PSI - Issue 23

A. Karolczuk et al. / Procedia Structural Integrity 23 (2019) 69–76 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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parameters between various stress or strain parameters forming the final scalar parameter are fixed over all evaluated lifetimes (e.g. Wang & Brown (1993)). However, there are a few exceptions applying material parameters being the function of lifetime, Slámečka et al. (2013), Carpinteri et al. (2013), Karolczuk et al. (2016). The concept of life dependent material parameters improves the lifetime prediction, Karolczuk (2016), and must be disseminated. The aim of the paper presented here is to show that the effectiveness of even well-known multiaxial fatigue criteria can be enhanced, if the fatigue criteria use weight functions between individual parameters which are life-dependent. 2. Multiaxial fatigue criteria Multiaxial fatigue criteria can be divided into three groups, according to the physical quantity applied for the damage state estimation: (i) stress-based criteria, (ii) strain-based criteria and (iii) energy-based criteria, Ellyin (1997), Karolczuk & Macha (2005). Each group is applicable for a different fatigue life domain, and each requires specific material parameters. The appropriate material parameters will be summarized below, together with some detailed examples. (1) where is the function for reducing the multiaxial stress tensor history ( ) to the scalar equivalent stress, which in the case of critical plane methods is dependent on the plane orientation set by the normal vector n ; is the set of material parameters, and q is the fatigue limit. The stress tensor history is projected on to the evaluated planes, and the amplitudes and the mean values of the related stress components form the left-hand side of Eq. (1). The individual stress parameters there are weighted by K set of parameters. For the simplest two-parametric models, at least one weight derived from two fatigue limits (  af under fully-reversed axial loading,  af under fully-reversed torsion) is necessary. If an estimate of the fatigue limit is replaced by an estimate of the fatigue life, a dependency between stresses and number of cycles can be observed on both sides of Eq. (1). The material parameter q switches to material function q ( N f ), i.e. to a description of the S-N curve under fully reversed axial  f ( N f ) or torsion loading  f ( N f ). If the set of K parameters has been derived from at least two fatigue limits, the transition to the fatigue life computation leads to the K ( N f ) set of material functions derived from at least two fatigue curves  f ( N f ) and  f ( N ), usually depending on the number of stress parameters that are used: ( ( ), , ( ( ), ( ))) ≤ ( ) (2) 2.1.1. Example – Matake criterion The Matake criterion, Matake (1977), is one of the widely cited critical plane criteria. It is based on the idea that the critical shear stress range or amplitude for crack initiation is reduced linearly by the normal tensile stress amplitude (Matake ’s original proposal from 1977 did not include the mean stress effect) . The Matake proposal is similar to the criteria presented by Stulen & Cummings (1954) and by Findley (1959), and it takes the following form { , } + , ≤ ǡ (3) where the stress components are calculated on the plane with the maximum shear stress. The constant k M is determined from the load case of fully-reversed axial loading  af . It yields , = 0.5 and , = 0.5 on the plane of the maximum shear stress. Finally, after these values have been used in Eq. (3), the constant k M is obtained as = 2 − 1 Ǥ (4) Considering the left-hand side of inequality (3) as the equivalent stress, the constant value of k M weight can be replaced by a function depending on the S-N curves in the torsion and axial load cases. The fatigue life can be calculated using the following equation 2.1. Stress-based criteria Stress-based critical plane multiaxial fatigue limit criteria have the following general form ( ( ), , ) ≤