PSI - Issue 43

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

285

4

( ) 1 ( ) t a h h h y h h D E E   − +     = − − , 1 3 2 2 2 1 ( ) ( )    + + ( ) ( ) 2 2 3 2 ( ) 1 3 1 2 2 1 3 6 1

(6)

3

( 2 2 h D E  = − The homogeneous Eq. (4) has a characteristic bi-quadratic equation of fourth order in respect to unknown stress function 1  (Petrova et al. 2022a). The corresponding quadratic equation discriminant 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 materials from which they are made, e.g. from the coefficients 1 5 , i D i =  . It does not depend on the external tensile load 0 2 P h  = . After determining of the function 1  and it’s derivatives, all stresses in the layers of the considered nanocomposite structure can be obtained, using Eqs. (1-3). The axial stress in the WS 2 layer (the general analytical solution in the case (noted here as Case 1) of four real roots λ i (Petrova et al. 2022a), and its first derivative are presented by: ( ) ( ) ( ) ( ) 1 1 1 2 2 3 3 4 4 x exp exp e p exp C x C x C x C x A      = + + + − (7) ( ) ( ) ( ) ( ) 1 1 1 2 2 2 3 3 3 4 4 4 1 ' exp exp exp exp C x C x C x C x          = + + + (8) Eqs. (9) and (10) represent the general solution in the case (noted here as Case 2) of the complex roots i    and its first derivative: 1 1 2 3 4 exp(- )[ cos( ) sin( )] exp( )[( cos( ) sin( )] x M x M x x M x M x A        = + + + − (9) 1 1 2 3 4 ' exp(- )[ (- cos( ) - sin( )) ( cos( ) - sin( ))] exp( )[( ( cos( ) - sin( )) ( sin( ) cos( ))] x M x x M x x x M x x M x x                    = + + + + + (10) The constant A=D 5 /2D 1 is the solution of the non-homogeneous Eq. (4) and depends on the external static load and C i and M i are integration constants in the model solutions, determined by the respective boundary conditions. The delamination in the structure depend on the value of the model ISS (third equation in Eqs. (2)) in the adhesive layer. If this shear stress is greater than an ultimate shear strength (USS) in the adhesive layer SU-8 (30 MPa), the delamination takes place for some values of the external load and the geometrical parameters of the considered nanocomposite structure. Having both solutions and their first derivatives, it is easy to obtain the model ISS in both cases. A theoretical non-linear equation for the length of debonding in the structure is presented, where ( ) a USS  is the ultimate shear strength of the adhesive: ( ) ( ) ( ) a a xy USS x   = (11) The value of x for which Eq. (11) holds, is the debond length of delamination. Graphically, it represents the intersection of the ISS model curve with the straight horizontal line corresponding to the value for the ultimate shear strength. The next section presents a parametric analysis of each of the obtained solutions for the ISS at constant values of the structure layers’ material properties and constant thickness of the WS 2 layer. The following parameters will be changed: 0  - the value of the axial load; l - the length of the RVE; 2 h - the thickness of the substrate layer; a h - the thickness of the adhesive layer. The aim of this parametric analysis is to obtain theoretically the critical stress values of loading, for which delamination appears, at different layers’ geometry for the considered structure. ) E E + ( ) 1 ( ) ( ) 2 ( ) ( ) a a h h , ( ) 2 1 1 2 4 1 2 1 2 3 2 3 h D h h + E + = ( ) 1 0 5