Determination of the Inhomogeneous Preliminary Stress–Strain State in a Piezoelectric Disk

The problem of steady radial vibrations of a thin electroelastic hollow disk in the presence of a preliminary inhomogeneous plane stress–strain state is solved. Vibrations are induced by applying a potential difference across the electrodes placed on the end surfaces of the disk. Equations of the vibrations and boundary conditions are formulated. The preliminary stress state corresponding to the solution of the Lam´e problem was investigated. The direct problem of determining the displacement function is solved numerically by the shooting method. The inverse problem of determining a pre-stress parameter from the change in the natural frequency of the disk is formulated and solved. The accuracy of determining the prestressed state for initial data specified with an error is analyzed.     

The equilibrium state of a structure subject to frictional contact

A structure in frictional contact subject to static loads has not, in general, a unique static equilibrium state. This is because the state, displacements and contact forces, depend on the load history of the structure.In cases where the exact load history is not known it would be of interest to find a state that is in some sense likely and define this as the equilibrium state. In this paper, it is assumed that the state with the smallest potential energy is the most likely one. The implication of this definition of likely state is analysed and shows that the resulting problem basically can be seen as a generalization of the frictionless contact problem to structures where no frictionless state is possible, i.e. structures where non-zero friction forces are necessary to satisfy force equilibrium.The results of several numerical experiments are given. The structures in the experiments are trusses and structures modelled by the finite element method. Both a sequential quadratic programming method and an enumeration method are used to solve the likely-state problem.     

非线性最小二乘跟踪的对偶变量变分方法

最小二乘跟踪方法是近几年提出的一种计算动力系统跟踪轨迹的方法.基于最小二乘跟踪的灵敏度分析算法可以有效避免传统的非线性系统灵敏度分析方法中的病态初值问题,因此其在混沌系统灵敏度分析方面有着重要的应用.针对非线性的最小二乘跟踪问题,首先将其重新描述为带有约束的非线性最优控制问题,引入协态变量并将系统的哈密顿函数表示为关于状态变量和协态变量的函数.然后将目标函数的积分时间离散化,根据对偶变量变分原理,以离散区间两端的状态变量作为独立变量,用Lagrange插值多项式近似离散区间内的状态变量和协态变量,进而将非线性最优控制问题转化为求解非线性方程组问题.这种算法无需对原问题做线性化处理,避免了复杂的线性化过程以及可能因此造成的误差,同时为求解非线性最小二乘跟踪问题提供了新的思路.根据最小二乘方法可以得到两条设计参数有微小变化的状态轨迹,基于这两条状态轨迹可进一步计算出系统关于设计参数的灵敏度,范德波振子作为数值算例验证了该方法在求解最小二乘跟踪问题以及计算非线性系统灵敏度时的有效性.      

Precise integration method for LQG optimal measurement feedback control problem

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Optimal control of structures governed by variational and hemiva-riational inequalities and applications

The optimal control problem for broad classes of structures is studied, including those structures having as state relations variational equalities, variational inequalities and hemivariational inequalities. The optimal control problem consists in the minimization of a functional (performance index) having the state relation, enlarged by the control actions, as side condition. Certain new results are given of the optimal control of structures governed by variational and hemivariational inequalities.Some propositions are proved on the existence and the approximation of the solution of the static optimal control problem of structures having a variational inequality as state relation. Then a regularization procedure is proposed for the treatment of corresponding dynamic problem, as well as for the case of hemivariational inequalities. The theory is illustrated by applications concerning convex elastoplasticity and convex and nonconvex unilateral contact problems.     

Surface Waves for a Compressible Viscous Fluid

In this paper, we are concerned with free boundary problem for compressible viscous isotropic Newtonian fluid. Our problem is to find the three-dimensional domain occupied by the fluid which is bounded below by the fixed bottom and above by the free surface together with the density, the velocity vector field and the absolute temperature of the fluid satisfying the system of Navier-Stokes equations and the initial-boundary conditions. The Navier-Stokes equations consist of the conservations of mass, momentum under the gravitational field in a downward direction and energy. The effect of the surface tension on the free surface is taken into account. The purpose of this paper is to establish two existence theorems to the problem mentioned above: the first concerns with the temporary local solvability in anisotropic Sobolev-Slobodetskiĭ spaces and the second the global solvability near the equilibrium rest state. Here the equilibrium rest state (heat conductive state) means that the temperature distribution is a linear function with respect to a vertical direction and the density is determined by an ordinary differential equation which involves equation of state. For the proof, we rely on the methods due to Solonnikov in the case of incompressible fluid with some modifications, since our problem is hyperbolic-parabolic coupled system. Dedicated to Professors Takaaki Nishida and Masayasu Mimura on their sixtieth birthdays     

