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

75 5

3. Results of Numerical Investigations and Their Analysis A set of calculation experiments was performed to determine the influence of the notch of width s = 5 mm on the principal frequency of flexural vibrations for the composite beam of l = 1 m with its cross-section of b × h = 50 × 25 mm using the developed FE model of the beam. The following properties of the components of the carbon fiber-reinforced plastic, namely, its matrix ( m ) from polyimide and carbon fibers ( f ), are taken as in Krawczuk and Ostachowicz (1995): E m = 2.756 GPa, E f = 275.60 GPa, μ m = 0.33, μ f = 0.20, G m = 1.036 GPa, G f = 114.80 GPa, ρ m = 1600 kg/m 3 , ρ f = 1900 kg/m 3 . The effective elastic moduli and effective density of the composite material were determined from Eq. (1) for the volume fraction of the fibers ν f = 0.5. The investigations on the determination of the influence of the fibers ’ orientation on the frequency variation ( ) 0 1 100% d d p p p  = −  were performed for the principal mode of vibrations of the damaged beam, where p d is the natural frequency of vibrations of the damaged beam. The calculations were carried out for the values of relative depth C a h h = = 0.2; 0.4 and 0.6 with its relative position C C x x l = = 0.1. Figure 3 illustrates the dependencies of the principal frequency variation Δ p d of flexural vibrations for the damaged beam on the angle of the fibers θ . The results of their analysis indicate that the maximum reduction of the principal frequency of vibrations is observed at θ = 0 deg, which is consistent with the largest flexural stiffness of the beam. Its lower flexural stiffness is at θ = 90 deg. However, the obtained calculation data indicate that the minimum reduction of the principal frequency of flexural vibrations is at θ ≈ 50 deg. The possible reason for such phenomenon is the influence of shear deformation in the composite material since the values of elastic moduli of the matrix and fibers of carbon-fiber-reinforced plastic differ significantly. To verify it, the dependencies of the principal frequency variation Δ p d of flexural vibrations for the beam with 10 times increased shear modulus of the matrix have been determined. Figure 4 shows the dependencies of the principal frequency variation of flexural vibrations on the fibers ’ orientation of the damaged ( a = 0.6; C x = 0.1) beam at the initial value and 10 times increased values of shear modulus of the matrix. Such increase defines the shear compliance reduction of the matrix of composite material. The analysis of the presented data implies that the minimum value of the principal frequency variation Δ p d of flexural vibrations at the increased shear modulus of the matrix is practically similar as for the case with its initial value but it is attained at the angle of the fibers θ = 90 deg. Moreover, with 10 times increase of the shear modulus of the matrix, i.e. composite material shear compliance reduction, the character of the specified dependence is similar to the obtained ones in Krawczuk and Ostachowicz (1995), Song et al . (2003), and Kisa (2004) using the beam models built without considering the shear deformation employing the classical Euler Bernoulli hypothesis.

Fig. 3. Dependencies of the principal frequency variation Δ p d of flexural vibrations of the damaged beam on the angle of the fibers θ for a = 0.2 (1); 0.4 (2) and 0.6 (3) at C x = 0.1.