Issue 38

G. S. Serovaev et alii, Frattura ed Integrità Strutturale, 38 (2016) 392-398; DOI: 10.3221/IGF-ESIS.38.48 393 The methods of numerical simulation allow us to investigate the processes occurring in structures during application of different damage detection methods and receive the answers to many questions without performing expensive experiments. Moreover, it is due to numerical methods that we can track the evolution of different parameters with a change of the defect size. Studies of objects with delamination type defects have received considerable attention in scientific literature. Analytical description of the free mode model of delamination in a beam, which allows the mutual penetration of volumes in the zone of delamination, is given in [6]. This disadvantage is eliminated in the constrained model where areas in the delamination zone have equal vertical displacements [7]. Only few low frequencies are considered in these studies. The presence of multiple delaminations is discussed in [8]. Experimental investigations of vibrations of delaminated structures are performed in [9, 10]. Excitation of vibrations and measurement of the signal must be implemented using actuators and special measurement devices (sensors). The capabilities of using piezoelements for registration of high frequency vibrations are described in [11]. The Authors of this research capture vibrations of the plate up to 40 kHz with the help of a piezoelectric sensor which proves the possibility of using such devices for measuring high frequency vibrations. The method of electromechanical impedance (EMI), which is one of the vibrational methods of damage detection applied to delaminated composite beam, is given in [12]. The need to place the actuator close to the defect and hence the requirement of arrangement of a dense grid of piezoelectric devices or prior knowledge of the location of defect is the main disadvantage of such a kind of damage detection method. On the other hand natural frequencies give integral characteristic of the object of research. In [13-15] vibrations of delaminated structures with different geometries such as beam, cylindrical and conical shells are studied. A numerical study of the dynamic parameter response to defects of different sizes is carried out in the framework of three models of delamination in a composite structure, the advantages and drawbacks of each model are estimated in the process of simulation. First, we consider a free mode model of delamination, in which the adjacent volumes in the zone of the defect are not coupled with each other and therefore they are assumed to be mutually penetrable. In the second, the so-called constrained model, the coincident nodes in the zone of delamination are coupled by one component of the displacement, while other components of the displacement remain independent. The last one is the model that takes into account the contact forces. The free mode and constrained models allow us to perform a modal analysis for computing the eigenfrequencies of the structure. With contact forces taken into account the problem becomes nonlinear. The algorithm for calculating the spectrum of eigenfrequencies in the nonlinear problem includes the following steps: setting of impact load, transient analysis, measurements of signal at a certain point of the structure with a subsequent conversion of the received response of the structure from the amplitude-time to amplitude-frequency domain with the help of the Fourier Transform N UMERICAL MODELING numerical study was performed on a square plate of size L = 0.15х0.15 m, made of a layered composite material. The thickness of one layer is h 1c = 0.0003 m. There are a total of n = 15 layers across the thickness of the plate therefore the total thickness is h 1c *n = 0.0045 m (Fig. 1). The square shape delamination of size Ld m is located between 6 and 7th layers. The center of delamination coincides with the center of the plate. The composite material is modeled as a homogeneous body with the following orthotropic effective mechanical properties (reinforcing direction is not taken into account). E x = 24 GPa, E y = 18 GPa, E z = 6 GPa, G xy = 4 GPa, G yz = 3 GPa, G xz = 3 GPa, v xy = 0.15, v yz = 0.18, v xz = 0.42, ρ = 1800 kg/m 3 . The presence of composite layers is taken into account only by geometrical dimensions. Finite elements with quadratic approximation providing more accurate results in terms of out of plane strains were used in the simulation. The principle of virtual work is used for the mathematical formulation of the problem i i i i i ij ij V S V u F u dV p u dS dV t 2 2 (               (1) where i u - components of the displacement vector, j i ij j i u u x x 1 ( ) 2        - components of strain tensor, ij ijkl kl C    - components of stress tensor, ijkl C - stiffness tensor,  - material density, i F - components of body forces vector, i p - components of surface tractions vector. A