Issue 41

S. E. Ferreira et alii, Frattura ed Integrità Strutturale, 41 (2017) 129-138; DOI: 10.3221/IGF-ESIS.41.18 133 increase the tensile flow stress S FL in the unbroken elements along the plastic zone during the loading. Hence, this constraint factor should vary from   1 for plane stress conditions in thin plates, to up to   1 /( 1  2  )  3 for plane strain limit conditions in thick plates, where  is Poisson's coefficient (but in practice it is often used as an additional data- fitting parameter). Since there is no crack-tip singularity when the crack closes, this constraint factor is not used to modify the compressive yield strength during unloading, assuming the conditions around the crack tip tend to remain uniaxial. Eq. (7) presented below governs the SYM behavior by requiring compatibility between the LE part of the cracked plate and all bar elements along the crack surfaces and inside the pz ahead of the crack tip. When the length of the wake elements L j is larger than their displacement V j under  min , they contact and induce a stress  j needed to force V j = L j . The influence functions f ( x i ) and g ( x i , x j ) used in Eq. (7) are related to the plate geometry and its width correction, and are calculated like described in [6]. (7) To improve the resolution of the SYM used here, the pz ahead of the crack tip is divided into 20 bar elements, or twice the number of elements used in the original Newman’s model [6]. After calculating the pz size induced by the peak stress  max applied in the current cycle, the widths of the bar elements inside the plastic zone are calculated by Eq. (7), replacing the terms n by 20 and  j by  S F , see Fig. 2a. When the plate is unloaded to the minimum load  min (Fig. 2b), the bar elements inside the pz are also unloaded until some of them near the crack tip start to yield in compression, because they try to reach a stress  j ≤  S F . The size of this reverse plastic zone pz r depends on the amount of crack closure and on the transversal constraints induced by the plate thickness. The broken elements located inside the plastic wake formed along the crack surfaces, which store residual deformations, may come into contact and carry compressive stresses as well. Some of these elements may also yield in compression, if they try to reach  j ≤  S F . The stresses  j at each of the n elements inside the plastic zone and along the plastic wake that surrounds the crack surfaces are calculated solving the system of equations from Eq. 7 using V i  L i (at  max ) and  n  min . Crack opening loads and residual plastic deformations are calculated considering contact stresses. During the crack propagation stage, the opening stress is kept constant during a small arbitrary crack increment to save computer cost. The number of load cycles  N required to grow the crack by this increment is calculated by by Eq. (8) [27], in which the parameters C n , m , p and q are data-fitting constants, K c is the material toughness, and the FCG threshold  K th can be estimated using a procedure described in [27]. (8) The combination of CDM with SYM procedures uses the same strip-yield description adopted by the SYM to calculate the plastic strain ranges and the consequent fatigue damage distribution ahead of the crack tip. This replace the displaced HRR strain field used by the original critical damage model [19]. This model estimates the FCG increments in a cycle-by- cycle basis by a gradual damage accumulation process, but considering possible crack closure effects on the cyclic strain field ahead of the crack tip. The pz ahead of the crack tip is divided into small bar elements with constant width, whose quantity varies between 150 and 550, depending on the loading conditions. The plastic displacements within the SYM strip-yield are transformed into plastic strains using a solution proposed by Rice [30] to estimate the strain field based on CTOD variations, properly modified to consider each bar element displacements by (9) The displacements L max and L min of the i th element inside the pz are calculated at the maximum and minimum applied stresses. The position of the elements starting from the crack tip, x ct (i) , is located at the center of each element. The strain range  y that acts at each element center can be correlated to the number of cycles that would be required to break that element by the plastic part of Coffin-Manson’s rule (C&M) Eq. (10), or else by SWT rule Eq. (11), to consider peak stress effects. Notice that only the plastic part of the strain range can be considered by this SY-CDM, because the strain ranges estimated from the SYM displacements are calculated assuming rigid-perfectly-plastic bar elements. The total damage accumulated by each element is evaluated by Palmgren-Miner’s rule Eq. (1).     1 , n i n i j i j j V f x g x x               1 1 m p q n eff th max c da dN C K K K K K            ( ) ( ) ( ) 2 2 ( ) ( ) y max ct min ct log L x L i i i i i x         