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A unified approach to the analysis of the nonstationary vibrations of piezoelectric ceramic plane layers, cylinders, and spheres taking into account the functional inhomogeneity and viscoelasticity of the material is proposed. The standard model of vibration damping and the Kelvin–Voigt viscoelastic model are considered. The proposed approach makes it possible to study the transition of the transducer to a static state or to a steady-state vibration mode under nonstationary perturbations. The effect of functional inhomogeneity on the nonstationary vibrations of a piezoelectric element is analyzed. The damping of the vibrations and viscoelastic axisymmetric vibrations of a radially polarized cylinder under electrical and mechanical perturbation using the Heaviside function is calculated. The dynamics of damping is analyzed, and the obtained results are compared with the static solution to validate the results. The generation of voltage by a piezoceramic cylinder under mechanical axisymmetric loading is also studied taking into account the viscoelastic properties of the material. It was found that the time to reach the steady-state mode for mechanical and electrical loads is almost the same. [ABSTRACT FROM AUTHOR]

The problem of analysis of creep strains in thin-walled tubular elements made of linearly viscoelastic materials under uniaxial tension, pure torsion, and combined tension with torsion is solved and experimentally validated. The analysis is based on the use of viscoelasticity models in the form of proportionality of deviators. The creep kernels are represented by fractional-exponential functions. The longitudinal, transversal, shear, and volume creep strains are calculated. [ABSTRACT FROM AUTHOR]

A combined numerical-and-analytical technique for studying the forced geometrically nonlinear vibrations of laminated shells of revolution made of piezoelectric viscoelastic materials under electrical and mechanical loadings with allowance for transverse shear strains and temperature of dissipative heating is proposed. The technique is based on the finite-element method in the variational statement and the harmonic linearization method. The results of numerical modeling of a hinged three-layer cylindrical panel consisting of similar outer piezoelectric layers and inner passive viscoelastic layer are presented. The behavior of the deflection in the neighborhood of the first resonance for cylindrical panels of different thickness is studied. The temperature field of vibroheating of the viscoelectroelastic cylindrical panel under electrical loading is analyzed numerically. [ABSTRACT FROM AUTHOR]

The problem of forced resonant vibrations and dissipative heating of a hinged viscoelastic elastomeric plate under membrane preloads is given. The case of square and rectangular plates is considered. The effect of elastomer properties, preload, and heat exchange conditions on the amplitude- and temperature–frequency characteristics of the forced vibrations of the plate and the achievement of critical temperature at which the material softens is studied. [ABSTRACT FROM AUTHOR]

The problem of creep strains and stress relaxation analysis in thin-walled tubular members made of linear viscoelastic materials under combined tension and torsion loading is solved. The solution is obtained using viscoelastic models in the form of superposition of shear and volume creep. The creep and relaxation kernels are represented by fractional-exponential functions. The solutions are validated experimentally by analyzing longitudinal, transverse, and shear creep strains as well as the relaxation of normal and tangential stresses in thin-walled tubular members made of organic glass and high-density polyethylene. [ABSTRACT FROM AUTHOR]

The problem of the forced resonant vibrations and dissipative self-heating of a hinged flexible beam with piezoelectric sensors and actuators is formulated taking into account the in-plane shear strain. The effect of geometrical nonlinearity, in-plane shear strain, and heat transfer conditions on the beam surfaces on the amplitude- and temperature-frequency characteristics of the beam as well as on the thermal failure of the system is studied. The possibility of active damping of the flexural vibration mode with piezoactuators and sensors is examined. [ABSTRACT FROM AUTHOR]

The opening of a mode I crack in a polymeric composite with viscoelastic properties subject to tensile loads is solved within the framework of nonlinear fracture mechanics. The modeling of the viscoelastic properties of the composite and crack is analyzed. Techniques for studying the deformation of a cracked composite and determining safe loads under which the crack is at rest are proposed. The efficiency of the techniques is evaluated by comparing the computed results with experimental data for polymeric composites used in modern aircraft. [ABSTRACT FROM AUTHOR]

