13th International Conference on Fracture June 16–21, 2013, Beijing, China 3 The tests were performed in air at room temperature with sine waveform at a frequency of 8 Hz. The fatigue crack growth rate was measured by using the compliance method and also creating beachmark to record the crack front as displayed in Figure 2. 3. Cohesive zone model 3.1. Kinematics and constitutive relation under quasi-static and cyclic loading Taking the principle of virtual work of the boundary value problem into account, the weak form of the equilibrium equation without body forces can be written as : d d d V S V V S s δ δ δ • • ∂ ′ + = ∫ ∫ ∫ σ ε T Δ T u (1) with σ as the Cauchy stress tensor, T´ denotes the external traction vector and <δε , δΔ ,δu> are the admissible displacement fields. The cohesive traction T is taken to have the following form, ), ( 2 s n T s n n s n s T T T Δ+ Δ Δ = + = ξ (2) where Ts and Tn are the cohesive traction components in shear and normal direction, respectively. Following the Ortiz et al. [1] and Ural et al. [17], the effective displacement jump Δ is a scalar parameter defined as , 2 2 2 n s Δ+ Δ =Δ ξ (3) where Δs=| Δs | and Δn are the displacement jumps in shear and normal directions and ξ weights the shear displacement jump over normal one. T is a scalar effective traction relates to the effective displacement jump Δ in the process zone where a crack may arise. The formulation of T used in the present study refers to a coupled damage model defined as Figure 3. Illustration of the cohesive zone model under monotonic loading.
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