PSI - Issue 59

3

Ivan Shatskyi et al. / Procedia Structural Integrity 59 (2024) 407–412 Shatskyi et al. / Structural Integrity Procedia 00 (0000) 000 – 000

409

Let us introduce a system of Cartesian coordinates with the origin on the surface and the axis y directed deep into the half-space (Fig. 1). Under the assumptions made, we write the boundary value problem for a multilayer piecewise homogeneous plate on an elastic foundation (Reddy (2004)):

4 dx d u dx d u 2

y

( ) k u P x y y 

D

,

( , )    x ;

(1)

 

4

3 dx d u

y

y

( ) 0 2  

( ) 0  

D

D

(2)

,

.

3

Here y u is a component of the elastic displacement vector of the neutral surface of the plate;

3

3

    

    

H

1

   

   

   

   

 N 3 1 1

m

m



c

E

2 ( ) 

E y

    C i h y

 0

2

m v

(

 y y dy )

,

D

 h y





C

i

C

m 2

1

y ( )





1

1

1

m

i

i







2

    

    

1 2

   

   

   

   

 N 2 1 1

m

m



H

E

   i h

c

( )

E y

m v

h

 0 H

ydy

i

m 2



2

1

y ( )





1

1

1

m

i

i







,

y





C

N

E

c



( )

E y

m v

 0

h

dy

m

m 2

1



2

1

y ( )





1

m



are the bending rigidity and the ordinate of the location of the physically neutral surface;

          m i m i i 1 1 1 ,

          m i m i i 1 1 1 ,

   

   

 E y E y , ( )

y ( )

m y   ,

h h

h h



, m N 1,  ,

,

m

i

i

are piecewise stable functions, m E are Young's modules and m  are Poisson's ratios of coating materials, m h are thicknesses of layers of coating,   m i i h 1 are ordinates of interfaces               c N i i i i h H h 1 0 1 0, ; h E k y 1 1 2    is the coefficient of rigidity of the soft layer with constants of elasticity E ,  and thickness h ; ( ) x  is Dirac function.

3. Results and discussion 3.1. Solution The solution to problem (1), (2) was found in the form:

P

| | x



y 

( )

(cos

sin

| |) x

u x y

e

x

,









y

y

3 

8

D

y

where is a pinching coefficient with dimension, opposite to the length. Stresses in the layered coating correspond to the transversal displacement and calculate according to the formula: 4 /(4 ) k D y y  