Projecting the Long-Term Life of SiC Fibers to Low Stresses: The Competition Effect Between Slow Crack Growth and Oxidation Embrittlement

Abstract

The delayed failure of SiC fibers is commonly described by a power law relating the growth rate to the stress intensity factor K_I, itself following the classical fracture mechanics law with a constant geometrical factor. For low stress levels, relevant for ceramic matrix composite (CMC) applications, this model predicts crack lengths exceeding the specimen size and unrealistic times to failure. Indeed, discrepancies between this model prediction and experiments have been reported. This paper proposes a model improvement with a simple and accessible analytical solution to work around this shortcoming. First, a more accurate description of fracture mechanics is introduced which yields physically reasonable estimates of the crack size at failure. Then, the contribution of silica scale formation to oxidation embrittlement (OE) is evaluated. If the corrected slow crack growth (SCG) model and the OE model are irrelevant when taken separately, their simultaneous presence accurately depicts the observations: OE prevails under low stresses, resulting in a finite lifetime below 150 MPa, whereas SCG takes over above 800 MPa. This result brings new insight for the design of CMC and may as well apply to other types of materials, prone to environment-assisted and stress-accelerated degradation.

1. Introduction

The durability of structural parts in an aggressive environment is a key design criterion. For instance, the slow crack growth (SCG) mechanism is responsible for a delayed failure under the combined effect of subcritical stresses (i.e., stresses beneath the one required for immediate failure) and environmental attack (i.e., chemical reaction with the material surface). This mechanism has been studied on many materials, including glasses [1,2], ceramics [3], metals [4], and polymers [5]. Models were proposed to depict the failure times observed in laboratory experiments [6], relating the crack growth rate (v) to the stress intensity factor (K_I):

(i)

the Erdogan–Paris power model, which is a purely empirical power law [7];

(ii)

the chemical kinetic model proposed by Wiederhorn [8,9];

(iii)

the atomistic bond energy proposed by Lawn [10].

Those models use the fracture mechanics law to depict the progressive loss of strength leading to the failure. The present work focuses on the practical case of small-diameter brittle SiC fibers reinforcing Ceramic Matrix Composite (CMC) parts [11], in particular the first-generation Si-C-O fibers [12] thanks to their lower cost, high strength and stability at intermediate temperatures (<1000 °C) [13]. Such composites display high strength, high toughness and low density even at elevated temperature, making them suitable for applications in nuclear power plants [14], solar energy [15] or aircraft engines, for instance [16,17].

The above-mentioned applications target several years of service life, hardly achievable in laboratory experiments. Therefore, the expected service life relies on extrapolations from one of the above models fitting laboratory accelerated experiments. Depending on the model selected, the projected life can greatly differ [6,18], which brings some understandable questioning on the prediction reliability. The power law was shown in a previous report to fit adequately the failure time for multifilament SiC fiber bundles [19], whereas other authors preferred the atomistic model [20]. Extrapolating those models toward low stress levels or close to the fatigue limit [1,21,22,23] presents two weaknesses here reviewed. First, they predict that a filament would fail once a crack has acquired a length a superior to the specimen diameter, which is physically impossible [20]; this has been attributed to a bias in the stress intensity factor. Second, the lifetime tends to an infinite value at vanishing applied stress, disregarding the oxidation embrittlement (OE) phenomenon [24,25,26,27,28], which is attributed to the oxide scale growth on the fiber surface [29,30] and may in turn pilot the failure of SiC fibers when the scale has reached a critical thickness [31,32].

The present work reviews the static fatigue results for two types of Si-C-O fibers, emphasizing the trend at low stresses. A workaround, proposed to solve the above-mentioned model inconsistencies, is developed in three steps. First, a numerical solution integrating the evolution of Y for large cracks to the power law was identified. Second, the life prediction by the OE mechanism is reviewed to introduce an upper life limit. Finally, a competition effect between SCG and OE models is proposed and compared to experimental data. The resulting empirical model uses simple and accessible parameters (power law coefficients and oxidation rate constants) to demonstrate how the oxidation can trigger the SCG. This work brings new insight into the life prediction of CMCs and, by extension, could apply to any type of material suffering from an environmentally assisted embrittlement accelerated by external stresses.

2. Materials and Methods

Since their first commercialization in the 1970s [33,34], three generations of SiC fibers were developed, with increasing thermal stability. This work focuses on Nicalon^® NL207 (NGS Advanced Fibers Co., Ltd., Toyama, Japan) and Tyranno Grade S (UBE Industries Ltd., Tokyo, Japan), respectively referred to as NL and TS hereafter. The properties of these fibers are gathered in

Table 1. Characteristics, life prediction [36], and oxidation rate constants [19,27,38] for NL and TS filaments. Toughness values were extracted from [39] assuming Y = 2/π.

Parameter	NL	TS
Nt	500	1600
ρ (g cm−3)	2.58	2.35
σf (MPa)	2840	3150
Ef (GPa)	210	180
KIC (MPa·m1/2)	1.2	1.1
m	5.05	8.17
σ0 (MPa)	3090	3550
αt (%)	14.7	8.54
n	10.0	7.4
A1 650 °C (m1−n/2 MPa−n s−1)	1.3 × 10−9	1.5 × 10−9
A (nm)	39	750
B (nm² s−1)	7.3 × 10−2	7.1 × 10−1
δ	1.49	1.18

. Their time to failure was extensively investigated by static fatigue on multifilament tows containing 500 (NL, 14 µm diameter) or 1600 (TS, 8.5 µm diameter) filaments as described in past reports [13,35,36]. Tows were hot gripped to alumina tubes to ensure the temperature at failure location is known and controlled [37]. Once positioned in a dedicated electrical vertical oven, a dead weight was suspended to the lower grip. The oven was then heated to 650 °C and maintained at this temperature until the specimen had failed, triggering a timer and automatically recording the failure time. No test was interrupted before failure.

