Introduction

Since early work in the 1970s [1], silicon carbide (SiC) has been proposed as an irradiation tolerant material for nuclear fusion and fission applications [1, 2]. Monolithic SiC is brittle, but the combination of SiC-fibre and SiC matrix in a composite offers non-linear deformation and damage tolerance [3, 4] with relatively high toughness achieved by introducing an “interphase” between the fibre and the matrix [5,6,7,8]. The early development of SiC-based composites for nuclear applications was driven by the international fusion research programs, aimed at possible application in the breeder blanket [9,10,11,12]. Advanced SiC fibres [13] have also been developed for high temperature structural applications in the aerospace industry [14]. Potential applications in fission reactors include guide tubes and channel boxes for the fuel assembly [15, 16], and within the European nuclear fission programme a tubular fuel cladding has been designed for gas-cooled high temperature reactors [10, 17]. A similar clad geometry has also been proposed for accident tolerant fuel (ATF) in light water reactors (LWRs) [18].

The current SiC/SiC composites proposed for nuclear fission applications consist of near-stoichiometric beta-phase SiC for both fibres and matrix and a pyrolytic carbon (PyC) interphase. The composite architecture affects the mechanical properties, and a two-dimensional (2D) woven architecture is often employed, commonly plain-weave or satin-weave [16]. Both offer similar properties, but for the tubular structure of fuel cladding a satin-weave with ± 45° or ± 30° stacking is commonly reported [9, 16].

The cladding contains the fuel whilst providing an impermeable heat-transfer medium that separates the fuel from the coolant. The cladding is thus one of the most critical components in a fission reactor. During operation, due to the potential strains of fuel pellet swelling and dimensional change from thermal and irradiation gradients, a complex multi-axial stress state may develop in the cladding [19, 20]. In some reactors, such as LWRs, additional stresses may come from internal gas pressure [17], and the rapid change of clad surface temperature in some accident scenarios would induce substantial strains [21, 22]. Reliable application of SiC/SiC composites in nuclear fuel cladding requires understanding of their mechanical and thermal properties, as well as their resistance to transport of gas and fission products, since these may be affected by mechanical damage.

Mechanical damage development in SiC/SiC composites has been studied in unidirectional minicomposites [23,24,25] that represent the microstructure as a single fibre tow, with early observations performed ex situ or after failure [23, 26]. The first in situ investigations that characterized damage in the matrix and fibres at the local scale were carried out by Chateau et al. [27, 28], and also Bale et al. [29]. These found cracks initiated in the SiC matrix, and propagated perpendicular to the tensile stress direction. The frequency of matrix cracks was observed to increase with strain, with saturation of the matrix crack density above a global strain of 0.3% (failure occurred at a global strain of ~ 0.7%) [27, 28]. The development of significant matrix microcracking is coincident with the proportional limit stress (PLS), which describes the onset of non-linearity in the stress-strain curve. The measurement of the PLS is influenced by the composite microstructure and testing methods [30], with typical PLS values for 2D CVI-composites in the range of 100–150 MPa [9, 16, 31, 32]. Comprehensive studies of multiaxial loading of tubular composites have shown the PLS depends on the loading state [33,34,35].

The role of the composite microstructure is important. Droillard et al. demonstrated that adopting a stronger interphase significantly increased the PLS [36], and Katoh et al. suggested that reduced porosity might be associated with increased PLS [16], which was supported by observations that cracks in the SiC matrix were associated with larger pores [37]. However, while there have been several ex situ studies of damage in SiC/SiC composites [25, 37, 38], few have observed the development of damage in situ. Some early in situ studies utilized surface observations [33, 34], and others examined the loading-unloading hysteresis loop to indirectly quantify the effects of damage at the macroscale [16]. To observe the damage phenomena within the material, techniques such as X-ray computed tomography (XCT) are necessary. Early XCT studies made ex situ observations [39, 40], but with the development of high resolution micro-XCT in situ observations of damage in SiC/SiC composites have been achieved [27, 29]. Image post-processing with digital volume correlation (DVC) to measure the three-dimensional displacement field has been applied to augment in situ observations of damage development in SiC/SiC composites [41,42,43]. For example, DVC analysis of XCT observations [44] of an aerospace-grade melt-infiltrated SiC/SiC composite, studied in tension at room and high temperature, showed how the initiation and propagation of matrix cracking was affected by the composite microstructure. Studies of tensile loaded [45, 46] and internally pressurized [47] CVI-SiC composite tubes have also applied in situ XCT/DVC and observed how matrix cracking was influenced by the stress concentrations from macropores at fibre tow intersections [45, 46] and inter-tow contacts [47].

