PSI - Issue 43

Tatyana Petrova et al. / Procedia Structural Integrity 43 (2023) 83–88 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

85 3

2. Problem statement and obtained analytical solutions In Fig. 1 a representative volume element (RVE) of three-layer WS 2 /SU-8/PMMA nanocomposite structure is shown. The axial tensile force P (N.m) is applied to the PMMA layer, where 0 2 P h  = . The coordinate 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 nanocomposite structure, the two-dimensional stress-function method has been applied (see Kirilova et al. 2019 and Petrova et al. 2022 for more details) and as a result, two different analytical solutions (noted here as Eq. (1) and Eq. (2)) for the axial stress noted as ( ) (1) 1 1 xx x    = = in the first layer WS 2 , have been obtained. For brevity, in this paper only most important equations are given.

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

( ) 1 C x C  + 1 exp

( ) 2 

( ) 3 x C x C  + + 3 exp

( ) 4 

exp

e p x

x A −

1 

=

(1) (2)

2

4

exp(- )[ cos( )  

sin( )] exp( )[( cos( ) +   

sin( )] x M x M x A + − 

x M x M x +

1 

=

1

2

3

4

The first solution, Eq. (1) is obtained on the base of 4 real roots 1,2,3,4  of the characteristic equation, corresponding to the 4 th order ordinary differential equation (ODE) in respect to axial stress, which has been derived in Petrova et al. 2022. The second one (Eq.(2)) has been derived (Kirilova et al. 2019) on the base of 4 complex conjugate roots ( ) i     of the same characteristic equation of the ODE. The difference between solutions comes after the sign of discriminant of the quadratic equation, corresponded to the abovementioned characteristic one; it can be positive or negative, so its roots can be real, complex or mixed. The sign of this discriminant depends on the thicknesses of the layers and the properties of the layers’ materials. The constants C i and M i are integration constants in the model solutions, determined from respective boundary conditions. In the following Eqs. (3) to (5), all two dimensional stresses - axial ( i ) xx  , shear ( ) i xy  , and normal ( ) i yy  in each layer ( 1, 2, ) i a = of the considered nanocomposite structure are presented . 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: ( ) ( ) ( ) 2 1 1 1 1 1 1 1 ( ) ( ) '' ( ) ' xx yy t xy t x , y y , y y        = = = − = − (3)

1 2

  

  

2 h h c y

(

)

( ) a

( ) a

'' ( ) a    = ' 1 1 , xy h 1

0,



1 = + −





(4)

1

xx

yy

2

h

 − 

(

)

(2)     = − (5) Eqs. (3-5) 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   . The constant ( ) ( ) ( ) ( ) 1 1 2 0 A E E E   = + in the solutions depends on the value of external static load 0  and Young modulus of the first and third layer in the structure in Fig.1. In the next section all stresses are calculated and presented in 3D plots for both solutions, named here as Case 1 (Eq. (1)) and Case 2 (Eq.(2)). (2) 2 y y y h − + '' (2)     = ' 1 0 1 1 1 2 , , , where 2 xx yy t a xy y h    = = 