The apparent stress acting on filaments is given by [40]:

$${\sigma }_{app.\gamma }={\displaystyle \frac{w_t}{S_t\left(1-\gamma \right)}}$$

(1)

$$\mathrm{with} S_t={\displaystyle \frac{m_0}{L_0\rho }}$$

(2)

$$\mathrm{and} \gamma =1-{\displaystyle \frac{N_0}{N_t}}=1-{\displaystyle \frac{E_t}{E_f}}$$

(3)

where m₀ is the mass of a tow with length L₀ of 300 mm, S_t its section, and ρ the fibers density (Table 1); w_t is the applied force, γ the unloaded section fraction (Equation (3), averaged to 11% on NL and 16% on TS [35]), N₀ and N_t the initial number of continuous filaments and the total number of manufactured filaments in the tow (including the broken ones), respectively, and finally E_t and E_f the tow and monofilament Young moduli, respectively.

3. Theory

We first recall the equations describing the Paris–Erdogan Power Law (PL):

$$v=A_1{K}_I^n$$

(4)

in which the stress intensity factor (SIF) $K_I$ is classically evaluated as:

$$K_I=\sigma Y\sqrt{\pi a}$$

(5)

where a is the size of the growing defect or crack.

3.1. Oxidation Embrittlement

When SiC is subjected to passive oxidation conditions, a silica scale grows on its surface according to the following balance equation [41,42,43]:

SiC (s) + 2 O₂ (g) → SiO₂ (s) + CO₂ (g),

(6)

following a linear–parabolic kinetic law [44,45,46]:

x² + Ax = Bt.

(7)

It is usual to simplify this model as purely linear (x = k_l × t) when the rate-limiting factor is the reaction with the substrate (x << A), and purely parabolic (x² = k_p × t) for high x when the mass transfer of oxidative species through the oxide scale becomes limiting (x >> A). This reaction is accompanied by a volume expansion by a coefficient δ corresponding to the ratio between silica molar volume and the fiber molar volume [47] (Table 1). It should be emphasized that the Deal–Grove kinetic model was developed for flat plate geometries. This model was adapted to small-diameter fibers by Zhu et al. [46,48] and further adjusted by Hay et al. [30] to account for δ. The full oxidation of filament is reached for thinner scales, thus shorter times. For the sake of simplifying further developments below, this effect will be left aside for the rest of this work, which also means the duration for a full oxidation will be overestimated by a factor δ − 1, i.e., 50% for NL and 20% for TS.

The oxide scale growth and its crystallization generate stresses (longitudinal and hoop) which further explain the fiber strength loss [29,30,49]. This oxidation embrittlement phenomenon (hereafter called OE) is empirically related to the inverse square root of the scale thickness [50] as depicted by:

$${\sigma }_f={\displaystyle \frac{C}{\sqrt{x}}}.$$

(8)

where the constant C can be approximated by K_IC/(Y π^1/2). This empirical expression only stands for scales exceeding the flaw size, as shown in [27,28]. Several authors associated this expression with the fracture mechanics law (Equation (5)) with x = a [25,26]. This convenient relationship (named FSO model [27]) might be a fortunate coincidence because flaws in the scale are unlikely to propagate through the fiber (mismatch of Young’s moduli). Lara-Curzio et al. combined this expression with the oxidation rate (assumed parabolic) [31,32,51], i.e.,

$${\sigma }_f={\displaystyle \frac{K_{IC}}{Y\sqrt{\pi \sqrt{k_{p}t}}}}.$$

(9)

to propose the first life prediction model:

$$t_f=\beta {\sigma }^{-4}$$

(10)

with β a constant depending on K_IC, Y and k_p. Again, this expression applies only for durations long enough to obtain x > a.

3.2. Slow Crack Growth

Later, static fatigue results showed the delayed failure of SiC fibers follows the classical Paris–Erdogan power law [52] (Equation (4)), which can also be formulated as:

$$t_f={\int }_{a_0}^{a_c}{\displaystyle \frac{da}{A_1{K}_I^n}}$$

(11)

Combined with Equation (5), this yields:

$$t_f={\displaystyle \frac{1}{A_1{\sigma }_{app}^n}}{\int }_{a_0}^{a_c}{\displaystyle \frac{da}{{\left(Y\sqrt{\pi a}\right)}^n}}$$

(12)

where the initial flaw size a₀ is dependent on the strength:

$$a_0={\displaystyle \frac{K_{Ic}^2}{{\pi \sigma }_f^2{Y}^2}}$$

(13)

and the critical flaw size a_c depends on the applied stress as:

$$a_c={\displaystyle \frac{K_{Ic}^2}{{\pi \sigma }_{app}^2·Y^2}}$$

(14)

Compared to the above model, the stress exponent n is higher than 4, which is representative for an SCG mechanism. Equation (12) can be solved as:

$$t_f={\displaystyle \frac{2}{A_1{\sigma }_{app}^n{Y}^n{\pi }^{\frac{n}{2}}\left(2-n\right)}}{\left[a^{1-{\displaystyle \frac{n}{2}}}\right]}_{a_0}^{a_c}$$

(15)

which further develops into:

$$t_f={\displaystyle \frac{2}{A_1{\sigma }_{app}^n{Y}^n{\pi }^{\frac{n}{2}}\left(n-2\right)}}\left(a_0^{1-{\displaystyle \frac{n}{2}}}-a_c^{1-{\displaystyle \frac{n}{2}}}\right)$$