Materials testing is crucial for the evaluation of structural integrity in engineering applications, and is also needed to qualify and optimise manufacturing processes [18]. SiC/SiC composites with unidirectional fibres and 2D woven architectures have been studied in a planar geometry [32, 48], for which standardised tests are available [49]. However, the geometry of the composite components affects the manufacturing process and the microstructure, such as the fibre architecture [27, 36], with consequences for mechanical behaviour. Testing of components in the as-manufactured geometry of fuel cladding under realistic loading is therefore necessary. Axial tests of tubular components are relatively straightforward [33, 50], and C-ring [51,52,53,54] and O-ring [55] tests have been used to measure mechanical properties at ambient and elevated temperatures [54]. The stress gradients and stress states in such tests, however, do not replicate well those that may be experienced in service.

The ASTM standard C1819 [56] provides a method to assess the hoop tensile strength of continuous fiber-reinforced ceramic composite tubular test specimens at ambient temperature. The radial expansion of an elastomer under uniaxial compression induces internal pressurisation of the tube, and a relation between the hoop stress and hoop strain is obtained from which the elastic modulus and PLS can be derived. Strain measurement is recommended by surface observations (optical or physical dilatometers or strain gauges). Such testing of a filament wound triplex ATF clad with monolithic inner and outer layers was done to observe the effect on the PLS of fibre volume fraction and winding angle [57], and similar tests have used oil pressurisation to induce the internal loading [58]. A recent comprehensive study compared elastomer inserts and pressurisation by an oil-filled bladder – the bladder is more suitable for longer tubes as it avoids the risk of buckling of the elastomer [34]. That study also used digital image correlation of optical images to examine the surface displacements, and observed a non-uniform distribution of hoop strains, including localised vertical strain features, that showed an influence of the composite structure on deformation and damage development.

Such macroscale tests provide valuable data for engineering design. They may also be used to evaluate variations in properties due to fabrication, such as the effects of the fibre textile weaves on the onset of the damage development and final failure [38, 59]. However, macroscale tests can only describe an average response of the tested specimen. The heterogeneous nature of SiC/SiC composites means their mechanical behaviour is sensitive to the microstructure, and reliable methods to investigate the local relations between microstructure damage and deformation are needed.

The motivation for this study was to investigate an experiment methodology to measure the evolution of the hoop deformation of an internally pressurised SiC/SiC composite tube. A previous study [47] showed that DVC of in situ high resolution X-ray tomographs could detect the local deformations due to damage under this type of loading, but their relation with the overall deformation of the composite tube was not investigated. This paper describes a simple internal pressurisation experiment on a nuclear-grade composite, using the radial expansion of a compressed elastomer insert, that was observed in situ by high resolution (synchrotron) X-ray tomography. Measurement of the full field three-dimensional displacements by digital volume correlation provided maps of the relative radial and circumferential displacements. This allowed the hoop strain and its spatial variation to be determined as a function of the applied hoop stress. These local measurements could then be related to the quantitative observations of matrix cracking, and also compared with the location of the critical crack that caused rupture.

