One method of solution of problems in the linear theory of viscoelasticity for aging anisotropic materials

A method of solution of problems in the linear theory of viscoclasticity for aging anisotropic materials is discussed. The method is based on evaluation of irrational functions of nondiffcrence viscoelasticity operators using the theory of continued fractions. It is shown that operator-valued continuedS-fractions converge for a wide class of integral Voltcrra operators. Application of the method is illustrated by the evaluation of an irrational function of a linear combination of nondiffcrence-type resolvent operators obtained in the solution of a concrete problem of failure of an aging anisotropic viscoclastic body. Translated from Prikladnaya Mekhanika, Vol. 34, No. 11, pp. 60–65, November, 1998.     

Delayed fracture of a transversally isotropic viscoelastic material with a crack under monotonically increasing loading

The delayed fracture of a transversally isotropic viscoelastic material due to slow subcritical growth of a flat normal-fracture macrocrack with a circular cross-section under monotonically increasing load is examined. The calculations employ the modified δ_C of fracture, which is based on the concept of constancy of the prefailure region. The investigation is carried out within the framework of the Boltzmann-Volterra theory for difference-type bounded resolvent operators, which describe the transversal isotropy of the viscoelastic deformational properties of the material. To find the analytical form of the kernel of an irrational function of a linear combination of the above-mentioned integral operators, the method of operator continued fractions is used. Analytical and numerical calculations are carried out for difference-type bounded resolvent operators with the kernel in the form of Rabotnov's fractional-exponential function. S. P. Timoshenko Institute of Mechanics. National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 54–60, October, 1999.     

Viscoelastic Deformation of a Reinforced Plate with a Crack

The deformation of a viscoelastic reinforced composite is studied. The composite has an axis of elastic symmetry and consists of transversally isotropic fibers and a viscoelastic matrix, which differ by the volume concentration and mechanical characteristics. The material is modeled by a transversely isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. A plate fabricated from the composite in question is weakened by a through mode I crack and is subjected to constant tensile forces. The viscoelastic properties of the matrix material are described by a convolution operator. The Volterra principle is used to derive expressions for the viscoelastic characteristics and crack opening. The irrational function of the integral operator that describes the crack opening is expanded into an operator continued fraction and is represented as the sum of base operators     

One method of solving boundary-value problems of the linear theory of viscoelasticity

A form of integral resolvent viscoelastic operators with arbitrary analytic difference-type kernel is found using an improved method of calculating irrational functions. The approach is illustrated by determining the viscoelastic characteristics of the plane stressed state of an orthotropic plate. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 77–85, December, 1998.     

Stable growth of penny-shaped crack in viscoelastic composite material under time-dependent loading

Considered is the long-term cracking of the three-dimensional fiber-reinforced viscoelastic composite with a plane penny-shaped crack under time-dependent loading. The composite has a hexagonal structure and consists of elastic isotropic fibers and viscoelastic isotropic matrix. The material is modeled by transversally isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. The crack propagation planecoincides with the plane of isotropy. A ring-shaped yield zone in front of the moving crack is modeled as a Dugdale's zone with time-dependent stresses. Crack growth under deformation of the composite occurs by application of a slowly increasing tensile load; it is normal to the plane of crack propagation. A convolution-type time operator describes the viscoelastic properties of the matrix material. Use is made of the Volterra principle and the theory of long-term cracking of viscoelastic bodies. The irrational function of integral operator associated with the viscoelastic crack opening expression is expanded into a continued fraction of operators. The solution is reduced to the nonlinear integral equations of crack growth. Numerical results are obtained for a specific material. Crack growth kinetics is discussed in connection with the onset of stable crack growth and crack border stress intensity factor.     