(16)

The substitution of (13) and (14) into (16) yields the life prediction model developed by Forio et al. [53] and Gauthier et al. [54] to depict the time to failure of filaments or tows. This model considers that failure occurs when the strength has reached the applied stress, leading to an upper stress limit (σ_f) for which the lifetime is null. Hence:

$$t_f={\displaystyle \frac{2K_{IC}^{2-n}}{A_1{{\sigma }_{app}^{n}Y}^2\pi \left(n-2\right)}}\left({\sigma }_f^{n-2}-{\sigma }_{app}^{n-2}\right)$$

(17)

It is worth keeping in mind that the power law is empirical despite some attempts to justify it [55,56]. A Weibull statistical law is commonly used to model the fiber strength distribution:

$$P_{fi}=1-e^{-{\left({\displaystyle \frac{{\sigma }_f}{{\sigma }_0}}\right)}^m}$$

(18)

where m is the Weibull modulus, σ₀ the characteristic strength, and P_fi is the probability of failure for the ith rank, usually computed as:

$$P_{fi}={\displaystyle \frac{i-0.5}{N}}$$

(19)

in which N is the size of the considered statistical data set [57]. The lifetime distribution follows the same type of statistical law [35], with P_ti the lifetime probability.

As mentioned in the “methods” section, static fatigue tests were performed on multifilament tows containing hundreds of filaments. Models to describe the failure of bundles during a force-controlled tensile test (representative for static fatigue) were initially proposed by Daniels [58] and Coleman [59]. According to these models, the force born by a failing filament is globally or locally shared between surviving fibers, which can in turn fail. There exists a critical fraction of failed fibers that triggers the cascade failure of the surviving elements. Under static fatigue condition, this critical fraction is given by [60]:

$${\alpha }_t=1-e^{\left(-{\displaystyle \frac{n-2}{nm}}\right)}$$

(20)

The time to failure of a bundle can hence simplify to the failure of this critical filament, characterized by a tensile strength and an applied stress.

Despite the huge variability characterizing the lifetime assessment (for a given condition, it is usual to observe a factor 100 separating the shortest and the longest lifetimes) [61], the power law (Equation (17)) was shown to best fit the life vs. stress trend for tows [35], at least better than exponential models [19]. It is usual to assume a homogeneous load bearing between all fibers of the tow, which constitutes a strong shortcoming [62]. Indeed, to interpret the lifetime variability, the fiber strength distribution is insufficient [63] and additional extrinsic and discrete phenomena were introduced, such as fiber bridging or local load sharing [13]. Alternatively, the introduction of a stress distribution was shown to adequately depict the lifetime variability [36,62]. Indeed, the filaments in a tow are barely aligned, which causes a delayed strain activation [64] and an inhomogeneous loading [65]. The filament load distribution was extracted from the initial loading branch of a bundle tensile strength in [35]. A Monte Carlo simulation tool was built, attributing to each of the N_t filaments a strength and a stress (randomly selected among the respective distributions) further used to assess their individual time to failure. The latter values were finally ranked to assess the tow lifetime (Equation (20)). Each simulation being unique, those lifetimes are broadly dispersed, in agreement with test results [62]. With this approach, it was shown that the discrepancy between the stress applied to the tow (average) and the critical filament rank (overloaded as α_t < 50%) generates a scale effect [66] affecting the stress exponent (n_filament > n_tow) [67] identified elsewhere on Hi-Nicalon^® fiber type [62,68].

3.3. Extrapolation to Low Stresses

In a recent paper, Hay et al. have pointed out the following inconsistency when life prediction models are projected to low stresses [20]: according to Equation (5) with constant Y, a filament under low stress would fail once a crack has acquired a length a superior to the specimen diameter (D), which is physically impossible. In reality, the stress intensity factor varies over the path of the crack (averaged in this work) and depends, among others, on the crack length to account for the reduction of the cross section as the crack grows [69,70]. As a consequence, the geometrical factor Y can be taken as a function of a. This phenomenon has been observed by different research teams on various materials, including small-diameter brittle fibers [71,72,73,74,75,76]. To distinguish this approach from the above (where Y = 2/π), this corrected geometrical factor is afterward named Y*. A polynomial expression relating Y* to a/D (up to a/D = 0.6) is commonly reported [70,77,78]. Typically, the geometrical factor is observed to significantly increase when a/D > 0.15. The present study is based on the expression identified by Van Sant et al. for a penny-shaped crack, valid up to a/D = 0.45 [72]:

$$Y^{*}=0.7+1.38{\left({\displaystyle \frac{2a}{D}}\right)}^2-2.16{\left({\displaystyle \frac{2a}{D}}\right)}^3+2.93{\left({\displaystyle \frac{2a}{D}}\right)}^3$$

(21)

Based on the assumption that Y* should tend to an infinite value when a → D so a cannot exceed D, Hay et al. [20] proposed an alternative formulation fitting the dataset of [72]:

$$Y^{*}={\displaystyle \frac{0.29}{{\left(1-\frac{a}{D}\right)}^{3.26}}}+0.41{\left(1-{\displaystyle \frac{a}{D}}\right)}^{2.13}$$

(22)

This expression was integrated to numerically simulate the slow crack growth of SiC tows. On the other hand, to the author’s knowledge, no analytical solution exists for the integration of (12) when substituting Y by Y* in (22).

