Three-dimensional stability of a rectangular plate under uniaxial tension

The paper studies the three-dimensional stability of an isotropic, linear elastic, rectangular plate under a uniform tensile load applied to its sides. The concept of free strains is used to reduce the three-dimensional problem to a two-dimensional one. It is solved using the three-dimensional linearized theory of stability. An approximate solution of the buckling problem is obtained by the finite-difference method. Numerical results are presented __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 116–123, July 2006.     

Three-dimensional theory of stability of a carbon nanotube in a matrix

The three-dimensional theory of stability of a carbon nanotube (CNT) in a polymer matrix is presented. The results are obtained on the basis of the three-dimensional linearized theory of stability of deformable bodies. Flexural and helical (torsional) buckling modes are considered. It is proved that the helical (torsional) buckling modes occur in a single CNT (the interaction of neighboring CNTs is neglected) and do not occur in nanocomposites (the interaction of neighboring CNTs is taken into account) __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 23–37, January 2006.     

On two models in the three-dimensional theory of stability of composites

A comparative analysis is made of the infinite-fiber and finite-fiber models in the three-dimensional theory of stability of composites. The results analyzed have been obtained using the three-dimensional linearized theory of stability of deformable bodies. A historical sketch is given of the theory of stability for and approaches used in the mechanics of laminated and fibrous composite materials Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 3–31, August 2008.     

Establishing the Fundamentals of the Theory of Stability of Mine Workings

The fundamentals of the theory of stability of mine workings are analyzed. The theory is based on the linearized three-dimensional theory of stability of deformable bodies. Some results on horizontal and vertical mine workings and on underground closed cavities are analyzed     

Nonlinear dynamical stability analysis of the circular three-dimensional frame

The three-dimensional frame is simplified into flat plate by the method of quasiplate. The nonlinear relationships between the surface strain and the midst plane displacement are established. According to the thin plate nonlinear dynamical theory, the nonlinear dynamical equations of three-dimensional frame in the orthogonal coordinates system are obtained. Then the equations are translated into the axial symmetry nonlinear dynamical equations in the polar coordinates system. Some dimensionless quantities different from the plate of uniform thickness are introduced under the boundary conditions of fixed edges, then these fundamental equations are simplified with these dimensionless quantities. A cubic nonlinear vibration equation is obtained with the method of Galerkin. The stability and bifurcation of the circular three-dimensional frame are studied under the condition of without outer motivation. The contingent chaotic vibration of the three-dimensional frame is studied with the method of Melnikov. Some phase figures of contingent chaotic vibration are plotted with digital artificial method.     

Studies on nonlinear stability of three-dimensional H-type disturbance

The three-dimensional H-type nonlinear evolution process for the problem of boundary layer stability is studied by using a newly developed method called parabolic stability equations (PSE). The key initial conditions for sub-harmonic disturbances are obtained by means of the secondaryinstability theory. The initial solutions of two-dimensional harmonic waves are expressed in Landau expansions. The numerical techniques developed in this paper, including the higher order spectrum method and the more effective algebraic mapping for dealing with the problem of an infinite region, increase the numerical accuracy and the rate of convergence greatly. With the predictor-corrector approach in the marching procedure, the normalization, which is very important for PSE method, is satisfied and the stability of the numerical calculation can be assured. The effects of different pressure gradients, including the favorable and adverse pressure gradients of the basic flow, on the “H-type“ evolution are studied in detail. The results of the three-dimensional nonlinear “H-type“ evolution are given accurately and show good agreement with the data of the experiment and the results of the DNS from the curves of the amplitude variation, disturbance velocity profile and the evolution of velocity.     

Influence of Geometrical Parameters on the Critical Load in Three-Dimensional Stability Problems for Rectangular Plates and Beams

Two- and three-dimensional stability problems for a rectangular plate under uniaxial compression are solved by the mesh method. It is shown that the difference between the critical loads obtained in the three- and two-dimensional problems is not greater than 5% for arbitrary fixation conditions and arbitrary length and thickness ratio of the plate     

Effect of a Standing Acoustic Wave on the Development of Large-Scale Convection in a Horizontal Layer

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.     

PSEUDOSPECTRAL－FINITE DIFFERENCE METHOD FOR THREE－DIMENSIONAL VORTICITY EQUATION WITH UNILATERALLY PERIODIC BOUNDARY CONDITION

ＰＳＥＵＤＯＳＰＥＣＴＲＡＬ－ＦＩＮＩＴＥＤＩＦＦＥＲＥＮＣＥＭＥＴＨＯＤＦＯＲＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＶＯＲＴＩＣＩＴＹＥＱＵＡＴＩＯＮＷＩＴＨＵＮＩＬＡＴＥＲＡＬＬＹＰＥＲＩＯＤＩＣＢＯＵＮＤＡＲＹＣＯＮＤＩＴＩＯＮＸｉｏｎｇＹｕｅ－ｓ...     

