PSI - Issue 43

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

253

2

1. Introduction The thin, infinite plate is observed, in which, the straight crack of the length 2 a, [m], is built-in. The plate is loaded in-plane, with two pairs of the concentrate forces, F, [N/m], which act on the crack surface and open it, Fig. 1a. It is assumed a plane state of stress. Only the stress components, , and x y xy    , [MPa], will occur. All other components of stress tensor will be equal to zero. The forces are acting on a distance, c , [m], in relation to the crack center, symmetrically to the axis, y. All other edges of the plate, at the infinity, are free of loading. In a case of plate with finite dimensions, there will be a significant influence the free boundaries on a stress distribution in a plate and around crack tip, also. This phenomenon was analyzed in the paper of Chen et al. (1992). A serious research how the plastic constrains, in- and out-of-plane, influence to the stress distribution around crack tip, magnitude of plastic zone and crack tip opening displacement was described in the papers of author Neimitz, A., et al. (2000, 2004 and 2008). The forces, F , are monotonously increased. The small plastic zones around crack tips will appear with increase of the external loads, as it is schematically presented in Fig. 1a and 1b. In accordance with Dugdale´s idea, those small plasticized ranges, together with the real, physical crack, make the fictitious elastic crack, of length Nomenclature

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

a



longitudinal strain, -

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

0 

b

distance of the forces, F , in relation to the axis of symmetry, y , m plastic zone magnitude around crack tip the two pairs of the concentrate forces acting on the crack surface, N/m non-dimensional loading of the crack, - stress intensity factor, (SIF), from the external loading, MPa m stress intensity factor from the cohesive stresses, MPa m thickness of a plate, m

c h

E

material parameter, -



strain hardening exponent, -

r p F

n

cohesive stress, MPa

p (x)

(

)

0 F a  

independent variable, m

x

K ext

m (x, b)

Gr een´s function,

1 2 m −

( ) x 

Gamma function, -

K coh

(

)

yield stress, MPa

0  y 

; ; ; z   

2 1 F

Hypergeometric function, -

independent variable in which the plastic zone is incorporated, -

(

)

2   =   +  a r

normal stress, MPa

P r

p

p

(

) p

2 , y  at the tip of that fictitious elastic crack, will not be singular, but it will obtain an ultimate value. In the other words, it means, that the stress intensity factor, K , at that point, will be equal to zero. That condition can be written, analytically, in the form, as it was pointed out in the papers of Chen et al. (1992), Guo (1993), Neimitz (2000) , Pustaić et al. (2019) and so on ( ) ( ) ( ) p ext p coh p 0. K a r K a r K a r + = + + + = (1) The plate is made of ductile metallic material which, when plastic deforms, is isotropic and non-linear hardened. The non-linear dependence between the true stress and the true strain in the hardening range can be well described by Ramberg- Osgood´s analytical e xpression, Chen et al. (1992) ( ) 0 0 0 . n        = +  (2) 2 , a r b  + = Fig.1b and 1c. The normal stress,