Data on SiC-based bundle lifetime variability: The insufficiency of external phenomena affecting the flaw size

A broad variability characterizes the lifetime of SiC-based bundles under static fatigue conditions at intermediate temperature and ambient air, challenging the accuracy of its prediction. The same is true, in a lower extend, with tensile properties, in apparent discrepancy with the bundle theory based on weakest link theory. The data presented here focus on lifetime scattering, evaluated on different fiber types (6 in total, Nicalon® or Tyranno®). It is hosted at http://dx.doi.org/10.17632/96xg3wmppf.1 and related to the research article “Static fatigue of SiC-based multifilament tows at intermediate temperature: the time to failure variability” (Mazerat et al., 2020) [1]. The insufficiency of classically invoked external and discrete bias (fiber sticking phenomenon for instance) was compared to a devoted Monte Carlo algorithm, attributing to each filament a strength (random) and a stress (homogeneous). Introduction of a stress inconsistency from tow to tow, experimentally observed through section variability, was revealed to overpass such biasing approach. This article can be referred to for the interpretation or prediction of CMC lifetime to guaranty long term performances over the broad offered application field.


Specifications
Material Science Specific subject area Delayed failure of SiC/SiC ceramic matrix composite Type of data Figure Table  How data were  acquired Static fatigue data were acquired suspending a dead weight to a bundle, placed in a resistive furnace opened to ambient atmosphere, and measuring the time before its failure. Data

Value of the Data
• This dataset is valued because it gives some insight onto the sources of SiC-bundle lifetime dispersion and limitation of some approaches invoked to interpret it. • The data can be used for comparative and comprehensive works on static fatigue behavior of SiC filaments or tows. • The dataset can assist the understanding of bundle strength and lifetime variability sources.
It may also argue the selection of a reinforcement type for a given application (design on purpose). • These data bring new insight onto the interpretation of such variability, introducing the uncertainty on applied stress in a Monte Carlo based simulation model.

Data Description
The dataset described herein analyzes the scatter experienced by extensive tensile or static fatigue testing of SiC-based bundles. A total of 6 fiber types were investigated: Nicalon® NL207, Hi-Nicalon®, Hi-Nicalon® Type S, Tyranno® Grade S, Tyranno® Lox-M and Tyranno® ZMI. It was deemed necessary to build supplementary figures and share the raw data to highlight the relevance of observations done in Ref. [1] . Raw data and the algorithm are available in Mendeley data repository under the following identifier DOI: 10.17632/96xg3wmppf.2 . As a preliminary study, bundle tensile behavior was investigated. The ruin under monotonous strain rate was therefore considered to be governed by a critical filament of rank α c (cf. method section, Eq. (5) ). The stochastic character of strength ascribed to each of the N 0 filaments was repeated 10 0 0 times to estimate the distribution parameter for virtual bundle strength. The latter were finally compared to experimental datasets from [2] ( Fig. 1 ). A drastic underestimation of the tow strength and its dispersion can be noticed. The Weibull modulus calculated on virtual bundle exceeded (2 to 10 times larger) the actual ones as shown in Table 2 .
A similar work was then performed under static fatigue conditions, with an invariant γ value and taking only the filament strength variability into account. Here, the bundle ruin was considered to happen when a critical fraction α t Eq. (16) , gathered in Table 3 ) of fibers had failed by subcritical crack growth. An example of this simulation tool applied to NL207 type ('Monte Carlo simulation NL207.xlsx') are given in the supplementary file. A Set of 50 bundles are randomly generated, each one having a unique architecture. The lifetime variability extracted for this algorithm was drastically underestimating (Weibull moduli > 5 times larger) tests results as shown on Weibull diagrams ( Fig. 2 ) and as mentioned in [1] . The computed and experimental raw data used to build these diagrams and assess the virtual distribution can be found in the supplementary file 'Variability comparison.xlsx' for all fiber types. The first attempt to interpret this scatter underestimation is to consider a fiber sticking-induced bias on α t fraction as described in [3] . Therefore, the upper lifetime limit was fitted by the prediction model ( Eqs. (16) , ( (18) with parameters given in Table 3 ) and lower bonds considered piloted by the weakest filament of the tow (bonded to a critical amount of fibers, α t = 1/ N t ). This approach could however