4. Results

4.1. Time to Failure for High Stresses

NL and TS fibers display an average strength close to 3000 MPa with a Weibull modulus (Equation (18)) below 10 and a moderate elastic Young’s modulus (close to 200 GPa). Table 1 gathers the lifetime prediction coefficients (Equation (17)) for filaments at 650 °C. This temperature was selected because of the large amount of static fatigue tests performed, highlighting the scatter of times to failure. As mentioned above, fibers in tows are not perfectly aligned; part of them are even never loaded. The delay in strain activation leads to a stress distribution: this has been integrated in a Monte Carlo simulation showing the lifetime scatter [36]. Thousands of tows were simulated this way (Figure S1 available online). Figure 1 is a simplified representation of this result, showing the lifetime envelope embedding the dataset. For stresses below 600 MPa on NL and below 750 MPa on TS, the upper lifetime limit is not reached anymore. It is worth noting that static fatigue tests conducted under such low stresses were not stopped (no restriction in test duration). This observation seems to indicate that the projection mode of the model is somehow biased. This constitutes the starting point of the present investigation.

To better understand this behavior, it is important to keep in mind that, theoretically, the tow failure is piloted by the critical filament triggering the cascade fiber failure. It could hence be assumed that the critical filaments of the “best” tows (high P_t.i) failed prematurely (earlier than predicted). From a statistical point of view, those filaments should be underloaded (good fiber alignment, narrow fiber loading) and/or strong [66]. To better understand the stress–strength balance affecting the life variability through Equation (17), simulation results were further examined: in addition to the failure time, the strength and the stress applied to the critical filament were extracted. This was repeated 1000 times for four tow stresses (σ_{app.t(γ = 0%)}): 300, 600, 900 and 1200 MPa. The main driver for the failure appears to be the stress applied to the critical filament (Figure 2a,b), whereas its strength looks randomly distributed (no trend towards lower t_f when σ_f decreases or vice versa).

As a consequence, it is possible to link probability of P_t.i of t_f to σ_app(αt) by a linear regression of the Weibull plots (Equation (18)), considering this critical fiber has a constant strength: 2400 MPa for NL and 2900 MPa for TS (Figure 2c,d). From this result, it is finally possible to attribute to each experienced time to failure a probability and hence a value for σ_app(αt) as shown in Figure 3. At the filament scale, the data tend to diverge from the power law when the stress becomes lower than 700 MPa on NL and lower than 900 MPa on TS. The crack sizes required to achieve such strengths are close to 15% of the average fiber diameter: 2.2 µm on NL (14 µm diameter) and 1.2 µm on TS (8.5 µm diameter). As mentioned above, the geometrical factor starts to increase significantly for the same ratio a/D [75,76]. It is therefore interesting to investigate the consequence of integrating the evolution of Y* with the crack size to understand if this behavior could result from the overestimation of a_c. Therefore, a life prediction model that can analytically be solved has been developed.

4.2. Time to Failure for Low Stresses

As a prerequisite for the lifetime model adjustment, the expression of Y*(a) (Equation (22)) had to be modified to find an easy analytical solution. The solution used for this work consists in separating the a/D domain in two parts (Figure 4a):

• Domain I, for a < a_t = 0.15 D, where Y* is constant;
• Domain II, for a > a_t = 0.15 D, where Y* progressively increases.

A transition stress can alternatively be used instead of a_t, inferred from the fracture mechanics law: σ_t = 720 MPa on NL and 875 MPa on TS. In domain II, several relationships were tested to replace (22). Finally, a power-type equation was identified between σ_f and (1 − a/D), as shown in Figure 4b:

$${\displaystyle \frac{{\sigma }_f\sqrt{\pi D}}{K_{IC}^{*}}}={\displaystyle \frac{1}{Y^{*}\sqrt{\frac{a}{D}}}}=e^Q{\left(1-{\displaystyle \frac{a}{D}}\right)}^P$$

(23)

which can also be written as:

$$ln\left(Y^{*}\sqrt{{\displaystyle \frac{a}{D}}}\right)=-Q-Pln\left(1-{\displaystyle \frac{a}{D}}\right)$$

(24)

with constants P and Q given in

Table 2. Corrected life prediction parameters considering $Y^{*}$ = $Y_{lim}^{*}$ = 0.86 and correction K_IC and A₁ into K_IC* and A₁*, respectively.

Parameter	NL	TS
KIC* (MPa·m1/2)	1.59	1.50
σt (MPa)	721	875
n	10.0	7.4
A1* @ 650 °C (m1−n/2 MPa−n s−1)	7.52 × 10−11	1.63 × 10−10
P	3.40
Q	1.70
A1′ (m1−n/2 MPa−n s−1)	1.20 × 10−9	1.61 × 10−9

.

For the sake of continuity between domains I and II, Y* in domain I is taken as the lower limit of Y* in domain II, reached when a = a_t ($Y_{lim}^{*}$ = 0.86). The toughness (K_IC) originally given in Table 1 [39] using Y = 2/π was consequently corrected ($K_{IC}^{*}$, Table 2). It is worth noting that for a/D > 0.45, the extrapolation could be considered doubtful because it is based on an arbitrary extrapolation of Van Sant et al. data [72].