An algorithm for studying the subcritical growth of a crack in a viscoelastic body with a small cohesive zone is constructed. The algorithm is illustrated by numerical solutions of several problems. The problem statement takes into account concentrated forces acting along normals to the crack. These forces can change the crack growth rate and cause the contact of the crack faces, which strongly affects kinetic curves. It is studied also how the shift of the line of application of the compressive forces relative to the central normal to the crack influences the kinetic curves. [ABSTRACT FROM AUTHOR]

The problem of determining the parameters of fractional-exponential heredity kernels of nonlinear viscoelastic materials is solved. The methods for determining the parameters that are used in the cubic theory of viscoelasticity and the nonlinear theories based on the conditions of similarity of primary creep curves and isochronous creep diagrams are analyzed. The parameters of fractional-exponential heredity kernels are determined and experimentally validated for the oriented polypropylene, FM3001 and FM10001 nylon fibers, microplastics, TC 8/3-250 glass-reinforced plastic, SWAM glass-reinforced plastic, and contact molding glass-reinforced plastic. [ABSTRACT FROM AUTHOR]

The relationships between the hereditary and creep kernels are established. The hereditary kernels define the scalar properties of isotropic linear viscoelastic materials in a combined stress state. The creep kernels are obtained in uniaxial-tension and pure-torsion tests. The constitutive equations are chosen so as to meet the hypothesis of proportional deviators. The problems of analyzing the creep deformation and stress relaxation of thin-walled tubular specimens under combined tension and torsion are solved and tested experimentally [ABSTRACT FROM AUTHOR]

The forced resonant axisymmetric vibrations and self-heating of a closed spherical shell, an infinitely long cylindrical shell, and a ring with piezoelectric sensor and actuator are studied. The effect of the temperature dependence of the complex characteristics of the passive material on the vibration amplitude and self-heating temperature is analyzed. The possibility of active damping of these vibrations with piezoelectric sensor and actuator is shown [ABSTRACT FROM AUTHOR]

A method for determining the parameters of the hereditary kernels for nonlinear viscoelastic materials is tested in conditions of pure torsion. A Rabotnov-type model is chosen. The parameters of the hereditary kernels are determined by fitting discrete values of the kernels found using a similarity condition. The discrete values of the kernels in the zone of singularity occurring in short-term tests are found using weight functions. The Abel kernel, a combination of power and exponential functions, and a fractional-exponential function are considered [ABSTRACT FROM AUTHOR]

The problem for a fluid-saturated porous viscoelastic material is a direct analogy of the problem for a saturated elastic material solved earlier. The material is theoretically described using Biot's classical linear model. The system of coupled integro-differential equations for the displacements of the skeleton and fluid is derived using the Volterra principle and Rabotnov's operator method. The propagation of linear elastic waves in a half-space subject to a short harmonic pulse of given frequency is studied. The problem is solved using the Laplace transform method. The analytical solution in the frequency domain is obtained taking into account the viscoelastic properties of the skeleton. The inverse transformation procedure is described in detail [ABSTRACT FROM AUTHOR]

The forced resonant vibrations and self-heating of a clamed thermoviscoelastic beam with piezoelectric actuators are studied considering geometrical nonlinearity of the second order. The viscoelastic behavior of passive (no piezoelectric effect) and piezoelectric materials is described in terms of long-term and complex moduli. To solve the nonlinear problem of electroelasticity and heat conduction, the quasilinearization method is used in combination with the discrete-orthogonalization and finite-difference methods. The effect of geometrical nonlinearity, boundary conditions, and heat-transfer conditions on the dynamic characteristics, self-heating temperature, and active damping of the beam with actuators is studied [ABSTRACT FROM AUTHOR]

