Issue 78

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

N UMERICAL RESULTS AND DISCUSSION

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ne of the key features of this study is the evaluation of coupling between biaxial loading conditions and crack tip shape effects on crack tip stress fields. To this end, a full range of biaxial nominal stress state from equibiaxial tension-compression ( η = -1) up to equibiaxial tension ( η = +1) and two types of crack tip configurations, i.e. the mathematical notch (MN) and the finite radius of curvature (FRC) are considered in our numerical calculations. In the case of the MN, the crack tip radius of curvature was equal to zero (  = 0 ), while for the CCP with the FRC, the radius of curvature was taken as  = 10 -3 a . In this study the J-integral as well as the plastic SIF are used to characterize the elastic– plastic stress fields near the crack tip. Special emphasis in our study is given to behavior of the J-integral and the plastic SIF under small-scale yielding and extensive plasticity. For this purpose, FEM analyses are performed in a wide range of remote nominal applied stress levels 0.1 ≤ ( σ yy / σ 0 ) ≤ 0.4. Here, the remote nominal applied stresses σ yy are normalized by the yield stress σ 0 . Recall that in our study, formulation of the plastic SIF for SSY is given in the form Eq. 14 while for extensive plasticity plastic SIF is calculated by Eq. 22 using the numerical procedure to determination of the J-integral. FEM analysis for biaxially loaded CCP is restricted to plane strain Mode I conditions. For comparative analysis numerical results are presented for different crack tip distances normalized by crack length r r a  . Numerical results in the form of J-integral and plastic SIF distributions as a function of loading biaxiality and applied stress levels will be presented below. Stress fields as a function of crack tip configuration In this section of the present paper numerical results in the form of radial distributions of the elastic-plastic parameters mentioned above ahead of the crack tip are presented. For all parameters considered here numerical results are given for equibiaxial tension-compression ( η = -1), uniaxial tension ( η = 0) and equibiaxial tension ( η = +1).

Figure 2: Stress distributions as a function of dimensionless distance from the crack tip. Fig.2 shows the radial distributions of the dimensionless tangential stresses ahead of the crack tip normalized by the yield stress σ 0 as a function of the biaxial stress ratio. Tangential stresses σ θθ distributions are presented depending on the distance from the crack tip normalized by (J/ σ 0 ) . As expected, due to the stress concentration at the crack tip the stress values for the MN are higher compared to the stresses at the crack tip with the FRC at the same nominal applied stress level. Fig 3. represents the effect of normalized distance ahead of the crack tip on the J-integral distributions. As it follows from these results, for the MN type crack as well as for the crack tip with the FRC, the J-integral distributions have significant gradients in the range of normalized crack tip distance 3 <( r σ 0 /J ) < 12. For the CCP with the MN crack tip the J-integral has higher values under the equibiaxial tension ( η = +1) with respect to the equibiaxial tension-compression ( η = -1). Contrary to that in the case of the FRC crack tip, maximum values of the J-integral is observed for the equibiaxial tension compression ( η = -1).

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