PSI - Issue 61

Aliyye Kara et al. / Procedia Structural Integrity 61 (2024) 98–107 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 After discretization, in a second step, the stress amplitude values, , are projected onto the Ramberg-Osgood curve with the help of Neuber’s hyperbola , yielding the corresponding transformed strain amplitudes, = ( ) , where (−) denotes Neuber’s rule. Within this transformation, each stress amplitude , , is shifted to the strain amplitude, , while keeping its corresponding area, , unchanged. The strain amplitudes, , represent the central values of the bins in the discretized strain amplitude PDF. Note that, while the stress amplitudes, , are evenly spaced, the strain amplitudes are not, due to the nonlinear nature of Neuber’s transformation. At the final step, once all have been transformed, the strain amplitude PDF, ( ) , can be recovered based on the relationship = ( ) ∙ ( ) = ( ) ∙ ( ) , where is the bin width. The previous equality simply states that the number of cycles in the i -th bin obviously remains the same regardless of whether such cycles are characterized by elastic stress or elasto-plastic strain amplitude. This is basically a reallocation of cycles in accordance with the strain amplitudes. The obtained PDF for elasto-plastic strain amplitudes is generally distorted from the original stress amplitude PDF, the degree of distortion being proportional to the amount of plasticity. It is not superfluous to emphasize that, according to the above procedure, the obtained strain amplitude PDF is inherently a discrete function defined by the pairs of values , and ( ) . Although numerical interpolation can be used to obtain the continuous probability distribution, ( ) , the overall level of accuracy strongly depends on the proper choice of the initial stress bin width, , i.e., too large values may eventually lead to a rough, or even inaccurate, approximation of ( ) . This constitutes the main drawback of the procedure. Once the strain amplitude distribution, ( ) , has been obtained, fatigue damage can be calculated as for Eq. (4), where the number of cycles to failure follows from the Manson-Coffin equation: = ′ (2 ) + ′ (2 ) (5) Similar to the approach described above, Böhm et al. (2020) recently published a study on strain-based spectral fatigue that compared the fatigue life estimations of different PDF models. This study used Neuber’s rule and the Ramberg-Osgood constitutive model to obtain the PDF of strain amplitudes, as in the study of Rognon et al. (2011, 2014). As a result, the fatigue life estimation obtained by considering local elasto- plasticity using Neuber’s rule is closer to the experimental fatigue life results than the fatigue life estimation obtained without taking plasticity into account. It has to be noted that the aforementioned strain-based approach is based on some implicit assumptions. Firstly, it applies the Neuber’s rule directly to stress amplitudes (after rainflow counting) and not to actual stress/strain values of the random loading; it therefore neglects the path-dependent nature of plasticity. Secondly, as stated by Böhm et al. (2020), the approach does not include strain-hardening. These two aspects motivate the comparison with time domain simulations, considering actual stress-time histories and the strain-hardening model, described in Section 4. 3.2. Reformulation based on CDFs The PDF transformation procedure described so far is herewith reformulated by using a more rigorous definition based on cumulative distribution functions (CDFs). The new formulation is not only formally more correct, but it also allows solving the numerical discretization error inherent in the original transformation procedure based on bins. Let σ ( σ a ) be the CDF of stress amplitudes corresponding to ( ) , as shown in Fig. 1(b ). Neuber’s rul e is used to transform a certain stress amplitude, say σ ∗ , into a corresponding strain amplitude, ∗ = ( ∗ ) , while keeping the corresponding probability unchanged. More specifically, the probability of the stress amplitude being equal to or lower than σ ∗ is equal to the probability of the strain amplitude being equal to or lower than the transformed strain amplitude, ∗ . In symbols: ( ∗ ) = { ≤ ∗ } = { ≤ ∗ }= ( ∗ ) (6) where {−} means ‘ probability ’ . This relationship has general validity regardless of the specific form of the stress amplitude distribution σ ( σ ) and of the non-linear transformation ∗ = ( ∗ ) ; furthermore, it does not involve any 101 4