Model studies of stress-intensity factors near cracks interacting with fibers

The problem of determining the stress-intensity factors near cracks interacting with fibers is solved for the first time using data from optical-polarization measurements. Simplified models of composites are investigated — plates reinforced with single short fibers with cracks assigned on one of the ends of the fibers and tensioned in the direction of the fibers. The plates (dies) were fabricated from a photosensitive material, and the fibers were modeled by bars of steel, glass, and polymer. The stress-intensity factors were determined from Eq. (5) using data derived from optical-polarization measurements at several points by the nonlinear method of least squares. We investigated the influence exerted by several physical and geometric parameters on the stress-intensity factors. It was established that the stress-intensity factor K₁ near the end of a crack interacting with a fiber is higher than the factor K⁰ ₁ near a crack of corresponding length in the unreinforced plate. The ratio K₁/K⁰ ₁ depends on the mechanical properties of the fiber and die materials and the geometric dimensions of the crack and fiber. Despite the fact that these results were obtained using simplified models of composites, they are of interest for evaluation of the failure and serviceability of real composites.Scientific-Research Institute of Mechanics, M. V. Lomonosov Moscow State University, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 4, pp. 493–501.     

Various methods of determining the natural frequencies and damping of composite cantilever plates. 2. Approximate solution by Galerkin's method for the trinomial model of damping

The application of Kantorovich's method to a trinomial model of deformation taking into account transverse bending of a plate leads to a connected system of three ordinary differential equations of fourth order with respect to three unknown functions of the longitudinal coordinate and to the coresponding boundary conditions for them at the fixed end and on the free edge. For the approximate calculation of the frequencies and forms of natural vibrations Galerkin's method is used, and as coordinate functions we chose orthogonal Jacobi polynomials with weight function. The dimensionless frequencies depend on the magnitude of the four dimensionless complexes, three of which characterize the anisotropy of the elastic properties of the composite. For the fibrous composites used at present we determined the possible range of change of the dimensionless complexes d₁₆ and d₂₆ attained by oblique placement. The article examines the influence of the angle of reinforcement on some first dimensionless frequencies of a plate made of unidirectional carbon reinforced plastic. It also analyzes the asymptotics of the frequencies when the length of the plate is increased, and it shows that for strongly anisotropic material with the structure []_T the frequencies of the flexural as well as of the torsional vibrations may be substantially lower when flexural-torsional interaction is taken into account.For Communication 1 see [4].Institute of Engineering Science of the Russian Academy of Sciences, St. Petersburg, Russia. St. Petersburg State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1. pp. 23–33, January–February, 1997.     

On the minimal global attractor of a system of phase field equations

The unique global solvability of the initial-boundary value problem (1)–(3) is proved for the system of phase field equations (1), (2). It is shown that the problem (1)–(3) generates a continuous compact semigroup V_t, t>0, for which there exists a minimal global B-attractor.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 188, pp. 70–86, 1991.     

Total gluon propagator in light-cone gauge and the transversality problem of the polarization operator in the infrared region. II

A three-gluon vertex that depends on a gauge vectorn and is free of kinematic and spurion (np)^–1 singularities in the physical region is constructed. It is shown that in the case of integer power-law asymptotic behavior of the gluon propagator there is separation of the singular contributions of the single- and two-loop diagrams in the Dyson—Schwinger equation. Necessary conditions for realization ofp ^–4 asymptotic behavior are obtained, and it is shown that an important role in their fulfillment is played by the parts of the three-gluon vertex that depend on the gauge vector. A solution of these conditions is found; when they are fulfilled, they lead to infrared asymptotic behaviors of the complete Dyson—Schwinger equation and its single-loop approximation that are mutually compatible.Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 1, pp. 14–23, July, 1994.     

Groups of classes of pseudohomotopic singular links. I

By a singular link of type (p₁, p₂) in Sⁿ we mean a pair of continuous mappings with disjoint images. In the paper the concept of the pseudohomotopy of singular links is defined, similar to the concept of concordance of classical links, and it is proved that for n>p₂+2 the set of the classes of pseudohomotopic singular links of type (p₁, p₂) in Sⁿ forms an Abelian group with respect to a componentwise connected summation. This group has been obtained in case n2p₂+1–max{n–p₁–2, 0}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 114–124, 1988.     

