Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Stability of the momentum state of a three-layer orthotropic cylindrical casing at uniform and nonuniform external pressures

The effect of the momentum subcritical state on the stability of three-layer orthotropic cylindrical casings at external pressures was investigated. The problem is solved by means of equations that take into account transversal shear. It is shown that in some cases the effect of the subcritical state momentum on the critical load parameter is significant.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 874–879, September–October, 1972.     

Flow of a nonlinear viscoelastic liquid in cylindrical channels

The problem of nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in an arbitrary cylindrical channel is examined. On the assumption that the cross flows are insignificant as compared with the longitudinal flows an equation of state is derived for the flow regime in question. A variational principle established for steady-state flows of the investigated media is proposed as the basis of a method of solving problems of the flow of polymer materials in arbitrary cylindrical channels. The flow of a polymer solution in rectangular channels is investigated.Institute of Mechanics, AS UkrSSR, Kiev. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1103–1111, November–December, 1968.     

Variational principles and mathematical methods of solving boundary-value problems for piecewise uniform nonlinear media

We give a variational formulation and a scheme for numerical solution of boundary-value problems for piecewise-uniform nonlinear media taking account of frictional forces. We solve the problem of determining the stressed state of a thick three-layer cylinder of elastoplastic material with linear reinforcing under the action of internal pressure and an axially symmetric temperature field.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 64–67.     

Some problems of simple finite deformation of viscoelastic bodies

Boltzmann's superposition principle is extended to large strains in the case where the principal strain axes do not change with time and are identically directed at all points of the body. The relations obtained are confirmed by experiment. On the basis of these relations the author examines the problem of large strains of a heated viscoelastic cylinder subjected to the action of time-dependent internal and external pressures, and the analogous problem in the presence of a reinforcing cylindrical shell. The solution of the solving nonlinear integral equation of the latter problem is unique and is obtained in the form of a convergent infinite series.Mekhanika Polimerov, Vol. 2, No. 4, pp. 508–518, 1966     

Optimization of a three-layer metal — Composite shell with stochastic properties of the composite by heuristic methods

Conclusions Heuristic algorithms yield solutions to optimization problems involving structural elements made of materials with stochastic properties. This has been demonstrated on the example of an optimally designed three-layer metal-composite cylindrical shell under axial compression. A solution of the optimization problem in a stochastic formulation obviates the necessity of stipulating the values of structural parameters without justification from the standpoint of reliability requirements.Central Scientific-Research and Design-Experimentation Institute of Automated Systems in Construction Industry. All-Russian Central State Construction Office, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 683–689, July–August, 1978.     

Polymer flow in cylindrical channels

The author examines the problem of flow of a polymer melt in a cylindrical channel of arbitrary cross section. It is assumed that the polymer is an isotropic viscoelastic medium. All the coefficients of the equation of state obtained for the flow in question are functions of the second invariant of the strain rate tensor only and can be determined experimentally in simple types of flow. A perturbation method is proposed for the solution of specific problems, the Dirichlet problem for Poisson's equation being solved in each perturbation step.Mekhanika Polimerov, Vol. 2, No. 3, pp. 421–428, 1966     

The effect of the nonuniformity properties of a material in problems of elastic equilibrium and stability of shells

On the basis of two improved models of shells that are nonuniform by thickness (based on assumptions of Timoshenko type stated separately for each layer and assuming a uniform stress-deformed state of a thin-walled element of the layered structure) we study the stressed state and stability of three-layer cylindrical shells with a nonsymmetric packet. We show the equivalence of the two models in describing the reaction of layered shells to uniform external actions. We give an analysis of the dependence of the stress-deformed state and the reserve stability of shells on the structure of the packet.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 89–93.     

Optimization of reinforced cylindrical shells with nonuniform thickness

An optimum multilayer shell is designed whose stack of elementary layers has a nonuniform thickness. This optimization problem is solved numerically for the special cases of three-layer cylindrical shells with dynamic and static stability. The optimum variants of layer distribution in this model are compared with the optimum solutions in [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 298–303, March–April, 1976.     

Dynamic stability of a viscoelastic orthotropic cylindrical shell

A method based on the use of Laplace transforms has been developed for reducing the system of equations of motion of a viscoelastic orthotropic cylindrical shell to a single integro-differential equation. The effect of the viscous components on the regions of dynamic instability is investigated (creep due to the action of the shear stresses is taken into account).Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 714–721, July–August, 1973.     

