13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 2 ε ε ε − ∇ = p p p c , where the local hydrostatic part of the plastic strain rate pε acts as a source term. The constant c can be regarded as an internal length parameter that controls the influence of the surrounding material. To complete the implicit gradient formulation, homogeneous Neumann boundary conditions are chosen at all boundaries to solve the averaging equation. Details of the non-local damage model and its numerical implementation into the commercial FEM software ABAQUS can be found in [16]. 2.2. Modified Beremin-Model In the transition region both ductile damage and brittle fracture occur. Therefore, material models are needed that represent both micromechanical processes independently. The combined use of a non-local GTN-model together with the Beremin model for the calculation of the probability of cleavage fracture is conceivable in principle. However, the problems that accompany the original Beremin are getting worse when ductile damage is considered, i.e. dependence of the Weibull parameters with regard to sample shape, temperature and strain rate, difficulties in the iterative determination of the Weibull parameters, and large variations in the calculated probabilities of cleavage fracture. Bordet et al [17] discussed that the above problems are a result of a oversimplified description of local cleavage in the Beremin model. Already in [7], a proposal for the consideration of the influence of plastic strain is made. Further modifications of the Beremin model, mainly concerning the calculation of the Weibull stress can be found in [18]-[23]. In this paper, we apply a modification proposed by Bernauer et al [24]. This modification takes into account that the nucleation of voids is promoted by the presence of carbide particles, either by the detaching of the surrounding matrix or the breaking of the particles. Both lead to the formation of cavities, whereby the initiation of cleavage fracture is no longer possible at this point. Therefore, the number of active micro-cracks is variable in the modification proposed by Bernauer. The modified Weibull stress is calculated numerically as ( ) { } 1 2 1 0 max τ σ σ σ ≤ = = ⋅ + ∑ pl N B i m W B B t i V t V with ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 1 τ τ σ σ τ σ σ ≤ ≤ ∆ − = ⋅ − ⋅ ∆ = ∑ ∑ j j m i n i j N B I n t m i i n j j I N B t n c t f f c t t f f . As in the original model, the probability of cleavage fracture is determined according to ( ) ( ) 1 exp σ σ = − − m W f u L P L . In conjunction with the non-local ductile damage model, the nucleation of voids and the associated reduction of cleavage initiation points can be considered.
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