Issue 33

Y. Wang et alii, Frattura ed Integrità Strutturale, 33 (2015) 345-356; DOI: 10.3221/IGF-ESIS.33.38 346 becomes clear that there are different ways to define the critical plane itself. However, it is not an easy task to determine the orientation of the critical plane when the applied multiaxial load history varies randomly. Several investigations [5-9] have been published by focusing attention on the determination of the critical plane under multiaxial random loading. Susmel [9] have recently formalized a novel technique, the Shear stress-Maximum Variance Method (  -MVM). The Maximum Variance Method (MVM) postulates that the critical plane can be defined as that plane containing the direction (passing through the assumed critical point) that experiences the maximum variance of the resolved shear stress. The peculiarity of the MVM is that it is not necessary to calculate the stress/strain components relative to any plane passing through the assumed critical point to determine the critical plane itself. Therefore, from a computational point of view, it is more efficient than the other existing methods. In order to take into account cyclic hardening or softening in the fatigue failure criteria, both stress and strain components are recommended to be used to estimate the fatigue damage extent. As to the in-field usage of the available VA multiaxial fatigue life estimation techniques, correctly performing the cycle counting under such complex loading conditions is one of the trickiest aspects [10]. Even though there are a few methods suitable for counting the cycles under uniaxial loading, among these the rainflow cycle counting method [11] has been most widely and successfully used. The number of published papers dealing with VA multiaxial loading histories is small [12]. Bannantine and Socie [13] proposed a method based on the critical plane concept and the rainflow cycle counting. Wang and Brown [14] proposed a cycle counting method based on the rainflow and a modified von Mises equivalent strain. The accuracy of the results predicted by using the critical plane approach to a great extent relies on the determination of the amplitude and mean value of the normal and shear stresses and strains acting on a particular plane [15]. For a particular plane under complex loading, the direction of the normal stress/strain does not change over time, since just the magnitude of the normal stress/strain vector varies. However, in the most general case, the shear stress/strain vector on this particular plane changes in both magnitude and direction. Therefore, the tip of the shear stress/strain vector draws an imaginary curve on that plane. In this situation, it is not an easy task to calculate the amplitude and mean value of the shear stress/strain on a given plane. There are several techniques proposed to address this problem, namely the Longest Chord Method, the Longest Projection Method, the Minimum Circumscribed Circle concept, and the Minimum Circumscribed Ellipse Method. These methods have been discussed in the Refs [9, 15]. The present paper summarizes an attempt of extending the use of the Modified Manson-Coffin Curve Method (MMCCM) to those situations where complex variable amplitude loadings are involved. By making the most of our previous experience [10, 16-18], now we intend to use the stain-based MVM in conjunction with MMCCM to predict fatigue lifetime under VA multiaxial fatigue loading. In more detail, the MMCCM is suggested here as being applied in conjunction with the maximum variance method (MVM) to estimate fatigue lifetime by directly post-processing the strain state relative to that material plane containing the direction along which the variance of the resolved shear strain is maximized. Since, by definition, the resolved shear strain is a monodimensional quantity, the rainflow method can directly be used to count the loading cycles. After counting the cycles and estimating the fatigue damage for each counted cycle, fatigue lifetime can be predicted directly by using the Palmgren-Miner’s linear damage rule. Finally, the validation of the new approach is presented. T HE SHEAR STRAIN _ MAXIMUM VARIANCE METHOD (  -MVM) he shear strain-Maximum Variance Method (   MVM) assumes that the critical plane can be defined as that plane containing the direction experiencing the maximum variance of the resolved shear strain. In order to calculate the shear strain relative to a generic material plane,  , and resolved along a generic direction, q , consider a body subjected to an external system of forces resulting in a triaxial strain state at point O (Fig. 1a). Point O is taken as the center of the absolute system of coordinates, Oxyz . The time-variable strain state at point O is defined through the following strain tensor:   x xy xz x xy xz yx y yz xy y yz zx zy z xz yz z 1 1 ε (t ) (t ) (t ) 2 2 ε (t ) ε (t ) ε (t ) 1 1 ε t ε (t ) ε (t ) ε (t ) γ (t ) ε (t ) γ (t ) 2 2 ε (t ) ε (t ) ε (t ) 1 1 γ (t ) γ (t ) ε (t ) 2 2                                     (1) T