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The results of experimental investigations using laser speckle interferometry on small size three-point bending notched beams and using photoelastic coating and the strain gauges on very large size compact tension specimens of concrete are presented in detail. The investigations showed that there exists a stage of stable crack propagation before unstable fracture occurs. The results are in agreement with other researchers' investigations using moire interferometry, holographic interferometry, dye-impregnation method and microscope. Further detailed study shows that the three different states, i.e., crack initiation, stable crack propagation and unstable fracture can be distinguished in the fracture process in concrete structures. In order to predict the crack propagation during the fracture process in quasi-brittle materials a double-K criterion is proposed. The double-K criterion consists of two size-independent parameters. Both of them are expressed in terms of the stress intensity factors. One of them reflects the initial cracking toughness, denoted with Kcini, which can be directly evaluated by the initial cracking load, Pini, and the precast crack length, a0, using a formula of LEFM. The other one refers to the unstable fracture toughness, denoted with Kcun, which can be obtained inserting the maximum load, Pmax, and the effective crack length, a, into the same formula of LEFM. The values of the two parameters, KIcini and KIcun, obtained from the small size three-point bending notched beams and the large size compact tension specimens show that KIcini and KIcun are size-independent. Evaluating with the KR-resistance curves obtained from the same test data, it is found that the proposed double-K criterion is equivalent to it in basic principle, but, the double-K criterion can be applied more easily than the KR-resistance curve. Finally, as a practical example, the application of the double-K criterion to the prediction of the crack propagation in a concrete dam is discussed.