Temperature-time analogy applied to thermorheologically complex polymer materials. 2. Mixtures of thermoplastic and thermosetting polymers (polymer-oligomer composites)

Conclusions 1. Stress-relaxation curves have been obtained for polymer-oligomer composites over a wide temperature range, and the temperature-time characteristics of properties of mixtures have been analyzed. The effect of each component on the mechanical characteristics of a mixture has also been examined.2. We have studied the temperature depenence of viscoelastic properties of mixtures of tetrafluoroethylene and vinylidene fluoride copolymer with epoxide-diphenylene propane resin. We have demonstrated that polymer-oligomer composites do not conform to the concept of thermorheologically simple behavior.3. The parameters of the temperature-time analogy were calculated with the aid of a computer for polymer-oligomer composites, examples of thermorheologically complex materials. It has been established, as a result, that the reduction factor depends strongly on time and the relaxation function depends strongly on the ratio of components.For communication 1, see [7].Okhtinsk Scientific-Industrial Combine, Plastpolimer, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 430–434, May–June, 1978.     

Periodicity of residues of the number of finite labeled To-topologies

Let T₀(n) be the number of labeled topologies on points satisfying the T_o separation axiom. It is proved that for any prime p the sequence of residue classes T_o(n) mod p is periodic with period length p–1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 32–36, 1982.     

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.     

Effect of filler on the viscosity and viscoelasticity of melts of low-density polyethylene. Communication 2

Conclusions 1. A filler causes a change in the viscoelastic functions of melts, which in general cannot be described in linearly transformable coordinates.2. With rising filler concentration, the maximum of the frequency of the relaxation spectrum N(s) increases, but the frequency corresponding to the maximum decreases.3. The dynamic viscosity and rigidity of matrices and composites as functions of frequency are satisfactorily described by a three-parameter relationship. The relationship derived contains parameters of the degree of filling. Using the dimensionless viscosity '/_o and deformation frequency '₁ it was possible to obtain a relationship which is invariant with respect to the filler in a limited range of change in and which can be used for approximate calculations.4. The initial shear modulus of a composite with a filler concentration corresponding to formation of a secondary network can be lower than the shear modulus of the matrix.For communication 1 see [13].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 122–126, January–February, 1977.     

A. F. Lavrik's truncated equations

By a new method, we obtain some known results of A. F. Lavrik (Dokl. Akad. Nauk SSSR,171, No. 2, 278–280 (1966); Mat. Zametki,2, No. 5, 475–482 (1967); Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 2, 433–448 (1966)) regarding the truncated functional equations of various L-functions. As an application, we give an estimate of Dedekind's zeta-function of an algebraic number field K of degree n4, _k(1/2+it)t^n/6log^ct,t>1 and a similar estimate for L-series with grössencharacters. The method of the paper allows us to consider fields of degree n12.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 124–158, 1978.     

Physicomechanical characteristics of a disperse-filled polytetrafluoroethylene

Composite polymer materials based on PTFE containing ultradisperse -sialon (1–10 wt.%) as the cross-linking agent are studied. It is shown that the injection of small amounts of fillers (1–2 wt.%) increases the degree of composite crystallinity. A correlation between the structure, element distribution on the surface layers of samples, and tribotechnical characteristics of composites is found. The change in the physicomechanical characteristics is associated with the effect of the interstructural plasticization.Ammosov Yakutsk State University, Yakutsk, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 797–806, November–December, 1999.     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

Stress-strain state in the matrix of stochastically reinforced composites

Conclusion We proposed a method of studying the concentration of stresses and strains in the matrix of composites with a stochastic structure in a three-dimensional formulation. The method is based on the use of tensor operators assigned at the inclusion-matrix interface and results from the theory of effective moduli of stochastically reinforced composites.Advantages of the proposed approach include its relative simplicity and clarity, as well as the fact that it can be used to analyze the three-dimensional stress and strain concentration in the matrix of composites with components having very different properties. However, for high values of the volume concentration of reinforcement c₁ > 0.6, it is necessary to use the results of exact solutions obtained, for example, within the framework of deterministic models. The correction that is introduced here is connected with the average of the stresses and strains over an inclusion. The character of their distribution over the interface remains the same as before.The numerical results obtained here show the significant effect of the relative dimensions of the inclusions on the effective elastic properties and the stress concentration in the matrix. Comparison of theoretical values of the longitudinal elastic modulus with experimental results can serve as grounds for validating the proposed variant of choosing the tensor L₀ in the determination of the corrected characteristics and stress-strain state of the matrix.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 396–402, May–June, 1989     

Some properties of solutions of linear homogeneous differential equations with continuous coefficients

An order-reducing formula is obtained for the Wronskian W[y₁,..., y_n] of a system of (n–l)-times continuously differentiable functions. This formula is used to prove that the condition W[y₁,..., y_n] 0 (x 1) is necessary and sufficient for linear independence on I of the solutions y₁,..., y_m of the equation ..... of order n m. A formula isderived which, given a system of n – 1 linearly independent solutions Y_1,..., Y_n–1 of Eq. (1) constructs in explicit forma solution Y₂ linearly independent with Y_1,..., Y_n–1 (a generalization of Abel's formula).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 51–56, 1985.     

