A method for the 3D quantification of bridging ligaments during crack propagation

Abstract
This paper shows how a Hole Closing Algorithm can be used to identify and quantify crack-bridging ligaments from a sequence of X-ray tomography images of intergranular stress corrosion cracking. This allows automatic quantification of the evolution of bridging ligaments through the crack propagation sequence providing fracture mechanics insight previously unobtainable from fractography. The method may also be applied to other three-dimensional materials science problems, such as closing walls in foams.

1. Introduction
In the last decade, the focus on mapping damage processes in engineering materials has increased steeply, due in part to the power of emerging analytical techniques such as X-ray microtomography and partly to the emergence of new damage tolerant microstructured materials. As a non destructive method of observing structure in 3D, tomography is well suited to the study of damage accumulation, as detailed in several review articles [1, 2]. Examples include fatigue crack growth in aluminium alloys [3-5], and Ti/SiC fibre composites [6-8] as well as intergranular stress corrosion cracking (IGSCC) in austenitic stainless steel [9-11]. In each case, tomography allows study of the interaction between cracks and their surrounding microstructure. Further the future design and optimisation of exciting new materials such as self-repairing materials and hierarchical biomimetic materials will require better knowledge about the crack shielding mechanisms.

In each case of damage accumulation the crack path is interrupted, deflected and bridged. These slow down crack propagation and can increase the materials lifetime. Quantifying the effects of these bridging zones in 3D is therefore becoming increasingly important. Isolating such features from the rest of the microstructure is key to quantifying their morphological properties (e.g. shape, size, orientation) and their microstructural properties (e.g. crystallographic orientation). This information can then be incorporated in models, such as the FEM approach developed by Jivkov et al. [12] to mimic the real 3D behaviour of cracks and to predict crack growth rates. The reliable extraction of bridging ligaments from tomography data requires an image processing strategy. Standard methods, such as histogram-based segmentation, will not isolate the bridges from the material. Indeed, after segmentation of the crack, bridges correspond to holes (or tunnels) in the crack and are part of the surrounding background. Since algorithms to close holes in 3D objects exist, these may be used to extract bridges. In this paper such methods are extended to quantify bridging ligaments in tomograms of IGSCC.

2. Methodology
The problem of hole-closing is considered in the literature within the general topic of repairing 3D models. Several methods have been developed recently to correct defects in isosurfaces and 3D meshes, as surveyed in [13]. One approach, called the Hole Closing Algorithm (HCA), is particularly well suited to finding holes in a 3D object, such as a bridge across a crack [14]. The method is based on topological considerations, such as topological number, topology preservation and simple points that are common concepts used in thinning algorithms [15-17]. General details of HCA are given below, supported by a 2D illustration of a crack shown in Figure 1 with two branches (in grey) and their respective simplified skeleton (in black). HCA considers a bounding box X containing an object of interest OOI (e.g. a portion of a crack containing a bridge as in Figure 1). The method processes the points of X that do not belong to the OOI. Topological number plays a fundamental role in this approach and can be described as follows: it is a calculation of the number of connected components surrounding a particular voxel p in a 3x3x3 cube surrounding p. This number is calculated both for an object O (Tp,O) and the background (or complementary object) B (Tp,B). It is illustrated in Figure 1 where each pixel (or point) in the images, excluding those of the skeleton, is represented by the corresponding topological numbers with the following notation:  , where D is the squared Euclidean distance from the closest point of the OOI (here the skeleton). It should be noted:
points that have Tp,B = 0 are called interior points since they have no connection to the background. If an interior point is deleted, it leaves a cavity in the object.
a point on a line is a 1D-isthmus while an interior point on a plane is a 2D-isthmus, these have Tp,O larger than 2 and Tp,B larger than 2, respectively. In other words, if a 1D-isthmus is deleted, it cuts a line into two parts. If a 2D-isthmus is withdrawn, it merges locally at least two connected components of the background.
points which have Tp,O = 1 and Tp,B = 1 are called simple points because when withdrawn, they do not change the topology of O.
In the initial map (Figure 1a), only simple points and interior points are present. The core part of HCA is then processed as follows. Points of X that do not belong to the OOI and are neither interior points nor 2D isthmuses are deleted iteratively, from the furthest to the closest to the OOI. After deletion of one point, the topological numbers of its neighbours are recalculated. For instance, Figure1b shows that after deletion of simple points with distance equal to 18, 13, then 10, one point at distance 9 becomes a 2D isthmus since Tp,B = 2 and it therefore cannot be deleted. The deletion procedure continues until all the points connecting the two parts are 2D isthmuses (Figure 1c). The final output is a set of voxels that fill cavities in the OOI, but more importantly, a set of 1-voxel thick patches that close all the holes in the OOI.

