PSI - Issue 72

Liubomyr Ropyak et al. / Procedia Structural Integrity 72 (2025) 20–25

22

Under the accepted assumptions, we write the equation of equilibrium of a flexible plate (hard part of heterogeneous coating) on a soft elastic foundation with fading forces and moments at the infinity (Reddy (2004)):

4 dx d u

2 dx d u 2

( ) k u P x y y y    ,

,

x x x   

( , )    x ;

D

(1)

( ) k u fP x

B

4

2 dx d u

3 dx d u

du

y

y

x

( ) 0  

( ) 0  

D

D

,

(2)

,

.

( ) 0  

B

2

3

dx

In equations (1), (2) x u , y u are the displacement vector components of the neutral surface of the functionally

H   1

H

( )

( )

E y

E y

2

 

gradient coating; – are the functionally gradient elastic plate tension and bending rigidity; ( ) E y , ( ) y  are piecewise continuous functions for Young’s modulus and Poisson’s dy y 2 ( ) B  0 , y y dy )  y ( ) D C 0 2 ( 1 

1



   

   

0  H

0  H

( )

( )

E y

E y

ratio of the coating material;

– is the coating neutral surface ordinate;

y

ydy

dy



C

2

2

1

( ) y

1

( ) y





1

1





   

   

   

   

 H h

 H h

2 ( )

( ) 2(1 ( ))  E y y

1

( ) y



 

 

,

– are the stiffness coefficients of the non-homogeneous

k

dy

k

dy

y

x

E y

H

H

foundation; H , h – are the thicknesses of the hard and soft layers, respectively; ( ) x  – is Dirac function; f – is friction coefficient of the abrasive-coating pair. 3. Results and discussion 3.1. Solution The solution to problem (1), (2) was found in the form:

P

fP

| | x



| | x

x e 

( )

(cos

sin    x

| |) x

y

u x y

e



( )

,

(3)

x u x

,



y

y

3

2

B



8

D



x

y

/(4 ) k D y

y  

,

are pinching coefficients with the dimension inverse to the length.

where

k B x /

 

4

x

Using solution (3) we obtained membrane force and bending moment in the coating:

2 dx d u

fP

P

y

| | x



| | x x x e 

( )

) sgn(

( ) M x D 

( ) x

N x



(cos

sin    x

| |) x

y

e

,

,



y

y

2

2

4



y

and contact stresses on the coating interface between hard part of coating and soft layer:

fP

P

| | x





| | x

x e  

( ) x k u x  ( ) x x 

( ) x k u x y y  ( ) 

(cos

sin

| |) x

e

x





y











,

.

x

y

y

y

2

2

Stresses in the functionally gradient coating are calculated according to the formulae: