PSI - Issue 16

Volodymyr Panasyuk / Procedia Structural Integrity 16 (2019) 3–10 Volodymyr Panasyuk / Structural Integrity Procedia 00 (2019) 000 – 000

6

4

Using the above approach Andreikiv (1982) received equations similar to (3) and (4) for the three-dimensional problem of the theory of cracks. Ivanitskii and Kun (2013) generalized Irwin formula (2) for the case of three-dimensional as formula (5). If in this equation it is assumed that n 1 = n 2 = n 3 = 4 than in this case the active load from (5) and experimental data agree satisfactorily.

n

n

n

1                c c K K K K 2 I θ IIθ I II    

  



K K

3

,

(5)

1



IIIθ

c

III

where n i (i = 1, 2, 3) are constant. So, the linear fracture mechanics already have a definite completion. Now the main task in this field of science is the development of effective methods for the experimental determination of the static fracture toughness of structural materials ( K I с , K II с , K III с ).

4. Non-linear fracture mechanics,  c -model

If the characteristic linear size of the prefracture zone  l in Fig. 4 a is commensurate with the crack initial length l 0 than the application of linear fracture mechanics concepts is no longer valid. Then deformational criteria in particular  c -model should be used. The essence of the  c -model is the following. Zones of non- elastic deformations (∆ l ) of the cracked deformed body that appear near the crack tip are replaced by cuts of ∆ l value. The opposite edges of these cuts are attracted by stresses  0 (  0 =  T or  0 = ½(  T +  B ), where  T and  B are material yield strength and strength limit respectively). Outside of these cuts, the material is considered as elastic. The fracture takes place when the crack tip opening displacement reaches value I δ δ   p c , where I δ c is a constant value for the given material. This value and stresses  0 are connected with the surface energy of the body (  ) by the equation σ 0 δ I c = 2γ.

Fig. 4. A plate with central crack (2 l 0 ) and model plastic zones (  l );  р is the crack tip opening displacement under load p (a); dependence of p * /  0 on l 0 / d * : 1 – by formula (6); 2 – by Griffith formula (b).

The solution of Griffith problem (Fig. 4 а ) according to  c -model is as follows:

 

 

d

0 2 σ arccos exp π    

p

,

,

(6)

  

  c d E



I 0 ( π δ ) /(8σ )

  



l

0