An improved version of the algorithm has been proposed by Janaszewski et al. [18]. It only detects holes from the Euclidean skeleton of the OOI and processes the skeleton to remove branches around the hole. This is important since their presence can affect the Euclidean distance map, and consequently, the final extraction of the bridge. Using this method, one may obtain unique information about the bridges such as their morphological properties and also follow their evolution as the crack propagates. This is illustrated in this letter.

3. Results
The use of our HCA algorithm is applied to an intergranular stress corrosion crack in a sensitized austenitic stainless steel specimen. Details of the in-situ X-ray microtomography corrosion experiment can be found in [10, 11]. The spatial resolution (here taken as the voxel size) was 1.4 micrometer after data binning. The tomogram in Figure 2 is one of a sequence of eight (scan #3) obtained as the crack propagated towards final specimen fracture (scan #8). This reconstructed slice (Figure 2a) shows the crack path (black) is discontinuous in 2D (although continuous in 3D), with unbroken crack bridging ligaments of material. Closed holes in the crack (i.e. bridging ligaments) are coloured red in Figure 2a. A clearer visual representation of the bridges is shown on the corresponding 3D rendering image (Figure 2b). From this it is noticeable that the bridges tend to have characteristic shapes/locations. For example, bridge C has a trapezoidal-like shape within a grain facet, and bridges A and B appear to be located at grain facet corners. The sensitization to IGSCC depends on the structure of the grain boundary plane, and it has been proposed [9-11] that bridges are associated with annealing twin growth. These shapes are consistent with the intersections of annealing twins with grain boundaries.

As the bridges correspond to real 3D objects in the volumetric image, it is now possible to characterize them precisely. In this manner it is possible to follow the failure sequence of bridges, for example consider the three bridges (A, B and C) over successive tomography datasets (i.e. from scan #2 to scan #4). This process is quantified in Table 1 in terms of their orientation and area. The principal orientation theta of a given bridge is obtained from its moment of inertia tensor and is defined as the angle between the loading direction and the eigenvector of the largest eigenvalue of the tensor. This eigenvector corresponds to the normal to the bridge (Figure 2a), which in most cases has an approximately flat morphology. Table 1 shows that the normal axes to the three selected bridges are inclined at more than 60° to the tensile axis (y-axis on Figure 2). Large angles may indicate bridges that develop simply due to the physical orientation of the grain boundary and not necessarily from the boundary having special properties or structure; as it is difficult for a crack to propagate when it is aligned close to the loading direction. However, this needs to be statistically verified on a larger number of bridges. This is work in progress.
It is clear from Table 1 that once formed the area of the three bridges sb decrease significantly in size between successive scans as they fail. The bridge failure rate would be expected to depend on the crack opening displacement, as the bridges fail in a ductile manner by tensile overloading [11]. The crack opening displacement (as a first order approximation) should be obtainable from the volume defined by the area of the bridge and by the surrounding thickness of the crack. This may be obtained by the Hole Filling Algorithm (HFA) [18], which propagates the surrounding OOI thickness, defined by the Euclidean skeleton, to the segmented hole. However, despite the robustness of the approach, which is based on well-defined mathematical considerations, it does not properly recover the surrounding crack width of the bridge because of the narrowness of the tapered crack opening close to the bridge (see Figure 2a). Bridges with corresponding filling volumes that are rarely larger than 3-4 voxels in thickness (i.e. 4-6 micrometers m) are obtained. However, crack opening can be significantly larger, as seen in Figure 2a (~15micrometers local to bridge A). The development of an alternative approach to circumvent this drawback is currently in progress. It will address the tapering of the crack tip. It involves applying iterative geodesic dilations of the segmented hole into the Euclidean skeleton until the average thickness given by the new contour of the hole does not increase significantly. The local thicknesses obtained from the contours are then used to generate the hole filling propagation, as discussed above. 
	
