Abstract
A mathematical model for the embrittlement of a long elastic-plastic crack by a relatively small, misfitting inclusion is presented. The model makes direct contact with the Dugdale–Bilby–Cottrell–Swinden model as a limiting case. The particular case of an oxide inclusion with a triangular cross-section at the tip of an intergranular crack in the Ni-based superalloy RR1000 at 650,^{circ }hbox {C} is considered. The positive misfit of the intrusion provides an additional tensile load on the crack tip and on the plastic zone, raising the local stress intensity factor k_I and the crack tip opening displacement {varDelta } u above those when the inclusion is replaced by a dislocation-free zone of the same length. It is shown that for a given misfit strain and inclusion shape, the enhancement of k_I and {varDelta } u is controlled by a dimensionless parameter omega = (sigma /sigma _1)sqrt{c/(2l)} where sigma is the applied stress, sigma _1 is the yield stress, c is the crack length and l is the length of the inclusion. The anti-shielding effect of the intrusion is significant only when omega lesssim 6. As a result of the anti-shielding effect of the intrusion, the stress singularity at the crack tip always exceeds the compressive normal stress that exists within the thickest part of the intrusion when it is isolated. It is also shown that the gradient of the hydrostatic stress within the intrusion subjected to different applied stresses drives the oxygen diffusion and, hence, assists the oxidation at the grain boundary. The fracture toughness is considerably greater than that of a bulk sample of the oxide particle, which we attribute to the plastic zone.
Highlights
When a brittle inclusion forms at a loaded crack tip it may enhance the growth of the crack by providing a path of relatively low fracture energy
Crack growth is enhanced through fracture of the oxide intrusion (McMahon and Coffin 1970; Andrieu et al 1992; Cruchley et al 2015a, b; Németh et al 2016, 2017)
In this paper we find the assumed variation in Evans et al (2013) differs significantly from the profile we derive, most notably we find the intrusion adjacent to the crack tip is always loaded in tension for experimentally observed intrusion geometries
Summary
When a brittle inclusion forms at a loaded crack tip it may enhance the growth of the crack by providing a path of relatively low fracture energy. At temperatures around 700 ◦C in an oxidising environment, the mode of fracture changes to intergranular In this case oxygen enters the crack tip, diffuses along a γ /γ or γ /γ interface and reacts to produce an oxide particle called an intrusion (McMahon and Coffin 1970; Andrieu et al 1992; Bache et al 1999; Knowles and Hunt 2002; Li et al 2015; Cruchley et al 2015a, b, 2016; Németh et al 2016, 2017; Hörnqvist et al 2014; Viskari et al 2013; Pineau and Antolovich 2009). The stress field of a loaded intergranular crack has been widely assumed to enhance diffusion of oxygen along the grain boundary This mechanism of failure of Ni-based superalloys at high temperatures has been called stress assisted grain boundary oxidation (SAGBO). It shows how the applied normal stress, the yield stress, the crack length, and the length of the intrusion conspire to control the influence of the intrusion on the embrittlement of the medium
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