Method of determining the volume and shear characteristics of elastico-viscous hereditary media from uniaxial-tension (compression) experiments

A method is proposed for determining the elastic constants — instantaneous modulus of elasticity, Poisson's ratio, shear modulus, bulk modulus, and the shear and volume influence functions — the shear creep kernel, the shear creep rate kernel, and the corresponding relaxation kernels from the data of creep or relaxation tests.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 754–758, July–August, 1969.     

Viscoelastic characteristics of the interaction between structures and soils

The shear coefficients of a body relative to the soil and the shear creep and relaxation kernels are determined from the data of creep and relaxation tests.Urazbaev Institute of Mechanics and Earthquake Resistance of Structures, Academy of Sciences of the SSR, Tashkent. Translated from Mekhanika Polimerov, No. 2, pp. 207–211, March–April, 1973.     

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.     

Ultrasonic method of determining the elastic characteristics of anisotropic materials

An ultrasonic method of determining the elastic characteristics, developed for isotropic materials and based on the measurement of the angles of total internal reflection of ultrasound at a liquid-solid interface, is applied to an anisotropic Compreg of the DSPA type. The results obtained for the shear modulus using the ultrasonic method at 2.5 MHz and an independent method based on measuring the frequency of the damped torsional vibrations of a prismatic bar at 10–30 Hz are compared. It is shown that the shear modulus varies only slightly (10–12%) on the frequency interval investigated.Leningrad Kirov Forest Engineering Academy. Translated from Mekhanika Polimerov, No. 4, pp. 728–731, July–August, 1969.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.     

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.     

Numerical methods in the theory of viscoelasticity

Formulas are given for calculating the coefficients of differential operators of defining equations on the basis of given approximations of the relaxation kernels in the form of the sum of exponential curves. As the defining equations it is suggested to use quadrature formulas into which are substituted the relaxation kernels found experimentally without preliminary analytic approximation. A three-dimensional difference problem of the linear isotropic theory of viscoelasticity is formulated. The direct and inverse -transformation establishing the correspondence between the viscoelastic and elastic difference problems is introduced. The specific characteristics of the use of the net, Ritz, finite-element, and variation-difference methods in solving problems of viscoelasticity theory are examined. A method facilitating the arrangement of the information on relaxation kernels in a computer memory is indicated.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 417–428, May–June, 1973.     

Discrete Wavelet Petrov–Galerkin Methods

In this paper, we develop a discrete wavelet Petrov–Galerkin method for integral equations of the second kind with weakly singular kernels suitable for solving boundary integral equations. A compression strategy for the design of a fast algorithm is suggested. Estimates for the rate of convergence and computational complexity of the method are provided.     

The S′-convolution with singular kernels in the Euclidean case and the product domain case

We characterize those tempered distributions which are S′-convolvable with a given class of singular convolution kernels. We study both, the Euclidean case and the product domain case. In the Euclidean case, we consider a class of kernels that includes Riesz kernels, Calderón–Zygmund singular convolution kernels, finite part distributions defined by hypersingular convolution kernels, and Hörmander multipliers. In the product domain case, we consider a class of singular kernels introduced by Fefferman and Stein as a generalization of the n-dimensional Hilbert kernel.     

Material stability analysis of particle methods

Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with reasonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz–Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability, so that material instabilities can occur in stress states that should be stable. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material stability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations. AMS subject classification 74S30     

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.     

Mechanism of failure of reinforced beams in bending

The mechanism of shear failure is investigated. A theoretical analysis makes it possible to determine the limits of shear failure of reinforced beams in relation to the geometric parameters of the beam, the mechanical properties of the reinforcement and the resin, their volume content, and the loading and support conditions. The results obtained are consistent with the experimental data of [1]. It is shown that in the process of shear failure the axial displacement distribution is modified and that the shear failure mechanism depends on the type of loading and the support conditions.Institute of Hydrodynamics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. Translated from Mekahnika Polimerov, No. 4, pp. 698–709, July–August, 1973.     

Nonlinear creep of TS8/3-250 glass-reinforced plastic

The creep and recovery of a glass-reinforced plastic are investigated within the frame-work of a nonlinear hereditary model. The material characteristics are determined for the three principal directions. The proportionality of the kernels is established.State Scientific-Research Institute of Machine Science, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 391–397, May–June, 1971.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

Equations of state of viscoelastic isotropic media

The equations of state of viscoelastic isotropic media are investigated in the form of a sum of integrals of increasing multiplicity. An expression is given for the second- and third-order kernels for an isotropic medium in terms of the metric tensor. The corresponding resolvents are found from the given first- and second-order kernels, and a method of obtaining the third-order resolvents is described. By means of the isotropy postulate of Il'yushin, the equations of state are simplified, and it is shown that they cannot contain scalar powers of tensors higher than the second. The form of these equations is written out.Mekhanika Polimerov, Vol. 3, No. 4, pp. 645–651, 1967     

On the theory of non-Newtonian polymer flow

The Eyring-Frenkel theory of viscosity of low-molecular liquids has been extended to solutions of high-molecular compounds. It is shown that there are flow units of different sizes in the system, their mean size being proportional to the molecular weight of the polymer. An expression is obtained for the non-Newtonian viscosity of polymer solutions. In the limiting case of high shear rates the viscosity of the solution coincides with that of the solvent. At low shear rates Flory's empirical relation for the viscosity of polymer solutions is theoretically obtained.Mekhanika Polimerov, Vol. 2, No. 5, pp. 779–784, 1966     

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.     

Analysis of certain methods of determining shear moduli 1. Testing composites uniform over the thickness

Four methods of determining shear moduli on flat specimens are discussed. For each method the effect of the geometric parameters and the load range on the values of the shear moduli determined is estimated. Experimental values of the shear moduli obtained by each of these methods for the same materials are compared with the calculated values.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 133–140, January–February, 1976.     

Theory of fiber-matrix delamination in sphero-fibrous plastics

A method for calculating the deformed state and the effective characteristics of sphero-fibrous composites with a three-dimensional structure and local delaminations at the fiber-matrix interface is proposed. It is assumed that the interfacial cracks take place along the whole length of fibers within the representative volume. The effect of the interface crack dimension on the variations in shear moduli of the 3Dm composite is investigated in detail. It is shown that small cracks have practically no influence on the integral characteristics of the composites.Blagonravov Machine Building Institute, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 644–650, September–October, 1997.     

Viscoelastic parameters of certain polymers in the temperature interval 4.2–240°K

The principal elastic constants of Nylon 6, Nylon 7, Nylon 6.10, Nylon 11, Nylon 12, PVC, PMMA, polystyrene, and polyvinylfluoride have been calculated from data on the velocity of ultrasonic longitudinal and shear waves measured at a frequency of 5 MHz on the temperature interval 4.2–240°K. It is shown that at cryogenic temperatures the dynamic moduli of elasticity are, to a considerable extent, determined by the characteristics of the chemical structure of the polymers. It is suggested that the Poisson's ratios of the polymers depend on their free volume.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 604–609, July–August, 1973.