Hereditary Viscoelasticity Research Articles

Eigenwaves propagation in three-layer cylindrical viscoelastic shells with a filler non-uniform in thickness

The paper provides the statement of the problem, the development of a calculation method and an algorithm to solve the problems of propagation and absorption of non-axisymmetric natural waves in layered dissipative-inhomogeneous viscoelastic three-layer cylindrical bodies. A detailed analysis of well-known publications devoted to this problem is given. The paper posed the problem of non-axisymmetric natural wave propagation in three-layer cylindrical shells with fillers, taking into account the theory of hereditary viscoelasticity. Dispersion equations were obtained and the dependence of the phase velocity on the wavenumber for the three-layer shells was constructed. A calculation method was developed based on the Muller, Gauss, and orthogonal sweep methods. The program (in C++ language) was compiled based on the developed algorithm. Numerical results were obtained for the complex phase velocity depending on various wavenumbers and parameters of an axisymmetric cylindrical mechanical system for the Kirchhoff-Love and Timoshenko hypotheses. The change in the real and imaginary parts of the complex phase velocity under various parameters of the system was investigated for structurally homogeneous and inhomogeneous mechanical systems. The results obtained by the Kirchhoff-Love and Timoshenko hypotheses are in good agreement but differ in the wave absorbing abilities. It was found that a decrease in filler thickness at a rigid inner shell leads to a rapid increase in the real and imaginary parts of the vibration frequencies.

К ВОПРОСУ ОБ ОБОСНОВАННОЙ ФОРМЕ НАСЛЕДСТВЕННОЙ ВЯЗКОУПРУГОСТИ С ОДНИМ ЯДРОМ ПОЛЗУЧЕСТИ

К ВОПРОСУ ОБ ОБОСНОВАННОЙ ФОРМЕ НАСЛЕДСТВЕННОЙ ВЯЗКОУПРУГОСТИ С ОДНИМ ЯДРОМ ПОЛЗУЧЕСТИ

Energy flux and dissipation of inhomogeneous plane waves in hereditary viscoelasticity

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material which displays hereditary viscoelasticity. The energy density, energy flux and dissipation are quadratic in the small quantities, namely, the displacement gradient, velocity and velocity gradient, each harmonic with frequency ω , and so give rise to attenuated constant terms as well as to inhomogeneous plane waves of frequency 2 ω . The quadratic terms are usually removed by time averaging but we retain them here as they are of comparable magnitude with the time-averaged quantities of frequency ω . A new relationship is derived in hereditary viscoelasticity that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2 ω . It is shown that the complex group velocity is related to the amplitudes of the terms with frequency 2 ω rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.

Theoretical-Experimental Method for Determining the Short- and Long-Term Creep Parameters of Technical Rubber in Shear

A technique is developed for identifying the hereditary properties of technical rubber in short-term shear creep from the experimental shift the center of damped flexural vibrations of a vertically fixed three-layer test specimens of symmetrical structure with a technical rubber midlayer after their preliminary holding in a static bent state. This technique is based on the finite element method and integral equations of the hereditary viscoelasticity theory with the Koltunov–Rzhanitsyn heredity kernel. For identifying the rheological parameters of this kernel an objective function is constructed. To minimize the function, the direct zero-order search is used, which does not require the calculation of its gradient. The rheological parameters found are compared with experimental data of the long-term shear creep of the rubber. Noted is the unfoundedness of the direction, developed by some researchers, where only equations of the theory of hereditary viscoelasticity is used to model the vibration processes and the energy dissipation. The parameters of rubber creep kernel were identified in a long-term regime at a constant shear stress, and results showed that this type of testing is inapplicable to studying the short-term creep of rubber in shear.

Algorithm for solving problems related to the natural vibrations of electro-viscoelastic structures with shunt circuits using ANSYS data

ABSTRACTAn algorithm for numerical realisation of a mathematical statement of the natural vibrations problem for electro-viscoelastic bodies with passive external electric circuits (i.e. shunting circuits) with an arbitrary configuration using the finite element method is proposed in the present paper. The proposed algorithm allows considering the viscoelastic properties of materials using the model of linear hereditary viscoelasticity with complex dynamic moduli and is used to solve 3D solid structure problems that are compatible for ANSYS package element types. This technique implies the usage of the global assembled matrices of stiffness and mass, formed in the ANSYS package. The basis of the algorithm is a novel approach that allows performing decomposition of the global assembled stiffness matrix formed in the ANSYS software package into constituents that are needed for calculation of the natural vibration frequencies of the objects under study. These matrix components are used in the program that was written in FORTRAN (Formula Translation) language. This problem could be efficiently applied for analysis of the dynamic processes in smart systems based on piezoelectric materials and could also form a basis for the development of numerical finite element algorithms for optimization of the dissipative characteristics of electromechanical systems with shunted piezoelectric elements.