Iterative algorithm for solving the retrospective problem of thermal convection in a viscous fluid

A retrospective problem consisting in the reconstruction of an apriori unknown initial state of a highly viscous, incompressible fluid from its known final state is studied. The fluid dynamics model in the Boussinesq approximation is described by the equations of Stokes, incompressibility, and thermal balance with the corresponding initial and boundary conditions. The problem is solved in the inverse direction of time using a new, specially developed iterative method which makes it possible to reduce the original unstable problem to a set of stable problems. The algorithm based on this method is realized on a parallel computer using the OpenFOAM package for engineering calculations. The computational efficiency of the algorithm is analyzed.     

Symplectic solution for three dimensional couple stress problem and its variational principle

A new state vector is presented for symplectic solution to three dimensional couple stress problem. Without relying on the analogy relationship, the dual PDEs of couple stress problem are derived by a new state vector. The duality solution methodology in a new form is thus extended to three dimensional couple stress. A new symplectic orthonormality relationship is proved. The symplectic solution to couple stress theory based a new state vector is more accordant with the custom of classical elasticity and is more convenient to process boundary conditions. A Hamilton mixed energy variational principle is derived by the integral method.     

Deformation and fracture of crystalline materials under complex loading-program conditions

In predicting the deformation and conditions of fracture of materials under complex loading program conditions one must consider the entire duration of the deformation process.The problem becomes complicated in cases of a simultaneous operation of various mechanisms of deformation and fracture, e. g., when plastic deformation is superposed (once or repeatedly) on creep.A promising phenomenological approach to this problem may be based on concepts of the mechanical equation of state of materials. A hypothesis of the existence of the equation of state depending on a finite number of structural parameters was formulated by Kröner [1] for the case of the three-dimensional law of plasticity and by Rabotnov [2] for the case of creep and fracture under uniaxial stress state conditions.This article is concerned with the application of the hypothesis of the mechanical equation of state to the problem of deformation and fracture of materials (in the uniaxial case) under complex loading program conditions.     

Mie-Grneisen状态方程可压缩多流体流动的PPM方法

采用流体体积分数的混合型多流体数值模型，将piecewise parabolic method (PPM)方法应用于可压缩多流体流动的数值模拟，拓展了以前提出的模型和数值方法，使它能够处理一般的Mie-Grneisen状态方程。采用双波近似和两层迭代算法求解一般状态方程的Riemann问题；并根据多流体接触界面无振荡原则设计高精度计算格式，对典型的纯界面平移问题可以从理论上证明本算法在接触间断附近压力和速度没有振荡，而且数值模拟结果表明界面数值耗散也被控制在2~3个网格之内。模拟了多种复杂的可压缩多流体流动，算例结果表明本文方法可以有效地处理接触间断、激波等物理问题，且具有耗散小精度高的特点。     

索杆体系的机构运动及其与弹性变形的混合问题

基于有限单元平衡方程推导了索杆体系机构运动的通解、特解及其路径跟踪的迭代求解列式，以及机构运动与弹性变形混合问题的迭代求解列式。从机构运动和弹性变形问题的比较角度出发，解释了机构运动中的若干基本定义，对索杆体系的机构运动和弹性变形混合问题作了分类。数值算例表明：采用本文方法可以非常有效地跟踪求解索杆体系机构运动、弹性变形及其混合问题的全过程路径，得到机构运动的静定状态和混合问题的平衡状态。     

Stress Analysis of Corrugated Hollow Cylinders

The stress problem for corrugated hollow cylinders is solved in a three-dimensional formulation. Use is made of end conditions that make the problem two-dimensional. By applying discrete Fourier series, the problem is made one-dimensional and then is solved by the stable numerical method of discrete orthogonalization. The stress state of the cylinders is analyzed depending on their thickness and corrugation characteristics     

Control strategy of optimal deployment for spacecraft solar array system with initial state uncertainty

