Constructing convex limit surfaces in material mechanics

In our previous paper [3], a method of constructing convex surfaces in 6D space of symmetric second-rank tensors by means of deforming the unit sphere S⁵ into a conoidal surface has been considered. Now we extend this method to parabolic surfaces and to the case where the shape of the surfaces depends also on the third tensor invariant. The resulting equations can be utilized for specifying the limit surfaces in the mechanics of isotropic and anisotropic solids. Some examples on approximating data on the experimental strength are presented.Institute of Polymer Mechanics, Latvian Academy of Sciences. Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 339–349, May–June, 1996.     

Further studies on the strength geometry of anisotropic materials

The connection between the form of the strength surface of an anisotropic material and the procedure of experimentally determining the starting parameters is examined. Strength surfaces which were constructed for a flat sheet of constructional plywood in four octants of stress space and strength surfaces constructed in three planes of symmetry for two fiberglass plastics in the third octant are shown.S. M. Kirov Leningrad Wood Technology Academy. Translated from Mekhanika Polimerov, No. 2, pp. 269–278, March–April, 1976.     

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.     

Polymer fractography and fracture kinetics

The fractographic method makes it possible to determine the test temperature and time at which anomalies appear in the temperature-time dependence of the strength of polymethyl methacrylate and polycaprolactam by finding the conditions of disappearance of specular zones from the fracture surfaces of these polymers. For PMMA these values are –40°C and 10^–2 sec, for PCL –120°C and 10^–7 sec, respectively.For communication 1 see [2].Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 232–237, March–April, 1971.     

The stability of a minimal surface in a pseudo-Euclidean space

We prove that any piece of a space-like minimal surface of the form z=z(x, y), in the pseudo-Euclidean space E^3,1 with line element dx²+dy²–dz² has the greatest area among close space-like surfaces with the same boundary. For time-like surfaces the following assertion is proved. Let V² be a time-like minimal surface in E^3,1. Then for any domain on V², there exists a smooth variation in the class of time-like surfaces with fixed boundary that increases the area, and there exists a smooth variation in the same class of surfaces that decreases the area.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 41–45, 1990.     

Metric characteristics of Np-compacta and removable singularities for the spaces Lp 1, p ∃ (1,∞)

New metric characteristics of N_p-compacta and NC_p-sets (q^–1+p^–1=1) are established in terms of the q-moduli of composite surfaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Metamaticheskogo Instituta im. V. A. Steklova AN SSSR, pp. 162–171, 1991.     

Smooth convex limit surfaces in the space of symmetric second-rank tensors

Equations are proposed to describe smooth (regular) convex hexagonal and triangular curves in the deviatoric plane. The equations are used to construct smooth convex limit surfaces for media whose properties are arbitrarily anisotropic. The results supplement the findings in [A. Lagzdin' and A. Zilauts, Mekh. Kompozitn. Mater.,32, No. 3, 339–349 (1996)], where the third invariant of a tensor was introduced into the limit-surface equation by means of singular deviatoric curves.Institute of Polymer Mechanics of the Latvian Academy of Sciences, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 171–181, March–April, 1997.     

Korn’s Inequalities for Junctions of Elastic Bodies with Thin Plates

We prove the anisotropic Korn inequalities for an elastic junction of a massive body with thin plates clamped along parts of the lateral surfaces. The distribution of the weight factors in the norms under consideration depends essentially on the disposition of the plates, the way they are clamped to the body, and their relative rigidity (as compared with the body).Original Russian Text Copyright © 2005 Nazarov S. A.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00835).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 876–889, July–August, 2005.     

Application of the method of nondestructive control to investigations of thermal characteristics of crystals

Six crystal samples were examined in the form of anisotropic half-spaces with corresponding arrangement of the directions of anisotropy. The heat sources and measurement devices for the excess temperature were moved about the surfaces of the samples in the given directions with identical velocity. The coefficient of thermal conductivity and the bulk thermal capacity of the crystals possessing straight-line anisotropy were determined.Translated from Matematiceskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 12–16, 1989.     

Cold flow and melting of plastics under severe friction conditions

Dry friction of a series of crystalline polymers (polypropylene, nylon, PTFE) against steel over a wide range of sliding velocities (4.4–4.4×10^–4 cm/sec) and loads (P=7.5–360 kg) with almost complete mutual overlap of the friction surfaces is studied. It is shown that friction of polymer materials against steel may be accompanied by melting of the crystalline polymer (at high velocities) or cold flow (at low velocities) in the surface layers associated with an orientation effect and the appearance of anisotropy.Mekhanika Polimerov, Vol. 1, No. 5, pp. 95–100, 1965     

Application of soliton theory to the construction of pseudospherical surfaces in R3

This paper studies the geometry of pseudospherical surfaces from the point of view of Lorentz harmonic maps from the Minkowski plane into S². After giving appropriate definitions, it is shown that such a map is the Gauss map of a pseudospherical surface. A natural subclass of harmonic maps is isolated and studied using well developed techniques of soliton theory. Then follows a numerical investigation based on these techniques. Examples that fall outside of the aforementioned subclass are also considered.Supported by DFG grants Pi 158/2 – 1 & 158/2–2.Partially supported by DFG grants Pi 158/2–1 & 158/2–2.     

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.     

Basis of a test procedure for anisotropic annular samples

The mixed boundary problem for an anisotropic ring loaded by semidiscs or semirings is discussed. A numerical analysis of the solution is given. The results can be used for evaluating results of tests on annular samples manufactured from anisotropic materials.Moscow Energy Institute. Translated from Mekhanika Polimerov, No. 1, pp. 171–175, January–February, 1972.     

Refined linear theory of elastic anisotropic multilayer shells. Part 2

On the basis of the refined linear theory of elastic anisotropic multilayer shells of arbitrary shape derived in [1] it is established that a number of theorems of the linear theory of elasticity have analogues in the theory of multilayer anisotropic shells.For Part 1 see [1].Institute of Fluid Mechanics, Bucharest. Translated from Mekhanika Polimerov, No. 1, pp. 100–109, January–February, 1976.     

On some classes of algebraic surfaces with an infinite set of subspaces of skew symmetry

Special classes of (m – 1)-dimensional algebraic surfaces F ⁿ in a space E^m with inifinite set ofsubspaces of skew symmetry (in particular, orthogonal) are studied. It is assumed that directions of symmetry, as a rule, are asymptotic for F _n .Translated from Dinamicheskie Sistemy, No. 8, pp. 119–126, 1989.     

Regularity of convex surfaces with definite metrics in three-dimensional pseudo-Euclidean space

This paper presents an investigation of convex surfaces with definite metrics in E³in(2;1) and of the question of imbedding definite metrics in the form of convex surfaces.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 257–277, 1977.The author profoundly thanks É. G. Poznyak and I. Kh. Sabitov for help.     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.     

Unirationality of certain surfaces

Nonsingular cubic surfaces in P³ and nonsingular intersections of two quadrics in P⁴ are investigated. It is proved that if a k-point exists on a surface, there is a k-point not on a line; k is the field over which the surfaces are defined.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 155–159, February, 1969.The author expresses his gratitude to Yu. I. Manin, under whose direction this work has been written.