Resistance of three-layered structures to static and cyclic bending

Conclusions The above studies of two types of three-layer structural elements showed that the types have different resistances to static deformation in bending. Regardless of the materials, the use of structures which are symmetrical in regard to stiffness makes it possible to obtain a stiffness and strength for the structure which are 10–15% lower than the stiffness and strength of the external plates if the thickness of the latter does not account for more than 25% of the thickness of the structure. This finding, in turn, permits a substantial reduction in the weight of the structure by the use of a lower-density material for the internal layer. Resistance to static bending is determined mainly by the resistance of the structure to shear stresses. The mechanism of fatigue fracture differs appreciably from the fracture mechanism in static deformation. Regardless of the thickness of the structural elements, fatigue fracture for both types of structure occurs as a result of the acting normal compressive stresses. The endurance limit of the hybrid structure is determined by the fatigue resistance of the external layers, and its value is nearly equal to the resistance of the pure materials.Presented at the Sixth All-Union Conference on the Mechanics of Polymer and Composite Materials (Riga, November, 1986).Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 878–882, September–October, 1986.     

Analysis of the states of stress and strain of circular oriented-plastic plates

Retaining the fundamental assumption _z=0, the author investigates two-dimensional problems of the bending of cylindrically orthotropic circular plates in relation to the anisotropy parameter k=E/E_r and a generalized parameter depending both on the transverse shears and the dimensions of the plates h²/a²×E_r/G_rz. Equations are given for calculating certain quantities from the classical variant of the solutions. There are shown to be important differences in the solutions for boundary conditions of the clamped-edge type (when a horizontal or vertical element of the edge is fixed). The effect of the stress _z is estimated from the standpoint of the Ambartsumyan theory [2]. Two types of loading, a distributed load q=const and a concentrated load P applied at the center of the plate, are considered.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 899–908, September–October, 1969.     

Analysis of the state of stress of cylindrically orthotropic rings subjected to concentrated loads

The state of stress of rings of different thickness with high degrees of anisotropy ²=E/E_r, ²=E_gq/G_r is analyzed on the basis of the solution of the plane problem for a cylindrically orthotropic body. The accuracy of the stress determination in accordance with approximate theories of bending of curved bars is estimated. The radial tensile stresses that develop when rings are bent by concentrated loads are calculated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 445–452, May–June, 1976.     

Strength and deformation properties of fiberglass-plastic in the flat stressed state as a function of reinforcement structure

Conclusion Limiting strength values have been ascertained in the flat stressed state as a function of reinforcement structure. The change in each strength surface tensor component as a function of reinforcement intensity has been approximated by the piecewise-linear approximation method. A strength condition has been derived which can be used in optimization problems. The problem of the optimum reinforcement structure of a composite at various ratios of the stresses ₁₁, ₂₂, and ₁₂ has been examined. By using the strength condition, one can predict strength values for structures which appear in the class of materials in question with various reinforcement intensities. The procedure developed can be used in designing composite materials.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 848–859, September–October, 1978.     

Rheological characteristics of melts of low-molecular-weight branched polyethylene in the region of critical molecular weights

The viscosity and the first difference of the normal stresses have been measured at stationary shear rates from 5.68 to 4500 sec^–1 for melts of low-density polyethylenes obtained by thermal degradation of commercial polyethylene in a nitrogen atmosphere. In addition, the complex modulus has been measured on the angular frequency range from 0.396 to 198.5 sec^–1. Three regions of molecular weights differing with respect to the value of the exponent in the relation ₀ = kM have been established. At > normal stresses are observed in the region of shear rates corresponding to a linear dependence of the shear rate on the shear stresses.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga; Plastpolymer Research and Production Association, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 963–967, November–December, 1973.     

Problem associated with the efficiency of parametric methods used in evaluating the long-term strength of fiberglasses

Conclusions 1. The feasibility of using the Larson-Miller parameter for accelerated evaluation of the long-term bending strength of fiberglasses is analyzed. The inadequate effectiveness of this parameter for predicting _lt ^T is demonstrated by statistical processing of test results.2. The possibility of using the Goldfein parametric method to extrapolate long-term strength is studied. Values of the quantities T₀, c, andlog _st, which enter into the Goldfein parameter, are evaluated and defined more precisely.3. The efficiency of the Goldfein parametric method in predicting the long-term strength of various fiberglasses subjected to both bending and compression is demonstrated; this will make it possible to consider this method effective for accelerated evaluation of _lt ^T in cases where the binder plays an active role in supporting a load.All-Union Scientific-Research Institute of Fiberglasses and Glass Fiber, Moscow Region. Translated from Mekhanika Polimerov, No. 4, pp. 645–652, July–August, 1978.     

