- Email: [email protected]
Thermal residual stresses in ceramic matrix composites—I. Axisymmetrical model and finite element analysis
Actu melall. mater. Vol. 43, No. 6, pp. 2241-2253,1995 Copyright 0 1995 Elsevier ScienceLtd Printed in Great Britain. All rights reserved 0956-7151(94)00429-3 0956-7151195 $9.50+ 0.00
Pergamon
THERMAL RESIDUAL STRESSES IN CERAMIC MATRIX COMPOSITES-I. AXISYMMETRICAL MODEL AND FINITE ELEMENT ANALYSIS J.-L. BOBET and J. LAMONf Laboratoire des Composites Thermostructuraux (UMR 47 CNRS-SEP-UBl), 3, all&e de La Bo&tie, 33600 Pessac, France
Domaine Universitaire,
(Received 18 January 1994; in revised form 20 September 1994)
Abstract-The residual stresses induced in composites when cooling down from the processing temperature were determined using a cylinder model and using a finite element computer program. Various specimen geometries were examined: microcomposites, unidirectional composites and flat substrates coated with one or two layers. Various combinations were investigated involving MoSi, as an interphase, Sic as a fiber, a matrix, a substrate or an external coating layer and C as a fiber, a substrate, an interphase or an intermediate coating layer. The influence of factors such as interphase thickness and uncertainty in interphase properties (including Young’s modulus and coefficient of thermal expansion) was analyzed. It was shown that trends in distribution of thermal residual stresses (TRS) prevailing in 1D composites can be satisfactorily predicted using the analytical cylinder model. The presence of a MoSi, interphase induces the highest interfacial stresses but it relieves stresses in the matrix. The presence of a C interphase essentially reduces the interfacial stresses.
1. INTRODUCTION Continuous fiber ceramic matrix composites are promising candidates for many applications, particularly as structural components since they retain the advantages of ceramics while providing an enhanced degree of damage tolerance. Ceramic matrix composites and more particularly those made by Chemical Vapor Infiltration (CVI) of a fiber preform [l] are elaborated at elevated temperatures (of the order of 1OOO’C). Thus upon cooling from the processing temperature, thermal residual stresses (TRS) arise due to thermal expansion mismatch between constituents (fiber, interphase and matrix). Then, these TRS are superimposed upon the applied stressfield. TRS are influenced by various parameters including volume fractions and properties of constituents. Estimation of TRS is a prerequisite to the development and the safe use of high performance composites. This two-part paper has a dual intent: first, assess methods for the determination of TRS in CMC, and second provide data on the residual stresses in CMC having MoSi, or C interfacial layers. In Part I, residual stresses were calculated for various specimen geometries including microcomposites and unidirectional fiber reinforced composites and film-substrate assemblies consisting of one or two coating layers deposited on a flat substrate. The former geometries represent different scales of 2D woven composites. In Part II, TRS were measured on microcomposites tTo whom all correspondence
should be addressed.
and film-substrate assemblies using X-ray techniques [21. Microcomposites represent an elementary cell of the composite. They consist of a single fiber coated with an interfacial material and then embedded in a SIC matrix. Microcomposite test specimens are currently used for investigation of the mechanical behavior and interfacial failure of CMC processed by CVI [3,4]. Sic/Sic and C/Sic fiber-matrix or substrate-film combinations were examined. A compensating material of high thermal expansion coefficient (MoSi,) and a compliant one of very low modulus (Pyrocarbon) were selected as interfacial materials. Several analytical models [5-91 have been put forward to determine the elastic stressfield in a set of two or more coaxial cylinders subject to thermomechanical loading. The one developed by Mikata and Taya [8] has been established for a four cylinder structure including the fiber, the coating layer, the matrix and an infinite radius cylinder made of the actual composite. Although it seems physically sound to take into account the interacting effects with actual composite, the Mikata and Taya approach presents two major drawbacks: (i) the axial force balance imposed as the boundary condition becomes insensitive to the stresses in the fiber and the matrix, so that the predictions become somewhat inconsistent and (ii) equivalent properties for the homogenized actual composite are obtained by a weighting operation which is not rigorous. An analytical model derived from the Mikata and Taya one was used for computation of residual
2241
2242
BOBET and LAMON: RESIDUAL STRESSES IN COMPOSITES-I
stresses in the Sic/Sic and C/Sic microcomposite specimens [lo]. This model can take into account any number of concentric cylinders having a finite outer radius. Although analytical models are important in understanding the influence of constituents upon TRS, their usefulness is limited by the lack of account of the interacting effects of the fiber distribution in actual composites. Attempts in the literature directed at determining TRS in 1D composites considered an equivalent homogenized material [l I-131. In the present paper, residual stresses in practical 1D composites and in film-substrate assemblies were computed using a three-dimensional finite element method. Finite element analysis is appropriate for examining the influence of such factors as fiber spacing as well as fiber packing arrangement (cubic or hexagonal). The effect of various factors including interphase properties, interphase thickness, fiber packing arrangement was studied. 2. DETERMINATION OF THERMALLY INDUCED RESIDUAL STRESSES
base
Fig. 1. Schematic diagram showing the microcomposite and the stress components. where c refers to the stiffness tensor, Z to the strain and oi to the coefficient of thermal expansion (CTE) tensor. AT is the temperature change. The following boundary conditions were assumed (u is the radial displacement, w the axial displacement, the subscripts 1, 2 and 3 refer to matrix, interphase and fiber respectively):
2.1. Analytical cylinder model This analytical model is based upon stress-strain equilibrium equations between the constituents under given boundary conditions of applied deformations. The set of resulting equations is then solved by a standard iterative numerical procedure. A schematic diagram of the microcomposite showing orientation of TRS is given in Fig. 1. It was assumed that stresses are induced by a uniform change of the temperature field. This hypothesis is reasonable since the end of the cool-down step is of primary importance for residual stresses determination. But the small diameter of microcomposites (e20pm) relative to length and the slow rate of cooling down (several hours) provide additional reasons. The constituent materials were assumed to exhibit transverse isotropy with respect to the cylinder axis. Hoop, radial and axial TRS components in the fibre, the interphase and the matrix were obtained from the following basic equation:
A zero radial stress at the external surface of the matrix cylinder: oR(‘)=O Continuity of fiber-interphase boundaries:
at
R =R,.
displacement across the and the interphase-matrix
MI=
u2,
w,
=
w2
at
R = R,
u2 =
U3t
w2 =
w3
at
R=R,.
Continuity of radial fiber-interphase and the boundaries:
Materials Carbon fiber (T300) SIC fiber (Nicalon) PyC interphase MoSi, interphase SIC matrix Sintered SIC (SSiC substrate) Grmhite substrate The subscripts
of the constituents
Young’s modulus @Pa) JG ET 220 200 30 310 350 400
22 200 12 310 350 400
30
30
L and T stand for longitudinal
Shear modulus @Pa) G, 4.8 80 2 124 146 170 12.5 and transverse
stress at the interphase-matrix
R = R, R=R,.
The following equation isfied by axial stresses:
of equilibrium RI i-
Table 1. Prowrties
@yCDTMOW
of the various
o(‘)r :i dr = 0.
s R2
svstems examined
Poisson’s ratio in plane axial
CTE 10-6 “C-1
“12
“13
6
0.12 0.12 0.12 0.25 0.2 0.18
0.42 0.12 0.4 0.25 0.2 0.18
0 3 2 8.4 4.6 4.0
21 3 28 8.4 4.6 4.0
0.2
0.2
2.0
2.0
respectively.
is sat-
ET
BOBET and LAMON: Table 2. Dimensions
(pm)
Matrix thickness
Interphase thickness
Interactive effects from the actual composite were not taken into account in this analysis. Equations of the residual stresses may be expressed as follows:
+CI;‘F
-flI”‘AT
.,,=c,,[,n+~]+c~[A”-~] +CI;‘F
- fiI”‘AT
Qz: = 2CI;‘A, + Cll’ F - BY’AT where
Cfi), Cl”,‘, C# are the components of stiffness tensor of material n. n refers to the order of concentric cylinder (n = 1 for the matrix, n = 2 for the interphase and n = 3 for the fiber); E is Young’s modulus; AT is the temperature change (AT x 1OOO’C); coefficients A,, B, and F depend on materials properties (Young’s modulus, Poisson’s ratio and coefficient of thermal expansion) and radial position. Closed-form equations for A,, B, and F are determined by boundary conditions. For lengthy reasons the resulting equations for TRS will not be detailed in the present paper. They are available in [lo]. Four different microcomposite specimens were examined: SiCrjCjSiC,, SiC,/MoSi,/SiC,, C,/C/SiC, Properties of the constituents and C,/MoSi,/SiC,. are given in Table 1. Dimensions of microcomposites are detailed in Table 2. tDeveloped
by Casa
gifts
Microcomposites
(U.S.A.).
