PSI - Issue 43

Elisaveta Kirilova et al. / Procedia Structural Integrity 43 (2023) 282–287 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

284

3

system Oxy is placed at the left end of the structure with a length l , the y-coordinates for the layers are: 2 2 2 1 = = + = + + , , a t a b h c h h y h h h . For the considered in Fig. 1 polymer nanocomposite structure, the provided by Kirilova et al. (2019) and Petrova et al. (2022a) a two-dimensional stress-function method has been applied. As a result, two different analytical solutions (Case 1 and Case 2) for the interphase shear stress ( ISS) in the middle layer of the structure have been obtained.

Fig. 1. RVE of three-layer WS 2 /SU-8/PMMA nanocomposite structure.

The initial assumptions of two-dimensional stress-function method for the considered nanocomposite structure are following: 1. The axial stresses in the layers are assumed to be functions of axial coordinate x only. Physically, no load transfer occurs over the layers ends. 2. In the adhesive interface layer the axial stress 0   ( ) a xx is neglected. 3. All stresses in the layers (axial, normal (peel) and shear stresses) are determined under the assumption of the plane stress formulation (standard constitutive strain-stress equations from the two-dimensional elasticity theory). Eqs. (1) and (2) represent the 2D stresses ( ) i xx  , ( ) i xy  , ( ) i yy  in each layer 1 2 ( , , ) i a = of the considered nanocomposite structure . The latter are expressed in terms of a single stress potential function (the axial stress of the tungsten disulphide layer is noted with 1  , function only of x ) and its first and second derivatives:

x

) t y y

) y y

1 2

xx

yy

( xy t

( ) = = 1 ( )     ,

(

2

(1)

1 ( )

1 ( )

''   ,

1  '

= −

= −

1

1

1

( xy h h c y h     =   ) 1 1 1 1 '' ( ) ' , a

a xx

a yy

  

2

0

( )

( )

,



1 = + −





(2)

2

h h

( y y y h − + ) t a

1 y

where

 − 

xx

yy

xy

(3)

 

2 ( )     = − ,

2 ( )

2

2 ( )     = '' ' ,

,



=

=

1



0

1

1

2

2

Application of two-dimensional stress-function method and minimization of complementary strain energy leads to Eq. (4) which is a 4 th order differential equation, in respect to the axial stress of graphene layer 1  , with constant coefficients i D . The Eqs. (4)-(6) depend on the thicknesses of the layers and the properties of the materials from which they are made such as Young module and Poisson ratio, as well as the magnitude of the applied static tensile load - ( ) ( ) 0 , , , i i i h E   . ( ) 2 1 3 4 1 1 1 5 2 2 2 2 0 '' IV D D D D D    + − + + = (4) ( ) 1 2 1 2 1 2 ( ) h h D E E  = + , ( ) ( ) ( ) ( ) ( ) 2 5 2 2 2 2 2 1 1 1 2 2 2 2 1 1 1 2 2 6 15 10 3 6 4 20 120 12 a t a t a a a h h h h h D h h y h y h h h h h E E E     = + − + + + + + +     (5)