Table 1
Statistical parameters describing the distributions of unloaded section fraction ( γ ) and filament strength. 23.6 * * indicates the data that were offset as given in [2] . These values are duplicated from the related research article [1] .
not encompass the data points for all fiber types as shown on endurance diagrams in Fig. 3: on Hi-Nicalon and Grade S types a drastic underestimation is observed. NL207, Hi-Nicalon type S and ZMI however show better consistency. This is to be linked with Weibull statistical parameters for the fiber strength ( m f > 6 on TS and Hi-Ni, Table 1 ) [1] . Also, no particular tendency could be noticed when trial temperature was increased as shown on NL207 comparing the results at 650, 750 and 850 °C ( Fig. 3 a-c). Hi-Ni at 900 °C, (e). Hi-Ni-S at 600 °C, TS at (f). 550 °C (g). 750 °C and (h). ZMI at 750 °C. A 1 coefficients (respectively 38, 580, 2300, 5900, 0.18, 6.6, 220, 520 × 10 −12 m 1-n/2 MPa − n s −1 ) were taken so the upper limit fits experimental data.
Introduction of bundle tensile strength in place of the critical filament strength in Eq. (13) , based on the over-estimation of this latter as shown in Fig. 1 was investigated. Values for higher and lower tow strength were extracted from a previous data article [2] and summarized in Table 4 . Here, again, results on Grade S and Hi-Nicalon underestimate the experience scattering, accompanied by ZMI ( Fig. 4 ). In contrast, NL207 and Hi-Nicalon S are more consistent. With  Table 3 Parameters used to construct the Fig. 3 strength of fiber with rank αt or 1/ Nt and A 1 coefficient empirically adjusted to fit upper lifetime limit.
Upper lifetime Lower lifetime this approach however, the scatter would be expected to decrease at lower applied stress, which is not evidenced on test results ( Fig. 4 b).
Above theories neglect a key factor: the uncertainty on the stress applied to the tow probe, affecting the crack growth kinetic. The section of a tow is indeed strongly method related and varies from probe to probe [2] , with a stress misestimation up to several hundreds of MPa [ 1 , 2 ]. As a first step, this concept was used to build endurance diagrams as shown in Fig. 5 , from t f.t and γ ( Eq. (3) ) datasets given in [2] and gather in the supplementary file 'Distribution association.xlsx'. If the stress exponents estimated this way are globally overestimated as shown in Table 5 , the approach comprise the elegancy not invoking other variability sources (stick to the unbiased bundle model). On TS11 and ZMI type, values were close to the expected ones (respectively 6.1 and 9 against 5.8 and 9.1). Largest discrepancies were found for Hi-Ni under σ app. t = 1500 MPa ( n est = 16 against 8.4) and NL207 under σ app. t = 700 MPa ( n est = 16 against 7.2).
In these conditions, because some tows were failing during the loading step (strength close to the applied stress, Table 4 ) and consequently discarded, the t f.t dispersion is most likely biased. Moreover, it is worth reminding these statistical parameters were extracted from a limited dataset size (commonly 30 values) and thus do not depict its full range.
The approach was hence transcribed to the above algorithm. Each tow was given a structure, summarized by its effective section fraction ( S t x (1-γ ), Eq. (3) ), and each of the N 0 filaments was given a strength, randomly selected among the 2-parameters Weibull distribution ( Table 1 ) [1] . From this set of data, the virtual time to failure for each filament ( Eq. (13) ) was calculated and ranked. A weakest link approach finally gave the bundle behavior. Tables 6-8 show an dispersed as awaited after Fig. 1 , γ and the associated virtual time to failure. On this fiber type ( N t = 500, γ = 25% and α t = 8.22%), the critical filament rank equals 30. It can be noticed critical filaments are systematically weak, strength probability P i(30 ) < 0.15 and σ f.f < 2400 MPa, whatever the virtual performance. This latter looks better related to γ covering 3 to 4 orders of magnitude when γ varies from 5 to 95%. Moreover, null virtual lifetime were observed for P i30 > 0.98 and 0.95 at respectively 600 and 900 MPa (when the applied stress exceeds σ f.f( αt) ). Table 5 Comparison of stress exponents extracted from experimental datasets ( n true ) or from the association of lifetime and γ respective variabilities ( n est ) ( Fig. 5 ).  Table 6 Summary of 20 simulation runs for Hi-Ni-S tows under σ app.t( γ = 0%) = 300 MPa and 600 °C. Pi (30) describes the strength probability for the 30th fiber to fail ( αt fraction) and P γ the γ probability of the tow. The tremendous impact stress misestimation plays on time to failure dispersion is illustrated on Weibull diagrams ( Fig. 6 ). On Lox-M and ZMI types, the comparison of Fig. 2 d,e with Fig. 6 b,c highlights the increase of virtual lifetime range obtained, almost encompassing the experimental results. Extended to different applied stresses ranging from 100 to 1400 MPa, endurance diagrams were constructed displaying the median time to failure as well as its simulated range, then compared with experimental data points ( Fig. 7 ). This visualization helps to identify the stress for which loading failure would be expected (600 MPa on above Hi-Ni-S mentioned fiber type and above 700 MPa on the other types). With this approach NL207, Hi-Ni-S, TS and ZMI results are consistent to each other (no discrepancy in simulated range) unlike above approaches. A slight underestimation of the predicted scattering compared to the experienced one is to be noticed on all types. On Lox-M tows however, this computation clearly underestimate experimental values. Table 9 gathers the coefficients A for the median, the upper and the lower virtual lifetimes as extracted from simulations. Table 7 Summary of 20 simulation runs for Hi-Ni-S tows under σ app.t( γ = 0%) = 600 MPa and 600 °C. P i(30) describes the strength probability for the 30th fiber to fail ( αt fraction) and P γ the γ probability of the tow.  Table 8 Summary of 20 simulation runs for Hi-Ni-S tows under σ app.t( γ = 0% ) = 600 MPa and 600 °C. P i(30) describes the strength probability for the 30th fiber to fail ( αt fraction) and P γ the γ probability of the tow.  Table 9 Prediction parameters used as simulation input ( A 1 ) or describing the simulation output (median, minimal and maximal lifetime).