Microcomposites
632
Film-substrate assemblies
1D composites
graphite,/MoSi,/SiC graphite,/MoSi, graphite,/SiC
C&Sic, SiC,/MOSi,/SiC,
s-SiC,/MoSi,/SiC s-SiC,/MoSi,
SiC,/C/SiC,
s-SiC,/C/SiC
2.2. Finite element analysis of TRS The TRS in microcomposites, unidirectional composites and film-substrate assemblies were computed using the GIFTS finite element code (version 6.3).? Various fiber/interphase/matrix combinations and substrate/coating assemblies were examined, as summarized in Table 3. The thermal and mechanical properties assumed for the constituents are given in Table 1. A two-dimensional axisymmetrical analysis was conducted for the microcomposites, whereas for the other geometries three-dimensional analyses were preferred. The volume cells were modelled by two- or three-dimensional isoparametric elements as indicated in Table 4. The meshes constructed for the analysis are exemplified by Figs 2 and 3 which show those meshes constructed for the microcomposite and for the unit cell model of 1D material. The boundary conditions for this problem are straightforward. They must be such that the symmetry is retained, i.e. the symmetry planes must remain in-plane and the angles between planes must be retained. Therefore all the nodes at the symmetry surfaces are constrained to have the same normal displacement. Various fiber packing arrangements including cubic (Figs 4 and 5) and hexagonal patterns (Fig. 6) were studied. Fibers were either uniformly distributed (Figs 4 and 6) or in contact (Fig. 5). A cubic packing arrangement only was considered for this latter case simply because it was observed that fiber packing arrangement exerted a limited influence on TRS distributions. The unit cell repeats itself throughout the composite when fibers are assumed to be identically packed throughout the entire composite. The residual stresses in the
Table 4. Data on the meshes constructed Number of elements
systems
C,/MoSi,/SiC,
2.8 2.8 1.3 1.3
0.5 0.5 0.5 0.5
14 14 7 I
SiC,/C/SiC, SiCr/MoSi,/SiC, C,/MoSi,/SiC, C,/CISiC_
Table 3. Examined
of the microcomposites
Fiber diameters
Microcomuosites
2243
RESIDUAL STRESSES IN COMPOSITES-I
for the finite element analyses Element type
Type of analysis
8 node rectangles
2D axisymmetrical
ID composites
2352
27 node cubes 10 node tetrahedrons 15 node triangular prisms
3D
Film-substrate assemblies 1 coating layer 2 coatine lavers
994 1496
27 node cubes
3D
2244
BOBET
and LAMON:
RESIDUAL
STRESSES
IN COMPOSITES-I
try Axis
Spun
t
Fig. 4. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/Sic composite with a uniform distribution of fibers (cubic fiber packing arrangement).
Matrix
Fig.
2. Mesh used for the finite element calculation residual stresses in microcomposites.
Fig.
3. Example
Interphase
Fiber
of
composite can be found by an analysis of this representative volume cell. The coating layer (thickness = 0.5 pm) was modelled by several layers of elements. The sizes of the cells were chosen such that I’, = 0.5 for the microcomposites and the 1D composites. In the filmsubstrate assemblies, the substrate was always at least 10 times as thick as the CVD coatings (Fig. 7). The influence of coating thickness relative to the substrate one was examined for the graphite/MoSi,/SiC combination.
Matrix
Interphase
Fig. 5. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/Sic composite with fibers in contact (cubic fiber packing arrangement).
Fiber
of mesh used for the analysis unidirectional composites.
of the
Fig. 6. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/WC composite with a uniform distribution of fibers (hexagonal fiber packing arrangemerit).
BOBET
and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I
Axis 1
his2 tis3 .A?&4
Fig. 7. Schematic diagram showing the unit eel1 model selected for the finite element computation of TRS in film-substrate assemblies. The horizontal axes (2, 3, 4) indicate the direction of stresses used for the analysis of stress results. Two thermal loading states were prescribed. The unit cells were assumed initially at a uniform temperature of 1000°C (state 1). Then in state 2, the temperature was uniformly set to 0°C. 3. RESULTS AND DISCUSSION
3.1. Residual stresses in microcomposites Figure 8 shows the TRS determined using the analytical model for SiC,/MoSi,/SiC, and SiC,/C/SiC, microcomposites. The following interesting features may be highlighted: (i) significantly high axial TRS operate on the
400
Fiber
Interphase
and hoop tensile MoSi, interphase,
whereas they are comparatively low in the PyC interphase, (ii) low radial TRS are obtained in both interfacial layers: they are essentially tensile in the SiC,/C/SiC, microcomposites and compressive in the SiC,/MoSi,/SiC, one, (iii) the Sic fiber is under axial compression whereas the matrix is under axial tension. It is worth noting that the presence of a MoSi, interphase significantly decreases the stresses operating on the matrix. These results suggest that the MoSi, interphase is particularly prone to failure (under axial and hoop stresses), whereas matrix cracking and interfacial debonding are favored by the presence of a PyC interphase. Figure 9 shows different trends in the microcomposites reinforced with C fibers: (i) the hoop TRS are now interphase, (ii) significant tensile radial interfacial layers, (iii) comparable tensile TRS whatever the interfacial
very low in the MoSi, TRS prevail
in both
operate on the matrix materials is.
It can be expected from these latter results that both interphases now enable debonding due to high tensile stresses, and that the matrix is prone to failure even in the presence of a MoSi, interfacial layer. These results agree with available experimental data on C,/C/SiC, and SiCI/C/SiC, microcomposite
Matrix 1000-
N.___L-L__ $
2245
Fiber_lnterphase_Matrix_ ___ -_q_
2OQ-
(a)
B
o-’ ._______
(a)
I x ij-lOOOY
8,0
8,5
II -2oooo’ N 3,0 3,5 4,0 4‘5 5,0 Radial distance ( pm)
9,O
Radial distance ( pm) Fiber
Interphase
0”
Matrix
2ooo,
2000 7
Fiber
IrThase
Matrix
5,5
,
__-
6)
(b)
-loOOO
II ” 65I
70I Radial
75I distance
Fig. 8. TRS in (a) SiC,/C/SiC, microcomposites
80I
85I
90I
( Km)
and (b) SiC,/MoSi,/SiC, determined using the analytical cylinder model.
Radial distance
( pm)
Fig. 9. TRS in (a) C&/SIC, and (b) C$/MoSi,/SiC, microcomposites determined using the analytical cylinder model.
2246
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I Matrixunder compression 2 3 2 0,6’ n* 2 : . % 3 Matrixunder tension
Interphase thickness (pm)
Y I
0,5-
g z
:-I o’40
1
2
3
4
5
(a)
6
Matrix thickness (pm)
-looo~~..~!“‘~““~‘~~.I op 0,5 l,o 1,5 2,0 5
Interphase thickness (pm) Fig. 10. Influence of interphase thickness upon TRS in SiC,/MoSi,/SiC, microcomposites.
test specimens which showed the presence of matrix cracks [3,4] in C,/C/SiC, microcomposites after cooling down from the processing temperature and which indicated important interfacial debonding in both systems as the matrix crack proceeds under tensile loads [3]. The effects of matrix thickness upon TRS are in agreement with logical expectation. Results show that TRS decrease as the matrix is thicker. However, TRS dependence upon matrix thickness is not significant provided the volume fraction of matrix remains within reasonable bounds far from the limit of 1. Figure 10 shows that, as logically expected, TRS decrease with increasing interphase thickness. However, interphase thickness influence is limited except for the axial stresses operating on the matrix. Axial stresses thus become compressive as the interphase is thicker. Transition from tension to compression is thickness observed for a critical interphase ~9,:0, < 1 pm for MoSi, and 0, < 3.5 pm for PyC. 19, sensitive to volume is particularly the fraction of matrix for the low values of matrix volume fraction. As shown by Fig. 11, the Bcdependence upon matrix thickness agrees with logical
I,5
0
1
2
3
4
5
6
Matrix thickness (pm) Fig. 11. Influence of matrix thickness upon 0, in (a) C/MoSi,/SiC and (b) C/C/Sic microcomposites.
expectation. Thus thicker interphases are required to get compressive axial TRS in the matrix as the volume fraction of matrix increases. It is worth pointing out that 0, is less sensitive to matrix thickness in the presence of a MoSi, interphase. Moreover, very thin MoSi, interphases when compared with the C ones are able to relieve TRS in the matrix. The above trends are substantiated by experimental data on C,/C/SiC, microcomposites possessing a 2-3 pm thick matrix and a 1 pm thick interphase. These microcomposites generally possess cracks right after cooling down from the processing temperature, indicating that significant tensile TRS built up in the matrix. Figure 11 confirms that, for this matrix thickness, the interphase thickness is much lower than & and therefore the matrix is subject to tensile TRS. Table 5 and Fig. 12 show that the TRS computed using the finite element method are in excellent agreement with the previous ones determined using the analytical model. 3.2. TRS in the ID composites Generally, the fiber is in compression whereas the interphase and the matrix are in tension. Inspection of the distribution of TRS obtained by finite
Table 5. TRS (units: MPa) determined by finite element analysis in microcomposites
Fibre Microcomposites C,/MoSi,/SiC, C&/XT.. C,/MoSi,/SiC, C,/C/SiC_
.‘.
