Modelling the effect of oxidation on the creep behaviour of fibre-reinforced ceramic matrix composites

Acta Materialia 51 (2003) 3745–3757 www.actamat-journals.com

Modelling the effect of oxidation on the creep behaviour of fibre-reinforced ceramic matrix composites L. Casas a,b, J.M. Martı´nez-Esnaola a,b,∗ a b

CEIT—Centro de Estudios e Investigaciones Te´cnicas de Gipuzkoa, P. Manuel Lardizabal 15, 20018 San Sebastia´n, Spain Escuela Superior de Ingenieros, TECNUN, Universidad de Navarra, P. Manuel Lardizabal 13, 20018 San Sebastia´n, Spain Received 11 November 2002; received in revised form 3 March 2003; accepted 4 April 2003

Abstract A creep-oxidation model is presented for continuous fibre-reinforced ceramic matrix composites at high temperature. The model includes the effects of interface and matrix oxidation, creep of the fibres and degradation of fibre strength with time. In particular, the influence of the glassy phases resulting from the oxidation of certain types of SiC based matrices is discussed. Model predictions are presented for the case of a woven Hi-Nicalon/SiC composite and compared to experimental results at 1000 and 1100 °C. The fraction of broken fibres increases with time in an accelerated manner as a result of load transfer and fibre degradation. The model also predicts that a fraction of broken fibres of about 15% triggers the unstable failure of the composite.  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fibre-reinforced composites; Creep; Oxidation; Interfaces; Micromechanical modelling

1. Introduction Continuous fibre-reinforced ceramic matrix composites (CMCs) have been studied over the past decades because of their promising characteristics for high temperature structural applications. Compared to monolithic ceramics, CMCs present higher toughness and tolerance to the presence of cracks, which implies a non-catastrophic mode of failure. The development of fine diameter silicon carbide fibres synthesized from polycarbosilane [1] ∗ Corresponding author. Tel.: +34 943 212800; fax: +34 943 213076. E-mail addresses: [email protected] (L. Casas); jmesnaola@ ceit.es (J.M. Martı´nez-Esnaola).

has allowed the manufacture of woven microstructures. Creep resistance is one of the main requirements for these materials because potential applications of CMCs, for example as parts of gas turbines for aircrafts, require maintaining the material properties over long periods of time (thousands of hours) at high temperature. A number of works can be found in the literature regarding the creep behaviour of CMCs, see for example Refs. [2–8]. Many of these investigations have pointed out the critical influence of the interface in the behaviour of these materials. Carbon and boron nitride are the materials most commonly used as interface in silicon carbide based composites. The behaviour of the interface and of the other constituents under oxygen rich environments has been the

1359-6454/03/$30.00  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00189-7

3746

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

object of many investigations because of their critical influence on the properties of the material and their evolution with time. In this regard, much work has been done to describe and to model the oxidation of fibres, matrices and interfaces in the absence of loading [9–13]. Models have also been developed that describe the influence of loss of interface by oxidation on load transfer at intermediate temperatures where creep is not relevant [14–16]. The aim of the present work is to incorporate the effects of the oxidation of constituents into the mechanical response of the material accounting for the synergistic creep-oxidation interactions at high temperature. In this paper a creep-oxidation model for continuous fibre-reinforced CMCs is presented. The model is based on the effect of the oxidation of interface and matrix, and on the degradation of the mechanical properties of the fibres. The model accounts for the loss of fibre strength with time and the creep of the fibres, both of which increase the deformation of the material. At the same time, the number of intact fibres reduces during deformation until final failure.

2. Materials The material used in this work is the Cerasep410. It is a silicon carbide based CMC produced by chemical vapour infiltration (CVI) by the Socie´ te´ Europe´ ene de Propulsion (SEP, a division of Snecma, France). A multilayered ceramic matrix (made up of distinct layers of SiC, B4C and SiBC) is reinforced with a 2.5D architecture of bundles of SiC Hi-Nicalon fibres (produced by Nippon Carbon Co., Japan). Before matrix infiltration, the preforms are coated with a carbon interface to decrease the bonding between the fibres and the matrix. This multilayered SiC-based matrix is described as self-healing, since the initial matrix cracking generated on loading allows limited oxygen ingress, leading to the formation of glassy phases (borosilicates) that flow plugging the cracks and preventing further oxygen ingress. Creep tests were performed under load control with a servohydraulic testing machine at temperatures of 1000 and 1100 °C and stresses between

115 and 300 MPa. Different loading steps were applied in some tests in order to study the steady state creep strain rate, which is typically a very important parameter for the design of high temperature components. Table 1 summarises the testing conditions, including the temperature, the applied stresses and the corresponding hold times, together with the time to rupture, tr, the final creep strain, ec, and the final total strain, et. The experimental strain-time plot of each test will be shown as part of the results of this paper and compared to the predictions of the model. A more detailed analysis of the results of creep testing can be found in the work reported by Casas et al. [7,8,17,18].

