PSI - Issue 41

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

223

4

Fixed coordinate system O’x’y’z’ is associated with the moving coordinate system Oxyz by the following relations:

x x Vt  = + , y y  = , z z  =

(1)

In rolling contact the relative slip velocities within the contact region  are determined by the formula:

( ) , u x y u x y  ( ) 2 , 1



  

  

R 

(

)

( ) ,

,

(2)

2 x y V + − 2

s x y

=

−

− 

2

x

x





where  is the angular velocity of the ball rotation in respect to the y -axis (Fig. 1), ( , ) i u x y are the elastic displacements of the contacting bodies ( 1, 2) i = in the direction of x -axis due to deformation, R is the radius of the ball, Δ is the relative slippage:

V R V  −

(3)

.

 =

In rolling the contact region consists of the slip and stick subregions. In the stick subregion the shear stress ( , ) x y  is related to the contact pressure ( , ) p x y by the following inequality: ( ) ( ) , , x y p x y    . (4)

and the slip velocity ( , ) s x y (Eq. 2) is equal to zero: ( ) , 0 s x y = .

(5)

In the slip subregion in rolling and in sliding contacts the Coulomb-Amonton law is satisfied:

( ) , x y

( ) ,

(6)

.

p x y



 =

Outside the contact region, the normal and shear stresses are equal to zero. In cyclic interaction of the system of spherical indenters and the elastic half-space the mutual effect of indenters deforming the elastic half-space is neglected. 3. Method of solution The fatigue wear model is developed here based on a macroscopic approach, in which damage at some point of the material is described by a positive nondecreasing function that depends on the stress state at this point, the number of passed cycles, and parameters describing the strength properties of the material. 3.1. Calculation of contact and internal stresses The stress state of an elastic half-space is determined by its elastic characteristics, contact pressure and shear stresses. Since the materials of the contacting bodies are considered the same, shear stresses do not influence the distribution of a contact pressure. Therefore, the problem of the contact normal and shear stress calculation is solved in two stages: first, the contact pressure is calculated, and then the shear stress is identified under sliding and rolling conditions.