Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

Yield condition for circular cylindrical shells made of a fiber-reinforced composite

A yield condition is obtained for circular cylindrical shells made of a definite class of fiber-reinforced composite material whose components possess plastic properties. It is shown that, in the plane of generalized stresses — the axial bending moment and the circumferential force (when the axial force is absent) — the yield curve consists of two linear and four curvilinear sections. By approximating the curvilinear sections, we get a piecewise linear yield condition described by a hexagon in the plane indicated. The nonlinear equations and the corresponding piecewise linear equations of the yield condition for particular cases are given in the form of tables. In solving specific boundary-value problems, we consider a circular cylindrical shell simply supported at its ends and loaded with a uniform internal pressure, for which the load-carrying capacity is determined in relation to the mechanical properties of composite components and some characteristic geometrical parameters. The results of numerical calculations are represented in the form of graphs. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 655–666, September–October, 2006.     

Families of Okamoto–Painlevé pairs and Painlevé equations

In [as reported by Saito et al. (J. Algebraic Geom. 11:311–362, 2002)], generalized Okamoto–Painlevé pairs are introduced as a generalization of Okamoto’s space of initial conditions of Painlevé equations (cf. [Okamoto (Jpn. J. Math. 5:1–79, 1979)]) and we established a way to derive differential equations from generalized rational Okamoto–Painlevé pairs through deformation theory of nonsingular pairs. In this article, we apply the method to concrete families of generalized rational Okamoto–Painlevé pairs with given affine coordinate systems and for all eight types of such Okamoto–Painlvé pairs we write down Painlevé equations in the coordinate systems explicitly. Moreover, except for a few cases, Hamitonians associated to these Painlevé equations are also given in all coordinate charts. Mathematics Subject Classification (2000) 34M55, 32G05, 14J26     

Effect of viscosity of a thermoplastic prepreg and matrix upon winding of rings

The problem of compression of a unidirectional layer and shear of a polymer interlayer during winding of rings is considered. The equations determining the dependence of the layer thickness and stresses on the parameters entering into the power flow law for a prepreg and polymer matrix and on the basic parameters of the winding process—the initial tension of the prepreg, its placement rate, and the radius of a mandrel—are derived. The ring thickness measurements obtained at various temperatures and initial tension forces of plies confirm the adequacy of the model offered. It is found that the viscous properties of the prepreg and matrix upon winding affect the relative change in the layer thickness to a greater extent than the stresses in these layers. With increase in temperature and tension force upon winding, the effect of viscous deformations of the prepreg and matrix increases. A decrease in viscosity and an increase in the tension force of the tape lead to a higher strength of the ring in tension and interlaminar shear; however, the growing percolation of the polymer melt leads to a greater inhomogeneity of the structure of the composite in the ring and to a lower reinforcing effect of the factors mentioned. Presented at the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 419–428, 2000.     

Estimation of limit strains in disk-type flywheels made of a compliant elastomeric matrix composite undergoing radial creep

Fiber reinforced elastomeric matrix composites (EMCs) offer several potential advantages for construction of rotors for flywheel energy storage systems. One potential advantage, for safety considerations, is the existence of maximum stresses near the outside radius of thick circumferentially wound EMC disks, which could lead to a desirable self-arresting failure mode at ultimate speeds. Certain unidirectionally reinforced EMCs, however, have been noted to creep readily under the influence of stress transverse to the fibers. In this paper, stress redistribution in a spinning thick disk made of a circumferentially filament wound EMC material on a small rigid hub has been analyzed with the assumption of total radial stress relaxation due to radial creep. It is shown that, following complete relaxation, the circumferential strains and stresses are maximized at the outside radius of the disk. Importantly, the radial tensile strains are three times greater than the circumferential strains at any given radius. Therefore, a unidirectional EMC material system that can safely endure transverse tensile creep strains of at least three times the elastic longitudinal strain capacity of the same material is likely to maintain the theoretically safe failure mode despite complete radial stress relaxation. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 87–94, January–February, 2000.     

Isomonodromic deformations of Heun and Painlevé equations

Continuing the study of the relationship between the Heun and the Painlevé classes of equations reported in two previous papers, we formulate and prove the main theorem expressing this relationship. We give a Hamiltonian interpretation of the isomonodromic deformation condition and propose an alternative classification of the Painlevé equations, which includes ten equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 395–406, June, 2000.     

Dependence of thermooxidative aging parameters on composite properties

An analytical relationship between the thermooxidation rate constants and mechanical properties of composite materials under isothermal and dynamic conditions is obtained. With the example of epoxy-based composites, it is shown that the kinetic parameters of thermooxidation can be used to predict the internal stresses and breakdown voltage of coatings. The calculated drop in the impact toughness exceeds its experimental value by 30%, while the calculated relative breaking elongation is 1.5–2 times greater than the experimental one. A considerable decrease in these indices is observed at a loss of 0.1–1 wt.% of volatile products of thermooxidation. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 237–248, January–February, 2000.     