3. Description of the model Consider an ideal unidirectional composite. The fibres are assumed to be coated with a carbon layer, resulting in the formation of two interfaces of low fracture energy, one between the matrix and the carbon layer, and one between the coating and the fibre (Fig. 1). The composite is subjected to a remote tensile stress, sc, at high temperature. It is assumed that the loading is quasistatic so as to avoid dynamic effects, but, at the same time, fast enough to neglect creep effects during loading. When the applied stress is higher than the matrix cracking stress, a distribution of approximately parallel cracks perpendicular to the loading direction appears in the matrix. These cracks are assumed to be equally spaced at a distance lc (Fig. 1). After matrix cracking, the axial stress in the fibre will increase in the crack plane to support the entire stress of the composite. Away from the matrix crack plane, the fibre transfers its portion of load to the matrix, and the stress in the fibre, sf, decays with a slope: dsf 2t ⫽⫺ dz r

(1)

where r is the fibre radius, t is the local interfacial sliding stress and z represents the axial distance to the matrix crack plane. If we assume a constant sliding stress at the interface, the stress distribution along the fibre can be written as

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

3747

Table 1 Creep testing conditions and results Sample

T (°C)

s (MPa)

Hold time (h)

tr (h)

ec (%)

et (%)

515–1

1000

0.308

0.606

1000

517

0.175

0.441

530–5 530–6

1100 1100

265 397 111 130 0.1 (until failure) 148 168 197 4 (until failure) 226 (until failure) 139 150 127 16 (until failure)

903

530–7

115 170 220 250 300 150 180 210 240 230 150 175 200 225

226 432

0.57 0.383

0.846 0.61

sf(z) ⫽ S⫺

2tz lc , for 0ⱕzⱕ r 2

(2)

where S is the maximum axial stress in the fibre in the matrix crack plane and lc is the matrix crack spacing. The stress in the fibre will decrease up to a distance lc / 2 (see Fig. 1) unless the stress corresponding to the compatibility of displacements between fibre and matrix, sf0, is reached at a distance l∗ ⬍ lc / 2. In such a case the stress in the fibre decreases along a distance l∗, then maintaining a constant value, sf0, between l∗ and lc / 2. The equilibrium stress, sf0, is given by sf0 ⫽

Efsc EfVf(1⫺⌽) ⫹ EmVm

(3)

where Ef and Em are the Young’s moduli of fibre and matrix, respectively, Vf and Vm are the volume fractions of the constituents, and ⌽ is the fraction of broken fibres, which can be calculated as ⌽ = 1⫺(N / N0), where N is the number of surviving fibres and N0 the original number of fibres. Assuming global load sharing conditions, a simple force balance at the matrix crack plane provides the relationship between the stress applied to the composite, sc, and the maximum stress in the fibre, S,

冋

sc ⫽ Vf S(1⫺⌽) ⫹

册

2t l⌽ r

(4)

where l is the average fibre pullout. The first term on the right-hand side of Eq. (4) represents the load supported by the intact fibres. The second term represents the contribution of the broken fibres during deformation of the material and fibre extraction through friction along an average length l. As the stress in the whole composite remains constant, and the term associated to the sliding stress in Eq. (4) is smaller than te first term (as will be discussed later), the stress in the surviving fibres increases with successive fibre failure due to the global load sharing assumption. A two-parameter Weibull distribution [19] has been used to describe the strength of the fibres. Then, the probability of failure of the fibres is given by:

再冕 冉 冊 冎

⌽(t) ⫽ 1⫺exp ⫺

1 sf(z,t) m dA areaA0 s0(t)

(5)

where s0 is the characteristic strength for a fibre of surface A0 = 2prl0 under monotonic tension, and m is the Weibull modulus. The strength s0 will be considered to vary with time to account for the decrease of the fibre properties due to the different degradation processes at high temperature. Assuming a periodic matrix crack spacing, lc, the probability of failure of a fibre along the gauge length, L, of a monotonic tension specimen is

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

3748

Fig. 1. Schematic representation of the ideal unidirectional composite.

