13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- N N c N b w w a = = (4) with N equals to number of lines in the softening curve. Fig. 2 shows hinge model geometry, which is described by the half of the angular deformation φ and depth of the neutral axis y0. (ε) (w) σ ft σ y M M N N 2φ y 0 h s crack σ Figure 2. The hinge model and assumed stress distribution. Olesen [11] analytically determined crack opening w(y) for each point y for known stress σ(y). The mean of longitudinal strains, ε*(y) is then calculated as: ( ) ( ) * 0 2 / y y y s ε ϕ = − (5) Then deformation of an incremental strip of the hinge is given by u(y) = sε*(y), where s is the length of the hinge (s = 0.5h). Once crack occurs, u(y) can be computed as the sum of the elastic deformation and the crack opening according Eq. (6). ( ) ( ) ( ) ( ) ( ) * w w y u y s y s w y E σ ε = = + (6) The stress distribution equation can be obtained by combining Eq. (5) and (6) as follows: ( ) ( ) ( ) 0 2 1 i i w i y y E w y s ς ϕ β σ β − − = − (7) Solving Eq. (7) by introducing cohesive law (Eq.1) with respect to w(y) and σw(w(y)) the following solution is observed: ( ) ( ) ( ) 0 2 1 i i w i y y E w y s ς ϕ β σ β − − = − (8) ( ) ( ) 0 2 1 i i y y w y ϕ ς β − − = − (9) where ςi and βi are dimensionless factors: and t i t i i i f a s f bs E E β ς = = (10)
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