- Email: [email protected]
Dual-stage tertiary creep of a ceramic matrix composite
Materials Science and Engineering A250 (1998) 279 – 284
Dual-stage tertiary creep of a ceramic matrix composite F. Lamouroux a, J.L. Valle´s b, M. Steen b,* a
b
Laboratoire des Multimate´riaux et Interfaces, Lyon, France European Commission, Institute for Ad6anced Materials, PO box 2, 1755 ZG Petten, The Netherlands
Abstract The creep behaviour of an alumina fibre/silicon carbide matrix composite has been studied. The creep curves are characterised by a short primary stage followed by one or two tertiary stages. The secondary regime of this composite is limited to a single point. The occurrence of one or two tertiary stages in the creep behaviour is discussed, and some theoretical considerations are invoked to explain this behaviour. Creep in this composite is controlled by two mechanisms, namely viscoplastic creep of the alumina fibres and damage accumulation within the composite. The two tertiary stages differ in the damage mechanisms occurring, the first one being related to fibre–matrix debonding only, whereas successive fibre failure dominates in the second part. The second tertiary regime occurs only at low creep stresses, for which a non-catastrophic rupture of the composite is observed. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Creep; Dual-stage; Interfacial debonding
1. Introduction Continuous fibre-reinforced ceramic matrix composites (CMCs) present a much higher damage tolerance than monolithic ceramics and are hence promising candidates for high temperature load-bearing applications. Under uniaxial tensile loading, for instance, a non-linear non-brittle behaviour is observed, which can be understood by considering the activation of composite-specific damage mechanisms, such as fibre–matrix debonding originating at matrix microcracks. When designing for high temperature structural applications, the mechanical behaviour of the material under creep loading is also highly relevant, but less understood at this date [1]. Mechanisms other than those controlling the creep of monolithic ceramics occur, and need to be identified. Compared with ceramics, CMCs are characterised by a high toughness because of their possibility to accommodate damage. The damage state is expected to change with the level of applied load, as well as with time and environmental parameters. These influences on the damage state may affect the creep response under high temperature mechanical loading [2]. The * Corresponding author. Tel.: +31 224 565271; fax: + 31 224 532036; e-mail: [email protected] 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00602-9
damage mechanisms are related to a mismatch in the thermomechanical properties of fibres and matrix. For instance, differences in Young’s modulus, Poisson ratio or thermal expansion coefficient may induce matrix or fibre failure, and/or fibre–matrix partial debonding. A difference between the creep rate of the fibre and that of the matrix may also promote the activation of those damage mechanisms under high temperature static loading. In order to identify the possible interactions between damage and creep, a model material has been studied. At a temperature of 1100°C the investigated Al2O3(f)/ SiC composite is characterised by a creep mismatch ratio (CMR), defined as the ratio of the intrinsic creep rate of the fibre to that of the matrix, considerably larger than one. In such a case, substantial stress redistribution occurs between the fibres and the matrix within the composite, and the mechanisms controlling creep behaviour and the changes in creep mechanisms associated with the occurrence of damage are amplified [3].
2. Material The investigated CMC consists of woven alumina fibres (Sumitomo®) incorporated in a silicon carbide
280
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
matrix. The fibres making up the 2D plain weave structure contain 85 wt.% g-alumina and 15 wt.% amorphous silica. The maximum temperature of the fibre manufacturing process is 970°C (final heat treatment), which causes alumina to crystallise with a small grain size of around 12 nm. In a first step, the fibres are coated with a carbon layer (interphase) in order to promote the desired pseudo-ductile behaviour of the composite. The thickness of this carbon layer ranges from 0.5 to 1 mm. The silicon carbide matrix is subsequently formed in situ by isothermal chemical vapour infiltration (CVI). The as-processed composite presents a fibre volume fraction of 0.4 and : 20% volume of porosity. The observation of a polished section of the as-received composite shows the presence of microcracks in the matrix inside the fibre tows and partial debonding between fibre and matrix. The intratow matrix microcracks are oriented perpendicular to the local fibre axis. These cracks develop during the cooling down step from the processing temperature and are caused by the thermal expansion mismatch between fibres and matrix [3].
creases non-linearly with time, and which gives rise to composite rupture. The creep behaviour of the composite is thus characterised by a dual-stage tertiary domain at low stresses and by a single tertiary stage at high creep stresses. Creep tests incorporating unloading–reloading loops allows for the monitoring of the elastic response and thus the damage state of the composite during creep. As shown in Fig. 2, the longitudinal elastic modulus decreases as the inelastic strain increases, indicating that damage accumulates during creep. Since the decrease in elastic modulus decelerates as creep proceeds, the linear increase in creep rate during the first tertiary stage cannot be explained by fibre failure. In order to identify the damage mechanism, morphological investigations have been performed before and after tertiary creep. It has been observed that crack multiplication does not occur during tertiary creep. Hence, from the absence of fibre failure and additional matrix cracking, it was concluded that the main damage mechanism responsible for the first tertiary creep stage is progressive fibre–matrix debonding.
