Exact analytical and numerical solutions of stability problems for a straight composite bar subjected to axial compression and torsion

Based on linearized equations of the theory of elastic stability of straight composite bars with a low shear rigidity, which are constructed using the consistent geometrically nonlinear equations of elasticity theory for small deformations and arbitrary displacements and a kinematic model of Timoshenko type, exact analytical solutions of nonclassical stability problems are obtained for a bar subjected to axial compression and torsion for various modes of end fixation. It is shown that the problem of direct determination of the critical parameter of the compressive load at a given torque parameter leads to transcendental characteristic equations that are solvable only if bar ends have cylindrical hinges. At the same time, we succeeded in obtaining solutions to these equations in terms of wave formation parameters of the bar; these parameters, in turn, enabled us to find the parameter of the critical load at any boundary conditions. Also, an algorithm for numerical solution of the problems stated is proposed, which is based on reducing the problems to systems of integroalgebraic equations with Volterra-type operators and on solving these equations by the method of mechanical quadratures (finite sums). It is demonstrated that such numerical solutions exist only for certain ranges of parameters of the bar and of the parameter of torque. In the general case, they can not be obtained by the numerical method used. It is also shown that the well-known solutions of the stability problem for a bar subjected to torsion or to compression with torsion are in correct. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 167–200, March–April, 2009.     

三维薄区域上MHD方程的渐进分析

在强解全局存在的基础上, 得到了三维薄区域上MHD方程的解(u,h)对任意时间t≥ 0的渐进分析. 当区域厚度ε小时, MHD方程的强解(u,h)可形式展开为u=ū(t)+up+U, h=h(t)+hp+H, t ≥0或u=ū(t)+us+U＊,h=h(t)+hs+H＊,t ≥0,其中(u,h) 是2D-3C MHD 方程的解, (u_p,h_p) 是P-S MHD 方程的解, u,h 分别是两个Stokes方程的解, (U,H),(U＊,H＊)是仅依赖于初始数据的两个函数对. (U,H)和(U＊,H＊)关于区域厚度\varepsilon是小的, (u_p,h_p)和u,h更小;证明了上述形式展开的收敛性.     

High-order accurate equations describing vibrations of thin bars

A method for deriving one-dimensional wave propagation equations in thin inhomogeneous anisotropic bars based on the mathematical homogenization theory for periodic media is used to obtain equations governing the longitudinal and transverse vibrations of a homogeneous circular bar. The equations are derived up to O(ε⁸) terms and take into account variable body forces and surface loads. Here, ε is the ratio of the bar’s typical thickness to the typical wavelength.     

On the Explicit Solution of Certain First Order Systems of Partial Differential Equations

We consider two classes of systems of partial differential equations of first order. One consists of generalized Stokes-Beltrami equations $ Aw_z = w^*_z $ , $ \lambda Bw_{\bar z} -w^*_{\bar z} $ with square matrices A and B and a scalar factor u . The other may be written in matrix notation as $ v_{\bar z} = c{\bar v} $ where c denotes a square matrix. This system is known as a Pascali system. Both systems are in close connections to certain systems of second order for which the solutions can be represented using particular differential operators. On the basis of these relations we give the solutions of the first order systems explicitly.     

The equilibrium boundary element method in boundary-value problems of elasticity theory

An equilibrium boundary element method is proposed for solving boundary-value problems in the theory of elasticity, thermo-elasticity, the dynamical theory of elasticity, bar torsion calculations, and the bending of a plate. The idea is to use simultaneously the method of constructing bundles of functions which exactly satisfy the equilibrium equations, the boundary variational equations of mechanics, and the methods of discrete finite-element approximation. The variational method of constructing the resolving boundary equations ensures that the linear system is symmetric and easily coupled to the finite-element method. Since volume integrals are eliminated the dimensions of the problem are reduced by one, but, unlike the boundary element method, there is no need to know the fundamental solutions. The solution of some bar torsion and plate bending problems confirms the high numerical efficiency of the method.     

The dynamic behaviour of piezoelectric laminated bars

A theory of laminated electroelastic bars with layers arranged symmetrically about the middle plane of the bar is constructed. Particular attention is given to the influence of the electrical conditions on the faces of the piezoelectric layers on the equations of the theory of bars. Formulae are obtained which, after solving the problem of a laminated bar, enable one to transfer from one-dimensional required quantities to three-dimensional required quantities. As an example, the vibrations of a three-layer electroelastic bar are considered, the displacements, stresses and electrical quantities are calculated, and the dependence of the electromechanical coupling coefficient on the frequency of the vibrations and the thicknesses of the elastic and piezoelectric layers is investigated.     

Modeling of Transverse Shears of Piecewise Homogeneous Composite Bars Using an Iterative Process with Account of Tangential Loads. 2. Resolving Equations and Results

Using the variational method, a system of resolving equations and boundary conditions is deduced on the basis of the nonclassical model for a composite bar. The order of the equations depends on the step of the iterative process. For normal and tangential loads, these equations are realized in trigonometric series. The results are presented as a sum of the classical solution and an additional part, which is determined by the influence of the shear deplanation of cross sections and is taken into account by higher iterations. The problems for bars with various cross section variants are considered.     

Some iterative algorithms for positive definite solution to nonlinear matrix equations

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.     