Materials and Experimental Method

The SiC/SiC composite was provided by the MatISSe EU FP7 project [60] and was fabricated [61] by the CEA (French Alternative Energies and Atomic Energy Commission) using third generation HNS SiC fibres (Hi-Nicalon™ Type-S, Nippon Carbon Co.), with an average diameter of 12 μm. The composite tube architecture has three layers [62]. The interior layer is a filament wound layer with a ± 45° stacking configuration. The intermediate and outer layers are both 2 × 2 (± 45°) 2D braided structure. The fabric preform had been deposited with a 100 nm pyrolytic carbon (PyC) and then densified with SiC by chemical vapour infiltration (CVI). The tubes had a smooth ground interior surface with a diameter of 7.80 mm, and the outer surface, which was not ground, had a diameter of 9.75 mm (± 0.05 mm fluctuation due to the roughness of the braided structure). The anisotropic elastic properties are reported in [33], which also found the hoop failure stress of the composite tube was ~ 360 MPa. Tubes were cut into sections with a length of 10 mm using a diamond cutting saw. The cut surfaces were ground and examined in an optical microscope and by laboratory X-ray tomography (Zeiss Xradia Versa 510, voxel resolution 2.5 μm) to verify there was no visible machining damage.

The in situ study was performed on the I12 Joint Engineering, Environmental and Processing (JEEP) beamline at the UK Diamond Light Source synchrotron [63] (Fig. 1(a)). A loading jig (Fig. 1(b)), suitable for X-ray tomography, generated the internal pressurization of the tube. Opposing aluminum alloy punches (5000 series alloy, 2% proof stress > 130 MPa, Young’s modulus 68 GPa) elastically compressed a Viton® (DuPont-Dow extrusion resistant fluorocarbon elastomer, hardness 80 Shore A) insert with a diameter of 8 mm; this was selected for low X-ray attenuation, high compressive strength and isotopic properties with a large Poisson’s ratio of nearly 0.5. The test method was based on ASTM C1819 [56], which recommends a minimum insert length of approximately 3.5 mm for these tube dimensions. All loading was performed in displacement-controlled mode using a Shimadzu AGS-X precision (10 kN capacity) universal testing machine. Example displacement-force data for six tubes tested to failure (Fig. 1(c)) demonstrated the reproducibility of the jig performance. The mean failiure load was 3867 MPa (standard deviation 258 MPa, n = 6).

Fig. 1
figure 1

(a) universal test machine on the translation/rotation stage at the I12 JEEP beamline (Diamond Light Source, UK); (b) loading jig and specimen with schematic; (c) example crosshead displacement-load tests from a set of 6 trial specimens

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At each increment of the applied compressive displacement (Fig. 2(a)) sets of radiographs were collected at two different resolutions with a monochromatic X-ray energy of 60 keV; lower resolution (7.91 μm pixel size with 20 mm field of view, 0.05 s exposure each) and higher resolution (3.25 μm pixel size with 8 mm field of view, 0.8 s exposure each). Limited-angle tomography (2501 radiographs over 146° rotation) was necessary due to the opacity of the columns of the universal tensile machine that was transported to the beamline from the University of Oxford for the experiment (Fig. 1(a)). At each load, an overall (lower resolution) tomograph was recorded with four local (higher resolution, region of interest) tomographs, which overlapped by 1.5 mm by translation of the sample stage, to observe the entire specimen (Fig. 2(b)). [see Supplementary information for comparison]. Tomographs were recorded at the preload (50 N), 1000, 2000, 2800, 3400 and 3800 N, with negligible load relaxation during the tomographs (< 1% of applied load). The total tomography scanning tine at each load was approximately 45 min. The sample failed as the load exceeded 4050 N, which was 71% of one standard deviation from the mean of the preliminary studies (Fig. 1(c)).

Fig. 2
figure 2

(a) Displacement (crosshead) vs. load data for the in situ study; (b) schematic diagram (not to scale) of the regions observed by X-ray tomography using lower resolution (overall scan, 20 mm field of view) and higher resolution (overlapping scans, 8 mm field of view). The tube diameter is approximately 10 mm

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The reconstruction of the tomographs used the I12 beamline software, which implemented back-filtered projection with Fourier-wavelet ring artefact removal. No special adjustments were done for the limited angle of data collection. Post testing examination of specimen was done using a Zeiss Xradia Versa 510 X-ray microscope (60 keV, 3201 projections over 360° at pixel sizes of 9.8 μm and 3.1 μm). All reconstructed tomographs were post-processed in ImageJ [64] (noise filtering and cropping) and visualized using the Avizo Fire software.

