13th International Conference on Fracture June 16–21, 2013, Beijing, China the weighting prefactor of the Laplace-operator. Regarding the boundary conditions necessary in addition to the Helmholtz-equation the reader is referred to [9,10]. A one-parametric power-law is implemented for the matrix yield stress σM in such a way that in a uniaxial tension test with the compact material f=0 a relation y y N E ε s s s = (5) between true stress σ and logarithmic strain ε is obtained. Therein, E denotes Young’s modulus and σy and N are the initial yield stress and the hardening exponent, respectively. The following simulations are performed with σy/E=0.003 and N=0.1 and a Poisson ratio of ν=0.3 corresponding to typical medium strength engineering alloys (e.g. E=210 GPa, σy=500…700 MPa). The initial void volume fraction is assumed to be f0=0.01. The work of separation is taken as Γ0=1/3 σylnl which is realistic for the material which will be considered finally for a comparison between simulations and experiments. Due to the non-local formulation the boundary-value problem is well-posed also in the case of ductile softening. Thus, the combined model of cohesive zone and non-local GTN-model leads to a mesh-independent solution in the complete temperature range. Correspondingly, the local field quantities in the process zone are determined only through the material parameters and the loading. In particular, the size of the process zone is defined by the material parameters of unit length. For ductile failure this is the non-local length lnl whereas for cleavage the size of the process zone is related to the characteristic cohesive separation δ0 which is necessary to reach the cohesive strength σc. For the employed cohesive law this value amounts to δ0=Γ0/(σcexp(1)), compare Fig. 4. For the simulation of the crack propagation by means of the finite element method (FEM) the fields in the process zone need to be resolved sufficiently fine. For realistic values, δ0 is always smaller than lnl so that the maximum feasible element size is determined by δ0. According to a performed convergence study elements of edge length be=1.2 δ0…1.8 δ0 are placed at the ligament. A typical mesh at the crack tip is depicted in Fig. 5. Only a half-model needs to be meshed for symmetry reasons. The cohesive elements do not have a geometric thickness and are thus only sketched in the figure. Fig. 5: FE-mesh at the crack tip Fig. 6: FE-mesh for 3PB specimen -4-
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