Anisotropy of the low-cycle tensile fatigue strength and deformability of a glass-reinforced plastic

Experimental results on the anisotropy of the strength and deformability of a 1:1 SVAM oriented glass-reinforced plastic in low-cycle tension in the directions =0, 15, 30, and 45° are discussed. The loading frequency was 1–1.5 cycles/minute. A three-dimensional fatigue strength diagram is constructed in _max - - log N coordinates. It is shown that in the direction =0° the fracture strain depends on stress, whereas at 0° the glass-reinforced plastic fails at almost the same maximum strain, the value of which depends on , but is invariant with respect to _max. It is established that the rate of development of the strain extrema on the linear part of the (N) graph and the stress _max are related by a power-law dependence, whose parameters are determined by the anisotropy of the mechanical properties of the composite.Institute of Mechanics, Academy of Sciences of the Armenian SSR, Erevan. Translated from Mekhanika Polimerov, No. 3, pp. 425–429, May–June, 1976.     

Method of estimating the strength of glass-reinforced plastic shells

A combined method of estimating the strength of glass-reinforced plastic shells in the plane stress is described. The method consists in a combination of destructive and nondestructive methods of determining the necessary starting characteristics for use with a criterion of the strength of orthotropic materials in the form of a fourth-degree polynomial.Ritm Scientific-Production Association. Central Scientific-Research Institute of Shipbuilding Technology, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 571–573, May–June, 1972.     

Some more general bounds on the residual term of the Lagrange interpolation formula

Error bounds are derived for the Lagrange interpolation formula and for the k-th derivative of the residual term of this formula in terms of the Lipschitz constant of the n-th derivative for the case with (n+1) nodes and also for the case when the functions satisfy a special condition: G x, y, z [a, b]: ¦ (x)(z–y) +(y)(x–z)+(z)(y–x)¦G¦(x–y)(y–z)(z–x)¦.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 73, pp. 27–32, 1992.     

Effect of “rest” on creep of glass-reinforced plastic SVAM

The effect of rest on creep of glass-reinforced plastic SVAM is investigated. It is shown that for the cyclic loading regime used (24-h load, 24-h rest) the effect of rest is closely related with orientation of the specimen and with the magnitude of the load.Institute of Mechanics, Academy of Sciences of the Armenian SSR, Erevan. Translated from Mekhanika Polymerov, No. 3, pp. 547–550, May–June, 1974.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

The stressed state and certain fracture characteristics of glass-reinforced plastics loaded in tension at an angle to the direction of reinforcement

The stressed state of glass-reinforced plastic specimens with various structures is investigated experimentally in the case of tensile loads acting at 45° to the direction of reinforcement. A relation is established between the stress distribution over the cross section and the scale factor associated with the relative width of the understressed zones along the edges of the specimen. The fracture characteristics of GRP specimens with different structures are analyzed. Recommendations are given for testing GRP with different structures in tension at an angle of 45° to the direction of reinforcement.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1030–1034, 1967     

Asymptotics of the elastic solution of a plane problem near the crack tip terminating at a nonideal bimaterial interface (mode I and II)

The local stress-strain state in the vicinity of the crack tip in a composite is studied, taking into account the mechanical and geometric features of the nearest interface. The modeling of Mode I and II problems for a semi-infinite crack terminating normally at a nonideal interface in the bimaterial plane is considered. The constituents, of the composite are assumed to be elastic, homogeneous, and isotropic. The intermediate zone between the constituents is modeled by interfacial conditions in the form: [ _n ]=0, [u]=r _n , where [u] and [ _n ] are jumps of the vectors of displacements and tractions along the interface. The diagonal matrix with nonnegative components and the parameter, 0 are defined by the mechanical and geometric characteristics of the intermediate zone, respectively. Thus, the case =0 corresponds to the usual ideal contact conditions along the interface. Using the method of integral transformations, the corresponding problems are reduced to systems of functional equations, and later to systems of integral equations with fixed point singularities. The solvability of the systems of integral equations is proved and the asymptotics of their solutions is found. Based on these results, the local distributions of the displacements and stresses near the crack tip are obtained. It is shown that the interfacial parameters and greatly influence the stress not only qualitatively (the character of the stress singularity near the crack tip changes), but also quantitatively (number of singular terms in the asymptotics increases). The graphs illustrating these results are presented as the values of the interfacial parameters and , as well as the ratio of the shear moduli ₀/₁ of the constituents.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Polytechnical Institute Poland. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 621–642, September–October, 1998.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Conditional functional equations and orthogonal additivity