DVC was applied to the tomographs using the LaVision DaVis StrainMaster 8.2 software on a dedicated workstation (Intel Xeon E5-2699 v3, 2 × 18 cores @ 2.3 GHz, 512 Gb RAM). The analysis used the direct correlation mode, the reference image being the undeformed sample (50 N preload), with iterative reduction of the subset size to 64 × 64 × 64 voxels using 75% overlap and 2 passes at each stage. The required valid voxels per subset was 50%, and the correlation coefficient threshold was 90%. This analysis yielded a full field 3D displacement map at grid spacings of 126.5 μm and 52 μm for the overall and local tomographs respectively. DVC analysis of pre-loaded tomographs (lower resolution, overall scans), between which a physical translation was applied to the sample stage by combined horizontal displacements of 40 μm in X and Y and a vertical displacement of 80 μm in Z, found a standard deviation in the displacement magnitude of 0.4 μm.

Results

Microstructure and Damage Visualisation

The example horizontal (X-Y plane) sections of reconstructed in situ tomographs (Fig. 3) show some artefacts (at \(\theta\)~70° and 250°) in the reconstruction of the external geometry of the tube that are due to the limited angle tomography. The artefact locations are aligned with the positions of the columns of the universal test frame. The internal structure is well visualized by attenuation contrast (pores are dark), as is the external geometry of the tube after threshold segmentation using the greyscale representation of X-ray attenuation (Fig. 4). This allowed visualization of axially aligned cracks, on both the inner and outer surfaces, that were first observed at an applied load of 2000 N. The crack locations were coincident with fine streak artefacts in the tomograph slices (X-Y plane) (Fig. 5); these are caused by the partial volume effect [65] where edge features such as cracks have a stronger effect on attenuation in radiographs obtained when aligned with the X-ray beam [66]. These fine streaks allowed the identification and counting of narrow cracks that could not be visualized via image contrast if their crack opening was substantially smaller than the voxel size [67]. The cracks were counted and their locations around the circumference determined using the lower resolution overall scans, and this was verified by inspection of the higher resolution region of interest tomography scans.

Fig. 3
figure 3

Example in situ tomograph sections in the horizontal X-Y plane (Preload, 50 B): (a) overall scan (7.91 μm voxel); (b) higher resolution scan (3.25 μm voxel)

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Fig. 4
figure 4

(a) Example 3D visualization of tomograph (2000 N load, 7.91 μm voxel) showing axial cracks on (b) inner surface and (c) outer surface

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Fig. 5
figure 5

Observations (in situ tomographs, X-Y plane, 3.25 μm voxel) of streak artefacts associated with axial crack: (a) Preload (no cracks); (b) 3800 N. Zoomed images are shown to the right

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The development of axial cracking was visualized using the 3D maximum strain, which is the maximum principal strain that is calculated from the gradients of the DVC displacement field. This semi-quantitative analysis detects discontinuities in the displacement field, e.g., those due to crack opening displacements where larger opening displacements give a higher magnitude of apparent strain. Figure 6 presents data from the higher resolution tomography of a section of the tube, and shows that axial cracking became apparent as the load exceeded 2000 N; there were some indications of inhomogeneous deformation of the microstructure at 1000 N. The DVC analysis shows some of the displacement discontinuities were aligned with the braided structure, though no cracks were observed in the tomographs at these locations.

Inspection of the post-test tomographs (Fig. 7(a)) was used to identify the location of the final rupture of the tube (critical crack) for comparison with the in situ tomographs [details of the comparison are provided in Supplementary information]. This critical crack is located at the location of \(\theta\)~350°. Post-fracture tomography (Figs. 5(b) and 7(a)) and scanning electron microscope fractography (Fig. 7(d)) show axial cracks that initiated in the monolithic CVI-SiC layer were deflected by the fiber bundles with significant fiber pullout. The critical crack appears to be coincident with an axial crack that developed with increasing load (Fig. 7(c)).

