13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- estimate the stress-intensity factors (SIFs) taking into account the real geometry of slanted cracks, but not suitable for the simulation of cyclic plasticity and 2) an elastic-plastic analysis carried out in the framework of the classical finite element method with Cast3M software (www-cast3m.cea.fr/), therefore limited to flat tunnelling cracks. The stress-intensity factors issued from the X-FEM computations will be used to analyse the measured crack growth rates and to determine the parameter which best correlates the data. But the SIFs are considered useless for the prediction of crack paths, since none of the bifurcation criteria based on those parameters can predict that a fatigue crack loaded in mode I changes its path to grow in mixed-mode along a shear lip. In the present case, the crack paths are believed to be determined by the stress and strain fields just ahead of the crack front, which must be computed by elastic-plastic cyclic computations. 4.1 The X-FEM computation 4.1. a General framework The extended finite element method, introduced by Moes et al. [10], allows the simulation of complex crack shapes where the structural finite element mesh does not have to conform to the crack surface. The crack surface and its front are defined geometrically by two signed distance functions named “level sets”. In order to take into account the displacement jump due to the presence of the crack and the crack tip singularity the discretized displacement field is “enriched” with discontinuous shape functions. The displacement field is approximated as follows: + + = ∑ ∑ ∑ ∑ = ∈ ∈ ∈ 1,4 , ( ). ( ) ( ) ( ). ( ) ( ) 0 k i k k i I i i I i i i I i i x b u x N x u N x H x a N x H γ γ (1) In which H denotes the Heavyside function: > + − < = 1 0 1 0 ( ) if x if x H x (2) I0 is the set of the standard finite element nodes, IH the set of nodes whose support is completely cut by the crack and Iγ the set of nodes whose support contains the crack front, ai and b i,k, the corresponding additional degrees of freedom. Ni are the standard finite element shape functions, ui the nodal displacements and γk is the base of Westergaard’s solution representing the asymptotic displacement field at the crack tip of a semi-infinite crack in an infinite medium: = ) 2 )cos( sin( ), )sin( 2 sin( ), 2 cos( ), 2 sin( ( ) θ θ θ θ θ θ γ r r r r x k (3) 4.1. b Representation of the crack. One of the ways to generate the level–sets used to represent the crack is to mesh both the crack surface and the crack front. This mesh is used only for the geometrical description of the crack and not for the resolution of the problem. An algorithm was thus developed to turn the measured topographic data, that is: a set of (x, y, z) triplets plus polynomial fits of crack front markings, into a mesh representative of the crack. The first step is to fit a polynomial interpolation to the cloud of points extracted from the topography, using the least square method, in 3D. Then a regular, flat grid of points is deformed, using both the equation of the crack surface z(x, y), and the polynomial fit of its front, x(y) (Fig. 4). This grid is then meshed with quadrangles and the linear elements of the
RkJQdWJsaXNoZXIy MjM0NDE=