The discontinuity of Y*(a) expression brings different case scenarios. The life prediction model given in the theory section (Equation (17)) applies to domain I (when σ_app and σ_f are exceeding σ_t, $Y^{*}=Y_{lim}^{*}$ is constant), given the environmental constant A₁ is corrected to A₁* (Table 2) to incorporate the modification of Y*. When the stress applied to the specimen and its tensile strength are below σ_t (domain II), Y* can no longer be considered as independent of a. Using (23), Equation (12) can be rewritten as:

$$t_f={\displaystyle \frac{{De}^{nQ}}{A_1^{\prime }{\left(\pi D\right)}^{\frac{n}{2}}{\sigma }_{app}^n}}{\int }_{a_0}^{a_c}{\left(1-{\displaystyle \frac{a}{D}}\right)}^{nP}d\left({\displaystyle \frac{a}{D}}\right)$$

(25)

which can be integrated into:

$$t_f={\displaystyle \frac{D{e}^{nQ}}{A_1^{\prime }{\left(\pi D\right)}^{\frac{n}{2}}{\sigma }_{app}^n}}{\left[{\displaystyle \frac{{\left(1-\frac{a}{D}\right)}^{nP+1}}{nP+1}}\right]}_{a_0}^{a_c}$$

(26)

which in turn yields:

$$t_f={\displaystyle \frac{D{e}^{nQ}}{A_1^{\prime }{\left(\pi D\right)}^{\frac{n}{2}}{\sigma }_{app}^n\left(nP+1\right)}}\left({\left(1-{\displaystyle \frac{a_c}{D}}\right)}^{nP+1}-{\left(1-{\displaystyle \frac{a_0}{D}}\right)}^{nP+1}\right)$$

(27)

where $A_1^{\prime }$ is the environmental constant in this domain. Based on Equation (23) for σ_f and σ_app below σ_t, the flaw size can be expressed as:

$$a_0=D\left(1-{\left({\displaystyle \frac{{\sigma }_f\sqrt{\pi D}}{{e^{Q}K}_{IC}^{*}}}\right)}^{{\displaystyle \frac{1}{P}}}\right)$$

(28)

and:

$$a_c=D\left(1-{\left({\displaystyle \frac{{\sigma }_{app}\sqrt{\pi D}}{{e^{Q}K}_{IC}^{*}}}\right)}^{{\displaystyle \frac{1}{P}}}\right)$$

(29)

Substituting (28) and (29) into (27) yields:

$$t_f={\displaystyle \frac{D{\left(\pi D\right)}^{\frac{1}{2P}}}{A_1^{\prime }{e^{\frac{Q}{P}}\left(nP+1\right){\left(K_{IC}^{*}\right)}^{\frac{1}{P}+n}\sigma }_{app}^n}}\left({{\sigma }_f^{{\displaystyle \frac{1}{P}}+n}-\sigma }_{app}^{{\displaystyle \frac{1}{P}}+n}\right)$$

(30)

With this formulation, the time to failure in domain II remains related to ${\sigma }_{app}^n$, similarly to domain I. When σ_app → σ_f, the lifetime also tends to zero, giving the upper stress limit. Equation (30) strongly suggests that, in the present case for which n is much larger than 1/P, the most important contribution to lifetime comes from ${\left({\sigma }_f/{\sigma }_{app}\right)}^n$.

The last scenario appears when the tensile strength is higher than σ_t whereas σ_app < σ_t. This is of practical relevance for most applications targeting long life with such high-strength fibers. These conditions are indeed met when the lifetime, assessed experimentally under high stress levels in domain I, needs to be extrapolated to lower stresses. Here, it is considered the failure time corresponds to time for the crack to grow from a₀ to a_t (domain I, Equation (16)) plus the time required for the crack to grow from a_t to a_c (Domain II, Equation (27)). This can also be expressed using Equations (17) and (30) as:

$$t_f={\displaystyle \frac{2K_{IC}^{*2-n}}{A_1{\sigma }_{app}^n{Y_{lim}^{*}}^2\pi \left(n-2\right)}}\left({\sigma }_f^{n-2}-{\sigma }_t^{n-2}\right)+{\displaystyle \frac{D{\left(\pi D\right)}^{\frac{1}{2P}}}{A_1^{\prime }{\left(nP+1\right){e^{\frac{Q}{P}}\left(K_{IC}^{*}\right)}^{\frac{1}{P}+n}\sigma }_{app}^n}}\left({\sigma }_t^{{\displaystyle \frac{1}{P}}+n}-{\sigma }_{app}^{{\displaystyle \frac{1}{P}}+n}\right)$$

(31)

The value of $A_1^{\prime }$ is inferred from the following assumption: for σ_app slightly lower than σ_t, it is assumed the lifetimes estimated from Equations (17) and (30) are close enough to be considered equal (σ_app = 710 MPa for NL and 860 MPa for TS). This case scenario gives:

$${\displaystyle \frac{2K_{IC}^{*2-n}}{A_1{\sigma }_{app}^n{Y_{lim}^{*}}^2\pi \left(n-2\right)}}\left({\sigma }_f^{n-2}-{\sigma }_{app}^{n-2}\right)={\displaystyle \frac{D{\left(\pi D\right)}^{\frac{1}{2P}}}{A_1^{\prime }{\left(nP+1\right)e^{\frac{Q}{P}}{\left(K_{IC}^{*}\right)}^{\frac{1}{P}+n}\sigma }_{app}^n}}\left({\sigma }_f^{{\displaystyle \frac{1}{P}}+n}-{\sigma }_{app}^{{\displaystyle \frac{1}{P}}+n}\right)$$

(32)

from which $A_1^{\prime }$ is obtained by a 1st-order limited expansion considering ${\sigma }_t-{\sigma }_{app}\ll {\sigma }_{app}$, as:

$$A_1^{\prime }=A_1{\displaystyle \frac{{Y_{lim}^{*}}^2}{2P{e}^{\frac{Q}{P}}}}{\left({\displaystyle \frac{{\sigma }_{app}\sqrt{\pi D}}{K_{IC}^{*}}}\right)}^{2+{\displaystyle \frac{1}{P}}}$$

(33)

Values of $A_1^{\prime }$ are reported in Table 2.

