Issue 78

A. Zakharov et alii, Fracture and Structural Integrity, 78 (2026) 260-274; DOI: 10.3221/IGF-ESIS.78,16

Within the framework of nonlinear fracture mechanics, the MN is characterized by the Hutchinson–Rice–Rosengren (HRR) singularity [1–3], while the crack with FRC is associated with the blunted crack-tip field [4]. For each of these crack tip shapes, both analytical and numerical solutions exist, with their stress field amplitudes governed by fracture parameters such as the J-integral and the plastic stress intensity factor (plastic SIF). These parameters allow to relate the near-tip fields to the magnitude of the remote applied loading and stress biaxiality. Within the deformation theory of plasticity, the J-integral and the plastic SIF are uniquely related for the MN crack tip configuration. Nevertheless, significant attention from researches was given to the plastic SIF as an alternative parameter in nonlinear fracture mechanics following the pioneering work of Rice [5]. As shown by Hutchinson [2], the asymptotic crack-tip field for a stationary Mode I crack in a power-law hardening elastic–plastic material is governed by the plastic SIF, which serves a function analogous to that of the elastic SIF in linear elastic fracture mechanics (LEFM). Thus, the critical value of the plastic SIF at fracture initiation can be regarded as a measure of elastic–plastic fracture toughness, that can be determined through the combined experimental testing and numerical modeling for a given specimen geometry. Under small-scale yielding regime, a direct relation exists between the J-integral and the elastic SIF. Shih [6] extended the HRR solution to mixed-mode fracture and demonstrated that an additional parameter is needed to characterize the singular stress fields. Later, Shlyannikov and Tumanov [7] revisited the solutions proposed by Hutchinson [2] and Shih [6] for both pure Mode I and mixed-mode loading and developed a generalized analytical method for solving plane mixed-mode elastic– plastic fracture. In this regard, new formulations of the J-integral and the plastic stress intensity factor based on the constitutive equations of strain gradient plasticity presented by the authors [8–15] deserve attention. For the extensive plasticity regime, the evaluation of the J-integral becomes considerably more difficult to evaluate due to its dependence on geometry, loading conditions, and the nonlinear stress-strain behavior of the material. Despite this difficulty, establishing relationships between the J-integral and loading biaxiality remains essential for fracture assessment of engineering structures and test specimens. Hilton and Hutchinson [16], Hilton and Sih [17], and Hilton [18] investigated the behavior of plastic stress and strain intensity factors across the full spectrum of yielding conditions, from small-scale to extensive plasticity. Goldman and Hutchinson [19] later investigated fully plastic crack problems and established approximate functional relationships between the J-integral and the plastic SIF. Further, Hilton [18] and Lee and Liebowitz [20] examined the influence of biaxial loading on these parameters for Mode I cracked plate and reported that biaxiality effects grow in significance with increasing remote stress. More recently, Shlyannikov and co-authors [21–23] confirmed the potential of the plastic SIF as a suitable fracture resistance metric for both monotonic and cyclic loading conditions. All of the studies reviewed above were performed for idealized crack-tip configurations, modeled as MN with a zero-root radius of curvature. However, systematic computational data that address the combined influence of finite crack-tip radius and remote biaxial loading on near-tip stress fields and fracture resistance parameters remain scarce. Experimental data of fatigue crack growth provided by Brown and Miller [24], Tanaka [25], Dalle Donne and Döker [26], Shanyavsky et al. [27,28] and Shlyannikov et al. [29] clearly indicate that such interactions can significantly affect crack propagation behavior. This motivates the present numerical investigation, which aims to investigate the coupled effects of loading biaxiality and crack tip configuration using the J-integral and the plastic SIF as fracture characterization parameters. In the present work, an elastic–plastic FEA was performed for a plane-strain CCP subjected to Mode I biaxial tension/compression loading. Two crack-tip shapes were investigated: the mathematical notch (MN) and the crack with finite radius of curvature (FRC). The J-integral and the plastic SIF were evaluated as functions of loading biaxiality and applied stress level across the entire spectrum of biaxial loading conditions, from equibiaxial tension–compression to equibiaxial tension, covering both small-scale yielding and extensive plasticity regimes. Particular emphasis was devoted to the contrasting behaviour of the two fracture parameters with respect to the different crack-tip geometries. Finally, the suitability of the plastic SIF approach for fracture characterization under extensive plastic deformation was assessed.

T HE RELATIONSHIP OF THE J- INTEGRAL TO THE PLASTIC STRESS INTENSITY FACTOR

I

The material behavior constitutive equations n the present study, to represent the material behaviour in the singular crack-tip fields, the Ramberg–Osgood constitutive model is used:

            0 0 n E E 

(1)

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