0.4
%
-500 - 520 - 500 -150
700 650 -450 100
Interphase fJR
flA
650 630 -430 130
1800 500 2100 100
OH 300 550 2500 400
Matrix OR 630 580 -210 75
OA 100 300 200 300
QH
OR
-800 -900 200 -150
220 230 -100 35
BOBET and LAMON:
RESIDUAL
STRESSES
2241
IN COMPOSITES--I Interphase
Interphase
0 = 0”
-1000,
J+
8,5
$6
.J,i
Radial distance Fiber
Interphase
ii,0
$5
-1 oooo
.d,o
’ io
b.0
+
( Km)
’ill.0 Lo
io.0 ( pm)
Matrix
2000
Interohase Fiber i -
500~
h I
0.
element analysis showed a uniform axial and hoop stress field throughout the unit cells with non contacting fibers. The average values of these stress components are given in Table 6. Radial TRS depend upon the position. Radial TRS profiles are exemplified in Figs 13 and 14 for a SiC,/MoSi,/SiC, unidirectional composite. Table 6 shows that the TRS were not strongly affected by the fiber packing arrangement selected for the analysis. Except for the hoop stresses (bH) in the interphase of SiC,/MoSi,/SiC, composites, the TRS were generally 20% higher with the hexagonal arrangement. The origin of the significant scatter observed for uH in the interphase of the SiCI/MoSi,/SiC, 1D composite has not been elucidated yet. Apart from this specific discrepancy, TRS dependence upon fiber packing arrangement may be regarded as rather small. This explains why the problem of contacting fibers was examined only for a cubic arrangement. It is worth pointing out that TRS computed for the 1D composites with no contacting fibers compare fairly well with those obtained for the microcomposites (Table 5), except for the previously mentioned hoop stresses in the interphase of the SiC/MoSi,/SiC system. As previously it appears that the MoSi, interphases are subject to much higher stresses than the C ones. However the presence of a MoSi, interphase exerts a limited influence on the axial matrix stresses. These stresses are decreased only for those composite reinforced with C fibers. Nevertheless, the 1D composite may be regarded as an assembly of juxtaposed microcomposites with non-contacting fibers.
Interphase . Fiber
i
-500
( pm)
Fig. 12. Comparison of the distributions of TRS calculated using the axisymmetrical cylinder model and the finite element code for a SiC,/MoSi,/SiC, microcomposite.
Matrix
0=45”
4 z
Radial distance
6.0
Distance
-‘@Ti NII 7.0 ., 11.0 Distance
15.00 19.00 ( pm)
23.00
Fig. 13. Radial TRS distributions along the fiber radii obtained for a SiC,/MoSi,/SiC, unidirectional composite (cubic fiber packing arrangement, uniform distribution). The 0 = 0” direction is given by the line joining the centers of two neighboring fibers.
Generally, the fiber is again in compression and the interphase and the matrix are essentially in tension when fibers are in contact. However, Table 7 shows Interphase Fiber -
+
500‘
B z
Fiber
0. 8 = 0”
B YI -500 -
3
2 -loo’,_ ”I, 7.0
$ 2
7.5 8.0 8.5 Distance ( pm)
9.0
9.5
500
0
3v)
0=45”
3 -500 -z c?
-loooo
I
!.lj-
iO.0 Distance ( lrn)
Fig. 14. Radial TRS distributions along the fiber radii obtained for a SiC/MoSi,/SiC unidirectional composite (cubic fiber packing arrangement, fibers in contact). The fl = 0” direction is given by the line joining the centers of two neighboring fibers.
2248
BOBET
and LAMON:
Table 6. Axial and hoop components
RESIDUAL
STRESSES
IN COMPOSITES-I
of TRS (units: MPa) determined by finite element analysis for ID unit cells with uniformly distributed fibers Interphase
Packing arrangement C,/MoSi,/SiC, C,/C/SiC, SiC,/MoSi,/SiC, SiC,/C/SiC,
Fiber Fiber Cubic Hexagonal Cubic Hexagonal Cubic Hexagonal Cubic Hexagonal
Table 7. Axial and hoop components
Matrix materials
fJA
OH
fJA
OH
fJA
QH
- 500 -600 -550 -620 -400 -450 -150 - 200
700 800 680 790 -450 -475 100 110
1700 1900 450 520 2000 2100 125 I50
400 350 480 420 I700 1200 425 400
150 200 200 280 320 350 300 350
- 700 - 650 -770 -650 300 350 -175 -150
of TRS (units: MPa) determined unit cell with in contact fibers
by finite element analysis for a Matrix
Fiber CA C,/MoSi,/SiC, C&/Sic, SiCI/MoSi,/SiC, SiC,/C/SiC_
%I - 240 -250 -200 -100
Interphase CA
680 690 -420 140
materials *A
atI
GA
1000 210 1300 80
380 210 1240 260
Interior 220 150 500 200
Periphery - 100-200 -so/100 -lOO/-200 -5o/-100
significant differences with the previous case of noncontacting fibers. Thus axial stresses are tremendously reduced in the fiber and in the interphase, and also in the matrix in the presence of a C interphase. Stresses in the matrix are increased by the presence of a MoSi, interphase. This effect may be logically related to the high CTE of MoSi,. The matrix at the periphery of the unit cell is subjected to compressive stresses (Fig. 15). The following points are primarily important:
Then, these cracks may be arrested by the compressive stresses acting at the periphery. This phenomenon was observed within the bundles of a 2D woven Sic/Sic composite made by CVI [14] (Fig. 16). Finally, computations performed for various interphase thicknesses gave, as previously, results in agreement with logical expectation. It was thus observed that stresses in the matrix and in the interphase are increased as interphase (MoSi,) is thicker.
lfirst, even with contacting
3.3. TRS in thin jilms on jlat substrates
l
fibers, the analytical cylinder model predicts trends prevailing in the core of the 1D system, second, it may be foreseen that the presence of contacting fibers within 1D composites or within fiber bundles in woven composites will favor initiation of matrix cracks preferentially from the interior.
m
tensile stresses Peak
Compressive stresses
Fig. 15. Schematic diagram showing the distribution of axial residual stress in a ID SiC,/MoSi,/SiC, composite.
In those substrate/single coating layer combinations, the CTE is always smaller in the substrate than in the coating layer (Table 8). Results of TRS computations agree with logical expectation based upon the comparison of CTE. Generally, the substrate was found in compression whereas the coating layer is in tension. Then, TRS are smaller in the
Fig. 16. Optical micrograph showing matrix cracks arrest phenomenon in the longitudinal bundles of a SiC,/C/SiC, 2D woven composite made by CVI (zoom factor = 320) (courtesy of L. Guillaumat [14]).
BOBET and LAMON:
2249
RESIDUAL STRESSES IN COMPOSITES-I
Table 8. TRS (units: MPa) determined by finite element analysis in the film-substrate assemblies with a single coating layer Interface
GraphiteJMoSi, GraphiteJSiC s-SiCJMoSi,
Max
Min
200 120 200
0 -20 -60
Substrate
Coating Material
AV
Max
Min
AV
Max
Min
Av
95 45 45
60 20 120
-60 -30 -160
-40 -20 -110
1200 950 1600
-200 -90 -250
990 700 1210
graphite substrate than in the SIC substrate, and they are much higher in the coating layers than in the substrates. Distribution of TRS within the specimens is exemplified in Fig. 17 for the SiC,/MoSi, combination. Longitudinal stress components parallel to the diagonals (Axes 2, 3 and 4) as shown in Fig. 7 were considered for the analysis of results. Figure 17 shows that the peak stresses arose in the interior of constituents and then decreased in the vicinity of the surfaces. The stress field is mainly uniform in the interior. Therefore trends can be summarized using average values (Table 8). The maximum and minimum values were also given. They will be used for the analysis of the experimental data in Part II of the paper [2].
Trends are comparable to those observed previously with microcomposites and 1D specimens. Thus it can be seen from Table 8 that the higher TRS are observed in the MoSi, coating layer. The peak stresses are observed with the s-SiC,/MoSi, combination. By contrast, the TRS in MoSi, are significantly decreased in the presence of a graphite substrate. It is worth mentioning that TRS ranking is commensurate with quantity E*Acr, where Acr is the CTE mismatch. In the specimens with a double layer, predictions of the trends in TRS from the CTE mismatch between constituents is not straightforward. Although the CTE of the intermediate layer was either much higher (MoSi,) or smaller (C) than those of the substrate
(a)
Distance along axis 2 (mm)
Fig. 17. Vertical and horizontal distributions of TRS vs distance along the axes 1, 2 and 3 within a SiCs/MoSi, specimen.
2250
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I
Table 9. TRS (units: MPa) determined by finite element analysis in the film-substrate assemblies with two coating layers. Also given are the thicknesses of constituents (units:Nm) Substrate
Coating 1
Coating 2 Material7
Max
Min
AV
Max
Min
AV
Max
Min
Av
900 20 900
- 500 -80 -900
-100 -60 -70
1400 1400
- 700 0 -400
900 1050 700
200 100 400
-300 -400 -900
-50 -300 -200
s-SiCJMoSiJSiC 5/0.5/l
50
-50
-20
1400
400
1200
50
-450
-380
s-SiC,/C/SiC 5/0.5/l
50
-150
-90
170
150
30
-110
-90
Cg,/MoSi,/SiC s/0.1/1 5/0.5/l 5/0.5/0.1
1000
-90
tCg refers to graphite and s-Sic to sintered Sic
and the outer layer, the substrate and the outer layer are in compression whereas the intermediate layer is in tension (Table 9). Distributions of TRS within constituents (Figs 18 and 19) are similar to those obtained previously with the single coating layer specimens. Peak stresses arise also in the interior of constituents and then
decrease more or less rapidly near the surfaces. Stress gradients are commensurate with the CTE mismatch between neighboring layers. For example, the steeper gradients were observed between the C substrate and the MoSi, coating layer. Smoother profiles appeared at the boundary between SIC and MoSi,.
