PSI - Issue 17

Shahnawaz Ahmad et al. / Procedia Structural Integrity 17 (2019) 758–765 Shahnawaz Ahmad/ Structural Integrity Procedia 00 (2019) 000 – 000

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( i p N total life N initiation life N propagation life = (5) The step by step procedure for estimation of the Crack initiation life, is summed up as: with the nominal stress (from stress analysis) as input true stresses are obtained in the vicini ty of the defect using Neuber’s rule[8] using equation (7). This true stress is then used for solving elasto-plastic strains using iterative technique, coupling with material hysteresis curve an d Massing’s hypo thesis[9],equation (6). The initiation life of the blade is assessed by accounting for the mean stress using Morrow’s hypo thesis [10], equation (8). ( ) 1/ 2 2 2 ' n E K    =  +  ò (6) ∆ σ ∆ σ E + 2 ∆ σ 2 K ' ( ) 1/ n ( ) { } = K t ∆ S ( ) 2 / E (7) ( ) ( ) ( ) ' ' 2 ( ) 2 2 b c f m i f i E N N    = − + ò ò (8) Number of cycles required for propagation of a fatigue crack until it reaches its critical size is the fatigue crack propagation life p N . The crack growth behaviour of the material is represented by da dn vs. K  on a log-log plot (Figure 11), where K  is the amplitude of the stress intensity factor and da dn is the crack propagation rate. The sigmoidal curve is characterized by three distinct regions- region I, in which almost no crack propagation takes place, region II, where there is slow crack propagation and a linear log( K  ) – log( da dn ) relation holds, and, region III, where the crack growth rate curve rises and the maximum intensity factor max . K in the fatigue cycle becomes equal to the critical stress intensity factor c K , leading to catastrophic failure. ) ( ) + ( ) t

Figure 11 Typical fatigue crack growth behaviour in metals (source: Anderson, “Fracture mechanics” [11]) The crack growth rate, from an initial crack size i a to a final size f a , is written in terms of number of propagation cycles, by Paris formulation [12]; Miner’s rule [13] is then utilized to predict the fatigue life under variable stress amplitude loading. The Miner ’s rule states that if the average number of cycles to failure at the stress amplitude corresponding to the th i stress block, i S is i N , then the damage occurred to the component, i D , is;

i i i D n N =

(9)

where, i n is the number of cycles accumulated at the stress amplitude i S .

i D is the corresponding fraction of life

consumed,. The crack propagation life ( N p ) can be estimated as- ( ) a f m p ai N da C K =   (10) where, C and m are known as Paris constants [6]. The stress intensity factor K defines the magnitude of the local stresses around the tip of crack for a given nominal stress, S , near a crack of length a , and is given by; K FS a  = (11) In terms of stress amplitudes, the fluctuating stress intensity factor, K  , can be written as; (12) where F is the geometry correction factor and S  is the remote stress applied to the component. The initial crack length i a is the smallest detectable crack length, which is limited by the measuring techniques. Analysis has max min K K K F S a   = − = 