Issue34

T.-T.-G. Vo et alii, Frattura ed Integrità Strutturale, 34 (2015) 237-245; DOI: 10.3221/IGF-ESIS.34.25 240 the stress state at instant t crack that will be considered as the initial stress state in the crack propagation analysis. These values are extracted and passed into the second step. This step is based on an algorithm where the LSM is coupled with the X-FEM to introduce and represent the crack and perform the crack propagation (see Fig. 2). Step 1 Step 2 Irradiation and oxidation of graphite Crack propagation (XFEM) Introduction of the crack (XFEM) Extraction of: Material properties: E(x,y,z,t crack ) Stress state: σ_ij(x,y,z,t crack ) Non linear behaviour (UMAT) Crack propagates when G=Gc Automatic mesh refinement Figure 2 : Description of the analysis done for crack propagation in ageing AGR graphite bricks. The computation of the energy release rate G is done by using the G-theta method [7]. When G is smaller than G c (the critical strain energy release rate), there is no crack propagation. When G reaches G c , there is propagation. Heterogeneous Young’s modulus in space Regarding the heterogeneity of the Young’s Modulus, developments have been made in Code_Aster in order to enable the calculation despite some theoretical approximations: indeed, the spatial variation of Young’s Modulus in the component is taken into account but there is a missing term, in gradient of E, which is not calculated. In fact, the generic definition of G which is derived from the evolution of the potential energy (E p ) with the increase of the crack surface (A) with propagation is described in Eq. (6). p E G A     (6) Therefore, in a 2D model, by simply calculating the potential energy for a crack of length l and then for a crack of length l+dl (where dl is small compared to l ), it is possible to approximate G with Eq. (7). This value will be called G diff . ( ) ( ) p p diff E l dl E l G dl     (7) The previous results of Martinuzzi and al. [6] had shown that the missing term in the calculation of G appeared to be negligible in the considered case of AGR bricks. R ESULTS AND DISCUSSIONS Benchmark of crack propagation tools on virgin graphite bricks ig. 3 represents the geometry of graphite bricks used in AGR. Depending on the reactors, the dimensions and the geometry may slightly vary. A benchmark was conducted in order to test the ability of four potentially interesting methods – X-FEM with Griffith adapted criteria, Cohesive Zone Models (CZM) [8], Damage modelling (these three first methods are developed in Code_Aster) and Configurational Forces (developed at the University of Glasgow) – to model crack propagation. Fig. 4 shows the results of the crack paths obtained from these four methods on the first and simplest case: a 100 mm slice of F