Material
Polymer derived SiC-based fibers presented in this dataset were provided by Nippon Carbon Co. Ltd. or UBE Industries Ltd. Different processing routes, leading to different generations, were tested: the first oxygen cured generation (Nicalon® NL207, Tyranno® Grade S, referred as TS, Tyranno® Lox-M, and Tyranno® ZMI), the second electron-beam cured generation (Hi-Nicalon® named Hi-Ni) and the third generation which underwent a high temperature annealing treatment (Hi-Nicalon® Type S named Hi-Ni-S) [ 4 , 5 ]. Two different TS fiber diameters were studied (8.5 μm or 11 μm, the later named TS11). Their respective properties were given in [1] .

Method
The same bundle probes were used for tensile or static fatigue testing. Sized bundles of 300 mm length ( L 0 ) were weighted ( m 0 , Eq. (2) ) and positioned in alumina tube grips. To ensure probe alignment, a pre-load was applied and maintained by fugitive Loctite® glue. A solution of dissolved PMMA was applied on the 25 mm gage length separating the grips, to avoid capillarity cement transportation during curing. Tubes were finally filled with alumina based thermostructural cement (Ceramabond 503, Polytec PI) and cured at 370 °C for 2 h. The engineering stress applied to the bundle, corrected accounting for the fraction of unloaded fibers ( γ Eq. (3) ), inferred from Eq. (1) .
Where ρ is the fibers density, w t the applied force, N 0 the initial number of intact filaments and N t its total manufactured number (50 0, 80 0 or 160 0). E t and E f are respectively monofilament and tow Young's moduli.
The effective tow section (S t (1-γ )) differs from probe to probe. Its dispersion was satisfactorily fitted by a Weibull statistical distribution law ( Eq. (4) ) as shown in [2] , where m γ is the modulus, γ 0 the characteristic fraction and P γ the probability.
Tensile tests were carried out at a constant displacement rate of 50 μm min −1 . Under these conditions (no load sharing), the bundle model considers filaments break progressively and individually as the force applied to them reach a critical value [6][7][8]. The maximal force is met when the ratio ( α) of broken filaments ( N ) to the large total number ( N 0 ) reaches a critical ratio ( α c ) assumed from filament strength distribution (Weibull statistic with m f as modulus and σ 1.f as characteristic strength) as follows [9][10][11] ( Table 2 ): Static fatigue experiments were however conducted in a vertical resistive furnace opened to atmospheric environment suspending a dead weight (applying a constant force w t ) to the lower grip and initiating the heating up. Only probes that survived to loading step were considered. The automatic stop of timer when specimen failed gave the tow lifetime. The experimental setup was shown in [1] and [3] . Because force and strain could not be recorded, the actual γ value of the tested tow could not be estimated ( E t unknown). Its average value, extracted from tensile tests [2] , was hence used. Tests were performed at different stresses (11 tests per condition) to construct endurance diagrams. Some conditions were more extensively tested for scattering assessment purpose. Lifetime variability can nicely be described using the Weibull statistic ( Eq. (6) ) [ 14 , 15 ], where t f.t0 is the characteristic time to failure and m df.t the static fatigue Weibull modulus. All these statistical parameters were assessed by linear least square method applied to Weibull plots.