Fig. 6
figure 6

Visualisations of the 3D maximum strain, derived from the displacement field, with increasing applied load, superposed on a visualization of the test specimen. (There is no strain for the preload, which was the reference for the displacement field analysis by DVC)

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Fig. 7
figure 7

(a) Visualisation of post test (laboratory) tomograph (9.8 μm voxel) of the failed specimen, showing the location of the critical crack; (b) fracture surface visualization (laboratory tomograph, 3.1 μm voxel); c) in situ tomographs (local region of interest scans at higher resolution) at the location of the critical crack showing the outside surface and the evolution with load of the DVC-measured strain; d) scanning electron microscope image of the fracture surface (red box in a) and white box in b) indicate the location of the SEM image)

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The number of axial cracks, which each extended over a significant height of the tube length (Fig. 8), increased with increasing load and was higher on the inner surface. The maximum crack number density was ~ 4 cracks/mm. Cracks were only observed within the surface monolithic CVI-SiC layer and were not observed to propagate into the fiber bundles in any of the in situ tomographs [A series of high resolution tomograph slices at the location of the critical crack is provided in Supplementary information, which shows the increasing number of axial cracks]. The average crack number density was similar over most of the circumference, but there were significant minima in both inner and outer crack numbers, particularly at around \(\theta\)~165±15° and \(\theta\)~345±15° (Fig. 8(b) and (c)); these locations were not affected by artefacts from the limited angle tomography reconstruction.

Fig. 8
figure 8

Number density of axial cracks (cracks/mm of circumference) with applied load: (a) on outside and inside surfaces, averaged over the full circumference; (b) on outside surface and (c) on inside surface, averaged within 30° angular sectors (±15° of labelled \(\theta\))

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DVC Analysis for Measurement of Hoop Strain

The motivation for this study was to quantify the evolution of the deformation with increasing applied internal pressure. Careful registration of the DVC-measured displacement data was required to measure the radial and circumferential displacements of the sample, from which the hoop strains were calculated. The internally pressurised sample had no fixed point, and could undergo 3D rigid body translations and rotations that were not negligible in comparison to the displacements within the material. These were not removed with sufficient precision by the DVC software (see supplementary information), so a MATLAB code, based on Mostafavi’s method [68], was applied to correct for these movements. This allowed precise measurements of the radial and circumferential deformations.

The tomographs have a Cartesian coordinate system defined by the X-ray beam direction and sample rotation axis. The DVC results, with respect to these axes, are relative displacement vectors \(\left[{U}_{i}\right]= \left[{u}_{i}^{x},{u}_{i}^{y},{u}_{i}^{z}\right]\) at the central point (\(\left[{A}_{i}^{o}\right]= \left[{x}_{i}^{o},{y}_{i}^{o},{z}_{i}^{o}\right]\)) of each correlation subset. The rigid body translation (typically tens of voxel dimensions) was first removed as \(\left[{A}_{i}^{1}\right]- \left[{\stackrel{-}{U}}_{i}\right]\) (the average of all the displacement vectors). The rigid body rotation was then corrected to achieve a negligible rotation with respect to the central axis of the tube: the displacements were transformed to polar coordinates, and the Euler rotation angles were calculated by subdividing the translation correlated matrix and deformed location matrix. The rotation of the displacement field was corrected using Shoemake’s method [69], after which the residual Euler rotation angle was less than 0.3°.

The polar displacement measurements obtained from the DVC analysis of the lower resolution tomographs were used to map the hoop strains in the pressured tube. The hoop strain, \({\epsilon }_{\theta \theta }\), is determined by two displacement-dependent terms [70], where \({u}_{r}\) is the radial (axial) displacement (radius \(r\)) and \({u}_{\theta }\) is the circumferential displacement (equation (1)). The radial displacements, averaged over the wall thickness of the tube due to the relatively sparse grid of measurements, are presented in Fig. 9 for representative horizontal and vertical sections. The vertical section (Fig. 9(b)) shows that barreling occurred, with a displacement difference at the mid-height, with respect to the ends, of approximately 40 μm (~ 0.9% of the radius at the tube wall mid-section). For loads of 2000 N and above, the radial expansion was not uniform around the circumference – the peak displacements show increasing ovalisation with applied load (Fig. 9(c)). At 1000 N load, the radial displacement showed a uniform dilation of ~ 0.5 μm (Fig. 9(d)).

$${\epsilon }_{\theta \theta }=\frac{{u}_{r}}{r}+\frac{\partial {u}_{\theta }}{r\ \partial {\uptheta }}$$
(1)
Fig. 9
figure 9