Distance along axis 1 (P) -500: 200
MoSiz )
Dietan-&g
DistancTsg
axis 2
axis 3
100 7 50: )
Distanc~$l~g
axis 4
-lOOFig. 18. Vertical and horizontal
distributions of TRS vs distance Cs/MoSi,/SiC specimen.
along the axes (1, 2, 3 and 4) within a
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES--I
)
1
Oo
Distance along axis 2 (mm)
b Distanc;$zyg i
4
6
8
2251
axis 3
io i2
DietancT$;xg
axis 4
Fig. 19. Vertical and horizontal distributions of TRS vs distance along the axes (1, 2, 3 and 4) within a SiCs/MoSi,/SiC specimen.
Examination of the TRS profiles shown on Figs 18 and 19, suggests that the TRS field is dictated by the interaction of the substrate and the intermediate coating layer. However, TRS were higher in the previous single coating systems, indicating that the outer coating layer exerts a certain influence. Predominant contribution of the substrate may be attributed to its thickness relative to that one of coating layers. This is supported by results obtained when increasing thickness of the coating layers in the graphite/MoSi,/SiC system. Table 9 shows that stresses in the substrate were decreased, whereas they were slightly increased in the coating layers. Table 9 also shows that the highest stresses were obtained in both the intermediate and the outer coating layers of those systems involving MoSi,. However, the influence of the presence of MoSi, interlayer upon the substrate stress field does not appear clearly. Stresses in the substrate are mainly relieved when C is used as a substrate.
3.4. Influence of interphase properties The thermophysical properties used for TRS computations were reported by several authors in the literature. Data on fibers are now well-established. Moreover, at this stage a significant amount of work was aimed at determining matrix properties. Therefore, uncertainty in fiber and matrix data may be considered as rather limited. Conversely, determination of properties pertinent to coating layers is not straightforward. In the present paper these properties were approximated by those of the bulk material. This may provide a rough estimate of interphase properties. Dependence of computed TRS on uncertainty in Young’s modulus and CTE of the interphase was examined on the microcomposites of the present paper (dimensions are given in Table 2). Microcomposites involving a Sic fiber and a SIC matrix were considered for the analysis.
2252
BOBET and LAMON: RESIDUAL STRESSES IN COMPOSITES-I
It can be noticed from Fig. 20 that axial TRS in the matrix and hoop stresses in the interphase are the most sensitive to an uncertainty in the elastic modulus of interphase. Elastic modulus underestimation leads to underestimation of matrix TRS and of hoop stresses in the interphases, whereas all the other TRS components are overestimated. However, it is important to point out that the trend in TRS is not affected by interphase Young’s modulus misestimation. Tensile stresses remain tensile and the compressive ones remain compressive. Uncertainty in CTE of interphase affects significantly the TRS acting on the matrix and on the interphase whereas influence on the fiber is limited (Fig. 21). Overestimation of the CTE causes overestimation of the stresses in the fiber and in the interphase (except uR) whereas stresses in the matrix are underestimated. The trend in certain TRS components including hoop stresses in the matrix and radial and axial stresses in the interphase may be reversed by using misestimated data for the interphase (tension becomes compression and vice versa). In the present paper, the results of TRS computations were in agreement with the mechanical behavior observed experimentally on microcomposites and 1D composites. Additional confidence in the results is provided by X-ray measurements discussed in Part II
PI.
a interphase I a fiber
z
_?:=,,,,, , , ,,,, ,, ,, ,,,, ,, ,,,1, 9
2345678 a interphase
10
(xl06 )
6 5 4 3 2 1 0 -1 -2
-3 a interphase
(x10’
)
Fig. 2 1.Influence of interphase CTE uncertainty upon TRS in a microcomposite with a Sic fiber and a SIC matrix (interphase Young’s modulus = 270 GPa). E interphase I E fiber
2,
P:?
-1 a 1.4 I I 1.8 . . 2.2 * ’
?:6,
,3
4. CONCLUSIONS
o’..-
loo
..I....I...,I....I...,’ 400 200 300 E interphase
2
100
200
300
500
600
(GPa)
400
500
600
E interphase (GPa)
Fig. 20. Influence of interphase Young’s modulus uncertainty upon TRS in microcomposites with SIC fiber and a Sic matrix (interphase CTE = 5 x 10-6”C-‘).
The residual stress field induced in CMC by cooling down from the processing temperature was evaluated on microcomposites and 1D composites with various fiber packing arrangements. An analytical cylinder model derived from the model developed by Mikata and Taya was used for determination of TRS in the microcomposites. TRS in microcomposites and in 1D composites were then computed using the Gifts finite element computer code. TRS in thin films involving one or two coating layers upon a substrate were also examined, for comparison of results with data measured using X-ray techniques, as discussed in the companion paper (Part II) [2]. Various combinations of C, SiC and MoSi, were considered: MoSi, as a coating layer only, C as a fiber, a substrate or an interphase, and Sic as a fiber, a matrix or a substrate. Generally, tensile TRS were observed in the matrix and in the interphase of microcomposites, whereas the fiber was essentially in compression. This trend is enhanced by the use of a stiff interphase with an important CTE (MoSi,). The TRS field is affected by the thickness of the interphase. Thus, depending upon interphase thickness, TRS in the matrix may
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-l
These results were found in good agreement with available experimental mechanical behavior of practical microcomposites. The results may be influenced by a misestimation of interphase properties, including Young’s modulus and CTE. Calculations for microcomposites showed that the trend in TRS is not affected by uncertainty in interphase Young’s modulus, whereas opposite TRS in the matrix and the interphase may be obtained when CTE is misestimated. The TRS field in the 1D composites was found similar to that one determined on microcomposites. Thus the TRS computed for a 1D composite with no contacting fibers were in excellent agreement with those established for the microcomposite. The TRS were not significantly affected by the use of a cubic or hexagonal fiber packing arrangement. Some differences were observed in the presence of fibers in contact, since peak tensile stresses arose in the inner matrix volumes bounded by neighboring fibers whereas pe iphery was subject to significant compressive stresses. On the basis of this result it may be foreseen that cracks will initiate preferentially from the interior of 1D specimens. The possibility exists that they are arrested by the compressive stresses in the external surface. However, the analytical cylinder model was able to predict the trends for TRS prevailing in the case of this 1D composite. In film-substrate assemblies, the substrate was in compression whereas the intermediate coating layer was in tension and the outer coating layer in compression. Trends in the TRS stress field are determined by stiffness and CTE of the constituents. In most cases, stresses were enhanced by the presence of MoSi,, and they were relieved by the presence of c. turn compressive.
AM 4316-I
2253
authors acknowledge the support that they received from SEP and the Ministry of Research and Technology, through a grant given to J.L.B. They are Acknowledgements-The
indebted to J. P. Massaloux and A. Cassagne (SEP) for fruitful discussion about finite element model and also L. Guillaumat (LCTS) for optical micrograph of the composite. REFERENCES
4.
5. 6. 7. 8. 9. 10.
II. 12. 13. 14.
R. Naslain and F. Langlais, Muter. Sci. Res. 20, 145 (1986). J-L. Bobet, R. Naslain, A. Guette, N. Ji, and J-L. Lebrun Acta metall. mater. 43, 2255 (1995). J. Lamon, C. Rechiniac, N. Lissart and P. Corne, Determination of interfacial properties in ceramic matrix composites using microcomposites specimens, in Proc. 5rh ECCM (edited by A. R. Bunsell et al.), pp. 895-900. EACM, Bordeaux (1992). J. Lamon and N. Lissart, Micromechanical and statistical approach to the behavior of CMCs. Proc. 17th Annual Cony. on Comp. and Adv. Ceram., Coca beach, Fla., 1991, pp. 1115-1124. Ceramic Enginering and Science Proceedings. Am. Ceram. Sot. (1993). J. D. Eshelby, Proc. R. Sot., Lond. Ser. A 252, 561 (1957). T. Ishikawa, K. Koyma and S. Kobayashi, J. Comp. Maler. 12, 153 (1978). D. Iesan, J. Thermal Stress 3, 495 (1980). Y. Mikata and M. Taya, J. Comp. Mater. 19, 554 (1985). C. M. Warwick and T. W. Clyne, J. Marer. Sci. 26,3817 (1991). J-L. Bobet, Sur l’emploi de MoSi, comme interphase dans les materiaux composites a matrice SIC tlabores par CVDjCVI. Annexe no 2, These no 987, Bordeaux (1993). P. W. R. Beaumont, J. Strain Anal. Engng Des. 24, 189 (1989). T. Mori and K. Tanaka, Acta metall. 21, 571 (1973). C. H. Hsueh and N. Naito, J. Mater. Sci. 23, 1901 (1988). L. Guillaumat, Microfissuration des CMC. Relation avec la microstructure et le comportement mecanique. Thesis, Univ. of Bordeaux (1993).
