An approximate method of solving dynamical problems of thermoviscoelasticity

A generalization of Whitham's asymptotic method is proposed for solving systems of integro-differential equations describing the dynamic behavior of bodies composed of nonlinear viscoelastic materials. The possibility of using this approach to solve coupled problems of thermoviscoelasticity is indicated. A number of examples are considered including the propagation of a harmonic disturbance in a rod of nonlinear viscoelastic material under isothermal conditions and the coupled problem of the high-frequency vibration of a rod of linear viscoelastic material, whose properties depend on temperature in accordance with the time-temperature superposition principle.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 1, pp. 58–65, January–February, 1973.     

Effect of shear stiffness and concrete wedge cone failure on fracture of a fiber-reinforced plastic rod at an intersection with a concrete crack

Experimental results of model speciments in which FRP rods fractured due to local deformation at a crack intersection in a concrete member were analyzed by a 3D nonlinear finite element method in which orthogonal anisotropy of the FRP rod was considered. The analytical results indicated that accurate prediction of shear modulus of the FRP rod and size of concrete wedge cone failure around the FRP rod was significant to predict deformation and fracture of the FRP rod. FRP rods as reinforcement in concrete members, the small shear modulus, because of the orthogonal anisotropy and the wedge cone failure, may prevent the FRP rod from fracturing at a very low tensile stress due to the local deformation at the crack intersection.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 21, No. 2, pp. 158–166, March–April, 1996.     

Nonlinear longitudinal waves in inhomogeneous rods

The work contains two basic results. The first consists in the derivation of an equation for the longitudinal vibrations of a rod which accounts for both nonlinear and dispersion effects as well as effects caused by the inhomogeneity of the material of the rod. This equation is found to be a perturbed Korteweg-de Vries equation. The second result consists in the development of a perturbation method for solving the Cauchy problem for this equation. The solution found describes the deformation of a soliton under the influence of the inhomogeneity of the rod.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 64–73, 1980.     

Stress relaxation in a polymer disk

The stress relaxation in a polymer disk with a central hole, into which a rigid metal rod is press-fitted, is considered. The progressive redistribution of the stresses in the disk has been observed by a photoelastic method. In the theoretical solution of the problem the relaxation properties of the polymer are taken into account by means of a generalized nonlinear Maxwell equation with two terms of the relaxation spectrum. It is shown that the linear, as distinct from the nonlinear, theory is not capable of conveying the characteristics of the stress relaxation process in a rigid polymer disk.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1070, November–December, 1971.     

Displacement of a rigid rod in a viscous medium

The spatial displacement of an absolutely rigid vertical rod in a viscous medium is considered; the rod is hinged at the upper end to a platform moving on the surface of the viscous medium over a specified curvilinear trajectory. A load is attached to the lower end of the rod. Using a variational Lagrangian equation, a nonlinear system of ordinary differential equations in terms of the angles of rod rotation determining the position of points of this rod at any time is obtained. As an example, the problem is solved for conditions of acceleration, uniform motion, and deceleration of the reference point moving over a trajectory consisting of rectilinear sections and portions of circles. The differential equations obtained may be used in determining the position of rod-type elements suspended in a viscous medium.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 108–112, 1990.     

Oscillations of an elastic rod with nonlinear boundary conditions

We study the dynamics of a building with a nonlinear seismic insulation system whose motion is described by the equation of transverse oscillations of a rod. We obtain the connection between the displacement of the lower section of the rod and the forces and moments in the section. We propose a numerical procedure for solving a nonlinear Volterra integral equation of second kind with dry friction damping. We determine the region of variation of the parameters of the building and the seismic insulation system in which it is possible to use the rigidbody model of the building. Two figures.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 76–80.     

Stability of hinged viscoelastic rods under dynamic loading

Equations of Mathieu type describing the transverse vibrations of a straight viscoelastic rod compressed by a periodic longitudinal force are examined. The resonance regime is investigated by means of the method of averaging [2–4] in the case of a hinged viscoelastic rod acted upon by a periodic force.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1126–1129, November–December, 1973.     

Numerical solution of the impact problem for a rigid mass and a viscoelastic rod of finite length

The problem of the propagation and interaction of longitudinal waves in a rod struck by a rigid mass is considered. A numerical computer solution is obtained for the case when the mechanical behavior of the rod material is expressed by a rectangular relaxation spectrum. A numerical method of integral transform inversion is developed and applied to problems of polymer dynamics.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 450–456, May–June, 1971.     

Creep stability of polymer rods

The author examines the problem of the buckling of a hinger rod of rigid homogeneous polymer material under constant load. The results of an experimental investigation are presented. The theoretical calculations are based on the nonlinear generalized Maxwell equation. A numerical solution has been obtained on a computer. The results of this solution are compared with the experimental data. An analytic solution that makes it possible to estimate certain limiting values is obtained for the linearized equation.Mekhanika Polimerov, Vol. 4, No. 1, pp. 145–150, 1968     

Buckling of vertical rod loaded by dead weight and tensile force subjected to its top end (applied to marine vertical oil pipelines—Risers)

The problem of buckling of a vertical rod loaded by dead weight and tensile force to its top end was examined. Four variants for attachment of the top end of the rod upon clamping the bottom end were analyzed. The problem was solved by individual consideration of the stretched and compressed zones of the rod and their sewing together. Critical values of the axial force were obtained, which permit prevention of buckling of a long rod by the action of dead weight. Analysis permits selection of the required tension of vertical pipelines for deep-water oil drilling and demonstration of the advantages of using composites (especially, carbon-fiber-reinforced plastics) as materials for manufacture of the elements of such structures.Institute of Polymer Mechanics, Latvian Academy of Sciences, LV-1006, Riga, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 527–539, July–August, 1997.     

