PSI - Issue 59

Hryhorii Habrusiev et al. / Procedia Structural Integrity 59 (2024) 494–501 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

498

5

w



N

1 0   n n a 

             

  0 K J a d  q

,

1, q N 

q



 

(15)

;





n

q

k

1

q        a

a       

a

a

  0 r

q K r  

q w r  

;

.

J

r d

r

q J r

d

r

* 

0

0

0

Introduce the notation

1

(16)

a

a

.



*

n

n

2

Rk

1

Using relation (16) from (15) we will have the system of N linear algebraic equations for unknowns * n a We assume that the radius of contact zone ( a ) is known and use the indenter equilibrium condition

  ,0 a zz r r dr P     .

2

(17)

0

The last equation specifies relations between the focal parameter of parabola R and applied force P . Using (17), (16) and (14), find

1

1

1 k P



(18)

.



2 2 R

N 



* a K

n n

1

n



Taking into account (12), (18) and (16) with the help of (14), finally obtain the distribution of contact stresses under the indenter and vertical displacements of points at the upper boundary plane:

n        r a



N

N

* n   a



      J r d      

* n a J

1 k P

P

0

0

n

  ,0 r

  ,0

.

1

1

n

u r

(19)

n













,

0

zz

2

z

2

N 

N 





a

* n n K

* a K

n n

1

1

n

n





4. Numerical examples The coefficient

1 k characterizes the influence of initial strains on stresses and displacements (19) and depends on the structure of elastic potential of the prestressed plate. In particular, in the case of Bartenev – Khazanovich potential ( Guz’ and Rudnitskii , 2006),   7 2 2 1  , where  is Poisson’s ratio, and E is Young’s modulus of the layer material. As is seen from presented relation, 1 k  as 3 1 3 1 0    , i.e., as 1 0.693 cr     . The value of cr  corresponds to the surface in stability at uniform biaxial compression. Here, as follows from relations (16), (18), (19), the vertical displacements of points of the boundary plane of layer grow without bound, and there are no contact stresses. Hence, we observe here the following mechanical effect: as 1  approaches the critical value cr  , we observe in the layer phenomena of the “resonance character,” which were revealed by G uz’ in problems of the brittle fracture of materials with initial stresses. A similar effect is observed in bodies with a harmonic-type elastic potential ( Guz’ and Rudnitskii , 2006), for which the coefficient 1 k takes the form 1    1 3 3 1  1 k E 