Modulus of Rupture: Size Effect due to Fracture Initiation in Boundary Layer

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Abstract
The modulus of rupture of concrete, which characterizes the bending strength of unreinforced beams, is known to depend on the beam size. Because there is no large stable growth of a crack before the maximum load is reached, this size effect, unlike that in many other types of failure of concrete structures, cannot be explained by energy release due to fracture. Rather, this size effect must be explained by the fact that distributed microcracking and slips with strain softening take place in the boundary layer of the beam before the maximum load is reached. The beam is considered to fail before any macroscopic cracks are formed. A simple formula describing the size effect is derived. Asymptotic analysis of the strain softening in the boundary layer shows that the excess of the modulus of rupture over the direct tensile strength is inversely proportional to the beam depth and proportional to the thickness of the boundary layer, which itself is approximately proportional to the maximum aggregate size. The proposed formula agrees with the existing experimental data quite well. The formula is further generalized to describe the effect of the gradient of normal strains near the concrete surface. Finally, it is shown that approximate analysis of the size effect by linear elastic fracture mechanics yields similar formulas. Those formulas, however, have some questionable features; for example, they indicate the size effect magnitude depends on the span-to-depth ratio of the beam, which has not been observed in experiments.