Crack Spacing and ‘Width, with Application to Highway Pavements
The prediction of the spacing and opening of cracks in asphalt or concrete pavements of roads and runways is important for their durability assessment. This problem is similar to the ice plate—a plate resting on Winkler elastic foundation—but the foundation is much stiffen Similar to the problem of sea ice penetration (Section 9.6), the crack spacing also is important for the size effect.
One basic problem is the spacing s of parallel planar cracks initiating from a half space surface (Fig. 10.5.6a), which was solved approximately by Bazant and Ohtsubo (1977) and Bazant, Ohtsubo and Aoh (1979), and rigorously by U, Hong and Baz. ant (1995) (see also Bazant and Cedolin 1991, Ch. 13). The crack opening at the crack mouth is approximately w — —se°, where e° is the free shrinkage strain or thermal (cooling) strain (e° < 0).
The problem of crack spacing in pavements has been solved according to the theory of plate (beam) on Winkler elastic foundation (Fig. 10.5.6b) by Hong, Li and Bazant (1997). The calculated values of crack spacing were in relatively good agreement with the previously reported observations on asphalt concrete pavements.
The theory of initiation of parallel equidistant cracks from a smooth surface, developed in Li and Bazant (1994b) as an extension of the approximate crack spacing criterion proposed by Bazant and Ohtsubo (1977) and Bazant, Ohtsubo and Aoh (1979) (see also Bazant and Cedolin 1991, Ch. 13), was applied in the aforementioned study. Although the strength concept must be applied for the crack initiation stage, the cracks are considered simply as LEF. M cracks afterward. The theory, which was studied rigorously in Li, Hong and Bazant (1995), rests on the following three conditions:
1. Just before crack initiation from a smooth surface, the stress at the surface reaches the material strength limit, //.
2. After initial cracks of a certain initial length do form (by a dynamic jump), the energy release rate must be equal to the fracture energy of the material or the Д-curve value.
3. The total energy release due to the initial crack jump must be equal to the energy needed to produce the initial cracks, according to the fracture energy Gy or the Д-curve (an equivalent statement is that the average of energy release rate during the initial crack formation must be equal to the value of the fracture energy Gy or the average value of the Д-curve, as illustrated in Fig. 10.5.6c).
The problem can be solved if the stress intensity factor (or energy release rate) as a function of the crack length, the crack spacing and the load parameter (e. g., the penetration depth of the cooling or drying front) is known. For the elastic halfspace, the stress intensity factor has been solved from a Cauchy integral equation (Li, Hong and Bazant 1995). The solution of conditions 2 and 3 graphically represents the intersection of the curves giving the energy release rate and the average energy release rate (the intersection always exists if the fracture geometry is, or becomes, positive); sec Fig. 10.5.6c. All three conditions together allow solving three unknowns: the initial crack spacings, the initial crack length, and the load level (load parameter) at which the cracks initiate. Generalization to the full cohesive crack model is possible.
A different basic problem is how a system of parallel cracks evolves after it has initiated. Often it happens that every other crack stops growing and closes when a certain critical length acr is reached
Figure 10.5.7 Specimen used in studying cracking in keyed joints in segmental box girder sections (alter Buyukozturk, Baklioum and Beattie 1990).
(Fig. 10.5.6a). The value of acr is decided by stability and bifurcation analysis of lire interacting crack system (Bazant and Ohtsubo 1977; Bazant, Ohtsubo and Aoh 1979; Bazant and Cedolin 1991, Ch. 13; Bazant and Wahab 1979, 1980). The increase of spacing of the opened crack causes their opening width (due to shrinkage or strain) to increase. Although this problem has been analyzed only two-dimensionally so far, the crack pattern viewed orthogonally to the surface of halfspace is often hexagonal or random, calling for three-dimensional analysis.