⌽(t) ⫽ 1⫺exp

再

冕

lc

1 L ⫺2 m (s0(t)) l0 lc

2

0

冎

(sf(z,t))mdz

SiBC, to form basically silicon oxide, SiO2, and other glassy phases (borosilicates) [11]. These glassy phases act sealing the cracks, reducing oxygen ingress and delaying the oxidation of the composite. Moreover, when the glassy phases slip into the fibre-matrix interface (due to the carbon layer volatilisation or crack propagation through the interface) [20], the stress transfer between fibre and matrix is controlled by a sliding stress, tg, different from the initial interfacial sliding stress, t. This new sliding stress tg, characteristic of the glassy phases, is typically lower than t. Fig. 2 shows the fracture surface of test 530–5 (1100 °C, 230 MPa) where the glassy phases covering the fibre along the whole pullout length are clearly observed. As a result of interface oxidation the carbon layer disappears along the fibre length (and is occupied by the various glassy phases) at a certain rate dz / dt, where z is the length of carbon lost in each side of the crack. Therefore, the resulting stress distribution along the fibre can be more complex than the simple linear (or bilinear) one predicted by Eq. (2) and will depend on the time instant, the matrix crack spacing, the length of oxidized interface and the equilibrium stress, sf0. As the test evolves, the fraction of broken fibres, ⌽, and the interfacial oxidation length, z, increase with time. The implications of this evolution are as follows. As a result of the increase of ⌽ with time, both the equilibrium stress, sf0, and the fibre stress at the matrix crack plane, S, also get higher, as described by Eqs. (3) and (4), i.e., they also become time dependent. In addition, note that the

(6)

where the integration is performed only along the length lc / 2, using the symmetry in the fibre stress distribution in both sides of the crack plane. Let us consider that our ideal composite is heated up to temperatures above 1000 °C. Between 650 and 900 °C, the boron carbide reacts with oxygen to form boron oxide, B2O3, that limits further oxygen ingress, then decreasing the oxidation rate of the composite. At temperatures above 900 °C, the boron oxide vanishes and the sealant capability decreases. At these temperatures the oxygen reacts with the other components of the matrix, SiC and

Fig. 2. Micrograph showing the fibres covered by glassy phases in test 530⫺5 (1100 °C, 230 MPa).

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

oxidized length z is the length at which the sliding stress tg governs the load transfer between fibre and matrix; in the rest of the transfer length, fibrematrix interaction is still controlled by t. The total transfer length is always given by the minimum between lc / 2 and l∗, as described above. Therefore, for a given test condition, the stress profile along the fibre varies with time as a consequence of the time evolution of z, t, tg and ⌽ (and thus sf0 and S). Fig. 3 illustrates all the possible shapes of the stress profiles that can appear during a single test at different time instants. The stress profiles depicted in Fig. 3 can be easily described with piecewise linear functions. Introducing these stresses in Eq. (6), the fraction of broken fibres, ⌽, at each time step can be calculated. It is important to notice that one of the parameters that defines the stress distribution is the axial stress in the fibre in the matrix crack plane, S, which in turn depends on the fraction of broken fibres, ⌽ (see Eq. (4)). Then, ⌽ and S have to be obtained at each computing step with Eqs. (4) and (6). This represents a non-linear problem that has to be solved numerically. In practice, only a few iterations in ⌽ or S, taking as starting point the solution of the previous step, are necessary to solve the problem and to determine the stress state in the new time step. The evolution of ⌽ with time also provides the prediction of time to failure as the time at which ⌽ = 1. Once the evolution of the stress distribution in the fibre has been obtained, the strain of the composite can be determined. The average elastic strain of the composite, ee, can be calculated by integration of the axial strain in the fibre:

冕

lc

2 ee(t) ⫽ Eflc

2

sf(z,t)dz

3749

ing fibres. The creep rate of the fibres will be considered to be described by:

冉 冊

Q (s (z,t))n e˙ c(t) ⫽ A0exp ⫺ RT f

(8)

where A0 and n are constants, Q is an activation energy, R is the gas constant and T is the absolute temperature. Eq. (8) corresponds to a situation of steady state creep. Neglecting transient effects, i.e., primary creep associated to variations of the stress level, the average creep strain of the sample is given by

冉 冊冕 冋冕

2A0 Q exp ⫺ ec(t) ⫽ lc RT

t

0

lc 2

册

(sf(z,t))ndz dt

0

(9)

The integration along lc / 2 can be performed analytically because sf(z,t) is a piecewise linear function in z (see Fig. 3), and therefore the closed form integral of (sf)n is immediate in each step. However, the integral on the time domain has been calculated numerically using a simple trapezoidal rule. The time increment for the numerical integration has been selected by the simple method of repeating the calculations doubling each time the number of integration intervals until the difference between two successive calculations is negligibly small. The total strain, et, is the sum of the elastic strain and the creep strain: et(t) ⫽ ee(t) ⫹ ec(t)

(10)

The oxidation of the carbon layer and the existence of glassy phases in the fibre-matrix interface modify the stress transfer mechanism, resulting in a higher length of fibres subjected to stresses near the peak stress, S (Fig. 3). Thus, the probability of failure of the fibres increases with time.

