PSI - Issue 28

Carlos D.S. Souto et al. / Procedia Structural Integrity 28 (2020) 146–154 Carlos D.S. Souto et al. / Structural Integrity Procedia 00 (2020) 000–000

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mination of Stress Intensity Factors (SIF). With the current computational capabilities, it is possible to evaluate large structures using regular computers (Tavares and de Castro, 2011). In this study, the implementation of two techniques for the numerical determination of SIFs are discussed: (i) the J-integral technique and (ii) the modified Virtual Crack Closure Technique (mVCCT). Both of these techniques were integrated in a finite element software fully developed by the first author, labeled as Omicron. Older, well known techniques also exist, such as the displacement extrapolation method or the force extrapolation method. These extrapolation techniques are well suited to be applied as a stand-alone post-processing procedure, but in order to embed a solution for the numerical determination of SIFs in a software, the J-integral technique or the mVCCT are more robust options, since they are based on energy balance and they do not require special mesh arrangements for node positioning in the crack’s vicinity.

2. Numerical approaches for stress intensity factor determination

2.1. Modified virtual crack closure technique

The modified Virtual Crack Closure Technique (mVCCT) is a procedure used to determine the stress intensity factor based on the calculation of the strain energy release rate (Krueger, 2004). By definition, the strain energy release rate, G , is the decrease in strain energy, U , per increase in fracture surface area, A . Assuming a body with a constant thickness t , A = a · t , and so G can be written as: G = = (1) Alternatively, G is the quotient between the necessary work, E , to virtually close a segment of the crack and the corresponding surface, A , thus: G = − ∆ E ∆ A = − (2) δ U δ A δ U δ a · t ≈ U a − U a +∆ a ∆ a · t

∆ E ∆ a · t

Considering the generic case shown in Figure 1, ∆ E might be calculated as:

1 2 ( X j · ∆ u i + Y j · ∆ v i )

(3)

∆ E ≈

It should be noted that the internal forces X j and Y j are relative to the node j (crack-tip) while the displacements ∆ u i and ∆ v i are relative to the neighboring nodes i and i ∗ . This means that the work ∆ E is an approximation, since the internal forces and displacements are relative to di ff erent, yet very close, nodes. This is a particularity of the modified virtual crack closure technique which assumes that a crack extension of ∆ a does not significantly alter the state at the crack-tip.

Fig. 1: Modified virtual crack closure technique.