Design of closed thin-walled composite shapes for bending

A study is made of the possibility of dividing the complex bending of a structure into elementary components, and an examination is made of closed wing-type monocoque structures with a rigid contour. The structures studied are asymmetric with respect to their geometric and stiffness characteristics. They are subjected to bending without torsion and are referred to a cylindrical coordinate system (Z, S). The longitudinal displacements are determined on the basis of the method of conjugate displacements by integrating the Cauchy equation and circulation equation, with the displacement along the contour (due to its stiffness) being represented in the form of a series containing terms describing the bending of the structure. The shear strains are similarly represented. The resolvent equations are obtained by using the principle of the minimum potential strain energy of the contour.Deceased.Moscow State Aviation Institute (Technical University). Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 82–89, January–February, 1997.     

Optimal rigidity of a thin-walled orthotropic cylinder in combined tension and torsion

The problem of optimizing the reinforcement of an orthotropic cylinder in combined torsion and tension is considered in relation to the condition of minimum displacement. A numerical example is given. It is shown that the stiffness depends on the ratio of the normal and shear stresses, the elastic moduli, and the parameters.Moscow Aviation Technological Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1121–1123, November–December, 1973.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

The Harnack Estimate for the Yamabe Flow on CR Manifolds of Dimension 3

We deform the contact form by the amount of the Tanaka–Webstercurvature on a closed spherical CR three-manifold.We show that if a contact form evolves with free torsion from initial datawith positive Tanaka–Webster curvature, then a certainHarnack inequality for the Tanaka–Webster curvature holds.     

Torsion of multiconnected bodies with cracks

A general expression is obtained for a multiconnected anisotropic (isotropic) body with rectilinear cracks, for the complex torsion potential that exactly satisfies conditions on the cracks and contains unknown functions determined from the boundary conditions on closed contours. A solution is given for the torsion problem of an elliptical rod with a crack. Results are presented of investigation on the clarification of the influence of geometric and elastic characteristics of the rod on the magnitude of the stress intensity coefficient near an outer edge.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 34–38, 1987.     

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.     

Displacement structure approach to q-adic Chebyshev–Vandermonde-like matrices

In this paper, a new class of so-called q-adic Chebyshev–Vandermonde-like matrices over an arbitrary non-algebraically closed field is introduced. This class generalizes both the ordinary Chebyshev–Vandermonde-like matrices over the complex field studied earlier by Kailath and Olshevsky [T. Kailath, V. Olshevsky, Displacement structure approach to Chebyshev–Vandermonde and related matrices, Integral Equations Operator Theory 22 (1995) 65–92], and the classical q-adic Vandermonde-like matrices with respect to power basis by Yang and Hu [Z.H. Yang, Y.J. Hu, Displacement structure and fast inversion formulas for q-adic Vandermonde-like matrices, J. Comput. Appl. Math. 176 (2005) 1–14]. Three kinds of displacement structures and consequently, three kinds of fast inversion formulas are presented for this class of matrices by using displacement structure theory method, which generalize the corresponding results for Chebyshev–Vandermonde-like and q-adic Vandermonde-like matrices.     

Approximate torsion calculations for anisotropic members with a cross section in the form of an elongated profile

An approximate method is proposed for solving the problem of the free torsion of a prismatic anisotropic member with a narrow cross section. The material is assumed to be homogeneous with a special form of anisotropy (nonorthotropic). An arbitrary elongated section and special cases are considered. Experimental data are presented for DSP-B wood-resin laminate and the limits of applicability of the method are established.Mekhanika Polimerov, Vol. 3, No. 2, pp. 312–317, 1967     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

Continuous finite element methods for Reissner-Mindlin plate problem

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.     

Calculation of the long-term strength of thin-walled polymeric tubes under biaxial loading

The problem on calculating the failure time of thin-walled polymer tubes in creep under internal pressure, internal pressure with tension, pure torsion, and torsion with tension is solved. The solution is constructed based on the concept of equivalent stresses. A mixed long-term failure criterion taking into account the sings of principal stresses is used as the equivalent stress. The calculation results are in a satisfactory agreement with experimental data.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 811–826, November–December, 2004.     

Finite-difference method of solving the bending problem for glass-reinforced plastic plates with allowance for creep

A new finite-difference method is proposed for solving the bending problem as it applies to thin rectangular orthotropic plates of linear hereditary material. The proposed method makes it possible to obtain a closed approximate solution in explicit form for a series of finite-difference systems of equations corresponding to the problem of the bending of plates of orthotropic glass-reinforced plastic with allowance for the rheological properties of the material under various boundary conditions.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1075–1082, November–December, 1969.     

