Issue 43

L.C.H. Ricardo, Frattura ed Integrità Strutturale, 43 (2018) 57-78; DOI: 10.3221/IGF-ESIS.43.04 67  The similarity can be violated because the crack growth mechanism are no longer similar  The crack can be too small for adopting K as a unique field parameter   K eff and others conditions being nominally similar, it is possible that other crack tip aspects also affect crack growth, such as crack tip blunting and strain hardening, Schijve [36]. Newman and Armen [46-48] and Ohji et al. [49] were the first to conduct the two dimensional analysis of the crack growth process. Their results under plane stress conditions were in quantitative agreement with experimental results by Elber [45], and showed that crack opening stresses were a function of R ratio (S min /S max ) and the stress level (S max /  0 ), where  0 is theflow stress i.e: the average between σ ys and σ u . Blom and Holm [50] and Fleck and Newman [51-52] studied crack growth and closure under plane-strain conditions and found that cracks did close but the cracks opening levels were much lower than those under plane stress conditions considering same loading condition. Sehitoglu et al. [53] found later that the residual plastic deformations that cause closure came from the crack. McClung [54-56] performed extensive finite element crack closure calculations on small cracks at holes, and various fatigue crack growth models. Newman [57] found that S max /  0 could correlate the crack opening stresses for different flow stresses (  0 ). This average value was used as stress level in the plastic zone for the middle crack tension specimen, McClung [56] found that K analogy, using K max /K 0 could correlate the crack opening stresses for different crack configurations for small scale yielding conditions where K 0 =  o  (  a) . ( K -analogy assumes that the stress-intensity factor controls the development of closure and crack-opening stresses, and that by matching the K solution among different cracked specimens, an estimate can be made for the crack opening stresses.) Crack opening stresses The applied stress level at which the crack surfaces are fully open (no surfaces contact), denoted as S 0 , was calculated from the contact stresses at S min . To have no surface contact, the stress-intensity factor due to applied stress increment (S 0 - S min ) is set equal to the stress intensity factor due to the contact stresses. Solving for S 0 gives :               1 1 1 0 min 2 1 11 2 sen sen n j j S S B B (2.30) where                0 sen sen K K b W B c W for K = 1 or 2 (2.31) and c 0 is the current crack length minus  c*, where  c* is the crack growth increment over which S 0 is held constant. The analytical closure model provides no information about the amount of crack growth per cycle. Crack growth is simulated by extending the crack an incremental value and the moment of the maximum applied stress. The amount of crack extension (  c*), was arbitrarily defined as  c*,=0.05  max , where  max is the plastic zone size caused by the maximum applied stress occurring during the  c* growth increment. The increment in width of element n , and its significance is discussed in the next section. If  j = 0 for j = 11 to n – 1 at the minimum applied stress, then the crack is already open, and S 0 cannot be determined from Eq. (14). The stress  j at the crack tip changes from compression to tension when the applied stress level reaches S 0 , [56]. The crack growth equation proposed by Elber [45] states that the crack growth rate is a power function of the effective stress intensity factor range only. Later, Hardraht et al. [58] showed that the power law was inadequate at high growth rates approaching to fracture. The results presented herein show that it is also inadequate at low growth rates approaching threshold. To account for these effects, the power law was modified to:                    2 0 1 2 2 max 5 1 1 eff eff K K dc C K C dN K C (2.32)