Invariant subspaces of the shift operator. Axiomatic approach

There is axiomatically described the class of spaces (resp. ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) z f (z) (resp. the inverse shift f(z)z^–1(f(z)–f (0))) are constructed just like the Hardy space H². It is proved that as one can take, for example, the space H¹, the disk-algebra C_A, the space U_A of all uniformly convergent power series; and as the space of integrals of Cauchy type L¹/H _– ¹ , the space VMO_A. There is also obtained an analog for the space U_A of W. Rudin's theorem on z-invariant subspaces of the space C_A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 7–26, 1981.     

The infinitely differentiable extension of systems of functions

The best extension of systems of functions of real variables from an (n-1)-dimensional hyperplane E_n–1 to the whole of E_n is investigated. It is shown that extension can be realized to a function, infinitely differentiable outside E_n–1, whose derivatives have in a certain sense the best possible rate of growth close to E_n–1 functions (the B-class).Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 723–734, June, 1971.The author wishes to express his gratitude to L. D. Kudryavtsev for directing this work.     

Quantum diffusion in irregular crystals containing two-level tunnel systems

The quantum diffusion problem for admixture particles interacting with two-level tunnel systems (TLS) is investigated. It is shown that for the temperature range ₀ < T 1 K (where ₀ is the mean value of the energy level disturbance for admixture particles in the neighboring wells), and the TLS concentration is N_D 10²⁷ m^–3, the quantum diffusion coefficient for the admixture particles is determined by their interaction with the TLS's.Dedicated to the memory of Dmitri Nikolajevich ZubarevTranslated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 489–497, March, 1996.Translated by A. R. Shirikyan.     

Effect of modifying additions on the physicomechanical properties of composites based on polyethylene and wastes from linen yarn production

This article examines the feasibility of using coupling agents to alleviate the shortcomings characteristics of materials that contain natural fibers: low water resistance and a high degree of heterogeneity. A determination is made of the effect of additions of polyisocyanate and stearic acid on the fluidity of the melts and the strainstrength properties and water resistance of polyethylene composites containing mixtures of wastes from linen yarn production. It is shown that an addition of stearic acid significantly improves the dispersion of fibers in the composites, which in turn leads to a reduction in melt fluidity and an increase in elastic modulus in the high-filler-content region compared to composites that do not have additions of stearic acid. Additions of polyisocyanate appreciably increase the strength and water resistance of the given composites.Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 540–547, July–August, 1997.     

Modification of glass-filled polyamide 66 by reactive oligoorganosilane

Properties of glass-filled polyamide 66 modified by reactive oligoorganosilane were investigated. It was found that modification led to the improvement of the rheological properties of polyamide. The addition of the modifier decreased the glass transition temperature T_g of the polyamide from 60 to 50–59°C, without affecting the melting point. Composites modified by oligoorganosilane are characterized by higher (10–40°C) temperatures of onset and 50% weight loss as compared to the initial composite. It was found that chemical reaction of oligoorganosilane with polyamide and glass fibers took place during coextrusion of the modifier and polyamide, which formed firm chemical bonds between the polyamide and filler and thus favored a considerable improvement in the physicomechanical properties of the composite. The change in the structure and properties of the polyamide observed during modification by oligoorganosilane significantly affected its behavior during friction. The modification made it possible to increase the wear resistance of the composite 1.5 to 2 times and to decrease its friction coefficient from 0.38 to 0.27–0.33. It was also found that the ability of oligoorganosilane to react during its processing with water in the polyamide allowed for a significant decrease in the intensity of hydrolytic processes in the polymer. Because of this, the physicomechanical, rheological, and antifrictional properties of modified composites with an increased content of moisture (up to 3%) in the initial polyamide surpass similar characteristics of the composites containing no modifier, with not only enhanced but also optimum (0.2%) humidity of polymer granules.Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 545–553, July–August, 1998.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Let c_k, k=1,...,4, be arbitrary distinct points of . LetD be the family of all systems of simply connected domains in. By R(D_k, c_k) we denote the conformal radius of the domain D_k, relative to the point c_k. We prove that in the familyD one has the sharp inequality, (1) where a=(+i)/(–1), being the cross-ratio of the points c₁,c₂, c₃, c₄: E(–1, 1,a) is the continuum of least capacity containing the points –1,1,a. An explicit expression for capE(–1,1,a.) in terms of elliptic Jacobi functions has been obtained earlier by the author [Tr. Mat. Inst. Akad. Nauk SSSR,94, 47–65, 1968]. On the basis of the well-known properties of continua of least capacity, one shows that the largest value of the right-hand side of (1) is attained for a=± i3 and it is equal to 4^–8/3·3². One gives all the configurations for which equality prevails in the obtained estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 131–145, 1980.