3. Experimental procedure Uniaxial tensile and creep tests have been performed at 1100°C. Flat rectangular cross section specimens of 10×3 (mm2) with a total length of 200 mm are tested in a vacuum below 10 − 4 Pa in order to avoid oxidative attack of the interphases after the formation of matrix cracks. Strain is recorded by a clip-on extensometer equipped with silicon carbide rods. Strain and stress data are recorded digitally and stored for further evaluation. Creep tests have been carried out at stresses of 100 and 170 MPa. The tests are periodically interrupted by an unloading– reloading cycle in order to observe the changes in the elastic response of the composite. The unloading and reloading rate is around 2 MPa s − 1 which is high enough to limit recovery phenomena during the cycle.
4. First tertiary stage The creep curves (Fig. 1) present a primary stage in which the composite creep rate decreases with time. The secondary creep stage, corresponding to a constant creep rate, is limited to a single point in the creep curves. A first tertiary creep stage occurs immediately after it, and is characterised by a linearly increasing creep rate as shown in Fig. 1. This stage is observed at both high and low creep stresses. However, composite rupture occurs at the end of this domain only in the case of high creep stresses. For low creep stresses, a second tertiary appears in which the creep rate in-
Fig. 1. Evolution of the strain rate during creep tests at 1100°C under (a) 100 MPa and (b) 170 MPa.
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
281
rate o; f, the debonding process is governed by a constant rate b [4]. If, moreover, one assumes that the strain rate in a matrix block between two cracks can be approximated by that of the debonded part as long as complete debonding has not been attained, one can deduce that the microcomposite strain rate is given by
o; (t)= o; f · X0 +
Fig. 2. Longitudinal elastic modulus versus creep strain (the dotted line is a guide for the eye to show the expected behaviour until rupture; the horizontal line corresponds to a composite with totally debonded interfaces).
After matrix cracking the fibres located in the debonded zone at the matrix cracks take up the entire load. The creep rate in this zone is hence higher than that in the bonded part, where the fibre stress is lower and where the restraining effect of the matrix limits strain accumulation. During constant stress creep Sumitomo® fibres present necking while creeping stationary. This radial inelastic strain is induced by the high longitudinal creep strain and is expected to promote fibre–matrix debonding. Because secondary or stationary longitudinal fibre creep implies a constant radial strain rate of the fibre, it can be assumed that radial fibre creep induces a constant debonding rate at the fibre–matrix interface, which is enhanced by a high creep mismatch ratio between the fibre and the matrix. Consequently, the fibre length exposed to the full applied creep stress steadily increases as creep proceeds, thus causing a linear increase of the composite creep rate with time. At low stresses (100 MPa) the modulus continues to decrease down to the theoretical value EfVfl given by the rule of mixtures, where Ef is the elastic modulus of the fibre and Vfl is the fraction of fibres loaded in the axial direction, which corresponds to half of the nominal fibre volume fraction of the bi-directional composite (see Fig. 2). The attainment of this value implies that complete interfacial debonding is reached in the low stress range. At high stresses (170 MPa) composite rupture occurs before the fibres are totally debonded, and only a single tertiary creep stage is observed. In order to model the strain accumulation in the first tertiary stage, a microcomposite is considered, consisting of a single alumina fibre surrounded by a silicon carbide matrix mantle which exhibits microcracks perpendicular to the fibre axis (Fig. 3). Assuming that the fibre is creeping in its secondary regime with a constant
2b t p
(1)
where X0 = X(t= 0) is the initial debonded fraction at the onset of the tertiary stage, and p is the average matrix crack spacing. In order to predict the tertiary creep rate of the composite at a given applied stress, it is necessary to understand how the debonding rate and the initial state of damage change with stress. Because, as explained before, the debonding rate is proportional to the fibre creep rate, and as the latter follows Norton’s law, the stress dependence of the debonding rate also takes the form: b= b*
s s*
n
(2)
where n is the fibre creep stress exponent, and s* represents a normalising constant. The initial fraction of debonded fibre can be obtained from the elastic modulus through:
Eb −1 E0 X0 = 2.Eb −1 Ef.Vf
(3)
where E0 is the experimental value of the composite modulus at the onset of the tertiary stage, and Eb is the value given by the rule of mixtures. Uniaxial tensile tests including unloading–reloading cycles provide the longitudinal elastic modulus as a function of stress. Using Eq. (3), these results allow for the determination of the stress dependence of the initial average debonded fraction X0, characteristic of the damage state at the end of the primary stage. The function X0(s) is plotted in Fig. 4, where experimental results from tensile tests
Fig. 3. Representation of a block in the microcomposite model. It is delimited by two matrix cracks and composed of two distinct regions: (1) the perfectly bonded region and (2) the totally debonded region.
