3D Detection and Quantitative Characterization of Cracks in a Ceramic Matrix Composite Tube Using X-Ray Computed Tomography

Abstract

Cracks play an essential role in the degradation of the thermomechanical behavior of ceramic matrix composites. However, characterizing their complex 3D geometries within a complex microstructure is still a challenge. This paper presents a series of procedures, based on X-ray tomographic images, to evaluate the applied 3D strains, including their through-thickness gradients, and to detect and quantify the induced crack networks in ceramic matrix composites. Digital volume correlation and some dedicated image processing algorithms are employed. A novel method is proposed to estimate the opening, orientation and surface area of the detected cracks. The proposed procedures are applied to the images of a SiC/SiC composite tube that has been tested in situ under uniaxial tension with synchrotron X-ray computed tomography.

Appendix

We provide here a detailed demonstration that links up the definition of crack thickness in equation (8) and its diffuse evaluation in equation (10). Substituting $f\left(\underset{\_}{X}\right),{f}_S$ and f_V by the definition of equation (9), the right-hand side of equation (10) becomes

$$ {\int}_{M_0}^{M_1}\left(\frac{f\left(\underset{\_}{X}\right)-{f}_S}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X}={\int}_{M_0}^{M_1}{\int}_{V\left(\underset{\_}{X}\right)}K\left(\underset{\_}{y}-\underset{\_}{X}\right)\frac{\mu \left(\underset{\_}{y}\right)-{\mu}_S}{\mu_V-{\mu}_S}\mathrm{d}\underset{\_}{y}\mathrm{d}\underset{\_}{X}+{\int}_{M_0}^{M_1}\left(\frac{f^{\prime}\left(\underset{\_}{X}\right)}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X} $$

(22)

Using the change of variable $\underset{\_}{H}=\underset{\_}{y}-\underset{\_}{X}$ and applying Fubuni’s theorem, one obtains:

$$ {\int}_{M_0}^{M_1}\left(\frac{f\left(\underset{\_}{X}\right)-{f}_S}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X}-{\int}_{M_0}^{M_1}\left(\frac{f^{\prime}\left(\underset{\_}{X}\right)}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X}={\int}_{M_0}^{M_1}{\int}_{V\left(\underset{\_}{O}\right)}K\left(\underset{\_}{H}\right)\frac{\mu \left(\underset{\_}{X}+\underset{\_}{H}\right)-{\mu}_S}{\mu_V-{\mu}_S}\mathrm{d}\underset{\_}{H}\mathrm{d}\underset{\_}{X} $$

$$ ={\int}_{V\left(\underset{\_}{O}\right)}K\left(\underset{\_}{H}\right){\int}_{M_0}^{M_1}\frac{\mu \left(\underset{\_}{X}+\underset{\_}{H}\right)-{\mu}_S}{\mu_V-{\mu}_S}\mathrm{d}\underset{\_}{X}\mathrm{d}\underset{\_}{H} $$

$$ ={\int}_{V\left(\underset{\_}{O}\right)}K\left(\underset{\_}{H}\right)\cdotp \Lambda \left(\underset{\_}{c}+\underset{\_}{H}\right)\mathrm{d}\underset{\_}{H} $$

(23)

where $\Lambda \left(\underset{\_}{c}+\underset{\_}{H}\right)$ is the physical crack thickness quantified through an integration along a path normal to the crack mean surface and passing through the point $\underset{\_}{c}+\underset{\_}{H}$ (see the definition of equation (8)). Let us decompose the vector $\underset{\_}{H}$ along its components parallel and normal to the crack mean surface, $\underset{\_}{H}={\underset{\_}{H}}^{\parallel }$ + ${\underset{\_}{H}}^{\perp }$. It is clear that $\Lambda \left(\underset{\_}{c}+\underset{\_}{H}\right)=\Lambda \left(\underset{\_}{c}+{\underset{\_}{H}}^{\parallel}\right)$, as long as the segment M₀M₁, translated by the vector ${\underset{\_}{H}}^{\perp }$, is long enough to still fully encompass the crack. One finally obtains

$$ {\int}_{M_0}^{M_1}\left(\frac{f\left(\underset{\_}{X}\right)-{f}_S}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X}={\Lambda}^K\left(\underset{\_}{c}\right)+{\int}_{M_0}^{M_1}\left(\frac{f^{\prime}\left(\underset{\_}{X}\right)}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X} $$

(24)

where ${\Lambda}^K\left(\underset{\_}{c}\right)={\int}_{V\left(\underset{\_}{O}\right)}K\left(\underset{\_}{H}\right)\cdotp \Lambda \left(\underset{\_}{c}+{\underset{\_}{H}}^{\parallel}\right)\ \mathrm{d}\underset{\_}{H}$ is a “smoothed” crack thickness. This quantity is calculated as a weighted average of the local thickness over the width determined by the spatial resolution of the tomography device, which is usually of the order of a few voxels. As long as the latter is small with respect to the typical curvature radius of the real crack and to the typical length of the spatial variations of its thickness, the approximation $\Lambda \left(\underset{\_}{c}\right)\approx {\Lambda}^K\left(\underset{\_}{c}\right)$ holds.

We emphasize that this result does not require the crack thickness to be large with respect to the image spatial resolution. Indeed, the accuracy of the evaluation of $\Lambda \left(\underset{\_}{c}\right)$ can even be significantly better than voxel size and image resolution, and it is essentially governed by the last term in equation (24), which characterizes the signal-to-noise ratio of the image. Assuming ${f}^{\prime}\left(\underset{\_}{X}\right)$ to be a white noise at voxel scale, with zero statistical expectation, standard deviation σ^f, and correlation length equal to one voxel, it can be easily shown that ${\int}_{M_0}^{M_1}\left(\frac{f\left(\underset{\_}{X}\right)-{f}_S}{f_V-{f}_S}\right)\mathrm{d}\underset{\_}{X}$ is an unbiased evaluation of ${\Lambda}^K\left(\underset{\_}{c}\right)$ and that its standard deviation is given by

$$ {\sigma}^{\Lambda^K}=\frac{\sigma^f\sqrt{\left\Vert {M}_0{M}_1\right\Vert }}{\left|{f}_V-{f}_S\right|} $$

(25)

This relation suggests that for optimal results, the integration segment should not be taken too large, so that $\sqrt{\left\Vert {M}_0{M}_1\right\Vert }$ remains close to unity. The best option is to take it as close as possible to the diameter of the support of the kernel $K\left(\underset{\_}{y}\right)$. Rigorously speaking, the herein presented analysis ignores the periodic spatial dependence of the kernel K, which is induced by gray level interpolation. A more detailed, rather technical, analysis would be possible, but without significant change on the result, hence it is not presented here for conciseness.