Calculation of Dependent Elastic Constants of Plastic Foams with a Pronounced Strut-Like Structure

Seven dependent elastic constants of monotropic plastic foams with an expressed strut-like structure are calculated. For this purpose, the basic results of the previously elaborated mathematical model for light-weight plastic foams is used. The model includes a model cell of local structure for monotropic/isotropic plastic foams and an ensemble of structural elements, which allows one to calculate the seven dependent elastic constants, taking into account the pronounced polydispersity of the structure of plastic foams. The numerical values of the constants are compared with the available experimental data, and a satisfactory agreement is found to exist. As a final result, a full set of general expressions and numerical values are obtained for all 12 elastic constants of monotropic plastic foams.     

Modeling of the response of elastic plastic materials treated as a mixture of hard and soft regions

Inelastic materials that form dislocation cells on being deformed are modeled as a constrained-mixture of plastically hard and soft regions by associating different natural states with these regions. The deformation gradient from the reference configuration to the natural configuration is identified as the plastic deformation tensor and the stress is measured from a changing set of natural configurations. Two sets of natural configurations are introduced: one for the hard phase and the other for the soft phase. The full elastic response of the body is determined by elastic responses from different natural configurations. The energy stored in the dislocation networks is explicitly accounted for in the Helmholtz potential. Within a specialized constitutive set up, the soft phase is assumed to be non-hardening while the hardening response of the hard phase is dependent upon the response of both the hard and soft phases. These special forms are used to model the response of the material that forms cellular structures when subjected to cyclic loading.     

Monodromy Approach to the Scaling Limits in Isomonodromy Systems

The isomonodromy deformation method is applied to the scaling limits in the linear N×N matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves that describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated with the n-large asymptotics in the sequence of the biorthogonal polynomials with cubic potentials.     

On the determination of higher order terms of singular elastic stress fields near corners

Summary In two-dimensional elasticity stresses at reentrant corners exhibit singular behavior. The stress field is of the form , where (r, ) are polar coordinates centered at the tip of the corner, andf _i (; _i are smooth functions. For practical use of this series the eigenvalues _i (which are generally complex numbers) are required in order of ascending real part. The problem then is to find the roots of a transcendental equation (eigenequation) in the complex plane and arranged in order of ascending real part.A theorem is proved on the number, location and nature of the roots of this equation with the real part in fixed intervals of length . Excellent initial estimates of the real part of the complex roots become available, and so are bounds, within which single real roots exist. This enables the determination of any number of roots in ascending order of real part. The critical angles at which the eigenvalues change nature are also determined. It is shown that for certain cases and for the symmetric mode of deformation, the eigenvalue =1 does not represent a rigid body rotation, therefore it has to be included in the series representation of the stresses. The coefficientsK _i can be determined by recently developed extraction techniques, thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture, crack initiation, plastic zone estimation etc.Dedicated to Professor Ivo Babuka on the occasion of his 60th birthdayPresented at the Conference: The Impact of Mathematical Analysis on the Solution of Engineering Problems, 17–19 September 1986, University of Maryland, College Park, Maryland, USA     

Elastic constants of monotropic plastic foams. 1. Deformation parallel to foam rise direction. A mathematical model

A mathematical model of the deformative properties and structure of lightweight, monotropic (or isotropic in the limiting case) plastic foams with a pronounced strut-like structure has been elaborated in the linear theory of deformation. A selection of five independent elastic constants is described. For the integral characterization of the deformative properties of plastic foams as micrononhomogeneous composite materials, the elastic constants are introduced as the effective constants. In order to describe the plastic foam structure, a local model consisting of two parts is proposed, i.e., a model of a continuous medium for the calculation of stresses and a local structure model. Considering deformation parallel to the foam rise direction when the semiaxes hypothesis is assumed, the Young modulus and Poisson's ratio are determined.Institute of Polymer Mechanics. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 719–733, November–December, 1997.     

Large Elastic Strains of Plastic Foams

The elastic deformation of plastic foams with a low (< 6%) volume fraction of solid phase is described based on a 4-rod equivalent element. A criterion is proposed which allows one to determine the parameter of structure of this element. Based on an analysis of the equivalent element, a procedure is developed for constructing the compression diagram of plastic foams in the region of large (> 70%) strains. The calculation results are compared with data found in the literature and experimental results for polyurethane foams obtained by the present authors. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 5, pp. 619–632, September–October, 2005.     

Explicit Elastic Curves

The equilibria of thin rods are given by curves which are critical points of the modified total squared curvature. The critical curves are known as elastic curves. It is shown how all the elastic curves are given explicitly in terms of elliptic functions as soon as a certain set of three parameters is known. Every regular curve can be parametrized to have a constant speed but the parametrization is rarely known explicitly. Remarkably, all the elastic curves are here explicitly parametrized to have a constant speed. Curves with fixed distinct endpoints as well as closed curves are admitted. The tangent direction may be constrained at one, both, or neither of the endpoints. There are three major strands of formulas corresponding to: fixed length L, variable length without tension, and variable length with tension (let > 0 and add a term L to the total squared curvature). In the most complicated cases the three parameters are given as solutions to a non-linear system of three equations. In the least complicated case everything is given explicitly in terms of elliptic functions. If the length is variable and there is no tension, at least one of the parameters is completely determined (the elliptic modulus m = 1/2). Using the same set of parameters explicit formulas are given for: the length when it is variable, the total squared curvature, and the tangent angle along the elastic curve. A number of examples are presented which illustrate the full range of constraints.     

