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.     

A method of solving dynamic problems of viscoelasticity

The "direct" numerical method is extended to the calculation of wave processes and dynamic stresses in bodies composed of viscoelastic material.Baranov Central Institute of Aviation Engine Building, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 727–729, July–August, 1971.     

Experimental method of solving an integral equation of the theory of viscoelasticity

A solution of an integral equation of the theory of viscoelasticity is proposed. This solution, based on a creep test on a polymer specimen with an included elastic element, improves the convergence of the method of approximations.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 584–587, July–August, 1969.     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

New multiplier method for solving linear complementarity problems

A new multiplier method for solving the linear complementarity problem LCP(q, M) is proposed. By introducing a Lagrangian of LCP(q, M), a new smooth merit function ϑ(x, λ) for LCP(q, M) is constructed. Based on it, a simple damped Newton-type algorithm with multiplier self-adjusting step is presented. When M is a P-matrix, the sequence {ϑ(x ^k, λ ^k)} (where {(x ^k, λ ^k)} is generated by the algorithm) is globally linearly convergent to zero and convergent in a finite number of iterations if the solution is degenerate. Numerical results suggest that the method is highly efficient and promising. Selected from Numerical Mathematics (A Journal of Chinese Universities), 2004, 26(2): 162–171     

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.     

A thermal-hereditary theory for linear viscoelasticity

Summary A constitutive law is presented that predicts the mechanical response of a linear viscoelastic material in a time varying temperature field. The development is based upon a modification of the superposition theory and includes the hereditary effects of temperature; that is, the temperature at a previous time can influence the response at the present time. The mathematical structure of the two resulting constitutive equations is relatively uncomplicated and has the advantage of predicting the response of the material in a time varying temperature environment from data obtained in experiments run in time and spatially constant temperature fields.

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Zusammenfassung Es wird eine Stoffbeziehung abgeleitet, die das mechanische Verhalten eines linearen viskoelastischen Materials in einem zeitveränderlichen Temperaturfeld vorauszusagen erlaubt. Die Ableitung beruht auf einer Modifikation der Superpositionstheorie und schliesst die hereditären Wirkungen der Temperatur ein; d. h. die Temperatur in der Vergangenheit kann das gegenwärtige Verhalten beeinflussen. Die mathematische Struktur der zwei hieraus folgenden Stoffgleichungen ist verhältnismässig unkompliziert und hat den Vorteil, dass sie das Verhalten des Materials in einem zeitveränderlichen Temperaturfeld aus Daten zu berechnen erlaubt, die in raum- und zeitkonstanten Temperaturfeldern erhalten wurden.

     

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.     

Method of approximations for solving problems of the linear theory of thermoviscoelasticity

A general approximate method of solving problems of the linear theory of thermoviscoelasticity is proposed. Use is made of the Laplace transformation and certain properties of the dependence of solutions to problems of the theory of elasticity on Poisson's ratio that make possible a simple approximation. As a result, the inverse transforms become elementary and the general solution of the problem is expressed by creep and relaxation functions.Mekhanika Polimerov, Vol. 4, No. 2, pp. 210–221, 1968     

On a method of solving two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology

A two-dimensional integral equatin appearing in axisymraetric contact problems for bodies with complex rheology is studied. A method of constructing the solution of this equation in proposed, based on inspecting the non-classical spectral properties of an integral operator. A contact problem for a non-uniformly aging viscoelastic foundation is solved as an example.     

An efficient method for solving linear goal programming problems

This note proposes a solution algorithm for linear goal programming problems. The proposed method simplifies the traditional solution methods. Also, the proposed method is computationally efficient.     

On the growth behavior of one-dimensional acceleration waves in the linear theory of viscoelasticity

The growth behavior of one-dimensional accelaration waves which propagate in linear viscoelastic materials with time-dependent relaxation functions is examined. Such a time dependence arises either by linearizing the constitutive equation of a simple material about an arbitrary deformation history or by linearizing about a natural state when there is an aging process taking place in the body. Explicit results are obtained for simple types of time dependence and the results are then compared with those which follow in the usual theory of isothermal linear viscoelasticity.     

Random search method for solving series of linear programming problems

Translated from Issledovaniya po Prikladnoi Matematike, No. 16, pp. 120–124, 1989.     

A method of solving fully fuzzified linear fractional programming problems

In this paper, we propose a method of solving the fully fuzzified linear fractional programming problems, where all the parameters and variables are triangular fuzzy numbers. We transform the problem of maximizing a function with triangular fuzzy value into a deterministic multiple objective linear fractional programming problem with quadratic constraints. We apply the extension principle of Zadeh to add fuzzy numbers, an approximate version of the same principle to multiply and divide fuzzy numbers and the Kerre’s method to evaluate a fuzzy constraint. The results obtained by Buckley and Feuring in 2000 applied to fractional programming and disjunctive constraints are taken into consideration here. The method needs to add extra zero-one variables for treating disjunctive constraints. In order to illustrate our method we consider a numerical example.     

On a method for solving a two-dimensional nonlinear integral equation of the second kind

In this article, the existence of at least one solution of a nonlinear integral equation of the second kind is proved. The degenerate method is used to obtain a nonlinear algebraic system, where the existence of at least one solution of this system is discussed. Finally, computational results with error estimates are obtained using Maple software.     

A smoothing Newton method for solving a class of stochastic linear complementarity problems

In this paper, we consider a class of stochastic linear complementarity problems (SLCPs) with finitely many elements. We present a smoothing Newton algorithm for solving the SLCP. Under suitable conditions, we obtain the global convergence and locally quadratic convergence of the proposed algorithm. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed algorithm.     

Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems

In this paper, a two-dimensional quasilinear elliptic problem of the form -divF(x,▽u)=g(x)$-\mathit{divF}(\mathbf{x}\text{,}▽u)=g(\mathbf{x})$ is considered. This problem is ill-conditioned and we therefore propose a modified iterative algorithm based on coupling of the Sobolev space gradient method and WEB-spline finite element method. Applying the preconditioned iterative method, which has been already provided by Farago and Karatson (2001) [1] reduces the our considered problem to a sequence of linear Poisson’s problems. Then the WEB-spline finite element method is applied to the approximate solution of these Poisson’s problems. In this sense, a convergence theorem is proved and the advantages of this technique than the gradient finite element method (GFEM) is also described. Finally, the presented method is tested on some examples and compared with GFEM. It is shown that the gradient WEB-spline finite element method gives better test results.     

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.