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.     

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     

Finite deformations of an area-preserving material

A continuum theory of finite deformations of an area-preserving material is proposed. The material is assumed to deform preserving both volume and the area on the surfaces defined by unit normal vector field in the material; that is, the material has the internal constraints of incompressibility and inexpansibility. After a brief discussion of plane deformations, basic equations for axisymmetric deformations of rotationally symmetric bodies are derived; deformation gradients are explicitly expressed, and compatibility conditions and equilibrium equations are obtained. To illustrate the calculation of deformation and stress, a simple problem of forming a spherical cap from a circular disc is solved.

---

Zusammenfassung Es wird eine Kontinuumtheorie für finite Verformungen eines flächenbewahrenden Materials vorgeschlagen. Es wird angenommen, daß das Material sich verformt, während es das Volumen und die Oberfläche wie durch das Einheitsnormalvektorfeld im Material definiert beibehält, d. h. das Material hat den inneren Zwang der Nichtzusammendrückbarkeit und der Nichtausdehnbarkeit. Nach einer kurzen Diskussion der Ebenenverformungen werden grundlegende Gleichungen für achsensymmetrische Verformungen von rotationssymmetrischen Körpern abgeleitet, die Verformungsgradienten werden explizit ausgedrückt, Kompatibilitätsbedingungen und Gleichgewichtsgleichungen werden erhalten. Um die Berechnung von Verformung und Spannung zu illustrieren, wird das einfache Problem der Formung einer sphärischen Kappe aus einer kreisförmigen Scheibe gelöst.

     

The mathematical theory of classical thermodynamics

In this paper we systematically present a new approach to classical thermodynamics, using asymptotic distribution from number theory and generalizing the Bose-Einstein distribution. The phase transition from gas to liquid, the thermodynamics of fluids, as well as the behavior of liquids under negative pressure are elucidated.     

The mathematical theory of endosymbiosis I

This work presents a new model of the evolutionary process formulated by the Serial Endosymbiosis Theory represented by a succession of stages involving different metabolic and ecological interactions among populations of bacteria considering both the population dynamics and production processes of these populations. In such an approach we make use of systems of differential equations known as Volterra–Hamilton systems as well as some geometric concepts involving KCC Theory and the Projective Geometry of Berwald Spaces and also correct a statement of M. Matsumoto in the literature on this topic. We also recount in some detail previous work comparing production stability of Endosymbiosis Theory with that of Ancestral Commune Theory.     

Concentrators of deformations and fracture of plastic bodies

An approach to the investigation of shape discontinuity regions as strain concentrators is proposed. The near-concentrator strain fields are determined on the basis of the theory of ideal rigid-plastic body; under the condition of plane deformation, their determination is reduced to integration of ordinary differential equations. The deformation as a function of the location of the plastic region and its shape evolution in the process of plastic flow is studied. The plastic flow is demonstrated to be not unique (within the framework of solution completeness). A deformation criterion for the choice of the preferred plastic flow is suggested. The fracture of a V-notched strip is considered. On the basis of the solutions obtained, an approach to the investigation of the fracture processes for more complicated models is formulated.     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Finite homogeneous deformations of symmetrically loaded compressible membranes

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney–Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made.     

Distributions: The evolution of a mathematical theory

The theory of distributions, or generalized functions, evolved from various concepts of generalized solutions of partial differential equations and generalized differentiation. Some of the principal steps in this evolution are described in this paper.     

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.     

The shell theory including transverse shear deformations and thickness change

A variational method for refining the theory of shells based on power series expansion of displacements has been described. The particular case of a cubic approximation for the tangential displacements and a quadratic approximation for the deflections is considered in detail. A constitutive system of differential equations in the canonical form for the axisymmetrical deformation of cyclindrical shells is derived. As an example, axisymmetrical deformations of a cylindrical shell made of an orthotropic composite material are discussed.Martin Luther Universität Halle-Wittenberg, Fachbereich Werkstoffwissenschaften. Germany. Kharkov State Polytechnical University, Department of Dynamics and Strength of Machines. Ukraine. Published in Mekhanika Kompozimykh Materialov, No. 6, pp. 768–780. November–December, 1997.     

Applications of versal deformations to Galois theory

In this paper we study which solutions to an embedding problem can be constructed using a versal deformation of a group representation over an algebraically closed field of positive characteristic. This question reduces (at least stably) to finding which representations of finite groups have faithful versal deformations. We determine exactly when a versal deformation of a representation of a finite group is faithful in case the representation belongs to a cyclic block and its endomorphisms are given by scalar multiplications. Received: January 30, 2001     

Instability at the interface of bodies subjected to high-elastic deformations

The problem of instability at the interface of two elastic half-planes in compression is investigated within the framework of the plane problem of the stability of deformation of compressible and incompressible elastic bodies at finite subcritical strains [1, 2]. The results are obtained for the plane deformation of bodies with an arbitrary form of the elastic potential. Numerical examples are presented for high-elastic bodies with potentials of the Mooney and Treloar types.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 999–1002, November–December, 1971.     

Finite plane deformations of a three-phase circular inhomogeneity-matrix system

We consider finite plane deformations of a three-phase circular inhomogeneity-matrix system in which the inhomogeneity, the interphase layer and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but with each phase possessing its own distinct material properties. We obtain the complete solution when the system is subjected to general classes of remote (Piola) stress, specifically, remote stress distributions characterized by stress functions described by general polynomials of order n?1 in the corresponding complex variable z used to describe the matrix. As a particular case of the aforementioned analysis, we establish an Eshelby-type result namely that, for this class of harmonic materials, a three-phase circular inhomogeneity under uniform remote stress and eigenstrain, admits an internal uniform stress field when subjected to plane deformations.     

A mathematical theory of climbing

Is it possible to develop a theory of climbing? The answer isno. Mountains are so different in quality of rock and geometryof their faces that any wall would require a specific theoryon the best strategy for being climbed. For this reason, climbingschools, so precise in teaching belaying, rappelling and prusiking,are instead vague in instructing how one must progress alongan irregular wall. This paper suggests a rough model for graspingthis complex problem.     

The unity of physical theory and its mathematical formalism

The relation between the physical theory and its mathematical formalism is shown. Kharkov Physicotechnical Institute, Kharkov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1594–1600, December, 1997.