PSI - Issue 66

Carl H. Wolf et al. / Procedia Structural Integrity 66 (2024) 26–37 6 Carl H. Wolf, Sebastian Henkel, Christian Düreth, Maik Gude and Horst Biermann / Structural Integrity Procedia 00 (2025) 000–000

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2.3. Virtual measuring clip Information about the crack tip loading can be obtained from the movement of the crack edges in a cracked component under cyclic loading. For this purpose, the method to measure the crack opening displacement  5 presented by Schwalbe et al. [6] was extended [9]. In this procedure, a virtual measuring clip is continuously applied at the crack tip perpendicular to the crack path. Kolossov 's formulas are used to determine the stress intensity factors K . In addition to the measured crack opening displacements  T in crack growth direction and  S perpendicular to the crack growth direction using DIC and the virtual measuring clip, the knowledge of the Young's modulus E , Poisson ratio and the stress state are required to use these formulas to calculate the stress intensity factors or the energy release rates. In this investigation, the gauge length of the virtual measuring clip was set to 4 mm. The measurement method was approved for fatigue crack growth of phase-shift loaded cruciform specimen made of an austenitic steel and was used to explain the reasons of crack branching as well as kinking of the crack paths. An advantage of this measurement principle is that the determination of  K is based on measured values. In order to ensure comparability with other measured values, a conversion can be made from the stress intensity factor  K into the energy release rate  G using Equation (1) [4], so that the energy release rates can be calculated from the Mode I and Mode II loading, assuming isotropy. �� � ⋅ �� (1) To calculate the cyclic Mode I and Mode II stress intensity factors  K I and  K II as well as the Mode I and Mode II energy release rates  G I and  G II , the following steps have to be carried out in analogy to the references [13, 14]: (1) Determination of the range of the displacements measured with the virtual measuring clip and (2) calculation of the stress intensity factors  K I and  K II as well as energy release rates  G I and  G II using Kolosov ’s formulas. For the calculation of the cyclic equivalent stress intensity factor  K eq and the cyclic equivalent energy release rate  G eq , it is necessary that the time-dependent Mode I and Mode II stress intensity factors K I (t) and K II (t) as well as the time dependent energy release rates G I (t) and G II (t) have to be determined in accordance with the calculation of the Mode I and Mode II stress intensity factors, cf. [13, 14]: (1) Normalization of the measured displacements, (2) calculation of the time-dependent values for K I (t), K II (t), G I (t) and G II (t), (3) calculation of the time-dependent values for K eq (t) and G eq (t), (4) determination of the extreme values K eq, min (t), K eq, max (t), G eq, min (t), G eq, max (t), (5) calculation of  K eq and  G eq . The formula proposed by Richard et al. [3] was used to calculate the equivalent stress intensity factor in step (5). To calculate the equivalent energy release rate, this formula has to be extended for orthotropic materials, cf. Equation (2) [15]. eq � �� 0,5 ⋅ I � �� 1,334 ⋅ II � �� 0,5 ⋅� I � � ⋅ I ⋅� I � � I � 5,336 ⋅ II � � II (2) 2.4. Numerical investigation for uniaxially loaded specimens In order to determine the crack tip loading as a function of the crack propagation, FE calculations were carried out with several 2D-plain-strain models in Abaqus . The discrete crack lengths were modeled using a seams crack. A half model with an orthotropic material and a linear-elastic material model was used for the calculation. Up to four J integrals were evaluated at the crack tip up to a maximum distance after estimation of the plastic zone for this orthotropic material according to Irwin [5]. In addition, the friction across the contact surfaces was modeled with a coefficient of friction of µ = 0.15. The conversion between the stress intensity factor and the energy release rate was carried out assuming a Youngs’s modulus of E = 11.5 GPa, i.e. the material anisotropy was not considered. Due to the calculation as a half model, it was assumed that the two cracks occurring at any time have the same crack length