PSI - Issue 33

Domenico Ammendolea et al. / Procedia Structural Integrity 33 (2021) 858–870 Domenico Ammendolea et al./ Structural Integrity Procedia 00 (2019) 000 – 000

865

Notice that, the proposed strategy re-meshes the computational domain only when certain finite elements reach larger distortions, thereby saving notable computational resources. This makes the proposed strategy more competitive than traditional FEM approaches, which re-mesh for each crack advance, thus increasing the computational time and the possibility of non-convergence issues. Indeed, for each re-mesh event, the solution is projected into a novel mesh frame, thus increasing the insurance of convergence issues. The procedure continues to stretch the short segment until that the angle variation regarding the pre-crack reaches a tolerance value (Toll.  ). When  =Toll.  , the propagation process stops, and the procedure updates the geometry of the model by adopting the configuration achieved in the last step of the analysis. The new geometry represents a new starting point of the process. As a result, a new stretching segment is formed. Then, the new geometry is re meshed and boundary conditions are imposed. In particular, the boundary conditions correspond to the solution of the problem achieved by the solver during the last step of the analysis ( i.e. , the last step when  =Toll.  ). Starting from this new condition, the code re-starts the analysis and continues up non-convergence conditions are met, such as those corresponding to the collapse mechanisms. 4 Results This section reports numerical results aimed at checking the reliability and accuracy of the proposed method. Specifically, two cases of study are investigated, and the results are compared with data reported in the literature. For both cases, the geometries are discretized using 6-node triangular elements under plane strain conditions. In addition, it is assumed that materials behave as linear elastic. Simulations are performed by means of a step-by-step incremental static analysis consistent with a displacement-based approach (Lonetti and Pascuzzo (2016)). This strategy consists of evaluating the increment of the external actions to gain the prescribed displacement increment of a selected control point. The propagation process moves mesh points according to fracture criteria conditions. This process is managed by the ALE formulation presented in Section 2.2. The M -integral method serves to extract fracture variables at the crack front. These are mandatory to identify crack onset conditions and the direction of propagation. To this end, the maximum circumferential stress criterion proposed by Erdogan and Sih (Erdogan and Sih (1963)) is selected as fracture criteria. Fig. 4-a shows a rectangular plate of length 2L=200 mm and a height 2H=100 mm, having a center crack of length 2 a inclined of  =30° about the horizontal. The Young ’s Modulus, the Poisson’s ratio, and the coefficient of thermal expansion are equal to E= 218400 Pa,  =0.3, and  = 1.67e-5 1/°C, respectively. Externally, the bottom corners are constrained. In particular, the bottom-left corner is fixed, while the bottom-right can move horizontally only. The plate is subjected to a thermal gradient, induced by temperature T=+10°C and T=-10°C on the top and bottom boundaries. The handbook provides SIFs for different values of crack length (2 a ) and slope (  ). The present study has been developed increasing the crack length from a/L = 0.2 to a/L =0.6. Specifically, the central crack is extended by means of the moving mesh, and meanwhile, the M -integral extracts the SIFs during the motion. Note that, the fundamental handbook solutions have been used by many other researchers to validate their method for extracting SIFs at the crack front (see for instance (Chen et al. (2016), Duflot (2008), Wang (2015), Wang and Zhang (2011))). Fig. 4-b depicts the meshes adopted in numerical simulations. Two meshes arrangements are used, i.e ., a coarse, and a refined one. The coarse mesh comprises 798 triangular elements finely arranged around the crack tip and rough elsewhere. In particular, the short segment that stretches during the propagation has an initial length of 10 mm and it has been subdivided into 4 elements of variable size. The refined frame involves 3922 triangular elements, in which the stretching segment has 8 segments. Fig. 5-a compares the results relative to the meshes of Fig. 4-b with the fundamental solutions reported in (Murakami and Aoki (1987)). Notice that Fig. 5-a presents the values of the SIFs in a dimensionless form, i.e., ( ) ( ) ( ) 0 ( ) 2 I II I II F F K K T T L H E L    =  −  , being T 0 the referential temperature. 4.1 A validation case: a rectangular plate with a slant center crack under a thermal gradient The first case of study aims to assess the accuracy of the proposed scheme in SIFs extraction during the motion of the crack front. The study has been conducted using a benchmark case for which SIFs are available in the “Stress Intensity Factors Handbook” of Murakami and Aoki (Murakami and Aoki (1987)).