Pergamon
THERMAL RESIDUAL STRESSES IN CERAMIC MATRIX COMPOSITES-I. AXISYMMETRICAL MODEL AND FINITE ELEMENT ANALYSIS J.-L. BOBET and J. LAMONf Laboratoire des Composites Thermostructuraux (UMR 47 CNRS-SEP-UBl), 3, all&e de La Bo&tie, 33600 Pessac, France
Domaine Universitaire,
(Received 18 January 1994; in revised form 20 September 1994)
Abstract-The residual stresses induced in composites when cooling down from the processing temperature were determined using a cylinder model and using a finite element computer program. Various specimen geometries were examined: microcomposites, unidirectional composites and flat substrates coated with one or two layers. Various combinations were investigated involving MoSi, as an interphase, Sic as a fiber, a matrix, a substrate or an external coating layer and C as a fiber, a substrate, an interphase or an intermediate coating layer. The influence of factors such as interphase thickness and uncertainty in interphase properties (including Young’s modulus and coefficient of thermal expansion) was analyzed. It was shown that trends in distribution of thermal residual stresses (TRS) prevailing in 1D composites can be satisfactorily predicted using the analytical cylinder model. The presence of a MoSi, interphase induces the highest interfacial stresses but it relieves stresses in the matrix. The presence of a C interphase essentially reduces the interfacial stresses.
1. INTRODUCTION Continuous fiber ceramic matrix composites are promising candidates for many applications, particularly as structural components since they retain the advantages of ceramics while providing an enhanced degree of damage tolerance. Ceramic matrix composites and more particularly those made by Chemical Vapor Infiltration (CVI) of a fiber preform [l] are elaborated at elevated temperatures (of the order of 1OOO’C). Thus upon cooling from the processing temperature, thermal residual stresses (TRS) arise due to thermal expansion mismatch between constituents (fiber, interphase and matrix). Then, these TRS are superimposed upon the applied stressfield. TRS are influenced by various parameters including volume fractions and properties of constituents. Estimation of TRS is a prerequisite to the development and the safe use of high performance composites. This two-part paper has a dual intent: first, assess methods for the determination of TRS in CMC, and second provide data on the residual stresses in CMC having MoSi, or C interfacial layers. In Part I, residual stresses were calculated for various specimen geometries including microcomposites and unidirectional fiber reinforced composites and film-substrate assemblies consisting of one or two coating layers deposited on a flat substrate. The former geometries represent different scales of 2D woven composites. In Part II, TRS were measured on microcomposites tTo whom all correspondence
should be addressed.
and film-substrate assemblies using X-ray techniques [21. Microcomposites represent an elementary cell of the composite. They consist of a single fiber coated with an interfacial material and then embedded in a SIC matrix. Microcomposite test specimens are currently used for investigation of the mechanical behavior and interfacial failure of CMC processed by CVI [3,4]. Sic/Sic and C/Sic fiber-matrix or substrate-film combinations were examined. A compensating material of high thermal expansion coefficient (MoSi,) and a compliant one of very low modulus (Pyrocarbon) were selected as interfacial materials. Several analytical models [5-91 have been put forward to determine the elastic stressfield in a set of two or more coaxial cylinders subject to thermomechanical loading. The one developed by Mikata and Taya [8] has been established for a four cylinder structure including the fiber, the coating layer, the matrix and an infinite radius cylinder made of the actual composite. Although it seems physically sound to take into account the interacting effects with actual composite, the Mikata and Taya approach presents two major drawbacks: (i) the axial force balance imposed as the boundary condition becomes insensitive to the stresses in the fiber and the matrix, so that the predictions become somewhat inconsistent and (ii) equivalent properties for the homogenized actual composite are obtained by a weighting operation which is not rigorous. An analytical model derived from the Mikata and Taya one was used for computation of residual
2241
2242
BOBET and LAMON: RESIDUAL STRESSES IN COMPOSITES-I
stresses in the Sic/Sic and C/Sic microcomposite specimens [lo]. This model can take into account any number of concentric cylinders having a finite outer radius. Although analytical models are important in understanding the influence of constituents upon TRS, their usefulness is limited by the lack of account of the interacting effects of the fiber distribution in actual composites. Attempts in the literature directed at determining TRS in 1D composites considered an equivalent homogenized material [l I-131. In the present paper, residual stresses in practical 1D composites and in film-substrate assemblies were computed using a three-dimensional finite element method. Finite element analysis is appropriate for examining the influence of such factors as fiber spacing as well as fiber packing arrangement (cubic or hexagonal). The effect of various factors including interphase properties, interphase thickness, fiber packing arrangement was studied. 2. DETERMINATION OF THERMALLY INDUCED RESIDUAL STRESSES
base
Fig. 1. Schematic diagram showing the microcomposite and the stress components. where c refers to the stiffness tensor, Z to the strain and oi to the coefficient of thermal expansion (CTE) tensor. AT is the temperature change. The following boundary conditions were assumed (u is the radial displacement, w the axial displacement, the subscripts 1, 2 and 3 refer to matrix, interphase and fiber respectively):
2.1. Analytical cylinder model This analytical model is based upon stress-strain equilibrium equations between the constituents under given boundary conditions of applied deformations. The set of resulting equations is then solved by a standard iterative numerical procedure. A schematic diagram of the microcomposite showing orientation of TRS is given in Fig. 1. It was assumed that stresses are induced by a uniform change of the temperature field. This hypothesis is reasonable since the end of the cool-down step is of primary importance for residual stresses determination. But the small diameter of microcomposites (e20pm) relative to length and the slow rate of cooling down (several hours) provide additional reasons. The constituent materials were assumed to exhibit transverse isotropy with respect to the cylinder axis. Hoop, radial and axial TRS components in the fibre, the interphase and the matrix were obtained from the following basic equation:
A zero radial stress at the external surface of the matrix cylinder: oR(‘)=O Continuity of fiber-interphase boundaries:
at
R =R,.
displacement across the and the interphase-matrix
MI=
u2,
w,
=
w2
at
R = R,
u2 =
U3t
w2 =
w3
at
R=R,.
Continuity of radial fiber-interphase and the boundaries:
Materials Carbon fiber (T300) SIC fiber (Nicalon) PyC interphase MoSi, interphase SIC matrix Sintered SIC (SSiC substrate) Grmhite substrate The subscripts
of the constituents
Young’s modulus @Pa) JG ET 220 200 30 310 350 400
22 200 12 310 350 400
30
30
L and T stand for longitudinal
Shear modulus @Pa) G, 4.8 80 2 124 146 170 12.5 and transverse
stress at the interphase-matrix
R = R, R=R,.
The following equation isfied by axial stresses:
of equilibrium RI i-
Table 1. Prowrties
@yCDTMOW
of the various
o(‘)r :i dr = 0.
s R2
svstems examined
Poisson’s ratio in plane axial
CTE 10-6 “C-1
“12
“13
6
0.12 0.12 0.12 0.25 0.2 0.18
0.42 0.12 0.4 0.25 0.2 0.18
0 3 2 8.4 4.6 4.0
21 3 28 8.4 4.6 4.0
0.2
0.2
2.0
2.0
respectively.
is sat-
ET
BOBET and LAMON: Table 2. Dimensions
(pm)
Matrix thickness
Interphase thickness
Interactive effects from the actual composite were not taken into account in this analysis. Equations of the residual stresses may be expressed as follows:
+CI;‘F
-flI”‘AT
.,,=c,,[,n+~]+c~[A”-~] +CI;‘F
- fiI”‘AT
Qz: = 2CI;‘A, + Cll’ F - BY’AT where
Cfi), Cl”,‘, C# are the components of stiffness tensor of material n. n refers to the order of concentric cylinder (n = 1 for the matrix, n = 2 for the interphase and n = 3 for the fiber); E is Young’s modulus; AT is the temperature change (AT x 1OOO’C); coefficients A,, B, and F depend on materials properties (Young’s modulus, Poisson’s ratio and coefficient of thermal expansion) and radial position. Closed-form equations for A,, B, and F are determined by boundary conditions. For lengthy reasons the resulting equations for TRS will not be detailed in the present paper. They are available in [lo]. Four different microcomposite specimens were examined: SiCrjCjSiC,, SiC,/MoSi,/SiC,, C,/C/SiC, Properties of the constituents and C,/MoSi,/SiC,. are given in Table 1. Dimensions of microcomposites are detailed in Table 2. tDeveloped
by Casa
gifts
Microcomposites
(U.S.A.).