(7)

0

Both fibres and matrix present time dependent deformation (creep or stress relaxation) when subjected to high temperatures. After matrix cracking and the volatilisation of the carbon layer, the creep behaviour of the material is controlled by the creep of the fibres. This is illustrated in Fig. 4, which shows a distribution of parallel matrix cracks perpendicular to the loading direction without break-

4. Parameters of the model The model described in the previous section has been applied to the creep tests performed with Cerasep410. The material parameters are listed in Table 2 together with the literature sources from which they have been obtained. Some of these parameters are obviously dependent on the testing conditions. Values between 300 and 350 GPa have

3750

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

Fig. 3. Stress in the fibre between two matrix cracks. Each case represents one of the possible stress profiles that can appear during a single test at different time instants. The dashed line represents the equilibrium stress given by the fibre-matrix compatibility of displacements. The dotted line represents the theoretical stress controlled by the sliding stresses t and tg. The continuous lines represent the actual stress profiles. Note that the inequality z ⬎ lc / 2 in cases 4 and 5 is only used to indicate that the entire C interface has been consumed. In fact, when z = lc / 2 the C interface has disappeared.

been reported for the Young’s modulus of the SiC matrix [21,22]. In this work, the value of 400 GPa reported by Bikok et al. [23] for CVI SiC matrices of the last generation has been used. In any case, this must be regarded as an estimate because there are no available data for a multilayered matrix similar to that of the Cerasep410. For the fibres,

a Young’s modulus of 270 GPa has been considered [24,25]. The ranges of values listed in Table 2 can also be used to estimate the relative weight of each term in Eq. (4). For example, in test 530–5 (sc = 230 MPa), using l = 200 µm, ⌽ = 0.15 and a sliding stress of 7 MPa (note that the final fibre extraction

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

3751

of the carbon layer, the distribution and evolution of the fibre strength during the test and the sliding stress at the interface are discussed below. 4.1. Oxidation of the carbon interface For a Nicalon/SiC composite, Filipuzzi and Naslain [13] have measured and modelled the change in the length z of the carbon interface (0.1 µm thick) under a pressure of 100 kPa of oxygen assuming that oxidation occurs according to (Fig. 5): Fig. 4. Micrograph showing matrix cracking perpendicular to the loading direction and the bridging fibres in test 515–1 (1000 °C, 115–170–220–250–300 MPa).

will be along the glassy interface) gives that the contribution of the broken fibres to the total load is about 5%. This confirms the arguments regarding the increase of the fibre stress S at the matrix crack plane with successive fibre failure. Some of the parameters in Table 2 need a more detailed analysis. In particular, the oxidation rate

C ⫹ O2→CO2

(11)

In order to simplify the use of these data in the present model, the results reported in [13] have been fitted using an equation of the type z ⫽ q1(1⫺e⫺q2t)

(12)

where q1 and q2 are fitting parameters dependent on temperature. q1 and q2 have been obtained for each available temperature and then Arrhenius type laws have been used to describe the dependency

Table 2 Material parameters of the model Parameter

Symbol

Value

Ref.

Fibre radius Young’s modulus of the fibre Young’s modulus of the matrix Volume fraction of fibresa Volume fraction of matrixa Gauge length Initial fibre strength (Weibull)

r Ef Em Vf Vm L modulus, m characteristic strength, s0 reference length, l0 λ lc tg A0b n Q t z

7 mm 270 GPa 400 GPa 0.2 0.3 25 mm 6 2590 MPa 25 mm 137–240 mm 172–380 mm 2.2–13.5 0.0667 2.2 367 kJ/mol variablec function of timec

[24] [24] [23] [35] [35] [35] [36]

Fibre pullout Matrix crack spacing Sliding stress of glassy phases Creep of the fibres, Eq. (8)

Interfacial sliding stress Length of oxidized interface a b c

In loading direction. When e˙ is in s⫺1 and the stress in MPa. See discussion below.

[7] [7,8] [37,38] [8,29]

[8],34] [13]

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

3752

Fig. 5. Length of oxidized C interface in a Nicalon/SiC composite as a function of time: model of Filipuzzi and Naslain [13] and fit given by Eq. (12).

of q1 and q2 on temperature (Fig. 6). The resulting expressions are: q1 ⫽ 7.021 ⫻ 10⫺3exp

冉

q2 ⫽ 227.1exp ⫺

冉 冊 冊

17090 T

8231 T

fitting curves obtained with Eqs. (12), (13a) and (13b). The results of Fig. 5 do not consider the oxidation of the matrix, which produces a sealant glass that slips into the fibre-matrix interface and through interfacial cracks. This sealant glass decreases the oxygen activity and, consequently, the different oxidation processes in the fibre surface and in the matrix in contact with the fibres are delayed. Thermodynamic calculations carried out using the Thermo-Calc software [26,27] indicate that the deceleration of the oxidation phenomena (as a consequence of the reduced oxygen activity due to the diffusion through the glassy phases) can represent several orders of magnitude in the oxidation time scale. This effect has been incorporated into the model using a delay factor, b (discussed later), in Eq. (12), which becomes

冉

z ⫽ q1 1⫺e⫺ b .