Developing a Compressive Failure Theory for Nanocomposites

The paper addresses a compressive-failure theory for polymer-matrix nanocomposites in the case where failure onset is due to microbuckling. Two approaches based on the three-dimensional linearized theory of stability of deformable bodies are applied to laminated and fibrous nanocomposites. According to the first approach (continuum compressive-failure theory), nanocomposites are modeled by a homogeneous anisotropic medium with effective constants, including microstructural parameters. The second approach uses the piecewise-homogeneous model, three-dimensional relations for fibers (CNT) and matrix, and continuity conditions at the fiber-matrix interface. The compressive-failure theory is used to solve specific problems for laminated and fibrous nanocomposites. Some approximate failure theories based on the one- and two-dimensional applied theories of stability of rods, plates, and shells are analyzed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 3–37, March 2005.     

A new high-order accuracy explict difference scheme for solving three-dimensional parabolic equations

In this paper, a new three-level explicit difference scheme with high-orderaccuracy is proposed for solving three-dimensional parabolic equations. The stabilitycondition is r＝△t/△x² ＝△t/△y²＝△t/△z²≤1/4, and the trumcation error is O(△t²+△x⁴).     

Stable Equilibrium of the Multilayer Lining of a Vertical Mine Opening in Elastoplastic Rock Mass

Based on the three-dimensional linearized theory of stability, the stability problem is solved for the multilayer lining of a vertical circular mine opening in rock mass under inelastic conditions. The effect of the geometric and physicomechanical parameters of the lining and rock on the critical contact pressures is evaluated     

Linear stability of three-dimensional boundary layers

Stability of compressible three-dimensional boundary layers on a swept wing model is studied within the framework of the linear theory. The analysis based on the approximation of local self-similarity of the mean flow was performed within the Falkner-Skan-Cooke solution extended to compressible flows. The calculated characteristics of stability for a subsonic boundary layer are found to agree well with the measured results. In the case of a supersonic boundary layer, the results calculated for a Mach number M = 2 are also in good agreement with the measured spanwise scales of nonstationary vortices of the secondary flow. The calculated growth rates of disturbances, however, are substantially different from the measured values. This difference can be attributed to a high initial amplitude of disturbances generated in the experiment, which does not allow the linear stability theory to be applied. The evolution of natural disturbances with moderate amplitudes is fairly well predicted by the theory. The effect of compressibility on crossflow instability modes is demonstrated to be insignificant. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 3–14, March–April, 2008.     

钢管混凝土轴心受压构件极限承载力的有限元分析

采用三维虚拟层合单元法，对钢管混凝土和纯混凝土轴心受压构件在轴向荷载作用下产生的稳定承载能力和材料承载能力问题进行了分析比较，同时进一步分析了前者失稳破坏和材料破坏的界限值，模拟了前者第一阶失稳模态，结论与试验结果符合良好。     

Estimating the error of the beam approximation in the plane stability problem for a rectangular plate with a central crack

The plane stability problem for a rectangular, linearly elastic, isotropic plate with a central crack is solved. The dependence of the critical load on the crack length is studied using exact (the three-dimensional linearized theory of stability of elastic bodies) and approximate (beam approximation) approaches. The results produced by the beam approach are evaluated.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 117–126, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Near-surface instability of composites weakly reinforced with short fibers

The paper is concerned with loss of stability near the free flat surface of a weakly short-fiber-reinforced composite material under compression along the fibers. Emphasis is on the effect of the mechanical and geometrical parameters of the composite and the distance between fibers and to the free surface on the critical strains. The piecewise-homogeneous medium model and the rigorous three-dimensional linearized theory of stability of deformable bodies are used __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 28–35, June 2008.     

非定常激波三维双楔面反射的数值研究

采用基于MUSCL-Hancock插值的有限体积方法, 在非结构自适应网格上求解三维Euler方程, 研究了非定常激波在三维垂直双楔面上的反射现象. 研究结果表明, 由于三维效应的影响, 通过二维非定常激波反射理论预测的三维马赫反射区域范围存在着一定的局限, 在该区域范围内出现了一种非典型的三维突起结构, 即第二类三维马赫干. 另外, 对于不同的激波马赫数和楔面倾角组合, 非定常激波在三维双楔面上会形成具有四波结构的三维马赫反射或具有三波结构的三维规则反射.      

Allowing for the plastic properties of rock in stability problems for a stratified rock mass

The paper addresses the important issue of allowing for the inelastic properties of rock in stability problems for a stratified rock mass. A three-dimensional nonlinear problem statement is used. An exact method to study the surface instability of a regularly layered semi-infinite medium is developed. New numerical results are obtained __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 68–81, December 2007.     

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional (3D) boundary layers to predict the transition with the e ^N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation (3D-LPSE) approach, a 3D extension of the two-dimensional LPSE (2D-LPSE), is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation (DNS) rather than a 2D-eigenvalue problem (EVP) procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.     

Numerical investigation of the first transition in the three-dimensional problem of convective flow in a porous medium

The branching off of steady-state regimes from mechanical equilibrium is studied for the problem of filtration convection in a parallelepiped. The conditions for the geometric parameters under which stable continuous families of steady-state regimes develop are found. The stability of equilibria of the family with respect to three-dimensional perturbations is analyzed in a numerical experiment using a finite-difference method.