_applied_  
_sciences_  
_Technical Note_  
Equivalent Stress Intensity Factor: The Consequences of the  
Lack of a Unique Definition. _Applied_  
[PERSON] 1,2 and [PERSON] 3  
Footnote 1: email: _Applied_  
Footnote 2: email: _Applied_  
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Fatigue crack growth (FCG) modelling relies upon relationships between the fatigue crack growth rate and stress intensity factor \(K\), as in the Paris law applied to mode I situations [3]:  
\[\frac{da}{dN}=C(\Delta K_{I})^{m} \tag{1}\]  
where \(a\) is the crack length, \(N\) is the number of cycles, and the empirical constants \(C\) and \(m\) depend upon the material and particular testing conditions, e.g., load ratio \(R\) (\(R=K_{min}/K_{max}\)). Following the findings of Paris [4], it has been noted that _'Fracture Mechanics approach applied to life prediction has often been misused, or has ignored obvious characteristics which effect the precision and relevance of the predictions'_. This warning might be taken into account in mixed-mode situations, where instead of the mode I stress intensity factor \(K_{I}\), the concept of equivalent stress intensity factor \(K_{eq}\) is commonly used, and the Paris law is rewritten as:  
\[\frac{da}{dN}=C\big{(}\Delta K_{eq}\big{)}^{m} \tag{2}\]  
where the constants \(C\) and \(m\) are generally different from those relevant for mode I.  
Several alternative definitions of mixed mode \(K_{eq}\) coexist, raising the question of what the consequences of that variety of definitions are--are the consequences noteworthy, or are they negligible? An alternative way of formulating the research question is: in the general \(K_{I}\), \(K_{II}\) domain, how do different definitions of \(K_{eq}\) compare?  
This note considers a number of \(K_{eq}\) definitions hitherto not analyzed simultaneously. Experimental and modelling aspects of the mixed-mode-I-II fatigue crack growth problem are revisited, and the values of \(K_{eq}\) according to several criteria were compared through the determination of \(K_{eq}/K_{I}\) over a wide range of \(K_{I}/K_{II}\) or \(K_{II}/K_{I}\) values.  
Mixed-mode situations are found in planar geometries, e.g., biaxially loaded plates, asymmetrically loaded 3- or 4-point bending specimens, or cylindrical specimens subjected to torsion combined with uniaxial tensile or compressive loading. This note concentrates on planar specimens, with crack path predictions based on 2D finite element method (FEM) analysis and a consideration of \(K_{I}\) and \(K_{II}\) in the frame of suitable \(K_{eq}\) criteria.  
Experimental FCG results were available for Al alloy AA6082 T6 [5; 6]. Four-point bend specimens subjected to asymmetrical loading and compact tension (CT) specimens modified with holes were subjected to fatigue crack growth tests, where the fatigue crack path and fatigue crack growth rate were measured. The presentation of the FCG data was made using a Paris law based upon \(K_{eq}\), and the tests were modelled using the extended finite element method (XFEM) implemented in ABAQUS [7]. The present note builds upon that earlier work, expanding the number of \(K_{eq}\) definitions considered and quantifying the important differences found in the Paris laws corresponding to the alternative definitions of \(K_{eq}\). While considering a wide range of mixity values, it was found that seven different definitions for \(K_{eq}\) led to different results. A data-driven modelling methodology (see e.g., [8]) is suggested as a perspective to overcome this unsatisfactory situation, following encouraging applications to mode I FCG, e.g., [9; 10; 11; 12].