Issue 47

Z. Hu et alii, Frattura ed Integrità Strutturale, 47 (2019) 383-393; DOI: 10.3221/IGF-ESIS.47.28 385 is based both on a precise definition of the control volume and on the fact that the critical energy does not depend on the notch sharpness. The control radius R 0 of the volume is a material property, which involves the plain specimen’s fatigue limit and the threshold stress intensity factor range [10]. The approach was successfully used under both static and CA fatigue loading conditions to assess the strength of notched components subjected to uniaxial [12-14] as well as to multiaxial loading [15-19]. Even if a lot of theoretical and experimental work has been carried out to check the accuracy and reliability of the SED approach, surprisingly, so far no systematic attempt has been made to extend the use of this powerful method to those situations involving VA fatigue loading. Accordingly, the ultimate goal of the research work summarized in the present paper was to reformulate the SED approach to make it suitable for assessing fatigue lifetime of notched components subjected to VA uniaxial fatigue load histories, with this being done by using a suitable cycle counting method (i.e., the Rain-Flow method [20]) as well as a suitable cumulative damage model (i.e., Palmgren and Miner’s rule [1, 2]). E XPERIMENT DETAILS n a recent work, Susmel and Taylor [3] tested under uniaxial VA fatigue loading circumferentially notched specimens made of a medium-carbon steel C40 and containing three different geometrical features. The geometry of the tested specimens is reported in Fig. 1a along with details of the notches that were characterized by three different values of the notch tip radius and opening angle. Static tensile tests were performed in order to evaluate the mechanical properties according to the ASTM standard procedure. The yield stress, tensile stress and Young’s modulus of the investigated materials were 672 MPa, 852 MPa and 209000 MPa, respectively. The net stress concentration factor, K t , was calculated by solving standard linear-elastic Finite Element (FE) models where the mesh density in the notch region was gradually increased until convergence occurred. This standard numerical procedure returned a K t value under tension equal to 4.42, 2.2 and 1.66 for the specimens with a root radius of 0.225, 1.2 and 3 mm, respectively. Both CA and VA fatigue tests were carried out in axial load control under a nominal load ratio, R = σ min / σ max , equal to -1 at a frequency of 4 Hz. All the CA fatigue results are summarised in the Wöhler diagrams of Fig. 1b in terms of nominal stress amplitude evaluated onto the net cross-sectional area of the specimens. Fig. 1b shows the fatigue data generated by testing both the un-notched specimens and the V-notched samples with root radius equal to 0.225 mm ( K t = 4.42). The statistical analyses have been performed assuming a log-normal distribution of the number of cycles to failure for each stress amplitude. All data obtained from specimens characterized by a fatigue life between 10 4 and 2  10 6 have been taken into account in the statistical analyses. It is worth observing here that the scatter bands plotted in the chart of Fig. 1b were calculated for each stress level by assuming a confidence level equal to 95%. The results from the statistical re-analysis are summarised in Tab. 1 in terms of: nominal stress amplitudes ( σ A for the plain specimens and σ An for the notched samples) for a probability of survival P S =50% at a reference number of cycles to failure, N A , equal to 10 6 ; inverse slope, k , of the Wöhler curves and the scatter index T σ that provides the width of the scatter band between the curves with a probability of survival of 10% and 90% respectively (with a confidence level equal to 95%). Specimen Type r n ( mm ) d g ( mm ) d n ( mm )  ( ° )  k σ A , σ An *  ( MPa )  T  K t Plain - 12 6 - 9.4 292.8 1.211 1.0 Sharp 0.225 12 9.15 35 4.2 97.8 1.361 4.42 Table 1 : Summary of the experimental results generated under fully-reversed CA loading [3]. As to the performed VA tests, the specimens were tested under the two different random spectra (Fig. 2) having sequence length, n tot = Ʃ n i , equal to 1000 cycles. All the random spectra were repeated until failure of the specimens occurred. In particular, the Concave Upwards Spectrum (CUS) was derived from a conventional Rayleigh distribution, whereas the Concave Downwards Spectrum (CDS) was defined so that we could have few cycles with high stress ranges combined with a relatively large number of cycles with low stress ranges [21]. The aim of two different spectra of tests was to determine the effect of spectrum shape on the fatigue lifetime of the notched material being investigated and to determinate the fatigue lifetime under random block loading. All the results generated by Susmel and Taylor [3] are summarized in Tabs. 2-4, where the performed tests are described in terms of adopted spectrum, maximum force, F a,max , maximum amplitude in the spectrum of the axial, σ a,max , load ratio, R , and experimental value of the number of cycles to failure, N f . For comparison, Tab. 5 shows the fatigue results generated by testing plain samples under the two load spectra. Tab. 5 also reports the corresponding experimental values of the critical damage sum, D cr,exp , calculated by using experimental CA plain Wöhler curve. I