PSI - Issue 23

Dragan Pustaić et al. / Procedia Structural Integrity 23 (2019) 27 – 32 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3

The non-linear dependence among the true stress and the true strain in the hardening range can be well described by Ramberg- Osgood´s equation, Chen et al. (1992).   0 0 0 . n           (2) The strain 0 ,  in the equation (2), corresponds to the yield point 0 ,  according to the Hooke´s law 0 0 , E    while the marks  and n designate the material parameters. In these investigations the values of 0 700MPa, 200 GPa, =3 7 and 1 E       are taken. The non-linear distribution of the cohesive stresses within the plastic zone around the crack tip must have the structure according to the conditions of the HRR stress and strain fields. As the exact analytical solution is in advance unknown, one of the possible approaches to the problem could be the following. The cohesive stresses distribution can be determined, for example, by means of the finite element method, applied program package Abaqus. That procedure was performed by Pustaić and Lovrenić (2006). The stress distribution is approximated with an analytical expression, i.e. with the function which excellently coincided with the one obtained with the analytical expression suggested by the authors Hoffman and Seeger (1985)       1 1 0 p . n p x r x a          (3) In the expression (3), p(x) is a function of the two parameters, i.e. the plastic zone magnitude, r p, and the strain hardening exponent, n .

2. Determination of the stress intensity coefficient from the cohesive stresses

The stress intensity coefficient from the cohesive stresses, at the tip of fictitious elastic crack coh ( ) K b is determined by means of the Green´s functions method , as it can be seen in the paper Chen et al. (1992). It can be determined, knowing the distribution of the cohesive stresses p(x) according to the expression (3). So, it could be written

b

b



 1 2 2 2 

      a K b p x m x b x p x b b x           coh , d 2 π a

d . x

 

(4)

The expression (4) will be transformed in a way that, instead of the variable x the new independent variable  is introduced, related to the previous one, according to Chen et al. (1992):   p 1 . a x r     The expression (3) is, also, necessary to transform introducing the new variable .  In such way, it is obtained:       1 1 0 1 . n p         The aim of this transformation is to achieve the limits of integration from 0 to 1. Stress intensity coefficient, according to the equation (4), now can be expressed through the new independent variable ,  looking like

  1 1 n

 





1  0





1 2



  K b r  coh

  1    0

 

 

p 2 π

1 2

d . 

p r b

   



 





(5)

The solution of the above integral was obtained by means of the program package Wolfram Mathematica 7.0, (2017),               coh p 0 2 1 p 2 1 1 2 1 1 2; 1 2; 1 2 1 ; 2 K b r n n n n F n n r b                        . (6)