Recurrence relations in the theory of Hecke operators

In this paper one computes the–2 power of the Frobenius element of the Hecke ring of the subgroup _n,1(q) of a modular group of genus n+1, which is the semidirect product of the Heisenberg group and the modular group _n(q) of genusn. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 65–73, 1983.     

Certain results of an experimental study of the stress distribution at the end of a monofilament

The present article gives the results of a polarization-optical investigation of the stress pattern at the end of a round aluminum rod in a sheet of ÉD-6M epoxy resin. The problem was solved for a pseudoplane in the elastic approximation. The distribution pattern for the principal stresses and the stresses _xy, _x, and _y over the specimen is given. It was found that the points of maximum stress _xy in transverse sections of the specimen did not lie at the fiber surface. The radial stresses _y had a weakening effect on the bonding at the interface with the medium. The results are discussed and compared with those of other authors.S. P. Korolev Kuibyshev Aviation Institute. Translated from Mekhanika Polimerov, No. 2, pp. 262–266, March–April, 1972.     

Some features of bone tissue deformation under unloading and repeated laoding

Conclusions 1. There is a change in the initial elastic properties of compact bone tissue when a load is removed after repeated loading.2. The elasticity modulus after the load has been removed can be assumed to be independent of the level of stress _11k at which the load was removed.3. With repeat loadings, the tangential elastic modulus increases for the same level of stress when the off-loading stress _11k is increased.4. The value of the irreversible expended specific energy of deformation W increases sharply as the moment of destruction approaches, while its distribution through the zones of the cross section of bone provides good rheological adaptation of the bone to external loads in human walking.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 882–890, September–October, 1976.     

Contact problem for beams made of composite materials

Conclusions 1. In beam sections close to the line of action of a concentrated force the profile of shearing stresses departs appreciably from a quadratic parabola. The magnitude of maximum _xy increases with a lower modulus of elasticity E_y and with a higher shear modulus G_xy.2. With the given values of the anisotropy parameters, typical for glass-plastic and carbon-plastic materials, positive transverse stresses _y become negligible. The zone of local transverse compression extends (accurately within 10% of P/2bh) over the height of the beam section.3. Lowering the transverse stiffness of a material lowers the maximum tensile stresses _x.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 56–62, January–February, 1977.     

Two-parameter model of the failure of adhesive joints

Conclusions The proposed two-parameter model of adhesive failure makes it possible to study the limiting states of a composites with delaminations by constructing criterional relations of the type (4.5), (4.6). Within the framework of the above-stated hypotheses and results, the process of the adhesive failure of composites with delaminations can be represented as follows. The application of an external load results in the formation of zones of interfacial shear prefracture near the ends of the delaminations. The dimensions of these zones depend both on the geometry of the delaminations and on the external load. With a further increase in the load, microscopic interfacial defects accumulate in the region S_es ⁽⁰⁾, which leads to an increase in the size of this region in accordance with a relation of the type (4.3). Avalanche shear failure occurs when the external loads reach the limiting values established in accordance with criterion (1.2).The results obtained here are based on the assumption that the strength conditions for the constituent phases are satisfied. It should be noted that the proposed approach can be used to construct a model which will embrace the process of adhesive-cohesive failure of composites. To do this, it is necessary to also employ the corresponding strength hypotheses of the phases, which in turn increases the number of phenomenological constants (such as the critical stress intensity factors K_Ic, the yield point _y, the fracture stress f, etc.) In the last case, it is possible to qualitatively describe a range of composite failure processes with allowance for the interaction of brittle microcracks within the phases and delaminations of different types. It should also be noted that the results obtained here -particularly the model of failure of adhesive joints with allowance for finiteness of the interfacial stresses — justify the use of the methods of plate and shell theory [8] and other methods in the analysis of laminated composites.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1034–1039, November–December, 1988.     

Asymptotic behavior of the prediction error

Let {X_n} _– be a random process, stationary in the broad sense, with spectral density f() satisfying the singularity condition: · We denote _n ² the mean square prediction error at the prediction of _o by linear forms in X_–1, ... , X_–n. In the paper one investigates the rate of decrease of _n to zero.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 11–24, 1983.In conclusion, the author wishes to express his gratitude to I. A. Ibragimov for his constant interest and help.     

Estimates of the rate of stabilization of certain problems with free boundary

With the help of energy estimates we study the behavior of solutions of the Dirichlet problem and the Stefan problem under unbounded growth of time for the semilinear equation u_t – u_XX + u = 0, (0, 1), in the case of one geometric variable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1299–1306, October, 1992.     