Euler implicit/explicit iterative scheme for the stationary Navier–Stokes equations

In this paper, a new uniqueness assumption (A2) of the solution for the stationary Navier–Stokes equations is presented. Under assumption (A2), the exponential stability of the solution $(\bar{u},\bar{p})$ for the stationary Navier–Stokes equations is proven. Moreover, the Euler implicit/explicit scheme based on the mixed finite element is applied to solve the stationary Navier–Stokes equations. Finally, the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme.     

Equivalence and symmetries of first order differential equations

In this article, the equivalence and symmetries of underdetermined differential equations and differential equations with deviations of the first order are considered with respect to the pseudogroup of transformations . That means, the transformed unknown function is obtained by means of the change of the independent variable and subsequent multiplication by a nonvanishing factor. Instead of the common direct calculations, we use some more advanced tools from differential geometry; however, the exposition is self-contained and only the most fundamental properties of differential forms are employed. We refer to analogous achievements in literature. In particular, the generalized higher symmetry problem involving a finite number of invariants of the kind is compared to similar results obtained by means of auxiliary functional equations.     

THE REGULARITY OF SOLUTIONS FOR NONLINEAR DEGENERATE ELLIPTIC BOUNDARY VALUE PROBLEM

This paper is devoted lo the study of regularity of solutions for Dirichlet probelm ofnonlinear degenerate elliptic equations in two dimensional case. A sufficient condition fortheir solutions in C~(3+α)(Ω) to be certainly in C~∞(Ω) is given.     

The problem of the bending of thin orthotropic bars in threedimensional formulation

For the three-dimensional problem of the theory of elasticity of an orthotropic body we obtain solutions satisfying the equilibrium equations inside a bar and boundary conditions on its planar faces. These solutions make it possible to satisfy a given set of boundary conditions on lateral surface of the bar. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 27–34.     

关于一类非线性发展方程整体解的存在性问题

本文研究一类模拟非线性弹性杆的纵振动的非线性发展方程的初边值问题，证明了其整体解的存在性、唯一性、光滑性及在一定条件下，整体解的不存在性。     

Eine verallgemeinerte Darboux-Gleichung I

The paper is concerned with the elliptic equation $$\begin{gathered} w_{z\bar z} + \left[ {\frac{{n (n + 1)}}{{(z - \bar z)^2 }} - \frac{{m (m + 1)}}{{(z + \bar z)^2 }} + \frac{{q (q + 1)}}{{(1 + z\bar z)^2 }} - \frac{{p (p + 1)}}{{(1 - z\bar z)^2 }}} \right]w = 0, \hfill \\ n, m, p, q \in \mathbb{N}_0 . \hfill \\ \end{gathered} $$ General representation theorems for, the solutions are derived by differential operators if three parameters are different from zero or two parameters are equal. Some applications are given to pseudo-analytic functions and generalized Tricomi equations.     

Eine verallgemeinerte Darboux-Gleichung II

The paper is concerned with the elliptic equation $$\begin{gathered} w_{z\bar z} + \left[ {\frac{{n(n + 1)}}{{(z - \bar z)^2 }} - \frac{{m(m + 1)}}{{(z + \bar z)^2 }} + \frac{{q(q + 1)}}{{(1 + z\bar z)^2 }} - \frac{{p(p + 1)}}{{(1 - z\bar z)^2 }}} \right]w = 0, \hfill \\ n,m,p,q \in \mathbb{N}_0 . \hfill \\ \end{gathered}$$ General representation theorems for the solutions are derived by differential operators if three parameters are different from zero or two parameters are equal. Some applications are given to pseudo-analytic functions and generalized Tricomi equations.     

Generalized three-point difference schemes of high-order accuracy for systems of second-order nonlinear ordinary differential equations

For systems of second-order nonlinear ordinary differential equations with the Dirichlet boundary conditions, we develop generalized three-point difference schemes of high-order accuracy on a nonuniform grid. The construction of the suggested schemes requires solving four auxiliary Cauchy problems (two problems for systems of nonlinear ordinary differential equations and two problems for matrix linear ordinary differential equations) on the intervals [x _j−1, x _j ] (forward) and [x _j , x _j+1] (backward) at each grid point; this is done at each step by any single-step method of accuracy order $ \bar m $ \bar m = 2[(m+1)/2]. (Here m is a given positive integer, and [·] is the integer part of a number.) We prove that such three-point difference schemes have the accuracy order $ \bar m $ \bar m for the approximation to both the solution u of the boundary value problem and the flux K(x)d u/dx at the grid points.     

Equivalence and symmetries of second-order differential equations

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations = φ(x), ȳ = ȳ() = L(x) + y(x), = () = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed. This research has been conducted at the Department of Mathematics as part of the research project CEZ: Progressive reliable and durable structures, MSM 0021630519.     

The $$\bar \partial $$ -problem with support conditions on some weakly pseudoconvex domainswith support conditions on some weakly pseudoconvex domains

We consider a domain Ω with Lipschitz boundary, which is relatively compact in ann-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the\(\bar \partial \)-equation with exact support in ω admits a solution in bidegrees (p, q), 1≤q≤n?1. Moreover, the range of\(\bar \partial \) acting on smooth (p, n?1)-forms with support in\(\bar \Omega \) is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flatCR manifolds of arbitrary codimension.     

Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the\bar \partial and\bar \partial _b equations

An intrinsic definition of Lipschitz classes in terms of vector fields on man-ifolds is provided and it is shown that it is locally equivalent with a more classical definition. A finer result is then proved for strongly pseudo-convex CR manifolds and applications of the theorems are given to smoothness of holomorphic functions and estimates for the \(\bar \partial \) and \(\bar \partial _b \) . equations.