The basic results of establishing the foundations of the mechanics of fracture of homogeneous materials compressed along cracks and inhomogeneous (composite) materials compressed along interface cracks are analyzed. These results were obtained using elastic, plastic, and viscoelastic material models. This review consists of three parts. The first part discusses the basic concept that the start (onset) of fracture is the mechanism of local instability near the cracks located in a single plane or parallel planes. The fracture criterion and the basic problems arising in this division of fracture mechanics are also formulated. Two basic approaches to establishing the foundations of the mechanics of fracture of materials compressed along cracks are outlined. One approach, so-called beam approximation, is based on various applied theories of stability of thin-walled systems (including the Bernoulli, Kirchhoff-Love, Timoshenko-type hypotheses, etc.). This approach is essentially approximate and introduces an irreducible error into the calculated stresses. The other approach is based on the basic equations and methods of the three-dimensional linearized theory of stability of deformable bodies for finite and small subcritical strains. This approach does not introduce major errors typical for the former approach and allows obtaining results with accuracy acceptable for mechanics. The second part offers a brief analysis of the basic results obtained with the first approach and a more detailed analysis of the basic results obtained with the second approach, including the consideration of the exact solutions for interacting cracks in a single plane and in parallel planes and results for some structural materials. The third part reports new results for interacting cracks in very closely spaced (or coinciding, as an asymptotic case) planes. These results may be considered a transition from the second approach (three-dimensional linearized theory of elastic stability) to the first approach (beam approximation). This is how the accuracy of results produced by the first approach is evaluated and the boundary conditions near the crack tip are established in the second approach. [ABSTRACT FROM AUTHOR]

Equations describing the growth of a mode III crack in a composite material with linear viscoelastic components are derived within the framework of nonlinear fracture mechanics. Two assumptions are made: the length of the fracture process zone is constant or the stresses in this zone are constant. Theproblem-solving procedure developed is used to find a numerical solution in the form of kinetic curves. The results obtained are analyzed [ABSTRACT FROM AUTHOR]

A method of determining the parameters of the hereditary kernels in a viscoelastic model with time-independent nonlinearity is tested. The parameters are determined by fitting the discrete values of the kernels that are obtained considering the similarity of isochronous creep curves and instantaneous deformation curve. The discrete values of the kernels in the zone of singularity are found using weight functions. The Abel function, a combination of power and exponential functions, and fractional-exponential functions are used as hereditary kernels. The method is tested by analyzing the creep, creep recovery, and stress relaxation in laminated composites, polymeric binders, and fiber reinforcements under uniaxial tension [ABSTRACT FROM AUTHOR]

The active damping of the resonant vibrations of a hinged flexible viscoelastic rectangular plate with distributed piezoelectric sensors and actuators is considered. It is shown that it is possible to considerably decrease the amplitude of resonant vibrations by choosing the appropriate feedback factor. The collective effect of geometrical nonlinearity and dissipative properties of the material on the effectiveness of active damping of the resonance vibrations of rectangular plates with sensors and actuators is analyzed [ABSTRACT FROM AUTHOR]

The thermal instability of a rectangular prism under vibrational heating is studied solving a coupled problem of thermoviscoelasticity and using finite-element modeling. The prism has copper and polyethylene or polymethylmethacrylate layers and undergoes high-frequency mechanical or kinematic compression. It is established that in the case of polyethylene, thermal instability occurs under mechanical loading and does not occur under kinematic loading. In the case of polymethylmethacrylate, thermal instability occurs under both kinds of loading. This is because of the increase in the shear and bulk loss compliances with temperature for each of the polymers. At the transformation temperature, shear dissipation goes over into bulk dissipation, which is predominant in rather thin polymeric layers. Dynamicity influences the critical loads both qualitatively and quantitatively [ABSTRACT FROM AUTHOR]

new approach is followed to study the effect of mixed mechanical boundary conditions on the effectiveness of active damping of the forced resonant vibrations of thermoviscoelastic orthotropic plates. The problem is solved by the Bubnov-Galerkin method. Formulas for the voltage that should be applied to the actuator to damp the first vibration mode are derived. It is shown that the mechanical boundary conditions, the dissipative properties of the material, and the dimenstions of the sensors and actuators have a strong effect on the effectiveness of active damping of the vibrations of plates [ABSTRACT FROM AUTHOR]

The well-known equations of photoelasticity of linear viscoelastic bodies are used to describe the photoelastic behavior of a viscoelastic orthotropic plate with a crack. Expressions for the stress intensity factors (SIFs) at the crack tip are obtained using photoelastic measurements. The time dependence of the SIFs is analyzed and shown to be determined by the angles between directions of the crack and tension [ABSTRACT FROM AUTHOR]

