Combined bending and compression of circular cylindrically orthotropic plates with low shear strength

Equations are derived for the combined bending and compression of cylindrically orthotropic plates with allowance for shear and without postulating a shear distribution lawf/it(z). Solutions of the equations obtained are given for various boundary conditions. The error introduced by postulating the distribution lawf/it(z) is estimated. The effect of the ratio of the Young's moduli in the circumferential and radial directions E/E_r and the ratio of the Young's modulus E_r and the shear modulus G on the deflection w and the critical load T_cr is investigated. It is shown that for plates made of materials such as oriented glass-reinforced plastics the shear correction is very important. It is noted that buckling (due to shear) can occur even when the Young's moduli of the material are infinite.Mekhanika Polimerov, Vol. 4, No. 1, pp. 116–123, 1968     

Study of microbuckling of fiber-reinforced composite materials

On the basis of the theory of microbuckling of lamina-reinforced composites and formula predicting the critical composite stress for microbuckling in the shear mode _cs published in the literature, a FORTRAN program for study of the behavior of microbuckling of fiber reinforced composites has been developed. Some types of composite materials (reinforcement of different fibers and epoxy matrix) have been studied. Graphics and curves, accounting for the dependences of the compressive stress at failure _cs from the reinforcement volume k, specimen length L, and shear modulus of resin G_r have been obtained. The comparison of the theoretical diagrams presented here and experimental and theoretical results, published in the literature shows good agreement. The basic conclusion of the work presented here is that the study could be used for other fiber reinforced composites (with different mechanical properties of matrices and fibers).Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria. Published in Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 531–538, July–August, 1996.     

Elastic constants of monotropic plastic foams. 3. Deformation parallel to foam rise direction. Analysis of the results

Based on an elaborate mathematical model shaped like an ellipsoidal cell, the Poisson's ratio v ₃₁ ^* and Young's modulusE ₃ ^* are calculated for monotropic (isotropic in the limiting case) plastic foams when loading parallel to the foam rise direction is considered and the hypothesis of half-axis is assumed. The effect of the state of the strut system on the calculation results is studied. The dependence of the calculated elastic constants on the characteristics of plastic foams such as the space filling coefficient, degree of anisotropy and knot parameter is analyzed. The theoretical results are compared with the experimental results as well as the results of other authors.Institute of Polymer Mechanics, University of Latvia, 23 Aizkraukles St., Riga, LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 823–838, November–December, 1998.     

Effect of fiber curvature on the modulus of elasticity for unidirectional glass-reinforced plastics in tension

The effect of curvature of the fibers on the modulus of elasticity of unidirectional glass-reinforced plastics (GRP) in tension has been investigated on the basis of the theory of layered reinforced media with random initial irregularities [1,2]. It is shown experimentally that relatively minor distortion of the fibers during manufacture may result in important changes in the modulus of elasticity. A model is proposed for determining the effect of fiber curvature on Young's modulus; experiments on specimens with a given regular fiber curvature indicate good agreement with the theory. The effect of prestressing the fibers on the modulus of elasticity has been studied using commercial AG-4S material and the optimum prestress has been established.Mekhanika Polimerov, Vol. 3, No. 2, pp. 243–249, 1967     

Experimental study of young's modulus of glass-plastics

The results of study of Young's modulus of glass-plastics in the 10^–1–10⁵-Hz frequency range are presented. A stepwise variation in the modulus, which governs the relaxation process corresponding to a low-temperature internal-friction peak in the filler, is obtained.Rybinsk Institute of Aircraft Technology. Translated from Mekhanika Polimerov, No. 6, pp. 1089–1091, November–December, 1975.     

Effect of hydrostatic pressure on the deformation properties of polymer materials

It is suggested that the bulk modulus of homogeneous polymer materials is a function of the specific volume only. The dependence of the bulk modulus and Young's modulus on temperature and hydrostatic pressure is determined on the basis of this assumption. It is shown that the Young's modulus must be higher in compression than in tension. The experimental data are confirmed by the relations obtained. The effect of hydrostatic pressure on the relaxation properties of materials is discussed.Institute of Polymer Mechanics, ASLatSSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 986–991, November–December, 1968.     

