PSI - Issue 2_B

Jaime Planas et al. / Procedia Structural Integrity 2 (2016) 3676–3683

3677

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J. Planas et al. / Structural Integrity Procedia 00 (2016) 000–000

Fig. 1. Crack patterns and maximum principal stress distribution around a corroding steel bar asymmetrically cast into a concrete prism (Sanz et al., 2013). Dark red correspond to stresses equal to the tensile strength of concrete and dark blue to zero stress. From left to right the pictures correspond to sates with radial expansions of the bar equivalent to, respectively, 4, 7 and 40 microns. In the first state, all the cracks are distributed and the main crack has not been formed. In the middle image the main crack just jumped across the bottom cover of the bar. The last picture shows a complete localization of the main crack and four fully localized secondary cracks;a few zones with barely open, distributed cracks still remain. Although the determination of crack spacing, and, thus, of the mean crack opening, is of essential engineering interest, there is the complementary challenging question of how the crack system evolves from a distribution of barely open, densely distributed cracks to one of widely open, sparsely distributed cracks. Even in the case of a quasibrittle beam subjected to three-point bending, which is known to fail through a single main crack, there is necessarily a zone of di ff use cracking before the main crack becomes dominant (Planas et al., 2003). When no sharp notches or macro-cracks exist in the structure, or, more generally, when, in the elastic range, the gradient of the maximum principal stress in the direction normal to the potential crack planes is small, a transition from distributed to localized cracking will take place. Shrinkage of slab is a limiting case in which such gradient is ideally zero, but similar situations may occur in presence of expansive phenomena, such as corrosion of a rebar, as illustrated in Fig. 1 —a selection of images from the work of Sanz et al. (2013)—. Closely spaced cracks at early stages of aggregate expansion were also reported by Idiart et al. (2012).

Nomenclature B

beam width

D E

beam or slab depth

elastic modulus

f t tensile strength G F fracture energy of a cohesive crack model 2 second characteristic length ch Hillerborg’s characteristic length n unit normal vector to the crack surface P load u horizontal component of displacement vector t cohesive crack traction vector w cohesive crack width in pure opening mode w 1 horizontal intercept in the initial linear approximation of the softening curve (Fig. 2) ˜ w e ff ective cohesive crack opening w cohesive crack displacement vector σ cohesive stress in pure opening mode

σ N nominal structural stress σ S nominal shrinkage stress