13th International Conference on Fracture June 16-21, 2013, Beijing, China 2. Theory 2.1. Cohesive Zone Model Cohesive zone model is frequently used to analyze crack propagation. In this modeling, a cohesive zone is placed between bulk elements as shown in Fig.1. Fracture can be regarded as elements separating along the cohesive surface resisted by cohesive traction[2][3]. As the separation occurs, the traction increases to a maximum and then falls back to 0. The procedure means the element separates completely, and the area under the traction-separation curve corresponds with the effective surface energy that needed for complete separation. Figure 1. Cohesive zone model placed between bulk elements 2.2. Cohesive Zone Law The traction vector T =(Tn, Tt) can be derived from the effective surface energyφ(∆).(E.g.[4][5]) T =− ∂φ(∆) ∂(∆) (1) where ∆=(∆n, ∆t). The components Tn, Tt and ∆n, ∆t are the normal and tangential components of traction vector and relative displacement vector, respectively. The surface energy [4] can be written as: φ(∆n, ∆t) =φn +φn exp(− ∆n δn )([1−r + ∆n δn ] 1−q r −1 −[q+( r −q r −1) ∆n δn ]exp(− ∆2 t δ2 t )) (2) where δn and δt represent the character length that satisfy Tn(δn) = σmax, Tt(δt/√2) = τmax. Stress σmax and τmax represent the maximum of normal traction and tangential traction, respectively. Pameters are defined as q = φt/φn, r = ∆0n/δn, where ∆0n represents the value of ∆n when Tn = 0. φn and φt are the areas under the normal traction-separation curve and the shear traction-separation curve representing the surface energy for complete separation respectively and can be calculated as: φn =σmax exp(1)δn, φt = √exp(1)/2τmaxδt (3) From Eqs.(1) and (2),the normal and shear traction can be obtained: Tn = φn δn exp(− ∆n δn )( ∆n δn exp(− ∆2 n δ2 n ) + 1−q r −1 [1−exp(− ∆2 n δ2 n )][r − ∆n δn ]) (4) -2-
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