282
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
theory developed for bundles of threads [5]. Because at the end of the first tertiary stage the fibres are totally debonded from the matrix (see Fig. 4), the composite behaves as an ensemble of equivalent bare fibres, and the theory of bundles of threads can be applied. After fibre failure, the load is assumed to be equally redistributed among the remaining fibres. The stress applied to the bundle is then given by: n0 n
s = s0.
(4)
where s0 is the stress before the first fibre failure and n0 and n are the number of initial and remaining fibres respectively. Consequently, the fraction of broken fibres, F(s), can be determined as: Fig. 4. Log – log plot of the stress dependence of the initial fraction of debonded fibre in the composite. The effect of temperature on the fibre – matrix (F – M) load transfer condition is indicated.
at room temperature and at 1100°C are shown. The two curves evidence the effect of temperature on the load transfer condition and thus on the damage rate. Since the thermal expansion coefficient of the fibre is higher than that of the matrix, a temperature increase enhances load transfer, and results in a higher resistance to debonding. This is shown in Fig. 4 by the reduced stress sensitivity of damage at high temperature, as manifested by the smaller slope of X0(s) versus s. At low temperatures the weaker fibre – matrix bond promotes a more sudden debonding of the fibre from the matrix. The lower applied stress required for the onset of interfacial debonding at 1100°C results from the axial residual tensile stress in the matrix which causes matrix cracking at lower stress than at room temperature (ref.).
F(s) =1−
n s = 1− 0. n0 s
(5)
Eq. (5) applies as long as the stress carried by the failed fibre is redistributed uniformly over the remaining unfailed fibres, i.e. as long as catastrophic failure does not occur. The probability of fibre failure at an applied stress s, P(s), is given by the Weibull expression: P(s)=1− exp(− as m)
(6)
where m represents the Weibull modulus of the fibres, and a is a constant. Eqs. (5) and (6) have been plotted in Fig. 5, where F(s) and P(s) are shown as a function of the inverse of the stress applied to the bundle. For a given applied stress, represented by point Q on the abscissa, survival of the bundle is possible when the fraction of fibres which should be broken according to Weibull statistics, P(s), is smaller than or equal to F(s). Consequently, the applied stress range where non-catastrophic failure is possible is bounded by the applied stresses associated to points A and B. However,
5. Second tertiary stage The first tertiary stage ends when the first fibre failure occurs. When a fibre fails, a stress redistribution takes place in the composite. At high stress levels, the associated sudden overloading of the other fibres can cause avalanche fibre failure leading to composite rupture. For a low creep stress, the first fibre failure does not induce severe overloading of the unfailed fibres. In this condition, the stress is redistributed, and the creep rate increases with the increase in fibre stress. Such a behaviour is observed in Fig. 1(a) at the end of the linear part. This second tertiary stage involves noncatastrophic, time-differed rupture of the composite. A threshold stress exists which separates the catastrophic from the non-catastrophic composite creep rupture mode. It can be determined using a statistical
Fig. 5. Graphic analysis of the load redistribution in a bundle of alumina fibres after first fibre failure.
