Resonance methods for determining the complex shear moduli of orthotropic composites

A survey of various methods for determining the complex elasticity and shear moduli from the resonant frequencies of flexural and torsional vibrations of rectangular rods cut out from a plate of an orthotropic composite is presented. The errors in the computed values of dynamic shear moduli caused by inaccuracies in the experimental determination of resonance frequencies are examined. A new variant of the resonance method is developed, which permits one to find three complex shear moduli of a composite from the resonant frequencies and the damping of torsional vibrations of three rods oriented along three symmetry axes of the material. For computing the moduli in the case of an overdetermined system, an algorithm of nonlinear optimization based on the least-squares method is recommended. From the results obtained it follows that, for determining the interlaminar shear moduli with a necessary accuracy, the rods must be sufficiently thick. It is shown that a good agreement alone between calculated and experimental frequencies of flexural and torsional vibrations of rods does not ensure a reliable determination of the moduli of interlaminar shear if experiments are carried out on wide test specimens cut out from a thin plate. Recommendations are given for the choice of geometrical sizes of test specimens for resonance experiments. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 721–744, November–December, 2007.     

Analysis of methods of determining the interlaminar shear strength of composite materials

Conclusions The range of geometrical dimensions of a sample in which the values of the interlaminar shear strength determined by four of the five methods discussed are practically constant has been established. Good agreement is shown in the values of the characteristic being determined which are obtained by each of the four methods. It has been established that the experimental data obtained by the method of three-point deflection of short beams does not characterize their interlaminar shear strength. It is shown that the interlaminar shear strength of glass-plastic and the shear strength of the adhesive with which it is prepared are very similar in value.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 640–648, July–August, 1976.     

On the structure of certain natural cones over moduli spaces of genus-one holomorphic maps

We show that certain naturally arising cones over the main component of a moduli space of J₀-holomorphic maps into Pn have a well-defined Euler class. We also prove that this is the case if the standard complex structure J₀ on Pn is replaced by a nearby almost complex structure J. The genus-zero analogue of the cone considered in this paper is a vector bundle. The genus-zero Gromov-Witten invariant of a projective complete intersection can be viewed as the Euler class of such a vector bundle. As shown in a separate paper, this is also the case for the “genus-one part” of the genus-one GW-invariant. The remaining part is a multiple of the genus-zero GW-invariant.     

Variation of certain transverse characteristics over the thickness of wound glass-reinforced plastic rings

The variation of certain transverse characteristics over the thickness of wound glass-reinforced plastic rings has been experimentally determined under normal conditions and at the polymerization temperature of the material investigated. The characteristics in question are used in investigating the state of stress of the ring and the initial fabrication stresses.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 165–167, January–February, 1970.     

Completions of certain modules over quantum groups 1

Completion is a procedure of constructing new representations for Lie algebras from given ones. This paper shows that a similar procedure exists for quantum groups.     

Analysis of elastomeric composites based on fiber-reinforced systems. 1. Development of design methods for composite materials

A method for calculating the elastic properties of fiber-reinforced composites is discussed. The method is based on the structural macroscopic theory for reinforced media [1, 2], which can be used for analysis of stiff and soft composites. As a measure of the elastic properties of composites, the parameters of macroscopic deformations of the base system of Cartesian coordinates are used, with the axes oriented in a certain direction relative to the general reinforcement and loading field. The corresponding macrostresses in the loaded composites are found by a solution of the microboundary problem for a composite macroelement with sides parallel to reinforcement planes of the system. The microboundary-value problem is multiply connected and is formulated based on the information about the homogeneous field of macroscopic displacements specified by the parameters of macroscopic deformation. The problem is solved using the local system of coordinates whose axes are directed along some of the reinforcement trajectories.State Metallurgical Academy of Ukraine, Dniepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 733–745, November–December, 1998.     

The embedding dimension of the formal moduli space of certain curve singularities

For curve singularities which can be deformed into complete intersections the vector space dimension of T¹ is estimated. Thus in the case that T² is trivial we prove a formula of Deligne on the dimension of smoothing components with purely local methods.The author was supported by a grant of the DAAD (Deutscher Akademischer Austauschdienst)     

Testing convexity of figures under the uniform distribution

We consider the following basic geometric problem: Given , a 2‐dimensional black‐and‐white figure is ?‐ far from convex if it differs in at least an ? fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is ?‐ far from convex are needed to detect a violation of convexity with constant probability? This question arises in the context of designing property testers for convexity. We show that uniform samples (and the same running time) are necessary and sufficient for detecting a violation of convexity in an ?‐far figure and, equivalently, for testing convexity of figures with 1‐sided error. Our algorithm beats the lower bound by Schmeltz [32] on the number of samples required for learning convex figures under the uniform distribution. It demonstrates that, with uniform samples, we can check if a set is approximately convex much faster than we can find an approximate representation of a convex set.     

$Omega$-admissible theoryII. Deligne pairings over moduli spaces of punctured Riemann surfaces

In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics. This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric on the moduli space is algebraic. Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001     

On the spectrum of shear flows and uniform ergodic theorems

The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both L²$L^2$ spaces and weighted-L²$L^2$ spaces. As a consequence, an example of a flow admitting a purely singular continuous spectrum is provided. For flows admitting more regular spectra the density of states is analyzed, and spaces on which it is uniformly bounded are identified. As an application, an ergodic theorem with uniform convergence is proved.