13th International Conference on Fracture June 16–21, 2013, Beijing, China by damage models like those after Gurson or Rousselier or modifications thereof. With such models all stages of crack initiation and propagation can be simulated. In contrast, cleavage is typically associated with unstable crack propagation. For this reason a possible cleavage initiation is usually evaluated by stress-based criteria like those after Ritchie-Knott-Rice (RKR) [2] or Beremin [3]. Especially the Beremin model and its numerous modifications enjoy great popularity as those models are probabilistic and the prediction of failure probabilities corresponds to the experimentally observed large scatter of measured fracture toughness values. The failure probabilities rely on the so-called weakest-link theory where the failure of an element is assumed to coincide with the failure of the complete structure. Comparisons with experiments show that models of Beremin-type allow realistic predictions in the lower ductile-brittle transition region. However, problems arise in the upper ductile-brittle transition region even if the ductile crack propagation is incorporated as sketched in Fig. 2. Furthermore, the weakest-link theory predicts a vanishing survival probability of very large specimens at arbitrary small loadings. Anderson, Stienstra and Dodds [4] argue that this prediction is physically questionable since some work is always necessary to separate the crystallographic planes. Experiments confirm the existence of a corresponding lower-bound fracture toughness, see e.g. [5]. The problem of models of Beremin-type is the identification of cleavage initiation with immediate unstable crack propagation since the stability of crack propagation is in general a matter of the interaction of material properties, specimen type, specimen size and loading. In the present study cleavage is modeled by a cohesive zone and ductile damage is described by a non-local Gurson-model. With this modeling of both failure mechanisms the fracture initiation and propagation can be simulated in the whole brittle transition region. The stability is an outcome of the simulations. 2. Model Concretely, cleavage is modeled by a so-called cohesive zone. In a model of this kind it is assumed that the normal stress σ(x) transmitted at each point x in the process zone depends on the local separation δ(x) of the crack edges as sketched in Fig. 3. The particular relation between σ and δ is described by the cohesive law. In the present study the exponential law after Xu and Needleman [6] is employed. The maximum transmittable stress σc is termed as cohesive strength. The area Γ0 is the work of separation and corresponds to the work which is necessary for the formation of new surfaces. Correspondingly, Γ0 has the unit of work per area. Modeling cleavage by means of a cohesive zone captures phenomenologically two essential Fig. 3: Cohesive zone at the crack tip Fig. 4: Exponential cohesive law -2-
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