The effect of applied load on the radial displacements within example sections that are marked in (a); (b) vertical section (Z-r); (c) horizontal section (r-\(\theta\)); (d) data for the horizontal section at 1000 N only with expanded displacement scale

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Maps of the radial and circumferential displacements (averaged through the wall thickness) are shown in Fig. 10 as a function of applied load. The radial displacement map at 1000 N has a sinusoidal variation with small magnitude that is judged to be due to the residual uncorrected rotation. At 2000 N and above, the displacements are non-uniform due to barreling and ovalisation. Figure 10c shows an example map for the hoop strains for loading at 3400 N; the total hoop strain is obtained from the contributions of the radial and circumferential displacements (equation (1)). Significant tensile hoop strains developed over the full height of the tube, and the peak tensile strains at \(\theta\)~60° and \(\theta\) ~240° are approximately diametrically opposed. There are minima in the hoop strains around \(\theta\)~160° and \(\theta\) ~340°.

Fig. 10
figure 10

Maps of (a) displacement in the radial axis and (b) displacements in the circumferential direction, as a function of the applied load. The displacements are averaged through the wall thickness and are relative to the pre-loaded state; (c) Example map of the hoop strains at 3400 N load; the total strain (C) is the sum of the strains from the (A) radial and (B) circumferential terms

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Discussion

The precise registration of the DVC-measured displacement fields allowed accurate measurement of the radial and circumferential displacements during a pressurization test for the first time. The observation that the internally pressurized tube deforms radially with a slight barreling (Fig. 9(b)) is not unexpected, as the ends of the tube are not fully loaded by the insert [56]. Nonetheless, its effect on the axial (Z) variation of the hoop strains in the observed region is not significant (Fig. 10(c)). The progressive ovalisation with increasing load (Fig. 9(c)) is more significant and leads to local and diametrically opposed peaks and troughs of the hoop strain around the circumference of the tube.

The hoop strains, after being averaged over the observed circumference, are presented in (Fig. 11(a)) as a function of the average hoop stress, \(\stackrel{-}{{\sigma }_{h}}\), which was calculated by treating the tube as an thick-walled cylinder using the Lamé equation, with radial integration of the hoop stress, \({\sigma }_{h}\),

$${\sigma }_{h}=\frac{{r}_{i}^{2}P}{{r}_{0}^{2}-{r}_{i}^{2}}\left(1+\frac{{r}_{0}^{2}}{{r}^{2}}\right)$$
(2)

where \(r\) is the radius, \({r}_{i}\) is the interior radius, \({r}_{i}\) is the outer radius and \(P\) is the internal pressure that is calculated from the applied axial load and internal cross-sectional area. At the failure load, the average hoop stress was 350 ± 10 MPa, with an (extrapolated) average hoop strain of ~ 0.15%. There is a non-linear relation between the average hoop stress and average hoop strain, above an estimate proportional limit stress (PLS) of ~ 150 MPa. These measurements of the PLS, the average strains at discrete stresses and the ultimate failure strain under internal pressurization are quite comparable to literature data for similarly manufactured composites that underwent monotonic internal pressurization to failure [33, 34]. This demonstrates that measurement of the radial and circumferential displacements deformation by DVC of tomographs provides a reliable assessment of the deformation, which is achieved over the entire circumference. A more sophisticated analysis could consider the elastic anisotropy of the layered structure (e.g. [71]), and might be used to then explore the effects of braid angle [46], but this would require higher spatial resolution observations of the deformation gradient through the wall thickness.

Subcritical axial cracks, which initiate in the monolithic surface SiC (Fig. 6), have been detected in previous surface observations of braided SiC-SiC composite tube [33]. They arrest due to deflection by the weak interface of the fibers [5,6,7,8], and may eventually propagate through the fibre bundles with increasing strain [42, 46, 72]. They are important for the hermetic properties of the material as they provide a potential pathway for transport of fluids (e.g. oxidizing species from impure coolant) into (and potentially through) the cladding [73]. The maxima and minima of the hoop strain (Fig. 10(c)) around the circumference are coincident with the maxima and minima in number density of subcritical axial cracks (Fig. 8(b) and (c)); these were detected due to their local effects on the reconstructed X-ray tomograph (Fig. 3(b)). The average number densities of subcritical cracks differ on the inner and outer surfaces (Fig. 8(a)), but by considering the radial variation of hoop stress that can be calculated at the respective surfaces using (equation (2)), a common dependence can be observed between the subcritical axial crack density and tensile hoop stress at the inner and outer surfaces (Fig. 11(b)).

