大学物理《力学》中的自由度分析

对《力学》中的物体自由度进行多方面分析,以深化教学、提高学生正 确分析物理问题的能力.使用实际教学分析的研究方法,在《力学》范围内讨论自由度与坐标、 自由与约束的关系并得以下结论: (1) 同一物体的自由度随其所在的``空间'不同而不同, 不因坐标系的选取不同而 异, 在同类参考系中不因参考系的动静而有别;(2)自由度遵循叠加原理. 讨论了质点系的总自由度及相关计算问题,并指出研究《力学》中自由度的意义.     

Method of molecular dynamics in mechanics of deformable solids

The basic principles of the method of molecular dynamics are analyzed. Symplectic difference schemes for the numerical solution of molecular dynamics equations are considered. Stability is studied, and the errors in the energy conservation law, which are induced by using these schemes, are estimated. Equations of mechanics of continuous media are derived by means of averaging over the volume of an atomic system. Expressions for the stress tensor are obtained by using the virial principle and the method of averaging over the volume. The principles of construction of EAM and MEAM potentials of atomic interaction in crystals are analyzed. Two problems of fracture of copper-molybdenum composites are solved by the method of molecular dynamics.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.     

圆柱壳的非线性自由振动分析

本文分析了各向同性封闭圆柱壳的非线性自由振动。文中采用经典的非线性弹性力学方法推导了圆柱壳的大振幅运动方程,这些方程的静态形式与冯·卡门的板理论方程具有同样的精度。文中讨论了四种基本振动模态,并且还以数学公式的形式给出了一般的最终结果,一些例子以曲线给出结果,并进行了比较。结果还表明线性振动可以作为非线性振动的一种特例。     

Analysis of Isolation Effectiveness of Shear Waves by a Row of Hollow Pipe Piles in Saturated Soils

Wave numbers about the three types of waves in saturated soils are firstly given in this paper. The lengths of the pipe piles are much larger than their diameters, so the isolation problem about SV waves by discontinuous barriers composed of a row of pipe piles can be simplified as a two-dimensional scattering problem. The expansion method of wave functions is adopted, the stresses and displacements at the boundaries between the pipe piles and adjacent soils are considered as continuous and the inner sides of the pipe piles are free, and then the theoretical solutions are obtained about this two-dimensional scattering problem. Normalized displacements are introduced, which are the displacements behind the barriers caused by both the incident and scattered waves to those only by the incident SV waves, contours and curves of the normalized displacements are drawn, and the influences of wall thickness of pipe piles, modulus ratio of pipe piles to soils, spacing distance between the pipe piles and pipe pile numbers on the isolation effectiveness are analyzed.     

Approximate symmetries of creeping flow equations of a second grade fluid

Creeping flow equations of a second grade fluid are considered. Two current approximate symmetry methods and a modified new one are applied to the equations of motion. Approximate symmetries obtained by different methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

Comparison of methods for calculating the shock hugoniot of mixtures

Various methods used to determine the shock Hugoniot of condensed phase multi-component mixtures are reviewed and compared to available experimental data. The assumptions inherent in the different models are presented in this overview and their implications are discussed. The comparisons of the various models demonstrate that the predicted shock Hugoniots are in good agreement with published data despite the simplifying assumptions that are associated with the models. Averaging models are shown to be among the simplest methods to implement and result in the closest agreement with experimental data.     

Direct modeling of flow of FENE fluids

Direct simulations of macromolecular fluids are carried out for flows between parallel plates and in expanding and contracting channels. The macromolecules are modeled as FENE dumbbells with soft disks or Lennard-Jones dumbbell-dumbbell interactions. The results are presented in terms of profiles and contour plots of velocity, pressure, temperature, density, and flow fields. In addition the data for potential energy, shear stress, and the normal components of the stress tensor are collected. In general, an excellent agreement is found between the simulated profiles and the well-known flow structures, such as flow separation and formation of viscous eddies, indicating that micro-hydrodynamics is a viable tool in linking macroscopic phenomena with the underlying physical mechanisms. The simulations are performed in the Newtonian regime, for medium-size systems comprising up to 3888 dumbbells. This number is sufficiently large to control boundary and particle number effects. The flow is induced by gravity. The traditional stochastic (thermal) and periodic boundary conditions are employed. Also, diffusive boundary conditions, which could include a stagnant fluid layer and repulsive potential walls, are developed. The scaling problems, which are related to the application of a large external force in a microscopic system (of the size of the order 100 Å), result in extreme pressure and temperature gradients. In addition, the viscosity and thermal conductivity coefficients obtained from velocity and temperature profiles of the channel flow are presented. These results are confirmed independently from modeling of Couette flow by the SLLOD equations of motion and from the Evans algorithm for thermal conductivity.     

Mesh effects on the computation of non-linear gravity waves of arbitrary discharge

Numerical computations of non-linear gravity waves are presented and the effects of mesh variations on the results are discussed. Waves are regarded as two-parameter families (λ,A)_Q of arbitrary discharge Q, and computations are carried out using a new Kantorovich algorithm. Mesh effects are to a large extent dependent on the particular wave region under consideration. Three such regions are identified and typical examples are computed and discussed.     

Numerical study of self-induced wing rock of a low-aspect-ratio delta wing in the presence of external disturbances

Self-induced wing rock of a delta wing, in particular, in the presence of external disturbances are studied by means of numerical simulations of a separated flow of an ideal incompressible fluid around a delta wing. The results obtained are compared with experimental data. The vortex nature and the mechanism of self-induced oscillations are studied. Regions of synchronization of the aerodynamic self-oscillatory system in the presence of external disturbances are identified. Methods of suppression of self-induced wing rock are proposed.     