The paper addresses subcritical growth of a crack in a thin isotropic plate made of an aging viscoelastic material with time-dependent elastic modulus. The behavior of the material is described by Arutyunyan's creep theory. To simulate fracture, a modified Leonov-Panasyuk-Dugdale model and a critical crack opening displacement criterion are used. An equation describing the subcritical growth of the crack is derived assuming that Poisson's ratio is constant. As an example, the critical loads are determined, and curves of subcritical crack growth are plotted for a specific material. The results are compared with the case of constant elastic modulus [ABSTRACT FROM AUTHOR]

The flutter of a viscoelastic plate in a supersonic gas flow is studied. A technique and algorithm for numerical solution of nonlinear integro-differential equations with weakly singular kernels are elaborated. The critical flutter speed of viscoelastic plates is determined [ABSTRACT FROM AUTHOR]

The forced monoharmonic vibrations and self-heating of a circular thermoviscoelastic plate with piezoelectric sensor and actuator are studied. The viscoelastic behavior of the passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli. The problems of electroelasticity and heat conduction are solved numerically, assuming that the mechanical load is unknown. The effect of self-heating on the active damping of the vibrations of the plate is analyzed [ABSTRACT FROM AUTHOR]

The relaxation properties of a two-component material are determined depending on time, volume fraction, and type of reinforcement, and the relationship among them. The type of reinforcement is determined by the aspect ratio of the ellipsoid of revolution that models the inclusion. The effective moduli of the composite are determined from the relaxation properties of the components. It is assumed that the composite components are made of isotropic viscoelastic materials with volume expansion and shear characteristics described by two Rabotnov’s fractional-exponential functions with different orders of fractionality. To obtain the solution in the time domain, its fractional rational representation in the frequency domain is used. Optimizing the parameters of this representation and transforming the parameters of the solution to the time domain make it possible to obtain solutions in compact form in terms of relaxation kernels [ABSTRACT FROM AUTHOR]

The creep strains in linear viscoelastic materials under nonstationary loading of various types (incremental loading, complete unloading, and cyclic loading) are determined. Boltzmann-Volterra hereditary theory with fractional exponential kernel is used. Nonstationary loads are specified by Heaviside functions. The calculated results are validated by experimentally determining nonstationary creep strains of glass-reinforced plastic, plastic laminate, polymer concrete, duralumin, and nylon. [ABSTRACT FROM AUTHOR]

The relaxation moduli of a composite are determined. The relaxation of its components is described by various few-parameter kernels: Mittag-Leffler functions of different orders and Rzhanitsyn kernel. It is assumed that the composite components are made of model materials with volume relaxation. The Laplace transform and fractional rational approximation are used to develop an algorithm for reducing the relaxation functions of the composite to one class (series of decreasing exponents or exponents of fractional order). The relaxation moduli of a unidirectionally reinforced composite are determined as an example. [ABSTRACT FROM AUTHOR]

The vibration and vibrational heating of a rectangular prism with copper and polyethylene layers is studied by solving numerically a coupled problem of thermoelasticity. The cases of kinematic and mechanical harmonic shear loads on a section of the prism surface are examined. Local heating regions are revealed. They are due to the stress fields in the neighborhood of the points at which the type of boundary conditions changes. The temperature–time curves have preresonance, resonant, and postresonance sections. The heating process reaches a steady thermal state under kinematic loading and may become avalanche-like (which is typical of thermal instability) under mechanical loading [ABSTRACT FROM AUTHOR]

The basic equations for thin-walled thermoviscoelastic plates with distributed piezoelectric actuators under monoharmonic mechanical and electric loads are derived. The thermomechanical behavior of materials is described using the concept of complex characteristics. Variational methods of solving nonlinear problems of active damping of the bending vibrations of plates are considered. The effect of dissipative heating on the damping of the axisymmetric bending vibrations of a circular plate is examined as an example [ABSTRACT FROM AUTHOR]

