PSI - Issue 50

D.A. Oshmarin et al. / Procedia Structural Integrity 50 (2023) 212–219 Oshmarin D.A., Iurlova N.A., Sevodina N.V. / Structural Integrity Procedia 00 (2019) 000 – 000

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technology is the creation and application of smart-structures. Their main feature is the ability to react to external influences by changing the characteristics of the original object. There are two main strategies for controlling the mechanical behavior of structures: passive and active. Smart systems that have only sensors in their composition are called passive. The embedding of sensors allows monitoring the condition of the structure. For the manufacture of actively-controlled or adaptive smart structures, actuators are needed that can cause deformation of the main structure. Currently, a large number of functional materials are used as actuators, among which piezoelectric materials occupy one of the central places (Tani J Takagi T., Qiu J., 1998). The widespread use of piezoelectric materials, especially for controlling the mechanical behavior of structures, is explained by the presence of direct and inverse piezoelectric effects, which allows them being used as sensors as well as actuators. The second reason is in a possibility of creating an electrically conductive surface, technologically realized for piezoelectric materials, allows connecting various options of electrical circuits to the smart-structure (Ayres J.W., et al., 1996). There exist fundamentally different approaches to the active control of structural vibrations, which are discussed in detail, for example, in monographs (Fuller C.R., et al., 1997; Preumont A., 2011). Basically, the attention of researchers is focused on the development of the control system which receives a signal generated by the sensor, implementing its change according to a given law and supplying a modified and amplified signal to the actuator. From the other hand, researchers practically do not pay attention to the factors that influence the required response of a system and affect the efficiency of the control law. In some cases, knowing the mechanisms that condition relations between applied impact and resulting response can ease the way of constructing control design and decrease its hardware requirements. This work is devoted to determining the factors that govern mechanical response of a structure with a single piezoelectric element to external mechanical and electrical impacts. 2. Mathematical statement of the problem 1 2 3 V V V V    , in which parts 1 V and 2 V consist of elastic and viscoelastic elements, and part 3 V is the element with piezoelectric properties. All elements are perfectly bonded to each other. A part of the surface 3 el S of 3 V volume is electrode, i.e., covered with negligibly-thin conductive coating. The variational equation of motion of a such body is formulated based on relations of the linear elasticity theory, the linear viscoelasticity theory and quasi-static Maxwell's equations (Iurlova et al., 2019):       1 2 3 3 1 2 3 i i el ij ij i ij ij i V V ij ij i i i i e i i V S S u u dV u u dV DE uudV q dS pudS                               (1) We consider a piecewise-homogeneous body of volume ij  i j  are the components of the symmetric Cauchy stress tensor and the linear strain tensor; i u are the components of the displacement vector; 1 2 3 , ,    are the specific densities for materials of elastic, viscoelastic and piezoelectric parts of V ; S  is the part of the whole body surface, where components of surface load vector i p are applied;  is the electric potential, e q is the surface density of free charges at the electroded part 3 el S of the surface of piezoelectric elements. The elastic elements behavior is governed by Hooke’s law : The following notations are introduced: , i i D E are the vector components of the electric flux density and electric field intensity;

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ij ijkl kl C   