Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 2. Toroidal bodies under the action of centrifugal forces and torsion

A technique of numerical solution of boundary-value problems is expounded for elastic homogeneous fiber-reinforced bodies of revolution, which is based on equations obtained previously and employs the methods of finite differences and continuation of solutions with respect to loading parameters of the bodies. Results for the torsion of highly elastic toroidal clutches with a semiannular profile are presented. The paper continues the investigation on large deformations of all-rubber and rubber-cord clutches initiated in its first part. The deformations are caused by centrifugal forces in their rotation around the symmetry axis and by the rotary movement in a combination with torsion caused by a relative rotation of their butt-end sections around the axis. The effect of imposition sequence of butt-end torsion and rotary movement on the deformational behavior of the clutches is evaluated.     

Axisymmetric deformation of cylinders made of homogeneous and fiber-reinforced elastic materials in butt-end torsion

The equations of axisymmetric deformation of a circular cylinder made of homogeneous and weakly fiber-reinforced materials in a macroscopically two-dimensional problem statement are presented. The fibers are placed in the axial, circumferential, and radial directions and along spirals in the cylinder wall. They also can lie along opposite spirals in adjacent cylindrical layers, forming an angle-ply reinforcement scheme. At a low content of fibers, it is assumed that they are in the uniaxial stress state, determined by the axial tension or constraint compression in the deformed cylinder. Based on this mathematical model, the deformation of homogeneous and fiber-reinforced elastic cylinders of various length, with free outer surfaces, in butt-end torsion is investigated. The effect of length of the cylinders on their deformation character in torsion is evaluated. The torsion of the cylinders under the conditions of sliding fit along the inner surface and under the action of pressure on the surface, when the initial and deformed configurations are practically congruent, is considered.     

Large deformations of homogeneous and fiber-reinforced cylinders under the action of centrifugal forces

The problem on large deformations of round cylinders made of homogeneous and fiber-reinforced elastomeric materials under the action of inertia forces caused by rotatory motion is solved. The solution is given in a plane axisymmetric statement, where the deformation parameters of the cylinders depend only on the radial coordinate. Based on the mathematical model presented, the deformation of revolving cylinders made of homogeneous elastomeric materials and reinforced with fibers in the axial, circumferential, and radial directions is investigated. The ultimate rotational speeds for cylinders with different reinforcement schemes and tight and loose fits are found at which their limiting configurations are reached. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 347-366, May-June, 2009.     

Large deformations of a new class of incompressible elastic bodies

The consequences of the constraint of incompressibility is studied for a new class of constitutive relation for elastic bodies, for which the left Cauchy–Green tensor is a function of the Cauchy stress tensor. The requirement of incompressibility is imposed directly in the constitutive relation, and it is not necessary to assume a priori that the stress tensor should be divided into two parts, a constraint stress and a constitutively specified part, as in the classical theory of nonlinear elasticity.     

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     

Iterative method of designing uniform and laminated shells of revolution made of highly elastic materials

Conclusion An iterative method was proposed for numerical solution of problems concerning the axisymmetric deformation of uniform and laminated shells of revolution made of highly elastic materials. Calculations performed for rubber, rubber-fiber, and rubber-layered shells demonstrate the reliability and effectiveness of the method.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 109–116, January–February, 1990.     

Finite elastic deformations of bodies with irregular shapes

Zusammenfassung In der Theorie endlicher Deformationen elastischer Festkörper sind jene Probleme, für die exakte Lösungen verfügbar sind, beschränkt auf Körper, deren gespannte und entspannte Zustände durch Koordinatenflächen eines der bekannten und allgemein verwendeten Koordinatensysteme begrenzt sind, und die daher symmetrische Eigenschaften in hohem Grade besitzen. Diese Abhandlung entwickelt eine Methode, mit deren Hilfe man Lösungen für endliiche Deformationen erhalten kann, und zwar für Körper, deren nichtdeformierte Form eine Perturbation der nichtdeformierten Form eines anderen Körpers ist, für den eine endliche Deformationslösung schon verfügbar ist. Diese Methode wird verwendet zur Bestimmung einer Lösung zu dem Problem der endlichen Streckung und Torsion eines elliptischen Zylinders, dessen Verhältnis von Hauptachse zu Nebenachse nahezu eins ist. Im Weiteren erleichtert diese Methode eine Analyse des Problems einer endlichen Ausdehnung für zwei Fälle achsensymmetrischer Körper: 1. Der Körper ist ein Kreiszylinder in allen Punkten ausserhalb eines Streifens endlicher Länge, wo seine nichtdeformierte Form eine Perturbation des Kreiszylinders ist. 2. Der Körper wird erhalten durch Rotation einer Sinuskurve um die Symmetrieachse.     

Torsion of prismatic elastic bodies containing screw dislocations

The problem of the stressed state of a prismatic anisotropic rod containing screw dislocations, the axes of which are parallel to the rod axis, is considered. Such defects may arise during the growth of filamentary crystals (metal “whiskers”), and may also exist in multiply connected cylindrical structures. The torsion of an anisotropic elastic bar with a multiply connected cross-section is investigated initially, assuming that the stresses and strains are single-valued but dispensing with the requirement that the warping function should be single-valued. The boundary-value problem is formulated in terms of the Prandtl stress function, which, unlike the warping function, is single-valued in a multiply connected region. A variational formulation of the boundary-value problem for the stress function is given. From the variational principle obtained a torsion boundary-value problem is formulated when there are lumped or continuously distributed dislocations. A modification of the membrane analogy for the torsion problem is proposed which takes into account the presence of dislocations. General theorems of the theory of the torsion of a rod containing dislocations are formulated. An effective formula is derived for the angle of torsion of a bar due to a specified dislocation distribution. Problems on dislocations in a thin-walled rod and a rectangular anisotropic bar are solved.     

