13th International Conference on Fracture June 16–21, 2013, Beijing, China experimentally observed issues. Firstly, the softening initiates (in the considered case of mode-I loading) if the maximum principal stress in the ligament reaches the value σc. Secondly, Γ0 is the minimum work necessary for the formation of free surface in absence of plastic deformations and corresponds thus to the lower-bound fracture toughness. In finite-element models implementing a cohesive zone, the results are expected to converge with an increasing mesh resolution. In the bulk material embedding the cohesive zone the ductile damage mechanism is described by the Gurson-model [7] in the established modification by Tvergaard und Needleman [8] (GTN-model). However, from this approach (and all other classical damage models) it is known that the results depend strongly on the element size and orientation. But for a consistent numerical implementation of the cohesive zone the stresses around the crack tip need to be resolved very fine. In order to overcome this problem the non-local extension of the GTN-model by Linse et al. [9] is employed for the following simulations. In classical constitutive theories the softening at a material point depends only on the internal state variables at this point. Within a non-local approach a region of finite size around the particular material point is incorporated for the softening. The dimension of this region is associated with a relevant microstructural length of the damage mechanism, for the ductile mechanism thus to the mean distance of the voids. Concretely, in the non-local modification by Linse et al. the flow potential ( ) 2 2 eq * * 2 h 1 1 2 M M 3 2 cosh 1 2 q q f q f s s Φ s s = + ⋅ − − (1) of the original GTN-model is retained. Therein, σeq and σh denote the Mises stress and the hydrostatic stress, σM is the equivalent yield stress of the matrix material and q1 and q2 are the fitting parameters after Tvergaard und Needleman. In the following the established values q1=1.5 and q2=1 are assigned. In the initial stage the effective void volume fraction f* corresponds to the actual value f. In the stage of void coalescence which initiates at f =fc, the effective value f* increases faster than f by the factor K: c * c c c f ( ) f f f f f f f K f f f ≤ = + − < ≤ with u c f c f f K f f − = − , u 1 1 f q = (2) At f = ff (corresponding to f*=fu) the material has lost its load carrying capacity completely. The non-local modification concerns the evolution law of f. In contrast to the classical GTN-model the non-local plastic volumetric strain εnl is used in this law: nl N (1 ) f f f ε = − + (3) The nucleation term fN is not considered in the following for simplicity, but all voids are assumed to be initially present and are thus captured with the initial void volume fraction f0. The non-local plastic volumetric strain εnl is defined in a so-called implicit-gradient formulation by an additional partial differential equation of Helmholtz-type 2 nl nl nl v l ε ε ε + ∆ = (4) wherein the local volumetric plastic strain εv forms the source term. A region around the current material point is incorporated through the Laplace-operator Δ. The intrinsic length scale lnl enters as -3-
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