Equation (2) assumes linear elasticity, and strictly a non-linear dependence between tensile stress and tensile strain [72] is needed to derive the actual strains and stresses at the inner and outer surfaces. However, the experimental data are sufficient to deduce that the tensile hoop strain controls the number of subcritical cracks that are aligned in the orthogonal axial direction. A higher resolution analysis that mapped the local strains across the wall thickness (e.g. as in [47]), rather than assessing the average hoop strain, could find a more accurate relation between the hoop strain and crack density. An inverse analysis of experimentally measured deformations and applied loading (e.g., [74, 75]) might also provide a means to derive the non-linear relations between stress, strain and damage, which could be used to predict the performance of fuel clad under more complex loading such as thermal gradients combined with internal pressurization and flexure.

Fig. 11
figure 11

(a) Average hoop strain and hoop stress compared with literature data for similar SiC/SiC composites tested by internal pressurisation from Bernachy-Barbe et al. [33] (oil pressure) and Shapovalov et al. [34] (elastomer insert and oil pressure (open-end) burst test); (b) relation between the average axial crack number density and the hoop stress calculated at the inner and outer surfaces, using (equation (2))

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The inner surface of the tube was ground smooth to achieve a circular cross-section at the central bore. The outer surface was not ground. The CVI process is applied to a fiber preform, which is woven onto a mandrel that is removed before full densification [61, 76]; the manufactured tube may not be perfectly circular and a systematic (small) variation in the tube wall thickness may occur around the tube circumference. Post-processing of the tomograph to measure the wall thickness (See Supplementary Information) confirms an approximately sinusoidal variation (amplitude 5.0% of the mean thickness) with maxima at around \(\theta\) ~140° and \(\theta\) ~340°, which are close to the locations of minimum circumferential strain (Fig. 10(c)). This would bias the hoop stresses at these diametrically opposed locations. Subcritical axial cracks nucleation, aided by local stress concentrations in the microstructure, would be slightly more probable in the regions of reduced wall thickness, and this is judged to be the cause of the oval deformation under pressurization. Grinding of both the inner and outer surfaces to achieve a constant tube wall thickness should give a more uniform distribution of the hoop strains and axial subcritical cracking.

It is noted that although the location of the burst failure of the tube (critical crack, Fig. 7) is coincident with an axial matrix crack, there is no relation to the hoop strain or crack density distributions – indeed the average hoop strain was only around 0.005 at this location compared with the peak values of 0.02. This emphasizes the importance of understanding how subcritical cracks transition to criticality. The critical fracture propagated from an axial crack in the surface SiC that became unstable due to factors that include its length, the local stress and the local resistance to propagation. Further understanding that could lead to a stochastic model for fracture, would require in situ observations of the interaction of cracks that initiate in the surface layers with other significant defects such as inter-tow pores, and numerical simulations of fracture in the complex composite microstructure. The analysis method presented here could be used to study this failure process, but this would require pressurisation with an incompressible fluid to achieve stable propagation.

Conclusion

A novel methodology to measure the deformation of an internally pressurised ceramic composite tube has been demonstrated in a digital volume correlation analysis of a burst test of a SiC/SiC ceramic composite tube that was observed in situ by high resolution (synchrotron) X-ray tomography. A precise rotation correction of the displacement field obtained the relative radial and circumferential displacements of the tube wall, from which the hoop strain and its spatial variations were determined as a function of the applied hoop stress. Ovalisation and barreling of the tube was observed, with local variations of hoop strain around the circumference. The quantity of subcritical matrix cracking increased with the tensile hoop strain, but the critical crack that caused rupture was not at the location of maximum tensile strain.