Identification of Parameters of Short-Time Creep of Plexiglas By Means of Studying Decaying Flexural Vibrations of Test Samples

A technique for identification of hereditary properties under conditions of short-time creep of Plexiglas is developed. The technique is based on experimental determination of shifting of the center of decaying flexural vibrations of vertically aligned test samples after their preliminary maintaining in a static bent state. Mathematically, the technique is based on using the finite element method and integral equations of the theory of hereditary viscoelasticity with the Koltunov–Rzhanitsyn hereditary kernel. An object function is constructed for identification of rheological parameters of this kernel. The minimum of this function is found by the direct search method, which does not require the function gradient to be known. The hereditary kernel of Plexiglas averaged on the basis of data for several test samples is obtained as a function of time.

Прогибы предварительно натянутой гибкой весомой нити, закрепленной в двух произвольных точках

In this paper, it is obtained the differential equation of sagging of flexible heavy rope stretched at an arbitrary angle to the horizon and its solution in the case of small deflections. It is shown that the maximum deflection is always achieved in the middle of rope projection, and its magnitude does not depend on the position of supports. For the first time in the formulation of the problem, it is assumed that the rope is stretched under the influence of two independent factors: deformation of the prestress and deformation due to gravity, and each of the factors may not be taken into account. It is considered the case of rope deformation in a homogeneous temperature field, and it is established that the presented model becomes incorrect in the case of small pretension in comparison with thermal deformation. For cases of absence of pretension and the equality of temperature deformation and pretension, solutions to the problem are represented for the linearly elastic and rheologically active materials. It is noted methodical features of solving the problem for different stages of creep using the Kelvin-Voigt equation. Solutions are given for the combined use of the Kelvin-Voigt model and linear hereditary viscoelasticity for the primary and secondary creep stages, respectively. The problems of modeling drying and reduction of biological filament are considered.

Criteria for long-term strength of compressible viscoelastic medium aging

The paper considers problem of damage and long-term strength of polymeric and composite materials with a ductile-brittle mechanical properties. The modified Maxwell equation written in the scale of the effective time and Boltzmann-Volterra equation of hereditary viscoelasticity are used. Parameter of continuity (of damage) is determined by the relative changes of density, which is an integral measure of the accumulation of structural microdefects in the long process of loading. The kinetic equation for continuity parameter is formulated and interrelated system of equations of creep and damage is analyzed. Analytical relations for creep, continuity parameter and criteria of long-term strength are obtained.

Positivity of Green's functions for a class of partial integro-differential equations including viscoelasticity

Positivity of Green's functions is established for a class of linear partial integro-differential equations in d ≤ 3 space dimensions with Volterra kernels satisfying appropriate conditions. In particular it is assumed that the Volterra kernel is completely monotone. This class includes the equations of motion of scalar hereditary viscoelasticity along with the extreme cases of elasticity and Newtonian viscosity. The positivity of Green's functions is also established under a less restrictive assumption. Positivity results are extended to a class of systems of partial integro-differential equations with completely monotone matrix-valued Volterra kernels. Positivity for longitudinal and transverse modes of 3D isotropic viscoelasticity is also established. Weaker results are obtained for higher space dimensions.

Numerical simulation of the rheological properties of granular composites by using a structural approach

Composites with an elastomeric matrix containing rigid particles of diameter 10–1000 µm are studied. One of possible mechanisms of the rheological behavior of such filled systems, related to the origination and growth of vacuoles near the rigid inclusions in a viscous matrix, is considered. For simulating the mechanism of formation of rheological properties of the filled elastomers, we use a structural cell in the form of an elastomeric cylinder, whose height and diameter are equal in magnitude, with a rigid spherical inclusion at its center. Deformation of the cells is examined with the observance of boundary conditions providing the preservation of their close packing. The inclusion is assumed to be rigid, and the matrix properties are described by equations of the linear hereditary viscoelasticity theory. The formation of vacuoles is described by using the approach suggesting that an initial debonding begins to propagate when the energy accumulated in the extended matrix reaches a value sufficient to create a new interface. The heterogeneity of the composite is simulated by taking into account the variability of the local filler concentration. Creep curves obtained for composite cells with different content of the solid phase are presented. Comparisons between the numerical and experimental results show a satisfactory agreement.

Predicting the nonlinear hereditary viscoelasticity of polymers

A mathematical model for the nonlinear hereditary viscoelasticity of polymer materials is proposed to predict deformation processes of various complexity — from simple relaxation and simple creep to complex deformation-relaxation and reverse relaxation processes with alternative loading and unloading.

Mathematical modeling of pressure on the strengthening vertical shaft support in “Mir” mine located in the Charsk saliferous rock series

The problem statement and the solution algorithm are presented for the case of contact between strengthening vertical shaft support and rocks. Hereditary viscoelasticity equations are used as constitutive relations for saliferous rocks. An analysis of the dynamics of the growing pressure on the support is performed versus the shaft cross-section occurrence depth, saliferous rock strength on the contact, and spacing between the rock contour and the support in the shaft.

