PSI - Issue 41

Irina Goryacheva et al. / Procedia Structural Integrity 41 (2022) 220–231 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

224

5

The contact pressure ( , ) p x y is calculated from the Hertz theory:

1

2 x y

2 2

 

+ 

( ) , p x y p

(7)

1 H =  −

 

2

a

1 3

1 3

(

) 2  − 

   

   

   

3 1 2 PR

2

3

PE

  

(8)

p

a

,

=

=

(

)

H

2

E

3 2

2

2

1

R 



−

H p is the maximum value of contact pressure, R is the

Here a is the contact radius, , E  are the elastic moduli,

ball radius. The shear stress ( , ) x y  in the contact region is calculated using a numerical-analytical approach based on the variational method (Goldstein et al. , 1982; Meshcheryakova and Goryacheva, 2021) which leads to minimization of the following functional: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , , , F x dxdy s p x y s x y x y s y        = −      , (9) where the slip velocity function is expressed as a function of shear stresses (Goldstein et al. , 1982). The internal stresses within the elastic half-space are calculated using the distributions of contact normal and shear stresses and the Boussinesq and Cerruti potentials (Johnson, 1985). 3.2. Modelling of damage accumulation and fatigue wear We assume that the cyclic loading of the elastic half-space is provided by identical rollers located at a sufficiently large distance from each other (which allows us to neglect their mutual influence on the distribution of contact and internal stresses), and the rate ( ) q z of damage accumulation at the depth z depends on the principal shear stress amplitudes ( ) max z   and is calculated by the formula (Collins, 1981):

m

( ) max z 

(

)

, Q z N





  

  

( )

(10)

q z

c

.

=

=

N

p



H

Here ( ) , Q z N is the damage accumulated over N cycles at the depth z, с and m are the parameters which describe the

strength properties of the material, and they are determined experimentally. The following relation is used to calculate the principal shear stress amplitude:



  

(

)

( ) z

(

)

(

)

,

(11)

, max 1 2 x y 

, , x y z

, , x y z



= 

−



1 

3 

max

where (

) 1 , , x y z  and ( ) 3 , , x y z  are the maximum and minimum values of the principal stresses at the point with ) , , x y z . Since the rollers are located at a large distance from each other, the principal stresses amplitudes

coordinates (

coincide with their maximum values at a fixed depth.