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
for applied stresses in the range A – C, i.e. higher than that corresponding to the tangent point C, F(s) increases faster than P(s), indicating unstable fracture. The applied stress associated to point C hence corresponds to the threshold stress sT, above which catastrophic rupture of the composite occurs. The threshold stress calculated for the alumina fibres in the composite studied here, and for which the statistical parameters m and a are known [4], equals 625 MPa for the fibre bundles, which translates into 125 MPa for the 2D woven composite with 20% of uniaxially loaded fibres and in a condition of full fibre–matrix debonding. This value of the threshold stress agrees with that observed in the creep tests: in a creep test at 100 MPa two tertiary stages are observed, whereas a creep stress of 170 MPa induces a catastrophic creep rupture during the first tertiary stage. A statistical approach which considers the failure probability for a time t, a fibre length L and a stress s, can give the time to failure of the first fibre. This time corresponds to the transition time between the two tertiary regimes, if it exists, or otherwise to the time for composite rupture. The approach assumes that fibre failure occurs in the zone which is exposed to the maximum stress from the beginning of the test. This part of the fibres is included in the initial X0 fraction of debonded fibres. From the Weibull expression,
P(s, t, L)=1−exp −
s s%
m
t t%
b
L L%
(7)
it follows that for a given failure probability, the rupture times scales as:
1
L b s t2 =t1. 1 . 1 L2 s2
m b
1
X b s =t1. 01 . 1 X02 s2
m b
(8)
where the ratio of lengths can be replaced by the ratio of initial fractions of debonded fibres. Knowing the time to rupture at a given stress characterised by a given damage state, it is thus possible to predict the time for first fibre failure for another applied stress and another initial damage state. At low stresses this approach only gives the time for the transition between the two tertiary regimes. The time for composite rupture at the end of the second tertiary regime is not studied in this work. An interesting route would be to consider the statistical distribution of the creep failure strain of the fibres. To our knowledge, this work has not been performed yet, but it could explain the extent of the second tertiary stage. In any case, the second part of the tertiary regime definitely warrants study because it represents the instability domain of the composite resulting in final rupture.
283
6. Discussion Creep of a ceramic matrix composite is controlled by two mechanisms: one related to viscoplastic creep of one of the constituents (fibre or matrix, as dictated by the CMR), and the other associated with damage accumulation. For CMCs with CMR\ 1, failure of the first fibre at low creep stresses induces a transition in the tertiary creep regime of the composite caused by a change in the damage mechanism. The first tertiary stage is controlled by stationary creep of the fibre combined with progressive interfacial debonding. In the second tertiary stage, the main damage mechanism is successive fibre failure with the fibres still creeping in their stationary regime. The combination of viscoplastic strain of one of the constituents and inelastic response associated to a particular damage process also applies to other CMCs, provided that some form of damage pre-exists. Taking into account that the main interest in the application of CMCs lies in their damage tolerance, it appears that this condition is fulfilled in practically every case. Hence, the approach outlined here applies to the majority of fibre reinforced CMCs. 7. Conclusion The tertiary creep of 2D Al2O3 –SiC composite has been studied. Two distinct parts are evidenced during this regime. During both tertiary stages, the fibres creep in their stationary regime. However, the two stages differ in the damage mechanisms which accompany the creep of the fibres. (1) During the first tertiary creep stage progressive fibre matrix debonding prevails. This damage mechanism is promoted by radial fibre shrinkage occurring during fibre creep. A micromechanical model is proposed to calculate the creep rate of the composite based on the creep of the debonded fibre coupled to the rate of the debonding process. (2) The second tertiary stage is observed only at low creep stresses and is related to successive fibre failures. (3) A threshold stress exists for the occurrence of the second tertiary stage. Below the threshold stress, failure of the first fibre induces delayed composite rupture, whereas above this stress, sudden composite rupture is observed. (4) The time to the first fibre failure and the threshold stress are predicted by a statistical model. Acknowledgements This work has been performed within the Research and Development Programme of the European Commission. The authors acknowledge Socie´te´ Europe´enne de Propulsion for providing the composite material.
284
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
References [1] J. Holmes, X. Wu, in: S.V. Nair, K. Yakus (Eds.), High Temperature Mechanical Behaviour of Ceramic Matrix Composites, Butterworth-Heinemann, London, 1995. [2] F. Lamouroux, J. L. Valle´s, M. Steen, Compos. Eng. 5 (1995)
.
1379. [3] F. Lamouroux, M. Steen, J.L. Valle´s, J. Eur. Ceram. Soc. 14 (1994) 529. [4] F. Lamouroux, J. L. Valle´s, M. Steen, J. Eur. Ceram. Soc. 14 (1994) 539. [5] H.E. Daniels, Proc. R. Soc. London A183 (1945) 405.