Microcomposites
632
Film-substrate assemblies
1D composites
graphite,/MoSi,/SiC graphite,/MoSi, graphite,/SiC
C&Sic, SiC,/MOSi,/SiC,
s-SiC,/MoSi,/SiC s-SiC,/MoSi,
SiC,/C/SiC,
s-SiC,/C/SiC
2.2. Finite element analysis of TRS The TRS in microcomposites, unidirectional composites and film-substrate assemblies were computed using the GIFTS finite element code (version 6.3).? Various fiber/interphase/matrix combinations and substrate/coating assemblies were examined, as summarized in Table 3. The thermal and mechanical properties assumed for the constituents are given in Table 1. A two-dimensional axisymmetrical analysis was conducted for the microcomposites, whereas for the other geometries three-dimensional analyses were preferred. The volume cells were modelled by two- or three-dimensional isoparametric elements as indicated in Table 4. The meshes constructed for the analysis are exemplified by Figs 2 and 3 which show those meshes constructed for the microcomposite and for the unit cell model of 1D material. The boundary conditions for this problem are straightforward. They must be such that the symmetry is retained, i.e. the symmetry planes must remain in-plane and the angles between planes must be retained. Therefore all the nodes at the symmetry surfaces are constrained to have the same normal displacement. Various fiber packing arrangements including cubic (Figs 4 and 5) and hexagonal patterns (Fig. 6) were studied. Fibers were either uniformly distributed (Figs 4 and 6) or in contact (Fig. 5). A cubic packing arrangement only was considered for this latter case simply because it was observed that fiber packing arrangement exerted a limited influence on TRS distributions. The unit cell repeats itself throughout the composite when fibers are assumed to be identically packed throughout the entire composite. The residual stresses in the
Table 4. Data on the meshes constructed Number of elements
systems
C,/MoSi,/SiC,
2.8 2.8 1.3 1.3
0.5 0.5 0.5 0.5
14 14 7 I
SiC,/C/SiC, SiCr/MoSi,/SiC, C,/MoSi,/SiC, C,/CISiC_
Table 3. Examined
of the microcomposites
Fiber diameters
Microcomuosites
2243
RESIDUAL STRESSES IN COMPOSITES-I
for the finite element analyses Element type
Type of analysis
8 node rectangles
2D axisymmetrical
ID composites
2352
27 node cubes 10 node tetrahedrons 15 node triangular prisms
3D
Film-substrate assemblies 1 coating layer 2 coatine lavers
994 1496
27 node cubes
3D
2244
BOBET
and LAMON:
RESIDUAL
STRESSES
IN COMPOSITES-I
try Axis
Spun
t
Fig. 4. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/Sic composite with a uniform distribution of fibers (cubic fiber packing arrangement).
Matrix
Fig.
2. Mesh used for the finite element calculation residual stresses in microcomposites.
Fig.
3. Example
Interphase
Fiber
of
composite can be found by an analysis of this representative volume cell. The coating layer (thickness = 0.5 pm) was modelled by several layers of elements. The sizes of the cells were chosen such that I’, = 0.5 for the microcomposites and the 1D composites. In the filmsubstrate assemblies, the substrate was always at least 10 times as thick as the CVD coatings (Fig. 7). The influence of coating thickness relative to the substrate one was examined for the graphite/MoSi,/SiC combination.
Matrix
Interphase
Fig. 5. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/Sic composite with fibers in contact (cubic fiber packing arrangement).
Fiber
of mesh used for the analysis unidirectional composites.
of the
Fig. 6. Example of finite element unit cell model for the analysis of TRS in a ID-Sic/WC composite with a uniform distribution of fibers (hexagonal fiber packing arrangemerit).
BOBET
and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I
Axis 1
his2 tis3 .A?&4
Fig. 7. Schematic diagram showing the unit eel1 model selected for the finite element computation of TRS in film-substrate assemblies. The horizontal axes (2, 3, 4) indicate the direction of stresses used for the analysis of stress results. Two thermal loading states were prescribed. The unit cells were assumed initially at a uniform temperature of 1000°C (state 1). Then in state 2, the temperature was uniformly set to 0°C. 3. RESULTS AND DISCUSSION
3.1. Residual stresses in microcomposites Figure 8 shows the TRS determined using the analytical model for SiC,/MoSi,/SiC, and SiC,/C/SiC, microcomposites. The following interesting features may be highlighted: (i) significantly high axial TRS operate on the
400
Fiber
Interphase
and hoop tensile MoSi, interphase,
whereas they are comparatively low in the PyC interphase, (ii) low radial TRS are obtained in both interfacial layers: they are essentially tensile in the SiC,/C/SiC, microcomposites and compressive in the SiC,/MoSi,/SiC, one, (iii) the Sic fiber is under axial compression whereas the matrix is under axial tension. It is worth noting that the presence of a MoSi, interphase significantly decreases the stresses operating on the matrix. These results suggest that the MoSi, interphase is particularly prone to failure (under axial and hoop stresses), whereas matrix cracking and interfacial debonding are favored by the presence of a PyC interphase. Figure 9 shows different trends in the microcomposites reinforced with C fibers: (i) the hoop TRS are now interphase, (ii) significant tensile radial interfacial layers, (iii) comparable tensile TRS whatever the interfacial
very low in the MoSi, TRS prevail
in both
operate on the matrix materials is.
It can be expected from these latter results that both interphases now enable debonding due to high tensile stresses, and that the matrix is prone to failure even in the presence of a MoSi, interfacial layer. These results agree with available experimental data on C,/C/SiC, and SiCI/C/SiC, microcomposite
Matrix 1000-
N.___L-L__ $
2245
Fiber_lnterphase_Matrix_ ___ -_q_
2OQ-
(a)
B
o-’ ._______
(a)
I x ij-lOOOY
8,0
8,5
II -2oooo’ N 3,0 3,5 4,0 4‘5 5,0 Radial distance ( pm)
9,O
Radial distance ( pm) Fiber
Interphase
0”
Matrix
2ooo,
2000 7
Fiber
IrThase
Matrix
5,5
,
__-
6)
(b)
-loOOO
II ” 65I
70I Radial
75I distance
Fig. 8. TRS in (a) SiC,/C/SiC, microcomposites
80I
85I
90I
( Km)
and (b) SiC,/MoSi,/SiC, determined using the analytical cylinder model.
Radial distance
( pm)
Fig. 9. TRS in (a) C&/SIC, and (b) C$/MoSi,/SiC, microcomposites determined using the analytical cylinder model.
2246
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I Matrixunder compression 2 3 2 0,6’ n* 2 : . % 3 Matrixunder tension
Interphase thickness (pm)
Y I
0,5-
g z
:-I o’40
1
2
3
4
5
(a)
6
Matrix thickness (pm)
-looo~~..~!“‘~““~‘~~.I op 0,5 l,o 1,5 2,0 5
Interphase thickness (pm) Fig. 10. Influence of interphase thickness upon TRS in SiC,/MoSi,/SiC, microcomposites.
test specimens which showed the presence of matrix cracks [3,4] in C,/C/SiC, microcomposites after cooling down from the processing temperature and which indicated important interfacial debonding in both systems as the matrix crack proceeds under tensile loads [3]. The effects of matrix thickness upon TRS are in agreement with logical expectation. Results show that TRS decrease as the matrix is thicker. However, TRS dependence upon matrix thickness is not significant provided the volume fraction of matrix remains within reasonable bounds far from the limit of 1. Figure 10 shows that, as logically expected, TRS decrease with increasing interphase thickness. However, interphase thickness influence is limited except for the axial stresses operating on the matrix. Axial stresses thus become compressive as the interphase is thicker. Transition from tension to compression is thickness observed for a critical interphase ~9,:0, < 1 pm for MoSi, and 0, < 3.5 pm for PyC. 19, sensitive to volume is particularly the fraction of matrix for the low values of matrix volume fraction. As shown by Fig. 11, the Bcdependence upon matrix thickness agrees with logical
I,5
0
1
2
3
4
5
6
Matrix thickness (pm) Fig. 11. Influence of matrix thickness upon 0, in (a) C/MoSi,/SiC and (b) C/C/Sic microcomposites.
expectation. Thus thicker interphases are required to get compressive axial TRS in the matrix as the volume fraction of matrix increases. It is worth pointing out that 0, is less sensitive to matrix thickness in the presence of a MoSi, interphase. Moreover, very thin MoSi, interphases when compared with the C ones are able to relieve TRS in the matrix. The above trends are substantiated by experimental data on C,/C/SiC, microcomposites possessing a 2-3 pm thick matrix and a 1 pm thick interphase. These microcomposites generally possess cracks right after cooling down from the processing temperature, indicating that significant tensile TRS built up in the matrix. Figure 11 confirms that, for this matrix thickness, the interphase thickness is much lower than & and therefore the matrix is subject to tensile TRS. Table 5 and Fig. 12 show that the TRS computed using the finite element method are in excellent agreement with the previous ones determined using the analytical model. 3.2. TRS in the ID composites Generally, the fiber is in compression whereas the interphase and the matrix are in tension. Inspection of the distribution of TRS obtained by finite
Table 5. TRS (units: MPa) determined by finite element analysis in microcomposites
Fibre Microcomposites C,/MoSi,/SiC, C&/XT.. C,/MoSi,/SiC, C,/C/SiC_
.‘.
