An array of cracks

The formation of a through-the-thickness crack in a film subjected to a residual or applied tensile stress relieves that stress in the film material at points adjacent to the crack path. At points in the film at some distance from the crack path, the stress remains unrelaxed due to the constraint of the substrate. Consequently, a long crack that is parallel to the first formed crack can also form. Indeed, an array of parallel cracks over the entire film surface is likely, and the point of the discussion in this section is to provide an estimate of the dependence of the spacing between cracks in such an array on the film thickness hf and the mismatch stress a. The discussion is limited to the case when the equi-biaxial mismatch stress is uniform throughout the film, the elastic properties of the film and substrate are nominally the same, and hg/hf is sufficiently large so that the behavior is insensitive to the substrate thickness hg. Furthermore, it is assumed that the cracks grow through the thickness of the film to the depth hj, but that they do not penetrate into the substrate. There is no fundamental barrier to relaxation of these limitations, but the relatively simple system is sufficiently rich in physical detail to reveal the principal features of behavior. 

The energy release rate per crack for growth of a periodic array of 

Once this function is known, the condition for minimum crack spacing Amin is determined by Wm = Wc- If the spacing falls below Amim then Wm Wc or there is insufficient energy available to drive each of the fractures without external influence. Thus, Amin must satisfy the condition 

When the approximation (4.71) is used, this equation can be inverted to yield the dependence of Amin/h on the normalized film characteristic stress 

To consider this sequential nature of crack formation, the condition that the propagating segments of the forming cracks advance in unison is relaxed. Instead, it is assumed that a periodic array of cracks first forms in the film at spacing 2A the mechanics of formation of this initial array is not an issue in this discussion. After these cracks are fully formed and their 

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If a thin film is bonded to a substrate, and if that film is subject to a residual tensile stress as a result of elastic mismatch with the substrate, then the stress can be partially relaxed by formation of cracks in the film. In this section, the behavior of through-the-thickness cracks within the film is considered. First, the behavior of an isolated, fuUy formed crack is examined and, subsequently, the mechanics of formation of an array of cracks is considered. 

Fig. 4.44. The minimum spacing possible for an array of cracks formed simultaneously, as in Figure 4.43, or sequentially, as an Figure 4.45, versus residual stress in the film. The arrow identifies the stress at which cracking first becomes possible as...

Consider a two-dimensional medium in (y, z) plane containing an array of cracks. The medium is loaded remotely in mode I sueh that the stress a-yy = corresponds to Kf and the opening gap between y = 0+ and y = 0 is 2uy z) = 8 z). Then (Tyy z) along z axis is obtained as... 

Cracks as a Superposition of Dislocations. A scheme that will suit our aim of building a synthetic description of cracks and any allied dislocations is to think of a crack as an array of dislocations. Indeed, the majority of our work has already been done earlier in the context of our consideration of dislocation pile-ups in section 11.4.2. In fact, our present analysis will do little more than demonstrate that the solutions written down there are relevant in the crack context as well. The more fundamental significance of the perspective to be offered here is that we will soon want to build up solutions in which cracks and dislocations are equal partners. 

Fig. 14.10. Observed cracks along with an array of AE sensors.

A comparison between FE simulations and FBG measurements was made. The result is shown in Figme 7 for a crack length of 7 mm. The BFS distribution measured by an array of FBG sensors and interpolated by spline interpolation has a similar trend as the FEM results, as shown in Figure 7. 

Figure 10. Static results of BFS distribution- by arrays of strain gauges for (a) no crack specimen, (b) specimen with 5.68 mm pre-crack by an array of FBG sensors for (c) no crack specimen, (d) specimen with 10 mm pre-crack.

Missaoui et al. 2000 Buzynin et al. 2008). The PS provides a large area composed of an array of voids and interconnected silicon clusters or rods. This spongelike structure could limit strain and crack development after the post-growth cooling (Matoussi et al. 2001). 

Hackle is very useful in fracture analysis because it is always oriented parallel to the direction of crack propagation (perpendicular to the crack front). Moreover, fine hackle features tend to coalesce into coarser features and then eventually disappear as the fracture segments converge into a common plane. That coalescence gives an array of hackle features the appearance of tributaries flowing into a river. Because of this appearance, hackle is often referred to as river lines. As the analogy implies, the direction of crack propagation corresponds to the direction of flow of the river. 

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