Dual-stage tertiary creep of a ceramic matrix composite F. Lamouroux a, J.L. Valle´s b, M. Steen b,* a
b
Laboratoire des Multimate´riaux et Interfaces, Lyon, France European Commission, Institute for Ad6anced Materials, PO box 2, 1755 ZG Petten, The Netherlands
Abstract The creep behaviour of an alumina fibre/silicon carbide matrix composite has been studied. The creep curves are characterised by a short primary stage followed by one or two tertiary stages. The secondary regime of this composite is limited to a single point. The occurrence of one or two tertiary stages in the creep behaviour is discussed, and some theoretical considerations are invoked to explain this behaviour. Creep in this composite is controlled by two mechanisms, namely viscoplastic creep of the alumina fibres and damage accumulation within the composite. The two tertiary stages differ in the damage mechanisms occurring, the first one being related to fibre–matrix debonding only, whereas successive fibre failure dominates in the second part. The second tertiary regime occurs only at low creep stresses, for which a non-catastrophic rupture of the composite is observed. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Creep; Dual-stage; Interfacial debonding
1. Introduction Continuous fibre-reinforced ceramic matrix composites (CMCs) present a much higher damage tolerance than monolithic ceramics and are hence promising candidates for high temperature load-bearing applications. Under uniaxial tensile loading, for instance, a non-linear non-brittle behaviour is observed, which can be understood by considering the activation of composite-specific damage mechanisms, such as fibre–matrix debonding originating at matrix microcracks. When designing for high temperature structural applications, the mechanical behaviour of the material under creep loading is also highly relevant, but less understood at this date [1]. Mechanisms other than those controlling the creep of monolithic ceramics occur, and need to be identified. Compared with ceramics, CMCs are characterised by a high toughness because of their possibility to accommodate damage. The damage state is expected to change with the level of applied load, as well as with time and environmental parameters. These influences on the damage state may affect the creep response under high temperature mechanical loading [2]. The * Corresponding author. Tel.: +31 224 565271; fax: + 31 224 532036; e-mail: [email protected] 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00602-9
damage mechanisms are related to a mismatch in the thermomechanical properties of fibres and matrix. For instance, differences in Young’s modulus, Poisson ratio or thermal expansion coefficient may induce matrix or fibre failure, and/or fibre–matrix partial debonding. A difference between the creep rate of the fibre and that of the matrix may also promote the activation of those damage mechanisms under high temperature static loading. In order to identify the possible interactions between damage and creep, a model material has been studied. At a temperature of 1100°C the investigated Al2O3(f)/ SiC composite is characterised by a creep mismatch ratio (CMR), defined as the ratio of the intrinsic creep rate of the fibre to that of the matrix, considerably larger than one. In such a case, substantial stress redistribution occurs between the fibres and the matrix within the composite, and the mechanisms controlling creep behaviour and the changes in creep mechanisms associated with the occurrence of damage are amplified [3].
2. Material The investigated CMC consists of woven alumina fibres (Sumitomo®) incorporated in a silicon carbide
280
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
matrix. The fibres making up the 2D plain weave structure contain 85 wt.% g-alumina and 15 wt.% amorphous silica. The maximum temperature of the fibre manufacturing process is 970°C (final heat treatment), which causes alumina to crystallise with a small grain size of around 12 nm. In a first step, the fibres are coated with a carbon layer (interphase) in order to promote the desired pseudo-ductile behaviour of the composite. The thickness of this carbon layer ranges from 0.5 to 1 mm. The silicon carbide matrix is subsequently formed in situ by isothermal chemical vapour infiltration (CVI). The as-processed composite presents a fibre volume fraction of 0.4 and : 20% volume of porosity. The observation of a polished section of the as-received composite shows the presence of microcracks in the matrix inside the fibre tows and partial debonding between fibre and matrix. The intratow matrix microcracks are oriented perpendicular to the local fibre axis. These cracks develop during the cooling down step from the processing temperature and are caused by the thermal expansion mismatch between fibres and matrix [3].
creases non-linearly with time, and which gives rise to composite rupture. The creep behaviour of the composite is thus characterised by a dual-stage tertiary domain at low stresses and by a single tertiary stage at high creep stresses. Creep tests incorporating unloading–reloading loops allows for the monitoring of the elastic response and thus the damage state of the composite during creep. As shown in Fig. 2, the longitudinal elastic modulus decreases as the inelastic strain increases, indicating that damage accumulates during creep. Since the decrease in elastic modulus decelerates as creep proceeds, the linear increase in creep rate during the first tertiary stage cannot be explained by fibre failure. In order to identify the damage mechanism, morphological investigations have been performed before and after tertiary creep. It has been observed that crack multiplication does not occur during tertiary creep. Hence, from the absence of fibre failure and additional matrix cracking, it was concluded that the main damage mechanism responsible for the first tertiary creep stage is progressive fibre–matrix debonding.