0.4
%
-500 - 520 - 500 -150
700 650 -450 100
Interphase fJR
flA
650 630 -430 130
1800 500 2100 100
OH 300 550 2500 400
Matrix OR 630 580 -210 75
OA 100 300 200 300
QH
OR
-800 -900 200 -150
220 230 -100 35
BOBET and LAMON:
RESIDUAL
STRESSES
2241
IN COMPOSITES--I Interphase
Interphase
0 = 0”
-1000,
J+
8,5
$6
.J,i
Radial distance Fiber
Interphase
ii,0
$5
-1 oooo
.d,o
’ io
b.0
+
( Km)
’ill.0 Lo
io.0 ( pm)
Matrix
2000
Interohase Fiber i -
500~
h I
0.
element analysis showed a uniform axial and hoop stress field throughout the unit cells with non contacting fibers. The average values of these stress components are given in Table 6. Radial TRS depend upon the position. Radial TRS profiles are exemplified in Figs 13 and 14 for a SiC,/MoSi,/SiC, unidirectional composite. Table 6 shows that the TRS were not strongly affected by the fiber packing arrangement selected for the analysis. Except for the hoop stresses (bH) in the interphase of SiC,/MoSi,/SiC, composites, the TRS were generally 20% higher with the hexagonal arrangement. The origin of the significant scatter observed for uH in the interphase of the SiCI/MoSi,/SiC, 1D composite has not been elucidated yet. Apart from this specific discrepancy, TRS dependence upon fiber packing arrangement may be regarded as rather small. This explains why the problem of contacting fibers was examined only for a cubic arrangement. It is worth pointing out that TRS computed for the 1D composites with no contacting fibers compare fairly well with those obtained for the microcomposites (Table 5), except for the previously mentioned hoop stresses in the interphase of the SiC/MoSi,/SiC system. As previously it appears that the MoSi, interphases are subject to much higher stresses than the C ones. However the presence of a MoSi, interphase exerts a limited influence on the axial matrix stresses. These stresses are decreased only for those composite reinforced with C fibers. Nevertheless, the 1D composite may be regarded as an assembly of juxtaposed microcomposites with non-contacting fibers.
Interphase . Fiber
i
-500
( pm)
Fig. 12. Comparison of the distributions of TRS calculated using the axisymmetrical cylinder model and the finite element code for a SiC,/MoSi,/SiC, microcomposite.
Matrix
0=45”
4 z
Radial distance
6.0
Distance
-‘@Ti NII 7.0 ., 11.0 Distance
15.00 19.00 ( pm)
23.00
Fig. 13. Radial TRS distributions along the fiber radii obtained for a SiC,/MoSi,/SiC, unidirectional composite (cubic fiber packing arrangement, uniform distribution). The 0 = 0” direction is given by the line joining the centers of two neighboring fibers.
Generally, the fiber is again in compression and the interphase and the matrix are essentially in tension when fibers are in contact. However, Table 7 shows Interphase Fiber -
+
500‘
B z
Fiber
0. 8 = 0”
B YI -500 -
3
2 -loo’,_ ”I, 7.0
$ 2
7.5 8.0 8.5 Distance ( pm)
9.0
9.5
500
0
3v)
0=45”
3 -500 -z c?
-loooo
I
!.lj-
iO.0 Distance ( lrn)
Fig. 14. Radial TRS distributions along the fiber radii obtained for a SiC/MoSi,/SiC unidirectional composite (cubic fiber packing arrangement, fibers in contact). The fl = 0” direction is given by the line joining the centers of two neighboring fibers.
2248
BOBET
and LAMON:
Table 6. Axial and hoop components
RESIDUAL
STRESSES
IN COMPOSITES-I
of TRS (units: MPa) determined by finite element analysis for ID unit cells with uniformly distributed fibers Interphase
Packing arrangement C,/MoSi,/SiC, C,/C/SiC, SiC,/MoSi,/SiC, SiC,/C/SiC,
Fiber Fiber Cubic Hexagonal Cubic Hexagonal Cubic Hexagonal Cubic Hexagonal
Table 7. Axial and hoop components
Matrix materials
fJA
OH
fJA
OH
fJA
QH
- 500 -600 -550 -620 -400 -450 -150 - 200
700 800 680 790 -450 -475 100 110
1700 1900 450 520 2000 2100 125 I50
400 350 480 420 I700 1200 425 400
150 200 200 280 320 350 300 350
- 700 - 650 -770 -650 300 350 -175 -150
of TRS (units: MPa) determined unit cell with in contact fibers
by finite element analysis for a Matrix
Fiber CA C,/MoSi,/SiC, C&/Sic, SiCI/MoSi,/SiC, SiC,/C/SiC_
%I - 240 -250 -200 -100
Interphase CA
680 690 -420 140
materials *A
atI
GA
1000 210 1300 80
380 210 1240 260
Interior 220 150 500 200
Periphery - 100-200 -so/100 -lOO/-200 -5o/-100
significant differences with the previous case of noncontacting fibers. Thus axial stresses are tremendously reduced in the fiber and in the interphase, and also in the matrix in the presence of a C interphase. Stresses in the matrix are increased by the presence of a MoSi, interphase. This effect may be logically related to the high CTE of MoSi,. The matrix at the periphery of the unit cell is subjected to compressive stresses (Fig. 15). The following points are primarily important:
Then, these cracks may be arrested by the compressive stresses acting at the periphery. This phenomenon was observed within the bundles of a 2D woven Sic/Sic composite made by CVI [14] (Fig. 16). Finally, computations performed for various interphase thicknesses gave, as previously, results in agreement with logical expectation. It was thus observed that stresses in the matrix and in the interphase are increased as interphase (MoSi,) is thicker.
lfirst, even with contacting
3.3. TRS in thin jilms on jlat substrates
l
fibers, the analytical cylinder model predicts trends prevailing in the core of the 1D system, second, it may be foreseen that the presence of contacting fibers within 1D composites or within fiber bundles in woven composites will favor initiation of matrix cracks preferentially from the interior.
m
tensile stresses Peak
Compressive stresses
Fig. 15. Schematic diagram showing the distribution of axial residual stress in a ID SiC,/MoSi,/SiC, composite.
In those substrate/single coating layer combinations, the CTE is always smaller in the substrate than in the coating layer (Table 8). Results of TRS computations agree with logical expectation based upon the comparison of CTE. Generally, the substrate was found in compression whereas the coating layer is in tension. Then, TRS are smaller in the
Fig. 16. Optical micrograph showing matrix cracks arrest phenomenon in the longitudinal bundles of a SiC,/C/SiC, 2D woven composite made by CVI (zoom factor = 320) (courtesy of L. Guillaumat [14]).
BOBET and LAMON:
2249
RESIDUAL STRESSES IN COMPOSITES-I
Table 8. TRS (units: MPa) determined by finite element analysis in the film-substrate assemblies with a single coating layer Interface
GraphiteJMoSi, GraphiteJSiC s-SiCJMoSi,
Max
Min
200 120 200
0 -20 -60
Substrate
Coating Material
AV
Max
Min
AV
Max
Min
Av
95 45 45
60 20 120
-60 -30 -160
-40 -20 -110
1200 950 1600
-200 -90 -250
990 700 1210
graphite substrate than in the SIC substrate, and they are much higher in the coating layers than in the substrates. Distribution of TRS within the specimens is exemplified in Fig. 17 for the SiC,/MoSi, combination. Longitudinal stress components parallel to the diagonals (Axes 2, 3 and 4) as shown in Fig. 7 were considered for the analysis of results. Figure 17 shows that the peak stresses arose in the interior of constituents and then decreased in the vicinity of the surfaces. The stress field is mainly uniform in the interior. Therefore trends can be summarized using average values (Table 8). The maximum and minimum values were also given. They will be used for the analysis of the experimental data in Part II of the paper [2].
Trends are comparable to those observed previously with microcomposites and 1D specimens. Thus it can be seen from Table 8 that the higher TRS are observed in the MoSi, coating layer. The peak stresses are observed with the s-SiC,/MoSi, combination. By contrast, the TRS in MoSi, are significantly decreased in the presence of a graphite substrate. It is worth mentioning that TRS ranking is commensurate with quantity E*Acr, where Acr is the CTE mismatch. In the specimens with a double layer, predictions of the trends in TRS from the CTE mismatch between constituents is not straightforward. Although the CTE of the intermediate layer was either much higher (MoSi,) or smaller (C) than those of the substrate
(a)
Distance along axis 2 (mm)
Fig. 17. Vertical and horizontal distributions of TRS vs distance along the axes 1, 2 and 3 within a SiCs/MoSi, specimen.
2250
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-I
Table 9. TRS (units: MPa) determined by finite element analysis in the film-substrate assemblies with two coating layers. Also given are the thicknesses of constituents (units:Nm) Substrate
Coating 1
Coating 2 Material7
Max
Min
AV
Max
Min
AV
Max
Min
Av
900 20 900
- 500 -80 -900
-100 -60 -70
1400 1400
- 700 0 -400
900 1050 700
200 100 400
-300 -400 -900
-50 -300 -200
s-SiCJMoSiJSiC 5/0.5/l
50
-50
-20
1400
400
1200
50
-450
-380
s-SiC,/C/SiC 5/0.5/l
50
-150
-90
170
150
30
-110
-90
Cg,/MoSi,/SiC s/0.1/1 5/0.5/l 5/0.5/0.1
1000
-90
tCg refers to graphite and s-Sic to sintered Sic
and the outer layer, the substrate and the outer layer are in compression whereas the intermediate layer is in tension (Table 9). Distributions of TRS within constituents (Figs 18 and 19) are similar to those obtained previously with the single coating layer specimens. Peak stresses arise also in the interior of constituents and then
decrease more or less rapidly near the surfaces. Stress gradients are commensurate with the CTE mismatch between neighboring layers. For example, the steeper gradients were observed between the C substrate and the MoSi, coating layer. Smoother profiles appeared at the boundary between SIC and MoSi,.
