PSI - Issue 43

Wilfried Becker et al. / Procedia Structural Integrity 43 (2023) 77–82 Author name / Structural Integrity Procedia 00 (2022) 000 – 000

79

3

The initial assumptions for application of 2D stress-function variation method for the considered nanocomposite structure consist in the following: 1) The axial stresses in the layers are assumed to be functions of axial coordinate x only; 2) In the adhesive interface layer the axial stress is neglected, or ( ) 0 a xx  = ; 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 2D elasticity theory). Of course, the same kind of analysis could be performed with a plane strain state. But this would not change the results significantly. Based on the abovementioned model assumptions, the following Eqs. (1) and (2) represent the model two dimensional stresses ISS - ( ) a xy  and IPS - ( ) a yy  respectively, in the interphase layer of the considered nanocomposite structure. In these equations the ISS and IPS are expressed in terms of a model solution for single stress potential function in Petrova et al. 2022 (the axial stress of the nanoclay layer is noted with 1  , function only of x ) and its first and second derivatives: ( ) ' 1 1 a xy h   = (1)

(  = + −     ) 2 h h c y 1 1 2 

( ) a yy

''  1



(2)

The axial stress 1  in the nanoclay layer (the general analytical solution of four real roots λ i in Petrova et al. (2022)) and its first and second derivative are given by:

( ) 1

( ) 2

( ) 3

( )

1 4 4 C exp x C exp x C exp x C exp x A      = + + + − 1 2 3

(3)

( ) 1

( ) 2

( ) 3

( ) 4

1 1 4 ' C exp x C exp x C exp x C exp x          = + + + 2 2 3 4 1 3

(4)

( ) 1

( ) 2

( ) 3

( )

2 4 4 4 x C exp x C exp x C exp x C exp           = + + + 2 2 2 1 1 1 2 2 3 3

(5)

(

)

The constant 0  and Young ’s modulus of the first and third layer in the structure in Fig.1, C i are the integration constants in the model solution, determined from the boundary conditions. The next section presents a parametric analysis of each of the obtained solutions for the ISS and IPS, when at constant other values of the geometrical parameters and applied load, the following parameters will be varied: ( ) a E - the value of the interphase Young modulus; l - the length of the RVE; a h - the thickness of the interphase layer. The aim of this analysis is to obtain results about the influence of the interphase properties on the interphase shear and peel stresses in it. 3. Results and discussion The geometrical dimensions and mechanical properties of the considered nanocomposite structure (Fig. 1) nanoclay-interphase-polymer are taken from Zhu and Narh, (2004) as: E (1) =178 GPa, E (2) =2.75 GPa, v (1) =0.2, v (a) =0.35, v (2) =0.35, σ 0 =350 MPa, h 1 = 1 nm, h 2 = 1 µ m. Here with superscript indices (1) - nanoclay, (a) - interphase and (2)- polymer, the respective layer number is noted. The interphase layer’s thickness and length as well as the values for interphase Young ’s modulus are varied according to Table 1, as the other properties and dimensions remain unchanged, unless otherwise is said. For the model stresses calculation and graphics representation, Mathcad Prime v.6.0 and Sigma Plot, v.13.0 have been used. ( ) 1 ( ) 1 ( ) 2 0 A E E E   = + in the solution depends on the value of external static load