PSI - Issue 36

O.L. Derkach et al. / Procedia Structural Integrity 36 (2022) 71–78 O.L. Derkach et al. / Structural Integrity Procedia 00 (2021) 000 – 000

73 3

The effective modulus concept was used to determine the elastic properties of the composite material. Therefore, the composite obtains the properties of a quasi-homogeneous anisotropic material with the averaged value of elastic moduli, which provide the fulfillment of the conditions of equivalency between stress and (or) strains in the quasi homogeneous and heterogeneous material. Due to the symmetry of the selected composite in the plane perpendicular to the fibers, the number of effective constants reduces from 9 to 6: E 11 , E 22 = E 33 are the elasticity moduli of the composite material along and across its fibers, respectively; G 12 = G 13 , G 23 are the shear moduli and μ 12 = μ 13 , μ 23 are Poisson’s ratios. The specified effective elastic constants and composite density are determined acco rding to Krawczuk and Ostachowicz (1995) from the known properties of the isotropic matrix ( m ) and fibers ( f ) at their volume fraction ν f : ( ) ( ) 11 22 , ; f m f f m f f m m m f m f f m E E E E E E E E E E E E E     + + − = + = + −

− + −

1

m E E

 

,

;

12

11

          = + = +

m

12

23

f

f

m m

f

f

m m

2    12 m m m

1

E E

− +

11

( (

) )

12 G G = m f G G G G G G G G         + + − + − − f m f m f f f

E

m

,

;

G

(1)

=

22

( 2 1

)

23

+



23

m

,

,

;

=

=

=

12

21

31

21 32

23

11 E E

22

1 . m f   = −

,

f m m      = + f

Hence, in the calculation investigations the elastic properties of the composite are considered for the quasi- homogeneous material model with the matrix of the effective elastic coefficients, which has the following form:

1

−

1

0 0 0 0 0 0 0 0 0

E

E

E E

−



− −

 

        

         

11

21 22

21 22

1

E E

E

− −

 

12 11

22

32 22

1

E

E

−



  C

12 11

23 22

22

.

(2)

= 

0 0 0

0 0 0

0 1

0 0

G

12

0 0

0 1

0

G

23

0 0 1

G

12

To develop the finite element (FE) model of the beam with local surface damage in the form of a rectangular notch of depth h C and width s , which is located at a distance x C from the attachment (Fig. 1), an 8-node brick finite element was used. The process of synthesis of the three-dimensional beam model was performed using the standard finite element method procedure. The determination of principal frequency p of flexural vibrations for both intact ( p 0 ) and damaged ( p d ) beams was made via the eigenvalue method, i.e. solution to the characteristic equation:     ( ) 2 det 0 K p M + = , (3)

where the mass [ M ] and stiffness [ K ] matrices are determined as:

      M N N d K   =  T   , e

        B T C T B d T T







=



,

(4)

e





e

e

here [ N ] is the matrix of the approximation functions of the finite element, [ B ] is the matrix of their first order partial