As mentioned above, the critical filaments considered have tensile strengths equal to 2400 and 2900 MPa for NL and TS, respectively. Both values lie above σ_t. As a consequence, this work considers Equation (17) when σ_app > σ_t or Equation (31) when σ_app < σ_t. Figure 5 compares the predicted lifetimes and crack sizes for NL at failure time according to Equation (17) over the full range of stresses to the new proposed model including Y* = f(a). The transition from Equations (17)–(31) at σ_t is smooth (Figure S3). There is clearly no appreciable difference between both models as far as the lifetimes are concerned. Indeed, the lifetime with the new model remains related to the applied stress elevated to the power n. The revised formulation does not show any curvature inflection of the endurance diagram when the flaw size approaches the fiber diameter. These results demonstrate that integrating the evolution of Y*(a) to the Paris–Erdogan model has no impact on the predicted lifetime. The same is expected for any strength or temperature. On the other hand, the crack sizes at failure time are neatly different in the low-stress region. For instance, under an applied stress of 100 MPa, Equation (31) predicts the failure of NL for a crack size equal to 7.3 µm, whereas Equation (17) predicts the same for a = 110 µm, largely exceeding the specimen diameter (D = 14 µm). In other words, the crack growth rate (nm s⁻¹) is not continuously increasing; it decreases when a/D > 15%. This can be explained by the different formulation for the crack growth (a⁻ⁿ in Equation (12) vs. (1 − a)^nP in Equation (25)).

4.3. Slow Crack Growth Mechanisms vs. Oxidation Embrittlement

4.3.1. SCG-OE Competition

The above model successfully solves the incoherency in flaw size required for the failure by SCG under moderate stress (Figure 5) but fails to predict the divergence from the power law observed in Figure 3. One possible interpretation for this discrepancy between the power law model and the experimental results when σ_app < σ_t would be the existence of a second embrittlement mechanism governing the failure when the SCG is very slow and gives excessive lifetimes. The oxidation embrittlement was therefore reviewed and compared to the power law prediction. An empirical relationship between the strength and the inverse square root of the scale thickness (Equation (8)) is set up to predict the failure time, following the initial proposition of Lara-Curzio et al. [31,32] while correcting two incoherencies. First, as the evolution of Y* with a was not integrated into (9), Lara-Curzio’s model predicts infinite lifetimes when σ_app → 0. Therefore, a is substituted by x (Equation (7)) in Equation (5) when a/D < 15% and in (24) when a/D > 15%. The second modification concerns the oxidation kinetic law, assumed to be purely parabolic by Lara-Curzio et al. whereas the linear parabolic model (Equation (7)) should be considered. Equation (7) is rewritten as:

$$x=\sqrt{Bt+{\displaystyle \frac{A^2}{4}}}-{\displaystyle \frac{A}{2}}$$

(34)

Actually, as shown recently on TS fibers, the linear oxidation regime in ambient air and intermediate temperatures is not restricted to thicknesses largely inferior to 100 nm as commonly assumed [19]. A similar behavior was noticed on NL [28].

Combining (34) with (8) or (23) yields:

$${\sigma }_f=\left\{\begin{array}{}\mathrm{(35a)}& {\displaystyle \frac{K_{IC}^{*}}{Y_{lim}^{*}\sqrt{\pi \left(\sqrt{Bt+\frac{A^2}{4}}-\frac{A}{2}\right)}}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} x/D<15\mathrm{\%}\hfill \mathrm{(35b)}& {\displaystyle \frac{K_{IC}^{*}{e}^Q}{\sqrt{\pi D}}}{\left(1-{\displaystyle \frac{\sqrt{Bt+\frac{A^2}{4}}-\frac{A}{2}}{D}}\right)}^P \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} x/D>15\mathrm{\%}\hfill \end{array}\right.$$

Or, assuming failure occurs when σ_f = σ_app:

$$t_f=\left\{\begin{array}{}\mathrm{(36a)}& {\displaystyle \frac{1}{B}}\left[{\left({\displaystyle \frac{K_{IC}^{*2}}{\pi {Y_{lim}^{*}}^2{\sigma }_{app}^2}}+{\displaystyle \frac{A}{2}}\right)}^2-{\displaystyle \frac{A^2}{4}}\right] \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} x/D<15\mathrm{\%}\hfill \mathrm{(36b)}& {{\displaystyle \frac{1}{B}}\left[D\left({1-\left({\displaystyle \frac{{\sigma }_{app}\sqrt{\pi D}}{e^Q{K}_{IC}^{*}}}\right)}^{{\displaystyle \frac{1}{P}}}\right)+{\displaystyle \frac{A}{2}}\right]}^2-{\displaystyle \frac{A^2}{4B}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} x/D>15\mathrm{\%}\hfill \end{array}\right.$$

The lifetime predicted from this model is shown in the Supplementary Figure S4a,b for the two types of fibers. The transition from (36a) to (36b) is clearly identified for σ_app = σ_t by a sudden change of stress exponent. Of note, Equation (36b) projected to very low stresses (less than 50 MPa) can exceed the theoretical limit (see Supplementary Figure S4c,d) because Equation (8) considers that σ_f = 0 MPa when x = D, whereas the fiber is actually taken as fully oxidized when x = δ D/2. As δ < 2, the full oxidation corresponds to x < D. Of note, as mentioned above, this value is itself overestimated for small-diameter fibers [46]. The exponent n sharply decreases below the transition stress (Figure S4a,b) because of the evolution of Y*(a). For higher stresses, this exponent continuously evolves because the oxidation moves from a parabolic regime for long duration (n close to 4) to linear (n close to 2) for shorter durations. This evolution of n shows that for stresses below 2000 MPa, NL follows the parabolic regime (n close to 4), whereas TS lies in the linear-parabolic transition (n close to 3). It should be reminded that the oxidation rate for NL (Table 1) was extracted from [38], in a reconstituted wet air (4% H₂O). On the other hand, the same parameters for TS were assessed from experiments conducted in ambient air [19], more representative of static fatigue conditions. The same kind of work is being conducted on NL and will be presented in a future communication.

