PSI - Issue 28

Alla V. Balueva et al. / Procedia Structural Integrity 28 (2020) 873–885 Author name / Structural Integrity Procedia 00 (2019) 000–000

876

4

found to absorb hydrogen at different rates. The authors highlighted a change from intergranular to transgranular cracking as the hydrogen concentration increased. Following a similar vein, researchers tested the effects of HE in a polycrystallographic nickle superalloy (Jothi et al., 2015). Hydrogen was most readily trapped at grain boundaries and junctions under tensile stress, according to their observations. Another research cohort showed that engineering the crystollographic structure could make a more HE-resistant metal (Masoumi et al., 2017). Their testing also revealed that the surface area of the grain size and the angle boundaries of the grain greatly affected how susceptible the solid was to HE. Small grain with large surface areas, as well as grains with relatively high angle boundaries trapped more hydrogen, increasing the effect of HE in those places. Several researchers made an important contribution to the theory of HE by recognizing the combined effects of brittle fracture under hydrogen-induced internal stress (HEDE) and ductility loss also caused by present hydrogen (HELP). As a result of their experiment, they also noted that under lower loads hydrogen undergoes a redistribution through trapping sites that increases its overall availability. They also noted that in steels where more hydrogen was diffused into the lattice and less into trapping sites, the effect of HE was not as strong. Finally, a very important observation was that the addition of specific precipitates decreased the general effects of HE significantly. 1.3. Modeling Techniques and Analysis Much research has been conducted qualitatively. Some recent papers describe a mathematical model that is compared to a physical specimen for accuracy. A few equations and physical laws governing some of the sub mechanism of HE are also becoming generally accepted. One team of researchers sought to identify the cause of two types of HE failures under the HELP mechanism (Yu et al., 2018). They constructed a number of unit cells with a micro-structural void, modeling different types of metals. They also employed an unusual softening function, opting for a sigmoidal function with the hydrogen concentration as the independent variable. This is as opposed to the typical linear function describing the softening effect. By doing this, the researchers were able to pick up the development of slip bands at very low stress triviality. 1.4. Purpose of the Present Work The work presented in this paper proceeds directly from that conducted previously (Balueva and Dashevskiy, 1999; Balueva, 2008). In the former work, an analytical model of crack growth based on the ideal gas equation was presented for both delamination and internal crack cases. Under the given framework, it was shown that crack growth rates stabilize linearly in both cases. Herein, a new model is proposed for gas under extreme pressures, where a real gas equation is taken into account. In this case, analytical calculations were much more difficult than for ideal gas equation. We present a model derived from the real gas equation and compare it to the former models. 1. Model of Crack Growth under Big Pressures We consider slow crack growth driven by hydrogen diffusion from the media with hydrogen concentration at infinity c 0 (Figure 2). The initial penny-shaped crack radius is a 0. The variable r represents the radial component of the cylindrical coordinates ( r , θ , z ). Gas diffuses into the crack void, and once the critical conditions are met, or when gas pressure becomes sufficiently large, the crack will begin to expand, and will reach a new radius, a ( t ). Once expanded, the crack will re-stabilize at its new radius, and the process will repeat. It is critical to note that the typical growth of Hydrogen Induced Cracks is extremely slow with respect to the time necessary to establish the equilibrium inside the crack, which gives us a good reason to consider the problem in quasi-stationary approximation, and solve the stationary diffusion problem for each moment of time.