PSI - Issue 39

Daniele Amato et al. / Procedia Structural Integrity 39 (2022) 582–598 Author name / Structural Integrity Procedia 00 (2019) 000–000

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components, it is strictly necessary to prevent their uncontained failure which may cause dramatic accidents. Due to its high kinetic energy, blisk failure represents a severe hazard for an airplane and its passengers; in fact, a crack initiated in the blade and subjected to centrifugal loading may propagate towards the disk and cause its breakage [1, 2]. Therefore, in such cases, crack propagation evaluations are required both in terms of Crack Growth Rate (CGR) and crack propagation direction, i.e., deflection angle. During a mission, intended as the time lapse during which a mechanical component experiences a certain spectrum load to be repeated periodically, a potential crack undergoes different mixed-mode states which depend upon the loading conditions encountered by the component itself. From a numerical point of view, a mission is discretized in several loading steps and for each of them a fracture analysis simulation is performed. According to Linear Elastic Fracture Mechanics (LEFM), each stress state separately results in a combination of the Stress Intensity Factors (SIFs) of all three fracture modes (I, II, III). For each sub-step in which the mission is divided up, a different stress state is configured with a corresponding crack deflection angle. Therefore, the problem is to individuate a way to go for the calculation of the deflection angle relatively to the entire mission cycle. During the years, researchers have attempted to validate criteria for the prediction of the crack growth propagation direction under proportional mixed mode loadings. In the case of plane loadings, the earliest criterion was postulated by Erdogan and Sih [2] in 1960s, which states that a mixed mode loaded crack extends towards the direction of the maximum tangential tensile stress ahead of the crack tip; in the same context, the Maximum Shear Stress criterion (MSS) [4,5] and other stress or energy-based fracture criteria should be considered. In the context of general three dimensional mixed-mode loading, Schöllmann et al. [6] developed the so called 1 ′ -criterion based on the MTS theory; Richard et al. [7] elaborated their empirical criterion, which excels because of its adaptability to different experimental results; Dhondt [8] developed his approach based on the principal planes of the asymptotic stress tensor at the crack tip. Theoretically, the criteria developed for the monotonic loading can also be applied to the cases of cracks undergoing cyclic proportional loads [9-12]. According to Bold et al. [13], this is due to the maximum mechanical parameter ranges (stress or strain) under proportional loading which are in direct proportion to the maximum value of these parameters in the crack tip region. This principle is no longer suitable in the case of non-proportional mixed-mode loadings. In fact, by observing the curve described by the tension vector on a fixed plane, it is a straight line in the case of proportional loads, whereas it is a generic closed curve, in the case of a non-proportional periodic loads. For the latter reason, a non-proportional time-varying stress path cannot be described by means of a single parameter. Complex loading conditions and non-proportional loads are very common circumstances in many engineering applications; for example, [1,14-16] have studied the case of a static load superimposed to a cyclic one; Sander and Richard [17,18] have studied the non-proportional mixed-mode loading condition generated by a Mode I base stress interspersed with mixed-mode overloads. Despite the large number of practical cases interested by non- proportional loading conditions, no unified criterion has been yet developed, also because the mixed-mode ratios vary continuously during the crack advancement. In the literature, two main ways of determining the crack propagation deflection angle are proposed in the case of a non-proportional loading condition. Both require the mission subdivision in several loading steps. The first proposes to combine all the angles related to each step in a weighted average having either the Equivalent SIF (ESIF), , or the crack growth rate, / , as weight. An alternative is the dominant step criterion: the mission deflection angle corresponds to the deflection angle of the most critical stress state, i.e., the step having the highest or CGR. Both the weighted average and the dominant step can use either the equivalent SIF or the CGR as main parameter for the mission deflection angle determination. The difference lies in the fact that by choosing / , also the temperature dependent material parameters are involved (e.g., by using a Paris-type crack propagation law), and a temperature dependent propagation may be modelled. This possibility results to be very powerful especially in modelling those cases in which thermal and mechanical dynamics are decoupled. An application of the aforementioned approach is available in [19]. In the present paper, the fatigue crack growth of a SEN-TC specimen, under non-proportional mixed-mode loadings [20-24], is numerically examined and compared with experimental results. The aim is to validate a numerical procedure to model those complex loading situations, which are very common in engineering applications. This paper focuses on the analysis of the general case of non-proportional, phase-shifted and mixed-mode loading. Beyond the phase shifting influence, also the mode-mixity and the stress ratio are considered as factors affecting the