Renormalization and homogenization of fractional diffusion equations with random data

 This paper presents a renormalization and homogenization theory for fractional-in-space or in-time diffusion equations with singular random initial conditions. The spectral representations for the solutions of these equations are provided. Gaussian and non-Gaussian limiting distributions of the renormalized solutions of these equations are then described in terms of multiple stochastic integral representations. Received: 30 May 2000 / Revised version: 9 November 2001 / Published online: 10 September 2002 Mathematics Subject Classification (2000): Primary 62M40, 62M15; Secondary 60H05, 60G60 Key words or phrases: Fractional diffusion equation – Scaling laws – Renormalised solution – Long-range dependence – Non-Gaussian scenario – Mittag-Leffler function – Stable distributions – Bessel potential – Riesz potential     

A limit surface formulation for plastically compressible polymers

The tightening of industrial safety standards for structures generates a need for refined computational methods, which, among other things, must be able to describe the yield surface and the deformation behaviour of non-reinforced thermoplastics. To describe the plastic behaviour of materials, a potential formulation is suggested. This formulation contains a number of known potentials as special cases. The parameters of the model, which are obtained from test data, are restricted by the convexity condition for the potential. The new model allows one to take into account effects of the second order, for instance, the unequal behaviour under tension and compression, the plastic compressibility, and the Poynting-Swift effect. For each particular choice of the parameters, the Poisson ratio in tension is computed. If the restrictions imposed on the Poisson ratio do not hold, the application of a non-associated flow rule is necessary. A simple non-associated flow rule with different values of Poisson ratio intension and compression is proposed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 367–384, May–June, 2007.     

Stability problem of a circular sandwich ring under uniform external pressure

The solution of the stability problem of a circular sandwich ring under uniform external pressure is given in a refined statement. The need to determinate the precritical stresses in load-bearing layers in the refined statement with regard to the transverse compression of the core is established, which is the basis for the detection of the mixed flexural buckling forms (BFs) with more than two half-waves along the circumferential coordinate (n>2). It is found that sandwich structures with a determining parameter of transverse compression corresponding to the limit of transition from the mixed BFs to synphasic ones are the most efficient from the weight viewpoint. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 317–328, May–June, 2000.     

Spatial braiding of cylindrical articles

The effective deformative characteristics of spatially reinforced composites made by spatial braiding along the generatrices of a one-sheet hyperboloid are analyzed. The geometrical relationships determining the structure of a unit cell of a braided composite are derived. The effective thermoelastic characteristics are calculated by the method of orientational averaging. The dependences of the bending and torsional stiffnesses of thick-walled cylindrical rods — made by the method suggested and by winding — on the braiding/winding angle are compared. The numerical estimations are given for rods made of carbon (CFRP) and aramid (AFRP) epoxy plastics. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompzitnykh Materialov, Vol. 36, No. 3, pp. 341–354, May–June, 2000.     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

Asymptotically recurrent solutions ofβ-differential equations

In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000.     

Refined Orthotropic Plate Motion Equations for Acoustoelasticity Problem Statement

The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk.     

Further PDE methods for weak KAM theory

We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a Hamilton–Jacobi type equation for a potential u and a coupled transport equation for a measure σ. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159–177, 2003) and demonstrate “approximate integrability” of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions–Lasry theory (Lasry and Lions in Japan J Math 2:229–260, 2007) of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory. Supported in part by NSF Grant DMS-0500452.     

Concentrators of deformations and fracture of plastic bodies

An approach to the investigation of shape discontinuity regions as strain concentrators is proposed. The near-concentrator strain fields are determined on the basis of the theory of ideal rigid-plastic body; under the condition of plane deformation, their determination is reduced to integration of ordinary differential equations. The deformation as a function of the location of the plastic region and its shape evolution in the process of plastic flow is studied. The plastic flow is demonstrated to be not unique (within the framework of solution completeness). A deformation criterion for the choice of the preferred plastic flow is suggested. The fracture of a V-notched strip is considered. On the basis of the solutions obtained, an approach to the investigation of the fracture processes for more complicated models is formulated.     

Discrete Approximations and Optimization of Evolution Inclusions

The paper studies discrete approximations of nonconvex valued evolution inclusions with the right-hand side satisfying Kamke condition which is more general than the Lipschitz one and more convenient than the variant of the one-sided Lipschitz condition used in Donchev et al. (J Differ Equ 243:301–328, 2007). We extend an interesting previous result of Mordukhovich to a large class of evolution systems appearing in the theory of parabolic partial differential equations. Examples of control systems governed by partial differential equations are provided.     

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804–828, June, 2006.     

On methods for the solution of integral equations of the theory of plasticity based on the concept of slip

We have proposed a modification of the methods for solving the system of integral equations [M. Ya. Leonov and N. Yu. Shvaiko, “Complex plane deformation,” Dokl. Akad. Nauk SSSR, 159, No. 2, 1007–1010 (1964); N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005)]. These equations describe the development of plane plastic deformation for simple and complex loading processes. A characteristic feature of these equations lies in the presence of unknown functions both under the integral sign and in the integration limits. We have written analytical solutions for monotone deformation and in a small neighborhood of an angular point of the loading trajectory. For arbitrary piecewise smooth trajectories, we have reduced this problem to the Cauchy problem for a first-order differential equation with known initial conditions. The results obtained simplify significantly the construction of constitutive equations [(s)\dot]_mn ~ [(e)\dot]_mn {\dot{\sigma }_{mn}} \sim {\dot{\varepsilon }_{mn}} and their use in applied problems of the theory of plasticity as compared with [N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005); N. Yu. Shvaiko, Complex Loading and Problems of Stability [in Russian], Izd. DGU, Dnepropetrovsk (1989)].     

Modeling the nonlinear deformation of composite laminates based on plasticity theory

The plasticity theory has been successfully used for describing the nonlinear deformation of laminated composite materials under a monotonically increasing loading. Generally, several tests are needed to determine the parameters of the plastic potential for a laminate. We explore an alternative approach and obtain the plastic potential by using theoretical considerations based on a laminate analysis. The model is shown to provide an accurate prediction for the response of a cross-ply glass/epoxy laminate under uniaxial tensile loading at different angles to the material orthotropy axes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 309–318, May–June, 2007.