Distance along axis 1 (P) -500: 200
MoSiz )
Dietan-&g
DistancTsg
axis 2
axis 3
100 7 50: )
Distanc~$l~g
axis 4
-lOOFig. 18. Vertical and horizontal
distributions of TRS vs distance Cs/MoSi,/SiC specimen.
along the axes (1, 2, 3 and 4) within a
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES--I
)
1
Oo
Distance along axis 2 (mm)
b Distanc;$zyg i
4
6
8
2251
axis 3
io i2
DietancT$;xg
axis 4
Fig. 19. Vertical and horizontal distributions of TRS vs distance along the axes (1, 2, 3 and 4) within a SiCs/MoSi,/SiC specimen.
Examination of the TRS profiles shown on Figs 18 and 19, suggests that the TRS field is dictated by the interaction of the substrate and the intermediate coating layer. However, TRS were higher in the previous single coating systems, indicating that the outer coating layer exerts a certain influence. Predominant contribution of the substrate may be attributed to its thickness relative to that one of coating layers. This is supported by results obtained when increasing thickness of the coating layers in the graphite/MoSi,/SiC system. Table 9 shows that stresses in the substrate were decreased, whereas they were slightly increased in the coating layers. Table 9 also shows that the highest stresses were obtained in both the intermediate and the outer coating layers of those systems involving MoSi,. However, the influence of the presence of MoSi, interlayer upon the substrate stress field does not appear clearly. Stresses in the substrate are mainly relieved when C is used as a substrate.
3.4. Influence of interphase properties The thermophysical properties used for TRS computations were reported by several authors in the literature. Data on fibers are now well-established. Moreover, at this stage a significant amount of work was aimed at determining matrix properties. Therefore, uncertainty in fiber and matrix data may be considered as rather limited. Conversely, determination of properties pertinent to coating layers is not straightforward. In the present paper these properties were approximated by those of the bulk material. This may provide a rough estimate of interphase properties. Dependence of computed TRS on uncertainty in Young’s modulus and CTE of the interphase was examined on the microcomposites of the present paper (dimensions are given in Table 2). Microcomposites involving a Sic fiber and a SIC matrix were considered for the analysis.
2252
BOBET and LAMON: RESIDUAL STRESSES IN COMPOSITES-I
It can be noticed from Fig. 20 that axial TRS in the matrix and hoop stresses in the interphase are the most sensitive to an uncertainty in the elastic modulus of interphase. Elastic modulus underestimation leads to underestimation of matrix TRS and of hoop stresses in the interphases, whereas all the other TRS components are overestimated. However, it is important to point out that the trend in TRS is not affected by interphase Young’s modulus misestimation. Tensile stresses remain tensile and the compressive ones remain compressive. Uncertainty in CTE of interphase affects significantly the TRS acting on the matrix and on the interphase whereas influence on the fiber is limited (Fig. 21). Overestimation of the CTE causes overestimation of the stresses in the fiber and in the interphase (except uR) whereas stresses in the matrix are underestimated. The trend in certain TRS components including hoop stresses in the matrix and radial and axial stresses in the interphase may be reversed by using misestimated data for the interphase (tension becomes compression and vice versa). In the present paper, the results of TRS computations were in agreement with the mechanical behavior observed experimentally on microcomposites and 1D composites. Additional confidence in the results is provided by X-ray measurements discussed in Part II
PI.
a interphase I a fiber
z
_?:=,,,,, , , ,,,, ,, ,, ,,,, ,, ,,,1, 9
2345678 a interphase
10
(xl06 )
6 5 4 3 2 1 0 -1 -2
-3 a interphase
(x10’
)
Fig. 2 1.Influence of interphase CTE uncertainty upon TRS in a microcomposite with a Sic fiber and a SIC matrix (interphase Young’s modulus = 270 GPa). E interphase I E fiber
2,
P:?
-1 a 1.4 I I 1.8 . . 2.2 * ’
?:6,
,3
4. CONCLUSIONS
o’..-
loo
..I....I...,I....I...,’ 400 200 300 E interphase
2
100
200
300
500
600
(GPa)
400
500
600
E interphase (GPa)
Fig. 20. Influence of interphase Young’s modulus uncertainty upon TRS in microcomposites with SIC fiber and a Sic matrix (interphase CTE = 5 x 10-6”C-‘).
The residual stress field induced in CMC by cooling down from the processing temperature was evaluated on microcomposites and 1D composites with various fiber packing arrangements. An analytical cylinder model derived from the model developed by Mikata and Taya was used for determination of TRS in the microcomposites. TRS in microcomposites and in 1D composites were then computed using the Gifts finite element computer code. TRS in thin films involving one or two coating layers upon a substrate were also examined, for comparison of results with data measured using X-ray techniques, as discussed in the companion paper (Part II) [2]. Various combinations of C, SiC and MoSi, were considered: MoSi, as a coating layer only, C as a fiber, a substrate or an interphase, and Sic as a fiber, a matrix or a substrate. Generally, tensile TRS were observed in the matrix and in the interphase of microcomposites, whereas the fiber was essentially in compression. This trend is enhanced by the use of a stiff interphase with an important CTE (MoSi,). The TRS field is affected by the thickness of the interphase. Thus, depending upon interphase thickness, TRS in the matrix may
BOBET and LAMON:
RESIDUAL STRESSES IN COMPOSITES-l
These results were found in good agreement with available experimental mechanical behavior of practical microcomposites. The results may be influenced by a misestimation of interphase properties, including Young’s modulus and CTE. Calculations for microcomposites showed that the trend in TRS is not affected by uncertainty in interphase Young’s modulus, whereas opposite TRS in the matrix and the interphase may be obtained when CTE is misestimated. The TRS field in the 1D composites was found similar to that one determined on microcomposites. Thus the TRS computed for a 1D composite with no contacting fibers were in excellent agreement with those established for the microcomposite. The TRS were not significantly affected by the use of a cubic or hexagonal fiber packing arrangement. Some differences were observed in the presence of fibers in contact, since peak tensile stresses arose in the inner matrix volumes bounded by neighboring fibers whereas pe iphery was subject to significant compressive stresses. On the basis of this result it may be foreseen that cracks will initiate preferentially from the interior of 1D specimens. The possibility exists that they are arrested by the compressive stresses in the external surface. However, the analytical cylinder model was able to predict the trends for TRS prevailing in the case of this 1D composite. In film-substrate assemblies, the substrate was in compression whereas the intermediate coating layer was in tension and the outer coating layer in compression. Trends in the TRS stress field are determined by stiffness and CTE of the constituents. In most cases, stresses were enhanced by the presence of MoSi,, and they were relieved by the presence of c. turn compressive.
AM 4316-I
2253
authors acknowledge the support that they received from SEP and the Ministry of Research and Technology, through a grant given to J.L.B. They are Acknowledgements-The
indebted to J. P. Massaloux and A. Cassagne (SEP) for fruitful discussion about finite element model and also L. Guillaumat (LCTS) for optical micrograph of the composite. REFERENCES
4.
5. 6. 7. 8. 9. 10.
II. 12. 13. 14.
R. Naslain and F. Langlais, Muter. Sci. Res. 20, 145 (1986). J-L. Bobet, R. Naslain, A. Guette, N. Ji, and J-L. Lebrun Acta metall. mater. 43, 2255 (1995). J. Lamon, C. Rechiniac, N. Lissart and P. Corne, Determination of interfacial properties in ceramic matrix composites using microcomposites specimens, in Proc. 5rh ECCM (edited by A. R. Bunsell et al.), pp. 895-900. EACM, Bordeaux (1992). J. Lamon and N. Lissart, Micromechanical and statistical approach to the behavior of CMCs. Proc. 17th Annual Cony. on Comp. and Adv. Ceram., Coca beach, Fla., 1991, pp. 1115-1124. Ceramic Enginering and Science Proceedings. Am. Ceram. Sot. (1993). J. D. Eshelby, Proc. R. Sot., Lond. Ser. A 252, 561 (1957). T. Ishikawa, K. Koyma and S. Kobayashi, J. Comp. Maler. 12, 153 (1978). D. Iesan, J. Thermal Stress 3, 495 (1980). Y. Mikata and M. Taya, J. Comp. Mater. 19, 554 (1985). C. M. Warwick and T. W. Clyne, J. Marer. Sci. 26,3817 (1991). J-L. Bobet, Sur l’emploi de MoSi, comme interphase dans les materiaux composites a matrice SIC tlabores par CVDjCVI. Annexe no 2, These no 987, Bordeaux (1993). P. W. R. Beaumont, J. Strain Anal. Engng Des. 24, 189 (1989). T. Mori and K. Tanaka, Acta metall. 21, 571 (1973). C. H. Hsueh and N. Naito, J. Mater. Sci. 23, 1901 (1988). L. Guillaumat, Microfissuration des CMC. Relation avec la microstructure et le comportement mecanique. Thesis, Univ. of Bordeaux (1993).