PSI - Issue 23

72 4 A. Karolczuk et al. / Procedia Structural Integrity 23 (2019) 69–76 A.Karolczuk, J. Papuga/ Structural Integrity Procedia 00 (2019) 000 – 000 { , } + ( ) , − ( ) = 0 Ǥ (5) The k M ( N f ) function can be derived similarly with the result of Eq. (4 . Finally, the Matake criterion applied to the finite lifetime calculation requires the following equation to be solved: { , } + (2 ( ) ( ) − 1) , − ( ) = 0 Ǥ (6) In Karolczuk (2016), Karolczuk et al. (2016), Kluger & Łagoda (2018) , this formulation improved the fatigue life calculation considerably. In a survey conducted by Papuga (2011), the Matake criterion was rated as one of the worst criteria as regards its prediction quality. It is used here as an example for several reasons. Firstly, it is very easy to handle and to compute. Secondly, the supporting calculations in Section 3, which prove the concept of life-dependent weight parameters, are based on data with no mean stress effect and with no load non-proportionality. Under these conditions, the Matake criterion provides sufficient prediction quality. 2.2. Strain-based criteria Strain-based criteria include a group in which the critical plane concept is applied. The general form of these criteria is similar to the stress-based criteria, hence { ( ( ), , )} = ( ) ǡ (7) where is a function for reducing the alternating and multiaxial strain state ( ) to a scalar strain value in this case. Strain-based fatigue criteria are mostly applied for the low-cycle fatigue regime, Ellyin (1997), i.e. for determining the number of cycles N f . The left-hand side of Eq. (7) is usually referred to as the fatigue damage parameter, Fatemi & Socie (1988). There are no significant differences in the identification procedure for material parameters , in comparison with the strategy presented by the stress-based criteria. The parameters must satisfy Eq. (7) both for push-pull load cases and also for torsion load cases. The most influential original papers by Fatemi & Socie (1988) and by Wang & Brown (1993) assume that the set of weight parameters K is constant over the whole range of fatigue lives. 2.2.1. Example – Fatemi-Socie damage model Fatemi & Socie (1988) assumed that the fatigue life is a function of the equivalent shear strain derived from the shear amplitude , , with a significant influence of the maximum normal stress , normalized by the yield tensile strength  y . The effect of the normal stress component is weighted by the material parameter k FS . The proposed damage model follows: , (1 + , ) = ′ (2 ) 0 + ′ (2 ) 0 Ǥ (8) The strain amplitude , and stress , components are calculated on the plane with the maximum shear strain range. When Eq. (8) is applied to uniaxial tension-compression loading, the ( ) parameter can be identified as presented in Karolczuk et al. (2019). Finally, the damage model Eq. (8) takes the following form , [1 + ( ′ (2 ) 0 + ′ (2 ) 0 (1+ ) ′ (2 ) +(1+ ) ′ (2 ) − 1) 2 , ′ (2 ) ] − ′ (2 ) 0 − ′ (2 ) 0 = 0 Ǥ (9) It should be noted that the yield strength is no longer needed. A numerical solution of Eq. (9) is presented in Karolczuk et al. (2019). 2.3. Energy-based criteria Energy-based fatigue criteria can be applied in the high- and low-cycle fatigue domains, because both elastic and