Reducing three-dimensional elasticity problems to two-dimensional problems by approximating stresses and displacements by legendre polynomials

Shell equations are constructed in orthogonal curvilinear coordinates using approximations of stresses and displacements by Legendre polynomials. The order of the constructed system of differential equations is independent of whether stresses and displacements or their combination are specified on the shell surfaces, which provides the correct formulation of the surface conditions in terms of both displacements and stresses. This allows the system of differential equations of laminated shells to be constructed using matching conditions for displacements and stresses on the contact surfaces. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 179–190, May–June, 2007.     

Equations of an Elastic Anisotropic Layer

Differential equations of an elastic orthotropic layer are constructed on the basis of expansion of the solutions of the elasticity theory in terms of the Legendre polynomials. The order of the system of differential equations is independent of the form of the boundary conditions on the layer surfaces, which allows a correct formulation of conditions on contact surfaces.     

Nonlinear laws of dry friction in contact problems of linear theory of elasticity

A problem of elastic body deformation with conditions of dry friction imposed at the boundary is considered. Various friction laws are studied, including linear oneparameter and nonlinear twoparameter laws. A general view of a nonlinear function with two constants is suggested, which determines the friction force as a function of normal pressure. The problem of elastic plate compression by rough infinite plates for a variety of friction conditions on the contact surfaces is solved. The plate equations are employed, which make it possible to specify arbitrary conditions on the front faces without reducing the order of differential equations. The unknown boundary of ideal contact zones and sliding zones is determined. Solutions obtained by using various friction conditions on the contact surfaces are compared.     

Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

Summary This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.     

Invariant Sets of Systems of Stochastic Differential Equations with Jumps

We introduce the notion of invariant surfaces for inhomogeneous stochastic differential equations with jumps. The results obtained enable one to determine invariant surfaces for stochastic differential equations of the type indicated. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 234–240, April–June, 2005.     

曲面物理和力学：最佳基本微分算子对

曲面物理和力学中有两个独立的基本微分算子(即"基本微分算子对"). 本文综述如下主题：在所有的基本微分算子对中,经典梯度▽(···) 和形状梯度▽ (···) 的配对[[▽,▽]] 是最佳的. 具体内容包括：(1)基本微分算子对的形式并不唯一;(2) 内积的可交换性确立了[[▽,▽]] 优于其他基本微分算子对的"最佳" 地位;(3) 基于[[▽,▽]] 可以最佳地构造曲面物理和力学的高阶标量微分算子,因而[[▽,▽]] 是构造曲面物理和力学微分方程的最佳"基本砖块";(4) [[▽,▽]] 在软物质曲面物理和力学中普遍存在.     

Analytical solution of the contact problem for a system of bodies under collective wear

The contact problem is considered for a system of bodies subject to wear on a common base. The deformation properties of the bodies and the base are described by the Winkler model. The problem is reduced to a system of ordinary differential equations and an integral equation of hereditary type with difference kernel. The solution of the problem is constructed with the use of the Laplace transform. The asymptotics of the solution at large times is studied. The continuity conditions for the contact of each of the bodies with the base are derived.     

Implicit difference methods for parabolic functional differential problems of the Neumann type

Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates for approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given. Published in Neliniini Kolyvannya, Vol. 11, No. 3, pp. 329–347, July–September, 2008.     

Periodic solution to a kind of singularly perturbed differential equations of parabolic type in chemical kinetics

This paper deals with the problem on the periodic solution for the singularly perturbeddifferential equations of parabolic type originating from chemical kinetics in stratifiedmedia,A uniformly valid asymptotic solution is constructed and the related asymptoticestimate is given.     

Nonlinear equations of elastic deformation of plates

A method for constructing nonlinear equations of elastic deformation of plates with boundary conditions for stresses and displacements at the face surfaces in an arbitrary coordinate system is proposed. The initial three–dimensional problem of the nonlinear theory of elasticity is reduced to a one–parameter sequence of two–dimensional problems by approximating the unknown functions by truncated series in Legendre polynomials. The same unknowns are approximated by different truncated series. In each approximation, a linearized system of equations whose differential order does not depend on the boundary conditions at the face surfaces which can be formulated in terms of stresses or displacements is obtained.     

