The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler — Jacobi — Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.     

Application of the variational-moment approach to problems of the theory of elasticity of shells

We consider a method of obtaining an approximate system of equations of elasticity theory for shells, based on representing the components of the stress tensor and the displacement vector as a sum of products of moment characteristics depending on the coordinates of the base surface of the shell and functions of the normal coordinate to the base surface.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 56–59.     

Mixed dynamic problem for a half plane and its application to rock mechanics

The mixed dynamic problem of the theory of elasticity is solved for an isotropic half plane. The dynamic equations are reduced to integration of fourth-degree equations in partial derivatives with constant coefficients, after whose solution, the components of the stress tensor and displacement vector are written in a form similar to that introduced by Lekhnitskii for an anisotropic body. The stress state of a rock mass subjected to rapid face advance in a seam is investigated using the solution obtained. The stress distribution is analyzed numerically. The existence of a critical rate at which the stress increases without restriction is demonstrated.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 56–61, 1990.     

<Emphasis Type="Italic">r</Emphasis>-Tuple almost product structures

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampère equations and Jacobi equations.     

Linear invariants of a Cartesian tensor

The number of linear invariants under SO(3) as well as SO(2)of a Cartesian tensor of an arbitrary rank is studied. A linearform is defined in terms of elements of a tensor. It is establishedthat the number of linear invariants of a tensor of rank n underSO(3) equals the dimension of the space of isotropic tensorsof rank n. Formulas for the number of invariants in the twocases are also derived. For the elasticity tensor, our analysisconfirms the results of Norris.     

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 347–358, May–June, 1999.     

Contact Problem for a Pneumatic Tire Interacting with a Rigid Foundation

Based on mixed finite-element approximations, a numerical algorithm is developed for solving a geometrically nonlinear contact problem for a prestressed multilayered Timoshenko-type shell undergoing arbitrarily large displacements and rotations. As unknowns, six displacements of faces of the shell are taken, which allows one to use principally new relationships for components of the Green–Lagrange strain tensor in curvilinear orthogonal coordinates, exactly representing arbitrarily large displacements of the shell as a rigid body. As an example, a tire interacting with a rigid foundation is considered.     

A Contact Linearization Problem for Monge-Ampère Equations and Laplace Invariants

We solve a problem of contact linearization for non-degenerate regular Monge-Ampère equations. In order to solve the problem we construct tensor invariants of equations with respect to contact transformations and generalize the classical Laplace invariants.      

On the applicability of two-dimensional applied theories to problems of the stability of axially loaded cylindrical shells made of materials with low shear stiffness

The applicability of two-dimensional applied theories of the Kirchhoff-Love, Timoshenko, and Ambartsumyan types to problems of the stability of shells with low shear stiffness [1] is discussed. The critical loads given by these theories are compared with those recently obtained [6–8] by solving the problem of the stability of a cylindrical shell on the basis of general solutions [3, 4] of the three-dimensional linearized equations of the theory of elasticity [5].Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 141–143, January–February, 1970.     

Asymptotic justification of the intrinsic equations of Koiter's model of a linearly elastic shell

We show that the intrinsic equations of Koiter's model of a linearly elastic shell can be derived from the intrinsic formulation of the three-dimensional equations of a linearly elastic shell, by using an appropriate a priori assumption regarding the three-dimensional strain tensor fields appearing in these equations. To this end, we recast in particular the Dirichlet boundary conditions satisfied by any admissible displacement field as boundary conditions satisfied by the covariant components of the corresponding strain tensor field expressed in the natural curvilinear coordinates of the shell. Then we show that, when restricted to strain tensor fields satisfying a specific a priori assumption, these new boundary conditions reduce to those of the intrinsic equations of Koiter's model of a linearly elastic shell.     

Stress-strain state in anisotropic multilayer shells with zones of nonideal contact between layers

Conclusions The theorem formulated here corresponds to the most general variational principle in the theory of elasticity. The equations and conditions derived from it constitute a complete system of relations necessary for defining and solving the problems which involve determining the stress-strain state in anisotropic multilayer shell structures. Assuming that some of the relations (2.2)–(2.9) are satisfied a priori, one can formulate other partial variational principles (Lagrange's, Reissner's, et al.).The result obtained here can be utilized for a correct derivation of two-dimensional equations for anisotropic multilayer shells of discrete structure, also as the starting point for devising approximate methods of solution of problems which involve determining the state of stress and strain in anisotropic multilayer shells.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 832–836, September–October, 1981.     

General criteria of the strength of isotropic media

The limit surfaces of isotropic materials are considered in the invariant spaces of the stress tensor. The general requirements that must be met by limit surfaces are formulated. A three-invariant strength criterion for isotropic media sensitive and insensitive to hydrostatic pressure is given in general form. The convenience of analyzing strength criteria in the two-dimensional space of the base invariants of the stress tensor is demonstrated.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 251–261, March–April, 1971.     