The parameters of the fractional exponential creep and relaxation kernels of linear viscoelastic materials are determined. Methods that approximate the kernel by using the Mittag-Leffler function, the Laplace-Carson transform, and direct approximation of the creep function by the original equation are analyzed. The parameters of fractional exponential kernels are determined for aramid fibers, parapolyamide fibers, glass-reinforced plastic, and polymer concrete. It is shown that the kernel parameters calculated through the direct approximation of the creep function provide the best agreement between theory and experiment. The methods are experimentally validated for constant-stress and variable-stress loading in the modes of additional loading and complete unloading [ABSTRACT FROM AUTHOR]

The paper studies the stress rupture behavior of a reinforced viscoelastic composite through which a penny-shaped mode I crack propagates under a constant load. The composite has hexagonal symmetry and consists of elastic isotropic fibers and viscoelastic isotropic matrix. The material is modeled as a transversely isotropic homogeneous viscoelastic medium with effective characteristics. The crack is in the isotropy plane. The ring-shaped fracture process zone at the crack front is modeled by a modified Dugdale zone with time-dependent stresses. The viscoelastic properties of the matrix are characterized using a resolvent integral operator. Use is made of Volterra's principle, the method of operator continued fractions, and the theory of precritical crack growth in viscoelastic bodies. The problem is reduced to nonlinear integral equations. Numerical results are obtained for certain components of the composite, constant volume fractions, and different fracture strengths [ABSTRACT FROM AUTHOR]

The nonlinear flutter of some aircraft elements is modeled. A viscoelastic model is used. Numerical algorithms for solving integro-differential equations are developed. The critical flutter speed and time for a viscoelastic plate are determined [ABSTRACT FROM AUTHOR]

The forced monoharmonic bending vibrations and dissipative heating of a piezoelectric circular sandwich plate under monoharmonic mechanical and electrical loading are studied. The core layer is passive and viscoelastic. The face layers (actuators) are piezoelectric and oppositely polarized over the thickness. The plate is subjected to harmonic pressure and electrical potential. The viscoelastic behavior of the materials is described by complex moduli dependent on the temperature of heating. The coupled nonlinear problem is solved numerically. A numerical analysis demonstrates that the natural frequency, amplitude of vibrations, mechanical stresses, and temperature of dissipative heating can be controlled by changing the area and thickness of the actuator. It is shown that the temperature dependence of the complex moduli do not affect the electric potential applied to the actuator to compensate for the mechanical stress [ABSTRACT FROM AUTHOR]

Investigations of viscoelastic composite materials and structural members carried out using the TDLTSDB and initial-imperfection method are reviewed. The investigations address the internal and surface loss of stability of layered and fibrous composites, the loss of stability of plates, and the delamination (buckling) of plates with cracks. Each of these problems is reviewed separately. New areas of further research are proposed. The review focuses on the investigations carried out by the author and his students [ABSTRACT FROM AUTHOR]

The subcritical growth of a mode I crack with nonsmall end zones in an aging viscoelastic plate under tensile loads applied at infinity is studied assuming that the fracture process zone is of constant length. The equations of incubation, transition, and major crack growth stages are obtained on the basis of the Volterra principle and critical crack opening criterion. The crack subcritical growth of a specific material is studied as an example [ABSTRACT FROM AUTHOR]

The vibrations and self-heating of a layered metal-polymer rectangular prism kinematically excited by a normally vibrating punch are analyzed numerically. The effect of reinforcement layers on the rate of variation in and the spatial distribution of the temperature field is examined. It is established that thin metallic layers in a polymer matrix change the dissipation mechanism from shear to bulk. The effect of contact stress concentration on the localization of temperature field is studied. It is established that the energy flux through the load application area decreases due to softening of the material. Several reinforcement effects are revealed. It is shown that they are determined by the value of the load parameter. The numerical results are in good agreement with experimental data on self-heating kinetics [ABSTRACT FROM AUTHOR]

The paper studies the subcritical stable growth of a penny-shaped mode I crack in an aging linear viscoelastic body with cylindrical anisotropy under tensile loads applied at infinity. The equations of quasistatic crack growth are derived using the Volterra principle for a modified Dugdale model and the critical crack opening displacement as a failure criterion. The kernel of the operator that describes the viscoelastic crack opening is found using operator-valued continued fractions. As an example, the subcritical stable growth of a crack in a specific material is studied numerically [ABSTRACT FROM AUTHOR]