(13b)

4.2. Mechanical properties of the fibres

where T is the absolute temperature, q1 is in mm and q2 in s⫺1. It can be noticed that q1 represents the asymptotic behaviour for long times, which decreases with temperature; the product q1q2 represents the initial oxidation rate, which is an increasing function of temperature. Both experimental observations are well described by the expressions used to fit q1 and q2. Fig. 5 shows the

Fit of parameters q1 and q2, see Eqs. (12), (13a) and

(14)

The loss of mechanical properties due to the fibre degradation at high temperature has also been considered. This results in a decrease of the characteristic strength s0 of the fibres with time and the consequent increase in the failure probability of the fibres (Eq. (6)). To the authors’ knowledge, only data about the evolution with time of the Nicalon NL201 fibre strength at 1000 °C have been reported [28]. Due to the superior creep behaviour of the Hi-Nicalon fibres compared to Nicalon fibres [29], the strength decrease will be assumed to occur in a slower manner. A delay factor of three with respect to the degradation of the Nicalon fibres has been found to give satisfactory results (see below). Fig. 7 shows the assumed evolution with time of the Hi-Nicalon strength, which can be described by s0 ⫽

Fig. 6. (13b).

冊

q2t

(13a)

再

for t ⬍ 30

2625

5510t

⫺0.218

for tⱖ30

(15)

where the time, t, is expressed in hours and the strength, s0, in MPa. As indicated in Eq. (15), a certain time interval is assumed during which the fibres show no sig-

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

3753

value ⫺0.218 obtained through the fitting used in this model (see Eq. (15) and Fig. 7). 4.3. Interfacial sliding stress, t

Fig. 7. Degradation of the strength of Nicalon NL201 fibres with time [28] and estimate for the Hi-Nicalon used in this work, assuming that degradation of Hi-Nicalon fibre is three times slower.

nificant decrease of their strength. This represents the existence of an incubation period before fibre damage starts to develop, similarly to the observations reported for Nicalon fibres [30–33]. At short oxidation times, the strength of the SiC fibres can even increase due to the sealing of surface defects with the oxidation products. However, at longer times, oxidation results in strength reduction. The present model assumes that the strength of the fibres remains constant until the oxide layer becomes larger than the critical defect size associated with the characteristic strength of the fibre in the initial non-oxidized state. After this time interval, the size of the oxide scales determines the new critical size of defect. The relationship between fibre toughness, KIC, and the strength, s0, is given by KIC ⫽ Ys0冑pa

We consider in this section the evolution of the interfacial sliding stress in the zone not affected by the glassy phases. An increase of the interfacial stress with the test time, irrespective of temperature, has been reported in results obtained using push-in tests [8,34] and through correlations with the width of hysteresis loops in loading-reloading cycles in creep tests of microcomposites [3] of similar (SiC/C/SiC) materials. Fig. 8 shows the results obtained by Casas et al. [8,34] in push-in tests using the nanoindentation technique on fibres of post-mortem specimens. The resulting values compare well with the range 5–55 MPa measured in-situ by Rugg et al. [3]. Due to the few available data, a linear interpolation between each pair of consecutive data points (see Fig. 8) has been applied to describe the evolution of t with time.

5. Results and discussion There is a lack of data in the literature to quantify the influence of the glassy phases in the fibrematrix interface both on the interfacial sliding stress and on the delay of the different oxidation phenomena. Therefore, simulations have been performed using different values of the oxidation

(16)

where Y is a shape factor and a is the defect (flaw) size. Assuming a parabolic oxidation kinetics, a⬀√t, we obtain KIC⬀Ys0冑pt1/4

(17)

And finally, assuming that the shape factor does not vary significantly with the defect size and that the fibre toughness remains constant, we get s0⬀t⫺1/4

(18)

The theoretical exponent ⫺1/4 is very close to the

Fig. 8. Evolution of interfacial sliding stress with test time. The plot shows a linear interpolation between consecutive data points.

