Calculation of thin-wall closed profiles of composite materials under a complex loading

Uniclosed caisson structures with deformable perimeters, which are asymmetric in terms of geometry and stiffness and which are subjected to a complex loading (bending in two planes and torsion with respect to the longitudinal axis) are examined in terms of a cylindrical coordinate system Z, S. The possibility of partitioning the general problem of the stress and strain state into elementary problems by fulfilling conditions of orthogonality is demonstrated. The coordinates of the center of rotation are determined. The need for consideration of the deformation of the cross-sectional perimeter, which defines the warping function and normal bitorque stresses under torsion is indicated. The law governing the distribution of tangential stresses, which contains both a constant component that corresponds to Bredt's theory, and also a part corresponding to Vlasov's theory, is derived.Moscow State Aviation Institute (Technical University), Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 3, pp. 349–359, May–June, 1997.     

Method of solving quasistatic boundary value problems of viscoelasticity for media with time-varying properties

It is shown that, if certain conditions are satisfied, the solution of a viscoelastic mixed boundary value problem can be obtained from the solution of the corresponding elastic problem by substituting time operators for the body and surface forces or given displacements. A similar assertion is also proved for media with time-varying properties. The applicability of the method to the theory of shallow flexible viscoelastic shells is demonstrated.Moscow Lomonosov State University. Translated from Mekhanika Polimerov, No. 6, pp. 987–993, November–December, 1969.     

Toward the rank-one singular perturbation theory of self-adjoint operators

The perturbation theory is developed for the case where an arbitrary positive self-adjoint operator is perturbed by the projector on a generalized vector. Similar to the well-known problem –+, we obtain, in the general case, explicit representations for singularly perturbed operators and their resolvents, and we find the point spectrum and an explicit form of the corresponding eigenvectors. Our approach differs from the usual ones and is based on the self-adjoint extension theory of semibounded operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1559–1566, November, 1991.     

State of stress of flexible viscoelastic plates with an opening

The problem of the state of stress of flexible plates with an opening is investigated with allowance for the viscoelastic properties of the material. The basic equations and the corresponding boundary conditions are formulated. The boundary value problem is solved by numerical integration. The effect of the creep properties of the material on the stress redistribution in the plate is investigated with reference to polymethyl methacrylate. It is shown that the angle of rotation, the ring moment, and the ring forces at the edge of the opening vary with time and as a function of the load.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 4, pp. 708–713, July–August, 1971.     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.     

Effect of the rate of deformation on the mechanical properties of compact bone tissue

Conclusions The results of this study show that the ultimate stress increases with increasing deformation rate from 10^–5 to 1 sec^–1 but the initial elasticity modulus remains virtually constant. A characteristic feature of bone tissue is the significant increase or even maximum in specific deformation energy for destruction in the range of deformation rates corresponding to normal physiological conditions for bone function. The deformation diagrams of human bone tissue for the same values of moisture content and deformation rate in tensile testing do not differ from the analogous curves for the bone tissue of cattle. Quantitatively, the ultimate stress ₁₁ and the initial elasticity modulus E₁ are 5–10% and 15–25% greater, respectively, than for cattle bone tissue, while the ultimate deformation eu is virtually the same. An increase in the moisture content of bone tissue leads to a more pronounced dependence of the mechanical parameters on the deformation rate.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 512–517, May–June, 1982.     

Properties of multicomponent polymers systems

In order to design materials having different mechanical properties combined with a permittivity of 2.2–2.6 and tg =2·10^–4-4.10^–4 at 20±1° C and 10⁶ Hz, it is possible to use compositions consisting of polystyrene, polyisobutylene and polyethylene. Projections of the three-dimensional triangular composition-property diagrams are presented for selecting the composition corresponding to spcified properties (hardness at 20 and 50° C and breaking stress and elongation).Central Scientific-Research Institute of Communications, Moscow. Moscow Krupskaya Regional Pedagogical Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 133–1135, November–December, 1968.     

The stressed state of glass disks during strength testing

By applying the improved theory of plates, which makes it possible to determine all components of the stress tensor, we study the stressed state of glass disks under axisymmetric bending. We obtain a closed-form solution of the corresponding boundary-value problem for systems of singularly perturbed equations. It is shown that the size of the zone of pure bending-the domain with uniformly stretched surface-differs from those known in the literature. The results obtained make it possible to determine the optimal geometric parameters of the punch-glass disk-support system in strength testing of the glass by the method of axisymmetric bending.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 73–79.     