Propagation of weak waves in elastic-plastic solids

Wave fronts admitting discontinuities only in the derivatives of the dependent variables are by convention called ‘weak’ waves. For the special case of discontinuous first-order derivatives, the fronts are customarily called ‘acceleration’ waves. If the governing equations are quasi-linear, then the weak waves are necessarily characteristic surfaces. Sometimes, these surfaces are also referred to as ‘singular surfaces’ of order r ? 1, where r stands for the order of the first discontinuous derivatives. This latter approach is adopted in this paper and applied to governing equations which form a set of first-order quasi-linear hyperbolic equations. When these equations are written in terms of singular surface coordinates, simplification occurs upon differencing equations written on the front and rear sides of the surface: a set of algebraic (‘connection’) equations is generated for the discontinuities in the normal derivatives of the dependent variables across the surface. When a similar operation is performed on the governing equations written for second-order derivatives, a set of first-order differential (‘transport’) equations is generated.     

Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.     

General analytical solution to bending of composite laminated beams with delaminations

Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delami- nated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the de- lamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamina- tion locations and sizes axe shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.     

叶片型线对离心油泵性能的影响

以65Y60型离心油泵为研究对象，研究了液体不同粘度下叶片型线对泵性能的影响。首先，利用本课题组独立开发的准三元叶片设计程序以反问题方式设计了两种型线。这两种型线叶片前半部分与原一元叶轮叶片的型线不同、后半部分与一元叶轮相同。然后在不同粘度下将三个叶轮进行性能对比实验。研究表明，叶片进口附近的型线的改变对泵性能有较大影响；适当增大叶轮后盖板流面上的叶片流体动力负荷有助于提高离心油泵输送粘油时的水力性能。     

多体系统动力学动态最优化设计与灵敏度分析

基于多体系统的动态最优化设计过程包括传统的多体系统仿真分析、系统设计灵敏度分析、 系统最优化设计等过程, 针对多体系统运动学、用二阶常微分方程和微分代数方程描述 的动力学,基于含设计参数的通用数学模型及通用的积分型目标函数,采用高效的系统灵 敏度分析伴随变量方法及易于实施的惩罚函数最优设计方法,建立了多体系统最优设计数学 模型和算法. 通过双摆系统、曲柄-滑块系统、弹簧/阻尼器-滑块系统3个算例对上述 算法的有效性进行了验证.     

Multiple scale perturbation for second-order non-linear differential equations

Asymptotic solutions of a class of second-order non-linear differential equations with variable coefficients are studied. For large values of the parameter, the differential equations are of the singular-perturbation type and approximations are constructed by the generalized method of multiple scales.     

Stress–Strain Analysis of Elastic Bodies on the Basis of a Heterogeneous Mathematical Model

A heterogeneous mathematical model is formulated. It permits us to use simultaneously the equations of the theories of elasticity and Timoshenko-type shells to describe different fragments of a structure. This model can be written as a closed system of differential equations of different dimensions with boundary conditions on the domain boundary and conjugate conditions on the surfaces where fragments are mated. A variational problem is formulated. The existence and uniqueness of the solution are analyzed. Numerical results demonstrate the efficiency of the approach     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Construction of piecewise-analytical gas flows joined through shock waves near the axis or center of symmetry

Analytical solutions of a quasilinear system of equations with partial derivatives are constructed in the case where the initial data for different functions are specified on different surfaces and the resultant problem has singularities of the form u/x and w/x. Conditions for existence and uniqueness of a solution in the form of formal power series for the problem posed and sufficient conditions for convergence of the series are indicated. A generalization of the problem considered is given. Results of the study are used to solve the problem of the focussing of a compression wave generated by a piston moving smoothly in a quiescent gas: a solution for t=0, including determination of the piston trajectory, and a solution for t<0, including unequivocal construction of the front of a reflected shock wave, are uniquely constructed from the distribution of gas-dynamic quantities for t>0. The solution of this problem is a generalization to the case of two independent variable self-similar Sedov's solutions. Ural State Academy of Communications, Ekaterinburg 620034. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 25–38, September–October, 1998.     

A multi-dimensional time-marching method of characteristics for viscous and heat-conducting flows

Summary A numerical scheme is presented which employs the characteristic surfaces in space-time for solving Navier-Stokes equations for compressible fluid flow. We consider the general case of a three-dimensional flow, a simplification of which yields the equations of the two-dimensional case. Emphasis is put on the method itself. We apply it to simulate a laminar hypersonic flow around a circular cylinder of a five-components gas mixture of nitrogen and oxygen with thermally perfect constituents and at chemical nonequilibrium. First, the partial differential equations are transformed into a standard form with directional derivatives, enabling to attain the compatibility conditions, including the viscosity terms. These conditions are discretized by approximating their integrals along the corresponding characteristic surfaces. The result is an explicit time-marching numerical scheme. Using a grid fitted between the shock and the cylinder, and starting from roughly estimated initial conditions, a steady solution is searched. A comparison is made with the solution obtained under the assumption of a perfect gas. Received 6 April 1999; accepted for publication 13 May 1999