On deforming of a linear viscoelastic orthotropic half-space under the turning rigid cylindrical roller

In this paper we consider the problem of rigid cylinder turning on a linear viscoelastic orthotropic half-space with Coulomb's friction acting along the contact area. Results for extents of contact area and pressure under the cylinder are obtained using Volterra's principle. The obtained functions of viscoelastic operators are interpreted by a method based on expansion of such functions in operator continued fractions. A solution is given for the general type of resolvent viscoelastic operators expressing rheological properties of half-space material. Algebra of resolvent Volterrian operators is used to facilitate the calculations. An example is given to illustrate the results for real viscoelastic material with the rheological properties expressed by the operators of Yu.N. Rabotnov.     

Development of a crack with a considerable prefracture zone in a viscoelastic orthotropic plate under variable loads

The study is made of the delayed fracture of a viscoelastic orthotropic plate caused by subcritical advancement of a rectilinear microcrack, which is located along one of the orthotropic axes. The crack develops because of stretching of the plate by uniformly distributed increasing and cyclic external forces perpendicular to the crack line. The investigation is carried out within the framework of the Boltzmann-Volterra theory for resolvent integral operators of difference type, which describe the deformation of a material with time-dependent rheological properties. The analytical form of the kernel of an irrational function of a linear combination of the above integral operators is determined by the method of operator continued fractions. Numerical calculations are conducted for resolvent bounded integral operators with a kernel in the form of Rabotnov's fractional-exponential function. The kinetics of growth of a crack with tip regions commensurable with the crack length is studied. A comparison with the results obtained within the framework of the concept of the thin structure of the crack tip is given. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 121–129, June, 2000.     

Solution of boundary-value problems of linear viscoelasticity by the method of operator continued fractions in the case of complex coefficients

The case is considered where complex coefficients of operators appear when irrational functions of integral resolvent operators of viscoelasticity are approximated by continued fractions. Approximating formulas in explicit form are derived for this case. The relations obtained, are applied to the solution of the problem on time redistribution of stresses in a viscoelastic transversal-isotropic hyperboloid. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 66–74, September, 1999.     

Subcritical growth of a crack in a transversally isotropic aging composite under long-term static loading

The problem of long-term fracture of an aging reinforced composite array having hexagonal symmetry and weakened by a flat circular macrocrack is considered based on the Boltzmann-Volterra principle and the theory of long-term fracture of viscoelastic bodies. The array is under the action of stationary tensile forces applied at infinity and normal to the crack plane. The Maslov-Arutyunyan operator is used to describe the aging strain properties of the array. The irrational functions of integral Volterra operators obtained during the solution are determined by expanding them into continued fractions. The crack growth equations derived are numerically solved for a specific material (ferroconcrete). Curves of the rupture life of the composite array, kinetics of crack growth, and safe loading are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 72–79, December, 1999.     

A Method for Solving Boundary-Value Problems of Linear Viscoelasticity for Anisotropic Composites

A new method is proposed to solve some problems of linear viscoelasticity for anisotropic bodies. The method uses a branching continued fraction to approximate an irrational multivariable function. Such an approach allows obtaining a linear operator as an approximation of a multivariable operator function. The deformation of a cracked composite body with a plastic matrix is analyzed as an example. Both composite components are assumed to exhibit viscoelastic properties     

On the application of branched operator continued fractions for a boundary problem of linear viscoelasticity

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 Published in Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 133–143, January 2006.     

The elastic–viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties

This paper revisits the elastic–viscoelastic correspondence principle for non-homogeneous materials. Several recent publications discussed this principle for functionally graded materials (FGMs) with time translation invariant viscoelastic properties. It was demonstrated that the correspondence principle is valid only for the FGMs with separable relaxation moduli (moduli in separable form in space and time). This paper reconsiders this issue. It shows that the correspondence principle is valid even for non-homogeneous materials with separable relaxation moduli even if the time-dependences of the relaxation moduli in shear and dilatation are not necessarily time translation invariant. The property of similarity of Volterra operators is used to obtain the corresponding elastic solution. The correspondence is established between the elastic solution and the operator-transformed viscoelastic solution. The transformation operators are combinations of the Laplace transform operator and additional integral operators.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