Deformation of a rib joined to a fiberglass casing when exposed to local stresses

The deformation of a rib joined to an orthotropic casing and exposed to a normal stress is investigated. The problem is solved by means of a semi-moment-free casing theory, taking into account shear deformation. Equations for bending moments and radial deflections in the rib are developed. Experimental verification of the functions obtained is in good agreement with the calculated results. The experiments were carried out on a casing with R/r=60.Translated from Mekhanika Polimerov, No. 5, pp. 854–860, September–October, 1972.     

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hookes law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 731–752, November–December, 2004.     

Internal instability of laminated composites with a metal matrix

Conclusions The constructed characteristic determinant for a three-dimensional nonaxisymmetric problem in the theory of stability of laminated composites with a metal matrix coincides with the characteristic determinant for the axisymmetric problem with the corresponding substitution of the wave parameters. The solution of the characteristic equation for a real material shows that loss of stability according to the different forms and for different values of deformation and the oscillation parameter as a function of the concentration of filler is possible.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1051–1056, November–December, 1990.     

Stress analysis for a rectangular thick plate of a composite material with a spatially locally curved structure under forced vibration

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially locally curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of the parameters of local curving on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three-dimensional FEM modelling.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 779–790, November–December, 2004.     

Stability of a transversely isotropic spherical shell coupled with an elastic foundation

The stability "in the small" of a flat spherical shell with elastic reinforcement is investigated. It is assumed that the shell is made of material (glass-reinforced plastic) with low shear resistance [7, 8], which determines the choice of calculation procedure: generalized applied shell theories of the Timoshenko and Ambartsumyan types [1, 3]. The results obtained are compared with the corresponding results of the Kirchhoff-Love theory.L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 129–131, January–February, 1970.     

The coupled thermoelasticity problem for thin plates

On the basis of the operator solution of the three-dimensional coupled thermoelasticity problem for a plate we state the main relations of a sharpened version of the theory of thermally stressed thin plates. These relations make it possible to determine all the components of the stress tensor and the corresponding temperature fields.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 1–6.     

Theory of spherofibrous composites. 1. Input relationships: Hypotheses and models

A theory is proposed for fibrous composites with a matrix reinforced with spherical hollow and solid inclusions based on an internal stress field and structural models. The problem solutions are obtained for the fiber-averaged matrix level. The matrix properties are determined assuming a regular distribution of the matrix inclusions. The problem of accounting for the scatter of inclusion properties on the effective composite parameters is examined.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia (October, 1995).A. A. Blagonravov Mechanical Engineering Institute. Russian Academy of Sciences. Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 32, No. 3, pp. 291–305. May–June, 1996.     

Photoelastic investigation of stress concentrations in model glass-reinforced plastics under static load

The title investigation was carried out using a double-beam polarimeter for studying the stress patterns in "sandwich" models of glass-reinforced plastics. The same experimental technique can be adapted to a wide range of reinforced materials. A comparison of the experimental results with calculations based on the theory for quasi-homogeneous anisotropic plates shows that it is not always permissible to assume that the laminated material has a quasi-homogeneous structure.Translated from Mekhanika Polimerov, No. 4, pp. 711–718, July–August, 1969.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.     

Photoelastic determination of polymer creep stresses and strains

A physical basis provided for the relation between the measured optical quantities and the unknown mechanical quantities in the photoelastic determination of stresses and strains. Questions relating to the "freezing in" of strains are considered in connection with creep tests on models made of optically sensitive polymeric materials. The determination of the stress concentration at the edge of an opening in a strip tested in creep is considered as an example of the application of the theory.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 739–746, July–August, 1970.     

Solution of a problem of nonlinear potential theory by the fixed domain method

A solution is derived for a one-dimensional boundary-value problem of nonlinear potential theory with one free end with a supplementary boundary condition. The problem is solved by a variant of the fixed domain method. In each fixed domain, the problem is reduced to the corresponding nonlinear difference problem, which is solved by Newton's method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 51–56, 1986.     

Resolvents of the theory of orthotropic shells with low shear stiffness in force-moments and stress functions

The problem of the theory of orthotropic layered shells with low shear stiffness is reformulated in force-moments and stress functions on the basis of new strain continuity relations constructed with allowance for transverse shears.Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, L'vov. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 950–952, September–October, 1970.