A control strategy combining feedforward control and feedback control is presented for the optimal deployment of a spacecraft solar array system with the initial state uncertainty. A dynamic equation of the spacecraft solar array system is established under the assumption that the initial linear momentum and angular momentum of the system are zero. In the design of feedforward control, the dissipation energy of each revolute joint is selected as the performance index of the system. A Legendre pseudospectral method (LPM) is used to transform the optimal control problem into a nonlinear programming problem. Then, a sequential quadratic programming algorithm is used to solve the nonlinear programming problem and offline generate the optimal reference trajectory of the system. In the design of feedback control, the dynamic equation is linearized along the reference trajectory in the presence of initial state errors. A trajectory tracking problem is converted to a two-point boundary value problem based on Pontryagin’s minimum principle. The LPM is used to discretize the two-point boundary value problem and transform it into a set of linear algebraic equations which can be easily calculated. Then, the closed-loop state feedback control law is designed based on the resulting optimal feedback control and achieves good performance in real time. Numerical simulations demonstrate the feasibility and effectiveness of the proposed control strategy.     

Tensile instability of nonlinear spherical membrane with large deformation

The problem on instability of nonlinear spherical membrane with large axisymmetric tensile deformations is investigated by using the bifurcation theory.It is proved that all singular points of the nonlinear boundary value problem must be simple limit points.The effect of loading and material parameters on the equilibrium state and its stability is discussed.     

Rayleigh-Benard problem for an anomalous fluid

The stability of the state of rest of a heated infinite horizontal layer of a viscous heat-conducting fluid (the Rayleigh-Benard problem) is considered. The equation of state for the fluid takes into account the nonmonotonic temperature and pressure dependence of water density. Instability of the mechanical equilibrium with respect to small monotonic perturbations is studied. The effect of the problem parameters on the Rayleigh numbers and their corresponding critical motions is investigated numerically using linear theory. Numerical investigation of the spectral problem is based on the Godunov-Abramov orthogonalization method. The calculation results are compared with the well-known results for the limiting case where the density is considered a quadratic function of temperature and does not depend on pressure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 27–38, March–April, 2007.     

On One Approach to the Solution of Stress Problems for Noncircular Hollow Cylinders

An approach is proposed to solve three-dimensional stress problems for noncircular hollow cylinders. The end conditions are such that the problem can be reduced to a two-dimensional problem. This problem is reduced to a one-dimensional problem by introducing additional functions into the resolvable system of equations. These functions are determined using discrete Fourier series. The one-dimensional problem is solved by a stable numerical method. As an example, the stress state of cylinders with an elliptic cross section is analyzed depending on their thickness and degree of ellipticity.     

The viscoelastic deformation of an orthotropic body (half-plane) by a rigid punch

The method of operator continued fractions is used to solve the problem on the stress state in a viscoelastic orthotropic half-plane loaded by a punch at the instantt=0. The pressure in the half-plane is determined on the basis of the Volterra principle and by solving the corresponding elastic problem. The influence of the rheological parameters on the stress state of the half-plane is shown by an example for a composite material. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 81–91, July, 2000.     

Monitoring of parallel galleries in the region of horizontal motion of lithospheric plates

The problem of estimating the stress-strain state of underground openings of solid minerals, which form long parallel underground galleries, is considered. Several papers studied the local causes of accidents that occur in underground galleries during mineral extraction due to variations in the stress-strain state of the medium. The influence of formation of new galleries, which violate the balance of vertical stresses in the walls between the galleries, is also studied. At the same time, the problem on the influence of horizontal displacements of lithospheric plates on the stress-strain state of parallel galleries is little studied. These displacements affect the horizontal components of the contact stress vector arising between the upper and lower layers and the walls between the galleries. In this paper, the theory of estimating the stress-strain state in underground openings with arbitrarily many parallel galleries of various dimensions is developed under the assumption that the tangential components of the stress vector exist in the contact regions between the layers and the walls. The study is based on factorization methods, block element method, and a topological approach.     

Problem of direct martensite transformation in a thick-walled cylinder made of shape memory alloy

The axisymmetric problem on the stress-strain state of a long thick-walled circular cylinder made of shape memory alloy experiencing the direct thermoelastic martensite phase transition under the action of constant internal and external pressures and a constant axial force is solved. The problem of limit loads in such a process is also considered.     

Solution of Plane Problems of the Three-Dimensional Stability of a Ribbon-Reinforced Composite

The plane problem of three-dimensional elastic stability is solved for a ribbon-reinforced composite under lateral compression if its initial state is nonuniform. The net approach is used to numerically solve the problem. The influence of the ratio of the elastic moduli of the matrix and the filler and the ribbon shape factor on the critical load of the material is studied