PSI - Issue 22

Yang Ai et al. / Procedia Structural Integrity 22 (2019) 70–77 Y. Ai et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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2. Proposed model for notch fatigue analysis under size effect 2.1 Equivalent highly stressed region

In general, the highly stressed volumes are generally defined as the critical region subjected to the maximum stress, rather than the equivalent region under similar local stress states [14], which means that the relationship of the highly stressed volume under different stress states is invariant. Therefore, the application of the HSV approach is principally limited to the prediction of fatigue strength or life distribution of geometrically similar specimens. Note from Kloos [19] that × is viewed as the local fatigue strength amplitude to describe the local stress state, in which is the theoretical stress concentration factor and is the maximum normal stress. Thus, for obtaining the equivalent highly stressed volume under the same local stress state, a dynamic relationship is elaborated as , × , = , × , (1) As aforementioned, the highly stressed volumes with the same maximum local stress are assumed in a similar local stress state, and fatigue strength of the structure decreases with the size of equivalent highly stressed volume. Based on the assumption of equivalent highly stressed volume, the effect of size and notch in a variety of specimens can be viewed as the statistical size effect of equivalent highly stressed volume to be analyzed [16]. 2.2 HSV-based probability model considering size effect A cumulative distribution function is used to define the component with the number of loading cycles to fatigue failure, ( ) , which represents the probability of no failure occurrence at the maximum normal stress and stress ratio R . As one of the most advantageous tools for the development of P-S-N diagrams and failure probability analysis [16], a novel Weibull model is developed as follows: ( ) = 1 − [−( ×( 0 ) − ) ] (2) where is the number of loading cycles at the maximum normal stress and stress ratio R . By utilizing Maximum Likelihood or Bootstrapping methods to consider the influence of statistical uncertainty [16], the shape parameter and scale parameter of Weibull distribution can be evaluated. 0 is the reference highly stressed volume of the tested specimen, and is the highly stressed volume of the specimen remained for assessment. Note that, establishes a relationship between different highly stressed volumes under the same maximum local stress and reflects the effect of volume change on the fatigue strength or fatigue life. Note from Susmel et al. [20] that a linear dependence between two variables can be mathematically formalized by linear equation with one unknown. Similarly, note that is a variable parameter related to the maximum local stress , . Therefore, can be defined by a quadratic equation of the maximum local stress , : ( , ) = 2 , + , + (3) Once the relationship of fatigue life under different highly stressed volumes are obtained, model coefficients , and can be calculated by the least square method. Moreover, when the Weibull scale parameter is taken as a constant 1.5 [21], the Weibull shape parameter ( ( ) = 10% , 50% , 90% ) is determined by the reference life of tested specimens 0 , which can be expressed as: = × 0 (4) where is an empirical constant 1.27551. Therefore, Eq. (2) can be transformed into: ( ) = 1 − [−( ( , ) × 0 ( , )×( 0 ) − ( , ) ) ] (5)