PSI - Issue 45

James Vidler et al. / Procedia Structural Integrity 45 (2023) 82–87 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 2. Mathematical model: (a) simplified plastic wake geometry; (b) its representation using edge dislocations. This equation can be partially analytically integrated and rewritten as − ( 2+ 1) + ln ( 2 2 − 2 ) + 2 2 − 2 =∫ ( ) − − . (3) The problem can be solved using different methods. However, the simplest approach would be to consider an embedded crack loaded by a pseudo-stress representing the left part of the last equation. Assuming no cohesive forces and smooth contact at the contact points, = ± , leads to vanishing the stress intensity factor, I (± ) = 0. From this condition, the following relationship linking the problem parameters can be obtained: ∫ (− ( 2+ 1) + ln 2 2 − 2 + 2 2 − 2 ) √ + − − = 0. (4) By introducing two dimensionless variables: = / , the ratio of the stretch to the half crack length and = / , the fraction of the crack that is open, Eq. (4) simplifies to: ∫ (ln 2 2 1− 2 2 + 1− 2 2 2 )√ 1+ 1 − = 2 ( 2 + 1) 1 −1 . (5) Analytical integration of Eq. (5) is difficult and would require the evaluation of polylogarithmic functions with lengthy arguments, therefore, the simplest way is to use numerical methods to evaluate the left part of Eq. (5). 3.1. Results Fig.3 shows as a function of the excitation loading a , material properties, and , and the ratio of the plastic stretch to the half crack length, = / . Several important observations can be made: in the absence of loading, i.e. when a = 0, more than half of the fatigue crack is open. Near this point, the half length of the open region, , is approximately a linear function of the applied stress, a . described by the following equation: ≈ 2 ( + 1) 40 +0.55. (6) At larger stresses, Eqs. (5) and (6) may become inaccurate as large applied compressive stresses can significantly change the shape of the wake of plasticity, which was initially assumed to be linear. Similarly, a large positive (tensile) excitation stress can violate another model simplification, as the direct and reverse plasticity regions can affect the crack opening.