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.     

Stress potential in the nonlinear theory of viscoelasticity

By introducing a functional derivative and the line integral of a functional the author supplies an energy proof of the nonlinear theory of viscoelasticity. The stress potential for small strains, analogous to the potential proposed by Coleman for large strains [7], is considered for comparison.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 633–642, July–August, 1970.     

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     

Equations of state in the nonlinear theory of viscoelasticity

The equations of state of nonlinearly viscoelastic media are investigated in the form of a sum of integrals of increasing multiplicity. The medium is assumed to be arbitrarily anisotropic, the deformation large. An inversion theorem is proved for the operator relating the stresses and strains.Mekhanika Polimerov, Vol. 3, No. 3, pp. 427–435, 1967     

Method of solving nonlinear dynamic problems in viscoelasticity

The authors propose a method of solving Volterra's system of nonlinear integrodifferential equations. This method is based on the use of a power series. As an illustration, the authors consider the vibration of flexible viscoelastic cylindrical shells under impulsive and periodic loads.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 554–558, May–June, 1973.     

On the solution of physically nonlinear quasistatic problems in viscoelasticity

The authors propose a method of solving a Volterra system of nonlinear integral equations. The method is based on Cauchy's expression for multiplication of a power series. As an illustration of the method, the authors consider the flexure of rectangular plates of physically nonlinear viscoelastic materials.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 558–561, May–June, 1973.     

Uniqueness of the solution of a nonlinear problem

Ukrainian Mathematical Journal -     

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.     

Quasistatic anti-plane motion in the simplest theory of nonlinear viscoelasticity

We consider a very simple model in the framework of differential viscoelastic materials which are isotropic and incompressible. In this model the Cauchy stress tensor is split in an elastic part and a dissipative part. The elastic part is derived from a strain-energy density function only of the first invariant of the Cauchy–Green strain tensor. The dissipative part is like the Navier–Stokes equations: linear in the stretching tensor with a constant viscosity parameter. For this model we provide some time and spatial estimates in the quasistatic approximations for the equations governing anti-plane shear motions. Several explicit examples for specific form of the strain energy are produced. Our results impose analytical restrictions on the mathematical properties of the strain energy to ensure a physical behavior in the creep and recovery experiments. Moreover, we show polynomial decay for the spatial behavior in the class of stress-hardening (or strain-stiffening) materials. For stress-softening materials a Phragmen–Lindelof alternative is proved.     

Formulation of the problem in the theory of geometrically nonlinear viscoelasticity

A variational formulation of the problem of geometrically nonlinear viscoelasticity is proposed here: the problem then reduces to a system of integrodifferential equations in functions of a single argument.     

The method of averaging in nonlinear dynamical problems occurring in the theory of viscoelasticity

The degenerate vibrations of an oscillator with slightly nonlinear elastic properties and a memory characteristic have been investigated with the aid of the averaging technique proposed in [2]. The vibrations have been studied in the case of fractional resonance and the principal resonance. A basis for the principle of averaging a nonlinear multidimensional system over an infinite time interval has been given.V. I. Lenin Tashkent State University. Translated from Mekhanika Polimerov, No. 1, pp. 39–44, January–February, 1972.     

Solution of some dynamic problems of viscoelasticity theory by the averaging method

Damping of longitudinal vibrations of a viscoelastic bar and transverse vibrations of a freely supported orthotropic plate is investigated by the averaging method. The problem of the forced movement of a viscoelastic cylinder enclosed in a thin elastic shell under the effect of an arbitrarily time-varying internal pressure is studied.Institute of Cybernetics at the Computer Center Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 246–252, March–April, 1970.     

Time-stress superposition in nonlinear viscoelasticity

Using low-density polyethylene, whose viscoelastic properties are nonlinear at high stresses, within reasonable limits of stress and time it is possible (under certain conditions) to use time-stress superposition, as well as time-temperature superposition, to predict the long-term creep of polymer materials. A variant of the compact approximation of nonlinear vicoelasticity effects in polymers is proposed.Mekhanika Polimerov, Vol. 4, No. 2, pp. 379–381, 1968Instantaneous elastic strains excluded.     

Viscosity solution theory of a class of nonlinear degenerate parabolic equations I. Uniqueness and existence of viscosity solutions

In this part we construct a unique bounded Hölder continuous viscosity solution for the nonlinear PDEs with the evolutionp-Laplacian equation and its anisotropic version as typical examples. The existence and properties of free boundaries will be discussed in part II.This research is supported by the National Natural Sciences Foundation of China.