We analyze approximate approaches to the modeling of the thermomechanical behavior of physically nonlinear materials under harmonic loading. The approaches are based on various harmonic-linearization schemes and the concept of complex moduli. Mechanical and mathematical features of various schemes are considered. Some modifications of the model are proposed to account for various aspects of material behavior under harmonic loading. The problems of vibration and dissipative heating of physically nonlinear bodies are formulated. The main thermomechanical characteristics are analyzed for some classes of problems.Introduction. Many structural elements and technological objects experience cyclic deformation during use or treatment [2,3,[30][31][32]61]. Loads are sometimes so high that the deformation process becomes nonlinear and mechanical elements may suffer low-cycle fatigue. Apart from purely mechanical fatigue failure, polymeric products may undergo thermal failure, i.e., softening or even melting due to vibrational heating, which is because of high hysteresis losses and low heat conductivity [32,59,60].In the cases mentioned, modeling the thermomechanical behavior of nonlinearly dissipative materials is one of the major tasks to be accomplished in evaluating the durability or working characteristics of cyclically deformed bodies.There are currently two approaches to solve such problems. One employs the constitutive equations valid for arbitrary or, at least, rather wide classes of loading histories. Quasistatic problems for inelastic bodies, specifically beams and plates, were addressed in [77,81,82]; and vibration problems for viscoplastic structures in [66,73,78,79].This approach was used to solve problems for thin-walled structural members (mainly beams) in a practically important formulation that allows for the elastoviscoplasticity and geometrical nonlinearity of the material. Such formulations in combination with well-tested computational schemes allow us to analyze mechanical effects of the interaction of physical and geometrical nonlinearities, specifically snap buckling of initially curved elements [92] and chaotic motions in such elements under harmonic loading [42,89].By using exact models under harmonic loading, we can study a number of effects accompanying vibrations such as snap buckling, drift of average plastic strain, dynamic buckling, chaotic motions, etc. Study of such processes is difficult because of

The problem of forced vibration of a hinged beam with piezoelectric layers is solved. Issues of mechanical and electric excitation of vibration and the possibility of damping mechanically induced vibration by applying a voltage to the electrodes of the piezolayers are studied. The effect of the physically nonlinear behavior of the passive layers on the response of the sensor layer and entire structure and the effect of geometric nonlinearity on the behavior of the structure and sensor layer are analyzed. The interaction of physical and geometrical nonlinearities for transient and stationary processes is studied Keywords: piezoelectric materials, physical nonlinearity, geometrical nonlinearity, forced vibration, layered beam, damping of vibration Introduction. The stringent requirements to modern complex devices have recently compelled many researchers to pay attention to the modeling and control of the vibration of flexible structures [2][3][4][12][13][14][15][16][17]. The rapid development of the modern technology necessitates changing over from the traditional methods of vibration control to new ones that allow implementing more complex and highly effective operating modes and observing numerous life and reliability criteria. As a result, the modeling of the vibration of flexible structures and their elements and study of the possibilities for controlling them have gained a new impetus and are aimed at the development of systems with improved or qualitatively new characteristics. The ultimate goal of such studies is to create a new generation of active adaptive materials capable of responding to an external load according to predefined criteria or a program and having the functions of self-checking, self-diagnostics, and self-restoration [19].Currently, structronics is the most intensively developing division of mechanics. It is concerned with electroelastic systems, active materials, and control of the vibration of structures made of various materials [22]. For example, there is a variety of materials having specific properties (piezoelectric materials, materials with shape memory, materials with electrostrictive, electromagnetoelastic, and other electrorheological properties) that are widely used for the active control of vibration [19]. Of them, the most popular are piezoelectric materials used as sensors or actuators due to the direct and inverse piezoelectric effects [5,6,17].Piezoelectric elements are most often used as distributed sensors or actuators (or their combinations) [1-4, 13-16, 21]. Modeling the vibration and control of elements with such distributed sensors and actuators is, as a rule, reduced to problems for layered beams, plates, or shells containing or partially or completely covered with piezoactive layers [1-4, 13-16, 21-23]. A piezoelectric layer can be used as a distributed sensor due to the direct piezoelectric effect, and by applying a high external voltage to this layer, we can excite or control (what an actuator does) vibration due to the inverse piezoelectric effect. These vibratio...

No abstract