For laboratory experiments conducted under high stresses, the above model is, however, too optimistic [54]: the times to failure are shorter than predicted. Nonetheless, a stress exists below which t_f values from Equations (36a) and (36b) become more severe than the PL, which brings us to propose the following dichotomy model (hereafter referred to as FSO_d): for σ_app > σ_t, t_f = min [Equations (17) and (36a)] and for σ_app < σ_t, t_f = min [Equations (31) and (36b)]. The FSO_d life prediction is shown in blue in Figure 6. The transition from SCG to OE is observed for applied stresses close to 450 MPa for both fiber types (when x/D > 15%, so n < 2). Let us now consider an application targeting a 1-year life under 400 MPa stress. For such stress level, the PL is thence too optimistic. Based on the FSO_d prediction, the NL filament (σ_f = 2400 MPa) could be used, whereas the TS fiber (σ_f = 2900 MPa) would be disqualified (Figure 6). Static fatigue results under such low stress were never reported, but the time to failure trend mentioned earlier (Figure 3) tends to highlight that FSO_d is still too optimistic. Thus, it does not seem that the oxidation embrittlement phenomenon can by itself explain the trend noticed for lower stresses.

4.3.2. SCG-OE Competition Effect

As mentioned above, the FSO_d is a dichotomy selecting the most critical mechanism (SCG modeled through PL or OE modeled through FSO). More realistically, the time to failure should result from a competition between both mechanisms. Indeed, the PL model can empirically be divided into two domains, as shown in Figure 7 for NL with σ_f = 2400 MPa:

(i)

an incubation domain, when the crack is <200 nm long and the crack growth rate is slow;

(ii)

a fast propagation domain, when the crack exceeds 200 nm.

In the first domain, the crack growth rate by OE could exceed the SCG rate and trigger a fast propagation. To exemplify this point, let us consider a NL fiber subjected to 400, 600 or 800 MPa. Figure 7 illustrates the evolution of the crack length from PL or FSO models. The 200 nm crack size triggering the propagation domain of SCG is reached sooner by the FSO model at 400 MPa after several days. It appears there that both models taken separately overestimate the time to failure. The same is true at 600 MPa, to a lower extent. At 800 MPa finally, the SCG is fast enough to reach 200 nm before the FSO (the oxidation is too slow).

To integrate this competition effect between the PL and the FSO models, let us now introduce x (Equation (34)) instead of a₀ in (12) when x > a₀; in other words substitute σ_f in (31) by (35) or (36) when it becomes lower than the initial strength and when, respectively, x/D < 15% (37), or x/D > 15% (38) (usual scenario, when σ_app < σ_t and σ_f > σ_t). The same approach can as well be used in other scenarios: introducing (35) in (17) if σ_app and σ_f > σ_t or (36) in (30) when σ_app < σ_f < σ_t. For x/D < 15%, one has:

$$t_f={\displaystyle \frac{2K_{IC}^{*2-n}}{A_1{\sigma }_{app}^n{Y_{lim}^{*}}^2\pi \left(n-2\right)}}\left[{\left({\displaystyle \frac{K_{IC}^{*}}{Y^{*}\sqrt{\pi \left(\sqrt{Bt+\frac{A^2}{4}}-\frac{A}{2}\right)}}}\right)}^{n-2}-{\sigma }_t^{n-2}\right]+{\displaystyle \frac{D{\left(\pi D\right)}^{\frac{1}{2P}}}{A_1^{\prime }\left(nP+1\right)e^{\frac{Q}{P}}{\left(K_{IC}^{*}\right)}^{\frac{1}{P}+n}{\sigma }_{app}^n}}\left({\sigma }_{app}^{{\displaystyle \frac{1}{P}}+n}-{\sigma }_t^{{\displaystyle \frac{1}{P}}+n}\right)$$

(37)

while in the converse case, one has:

$$t_f={\displaystyle \frac{2K_{IC}^{*2-n}}{A_1{\sigma }_{app}^n{Y_{lim}^{*}}^2\pi \left(n-2\right)}}\left[{\left({\displaystyle \frac{K_{IC}^{*}{e}^Q}{\sqrt{\pi D}}}{\left(1-{\displaystyle \frac{\sqrt{Bt+\frac{A^2}{4}}-\frac{A}{2}}{D}}\right)}^P\right)}^{n-2}-{\sigma }_t^{n-2}\right]+{\displaystyle \frac{D{\left(\pi D\right)}^{\frac{1}{2P}}}{A_1^{\prime }\left(nP+1\right)e^{\frac{Q}{P}}{\left(K_{IC}^{*}\right)}^{\frac{1}{P}+n}{\sigma }_{app}^n}}\left({\sigma }_{app}^{{\displaystyle \frac{1}{P}}+n}-{\sigma }_t^{{\displaystyle \frac{1}{P}}+n}\right)$$

(38)

