Quasilinear theory of viscoelasticity and the small parameter method

A closed quasilinear quadratic theory of viscoelasticity is proposed for bodies with physical nonlincarity. All the basic unique relations between the stress and strain tensors and time are established. Time relations between the invariant characteristics of combined loading processes are given. Integral equations relating the secondary kernels of creep and relaxation are obtained, and their connection with the corresponding kernels of the linear theory is determined. The singular properties of the kernels are elucidated, and their principal parts are isolated. Examples of possible expressions for the secondary kernels are given, and their general properties are examined.Mekhanika Polimerov, Vol. 2, No. 2, pp. 170–189, 1966Read at the Riga Conference on Polymer Mechanics, 11 November 1965.     

Quasilinear theory of viscoelasticity

By postulating equal contributions the number of kernels in the principal cubic theory of viscoelasticity and in the theory with regular kernels of two arguments is reduced to three. For certain quasilinear relations all the kernels and functions are determined from creep, relaxation, and simple loading and deformation tests. In the case of simple loading and deformation the problems for a viscoelastic incompressible material reduce to problems of the theory of small elastoplastic deformations of an incompressible material. Several problems relating to this case are considered.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 603–611, July–August, 1969.     

One of the simplest means of accounting for nonlinearity in viscoelastic media

The author examines a simple extension, to the nonlinear case, of memory-type theory based on the Boltzmann-Volterra superposition principle. It is shown that given certain assumptions the quasi-linear theory of viscoelasticity reduces to introduction into the equations of linear memory theory of a single stress- or strain-intensity function. This function is determined from creep or relaxation tests. A successive-approximation method is presented for solving problems of nonlinear viscoelasticity with the aid of the equations introduced. It is shown that in the case of simple loading the equations of the theory of small elastic-plastic deformations are an analog of the equations considered.Mekhanika Polimerov, Vol. 3, No. 2, pp. 207–212, 1967     

Use of the results of auxiliary experiments in solving problems in the linear theory of viscoelasticity

The approximate method of solving problems of the theory of linear viscoelasticity with arbitrary creep and relaxation kernels, proposed in [2], is substantiated and generalized. The essence of this method consists in the approximation of the functions depending on the Laplace — Carson transforms of the mechanical characteristics of a viscoelastic body by means of certain combinations of the transforms of the creep and relaxation kernels. The expressions obtained as a result of the approximation enable the inverse transforms of the unknown functions to be found without difficulty.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 963–969, November–December, 1968.     

Contribution to the nonlinear theory of viscoelasticity

The author presents a generalization of certain results obtained in [3] relating to the region of rigid behavior of plastics for which the nonlinearity of the viscoelastic properties is important even in the area of small strains. Certain questions connected with the derivation of the basic equations of viscoelasticity are considered for small strains. Formulas are obtained for the resolvents of kernels of arbitrary order. The general equations of the "principal quadratic theory of viscoelasticity" are derived.Mekhanika Polimerov, Vol. 2, No. 4, pp. 498–507, 1966     

Estimation of the upper and lower bounds of solutions of the integral equations of viscoelasticity

The different methods of solving problems of viscoelasticity for hereditary media that employ Laplace transforms or exact solutions of integral equations for inaccurately approximated kernels inevitably introduce errors associated with the approximation and the inverse transformations. Accordingly, it is necessary to estimate the accuracy of these methods. It is shown that the kernels of the integral equations of viscoelasticity permit the estimation, with a certain accuracy, of upper and lower bounds directly for the solutions of these integral equations. In cases when the accuracy of the estimate is sufficient, there is no need to employ other methods of solution.Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 976–985, November–December, 1968.     

Pair-interaction approximation in the equations of quantum field theory for a four-body system

It is shown that for the correct formulation of the pair-interaction approximation in the dynamical equations for a system of four and more particles in the various approaches of quantum field theory not only the sum of the pair-interaction potentials but also additional, two-pair kernels must be taken into account. Equations of two types with compact kernels that take into account these terms are obtained. The kernels of these equations are expressed in terms of the two-annd three-particle scattering amplitudes. The equations of the first type have a structure identical to their nonrelativistic analog, whereas the equations of the second type differ from the nonrelativistic equations even in the number of components. An advantage of these equations is that they can be generalized to the case of arbitrarily many particles.A. M. Razmadze Mathematics Institute, Georgian SSR Academy of Sciences, Tbilisi. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 95–112, January, 1992.     

Asymptotic methods in the nonlinear theory of viscoelasticity

Dynamic and quasistatic problems of the nonlinear theory of viscoelasticity are described by nonlinear integrodifferential and integral equations. Methods of averaging various classes of nonlinear integrodifferential and integral equations are described and asymptotic expansions of the solutions of these equations are given.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 221–229, March–April, 1974.     

Method of successive approximations in the nonlinear theory of viscoelasticity

A method of successive approximations, a generalization of the Il'yushin method of elastic solutions, is proposed for solving problems of the nonlinear theory of elasticity in which the stress-strain relation is given in the form of a time operator Frechet-differentiable in a neighborhood of zero. The nonlinear relaxation kernels are found from the given nonlinear creep kernels for the principal quadratic theory of elasticity. These relations make it possible to formulate the boundary value problem for this theory. By way of illustration the problem of the pressure exerted on a space by a sphere is examined within the framework of the developed theory. The question of the convergence of the method is discussed in relation to the quadratic theory of visco-elasticity.Presented at the Third All-Union Conference on Theoretical and Applied Mechanics, Moscow (January, 1968).Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 236–242, March–April, 1969.     

