13th International Conference on Fracture June 16–21, 2013, Beijing, China 4 ( ) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ Δ≤Δ < Δ Δ+ − Δ −Δ ≤Δ≤Δ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + Δ =Δ u f u f u f f n c c c T T T σ σ κ σ 0 , (4) where Δc is the onset of the displacement jump associated with initiation of the cohesive zone hardening, σc refers to the initial cohesive strength of the process zone corresponding to Δc and Δu is the final displacement jump which mirrors the complete loss of stress carrying capacity when the traction equals to zero. κ is a parameter which has the same unit as the displacement jump and n is the hardening exponent. Δf is the value of displacement jump after which the initiation of damage begins and the corresponding cohesive strength is σf , the effects of these parameters are shown in Figure 3. For the application of Eq. (4), the inequality must be satisfied that ), ( ) (1 D T C D f − = ≤ σ (5) where D is a scalar variable representing the overall damage in materials and C is identified with the cohesion as the function of the damage. The monotonic cohesive law is insufficient and the cyclic damage should be taken into account in the constitutive description. Should the cohesive model be applied for both fracture and fatigue crack growth, the material damage depends on both monotonic loading and loading cycles, i.e. dD dt = D= Ds + Dc (6) with Ds for damage under monotonic loading and Dc for damage under cyclic loading. Obviously, both damage variables have to be expressed in evolution equations. Under the monotonic loading condition, Eq. (4) denotes inelastic behavior of the material in the cohesive zone and Eq. (5) implies that the degradation of the material strength for Δ < Δf. For Δ>Δf , the damage grows and assumes to be expressed in a linear function of the displacement increment [17], DS = Δ−Δf Δu −Δf (7) under monotonic loading condition. For fatigue, the material should be damaged even under the cohesive strength, σf. The damage evolution equation proposed by Ural et al. [17] is used to characterize fatigue process reads Dc = αDc(T − βC) Δ T − βC>0, Δ>0 γDc(T − βC) Δ T − βC<0, Δ<0 0 (T − βC) Δ<0 λ T =C, Δ>0 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ , (8) where α and γ are two difference material constants used to conduct the damage accumulation during the reloading and unloading under the given conditions, respectively. The parameter β is adopted to denote the fatigue threshold in conjunction with the current cohesion C. The rate form of λ is a free variable is defined to represent the damage growth rate at a random stress state associated with the second equation of Eq. (4). Following the suggestion in [17], the unloading and reloading paths of the present study can be
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