Issue34

T.-T.-G. Vo et alii, Frattura ed Integrità Strutturale, 34 (2015) 237-245; DOI: 10.3221/IGF-ESIS.34.25 238 Code_Aster enable modelling of the behaviour of ageing graphite (spatial heterogeneity of the Young’s modulus and internal stresses). In this paper, crack propagation in AGR graphite bricks with ageing properties is studied using the X-FEM. The accuracy of the strain energy release rate computation in a heterogeneous material is evaluated using a finite difference approach. A parametric study for crack propagation, including the influence of different initial crack shapes and propagation criteria, is then conducted. The robustness and the consistency of the propagation are also presented. In Section 2, the implementation of X-FEM model is given. A description of the methodology for crack propagation in ageing graphite bricks is presented in Section 3. The benchmark study, the numerical simulation results for planar and non-planar crack problem on the graphite bricks appear in Section 4 with the discussions. And finally, in Section 5, some conclusions and prospects are provided. C RACK PROPAGATION USING X-FEM Basics of Level Set Method and X-FEM he Level Set Method (LSM) is the numerical scheme developed by Osher and Sethian [1] to track the evolution of interfaces and shapes. In the LSM the interface is represented as the zero level set of a function of one higher dimension. To describe a crack, two level set functions are required [2]. The normal level set (lsn) represents the signed distance to the crack surface and the tangent level set (lst) represents the distance to the crack front. The X-FEM allows modelling the crack growth without having to modify the original mesh. In cracking, the discontinuity of displacement due to the crack is introduced by a generalized Heaviside function and the addition of the asymptotic fields near the crack tip in order to improve the accuracy in elastic fracture mechanics. The formula used for the displacement approximation u h (x) at point x is given in Eq. (1). The first term is the standard one (continuous). The second term brings the discontinuity of displacement for the elements crossed by the crack (Heaviside enrichment) and the third term brings the singularities for the elements around the crack tip (asymptotic enrichment). Fig. 1a shows a representation of a crack with additional degrees of freedom for enriched elements. Note that the crack passes through the elements and that the crack is not explicitly meshed [4]. 4 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ( )) ( ) ( ( ), ( )) n n n h i i j j k k i N x j N x K k N x L u x a x b x H lsn x c x F lsn x lst x                    (1) where a i are the traditional displacement degrees of freedom associated with node i , Φ i is the linear shape function of node i , b j are the enriched degrees of freedom associated with node j , k c  are the enriched degrees of freedom associated with node k . The function H(x) is a Heaviside function, which is discontinuous across the crack surface. Consider an element that is crossed by the crack, H(lsn(x))=1 for the nodes located on the side where lsn(x)>0, whereas H(lsn(x))=-1 for those located on the other side where lsn(x)<0 . Only one layer of elements is enriched. Note that a geometrical (fixed area) enrichment can also be done. The expressions of the function F  , α = 1.4, are the following [3]:   sin , cos , sin sin , cos sin 2 2 2 2 F            (2) The local polar co-ordinates (  ,  ) can be expressed in terms of the level sets at the crack tip as (see Fig. 1b): 2 2 1 ; tan lst lsn lst lsn             (3) The propagation of crack thus consists in updating the level sets and determining the new crack front. The mesh is refined in the area near the crack front (the distance is evaluated by the values of the level sets) to improve the accuracy of the stress intensity factors and the strain energy release rate computation. The flowchart for a Code_Aster model with X- FEM crack propagation is presented in Fig. 1c. T