PSI - Issue 23

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

28

2

Nomenclature

a

n

half physical crack length, m half fictitious crack length, m

strain hardening exponent, - cohesive stress, MPa independent variable, m

b

p (x)

r p

x

plastic zone magnitude around crack tip, m

m (x, b)

remote loading of the plate, MPa

1 2 m 

yy     

Green´s function , weight function,

K ext



stress intensity coefficient from the external loading, MPa m stress intensity coefficient from the cohesive stresses, MPa m

new independent variable, -

  x 

K coh

Gamma function, -





0 

yield stress, MPa

Hypergeometric function, -

2 1 ; ; ; F z   

0 t    

0 

strain corresponds to the yield stress according to Hooke´s law Young´s modulus of elasticity, GPa

traction, ratio of the remote loading to the yield stress, - new independent variable in which the plastic zone magnitude is incorporated, it is a function of remote loading, -





E

p p 2 P r a r   



material parameter, -

1. Introduction A straight crack of length 2 a in a thin infinite plate is considered, Fig. 1a. The plate is loaded with uniformly distributed continuous loading yy      in the direction of the axis y , while the crack surface is free of loading, Fig. 1a. If the plate material is ductile metallic material, the small plastic zones around crack tips will appear and there will be no singularity in the stress distribution at the crack tip. It means that the elastic-plastic response of structure on the external loads cancels the stress singularity at the crack tip. The essence of the cohesive models lies in a fact that the stress   ,0 yy b  at the tip of fictitious elastic crack assumes an ultimate value, as it was pointed out in the papers of Chen et al. (1992), Guo (1993), Neimitz (2000) and so on.

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

  ,0 yy b  at the tip of fictitious elastic crack has an ultimate value could be

The condition that the resulting stress

written by means of an analytical expression       p ext p coh p 0. K a r K a r K a r      

(1)