermal expansion coefficients and axial stresses in 4-layered fiber-reinforced composite using TSC model in coated Nicalon-fiber/BMAS composites as function of thermal coating properties (ratio of coating thickness to radius of matrix outer cylinder equal to 0'05, !1T = rC)
v,
VI
Data for model composites
Table 2. Comparison of axial microstresses determined by the micromechanical cell model (MCM),8 effective modulus (EM) modef and TSC model, in Nicalon-fiber/BMAS composite (ratio of coating thickness to radius of matrix outer cylinder equal to 0·05, !1T = - rc)
N
U1 U1
Cj
~
~ §.
a
~
;::s
'"o·
;::s
-Gi:::l'"
!2..
~
'"~
;::s :;::,..,
i:::l
~
~
~
§.
:;!
~ ~
256
A. AgbosSOll, 1. Pastor 0.00
It
000
n t
1 -0.01
r
-001
a .. s •.
-0.02
rJ:J
-;
:t
:t
'IJ
~ .......
it
t
e ,
-0.02
:a
-0.03
~
=:
"C
...-; ~
N
-0.03
-0.04
E C
-0.05 075
Z
0.80
085
095
0.90
100
-0.04
Fiber
Matrix
'----=::::C:====:::;:::==::"'"
-0.05 L_--L.._ _.l-_~_ _- L - _ - - - ' -_ _--L-_ _ 0.0 0.2 0.6 0.4
0.8
1.0
Normalized Radial Distance from Fiber Center Fig. 2. Thermal radial stress in 4-layered fiber-reinforced composite: ( - - ) coating thermally isotropic, case (a) in Table 3; ( ... ) coating thermally inhomogeneous, case (b) in Table 3; and (- - -) coatmg thermally anisotropic, case (c) in Table 3. 1: Fiber domain; 2: first coating domain; 3: second coating domain: 4: matrix domain.
When the variation laws are not simple, the direct solution from the CCA model becomes complex. The advantage of the proposed n-Iayered model is that it is capable of treating any problem with continuous property variations by fine discretization.
4.2 Influencing of coating and fiber anisotropy in controlling thermal performance In this section a parametric study is done to examine the influence of coating properties on the thermal stresses and the coefficients of thermal expansion of composites. We focus our analysis on the effects of coating and fiber anisotropy on the thermal expansion coefficient for fixed values of fiber, coating and matrix volume fraction (VI = 0·500, V2 = 0·005 and VJ = 0·459 respectively). Two cases were analysed by using an isotropic matrix: (a) the coating and fiber properties are isotropic (i.e. fiber-glass/epoxy, see Table 4). The coating Young's modulus and thermal expansion coefficients vary over a wide range of values: and
(b) the coating properties are isotropic, while the fiber thermal and elastic properties are anisotropic (i.e. T300/epoxy, see Table 4). The coating Young's modulus and thermal expansion coefficients vary over the same range of values as in case (a). The composites are subjected to a uniform temperature change of - 1°C. Figure 5 represents, in three-dimensional (3D) diagrams, the normalized transverse expansion coefficients (CI'!p0m)/CI'J) of the above composites as a function of normalized coating Young's modulus (E~/ E,) and thermal expansion coefficient (CI'~/ 0'3)' It is observed that the fiber anisotropy significantly changes the transverse expansion coefficients. In case (a), the maximum thermal expansion coefficient occurs when 0")0'., is large and E~/EJ = 1: whereas in case (b) the maximum is reached when C1'~/CI'3 and E2/ E J are both high, which corresponds to perfect interface conditions. 19 However, in both case (a) and case (b) when E~/ EJ < 1 and 0'2/0'3 < 1, an increase in (E 2 /E 3 , C1'~/CI'3) yields a decrease in C1'!p0m)/CI'3' This implies that: (1) for lower coating properties, the fiber
Table 4. Properties of the composite constituents used in the parametric studies
Material Graphite T300 Ni AI 6061 Glass fiber Epoxy matrix
EL
(GPa)
ET
(GPa)
226·0 207·0 68·6
22·0 207·0 68·6
2·9
2·9
no
no
(hum)
(hom)
VLT
VTl
aL (10" eel)
(10-" eel)
0·30 0·31 0·345 0·22 0·35
0·42 0·31 0·345 0·22 0·35
-]·5 13·3 23·5 4·9 60·0
27-0 13·3 23·5 4·9 60·0
aT
257
Thermal stresses and thermal expansion coefficients 24'\0\\
r----r---,---,----,-------,r------,
4.3 Effect of interfacial imperfections on the micromechanical stress and thermal expansion coefficient Let us consider the proposed relationships 22 of properties in the interfacial region of polymer-matrix composite materials. The interphase follows an exponential law: E 2 (r) = E3
06
063
a 75
072
069
066
v 2 (r) =
078
VJ
+ (yE)
- E3)R(r)
(29)
+ (f3v I
-
v,)R(r)
(30)
where:
Normalized Radial Distance
Fig. 3. Interphase modulus variation for T300/N)/ AI-6061 according to Jayaraman et a/.17 (r2 - r\)/2r\ = 0'15, i.e. v2~0'248: - - (a, constant variation): - - - - (b, power variation): - - - (c, reciprocal variation); - - - (d, cubic variation).