This new life prediction model, hereafter named FSO_c (“c” standing for competition), diverges from the PL for a stress slightly lower than 800 MPa (when σ_app is close to σ_t), which seems to properly fit the experimental data shown above (Figure 3), at least more convincingly than other attempts (Figure 6). Let us again consider the above lifetime target (1 year at 400 MPa); the FSO_c is now disqualifying both fiber types. Moreover, the lifetime reaches a plateau below 150 MPa for a lifetime close to 25 months on NL and 40 days on TS, corresponding to x/D close to 15% (2.1 and 1.3 µm on NL and TS, respectively, fibers not fully oxidized). This is a consequence of the power law used for Y*(a): the critical flaw size is not continuously increasing when it reaches 15% of D. For these conditions, x becomes large enough to trigger the SCG propagation domain. The derivative of the FSO_c lifetime trend (in other words, the stress exponent) is shown in Figure 8. This value evolves in three domains:

• for high stresses (above 800 MPa in the present example), the PL model prevails (Figure 7c) with stress exponents equal to 10 on NL and 7.4 on TS (Table 2, Equation (17));
• for lower stresses, the embrittlement induced by the oxidation (Equation (37)) reduces n to 3.2 on NL and 2.4 (difference coming from the oxidation regime as mentioned above);
• when x becomes large enough (after 25 months on NL and 40 days on TS) to enter in domain II (Figure 4) for a stress lower than 200 MPa, n falls again down to 0.

The second domain is clearly identified from the experimental results (Figure 6). Given the limited time to failure for TS at even lower stresses, it could have been expected to see the third domain too. Nevertheless, it is important to remind those results stand for critical filaments, whereas tests were made on tows (the stress applied to the critical filament is unknown). Therefore, the Monte Carlo simulation tool was adapted to integrate the FSO_c model for each filament and simulate the tow behavior (Figure 9, deduced from the Supplementary Figure S5). As expected, this model diverges from the PL for low stresses, a phenomenon first noticed on longer lifetimes for stresses close to 700 MPa. Here, the third domain is not observed because critical filaments are systematically overloaded: for σ_app(γ) = 100 MPa, σ_app(αc) is already >300 MPa. Without surprise, the lifetime of tows appears lower than the life of critical filaments piloting their failure (Figure 8), because the considered applied stress (σ_app.t(γ)) underestimates σ_app(αt) [66]. Results properly encompass most of the data points and give an accurate prediction of the shift observed at low stress. Based on these results, the median lifetime of tows under 200–400 MPa would be less than 1 month on NL and 1 week on TS, i.e., significantly lower than initial expectations using the empirical PL model. This result is of prime importance for the design of CMC parts, piloted by the weakest tows for which the shifting is observed for stresses below 400 MPa (Figure 9).

As shown in the Supplementary Figure S6, the FSO_c model introduces an upper life limit in addition to the upper stress limit. The introduction of the FSO (simplification of the oxidation embrittlement phenomenon) into the life prediction model led to a rather simple analytical solution: it requires only the assessment of linear and parabolic oxidation rates, extensively reported in the literature for various temperatures and environments [42,78].

This work has shown how the introduction of the oxidation embrittlement can affect the life prediction for NL and TS fiber tows at 650 °C. These fibers and this temperature were selected because they have been extensively analyzed and they depart from the power law at low stresses. We found that the endurance diagram contains an SCG-dominated and an OE-dominated regime, with a transition at intermediate stress and time. Future work will focus on the oxidation of NL fibers in ambient air (more representative for present static fatigue test) to adjust the above results. The model presented here considers that the strength of the oxide scale can be neglected, a statement which could be challenged. Moreover, it assumes fibers have full access to the environment (worst-case scenario), neglecting the protective matrix [79]. Also, the FSO_c neglects the cross-section reduction as fibers get oxidized [27], which causes an increasing stress (so far, taken as constant) and should further reduce the predicted life. A perspective of this work is to investigate endurance at lower or higher temperatures on the same fibers or the endurance of other fibers at 650 °C, which could reveal a different balance between SCG and OE mechanisms, if ever no other additional phenomenon adds up. However, it is to be noted that addressing low stresses poses difficulties in terms of time constraints.

Finally, this life prediction model could be generalized to other types of materials for which the SCG mechanism can be combined with alternative environmental-induced embrittlement causes (oxygen, moisture, salts, UV, etc.), same or different from the one driving the crack growth, for long-term applications [80,81,82,83,84,85,86].

5. Conclusions

Most structural materials are subjected to SCG, a phenomenon that limits their lifetime under stress. However, when the stress becomes too low, the lifetime may be controlled by alternative embrittlement phenomena. Lifetime models based only on SCG can be unrealistic at low stresses; they should be corrected. The present work has exemplified this statement for two different SiC-based fibers at 650 °C in ambient air. Under low stresses, the time to failure was observed to progressively diverge from the Paris-Erdogan power law. Two different approaches were evaluated. First, the life prediction model was adapted to solve the following incoherency: for low stresses, the crack size required to break a fiber would exceed its own diameter. Therefore, two crack length domains were distinguished: (i) for a crack to diameter ratio smaller than 15%, the dimensionless stress intensity factor Y* is independent from a, whereas (ii) for a ratio larger than 15%, Y* is increasing and tends to infinity when a = D. With this alternative model, physically reasonable crack length values are calculated while the lifetime keeps its power form. The model was further adapted to integrate a second source of damage resulting from the oxide scale growth on the surface. Its behavior diverges from the power law model for moderate stresses, in coherence with experimental observations, and tends towards a finite asymptotic life for even lower loads. This result is of particular interest for the design of CMC parts aiming at extended life. This model can be further adjusted to integrate the cross-section reduction while fibers get oxidized. Further works are being considered to assess the oxidation kinetics of different fibers in ambient atmosphere (relevant for the static fatigue test method). The same approach could apply to other types of materials, such as ceramic, glass, metals, or polymers, suffering from environment-driven degradation that could prevail for long-term usage.