PSI - Issue 72

Thomas Steffen Methfessel et al. / Procedia Structural Integrity 72 (2025) 105–112

110

̅ I = 2 1 ∫ ( ( ) − ( ) ) 0 , ̅ II = 2 1 ∫ ( ( ) − ( ) ) 0

(11)

In order to determine the effective strength or failure critical force F c (or failure critical temperature load  T c ) the lowest load fulfilling the coupled criterion has to be determined. This can be formulated as the following optimization problem: F c = min{ F subject to f(σ)≥1 along Δ a and ̅ ≥ G c }. (12) Herein in addition to the critical load magnitude also the finite length Δ a of the crack is unknown but is obtained at the end of the optimization process. For the calculation of the incremental energy release rate the crack opening is needed for a potential debonding which has to be included into the sandwich-type model. This can be done by a subdivision of the adhesive joint into a debonded and an intact part as it is sketched in Fig. 6 taking into account crack opening displacements and continuity conditions of the deformation functions and the related dynamic cross-sectional quantities.

Fig. 6. Subdivision of the adhesive joint into cracked and uncracked parts.

By a respective extension this debonding scenario has been included into the existent MATLAB code so that this is also covered by the sandwich-type model. For the case of the already considered single-lap joint under a given load of 8kN and with a debonding length of 0.5mm Fig. 7 shows the crack opening displacements in comparison with respective finite element calculations. For the same joint Fig. 8 shows the resultant incremental energy release rates in dependence of the crack length. Obviously, in both outcomes there is a pretty good agreement. This is remarkable insofar, as the structural behavior around the crack tip is very complex and can only be taken into account in an approximate manner by the described closed-form analytical approach.

Fig. 7. Comparison of crack opening displacements.