PSI - Issue 54

Francisco Q. de Melo et al. / Procedia Structural Integrity 54 (2024) 585–592 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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the original crack profile. Each blade behaves as a spring having coupled membrane and bending flexibility, either for remote tensile loads or bending moments, being this method given the name of the Line Spring Model (LSM) by the authors. The structural model of a part-through crack is a 3D problem that, if addressed with finite element techniques, demands high-order multimode 3D finite elements, with considerable use of computation power for accurate results. At the time LSM was developed, the target was centered on the reduction of the complexity of numerical solutions with this 3D problem. It was indispensable to obtain flexibility or stiffness factors for a model of a small thickness plate, the line-spring element, with a side edge crack and subjected to tensile or bending loads at its ends. Their displacements and rotation are referred to as the relative opening and rotation of the crack mating faces. The flexibility factors of the line-spring elements obtained a valuable contribution by studies due to J. C. Newman and I. S. Raju (Newman Jr and Raju 1981), who developed accurate expressions for the stress-intensity factors of embedded elliptical cracks, semi-elliptical surface cracks, quarter-elliptical corner cracks and semi-elliptical surface cracks at a hole. For the purpose described, a 3D finite element analysis was performed, which produced extensive graphical data and practical expressions for the calculation of the SIF of part-through or embedded elliptical cracks existing in flat plates under uniform tensile load.

Figure 1: The line-spring as a substitution of the crack remaining ligament of a plate or shell.

1.2. Progress of Line-Spring by linear and non-linear approaches Since the initial contribution by Rice and Levy (Rice and Levy 1972) followed by 3D structure modelling by (Newman Jr and Raju 1981; Newman 1984), a considerable number of researchers maintained the upgrading of the line-spring technique, thanks to the development of more mathematical tools and computational potentialities. (Delale and Erdogan 1981), recovered the first solution by Rice and Levy, having repeated the study with a Reissner/Mindlin plate deformation model, which was physically more realistic than the Kirchhoff deformation model. These authors presented the flexibility terms of the cracked blade line spring via polynomial expression, now with 12 terms instead 8, as in Rice and Levy’s solution. (Langre et al. 1987) developed a line-spring element as an adjacent line connected to conventional multi-nodal plate or shell elements. Based on the same technique, independently of these authors, (Oliveira, de Melo, and de Castro 1991) developed a similar model for part-through cracks existing in plates or cylindrical shells, deforming according to the Reissner’s model, using as main working reference that of (Delale and Erdogan 1981). The evaluation of the SIF included Mode I and II over the crack plane. An alternative method to Finite Element Techniques, but still working with Line-Spring in the approach of the structural behavior of a part-through crack, (Zeng, Dai, and piping 1993) used