Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture

The crack extension resistance and fracture properties are studied in detail for quasi-brittle materials like concrete with a softening traction-separation law by investigating the complete fracture process. The computed samples are the three-point bending notched beams of concrete with different sizes tested by other researchers. The softening traction-separation law which was proposed by Reinhardt et al. based on direct tension tests for normal concrete materials was chosen in the computations. Different distribution shapes of the cohesive force on the fictitious crack zone were considered for the corresponding loading states. The computations were mainly based on the analytic solutions for this problem using Gauss-Chebyshev quadrature to achieve the integration which is singular at the integral boundary. The crack extension resistance curves in terms of stress intensity (K(R)-curves) were determined by combining the crack initiation toughness K(Ic)(ini) that is the inherent toughness of the material needed to resist the crack initiation in the case that is in the lack of an extension of the main crack with the contribution due to the cohesive force along the fictitious crack zone during the complete processes of fracture. The situation of crack propagation can be judged by comparing K(R)-curves of crack extension resistance with the stress intensity factor curves which were calculated using the lengths of the extending crack and the corresponding loads at each loading states, e.g., when the crack extension resistance curve (K(R)-curve) is higher than the stress intensity factor curve, the crack propagation is stable; otherwise, it is unstable. In the computation, the obtained relationship between the crack tip opening displacement CTOD and the amount of crack extension for the complete fracture process is in agreement with the testing results of other researchers.