Dynamic response of the piezoelectric polymer PVDF

The paper concerns the dynamic response of polyvinylidene fluoride (PVDF), a commercially available piezoelectric polymer currently used in numerous applications. The paper consists of two parts. The first part presents a summary review of recent experimental results regarding the time- dependent behavior of PVDF tested under combined static and cyclic loading conditions. It has been observed that the polymer has demonstrated accelerated creep rates due to cyclic loading effects. Cyclic creep acceleration in PVDF has been significant even in the range of stresses well below the viscoelastic linearity limit. This phenomenon, defined as "vibrocreep", appears to be essentially nonlinear, since the material response under combined static and cyclic loads did not represent a simple superposition of the responses to static and fully reversed cyclic loads applied separately. The second part of the paper provides a formulation of a nonlinear constitutive model, which describes the synergistic interaction between creep and cyclic damage evolution. The proposed constitutive model combines the concepts of linear hereditary viscoelasticity with damage characterization through a cyclic damage function. The latter is formulated using the basic principles of continuum damage mechanics. The potentials and limitations of the proposed constitutive model are discussed, and experimental results providing model validation are presented.

Mathematical Modeling of the Stress-Strain State in Rock and Artificial Masses During Slice Chamber Mining of Underpit Reserves in the “Internatsionalnaya” Kimberlite Pipe

The statement and solution algorithm are proposed for the problem on the stress-strain state of a kimberlite body, filling massif, and enclosing rock during slice chamber mining with filling. The equations of nonlinear hereditary viscoelasticity are used as the constitutive relations for saliferous rocks.

Numerical Solution of Linear Quasistatic Hereditary Viscoelasticity Problems

We give a space-time Galerkin finite element discretization of the linear quasistatic compressible viscoelasticity problem as described by an elliptic partial differential equation with a Volterra (memory) term. The discretization consists of a continuous piecewise linear approximation in space with a discontinuous piecewise constant or linear approximation in time. We derive an a priori maximum energy-error estimate by exploiting Galerkin "orthogonality" and the data-stability of a related discrete backward problem. Illustrative numerical experiments are also included, as also is a brief description of our first results on a posteriori error estimation. This allows for adaptive control of the space mesh but not of the time step.

Electro-mechanical properties of the piezoelectric polymer PVDF

A consistent experimental program has been developed to investigate the mechanical and electrical properties of polyvinylidene fluoride (PVDF). The results of the study indicate that this piezoelectric polymer can be characterized as an orthotropic, thermorheologically simple material with constant piezoelectric strain coefficients over the experimental range of stresses, frequencies and temperatures. The mechanical properties of PVDF are time-dependent and, under certain loading and temperature conditions, can be treated using the principles of linear hereditary viscoelasticity.

Mechanical phenomena accompanying the structure formation of reinforced polymer materials

A systemic approach to the processing of reinforced polymers into articles is outlined. The structuralstate parameter of the polymer matrix is introduced, and used to model mechanical phenomena, taking into account that the conversion fields are frontal for the whole technological cycle as a continuous process. By postulating the principles of temperature-time analogy and homogeneous equivalence, the problem is reduced to coupled boundary problems of thermokinetics and hereditary viscoelasticity.

Deformation of technical purpose woven tapes

An analogy has been found between the viscoelastic properties of yarns and fabrics which makes it possible to apply a previously developed rapid method of determining deformation characteristics to fabrics. The possibility of applying the nonlinear hereditary viscoelasticity equation for calculating the stretching diagrams of woven tapes, with allowance for the specifics of their deformation, has been established.

Buckling of viscoelastic beam columns

The paper presents a theoretical study of the creep buckling behaviour of viscoelastic beam-columns under general loading conditions. A detailed analysis is given for three particular cases, axially compressed columns with initial curvatures, laterally loaded beam-columns and beam-columns in bending. The creep buckling problem is formulated in terms of the constitutive equations of the linear hereditary viscoelasticity. It includes two types of viscoelastic materials, those with limited and unlimited creep. The general solution is derived by means of the quasi-elastic method and is examined in detail for two simple rheological material models. It is shown that the creep buckling behaviour of linearly viscoelastic beam-columns under various loading conditions is typically governed by the magnitude of the axial compressive force. For viscoelastic materials of the limited creep type there is a safe load limit below which the creep buckling characteristics of the structure are limited in time. The magnitude of the safe load limit as related to the Euler's elastic critical load solely depends upon the asymptotic value of the creep function of the material. Quasi-elastic approximations and the corresponding exact analytical solutions are compared in two simple problems. It is observed that the quasi-elastic technique is adequate for applications to the linear creep buckling analysis.

Vibrations of a shock-absorbing device on a nonlinear viscoelastic base

A method of calculating a viscoelastic shock-absorbing device or structure by the method of averaging systems of integro-differential equations is considered. The principal quadratic theory of hereditary viscoelasticity is used as the physical relationship between forces and displacements of the shock-absorbing base or structure.