It is helpful to analyse the bridges in relation to the macro crack, as it propagates. This can be directly related to post-test imaging of the fracture surface by scanning electron microscopy, thereby adding temporal information to conventional SEM-based fractography. Further it is the location and number of bridges active at any stage that provides some level of crack-tip shielding to the tensile load. This is most vividly shown by superimposing the position of the macro crack and the current active bridging ligaments for a given stage of crack growth (scan #) onto a 3D representation of the final fracture surface. This fracture surface is obtained from the last tomography scan (scan #8). The result is a sequence of frames, presented in Figure 3 as a movie (only scan #1 is included in the paper version), which shows how the main crack propagates with bridges remaining in the wake of the crack tip. The superposition requires that the same features exist in two successive images and there also exists an unknown rigid body motion between the first view (the scene) to the second view (the model) that can be calculated between each pair of successive images. This rigid body rotation occurs because the sample moves and deforms slightly between scans as the crack opens.

This correlation between successive images can be done using a method of 3D pose estimation based on point matching [19] that searches for the best affine transformation (i.e. rotation + translation matrices) between the corresponding points dataset. Since the crack grows and also opens between images, the point matching is done “backwards” in experimental time. The characteristic points at scan #i, which is set as the scene, are selected because they do exist at scan #i-1, which is set as the model. Typical points used for the pose estimation are the small number of void defects that can be present in the material, and also characteristic points from grain junctions that are revealed by the intergranular crack propagation. The main problem for pose estimation is that the hypothesis of a rigid body is not fully satisfied because the distortions in the volumetric image arise not only from crack growth but also from plastic deformation of the un-fractured ligament ahead of the propagating crack tip. This plastic deformation does not fully preserve the conservation of distances and angles between sets of corresponding points for 2 successive scans. Therefore, some manual correction is needed to label correctly each voxel into one of 3 classes: material, crack and bridge.
Because of the crack opening, it is also more convenient to display only the voxels of the crack that are found in contact with one specimen surface rather than the complete volume of the open crack. A simple 3D geodesic dilation of the material object into the crack object allows one to obtain a so-called “crack footprint”. In this way, the objects representing bridges and the crack footprint have a thickness of 1 voxel. Besides simplifying the visualisation, the crack footprint allows one to more precisely estimate the fractional crack area in each scan, relative to the final fracture surface. It also allows one to calculate the geodesic distance from the centroid of each bridge to the closest voxel of the crack front shown in Figure 3. The crack front is obtained as follows. The boundaries of the crack footprints and the fracture surface can be extracted based on topological number considerations (i.e. most of the points of these contours are simple points). The crack front is then obtained by a binary XOR operation between these boundaries.

The ratio between the crack footprint area and the final fracture surface area is used to define the crack expansion factor fc. The evolution of the average geodesic distance of the bridges to the crack front <dcf> and average size of bridges <sb> as a function of fc, is shown in Figure 4. The purpose of the analysis presented is simply to demonstrate the potential for quantifying bridge behaviour. The data show that both <dcf> and <sb> tend to increase with fc. This might indicate that there is a zone of crack bridging ligaments becomes larger as the crack develops. However, the results gathered here are not statistically significant and may be caused by local effects.

4. Conclusion
The example study presented in this letter shows that the hole closing algorithm (HCA) offers new possibilities to study the mechanisms by which microstructure affects crack propagation. For example, if combined with data on grain and grain boundary plane orientations (e.g. obtained by diffraction contrast tomography [11]) and observations of the failure mechanism from fractography of failed bridges, models for stress corrosion crack behaviour that are more microstructurally faithful may be developed and tested. The algorithms described here, i.e. HCA and HFA have been successfully tested in other materials science application such as investigation of weld profile in friction stir spot welding and study of deformation in auxetic polyurethane foam. In the former example, the technique helped to correct images and in the latter it facilitated microstructural visualisation, as shown in [20]. This approach may also be of interest in fibre composites and in many others cases where crack bridging occurs.

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Fig. 1: 2D illustration of HCA. 


Fig. 2: 3D rendering of bridges.


Fig. 3: 3D crack print on fracture surface.

Fig. 4: Evolution of bridge size vs. the crack expansion factor.