Shear creep of orthogonally reinforced spherofibrous composites

Our previous theory for the viscoelasticity of spheroplastics and two-phase structural models was used to construct stress creep and relaxation operators for shear of orthogonally reinforced spherofibrous composites. The operators were constructed using the Volterra principle, Rabotnov's fraction exponential kernels, and approximate analytical relationships for the integral composite characteristics. Operators were taken incorporating data on the rheonomic characteristics of the composite, components with hybrid, hollow, and other fiber types. Approximate formulas were obtained for operators convenient for studying stress creep and relaxation in elements of three-dimensional structures.A. A. Blagonravov Mechanical Engineering Institute, Russian Academy of Sciences, Moscow. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 6, pp. 770–779, November–December, 1996.     

The numerical solution of Fredholm integral equations of the second kind with singular kernels

A numerical method is given for integral equations with singular kernels. The method modifies the ideas of product integration contained in [3], and it is analyzed using the general schema of [1]. The emphasis is on equations which were not amenable to the method in [3]; in addition, the method tries to keep computer running time to a minimum, while maintaining an adequate order of convergence. The method is illustrated extensively with an integral equation reformulation of boundary value problems for u–P(r ²)u=0; see [9].This research was supported in part by NSF grant GP-8554.     

Solution of the problem of the plane deformation of a tube made of physically nonlinear quadratic viscoelastic material

An exact solution is obtained for the problem of the plane deformation of a tube of quadratically nonlinear viscoelastic material characterized by four kernels. For given boundary pressures the solution contains functions determined from simple creep and relaxation tests for stresses and strains lying in the region of linearity of the relation. In accordance with the method of approximations for nonlinear materials [9], an approximation is given for a series of convolution integrals of increasing multiplicity in transforms and inverse transforms together with an error estimate.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 1, pp. 117–123, January–February, 1973.     

Stability of the eigenvalues of certain types of singular integral equations

In the paper one finds necessary and sufficient conditions for the existence of nontrivial solutions of the equation (A+C)f=f for all compact (or small in norm) perturbations C; for the unperturbed operators A one considers the operators of dual and conjugate to dual integral equations with kernels depending on the difference of arguments, Wiener-Hopf operators, and singular integral operators. The answer is given in terms of the symbol S (A) of the operator A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 127–136, 1984.     

Nonlocal hydrodynamics in the quantum field modelϕ 4

The procedure of passing from quantum statistical mechanics to the hydrodynamics previously developed by the author is now applied to the quantum field model ⁴. For a certain class of external forces, the equations of many-body systems in quantum theory appear to be equivalent to the equations of nonlocal hydrodynamics. The hydrodynamic nonlocalities arising in constituent relations are expressed through the Green's functions for currents. Some properties of the nonlocal kernels, in particular, the conditions related to dissipation and T-invariance of the model ⁴ (an analogue of Onsager's relations), are deduced from the general symmetry properties. In hydrodynamics, nonlocality allows causality and dissipativity to be consistently combined. The connection between the classical transport coefficients and the hydrodynamic kernels is established. An algorithm for calculating constituent relations by perturbation theory, using the technique of temperature Green's functions, is described.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 50–68, July, 1996.     

Boundary problems of the theory of analytic functions on Riemann surfaces

The survey includes papers reviewed in RZh Matematika from 1954–1979. We consider the Riemann boundary problem on a compact Riemann surface and in the class of piecewise-meromorphic automorphic functions; singular integral equations with automorphic kernels and in the form of Abelian integrals; the method of symmetry in solving the problems of Hilbert (linear and nonlinear), Schwarz, Carleman, etc., in the case of a Riemann surface with boundary and in the case of a planar domain, bounded by an algebraic curve; and boundary problems on open Riemann surfaces.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 18, pp. 3–66, 1980.     

Creep on an initial small interval

The creep processes of hereditary media on a small initial interval are described, using as influence functions the resolvent functions for the generators of the Dirac function, which are employed as relaxation kernels. Equations are constructed for determining the instantaneous modulus and the parameters of these functions from experimental stress-strain diagrams obtained at three different constant loading rates.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 5, pp. 945–950, September–October, 1970.     

Temperature stresses in a long hollow viscoelastic cylinder with variable inner boundary

As distinct from existing solutions for the state of stress and strain, in addition to the stresses created by internal pressure acting on the moving inner surface, the thermal stresses created by a temperature field are also taken into account. A solution is obtained using tabulated singular kernels of the hereditary theory of viscoelasticity.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 219–226, March–April, 1969.     

On the order of the error in discretization methods for weakly singular second kind non-smooth solutions

In general, second kind Volterra integral equations with weakly singular kernels of the formk(t,s)(t –s)^– posses solutions which have discontinuous derivatives att=0. A discrete Gronwall inequality is employed to prove that, away from the origin, the error in product integration and collocation schemes for these equations is of order 2-.     

Defining relations for a viscoelastic medium with microrotation

The results of [1, 2] are extended to the case of a Cosserat medium with a memory (the force stress tensor and the couple stress tensor depend on the history of deformations and rotations of a particle in the medium). In the linear approximation the defining relations have the form of convolutions with some relaxation kernels with respect to time. Restrictions for the kernels are obtained, which follow from the general principles of thermodynamics. The propagation of weak perturbations is studied. The general functional form of the ken nels corresponding to experimental data on the viscoelasticity of rock formations is given.     

On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable -derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 893–903, July, 1994.