The growth of a straight mode II crack in a viscoelastic orthotropic plate is examined. The plate material is modeled by a viscoelastic anisotropic medium. The shear displacement in the fracture process zone is determined as a function of time using the corresponding elastic solution, the Volterra principle, and the method of operator continued functions. The time dependence of the crack length is constructed as integral equations of three phases of stable growth. The solution of these equations gives kinetic curves [ABSTRACT FROM AUTHOR]

The paper studies the stress-strain state of a deep cylindrical panel weakened by a hole and subjected to a tensile load at the outer boundary. A variational difference method is used. A numerical analysis is carried out for an orthotropic panel with low shear stiffness [ABSTRACT FROM AUTHOR]

The paper outlines a technique of determining the dynamic stress-intensity factors (SIFs) in structurally anisotropic composites from photoelastic measurement data. A composite is considered a linear elastic, homogeneous, orthotropic body. An equation is derived that relates the fringe order with the SIFs and the far-field stresses at the tip of a crack, which is parallel to the orthotropy axis [ABSTRACT FROM AUTHOR]

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated [ABSTRACT FROM AUTHOR]

In solving linear viscoelastic problems for composite materials, the problem arises of representing a multivariable operator function. To resolve this problem, the method of operator continued fractions is generalized to the case of a multivariable operator function. The method is based on the theory of branched continued fractions. Branched operator continued fractions are considered. Using the convolution theorem, fractions can be represented in terms of operators of basic class. This representation makes it possible to effectively solve boundary problems of linear viscoelasticity [ABSTRACT FROM AUTHOR]

An efficient method is proposed to determine the deformation function of a viscoelastic material from experimental data. The deformation function is assumed to be an integral operator with Rabotnov's fractional-exponential kernel or a sum of such kernels. This representation enables effective use of the method of operator continued fractions. To illustrate the method, deformation data for polymethylmethacrylate are used. The viscoelastic characteristics of a composite based on this material are obtained using the method of operator continued fractions [ABSTRACT FROM AUTHOR]

An efficient method for determining the deformation function of a composite is discussed. The method is based on a fractional exponential representation of the deformation functions of the composite components. The viscoelastic solution is obtained using the Volterra principle. The deformation function is represented as a function of a base operator. Thus, the problem is solved by approximating the deformation function by a continued fraction and applying the method of operator continued fractions. A computational procedure is detailed and illustrated using data on longitudinal relaxation of polymethylmethacrylate. As an example, the deformation of a polymethylmethacrylate-based fibrous composite with viscoelastic properties is analyzed [ABSTRACT FROM AUTHOR]

The surface instability of inelastic laminated coatings is analyzed using a piecewise-homogeneous model and the three-dimensional linearized theory of stability. The general formulation of the problem is given, and the characteristic equations are derived. Specific problems for some viscoelastic and elastoplastic models are solved [ABSTRACT FROM AUTHOR]

The flutter velocities of viscoelastic plates are determined. It is shown that the viscoelastic characteristics reduce them [ABSTRACT FROM AUTHOR]

The linearity domain for the viscoelastic properties of high-molecular organic fibers is determined. The linearity criteria are coincidence of experimental compliance curves and linearity of isochronic creep curves. Statistical criteria are used to establish linearity. The influence function in the constitutive equation of linear viscoelasticity is an Abel-type power kernel. The calculated and experimental creep strains are in good agreement both at the initial stage of deformation and after long-term loading [ABSTRACT FROM AUTHOR]

The vibrations and dissipative heating of an infinite viscoelastic beam under a moving load are studied on the basis of Timoshenko beam theory. The influence of transverse-shear strain and rotary inertia on the critical velocities of the moving load, the amplitude of bending vibration, and the temperature of dissipative heating is analyzed [ABSTRACT FROM AUTHOR]

The kinetics of crack growth in a viscoelastic orthotropic plate made of a composite material is analyzed. The external load is the superposition of a cyclic load and a constant tensile load. The level of the tensile load is assumed to exceed the amplitude of the cyclic load. The kinetic curves are compared with the constant tension curves. The effect of the cyclic component is studied. [ABSTRACT FROM AUTHOR]