Effect of thermal treatment temperature on the structure and properties of carbon fibers

Conclusions 1. It has been shown that the presence of a maximum in the dependence of strength on Young's modulus for carbon fibers made from PAN fiber may be explained by an effect of the process of temperature stress accumulation which takes place under the conditions of isometric heating. The start of this process, which causes a rearrangement of the internal structure of the high-modulus fiber, coincides with the start of the anomalous rise in fiber density.2. The interconnection between surface and internal defects and the elastic-strength properties of carbon fibers made in the temperature treatment range 600–3000°C has been studied.3. Original data on the elastic-strength properties of borided carbon fibers have been obtained; the structure of these is marked by a high degree of perfection. It has been shown that in boriding, which facilitates graphitization of the carbon, the process of regular reduction in fiber strength which is reached in the precrystallization stage is somewhat retarded.All-Union Scientific-Research Institute of Aviation Materials, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1036–1042, November–December, 1976.     

The effect of interphase on residual thermal stresses. 2. Unidirectional fiber composite materials

In real composite materials an additional phase may exist between the fiber and the matrix. This phase, commonly known as the interphase, is a local region that results from the matrix bonds with the fiber surface or the fiber sizing. The differing thermal expansions or contractions of the fiber and matrix cause thermally induced stresses in composite materials. In the present study, a four-cylinder model is proposed for the determination of residual thermal stresses in unidirectional composite materials. The elastic modulus of the interphase is a function of the interphase radius and thickness. The governing equations in terms of displacements are solved in the form of expansion into a series [1]. The effective elastic characteristics are obtained using the finite element approach. The effect of the interphase thickness and different distributions of the interphase Young's modulus on the thermal residual stress field in unidirectional composite materials is investigated.For Pt. 1, see [1].Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 200–214, March–April, 1997.     

On equivalence of moduli of smoothness of polynomials in ,

It is well known that for functions , 1p∞. For general functions fL_p, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the L_p space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with .     

Relative stiffness of irregularly perforated plates

Approximation of the average field is used in order to calculate the relative elastic properties of plates irregularly perforated with a large number of different hole. Relative properties are calculated for a plate perforated with a different kind of elliptical hole. It is shown that the Young's modulus and shear modulus of a plate perforated with identical round holes calculated in this approximation have values similar to those calculated accurately for the case of a regularly perforated plate.Translated from Dinamicheskie Sistemy, No. 5, pp. 3–11, 1986.     

Natural aging and dynamic mechanical properties of polymer matrices

The influence of natural six-month aging at 293 K of two kinds of epoxy resins with different network density on their dynamical properties was studied. All the samples were examined by forced resonance oscillations at a frequency of 200 Hz in the range of 120–520 K. The components of the complex Young's modulus E, E, tan , and low-frequency velocity of sound were measured. The basic temperature transitions including T_g and the network density were determined from the experimental data. It was established that after aging of epoxy resins cured by the cross-linking agent meta-phenylenediamine their network density decreased (cold degradation). By using the cross-linking agent 2,6-diaminopyridine, and T_g increased. An attempt is made to explain the nature of this phenomenon.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Moscow State Technical University, Russia. Translated from Mechanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 407–412, May–June, 1998.     

Elastic and Sorption Characteristics of an Epoxy Binder in a Composite During Its Moistening

Results of an experimental investigation into the elastic and sorption characteristics of a model composite material (CM) — epoxy resin filled with LiF crystals — during its moistening are presented. Properties of the binder in the CM with different filler contents (v _f = 0, 0.05, 0.11, 0.23, 0.28, 0.33, 0.38, and 0.46) were evaluated indirectly by using known micromechanical models of CMs. It was revealed that, for the CM in a conditionally initial state, the elastic modulus of the binder in it and the filler microstrain (change in the interplanar distance in the crystals, measured by the X-ray method) as functions of filler content had the same character. The elastic modulus of the binder in the CM with a low filler content was equal to that for the binder in a block; the elastic modulus of the binder in the CM decreased with increasing filler content. The maximum (corresponding to water saturation of the CM) stresses in the binder and the filler microstresses as functions of filler content were of the same character. Moreover, the absolute values of maximum stresses in the binder and of filler microstresses coincided for high and low contents of the filler. At v _f = 0.2–0. 3, the filler microstrains exceeded the stresses in the binder. The effect of moisture on the epoxy binder in the CM with a high filler content was not entirely reversible: the elastic characteristics of the binder increased, the diffusivity decreased, and the ultimate water content increased after a moistening-drying cycle.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 499–511, July–August, 2005.     