Solution of the problems of a perforated disk,a conical bar,and a flat wedge of nonlinear viscoelastic material in pure shear,torsion, and bending,respectively

Exact solutions and approximations have been obtained for the problems of a disk with an opening twisted by opposite moments uniformly distributed over the inner and outer surfaces and of a conical bar twisted by a moment applied at the vertex of the cone. An approximate solution has been found for the problem of a flat wedge bent by pressure uniformly distributed along one of its sides. The disk is made of nonlinear viscoelastic material. In [1] it was proposed that problems for such a material be solved by the method of approximations. The rheological law of the nonlinear viscoelastic material of the cone and the wedge in Laplace—Carson transforms is the relation of the theory of small elastoplastic deformations with a power law of strain hardening.Institute of Electronic Engineering, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1071–1076, November–December, 1971.     

Boundary Hemivariational Inequalities of Hyperbolic Type and Applications

In this paper we examine two classes of nonlinear hyperbolic initial boundary value problems with nonmonotone multivalued boundary conditions characterized by the Clarke subdifferential. We prove two existence results for multidimensional hemivariational inequalities: one for the inequalities with relation between reaction and velocity and the other for the expressions containing the reaction–displacement law. The existence of weak solutions is established by using a surjectivity result for pseudomonotone operators and a priori estimates. We present also an example of dynamic viscoelastic contact problem in mechanics which illustrate the applicability of our results.Mathematics Subject Classifications (2000). 34G20, 35A15, 35L85, 35L70, 74H20     

Mixed linear complementary formulation of lateral contacts among a system of parallel beams

A novel linear complementary formulation is developed in this paper to deal with multiple lateral contacts at a number of cross sections for a structure consisting of many parallel beams and having rigid body displacements. The proposed method can be used to handle a challenging situation where the number of potential contact pairs exceeds the total number of available lateral displacements at a cross section. The traditional displacement-based contact algorithms cannot be employed to solve this type of contact problem. In this paper, a mixed force–displacement scheme is developed. This scheme removes the restriction on the number of contact pairs at a cross section and is suitable for handling contacts among many parallel beams. It is proven that a unique solution to the linear complementary equations obtained using the mixed force–displacement scheme exists and can be obtained using the Lemke algorithm. Numerical results for an example application show that the scheme is indeed convergent and accurate.     

Additional Curvature of Spinning-Particle Trajectories Proportional to the Trajectory Torsion

In the geometric optics approximation, we predict the effect of an additional curvature of the trajectories of particles (an analogue of the inverse Magnus optical effect). The effect is determined by the polarization (chirality) and torsion of particle trajectories. We find that the considered effect is linked to the Berry phase. The effect is a consequence of the conservation law for the angular momentum of the particles. We show that the effect must result in ultracold neutrons deviating from the mirror reflection law.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 555–563, September, 2005.     

Progressive increase in the displacements of a glass-reinforced plastic helicopter blade under the influence of its own weight

The question of the progressive increase in the displacements of a glass-reinforced plastic helicopter blade under its own weight is examined. Integral operators, expressed in terms of fractional-exponential functions, are used for solving the flexural creep problem. The creep curve parameters for simple deformation (tension and torsion) are found for ÉF-32-301 glass-reinforced textolite.K. É. Tsiolkovskii Moscow Aviation Engineering Institute. Translated from Mekhanika Polimerov, No. 5, pp. 943–946, September–October, 1975.     

Evaluation of discrete relaxation-time spectrum in the torsion of a round polymer rod

The torsion of a round polymer rod is examined on the basis of the generalized Maxwell equation, taking into account two terms of the relaxation-time spectrum. In finding the relaxation constants corresponding to the younger term, we used experimental torsion diagrams obtained with thick cylindrical specimens having a significantly inhomogeneous stress pattern. It was shown that use of the generalized nonlinear Maxwell equation with two terms makes it possible to obtain satisfactory agreement between the theoretical and experimental torsion diagrams and between the theoretical and experimental creep curves over a considerable time interval. It was established that calculation of the maximum tangential stresses by use of experimental torsion diagrams yields values virtually identical to those calculated by the refined method. The polymer constants given, determined on the basis of experimental torsion diagrams, can be used to find a first approximation of the two polymer relaxation constants constituting the youngest component of the spectrum. The theoretical and experimental curves were compared.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 682–688, July–August, 1972.     

Optimal Control and Filtering of the Reproduction Law of a Branching Process

A finite horizon control problem for the reproduction law of a branching process is studied. Some examples with complete information are tackled via the Hamilton–Jacobi–Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and the observations and the observation process has unbounded intensity, a Girsanov-type change of probability measure can be set and the filtering equation for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, also for the Kushner–Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitting process and the unnormalized conditional distribution. By the strong uniqueness for the Zakai equation, equivalence between the partially observable control problem and the separated one is proved.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.