PSI - Issue 32

D.A. Oshmarin et al. / Procedia Structural Integrity 32 (2021) 158–165 Oshmarin D.A., Iurlova N.A., Sevodina N.V., Iurlov M.A. / StructuralIntegrity Procedia 00 (2021) 000 – 000

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viscoelastic layer in case of possibility of such a modification are considered as such approaches (Mead, 1999; Nashif et.al., 1985). Techniques of increasing dissipative properties of structures that based on application of viscoelastic materials are applied in aircraft, airspace and automotive industries. Theyareusedforpassivedampingofvibrations. However, an efficiency of damping depends not only on properties of a viscoelastic material but also on its volume which is constrained by requirements put on weight and dimensions of an original structure. Supplementing viscoelastic elements with additional layers that induce additional strains in structures at vibrations leads to increasing of dissipative properties. This technique is considered as a modification of conventional approach to damping of vibrations by means of structural elements made of viscoelastic materials. Layersmade of typical structural materials having no viscoelastic properties (steel, aluminum, etc.) can be used as such additional ones. However,theselayerscanbeperformedof functional materials (shape memory alloys, piezoelectric materials, electro strictive materials, etc.) and piezoelectric materials are the most widespread within the frameworks of the approach under consideration (Hagood and Von Flotow, 1991, Stanway et al., 2003). Dissipation of mechanical energy of vibrations transformed into electric energy is provided with the aid of passive external electric circuits connected to electrodes of a piezoelectric element and tuned on damping of vibrations at a corresponding frequency. Thiseffectthereforeresultsinincreaseofdissipativepropertiesof a structure. Inthiscasepiezoelectric elements can act as transformers of mechanical energy of vibrations into electric energy (due to direct piezoelectric effect) which is then dissipated as a heat or an electromagnetic radiation. Fromtheotherhandtheycanact as actuators which supplied with electric potential that induces deformations of piezoelectric elements during vibrations (due to inverse piezoelectric effect) (Benjeddou, 2001). The aim of the current paper is in demonstration of an influence of a joint application of a viscoelastic element and a shunted piezoelectric element in a structure on an increase of its dissipative properties. 2. Mathematical statement of the problem The object under study is a piece-wise homogeneous solid body of the volume 1 2 3 V V V V    , where 1 V is the volume of its elastic part, 2 V is the volume of viscoelastic part and 3 V is the volume of piezoelectric elements. In the examined body, the electrode-covered part of the surface 3 el S of volume 3 V can be connected in the general case to the electric circuit of arbitrary configuration, consisting of the resistive, capacitive and inductive elements. The variational equation of the motion for the object under consideration is formulated based on the relations of the linear theory of elasticity and Maxwell's equations for a medium in the quasi-static approximation (Iurlova et al., 2019; Karnaukhov and Kirichok, 1988; Parton and Kudryavtsev, 1988) and it takes the following form (1):       1 2 3 1 2 3 1 1 0 i i ij ij i ij ij i ij ij i i i i V V V L R C u u dV u u dV D E u u dV dtdt dt C L R                                          (1) Here , i i D E are the components of the vectors of electric induction and electric field strength, respectively; ij  are the components of the symmetric Cauchy stress tensor, ij  and i u are the components of the linear strain tensor and displacement vector, respectively; 1 2 3 , ,    are the specific density of elastic, viscoelastic and piezoelectric parts,  is the electrical potential; , , L R C       are the potential difference on the inductance, resistance and capacitance elements, respectively;  is the variation of the corresponding variable. The constitutive relations for each of the considered parts of piecewise homogeneous body can take the following form: for the elastic isotropic part of the volume 1 V