3754

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

delay factor, b, and the interfacial sliding stress in the presence of glass, tg. To illustrate the influence of these parameters, Figs. 9 and 10 show the straintime curves predicted by the model with different values of b and tg for the creep test of sample 530– 7 (1100 °C, 150–180–210–240 MPa). As illustrated in Fig. 9, as the stress tg increases, the time to rupture increases and the failure strain decreases. This is because the higher tg the shorter the length of the fibre subjected to high stresses (i.e., the stress profile in the fibre is lower), and therefore the probability of failure of the fibre decreases. Fig. 9 also shows that the first step of loading and a short transient regime following the

second step of loading are well captured by the model with tg = 8 MPa and b = 5000. Fig. 10 shows the influence of the oxidation delay factor, b. For low values of b, the volatilisation of the carbon layer occurs very early during the test, and the steady state of deformation controlled by the creep of the fibres is reached in a few hours. For higher values of b, the steady state is reached later because the carbon recession rate is lower and the fibre-matrix stress transfer is maintained for a longer time. Nevertheless the failure strain and the time to failure predicted by the model are practically independent of b. Fig. 10 also indicates that the best prediction for the first sep of loading is for b = 5000, whereas the value that gives the best prediction for the second and third steps of loading is b = 10000. This increasing value of b with time would represent the progressive delay of oxidation. The progressive oxidation of the matrix produces an increasing amount of glassy products that seal a larger number of cracks, then reducing oxygen ingress. Figs. 11–14 show the experimental strain-time curves and the predictions of the model for the creep tests listed in Table 1. The value of the oxidation delay factor is found to be in the range 5000–10,000 and the sliding stress of the glass between 3 and 11 MPa, which are consistent with values reported in the literature (see Table 2). Even though there are obvious uncertainties in these parameters, the model successfully predicts the

Fig. 10. Experimental curve and predictions of the model (continuous lines) for a sliding stress of the glass tg = 7 MPa and different values of the oxidation delay factor, b. Test 530– 7 (1000 °C, 150–180–210–240 MPa).

Fig. 11. Experimental curve and prediction of the model (total strain and evolution of ⌽) for the creep test of sample 515–1 (1000 °C, 115–170–220–250–300 MPa). The best prediction is given by b = 5000 and tg = 3 MPa.

Fig. 9. Experimental curve and predictions of the model (continuous lines) for an oxidation delay factor b = 5000 and different values of the sliding stress of the glass, tg. Test 530– 7 (1000 °C, 150–180–210–240 MPa).

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

Fig. 12. Experimental curve and prediction of the model (total strain and evolution of ⌽) for the creep test of sample 530–7 (1000 °C, 150–180–210–240 MPa). The best prediction is given by b = 10000 and tg = 7 MPa.

Fig. 13. Experimental curve and prediction of the model (total strain and evolution of ⌽) for the creep test of sample 530–5 (1100 °C, 230 MPa). The best prediction is given by b = 5000 and tg = 7 MPa.

general shape of the strain-time curves as well as the time to rupture and final failure strain. As described above, the model calculates the evolution of the fraction of broken fibres, ⌽, with time, and predicts the failure of the composite when ⌽ = 1. Figs. 11–14 also show the evolution of ⌽ with time. The model predicts an increasing slope in the ⌽-t curve. This reflects the acceleration of fibre damage with time as a consequence of load transfer to the intact fibres and loss of strength of the surviving fibres. The simulations also show that, in all the cases, final fracture of the composite occurs when the fraction of broken fibres is about 0.15 (the numerical results are between 0.13 and 0.17), indicating that the surviving fibres at this level of

3755

Fig. 14. Experimental curve and prediction of the model (total strain and evolution of ⌽) for the creep test of sample 530–6 (1100 °C, 150–175–200–225 MPa). The best prediction is given by b = 10000 and tg = 11 MPa.

damage are no longer able to withstand the applied load. The origin of the arrow in each plot represents the point (value of ⌽) that triggers the unstable final failure of the composite. A final remark is regarding the values of b and tg used in the model. As discussed above, the physical meaning of tg is clear, as it represents the modified sliding stress at the interface after volatilisation of the original carbon layer between fibre and matrix, which is then replaced by the various glassy phases resulting from the oxidation of the matrix. The composition of these glassy phases and their viscosity also vary with temperature [11]. The parameter b has been introduced to account for the slower oxidation rate of the interface in the present material compared to the experimental measurements available for a composite with a SiC matrix with no sealing effect [12,13]. However, it is important to note that, at the present stage of the model, both tg and b have been used as fitting parameters. The introduction of real measurements of tg and b (or, alternatively, the interface oxidation rate in this material) as parameters depending on temperature and evolving with time might further improve the model.