Delayed fracture of a transversally isotropic viscoelastic material with a crack under a cyclic load

The delayed fracture of a transversally isotropic viscoelastic material due to slow subcritical growth of a flat circular macrocrack of normal separation under a cyclic load is investigated. The analysis is based on the modified δC of fracture and the hypothesis of the constancy of the prefracture zone. The study is made within the framework of the Boltzmann-Volterra theory for bounded resolvent operators of difference type, which describe the transversal isotropy of the strain properties of the material. To determine the analytic form of the kernel for the irrational function of a linear combination of the above-mentioned integral operators, the method of continued fractions is used. Analytical and numerical calculations are carried out for the bounded resolvent operators of difference type with the kernel in the form of Rabotnov's fractional exponential function. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 123–129, January, 2000.     

Numerical-analytic investigation of the dynamics of viscoelastic and porous elastic bodies

This paper presents the results of mathematical and discrete modeling of linear dynamics problems for three-dimensional viscoelastic and porous elastic bodies. The employed methods and approaches are based on formulating boundary integral equations solved using boundary elements. The model of a standard viscoelastic body is employed as the viscoelastic model. The properties of porous elastic materials are described using the full Biot model with four basic functions. Examples of numerical solutions of the problems are compared with known results of solutions.     

Adequacy of a nonlinear theory of viscoelasticity

Some features of the behavior of viscoelastic materials whose existence leads to the choice of nonlinear constitutive relations are discussed. A classification of such constitutive relations is given and a number of requirements imposed by practice on their adequacy are formulated. A nonlinear theory of viscoelasticity is proposed; this theory offers the advantages over the theory in which stresses are expressed in terms of strains by integral operators of increasing multiplicity. By a one-dimensional example, it is shown that the constitutive operator relations are reciprocal.     

Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains

The state of stress and strain of previously loaded viscoelastic bodies with holes originating in them, successively or simultaneously, is analyzed under finite plane deformations. The problem statement and solution are based on the theory of repeatedly superimposed large deformations. The material mechanical properties are described using integral relations of the convolution type over time with a weakly singular kernel. The problem solving is based on the finite-element method. To calculate the integral of the convolution type, a recurrence formula is used that can be obtained by approximating the initial kernel with a linear combination of exponential functions (the truncated Prony’s series). The nonlinear effects and the effect of the interaction between holes on the stress concentration are analyzed. For the dynamic problems, the results for incompressible and weakly compressible materials are compared.     

A review of the rheo-optical properties of linear high polymers

Some principles ans laws, expressing the mechanical and optical behavior of linear viscoelastic materials, are reviewed. The mechanical properties of the polymers in the transition region may be represented by a condensed general method containing Ferry's modulus or compliance-reduction scheme, the time-temperature superposition principle and the Gauss error integral representation. The optical behavior of high polymers is expressed by the stress- and strain-optical coefficients in creep or relaxation, which relate birefringence to stresses or strains. It was recently shown experimentally that, instead of a pair of independent linear differential operator relations, which characterize the mechanical properties of the viscoelastic materials, only one operator relation is needed and the initial value of another at the glassy or rubbery state. Then, a single test is sufficient for the complete determination of the mechanical and optical viscoelastic behavior, provided the value of another elastic constant at the glassy or rubbery state is also determined and the variation of birefringence with time is simultaneously measured with the mechanical-characteristic quantities of the material.     

Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam

The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.     

Boundary-value problems for linear equations with a generalized invertible operator in a Banach space with basis

We consider linear boundary-value problems for operator equations with generalized invertible operators in Banach spaces that have bases. Using the technique of generalized inverse operators applied to generalized invertible operators in Banach spaces, we establish conditions for the solvability of linear boundary-value problems for these operator equations and obtain formulas for the representation of their solutions. We consider special cases of these boundary-value problems, namely, so-called n- and d-normally solvable boundary-value problems as well as normally solvable problems for Noetherian operator equations.