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L_∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O_P(n^−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.     

Constraints of the KP hierarchy and the bilinear method

We consider generalizedk-constraints of the KP hierarchy where the Lax operatorL is forced to satisfy L _– ^k =q^–1r. We study the effect of those constraints on the bilinear equations.     

Investigations of interlaminar fracture toughness of laminated polymeric composites

The behavior of interlaminar fracture of fiber reinforced laminated polymeric composites has been investigated in modes I, II, and different mixed mode I/II ratios. The experimental investigations were carried out by using conventional beam specimens and the compound version of the CTS (compact tension shear) specimen. In this study, a compound version of the CTS specimen is used for the first time to determine the interlaminar fracture toughness of composites. In order to verify the results obtained by the CTS tests, conventional beam tests were also carried out. In the beam tests, specimens of double cantilever beam (DCB) and end notched flexure (ENF) were used to obtain the critical rates of the energy release for failure modes I and II. The CTS specimen is used to obtain different mixed mode ratios, from pure mode I to pure mode II, by varying the loading conditions. The highest mixed mode ratio obtained in the experiment was G _I /G _II =60. The data obtained from these tests were analyzed by the finite element method. The separated critical rates G _I and G _II of the energy release were calculated by using the modified virtual crack closure integral (MVCCI) method. The experimental investigations were performed on a unidirectional glass/epoxy composite. The results obtained by the beam and CTS tests were compared. It was found that the interlaminar fracture toughness G _IC ^init of mode I at crack initiation and the corresponding value G _II ^Cinit of mode II obtained by the conventional beam and the CTS tests were in rather good agreement. The experimental results of interlaminar fracture of mixed mode were used to obtain the parameters required for the failure criterion. The two different failure criteria were compared. The best correlation with the experimental data was obtained by using the failure criterion proposed by Wu in 1967 containing linear and quadratic terms of the rates of the energy release.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 307–322, May–June, 1998.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.     

A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of S _n is derived from Young's seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Young's seminormal representations.     

The effect of interphase on residual thermal stresses. 1. Single fiber composite materials

Conclusions The residual thermal stresses in the constituents of a fiber-reinforced epoxy have been predicted using a concentric three-cylindrical (fiber-interphase-matrix) assemblage analysis. The interphase has been treated as a region with a variable Young's modulus — a direct consequence of the changes in the microstructure of the matrix near the fiber surface. The Navier equations of elasticity have been solved in series form solutions for each type of property variation.A parametric study is used to demonstrate the fact that changes in the interphase properties can drastically affect the residual stresses in the interphase.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 5, pp. 579–589, September–October, 1994.     

Effect of hydrostatic pressure on the tensile strength of polymer materials

The effect of hydrostatic pressure on the tensile strength of amorphous, crystalline, and thermosetting polymers has been investigated. The maximum hydrostatic pressure was 2000 kgf/cm². The surfaces of some specimens were protected from the ambient medium (oil). The tests showed that hydrostatic pressure improves the strength or high-elastic limit and Young's modulus of all the materials investigated. In the case of brittle materials, the increase in strength is greater if the surface is protected, whereupon the plasticity is also improved. Hydrostatic pressure produces important changes in the deformation behavior of crystalline polymers.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1043–1047, 1967     

Little q-Legendre Polynomials and Irrationality of Certain Lambert Series

Certain q-analogs h _p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erds (J. Indiana Math. Soc. 12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory 37, 1991, 253–259; Proc. Cambridge Philos. Soc. 112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln _p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of (2) and (3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln _p (2)). In this paper we show how one can obtain rational approximants for h _p(1) and ln _p (2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for h _p(1) and a better upper bound as the one given by Matala-aho and Väänänen for ln _p (2).