PSI - Issue 43

Dragan Pustaić et al. / Procedia Structural Integrity 43 (2023) 252 – 257 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

254

3

The mark  , [-], and n , [-], designate the material parameters. In these research, the parameter,  was varied among 3/7 and 1, while the strain hardening exponent, n , was quoted in the Abstract. At the same time, the value of the stress, 0 ,  amounts to 310 MPa and Young´s modulus of elasticity, E , was taken 210 GPa. Poisson´s ratio for this material amounts to, 0.3  = , [-], but in these investigation it wasn´t used because of assumption about plane state of stress. In case of plane strain condition (for example, like in case of crack in thick plate) its influence on a magnitude of plastic zone is significant what was discussed in details in the paper given by Guo, W., (1995).

Fig. 1. a) Thin infinite plate with straight crack of length 2 a loaded with two pairs of the concentrate forces; b) fictitious elastic crack including a small plastic zone around crack tip; c) variable cohesive stresses act on a part of fictitious elastic crack.

The cohesive stresses within the plastic zone around the crack tip are changed according to non-linear law. However, the distribution law of those stresses isn ´t, known, at the beginning. Therefore, the procedure for solving this problem is as follows: the stress distribution is determined numerically (for example, by means of the Finite Element Method, FEM, program package Abaqus ), and afterward that stress distribution is approximated with an analytical expression. The procedure the authors applied in the paper, Pus taić and Lovrenić (2006) turned out to be very accurate and reliable. Equally, in the article Pustaić and Lovrenić - Jugović (2019, MSMF9) the analyti cal expression was applied which the authors Hoffman and Seeger (1985) suggested and an excellent approximation of cohesive stress distribution was obtained. That expression can be written in a form

) ( ) 1 1 n + 

( )

(

0 p r x a  

.

(3)

p x

= 

− 

In expression (3), p(x), [MPa], is a function of the two parameters, i.e. the plastic zone magnitude, p r , [m], and the strain hardening exponent, n. 2. Stress intensity factors from the external loading and from the cohesive stresses The stress intensity factor (SIF) from the external loading, ( ) ext p , K a r + [MPa m ], at the tip of fictitious elastic crack, according to the Fig. 1, was derived in the PhD Thesis, Pustaić (1990), and it amounts to

  

   

2

F b c b c + −

F b

(

)

(4)

.

ext K a r p

+ = 

+

= 

2 2

b c b c − +

b c −

π

π

b  

The meaning of the variable c is seen in Fig. 1. It changes in a range 0 . c a   In this paper, only as a numerical example, it was calculated with c = (1/2) a and a = 10 mm. We could to take any arbitrary value of variable c and to investigate how a point of acting the forces F influences to magnitude of plastic zone. The new independent variable, ( ) p p 2 , P r a r   =   +  [-], will be introduced here, according to Pustaić and Lovrenić - Jugović (2019, MSMF9), in which the magnitude of plastic zone, p , r is incorporated. It is necessary, in order to transform the expression (4), as

Made with FlippingBook flipbook maker