Reduction of elasticity theory boundary problems for inhomogeneous media to sets of integral equations

The boundary problem of elasticity theory in stresses or displacements for materials which are continuously inhomogeneous along one coordinate is reduced by means of Laplace and Helmholtz equations to a set of four integro-differential equations, two of which are singular. Each of the equations contains integrals for the contour of the transverse section of a body which is assumed to be piecewise-smooth, and integrals for a region coincident with the section of the body.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 20–23, 1990.     

Review and complements on mixed-hybrid finite element methods for fluid flows

Mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems (linear elasticity, Stokes problem, Navier–Stokes equations, Boussinesq equations, etc.) have known a great development in the last few years. These methods allow simultaneous computation of the original variable and its gradient, both of them being equally accurate. Moreover, they have local conservation properties (conservation of the mass and the momentum) as in the finite volume methods.The purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of Stokes and Navier–Stokes equations, and the linear elasticity problem. Further developments for a quasi-Newtonian flow obeying the power law are presented.     

Spectra of the eigenfrequencies and regions of dynamical instability of cylindrical shells with elastic filler in the case of nonaxisymmetrical vibrations

Conclusions 1. The number of natural vibration frequencies in any frequency interval for an empty cylindrical shell increases in direct proportion to the second power of the interval size, and for a shell with filler — in direct proportion to the third power of it.2. The widest (and also located at the smallest frequencies) dynamical instability region of a cylindrical shell with elastic filler corresponds to nonaxisymmetrical modes of wave formation.3. The limiting transition in the equations of this paper in the case of the frequency of the driving force tending to zero results in an expression for the critical static force for a shell with an incompressible filler. Numerical calculations in this case show, in particular, an increase of the critical force upon an increase in the modulus of elasticity of the filler, which has been noted in a number of the papers of other authors.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. P. Stuchki Latvian State University, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 263–269, March–April, 1977.     

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.     

On the theory of layered anisotropic cylindrical shells stiffened with ribs

The deformation, stability and vibration equations for anisotropic cylindrical shells stiffened with individual longitudinal and circumferential ribs are derived without introducing the hypothesis of nondeformable normals. The more general assumption adopted for layered materials (for example, glass-reinforced plastics) of a linear variation of the displacements over the thickness of the shell and the height of the ribs is used; in this case for the points of contact of the shell and the ribs after deformation the common normals form broken lines. The solution of the problem of the stability of a cylindrical shell stiffened with circumferential ribs is examined. For a shell with different, arbitrarily located ribs the problem is reduced to a homogeneous algebraic system of equations equal in number to three times the number of ribs.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 647–654, July–August, 1974.     

The Algebra of the Riemann Curvature Tensor in General Relativity: Preliminaries

In a four-dimensional curved space-time it is well-known that the Riemann curvature tensor has twenty independent components; ten of these components appear in the Weyl tensor, and nine of these components appear in the Einstein curvature tensor. It is also known that there are fourteen combinations of these components which are invariant under local Lorentz transformations. In this paper, we derive explicitly closed form expressions which contain these twenty independent components in a manifest way. We also write the fourteen invariants in two ways; firstly, we write them in terms of the components; and, secondly, we write them in a covariant fashion, and we further derive the appropriate characteristic value equations and the corresponding Cayley-Hamilton equations for these invariants. We also show explicitly how all of the relevant components transform under a Lorentz transformation. We shall follow the very general and powerful methods of Sachs [1]. We shall not point out at every stage of the calculation which equations are due to Sachs, and which equations are new; this is easily ascertained. Generally speaking, however, the equations depending on the Einstein curvature tensor, and the emphasis placed on this tensor, appear to be new.     

The stress concentration in an elastic ball with nonconcentric spherical cavity

The method of asymptotic perturbation of the shape of the boundary is used to solve an axisymmetric problem of the theory of elasticity for a ball with a nonconcentric cavity loaded by a uniform pressure. Approximate analytic expressions are given for the components of the stress tensor at an arbitrary point of the elastic ball. Four figures. Bibliography: 6 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 37–41.     

Study of wave processes in shells and plates with regard to transverse normal and shear deformation

A variant of the theory of orthotropic plates and cylindrical shells taking account of transverse normal and shear deformation was examined. Independent approximations were adopted for distribution of displacements and stresses over the thickness of the shell. One of the requirements for constructing the theory is physical correctness, which is achieved by utilizing variational methods for formulating the mathematical model. The Reissner principle for dynamic processes was used for derivation of the equations. The elliptical part of the starting differential operator was shown to be symmetrical and positive in the space of the integrate of square functions. We examined the problem of the propagation of axially symmetric harmonic waves in the cylinder using the starting differential equations. These results were compared with those obtained equations derived in elasticity theory. Analysis of induced vibration was carried out for the case of a square plate upon the action of a suddenly applied load.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6. pp. 816–823, November–December, 1995.