Investigation of longitudinal wave propagation in a polyethylene rod

Longitudinal wave propagation in a polyethylene rod has been investigated. A computer has been used to obtain a numerical solution of the problem of wave propagation in a rod, whose material is described by the Boltzmann-Volterra equation. The results of the computations are compared with the experimental data. The method of characteristics is employed to construct the dynamic – diagrams for the case of wave propagation in a prestressed rod on the basis of the model of a viscoelastic standard solid.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 59–67, January–February, 1970.     

Deformability and load-carrying capacity of netted composite rod assemblies with metal fittings

A technology is proposed for making netted rod assemblies with metal joints 3–5 times lighter than existing comparable structures. A procedure was developed for winding assemblies out of carbon-fiber-reinforced plastic (UKN-5000 and binder EKhD-MK). Results are presented from a study of the strain characteristics and load-carrying capacity of assemblies with different reinforcement schemes for different conditions of static damage. It is shown that the design of netted rod assembly proposed here is significantly superior to all known analogs with respect to specific stiffness characteristics and provides great dimensional stability in the case of unilateral heating of the product. The method is most effective if used to develop framework-type structures for use in space — platforms, telescope and antenna supports, and masts.Paper to be presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 3, pp. 387–392, May–June, 1995.     

Stress distribution near a reinforced elliptical hole in a spherical shell at the elastoplastic stage of deformation

On the basis of Lagrange's variational equation the authors obtained nonlinear resolvent equations and coefficients taking into account the effect of a reinforcing element (a rod) on the state of stress and strain of a spherical shell weakened by a curvilinear (elliptical) hole. The article explains the method of numerical investigation of the inelastic state of the shell based on the application of the variational difference method in combination with the method of elastic solutions. The inelastic state of a shell with a reinforced hole was numerically investigated.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 80–83, 1990.     

A method for determining the dynamic Young's modulus of curvilinear specimens

On the basis of a numerical solution of the problem of the natural frequencies of a rod shaped as the are of a circle, the authors suggest a method for determining the dynamic Young's modulus of a curvilinear specimen. They derive an approximate expression suitable for practical use. The results agree well with experiment.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 335–339, March–April, 1974.     

Experimental investigation of the correspondence between quasistatic tension and longitudinal wave propagation in a polyethylene rod

The propagation of longitudinal waves created by the impact of a rigid mass against a polyethylene rod of finite length has been investigated. The experimental procedure is described and oscillograms of the process are presented. The relaxation spectrum is calculated from the experimental data on the mechanical behavior of the material over a broad range of strain rates. The results of the experiments are compared with a numberical solution of the problem of the impact of a rigid mass against a rod whose material is characterized by a relaxation spectrum.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 564–567, May–June, 1971.     

Nonlinear creep of plastics under random loads

A closed system of equations in moment functions is derived for the geometrically linear, but physically and statistically nonlinear, boundary value problem of the theory of creep of isotropic plastics, homogeneous in the starting state, in the presence of random loads with small variances (within the framework of the applied theory of random functions). The boundary value problem is solved by constructing successive approximations. The convergence of the approximations is illustrated with reference to stress relaxation in a rod of uniaxially reinforced plastic subjected to a random axial load.Mekhanika Polimerov, Vol. 4, No. 2, pp. 237–245, 1968     

Sudden buckling of the outer layer of a noncentrally compressed triple-layer rod

The stability of the outer nonmetallic layer of a triple-layer rod with foam core was studied both experimentally and theoretically by the energy balance method. The buckling of the outer layer occurs by a sudden transition to a nonadjacent equilibrium form and this process should be seen in regard to a set of deformational factors.Moscow State University for Ecology and Environmental Studies, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 525–530, July–August, 1996.     

Evaluation of discrete relaxation-time spectrum in the torsion of a round polymer rod

The torsion of a round polymer rod is examined on the basis of the generalized Maxwell equation, taking into account two terms of the relaxation-time spectrum. In finding the relaxation constants corresponding to the younger term, we used experimental torsion diagrams obtained with thick cylindrical specimens having a significantly inhomogeneous stress pattern. It was shown that use of the generalized nonlinear Maxwell equation with two terms makes it possible to obtain satisfactory agreement between the theoretical and experimental torsion diagrams and between the theoretical and experimental creep curves over a considerable time interval. It was established that calculation of the maximum tangential stresses by use of experimental torsion diagrams yields values virtually identical to those calculated by the refined method. The polymer constants given, determined on the basis of experimental torsion diagrams, can be used to find a first approximation of the two polymer relaxation constants constituting the youngest component of the spectrum. The theoretical and experimental curves were compared.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 682–688, July–August, 1972.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.