6. Conclusions A creep-oxidation model for fibre-reinforced CMCs has been presented. The model takes into

3756

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

account the oxidation of interface and matrix, the creep of the fibres and the degradation of the mechanical properties of the fibres with time at high temperature. Interface oxidation and loss of the carbon layer have been accounted for in combination with estimates of the influence of the sealing glassy phases resulting from the oxidation of SiBC type matrices. The model predictions illustrate the effect of oxidation on the creep life of the material and highlight the important role of carbon layer oxidation and matrix oxidation. The model shows that matrix cracking causes a progressive overloading of the fibres. This combines with the degradation of fibres with time to cause final failure. The theoretical loss of strength of the fibres predicted by the model is proportional to t⫺1 / 4, which is consistent with experiments. The model also predicts strain-time curves that compare satisfactorily with the experimental records. The computed fraction of broken fibres exhibits an increasing slope with time until failure of the composite. The model also indicates that when the fraction of broken fibres is about 15%, unstable fracture of the material occurs.

Acknowledgements This work is part of the Brite-EuRam project BE97-4020 with financial support from the European Commission, co-ordinated by Rolls-Royce plc, in collaboration with Rolls-Royce Deutschland, MTU, ITP, Ansaldo Ricerche, Qinetiq (former DERA), IE (former IAM)-JRC Petten and CEIT. The authors also thank Dr. T. Go´ mez-Acebo for his help with the thermodynamic calculations using the Thermo-Calc software.

References [1] Yajima S, Hayashi J, Omori M. Development of a silicon carbide fibre with high tensile strength. Nature 1976;261:683–5. [2] Darzens S, Chermant JL, Vicens J. First results on SiCfSiBC creep. Scripta Materialia 2000;43:497–502. [3] Rugg KL, Tressler RE, Lamon J. Interfacial behavior of microcomposites during creep at elevated temperatures. J Eur Ceram Soc 1999;19:2297–303.

[4] Se´ ller TE, Zok FW. Stress rupture of an enhanced Nicalon/silicon carbide composite at intermediate temperatures. J Am Ceram Soc 1998;81:2140–6. [5] Henager Jr CH, Hoagland RG. Subcritical crack growth in CVI SiCf/SiC composites at elevated temperatures: dynamic crack growth model. Acta Materialia 2001;49:3739–53. [6] Zhu S, Mizuno M, Kagawa Y, Cao J, Nagano Y, Kaya H. Creep and fatigue behavior in Hi-NicalonTM-fiberreinforced silicon carbide composites at high temperatures. J Am Ceram Soc 1999;82:117–28. [7] Casas L, Elizalde MR, Sa´ nchez JM, Puente I, Martı´nezEsnaola JM, Martı´n-Meizoso A, Fuentes M. Comportamiento a fluencia de un material compuesto tejido SiC/SiC. Anales de Meca´ nica de la Fractura 2001;18:316–20. [8] Casas L. Fluencia de CMCs reforzados con fibras largas, Tesis Doctoral, Escuela Superior de Ingenieros, TECNUN, Universidad de Navarra, San Sebastia´ n, 2003. [9] Takeda M, Urano A, Sakamoto J, Imai Y. Microstructure and oxidation behavior of silicon carbide fibers derived from polycarbosilane. J Am Ceram Soc 2000;83:1171–6. [10] Elizalde MR, Daniel AM, Puente I, Martı´n-Meizoso A, Martı´nez-Esnaola JM. Efecto de la oxidacio´ n en las propiedades meca´ nicas de la intercara de un material compuesto de matriz cera´ mica SiC/C/SiC. Anales de Meca´ nica de la Fractura 1997;14:355–60. [11] Viricelle JP, Goursat P, Bahloul-Hourlier D. Oxidation behavior of a multilayered ceramic-matrix composite (SiC)f/C/(SiBC)m. Composites Science and Technology 2001;61:607–14. [12] Filipuzzi L, Camus G, Naslain R. Oxidation mechanisms and kinetics of 1D-SiC/C/SiC composite materials: I, An experimental approach. J Am Ceram Soc 1994;77:459–65. [13] Filipuzzi L, Naslain R. Oxidation mechanisms and kinetics of 1D-SiC/C/SiC composite materials: II, Modelling. J Am Ceram Soc 1994;77:467–80. [14] Lara-Curzio E, Ferber MK, Tortorelli FF. Interface oxidation stress rupture of NicalonTM/SiC CFCCs at intermediate temperatures. Key Engineering Materials Vols 1997;127-131:1069–82. [15] Lara-Curzio E. Oxidation induced stress-rupture of fibre bundles. Journal of Engineering Materials and Technology 1998;120:105–9. [16] Lara-Curzio E. Analysis of oxidation-assisted stress-rupture of fiber-reinforced ceramic matrix composites at intermediate temperatures. Composites Part A 1999;30:549– 54. [17] Casas L, Elizalde MR, Martı´nez-Esnaola JM, Martı´n-Meizoso A, Gil Sevillano J, Claxton E, Doleman P. Behavior of a 2.5D woven composite material SiC/SiC. In: Krenkel W, Naslain R, Schneider H, editors. High Temperature Ceramic Matrix Composites, HT-CMC4. Weinheim: Wiley-VCH; 2001. p. 486–91. [18] Casas L, Martı´nez-Esnaola JM, Martı´n-Meizoso A, Fuentes M. Energy balance in the creep induced fracture of a 2.5D woven SiC/SiC composite. In: Neimitz A, Rokach IV, Kocanda D, Golos K, editors. ECF 14, Fracture