Summary Some examples of classes of conditional equations coming from information theory, geometry and from the social and behavioral sciences are presented. Then the classical case of the Cauchy equation on a restricted domain is extensively discussed. Some results concerning the extension of local homomorphisms and the implication -additivity implies global additivity are illustrated. Problems concerning the equations[cf(x + y) – af(x) – bf(y) – d][f(x + y) – f(x – f(y)] = 0[g(x + y) – g(x) – g(y)][f(x + y) – f(x) – f(y)] = 0f(x + y) – f(x) – f(y) V (a suitable subset of the range) are presented.The consideration of the conditional Cauchy equation is subsequently focused on the case when it makes sense to interpret as a binary relation (orthogonality):f: (X, +, ) (Y, +);f(x + z) = f(x) + f(z) (x, z Z; x z). A brief sketch on solutions under regularity conditions is given. It is then shown that all regularity conditions can be removed. Finally, several applications (also to physics and to the actuarial sciences) are discussed. In all these cases the attention is focused on open problems and possible extensions of previous results.     

On two mean value properties and functional equations associated with them

Summary Motivated by different mean value properties, the functional equationsf(x) – f(y)/x–y=[(x, y)], (i)xf(y) – yf(x)/x–y=[(x, y)] (ii) (x y) are completely solved when, are arithmetic, geometric or harmonic means andx, y elements of proper real intervals. In view of a duality between (i) and (ii), three of the results are consequences of other three.The equation (ii) is also solved when is a (strictly monotonic) quasiarithmetic mean while the real interval contains 0 and when is the arithmetic mean while the domain is a field of characteristic different from 2 and 3. (A result similar to the latter has been proved previously for (i).)     

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.     

Positive projections and conditional mathematical expectations

One indicates how to construct the reduction of an arbitrary positive projection in to a strictly positive one and one proves that the latter is almost the operator of the conditional mathematical expectation relative to some -sub-algebra. More exactly, one shows a method of selecting from a strictly positive projection an operator of conditional mathematical expectation type (this concept is introduced in the paper).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 172–174, 1974.     

Microwave determination of the mechanical strains in glass-reinforced plastic articles

The theoretical basis for the relation between the change in the electrical parameters of a glass-reinforced plastic article and the mechanical load is examined. Results of experiments confirming the dependence of on are presented.For communication 1 see [2].V. I. Ulyanov (Lenin) Leningrad Electrical Engineering Institute. Translated from Mekhanika Polimerov, No. 5, pp. 917–922, September–October, 1975.     

A class of non-Moufang Bol loops isomorphic to all their loop isotopes

For any two primes, , such that< and divides–1, it is shown that there exists a non-Moufang Bol loop of order ² which is isomorphic to each of its loop isotopes.     

A sandwich theorem and Hyers—Ulam stability of affine functions

It is shown that two real functionsf andg, defined on a real intervalI, satisfy the inequalitiesf(x + (1 – )y) g(x) + (1 – )g(y) andg(x + (1 – )y) f(x) + (1 – )f(y) for allx, y I and [0, 1], iff there exists an affine functionh: I such thatf h g. As a consequence we obtain a stability result of Hyers—Ulam type for affine functions.     

Fundamental equation of information revisited

Summary This paper presents a new, shorter and more direct proof of the following result of J. Aczél and C. T. Ng: IfM: J R (J =]0, 1[ ^k ) is both multiplicative and additive, then the general solution: J R of(x) + M(1 – x)(y/1 – x) = (y) + M(1 – y)(x/1 – y) (x, y, x + y J) is given by(x) = ifM = 0,(x) = M(x)[L(x) + ] + M(1 – x)L(1 – x) ifM 0,where is an arbitrary constant andL: J R is an arbitrary solution of the logarithmic functional equationL(xy) = L(x) + L(y) (x, y J). Also, some extensions of this result to fields more general than the reals are given.     

Weighted Discrepancy and High-Dimensional Numerical Integration

The concept of weighted discrepancy of sequences was introduced by Sloan and Woniakowski when they proved a general form of a Koksma–Hlawka inequality for the numerical integration of functions. This version takes imbalances in the importance of the projections of the integrand into account.In this paper we give estimates for the weighted discrepancy of several important point sets. Further we carry out various (high-dimensional) numerical integration experiments and we compare the results with the error bounds provided by the generalized Koksma–Hlawka inequality and by the estimates for the weighted discrepancy. Finally we discuss various consequences.     

Inversion and structure formulae for a generalized Laplace transform

, D_v(.) , .     

Sur les égalités du miroir et certaines formes faibles de la conjecture de Greenberg

Let p be an odd prime. For any CM number field K containing a primitive p^th-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality ^+– between the plus and minus parts of the -invariant of K. Greenbergs conjecture asserts that ⁺ is always trivial. We study here a weak form of this conjecture, namely ⁺=^– if and only if ⁺=^–=0.     

Final Lift Actions Associated with Topological Functors

For a topological functor U:EB, the fiber U ^–1(b), bB, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U ^–1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=, the subobject classifier, then B(,) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(,) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.