3. Experimental procedure Uniaxial tensile and creep tests have been performed at 1100°C. Flat rectangular cross section specimens of 10×3 (mm2) with a total length of 200 mm are tested in a vacuum below 10 − 4 Pa in order to avoid oxidative attack of the interphases after the formation of matrix cracks. Strain is recorded by a clip-on extensometer equipped with silicon carbide rods. Strain and stress data are recorded digitally and stored for further evaluation. Creep tests have been carried out at stresses of 100 and 170 MPa. The tests are periodically interrupted by an unloading– reloading cycle in order to observe the changes in the elastic response of the composite. The unloading and reloading rate is around 2 MPa s − 1 which is high enough to limit recovery phenomena during the cycle.
4. First tertiary stage The creep curves (Fig. 1) present a primary stage in which the composite creep rate decreases with time. The secondary creep stage, corresponding to a constant creep rate, is limited to a single point in the creep curves. A first tertiary creep stage occurs immediately after it, and is characterised by a linearly increasing creep rate as shown in Fig. 1. This stage is observed at both high and low creep stresses. However, composite rupture occurs at the end of this domain only in the case of high creep stresses. For low creep stresses, a second tertiary appears in which the creep rate in-
Fig. 1. Evolution of the strain rate during creep tests at 1100°C under (a) 100 MPa and (b) 170 MPa.
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
281
rate o; f, the debonding process is governed by a constant rate b [4]. If, moreover, one assumes that the strain rate in a matrix block between two cracks can be approximated by that of the debonded part as long as complete debonding has not been attained, one can deduce that the microcomposite strain rate is given by
o; (t)= o; f · X0 +
Fig. 2. Longitudinal elastic modulus versus creep strain (the dotted line is a guide for the eye to show the expected behaviour until rupture; the horizontal line corresponds to a composite with totally debonded interfaces).
After matrix cracking the fibres located in the debonded zone at the matrix cracks take up the entire load. The creep rate in this zone is hence higher than that in the bonded part, where the fibre stress is lower and where the restraining effect of the matrix limits strain accumulation. During constant stress creep Sumitomo® fibres present necking while creeping stationary. This radial inelastic strain is induced by the high longitudinal creep strain and is expected to promote fibre–matrix debonding. Because secondary or stationary longitudinal fibre creep implies a constant radial strain rate of the fibre, it can be assumed that radial fibre creep induces a constant debonding rate at the fibre–matrix interface, which is enhanced by a high creep mismatch ratio between the fibre and the matrix. Consequently, the fibre length exposed to the full applied creep stress steadily increases as creep proceeds, thus causing a linear increase of the composite creep rate with time. At low stresses (100 MPa) the modulus continues to decrease down to the theoretical value EfVfl given by the rule of mixtures, where Ef is the elastic modulus of the fibre and Vfl is the fraction of fibres loaded in the axial direction, which corresponds to half of the nominal fibre volume fraction of the bi-directional composite (see Fig. 2). The attainment of this value implies that complete interfacial debonding is reached in the low stress range. At high stresses (170 MPa) composite rupture occurs before the fibres are totally debonded, and only a single tertiary creep stage is observed. In order to model the strain accumulation in the first tertiary stage, a microcomposite is considered, consisting of a single alumina fibre surrounded by a silicon carbide matrix mantle which exhibits microcracks perpendicular to the fibre axis (Fig. 3). Assuming that the fibre is creeping in its secondary regime with a constant
2b t p
(1)
where X0 = X(t= 0) is the initial debonded fraction at the onset of the tertiary stage, and p is the average matrix crack spacing. In order to predict the tertiary creep rate of the composite at a given applied stress, it is necessary to understand how the debonding rate and the initial state of damage change with stress. Because, as explained before, the debonding rate is proportional to the fibre creep rate, and as the latter follows Norton’s law, the stress dependence of the debonding rate also takes the form: b= b*
s s*
n
(2)
where n is the fibre creep stress exponent, and s* represents a normalising constant. The initial fraction of debonded fibre can be obtained from the elastic modulus through:
Eb −1 E0 X0 = 2.Eb −1 Ef.Vf
(3)
where E0 is the experimental value of the composite modulus at the onset of the tertiary stage, and Eb is the value given by the rule of mixtures. Uniaxial tensile tests including unloading–reloading cycles provide the longitudinal elastic modulus as a function of stress. Using Eq. (3), these results allow for the determination of the stress dependence of the initial average debonded fraction X0, characteristic of the damage state at the end of the primary stage. The function X0(s) is plotted in Fig. 4, where experimental results from tensile tests
Fig. 3. Representation of a block in the microcomposite model. It is delimited by two matrix cracks and composed of two distinct regions: (1) the perfectly bonded region and (2) the totally debonded region.
