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Fatigue defect tolerant assessment of hydraulic cylinders
2004
Hydraulics cylinders are mechanical components subjected to fatigue. Indeed, to assess their fatigue strength is essential to consider the presence of inhomogeneities, for example inclusions or surface defects, since inhomogeneities reduce the fatigue strength. In order to address this problem, in the present paper the defects in a stress relieved steel are analysed by polishing sections and the largest defect is estimated by means of the statistics of extreme values, both for surface and internal defects. Besides, the so-called Kitagawa diagram is obtained through fatigue tests performed onto smooth and micro-notched specimens. This diagram gives the information about the detrimental effect of the defects. Then, both of these two information, Kitagawa diagram and maximum inclusions present in the material, are used in order to obtain the fatigue limit. Moreover, low cycles fatigue properties and cyclic properties of the material are analysed and a simplified model is proposed for estimation the S-N diagram for a notched-cylinder. In both cases, fatigue limit and fatigue life predictions show a good agreement with full scale evidences. INTRODUCTION The existence of crack-like flaws cannot be excluded in pressure vessels and piping. Also, as it will be shown in next section, inclusions or surface microfolds can act as small cracks when are subjected to fatigue loads. Thus, in order to provide service in a safe condition, it is important to perform fracture mechanics assessment. There are several papers dealing with fatigue crack growth for cylindrical vessels subjected to internal pressure starting from external or internal defects. Lin and Smith [1] have proposed a numerical analysis of fatigue growth of external surface cracks, using a multiple degrees-offreedom method, allowing to the crack to not have only a semi-elliptical shape, which is the main hypothesis of the method used by the ASME XI [2] code. Shlyannikov, in turn, has analyzed, using a fracture damage zone size model, the crack propagation of internal surface flows [3]. In his model, the flaw is considered with two degree of freedom, being of semi-elliptical shape. Koh and coworkers had analyzed the crack propagation from cracks starting at the external surface, due to the presence of grooves, in the case of autofrettaged thick walled vessels [4-5], using a two dimensional model. In last years, a lot of effort has been applied also to assess the conditions of leak before break (LBB): for example, Bergmand and Brickstad had analyzed the conditions of LBB for an initial circumferential surface crack, considering that the crack growth mechanisms could be either driven by fatigue or by stress corrosion [6]. Also Burande and Sethuraman had provided a criterion for LBB [7]. Regarding circumferential surface flaw growth, Carpintieri and Brighenti have proposed a threemodel parameter to describe the fatigue growth behavior [8]. Another important issue, the growth of multiple surface cracks, was modeled by Lin and Smith [9], between others. In the present paper, we will focus onto the relation between inhomogeneities and cracks starting from them. The fatigue assessment of the hydraulic cylinder will include, in this context, an appropriate research on inhomogeneities, in order to assess their dimensions and shapes. In particular, the importance of inclusions will be confronted with surface inhomogeneities. In the next section a brief summary of the relationship between defects and cracks will be presented. Then a full characterization of the material, in terms of Kitagawa diagram, research on defects and low cycle curves will be presented. Finally, the obtained data will be utilized in order to get accurate fatigue life and fatigue limit predictions. All the work will be developed thinking in pulsating loads, i.e. the stress ratio R will be in the order of 0 to 0.1, in agreement with [1, 3-5]. FATIGUE ASSESSMENT IN PRESENCE OF DEFECTS Fatigue strength is a complex damage process of a material which involves different phases: i) a “stage I” formation of short cracks within single grains and growth up to a dimension of a few “microstructural units” (this phase is also called “nucleation”); ii) “stage II” growth under the action of normal stresses up to a crack size which then leads to the final failure of the mechanical component [10]. Defects and inhomogeneities are detrimental to fatigue strength of steels. The behaviour of materials containing defects has been explained by Murakami and co-authors [11-12], who clearly showed that the fatigue limit of specimens containing micro-holes is characterised by the presence of non-propagating cracks. It can then be said that the fatigue limit is not a limit stress for “nucleation” but it rather is the threshold stress for non-propagation of the small crack emanating from the original defects. Because of the presence of non-propagating cracks, defects are similar to cracks (Fig. 1). Figure 1. Defects are similar to small cracks (Murakami [12]). / (a) (b) Figure 2. The Kitagawa-Takahashi diagram (Tanaka et al. [13]): a) dependence of fatigue limit on crack size; b) dependence of ∆Kth on crack size. atr a/a0 Fatigue strength in presence of defects is characterised by the so-called Kitagawa diagram, which takes into account the variation of fatigue strength with crack size, as shown in Fig. 2a. Its main features are: i) fatigue strength tends to increase by decreasing defect (or crack) size; ii) there is a critical size atr below which defects (or cracks) are non-damaging and fatigue strength corresponds to the limit stress amplitude of smooth specimens [13]. The peculiar shape of Kitagawa diagram is due to the fact that the threshold condition ∆Kth – for cracks and defects smaller than 1 mm is not constant, but it depends on crack size. In particular, ∆Kth increases with crack size and it tends to the threshold of long cracks ∆Kth,LC , where “LC” means long cracks (Fig. 2b). The stress intensity factor range (SIF) at the tip of defects with irregular shape can be calculated with the Murakami’s relationship [14] (Eq.1): where √area (square root of defect area projected onto a plane perpendicular to the applied stress) is the geometrical parameter expressing defect size [14]. Fatigue strength prediction can then be achieved with a model able to describe the relationship between ∆Kth and defect (or crack) size. One of the most widely used is the √area model by Murakami and Endo [11], which for surface defects predicts (Eq.2): where ∆Kth (MPa√m) is the threshold SIF, HV (kgf/mm2) is the Vickers hardness of the metal matrix, R is the stress ratio and √area (μm) is the defect dimension. Eq. (2) is valid for a variety of materials (especially steels and cast irons) for defect sizes in the range 100-1000 μm, with an error of ±20%. Combining Eqs. (1) and (2), fatigue limit stress amplitude can be estimated as (Eq. 3): Considering internal defects, Eq. (1) should be changed into (Eq. 4): and thresholds become (Eq. 5): The reason for the change of fatigue thresholds for internal defects (compare Eq. (5) with Eq. (2)) is due to the different “constraint factor” at the tip of the non-propagating cracks: it is easy to imagine that for an internal inclusion a state of stress close to “plane strain” is present and therefore ∆Kth decreases with respect to surface defects and cracks, where the higher plasticity enhances crack closure [15].The main problem of Murakami-Endo’s model is the fact that there is no distinction between “mild” and “hard” steels, while the crack size at which there is transition to ∆Kth,LC can be very different. Another model used is the El-Haddad model, which is historically the older model for describing the Kitagawa diagram and it is based on the so called "fictitious crack length" ao. El-Haddad et al. [16] proposed to model the Kitagawa diagram with the equation (Eq. 6) here ∆σwo is the fatigue limit stress range for smooth specimens without defects and F is the geometric factor for the crack. This model derived on 2D cracks, describes well the transition from LEFM to small-cracks threshold (cracks with depth ao
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