Issue 38

T. Lassen et alii, Frattura ed Integrità Strutturale, 38 (2016) 54-60; DOI: 10.3221/IGF-ESIS.38.07 57 Test series 2: Alternating compression and tensile testing at R=-1 For this series the actual stress situation will again be as shown in Fig. 3, but as this stress history is partly compressive the maximum hydrostatic stress decreases compared with test series 1. The following equations for the stress situation apply for the two test series: h a  ,    3 4         4   Test series 1 (2) h a  ,    2 3 4          Test series 2 (3) Due to the lower hydrostatic stress in test series 1 this series will sustain higher shear amplitude as predicted by Eq. (1) and illustrated in Fig. 2. All the tests were run to failure or stopped at N=2·10 7 cycles. Failure is defined by total fracture of the specimen. The fatigue limit is defined at N=10 7 cycles as is in accordance with rule and regulations. Typically applied normal stress for test series 1 was close to a range of 450 MPa, whereas the stress range for series 2 was close to 500 MPa. The applied loading was tuned in after the first tests have been completed. The RFLM model was applied to determine the fatigue limit, see description next section. Hence, the two test series gives us two points on the Dang Van design curve as shown in Fig. 2. Consequently, the slope of the curve denoted a DV is determined. T HE RANDOM FATIGUE LIMIT METHODOLOGY FOR DATA ANALYSIS he conventional statistical analyses of data points are based on linear regression of fatigue life data only, [5] or by the staircase method for the fatigue limit only, [6]. In the present case with large scatter in both fatigue life and the fatigue limit these methods are regarded as less appropriate. Due to the uncertainty and large scatter in fatigue life in the region close to 10 7 cycles, an S-N curve based on a Random Fatigue-Limit Model (RFLM) is adopted in the present work, [4]. The basic feature of the model is that both fatigue life and the fatigue-limit are treated as random variables simultaneously. The fatigue limit should not be treated separately as it is done for the bi-linear curves. The S-N curve obtained from the RFLM will not have an abrupt change from an inclined straight line to a horizontal line, but a gradually change in slope as stress ranges get very low. It then remains to be seen if the slope goes asymptotically towards a horizontal line as the number of cycles increases. If this is not the case the existence of a fatigue-limit should be rejected. We shall not elaborate the method in the present article but outline the most important characteristics. The method applies a Maximum Likelihood Method that gives a rational treatment of run outs. The basic equations are:     N S 0 0 ln ln          (4) where ln denotes the natural logarithm and γ = Δ S 0 is the fatigue-limit. The parameters β 0 and β 1 are fatigue curve coefficients. For given sample data w i and x i from various test specimens i = 1,…n, the model parameters can be determined by the Maximum Likelihood (ML) function:       n i i w i i w i i i L f w x F w x 1 1 ; 1 ;                Q Q Q (5) where δ i = 1 if w i is a failure and δ i = 0 if w i is a censored observation (run out). The vector Q contains the model parameters:   s 0 1 ,  , , ,          Q (6) where σ is the standard deviation for the natural logarithm to fatigue life, whereas µ γ and s γ are the mean value and standard deviation respectively for the fatigue limit ln γ . Once these parameters have been determined from optimization T