282
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
theory developed for bundles of threads [5]. Because at the end of the first tertiary stage the fibres are totally debonded from the matrix (see Fig. 4), the composite behaves as an ensemble of equivalent bare fibres, and the theory of bundles of threads can be applied. After fibre failure, the load is assumed to be equally redistributed among the remaining fibres. The stress applied to the bundle is then given by: n0 n
s = s0.
(4)
where s0 is the stress before the first fibre failure and n0 and n are the number of initial and remaining fibres respectively. Consequently, the fraction of broken fibres, F(s), can be determined as: Fig. 4. Log – log plot of the stress dependence of the initial fraction of debonded fibre in the composite. The effect of temperature on the fibre – matrix (F – M) load transfer condition is indicated.
at room temperature and at 1100°C are shown. The two curves evidence the effect of temperature on the load transfer condition and thus on the damage rate. Since the thermal expansion coefficient of the fibre is higher than that of the matrix, a temperature increase enhances load transfer, and results in a higher resistance to debonding. This is shown in Fig. 4 by the reduced stress sensitivity of damage at high temperature, as manifested by the smaller slope of X0(s) versus s. At low temperatures the weaker fibre – matrix bond promotes a more sudden debonding of the fibre from the matrix. The lower applied stress required for the onset of interfacial debonding at 1100°C results from the axial residual tensile stress in the matrix which causes matrix cracking at lower stress than at room temperature (ref.).
F(s) =1−
n s = 1− 0. n0 s
(5)
Eq. (5) applies as long as the stress carried by the failed fibre is redistributed uniformly over the remaining unfailed fibres, i.e. as long as catastrophic failure does not occur. The probability of fibre failure at an applied stress s, P(s), is given by the Weibull expression: P(s)=1− exp(− as m)
(6)
where m represents the Weibull modulus of the fibres, and a is a constant. Eqs. (5) and (6) have been plotted in Fig. 5, where F(s) and P(s) are shown as a function of the inverse of the stress applied to the bundle. For a given applied stress, represented by point Q on the abscissa, survival of the bundle is possible when the fraction of fibres which should be broken according to Weibull statistics, P(s), is smaller than or equal to F(s). Consequently, the applied stress range where non-catastrophic failure is possible is bounded by the applied stresses associated to points A and B. However,
5. Second tertiary stage The first tertiary stage ends when the first fibre failure occurs. When a fibre fails, a stress redistribution takes place in the composite. At high stress levels, the associated sudden overloading of the other fibres can cause avalanche fibre failure leading to composite rupture. For a low creep stress, the first fibre failure does not induce severe overloading of the unfailed fibres. In this condition, the stress is redistributed, and the creep rate increases with the increase in fibre stress. Such a behaviour is observed in Fig. 1(a) at the end of the linear part. This second tertiary stage involves noncatastrophic, time-differed rupture of the composite. A threshold stress exists which separates the catastrophic from the non-catastrophic composite creep rupture mode. It can be determined using a statistical
Fig. 5. Graphic analysis of the load redistribution in a bundle of alumina fibres after first fibre failure.