Large plane deformations of rectangular elastic sheets

The large deformation of a plane Mooney sheet is considered. The total potential energy of the deformed sheet is written in terms of the gradient of functions describing the deformed configuration. The minimum potential energy principle and the Ritz procedure are used to obtain the solution. Nonhomogeneous deformations of a rectangular sheet under uniaxial stretching are given as an example.

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Résumé Nous considérons ici la déformation extrême d'une feuille plate de Mooney. L'énergie potentielle totale da la feuille déformée peut se décrire au moyen du gradient des fonctions qui décrivent la configuration de la feuille. Le principe de l'énergie potentielle minimum et le procédé de Ritz nous permettent alors de résoudre le problème. Nous discutons, à titre d'exemples, des déformations nonhomogènes d'une feuille rectangulaire qui subit des tensions uniaxiales.

     

Superposition of large butt-end and coaxial torsional and axial shear deformations of homogeneous and fiber-reinforced thick-wall cylinders

A mathematical model of superposition of large butt-end and coaxial torsional and axial shear deformations of homogeneous and fiber-reinforced thick-wall cylinders is constructed. The macroscopic stresses of the reinforced material are additively determined by matrix stresses and by tensile or constrained compression stresses in the reinforcing fibers. The model is based on the numerical solution of two boundary-value problems, one of which corresponds to the butt-end torsion and the other to the coaxial torsion and axial shear. The boundary-value problem on joint deformations is solved with the use of the displacement field determined from the solution to the boundary-value problem on butt-end torsion. The results obtained by applying this method to homogeneous and axially-radially reinforced thick-wall cylinders subjected to butt-end torsion with subsequent coaxial torsion and axial shear are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 465–492, July–August, 2007.     

A study of the elastic deformations in a thermoelastic inhomogeneous solid of revolution

We construct a mathematical model for studying the elastic deformations in a thermoelastic inhomogeneous solid of revolution applicable to sliding bearings. The method of numerical solution is based on the grid method and the relaxation method. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 91–94.     

Large plane deformations of thin elastic sheets of neo-Hookean material

Zusammenfassung Es werden grosse ebene Verformungen dünner elastischer Scheiben aus Neo-Hookeschem Material betrachtet und eine Methode der sukzessiven Substitutionen entwickelt, um Probleme im Rahmen der zweidimensionalen Theorie endlicher ebener Spannungszustände zu lösen. Die erste Näherung wird durch lineare Randwertprobleme für zwei harmonische Funktionen bestimmt, und sie wird asymptotisch angenähert für sehr grosse Dehnungen in der Ebene der Scheiben. Die zweite und die höheren Annäherungen werden durch Lösung Poissonscher Gleichungen gewonnen. Es werden verschiedene Beispiele behandelt, und für rotationssymmetrische Verformungen wird gute übereinstimmung zwischen den Näherungen und den exakten Lösungen gefunden.     

Two displacement methods for in-plane deformations of orthotropic linear elastic materials

Two displacement formulation methods are presented for the plane strain and plane stress problems of orthotropic linear elastic materials having the three planes of symmetry at x₁ = 0, x₂ = 0 and x₃ = 0. The first method starts with solving the two governing partial differential equations simultaneously, while the second method begins with solving one equation and ends with enforcing the other. The former follows the approach of Eshelby, Read and Shockley, whereas the latter is based on an extended version of Green's theorem and thus has similarities with Airy's stress function method. The two displacement methods lead to the same characteristic equation that is identical to the one obtained by Lekhnitskii using a stress formulation method. The general solutions resulting from the two displacement methods can be used to solve plane elasticity problems of orthotropic materials with displacement or mixed boundary conditions.     

Spatial contact problems for rough elastic bodies under elastoplastic deformations of the unevenness

On the basis of /1, 2/, a model is constructed for the contact between a rigid stamp and a rough body taking elastoplastic deformations of the unevenness into account. The contact model for rough bodies with elastic deformations of the unevenness is a special case. A classical approach utilizing boundary integral equations is applied in the mathematical formulation of the contact problem. Under quite general assumptions (for instance, the multiconnectedness of the contact domain desired), the uniqueness and existence of the solution are investigated. A method is developed to determine the contact pressure, the closure of the bodies, and also the contact area which consists of two parts in the general case, a zone of elastoplastic deformation of the unevenness and a zone of their elastic deformation. The efficiency of the method is shown in examples of new contact problems. The solution is represented in a convenient form for analysing the influence of the roughness. This is of considerable value for material testing by a contact method. A fairly complete survey of research on contact problems for rough bodies can be found in /1–4/.     

Hamiltonian approach to studying thin shells of revolution made of composite materials

A method for constructing defining relations of the linear theory of shells of revolution in complex Hamiltonian form has been proposed. Based on the Lagrange variational principle, we have constructed a mathematical model of a multilayer orthotropic shell of revolution. We have obtained explicit expressions for the coefficients and right-hand sides of the Hamiltonian complex system of equations describing the statics of shells of revolution in terms of their rigid characteristics and acting loads. The Hamiltonian resolving system of linear differential equations, formulated in the axially symmetric case, has some specific properties facilitating both analytical studies and numerical procedures of their solution.