Issue 33

Y. Wang et alii, Frattura ed Integrità Strutturale, 33 (2015) 345-356; DOI: 10.3221/IGF-ESIS.33.38 351 To conclude the present section it is important to highlight that, in general, there exist two or more different directions which experience the maximum variance of the resolved shear strain, so that, it is always possible to locate, at least, two potential critical planes: amongst all the potential critical planes, the one which has to be used to estimate fatigue lifetime is that experiencing the largest value of ρ . T HE MMCCM TO ADDRESS THE VARIABLE AMPLITUDE PROBLEM The use of the MMCCM in those situations involving VA multiaxial load histories is based on the following assumptions: (i) initiation and initial propagation of Stage  cracks occur on that material plane containing the direction experiencing the maximum variance of the resolved strain, MV ( ) t  ; (ii) the fatigue damage depends also on the stress ratio ρ which is defined in the previous Section. Fig. 3 summarizes the approach proposed in the present paper to be followed to estimate fatigue lifetime of engineering materials subjected to in-field VA multiaxial fatigue loading. In more detail, consider a body subjected to a complex system of time variable forces resulting in a VA multiaxial strain state at the assumed crack initiation site (i.e. point O in Fig. 3a). Initially, by making use of the shear  -MVM, the orientation of the candidate critical planes can directly be determined through that direction, MV, experiencing the maximum variance of the resolved shear strain (Fig. 3b). After determining the direction of maximum variance of the shear strain, the shear strain resolved along this direction, MV ( ) t  , the shear stress resolved along the maximum variance direction, MV ( ) t  , and the stress normal to the critical planes, n ( ) t  , can be evaluated, at any instant, t , of the load history, through Eq. (14) to (16) (Figs 3c and d). Subsequently, the calculated values for n,m  (Eq. 17), n,a  (Eq. 18) and a  (Eq. 21) have to be used to estimate, according to Eq. (23), critical plane stress ratio ρ . According to the determined value for ρ , the profile of the corresponding modified Manson-Coffin curve can be described by using the following general relationship [18]:     ' ( ) f ( ) ' a f f f (2 ) ( ) 2 c b N N G           (24) where ' f ( )   , ' f ( )   , ( ) b  , ( ) c  are material functions, which can be determined through the following equations [18]:     0 0 b b b b b b       (25)     A,ref ' f ( ) A ( ) 2 b N       (26)       0 ' ' f A,ref A f A 2 (1 ) 2 2 b b N N          (27)     ' ' ' f p f f 1 ν (1 )            (28)     0 0 c ρ c c c c c      (29) where p  is Poisson’s ratio for plastic strain, N A is the reference number of cycles to failure. After determining the appropriate reference fatigue curve as above, by post-processing the shear strain resolved along direction MV, the classical rain-flow method allows the corresponding shear strain spectrum to be built directly (Figs 3g and h). Finally, from such spectrum fatigue strength under VA multiaxial fatigue loading can be predicted according to Palmgren-Miner’s rule (Fig. 3i). Here it is assumed that breakage takes place when the damage sum equals unity.