_)
R (r
1 - rexp(l- r) 1 - fi exp( 1 - fi)
=-------"----''-------
and K
anisotropy has no effect on the thermal expansion coefficient; and (2) E2/ E J = 1 may give the lower bound of the thermal expansion coefficient. Figure 6 presents, in 3D diagrams, the normalized radial stress determined at the fiber/coating interface (17 r (r = r)/E3aij.T) as a function of a normalized coating Young's modulus (E2/ E 3 ) and thermal expansion coefficient (a2/ (3)' It is observed that the fiber anisotropy has no effect on axial thermal stresses. However, as expected, an increase in the ratios E2/ E3 and a2/ a 3 yields an increase in interfacial stress. When E2/ E3 = 0, which corresponds to debonding at the interface, the axial stress at the fiber/coating interface goes to zero.
K
y=-+A f3
= 1- (E3/ E ) 1-(vdv3)
A = (E 3 /E I )
-
(VI/V3)
l-(v)/vJ)
In the light of the present state of technology and research on spatial property variations in the interfacial zone of polymer composites, eqns (29) and (30) are reasonable and suitable for parametric studies. One can see clearly the relationship between the material inhomogeneity, the fibre/matrix adhesion efficiency ( y) and the volume fractions of the constituents (v), V2)' The model of Kakavas et al. 22 depends directly on the parameters y and f3, and accounts for an abrupt variation of elastic properties at the fiber/interphase boundary (y = f3 = 1 corresponds to continuous variation of the elastic properties at the interface of fiber and interphase). 011 010
0)0
- (a) -- (b) ---- (e) -
009 008 007
..
006
;a
003
00
002
~
001
til til
(d)
<1/
005
ijj ';
=.::
o
<1/
005 004
.!::
0
..8
-002
'; -001 <:>
:z
-003 -004
-005
(d)
----------.... . •••. (e)
-005
(b)
-006 -007
o
02
04
06
08
Normalized Radial Distance
10
-008 0
(a) 0.2
04
06
08
10
Normalized Radial Distance (I)
(II)
Fig. 4. Comparison of the radial stress distribution along the r aXIs under a temperature change of -IT for T300/Ni/AI-6061 (v\ = 0·36: V2 ~ 0·248, (r2 - r\)/2r\ = 0·15): (1) n-Iayered thermal self-consistent model (/1 = 10): (II) direct solution using CCA model. 17
A. Agbossou, J. Pastor
258
atom) a3
6
atom) 20
ii3
4 10
2 100
E2 E3
100
aZ a3
E2 E3
0.01
0·2 1:1
3
0.01
(a)
(b)
Fig. 5. Normalized transverse expansion coefficients 0'~hom)/0'3 as a function of normalized coating Young's modulus E2/E3 and thermal expansion coefficient 0'2/0'3: (a) isotropic fiber composite; (b) anisotropic fiber composite (EU E~ ~ 10). VI = 0·50, V:, = 0·005 and v, = 0·495.
Figure 7 shows the vanatlOn of the Young's modulus and Poisson's ratio within the interphase of the analysed composites. The constituent volume fractions of these composites are VI = 0·60, V2 = 0·08 and V3 = 0·32. The study is conducted for a uniform temperature change of - 1°C.
Figure 8 presents the thermal stresses as functions of interphase inhomogeneity and adhesion efficiency. We note a strong effect of the adhesion parameter y on the stresses. A high stress variation was found to exist within the interphase. In the fiber, the radial and axial stresses decrease when adhesion between the
600 .r. .r.
