There are numerous approaches that have been attempted quantify a crack pattern. There are two approaches used- one approaches involves generating either a purely artificial network or an artificial network based on a real network and developing methods to analyse them. The second approach involves using real crack networks to define measures for a crack pattern.
Andresen et al. study fracture outcrops by representing them as an artificial networks and measuring network parameters [5]. Such an approach has also been used by Valentini et al. to analyse rock fractures networks [86]. Fracture outcrops are lines of cracks that are part
of exposed geological structures. Examples of these geological structures are large boulders or sedimentary rocks. Andresen et al. took various fracture outcrops and generated networks by labelling cracks as nodes. The intersection between the cracks are labeled as edges which connect the nodes. They measured the degree distribution, the clustering coefficient, the efficiency of the network and the characteristic path length. The degree k of a node is the total number of neighboring nodes. If the degree distributionP (k) follows a power law, then the network is considered scale free. The clustering coefficient measures the local connectivity of the network. The clustering coefficient has a value between 0 and 1 where values close to 1 represents a condition where two neighbours of a single node share an edge of the network.
The clustering coefficient is a means to measure the local connectivity of the network. Such local measures would be useful pattern seen in polygonal terrain where local structure exists with a larger crack pattern. The efficiencyE of the network is a measure of how well different parts of the networks are connected to each other, and this is a global measure of the pattern.
E is proportional to the inverse of the distance between two nodes in the network. It falls between 0 and 1 except for the case whereE =∞. This too can be generally applied to many crack patterns. For example, comparing the blood cracks and the memory paste cracks, E may be smaller for the blood cracks since very few radial cracks are connected to each other ( white regions in the figure 1.9 (b)) whereas in figure 1.8 a path between any two vertices of the crack pattern can be found by travelling along the crack pattern. The last parameter they mention is the characterstic path length L which represents the average distance between any two nodes in the network. The average path length is large for fracture networks that have a small E value since the network is not well connected. While the methods discussed by Andresen et al. work well at characterizing networks, they may not be easily applicable to real crack patterns. Firstly, they require a crack pattern with a large number of cracks, in order to plot any meaningful distributions of P (k). With systems like that of Nakahara et al., it would be difficult to get the adequate statistics. Secondly, the measures contain very little information about the substrate of the crack pattern. This is required initially to understand how a crack patterns evolve.
Hafver et al. [42] took a different approach where rather than using existing crack patterns as models they generated artificial crack patterns and made measurements on them. Their cracks were straight lines whose position was chosen according to a probability which was weighted by a distance map of the pattern, and whose the orientation was chosen randomly.
Once a line was placed, it was extended in both directions until it either met another line or hit a boundary. Hafver et al. had two control parameters for their patterns: γ which controls the homogeneity of the pattern andωwhich controls the topology of the pattern. γcould take the values−2≤γ ≤2 whereas 0≤ω≤1. For a value ofω = 0 lines form a tree like structure and forω= 1 the lines generate polygonal structures. A pattern with γ = 2 andω= 0 is an isotropic pattern where most lines have one free end. The free end refers to one end of the line not intersecting with the boundary, or with any other line. On the other extreme where γ =−2 and ω = 0 most of the lines in the pattern lie at the boundary; no lines are present in the center of the image. Increasing the value of ω to 1 with γ = 2, generates a pattern
where the majority of lines lie along the diagonal to the square boundary or perpendicular to the diagonal. Values ofγ = 2 andω = 1 generate an isotropic pattern where there are no free ends for any line, all lines are connected on both ends to either another line or to the boundary. Usingω, Hafver et al. define an order parameterR= (1−ω)/(1 +ω) which is the ratio of the free ends to the number of intersections of lines. Since replacing the lines with cracks does not affect the definition of the parameter, they propose thatω can be used as a measure of crack patterns as well. Another order parameter they define is the measured value ofγ. They measureγ based on the temporal hierarchy of the pattern. Both the parameters are applied to crack patterns in gelatin confined to a Hele-Shaw cell, ice fractures on Mars and weathering cracks on the surface roads. Similar to the parameters of Anderesen et al.
such a parameter does not characterise the symmetry of the pattern. A radial crack pattern can either have cracks extending radially outwards or cracks that lie parallel to boundary, the two parameters can be tuned such that in both those cases the values of ω and γ are the same. Furthermore, the parameterR is then dependent on the number of free ends in the crack pattern, hence if a crack pattern were allowed to evolve for long enough, R will drastically change. This is a benefit for time lapse imaging of a crack pattern but if the crack pattern has an overall directionality, which can be imposed by the substrate, thenR would not be able to capture the influence of the substrate.
Bohn et al. [16] took an experimental approach to defining an order parameter. They studied the temporal and hierarchical evolution of glaze in ceramics by analysing a crack pat-terns generated by drying starch slurries on a rectangular polymethylmethacrylate (PMMA) substrate. They show that orientation of the first fracture is non-deterministic for low layer heights and with increasing layer height, the orientation and structure of the first crack becomes deterministic. They quantify this using two order parameters, these are δ = |d1 +d2|/√
A and ∆ = p
d12+d22/√
A where d1 is the distance along the rectan-gle, between one end of the crack and the center of the left edge of the rectanrectan-gle, d2 is the same, however for the right edge. The definitions of these are shown in figure 1.10. Both parameters δ and ∆ yield 0 if a crack divides rectangular region into two equal halves. For a curved crack δ > 0 and ∆ >0. These parameters approach close to zero with increasing layer heights. The large spread in values ofδ and ∆ at low layer heights is what signifies the disorder whereas at large layer heights the standard deviation of both order parameters is small. Both these order parameters can be written according to the symmetry of the region bounding the crack pattern, however, since they pertain to only a single crack, they cannot be used to analyse current experiments because the information about the first crack is lost in a mature crack pattern.
The various methods to quantify crack patterns that have been presented above have a certain realm of applicability. The δ and ∆ parameters serve to quantify a crack in a rectangular domain, the network parameter presented by Andersen et al. are suited for crack patterns with large number of intersections, the parametersR andγ are best suited to study an evolving crack pattern. None of these parameters contain any information about the orientation or the symmetry of the substrate, hence it is difficult to apply them to the
(a) (b)
Figure 1.10: Definitions ofd1 and d2. Figure reproduced from [16]
current problem.