PSI - Issue 57

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 57 (2024) 271–279

277

K. Ka¨rkka¨inen et al. / Structural Integrity Procedia 00 (2023) 000–000

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Fig. 5: (a) Opening levels and (b) stress intensity factors comparing analyses considering an overload (OL), an underload (UL) and constant amplitude loading (CA).

can be clearly seen to take place from the crack profiles in Fig. 3. In the overload case this is expected, but a sim ilar occurrence is also observed with the underload due to compression and reversed plasticity of the plastic wake. Plasticity-induced closure strengthens heavily after the overload, which is in line with literature (Pommier and Bom pard, 2000; Baptista et al., 2017). It should be also noted that higher opening levels are obtained from the defect edge ( x / r = 0) than crack tip ( x / r ≈ a / r )when a / r ⪆ 0 . 3 in the constant amplitude case. Fig. 5a presents the opening levels of the first node behind the crack tip for all three cases. The Newman plane strain opening level is included for reference (Newman, 1984). A characteristic shape for kinematic hardening in plane strain is observed in the constant amplitude case (de Matos and Nowell, 2007; Antunes et al., 2015). To estimate the e ff ect of the over / underload to fatigue behavior, numerically obtained nominal and e ff ective stress intensity factors, ∆ K nom and ∆ K e ff , are presented in Fig. 5b. ∆ K e ff is derived by accounting for the obtained opening levels, as in Eq. 4 (Ka¨rkka¨inen et al., 2023), and is considered to reflect the crack driving force. An analytical result for ∆ K nom of a penny-shaped crack in an infinite medium, labeled ∆ K penny , is provided for reference (Anderson, 2017). ∆ K e ff = K max (1 − σ op /σ max ) (4) It should be noted that in the beginning of crack propagation ∆ K nom declines slightly; this is not due to plasticity e ff ects included in the nominal curve, as an elastic analysis produces an almost identical result. It is perhaps a product of stricter out-of-plane constraint as the crack tip proceeds away from the cavity and into the plane strain matrix. Although the final crack length is insu ffi cient to verify this, the e ff ect of the overload seems to extend roughly until the edge of the crack tip plastic zone surpasses the overload plastic zone, as proposed by Wheeler (1972). In the overload case, the distinctive sharp increase, followed by a deep decline, is present in the ∆ K e ff curve. This should yield analogous behavior in the crack growth rate, and possibly even crack arrest. Thus, in the simulated case considering plasticity-induced crack closure, an overload produces a response that is qualitatively similar to literature results from multiple di ff erent experimental configurations (Wheeler, 1972; Changqing et al., 1996; Song et al., 2001). As for thee ff ect of underload, the present model does not predict an increase in the crack growth rate that is usually observed in experimental works considering a single underload (Liang et al., 2022a), apart from the initial spike which is also observed with an overload. The behavior appears quite similar to the overload case, but the reduction in ∆ K e ff is not as strong. How the underload a ff ects plasticity-induced closure in this case is not entirely straight-forward, and based on further simulations conducted by the authors, the e ff ect can be separated into two parts. The underload causes reversed plasticity in the entire crack flank, as well as a compressive plastic zone near the initial defect. The reversed plasticity of the wake eases the x − directional material flow into the crack tip (which is actually the mechanism of plasticity-induced crack closure in plane strain, see McClung et al. (1991); Pippan and Hohenwarter (2017)) due to 3.3. Stress intensity factor