Faculty of Engineering and Physical Sciences School of Materials, Materials Performance Centre Computational studies of intergranular stress corrosion crack propagation and the role of bridging ligaments AP Jivkov, NPC Stevens and TJ Marrow Introduction Results High resolution X-ray tomographic observations of IGSCC in sensitised austenitic stainless steel provided evidence that non-sensitised special grain boundaries form crack bridging ligaments along the crack surface, Fig. 1. Existing models of intergranular stress corrosion cracking have used percolation-like approach with 2D geometries. These models cannot capture the formation of bridges and do not account for the mechanical crack driving force. The aim with the simulations was to answer the questions: To what extent the bridges formed by RB control crack growth and can cause crack arrest? How does this control vary with the strength and fraction of SB and RB? Experimentally studied microstructures have SB fractions 0.65 ≤ f ≤ 0.75. Results have been obtained for a wider range 0.5 ≤ f ≤ 0.8, and for a series of SB failure strains, esf = 0.1, 0.2, 0.3, 0.4 ey, and RB failure strains, erf = 5, 10, 50 ey. For each choice of these parameters, simulations have been performed for 30 random distributions of GB character and results have been averaged. Fig.3 shows a result of one crack propagation simulation. As demonstrated, crack advance is accompanied by formation of branches and bridges. These collectively create a crack shielding effect and their amounts increase with decreasing f. Figure 5. Development of crack driving force (a) and bridges retarding force (b) with crack extension. Figure 6. Shielding effect of microstructure with fraction f. (a) SB failure strain effect; (b) RB failure strain effect. Figure 1. Tomography data from in-situ stress corrosion cracking experiments (pH 2 tetrathionate solution, sensitised 302 stainless steel). B and C designate bridging ligaments. Clearly, the influence of SB failure strain in the range of f considered is negligible, while the effect of RB failure strain is more pronounced. Fig. 7 shows the contribution of all bridges (a) and yielding bridges (b) to the total shielding effect only for the three RB failure strains considered. The plots demonstrate the importance of erf for developing a crack retarding force in the microstructures of interest. 2D mechanical model A model is developed, based on the following assumptions: The microstructure is represented by a regular tessellation of space into identical hexagonal cells of diameter D. Grain boundaries (GB) belong to either of two classes – resistant (RB) and susceptible (SB) to corrosion. Material is allowed to fail via crack propagation along grain boundaries only. If ey and eu are the yield and ultimate tensile strains, SB fail at strain esf < ey and RB fail at strain ey < erf < eu. Assembly of grains filling the region {-50D ≤ X1 ≤ 50D and 0 ≤ X2 ≤ 50D} is considered (7740 grains and 22850 GB). The assembly is loaded via prescribed displacements parallel to X1 that introduce a homogeneous strain of e∞ = 0.5ey. An initial crack, extending along three grain boundaries, is introduced from the origin of the coordinate system. Figure 3. Crack propagated in a microstructure with f = 0.7. Susceptible boundaries in red, resistant boundaries in blue. Fig. 4 shows the projected length of all bridges (a) and only of yielding bridges (b) as function of crack extension for selected f. Crack extension is plotted in terms of number of grains, while bridges lengths are given as fractions of crack length. The expected upper limit for these fractions in Fig. 4(a) is f, so values above f indicate bridges formed along branches. The results show that the share of yielding bridges in the total length of bridges increases dramatically with f. Figure 7. Contribution of all bridges (a) and yielding bridges (b) in the microstructure shielding effect. Conclusions Numerical implementation The proposed model has the potential to simulate intergranular crack propagation in a realistic manner by including the phenomenon of crack bridging by ductile ligaments. External load magnitude and failure properties of susceptible and resistant boundaries are accounted for. Crack arrest was not observed but a significant degree of crack tip shielding was developed, which would be expected to reduce the crack propagation rate. Shielding is created by crack branches and bridges and increases with increasing fraction of resistant boundaries. The failure properties of susceptible boundaries are found to have negligible effect on crack shielding in the region of f considered. The bridges’ contribution to shielding is strongly dependent on the resistant boundaries’ failure properties. Comparing experimental data on bridges area (size and location of bridging zone behind the crack tip) with simulation results will help identifying the model parameter resistant boundaries failure strain. Crack growth is simulated as a sequence of equilibrium calculations and single-failure events. Fig. 2 illustrates the strategy for crack advance. For each surface boundary the difference, d, between its calculated strain and its failure strain is formed. If there are surface boundaries with d > 0 the one with largest d is failed. If there are no such boundaries, the procedure is repeated for the subsurface boundaries to account for the possibility that the crack bypasses a surface boundary (e.g. from A to B) in the real 3D set. Figure 4. Length of bridges with crack extension, all bridges (a) and yielding bridges only (b). The stress intensity factor, KI, calculated with respect to the tip of the main crack is used for crack driving force (CDF). Stress intensity factor Kbr, due to the bridge stresses measures the bridged crack retarding force (CRF). Fig. 5 shows the development of CDF (a) and CRF (b) as functions of crack extension. In Fig. 5(a) the theoretical CDF for a straight crack without branches and bridges is also plotted. The difference between the theoretical and calculated in the model driving forces represents the total shielding effect of the microstructure, Ksh. The dependence of Ksh on the three model parameters, f, esf and erf is demonstrated in Fig. 6, where the values are taken at selected crack extension a = 15 D. Figure 2. Search for failing boundaries. Crack surface in white, surface GB in red, sub-surface GB in yellow. Acknowledgments: The authors gratefully acknowledge the support of this work by Rolls Royce plc. Contact details: Dr A P Jivkov Tel: +44 (0) 161 306 3556 Fax: +44 (0) 161 306 4865 Email: andrey.jivkov@manchester.ac.uk