Determination of double-K criterion for crack propagation in quasi-brittle fracture, Part II:: Analytical evaluating and practical measuring methods for three-point bending notched beams

In this paper, an attempt is made to determine the double-K fracture parameters K(Ic)(ini) and K(Ic)(un) using three-point bending notched beams. First, based on the knowledge from extensive investigations which showed that the nonlinearity of P-CMOD curve is mainly associated with crack propagation, a linear asymptotic superposition assumption is proposed. Then, the critical effective crack length a(c) is analytically evaluated by inserting the secant compliance c(s) into the formula of LEFM. Furthermore, an analytical result of a fictitious crack with cohesive force in an infinite strip model was obtained. The double-K fracture parameters K(Ic)(ini) and K(Ic)(un) as well the critical crack tip opening displacement CTOD(c) were analytically determined. The experimental evidence showed that the double-K fracture parameters K(Ic)(ini) and K(Ic)(un) are size-independent and can be considered as the fracture parameters to describe cracking initiation and unstable fracture in concrete structures. The testing method required to determine K(Ic)(ini) and K(Ic)(un) is quite simple, without unloading and reloading procedures. So, for performing this test, a closed-loop testing system is not necessary.