The Classical and Generalized Love Problems in a Domain of Low-Frequency Interference Waves of Signal Type. II

Computations of low-frequency interference waves of SH type are performed for the generalized Love problem. The model of a medium consists of a thin elastic layer, a thick layer, and a half-space. The layers and half-space are in rigid contact with each other. It is assumed that the waves pass through the thick layer twice, there and back. A new method, suggested by G. I. Petrashen, for computing wave fields allows us to compute the fields of SH waves and to draw conclusions concerning their behavior. The computations are carried out for < 1 and > 1, where is the ratio of the velocities in the thin and thick layers. In either case, the thin layer significantly distorts a signal transmitted deep into the medium. Bibliography: 6 titles.     

Isotopies of Real Trigonal Curves on Hirzebruch Surfaces

The rigid isotopy classification of nonsingular real algebraic curves of bidegree on the Hirzebruch surface (the projective plane with a point blown up) is obtained. Consequences for the space of curves with a single node or a cusp on a hyperboloid and on are given. Bibliography: 15 titles.     

Wellensysteme in einem geraden dünnen elastischen Verbundstab mit drei Segmenten verschiedener Impedanz

Wave trains of diffraction of an incident pulse in a thin composite rod are determined by means of the well-known reflection and transmission coefficients of plane waves at plane interfaces. The discrete spectrum of the interface stresses of an elastic inclusion is compared with a corresponding continuous spectrum of interface stresses when a rigid but mobile inclusion is inserted. The background of the considerations is the problem of tensile failure of interface bonds in a composite body subjected to compressive pulse loads. The analogy in response found between stiff elastic and rigid inclusions in the one-dimensional case considerably simplifies the problems encountered in more complicated geometrical configurations as, e.g., the problem of a fiber reinforced infinite matrix body.

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Herrn Professor Heinz Parkus zum 60. Geburtstag gewidmet.     

Effect of Interruptions in Loading on the Durability and Deformability of Structural Plastics

An alternative way of estimating the durability of structural plastics under stresses = const 0.4-0.8_f(_f is the failure stress) based on test data obtained in multiple rectangular loading-rest cycles is proposed. It is also suggested to employ the decrease in the instantaneous elastic modulus as a criterion for evaluating the residual service life of plastic parts and structures and elucidating the reasons for their early failure. For the first time, a possibility of considerably increasing the durability and endurance of structural plastics under short pulsed loadings with relatively long interruptions at the initial stage of stress concentration is considered. The cases of a significantly increased endurance of plastics caused by long interruptions after some fatigue loading by high-frequency tension cycles with a zero maximum stress are explained. First experimental confirmations of an increased durability and endurance of microcomposites subjected to short pulsed loadings alternating with long interruptions are obtained. The evolution of the effect of the loading-rest modes on the durability of massive specimens, microcomposites, and dry fibrous reinforcing fillers is demonstrated with examples of a glass-fabric laminate, a microcomposite, and a nonimpregnated glass strand.     

Mechanical properties of polyethylene and teflon over a broad range of strain rates

Tensile load-extension diagrams have been obtained for low- and high-density polyethylene and teflon. Conventional notions concerning the effect of strain rate on the properties of solids are found not to apply. The unconventional distribution of the family of - curves plotted for different strain rates is caused by the combination of high-elastic and plastic deformation associated with the extension of crystalline polymers. An attempt is made to correlate the data on a narrow range of strain rates with the aid of a model of a viscoelastic solid with variable relaxation time, and on a broader range by means of an equation incorporating the limiting dynamic diagram.Mekhanika Polimerov, Vol. 4, No. 1, pp. 45–52, 1968     

Une interprétation du théorème de colonnetti

Summary Considering an elastic-plastic continuum, the plastic deformations of which are given, it is prooved in a general way thatColonnetti's function (4), where ( _x , ..., _xy , ...) is the stress tensor, ( ) the plastic-strain tensor and the elastic potential energy, differs only by a constant value from the fictitious strain energy evaluated from the total strain as if it were purely elastic. Consequently, both functions are simultaneously minima andColonnetti's theorem can be rephrased in terms of fictitious (elastic) strain energy.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

Geometrische Bedingungen für die Integrabilität von Vektorfeldern auf Teilmengen des ℝn

We give a geometric characterisation for those vectorfields on a subset X ⁿ, wich are locally integrable, that is, which locally have sufficiently many integral curves on X. From this we deduce, that integrable spaces X (where each field of a fixed class of differentiability is locally integrable) are rigid under differentiable deformations in the sense of Kodaira-Kuranishi. We give a general construction for integrable spaces and obtain, that analytic varieties induce integrable spaces for each class of differentiability. Compact analytic varieties are therefore C-rigid, which extends [4], 3,1.     

Deformability modelling of foam elongation

The relationships concerning the deformation of rigid cellular foams during elongation are reviewed under the conditions of a monotonous load increase. The cellular structure behavior is modelled by the operation of a certain piston structure. The proposed equation leads to a satisfactory approximation of the experimental stress-strain diagram for polystyrene and poly(vinyl chloride) foams.V. A. Kucherenko Central Scientific-Research Institute of Building Constructions, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 154–157, January–February, 1972.     

The Proalgebraic Completion of Rigid Groups

A finitely generated group is called representation rigid (briefly, rigid) if for every n, has only finitely many classes of simple representations in dimension n. Examples include higher rank S-arithmetic groups. By Margulis super rigidity, the latter have a stronger property: they are representation super rigid; i.e., their proalgebraic completion is finite dimensional. We construct examples of nonlinear rigid groups which are not super rigid, and which exhibit every possible type of infinite dimensionality. Whether linear representation rigid groups are super rigid remains an open question.