Expansion of automorphic functions on sl3(C>/SU<3)

One constructs the expansion for automorphic functions on SL₃,()/SU(3) using the method presented by the author in a previous paper [Zap. Nauch. Sem. Leningr. Otd. Mat. Inst.,116, 119–141 (1982)].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 144–153, 1983.     

Quasilinear operators and Hammerstein's equation

We describe the class of operators in a Hilbert space H, introduced by A. I. Perov, which can be represented in the form Ax = D(x)x, where D(x) is a self-conjugate operator satisfying the inequalities B_–D(x) B₊ (B_– and B₊ are fixed self-conjugate operators). As an application we obtain new theorems on the solvability of Hammerstein's equation.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 453–464, October, 1972.     

Effect of chemical nature of matrix on the strength of bonds with Armos aramide fibers

This investigation deals with adhesion between high-strength and high-modulus Armos aramide fibers (polyheteroarylene-co-p-phenyleneterephthalamide) and a series of different thermosetting matrices. The effect of the chemical nature of the matrix, time-temperature conditions of bond formation, and test temperature on the strength of the fiber-matrix interface was studied. Modified epoxy and heat-resistant matrices were used as adhesives. As a measure of adhesion, the shear adhesive strength ₀ determined by the fiber pull-out technique was used. It was found that both the adhesive strength and the fracture location in adhesive bonds depended on the nature of the matrix. At room temperature, chlorine-containing epoxy matrices provide the highest values of ₀, while the smallest strength of the interface is observed for bonds with heat-resistant (bismaleimide, oligomethacrylate) matrices. Fracture of adhesive bonds does not always occur at the fiber-matrix interface. A number of the specimens failed near the interface of the fiber. With temperature increase, the values of ₀ decrease. The adhesive strength falls especially drastically in the region of matrix softening. An advantage of heat-resistant matrices is that they retain 60–67% of ₀ value even at 250°C. The strength of unidirectional composites based on the investigated fibers and matrices was also estimated under different loading conditions such as tension, shear, compression, and bending. It was found that the strength in shear and compression did not correlate with the interface strength. The values of _c in bending and tension increased linearly with increase of ₀. The obtained dependences _c–₀ were compared with those of composites based on the SVM polyheteroarylene fibers determined by us earlier.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. 391–406, May–June, 1998.     

Dependence of the crystallization kinetics of cis-1,4-polybutadiene vulcanizates on the type and degree of deformation

The effect of tension and compression on the crystallization rate of cis-1, 4-polybutadiene vulcanizate has been investigated. The crystallization kinetics of deformed specimens were studied by measuring the stress relaxation. A linear relation, common to tension and compression, was observed between the logarithm of the crystallization half-period (log _1/2) and the true stress in the specimens before the onset of crystallization. The crystallization time of the unstressed specimen can be found from the intersection of the straight line with the log _1/2 axis. The dependence of log _1/2 on the degree of deformation is linear only in the region of relatively small strains.Lebedev All-Union Scientific-Research Institute of Synthetic Rubber, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 929–931, September–October, 1970.     

Deformation and strength of laminated quasiisotropic carbon-reinforced plastics

Conclusions The stress-strain state and strength of the quasiisotropic carbon-reinforced plastics depend strongly on the lay-up system of the adjacent plies because the magnitude of the interply stresses and their distribution greatly change when the reinforcement system is changed. The highest stress intensity is recorded at the free edge of the specimen. The magnitude of the residual thermal stresses is comparable with that of the mechanical stresses at the instant of formation of cracks in the weak plies of the composite. The first microcracks form inside the plies with the orientation 90 and ±45° in the direction of reinforcement. The IACs start to form at the free edges of the specimen and propagate into the material at strains from 12 to 39% of limiting strain _x ^l . Delamination of the laminated composite starts later than the formation of IACs (40–88% of _x ^l ) because the normal stress _x is higher than _z. The optimum lay-up system of the composite is [0/45/90/-45]_s because this system ensures higher stiffness of the material. The ±45 and 90° plies are not suitable for external surfaces of the composite since they start to crack very early. The quasiisotropic composites are less sensitive to the value of G_{2 3} than the composites with the ply lay-up [±]_s.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 449–454, May–June, 1985.     

Unitary point spectrum of almost unitary operators

In order that a set, be the unitary point spectrum of some almost unitary operator (i.e., of an operator of the form U +K, where U is a unitary operator and K₁) it is necessary and sufficient that E be a countable union of Carleson sets.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 143–149, 1983.     

A remark on the essential self-adjointness of a Schrödinger operator with a singular potential

We examine the operators=–+v, v L_{2, loe} (R ⁿ ), where S satisfies a natural additional condition of a local nature. If a condition of Titchmarsh type is fulfilled at infinity, then S is essentially self-adjoint in L₂(Rⁿ).Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 571–580, October, 1976.