A simplified approach to estimating the stress state of flexible cylindrical shells under nonaxisymmetric load

An approach is proposed to approximate numerical analysis of the stress state of a geometrically nonlinear cylindrical shell under an unsymmetric load. An approximate technique is also proposed to reduce the solution of a two-dimensional nonlinear problem of a cylindrical shell with unsymmetric parameters to the solution of axisymmetric problems. A numerical example is considered. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 61–66, 1986.     

Homogeneous solutions in the equilibrium problem of an elastic cylinder of finite length

The problem of the equilibrium of a finite elastic cylinder, under the action of axisymmetric normal and tangential loads on its surface, is considered. Within the framework of the method of homogeneous solutions, one establishes the connection between the representations of the displacement vector in the cylinder in the form of series in layer and cylindrical homogeneous solutions. The expansion coefficients in both representations are expressed in terms of the solution of an infinite regular system of linear algebraic equations.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 3–9, 1989.     

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.     

Solution of boundary-value problems of the statics of layered elastic bodies with nonrigid contact between the layers

A technique is proposed for solving three-dimensional problems of the stress-strain state of cylinders, spheres, and shallow elastic bodies with a rectangular projection which are composed of laterally nonhomogeneous anisotropic layers with nonrigid contact between the layers. The solution of the corresponding many-point boundary-value problem is reduced to solving a number of two-point problems by a known numerical apparatus. Solution results are reported for the strain of a three-layer spherical shell with slipping layers.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 62–68, 1985.     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004.     

Application of the technical theory to sandwich shells with a solid polymeric core

The possibilities of using the technical theory for analyzing cylindrical sandwich shells with a core of low-modulus polymeric material are considered. It is shown to be necessary to make assumptions concerning the distribution of the deformations over the elements of the three-layer section and to take account of the shear strains in the core, the flexural rigidity in the longitudinal direction, and the Poisson ratio in determining the forces and moments. The theoretical conclusions have been experimentally confirmed by static tests on a model.All-Union Structural Engineering Correspondence Institute, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 298–304, March–April, 1973.     

Control of the elastic properties of a shell working at stability

This examines a shell with elastic properties varying across the coordinates, which are prescribed by means of scalar functions of the invariants of the elasticity tensor. The basis of the arrangement of the tensor for the elasticity consists of q linear-independent tensors of the fourth range (q is the number of linear-independent components of the elasticity tensor) which are obtained by multiplying and turning the first tensor of the surface and the tensor characterizing the class of symmetry of the medium. The invariants of the elasticity tensor present in the stability equation and their derivatives are taken to be the equations and parameters for the state of the system (shell), and the problem is thus reduced to a problem of optimum equations. As an example we shall examine an orthotropic cylindrical shell with a model varying over the length under the action of external pressure.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 93–100, January–February, 1974.     

Stability of an orthotropic cylindrical shell reinforced by ring ribs with allowance for transverse shear

A detailed discussion is given on the stability of a freely supported orthotropic cylindrical shell reinforced by similar ring ribs (frames) placed at equal distances under the action of a uniform external pressure. The shear is taken into account both in the shell itself and in the frames. The solution obtained is illustrated by examples.Translated from Mekhanika Polimerov, No. 3, pp. 490–495, May–June, 1974.     

Study of the effect of asymmetric arrangement of the layers on the stability of three-layer cylindrical shells in axial compression

The stability of a three-layer cylindrical shell under axial compression is examined in a three-dimensional linearized formulation. The load-bearing layers are assumed to be orthotropic, while the binder is modeled by an isotropic body. The effect of asymmetric arrangement of the layers and the mechanical characteristics of the material on the critical strains is determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 103–106, 1988.     

Large deformations of thermorheologically simple bodies

The relation between the true stresses and true strains associated with the simple finite deformation of a viscoelastic body is modified for the case in which the properties of the material depend on temperature and the body is located in an inhomogeneous nonstationary temperature field. This modification is based on the time-temperature equivalence principle. The validity of this principle for the case in question is confirmed by the experimental data of Andrews, Hofman-Bang, and Tobolsky on the true stress relaxation at large deformations of a polyisobutylene specimen. The modified relations are used to solve the problem of the large deformations of a heated cylinder, made of a viscoelastic material whose properties depend on temperature, under the action of internal and external pressures that are functions of time. The case of polyisobutylene is analyzed as an example of a specific viscoelastic material. The analogous problem in the presence of a reinforcing cylindrical shell is also considered. In the latter problem the solution of the solving nonlinear integrodifferential equation is unique and is obtained in the form of an infinite convergent series.Mekhanika Polimerov, Vol. 3, No. 3, pp. 436–447, 1967