Issue 49

Yu. G. Matvienko, Frattura ed Integrità Strutturale, 49 (2019) 36-43; DOI: 10.3221/IGF-ESIS.49.04

proportional to the J -integral. In our opinion the parameter p p S A  is not suitable for using as a constraint parameter because plastic zone areas and J values are proportional. To avoid direct dependence of these parameters on each other, it is proposed to replace coordinate r (as well as x and y ) with dimensionless coordinate   0 / / r J    [8]. In this case, the area inside equivalent plastic strain becomes dimensionless   2 0 / / PEEQ PEEQ A A J   (11)

p A can be introduced into consideration by means of the following

Therefore, a normalized constraint parameter

expression

p p S A  , where

/ PEEQ ref A A A  . p

(12)

The parameter S should be less dependent on the J-integral. In this case, the separation of the constraint parameter and the J-integral improves the quality of a fracture criterion that is p A is denoted as p S . The modified constraint parameter p

based on these parameters. Behavior of the parameter

p S as a function of A/A SSY

is investigated using the J-A elastic-plastic asymptotic field. The

 is estimated with the following

area surrounded by the equivalent plastic strain contour with specified value of strain p

p  corresponding to specified plastic strain p

 can be represented as

algorithm [8]. Stress

n

n

1/

1/

 





E

  

  

  

  

p

p

p









(13)

0 



0

0

The rectangular area in the vicinity of the crack tip is divided into square cells. Equivalent stress is computed at each cell by means of the J-A elastic-plastic asymptotic field. If this stress is larger than p  , the cell area is added to PEEQ A . Computation of the plastic area is performed in coordinates 0 / x J  and 0 / y J  , so the resulting area is PEEQ A . Fig. 4 summarizes the computed results for plastic zones inside plastic strain contours corresponding to p  =0.1, 0.2 and 0.3 and relationships between parameters p S and A for a material with 1   and 5 n  . Smallest plastic zone corresponds to small scale yielding value of the constraint parameter A ( A SSY ( n =5) = 0.3803). Intermediate plastic zone is computed for the value of   max / 2 SSY A A  . Largest plastic zone is related to A max (shown in Fig. 4). It can be seen that there is a unique relationship between the constraint parameters p S and A in a wide range of specified values of strain p  . small scale yielding conditions. Constraint parameters A and Q show similar behavior. At the same time, it should be noted that small errors in the crack-tip stress obtained by finite element method lead to the significant error in the value of Q . The constraint parameter p p S A  based on the area surrounded by the equivalent plastic strain p  contour ahead of the crack tip is not suitable as a crack-tip constraint parameter because plastic zone areas and J values are proportional. To avoid direct dependence of these parameters on each other, the constraint characterization parameter p A is modified as R C ONCLUSIONS elationship between the crack-tip constraint parameters A , A 2 p S of elastic-plastic fracture mechanics is based on analysis of the stress fields in the vicinity of the crack tip. The following conclusions can be drawn. It was demonstrated that J-A and J-A 2 approaches are mathematically equivalent. The difference of these approaches is in distance scaling r . But, in contrast to the parameter A , the parameter A 2 does not have its value under , Q and

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