.r. '" Z
2
400
2
Z
OJ 400
~
-'3
":;J
~
~
":;J
":;J
'"
'"
e)
'-' .~
::::
OJ 200
-,; E
:i
z
E
200
':
0
0
001
001
(aJ
(bJ
Fig. 6. Normalized radial stress at the fiber/coating interface as a function of normalized coating Young's modulus E2/E3 and thermal expansion coefficient a2/0'3: (a) isotropic fiber composite; (b) amsotropic fiber composite (EU E~ ~ 10). VI = 0·500, V:, = 0·005 and V, = 0·495.
259
Thermal stresses and thermal expansion coefficients 0.4 60
0.3
40 0.2
20
0.1
o
0.775
r1
13
0.785
0.805
0.795
0.815
Normalized Interphase Radius
0.78
r
13
0.82
Normalized Interphase Radius
Fig. 7. Variation of the elastic properties within the interphase according to Ref. 22.
fiber and matrix is poor. One can also see that adhesion efficiency has no effect on the stresses in the matrix domain. Table 5 shows the thermal expansion coefficient as a function of the adhesion parameters. An increase in adhesion efficiency causes an increase in thermal expansion coefficients. The higher values of (XL when the adhesion efficiency increases favor the deformation of composite structures subjected to temperature changes.
5 CONCLUSION A thermal self-consistent model for n-layered fiber-reinforced composites subjected to a uniform temperature change has been investigated in this paper. On the basis of the energy balance criterion we have shown that, by using the thermal self-consistent model. the problem with no external load can be treated without having to determine the elastic
1.0
0.81
0.8
0.79
,.,===========,--.----,
= 0·60 and
VI
Ve
r
l':i
= o·om~.
characteristics of the homogeneous medium. Therefore we propose semi-analytical and analytical procedures for determining the thermal behavior. Both procedures, different in their implementation, give the same results. However, the analytical expressions seem to be more useful when the composite properties vary continuously. Owing to the lack of published results on thermal Table 5. Thermal expansion coefficients of the glassfiber/epoxy composite as a function of adhesion between fiber and matrix. The interphase properties follow the exponential law of variation presented in Ref. 22 (hnm)
(h(lm)
Adhesion efficiency (y)
O'L
O'T
(10 "oC I)
(lO-n0C- I )
6·671 7·677
28·94 29·01 29·34
0·1 0·5 1
9·079
10r------------------------..-----,
y =0.1
0.8
Y=0.5
y=l
o
=0.1 Y=0.5
y
0.6
-10
y=l
0.4 -20
0.2
-30~~--~~--~--~~--~~U-~~
0.2
0.4
0.6
Nonnahzed Radial Distance
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Nonnahzed RadIal Distance
Fig. 8. Radial stress and axial stress along the r axis under a temperature change of -1 DC for different values of the adhesion parameter in the composite with inhomogeneous interphase; V\ = 0·60, Ve = 0·008 and v, = 0·32. The interpha~e properties follow the exponential law of variation presented in Fig. 7.
260
A. Agbossou. J. Pastor
stresses in n-Iayered (n > 3) composites, we have compared the present results with other theoretical predictions in three-phase composites. The present model successfully reproduces the composite residual stresses calculated by similar studies using different micromechanical models. We have illustrated the application of the n-Iayered model in the evaluation of thermal stresses and thermal expansion coefficients in materials with spatial property variation. It is shown, based on the CCA model. that the thermal self-consistent (TSC) model can be used to treat any problem with continuous property variations by fine discretization. In the light of the present state of technology and research on spatial property variations in the interfacial domain of polymer composites, we have examined the influence of different coating properties on the thermal stresses and coefficients of thermal expansion of the composites. We found that fiber anisotropy and interphase imperfections significantly change the thermal expansion coefficients and the thermal microstresses. The thermal stresses in the fiber decrease when adhesion between fiber and matrix is poor. We also note a strong increase of the composite expansion coefficients with an increase of the fiber/matrix adhesion efficiency. showing that an increase in adhesion efficiency would lead to significant deformations of composite structures subjected to temperature changes. The proposed model could be useful for: 1. analysing the thermal behavior of materials with property gradients: 2. determining thermal expansion coefficients and thermal stresses in n-Iayered media such as gas pipes or tubing: 3. studying the effect of interfacial imperfections on thermal behavior; and 4. providing material guidance for controlling the micromechanical failure modes.
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