L. Casas, J.M. Martı´nez-Esnaola / Acta Materialia 51 (2003) 3745–3757

[19] [20]

[21]

[22]

[23]

[24]

[25]

[26] [27]

[28]

Mechanics Beyond 2000, Vol. I/III. Sheffield: EMAS Publications; 2002. p. 425–32. Weibull W. A statistical distribution of wide applicability. J Appl Mech 1951;18:293–7. Zhu S, Mizuno M, Nagano Y, Cao J, Kagawa Y, Kaya H. Creep and fatigue behavior in an enhanced SiC/SiC composite at high temperature. J Am Ceram Soc 1998;81:2269–77. Aubard X. Modelisation et identification du comportment mecanique des materiaux composites 2D SiC/SiC, PhD Thesis, Laboratoire des Composites Thermostructuraux, Domaine Universitaire, Bordeaux, 1993. Beyerle D, Spearing SM, Evans AG. Damage mechanisms and the mechanical properties of a laminated 0/90 ceramic/matrix composite. J Am Ceram Soc 1992;75:3321–30. Bikok M, Lavaire F, Forio P, Pailler R, Lamon J. Prediction of the lifetime of minicomposites under static fatigue at high temperature. LCTS Final Report, Bordeaux, 2001. Ichikawa H. Recent advances in Nicalon ceramic fibres including Hi-Hicalon Type s. Ann Chim Sci Mat 2000;25:523–8. Papakonstantinou CG, Balagaru P, Lyon RE. Comparative study of high temperature composites. Composites Part B 2001;32:637–49. Sundman B, Jansson B, Andersson JO. The Thermo-Calc databank system. Calphad 1985;2:153–90. Ansara I, Sundman B. The Scientific Group Thermodata Europe, Calphad. In: Computer handling and dissemination of data. Proceedings Xth CODATA Conference. Ottawa, July 1986: Elsevier Science; 1987. Tressler RE. Recent developments in fibers and interphases for high temperature ceramic matrix composites. Composites Part A 1999;30:429–37.

3757

[29] Bodet R, Bourrat X, Lamon J, Naslain R. Tensile creep behaviour of a silicon carbide-based fibre with low oxygen content. J Mater Sci 1995;30:661–77. [30] Cox BN, Sridhar N, Argento CR. A bridging law for creeping fibres. Acta Materialia 2000;48:4137–50. [31] Henager Jr CH, Jones RH. High-temperature plasticity effects in bridged cracks and subcritical crack growth in ceramic composites. Mater Sci Engng A 1993;166:211– 20. [32] Simon G, Bunsell AR. Creep behaviour and structural characterization at high temperatures of Nicalon SiC fibres. J Mater Sci 1984;19:3658–70. [33] Bunsell AR, Berger MH. Fine diameter ceramic fibres. J Eur Ceram Soc 2000;20:2249–60. [34] Casas L, Elizalde MR, Martı´nez-Esnaola JM. Interface characterisation and correlation with the creep behaviour of a 2.5D SiC/SiC composite. Composites Part A 2002;33:1449–52. [35] Casas L, Elizalde MR, Martı´n-Meizoso A, Martı´nez-Esnaola JM, Sa´ nchez JM. Development of a Lifing Methodology for Ceramic Matrix Composites in Aero Gas Turbine Engines. Brite-EuRam Project BE97-4020 CMC Lifing, CEIT Report MAT1226-PR4: San Sebastia´ n, 2000. [36] Youngblood GE, Lewinsohn C, Jones RH, Kohyama A. Tensile strength and fracture surface characterisation of Hi-Nicalon SiC fibres. J Nuc Mater 2001;289:1–9. [37] Elizalde MR. Efecto de la intercara en materiales compuestos de matriz cera´ mica y fibras continuas. Tesis Doctoral, Escuela Superior de Ingenieros, TECNUN, Universidad de Navarra, San Sebastia´ n, 1997. [38] Hsueh CH. Evaluation of interfacial properties of fiberreinforced ceramic composites using a mechanical properties microprobe. J Am Ceram Soc 1993;76:3041–50.