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
for applied stresses in the range A – C, i.e. higher than that corresponding to the tangent point C, F(s) increases faster than P(s), indicating unstable fracture. The applied stress associated to point C hence corresponds to the threshold stress sT, above which catastrophic rupture of the composite occurs. The threshold stress calculated for the alumina fibres in the composite studied here, and for which the statistical parameters m and a are known [4], equals 625 MPa for the fibre bundles, which translates into 125 MPa for the 2D woven composite with 20% of uniaxially loaded fibres and in a condition of full fibre–matrix debonding. This value of the threshold stress agrees with that observed in the creep tests: in a creep test at 100 MPa two tertiary stages are observed, whereas a creep stress of 170 MPa induces a catastrophic creep rupture during the first tertiary stage. A statistical approach which considers the failure probability for a time t, a fibre length L and a stress s, can give the time to failure of the first fibre. This time corresponds to the transition time between the two tertiary regimes, if it exists, or otherwise to the time for composite rupture. The approach assumes that fibre failure occurs in the zone which is exposed to the maximum stress from the beginning of the test. This part of the fibres is included in the initial X0 fraction of debonded fibres. From the Weibull expression,
P(s, t, L)=1−exp −
s s%
m
t t%
b
L L%
(7)
it follows that for a given failure probability, the rupture times scales as:
1
L b s t2 =t1. 1 . 1 L2 s2
m b
1
X b s =t1. 01 . 1 X02 s2
m b
(8)
where the ratio of lengths can be replaced by the ratio of initial fractions of debonded fibres. Knowing the time to rupture at a given stress characterised by a given damage state, it is thus possible to predict the time for first fibre failure for another applied stress and another initial damage state. At low stresses this approach only gives the time for the transition between the two tertiary regimes. The time for composite rupture at the end of the second tertiary regime is not studied in this work. An interesting route would be to consider the statistical distribution of the creep failure strain of the fibres. To our knowledge, this work has not been performed yet, but it could explain the extent of the second tertiary stage. In any case, the second part of the tertiary regime definitely warrants study because it represents the instability domain of the composite resulting in final rupture.
283
6. Discussion Creep of a ceramic matrix composite is controlled by two mechanisms: one related to viscoplastic creep of one of the constituents (fibre or matrix, as dictated by the CMR), and the other associated with damage accumulation. For CMCs with CMR\ 1, failure of the first fibre at low creep stresses induces a transition in the tertiary creep regime of the composite caused by a change in the damage mechanism. The first tertiary stage is controlled by stationary creep of the fibre combined with progressive interfacial debonding. In the second tertiary stage, the main damage mechanism is successive fibre failure with the fibres still creeping in their stationary regime. The combination of viscoplastic strain of one of the constituents and inelastic response associated to a particular damage process also applies to other CMCs, provided that some form of damage pre-exists. Taking into account that the main interest in the application of CMCs lies in their damage tolerance, it appears that this condition is fulfilled in practically every case. Hence, the approach outlined here applies to the majority of fibre reinforced CMCs. 7. Conclusion The tertiary creep of 2D Al2O3 –SiC composite has been studied. Two distinct parts are evidenced during this regime. During both tertiary stages, the fibres creep in their stationary regime. However, the two stages differ in the damage mechanisms which accompany the creep of the fibres. (1) During the first tertiary creep stage progressive fibre matrix debonding prevails. This damage mechanism is promoted by radial fibre shrinkage occurring during fibre creep. A micromechanical model is proposed to calculate the creep rate of the composite based on the creep of the debonded fibre coupled to the rate of the debonding process. (2) The second tertiary stage is observed only at low creep stresses and is related to successive fibre failures. (3) A threshold stress exists for the occurrence of the second tertiary stage. Below the threshold stress, failure of the first fibre induces delayed composite rupture, whereas above this stress, sudden composite rupture is observed. (4) The time to the first fibre failure and the threshold stress are predicted by a statistical model. Acknowledgements This work has been performed within the Research and Development Programme of the European Commission. The authors acknowledge Socie´te´ Europe´enne de Propulsion for providing the composite material.
284
F. Lamouroux et al. / Materials Science and Engineering A250 (1998) 279–284
References [1] J. Holmes, X. Wu, in: S.V. Nair, K. Yakus (Eds.), High Temperature Mechanical Behaviour of Ceramic Matrix Composites, Butterworth-Heinemann, London, 1995. [2] F. Lamouroux, J. L. Valle´s, M. Steen, Compos. Eng. 5 (1995)
.
1379. [3] F. Lamouroux, M. Steen, J.L. Valle´s, J. Eur. Ceram. Soc. 14 (1994) 529. [4] F. Lamouroux, J. L. Valle´s, M. Steen, J. Eur. Ceram. Soc. 14 (1994) 539. [5] H.E. Daniels, Proc. R. Soc. London A183 (1945) 405.