Issue34

Yu.G. Matvienko et alii, Frattura ed Integrità Strutturale, 34 (2015) 255-260; DOI: 10.3221/IGF-ESIS.34.27

I II     K   K

2 arctan



M

(4)

e



e M were varied through 1 (pure mode I), 0.75, 0.5, 0.25, 0 (for pure mode II).

The values of

According to the MATS criterion (Eq. (3)), four parameters I K , II K , T and d at fracture need to be known for each mode mixity. The local strength of this brittle material and the fracture process zone length d are treated as a function of the tensile strength [8] which is equal to 2 MPa. The fracture toughness mat K is 0.24 MPa m . These values were assumed to be constant for all mixities. The fracture trajectories predicted using the MATS criterion (Eq. (3)) are consistent with those observed in the broken specimens under various mode mixities (Fig. 1).

T HE SURFACE CRACK PATH UNDER ROLLING - SLIDING CONTACT LOADING

A

ssuming the singular and non-singular ( T -stress) terms are sufficient to characterize the crack tip stress under mode I/II loading in the case of rolling-sliding contact, the tangential stress   is written in polar coordinate as follows [10]





1

3

  

  

.

-   c

c   

2

2

2

  











( , ) r

K

K

T

(5)

cos

cos

sin

sin

sin 2

cos

I

II

xy

yy

2

2 2



r

2

c xy  and

c

yy  are defined at the crack tip and their distribution is smooth enough along the crack surface. This

Here, the tractions

c yy  is then equal to

  

equation is valid, if the crack surfaces are loaded with constant pressure ( p

x y

, )

. The stress component

the – y p . Averaging the tangential stress over the fracture process zone, which is characterised by a critical distance d , the maximum average tangential stress (MATS) criterion leads to the following mathematical expression p while c xy  =− x

0 







cos 2

4 3

8 3



c   d

c 





1)  

(   T

0 ) 2 cos sin 0 d   

0 

0 

K

K

(6)

sin

(3cos

2

0 

I

II

0





2

xy

yy

cos

2

The fracture process zone size d in the above-mentioned equation can be determined for critical state 0    

as follows

0        2

0       

  

  

3

2 

  0 

 sin 2



  0 

  0 

c    xy

c   

2

2

2

0 

0 











 







T

K

K

sin

cos

cos

cos

sin

(7)

yy

I

II

d

2 2

0  is dependent on the model of a solid and can be treated for plane strain according to

In general case, the local strength

 and the T -stress which quantifies constraint in different

von Mises yield criterion as a property of both the yield stress Y

geometries and type of loading [11, 12]



 



2

2

2   







T

1

/

1

Y   T   

T

1 4

Y

,

0 

  



  

(8)

Y





2

2



1 2 

where .  .is Poisson’s ratio. According to Zafošnik et al. [10], the real contact geometry (e.g. gear tooth flanks) can be transformed into a pair of equivalent contacting cylinders with the radii corresponding to curvature radii of analyzed mechanical elements. For small

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