Evolution of the Theory of Laminated Plates and Shells

The theory of laminated plates and shells is considered. Three-dimensional models of layered systems and methods of reducing them to two-dimensional models are elucidated. An analysis is made of how two-dimensional models are constructed by the method of hypotheses. Two basic approaches to the construction are presented: one leads to the discrete structural theory of laminated systems and the other to continuous structural theory. Attention is drawn to transverse shear and reduction in nonclassical theories of high approximation. The finite-element implementation of the theory is described. Examples of analysis by various models are given. Results of an applicability analysis of various theories and experimental data supporting them are presented. New research areas for the theory of laminated structures are pointed out.     

Basic properties of natural vibrations of an extended segment of a pipeline

Natural transverse vibrations of an extended segment of a pipeline containing a uniformly moving fluid are considered. The mechanical model under study takes into account the inertial forces of the pipe and environment and the moment of Coriolis and centrifugal forces arising because of the medium motion. It is proved that all natural frequencies of the pipeline rigidly clamped at both ends are real (and hence no flutter can arise in this model). For the first three modes, the dependence of the eigenvalues on the fluid flow velocity (varying from zero to the buckling velocity) are constructed, and their properties depending on the inertia parameter are studied. Families of vibration mode shapes of the pipeline are constructed and investigated.     

On a Model of Heterogenous Deformation of Elastic Bodies by the Mechanism of Multiple Appearance of New Phase Layers

A model describing a possible scenario of martensite type phase transformations is examined. A new phase is supposed to nucleate in the form of plane parallel layers. As the boundary condition, average strains are imposed. Then, the governing parameters of the two-phase structure are the concentration of new phase layers, their orientation and also the orientation of anisotropy axes. The parameters depend on the average strains and are determined by the requirement to minimize the average Helmholtz free-energy function. Once a general procedure has been discussed, average strain–stress diagrams are constructed for two cases. In the first case, for the simplicity sake, both phases are assumed to be isotropic. In the second case anisotropy is produced by a non-spherical phase transformation strain tensor. For both cases phase transition zones (PTZs) are constructed. The PTZ is formed in the space of strains by those which can exist on equilibrium interfaces. Loading and unloading paths, corresponding to uniaxial stretching and plane stretching/compression, are examined and related with the PTZ. Effects of internal stresses induced by the nucleation of new phase areas and the anisotropy of new phase are discussed.     

Identification of regions of fastest mixing in a system of point vortices

This paper describes a numerical method for efficiently identifying the regions of fastest mixing of a passive dye in a flow due to a system of point vortices. Results obtained from computations are presented for systems of three and four point vortices, both in the unbounded domain and inside a circular cylinder. The flow is two‐dimensional and the fluid is incompressible. The regions where mixing is possible are found by studying the largest Lagrangian Lyapunov exponent distribution with respect to various initial positions of tracer particles. The regions of fastest mixing are then identified from the Lyapunov exponent distribution at small times. The results of the method are verified by quantifying the mixing by using a traditional box counting technique. The technique is then applied to several different initial configurations of vortices and some interesting results are obtained. Some numerical findings about the nature of the exponents computed are also discussed. Copyright © 2002 John Wiley Sons, Ltd.     

Experimental validation of the constitutive equations of thermoelastoplasticity including the third invariant of the stress deviator

The equations describing thermoelastoplastic deformation along nonstraight paths and taking into account the third invariant of the stress deviator are experimentally validated. The equations contain two scalar functions that are specified in base tests on tubular specimens. The test data are tabulated. The values of the scalar functions for strains, temperature, and stress mode are found by using nonlinear interpolation of the data and the temperature similarity of the functions. The stresses in elements of the body are calculated from given strains by the method of successive approximations     

Integrable types of nonlinear ordinary differential equations of higher-orders

In this paper some integrable types of more general nonlinear ordinary differentialequations of higher-orders are proposed in virtue of Leibnitz formula,and formulas ofhigher-order derivatives of the composite functions as well as substitution vartables.Theexpressions for the general integrations of some of the equations are presented.The resultsobtained are the generalization of those in the references.Finally,some examples are alsogiven.     

Adequacy of a nonlinear theory of viscoelasticity

Some features of the behavior of viscoelastic materials whose existence leads to the choice of nonlinear constitutive relations are discussed. A classification of such constitutive relations is given and a number of requirements imposed by practice on their adequacy are formulated. A nonlinear theory of viscoelasticity is proposed; this theory offers the advantages over the theory in which stresses are expressed in terms of strains by integral operators of increasing multiplicity. By a one-dimensional example, it is shown that the constitutive operator relations are reciprocal.     

Measurement of the Viscoelastic Properties of Bitumen Using Instrumented Spherical Indentation

Indentation testing as a tool for determination of the viscoelastic mechanical properties of bitumen is examined in some detail using theoretical, numerical as well as experimental methods. In particular Brinell indentation is analysed and simple but rigorous formulae for a complete characterization of linear viscoelastic materials are presented. Numerical methods (finite element methods) are used in order to verify and substantiate these relations for an experimental situation. Indentation experiments are then performed on bitumen and special efforts are made in order to avoid size effects, i. e. anomalous results due to the fact that the indented specimens are too small and as a result, far field boundary conditions will influence the interpretation of the experimental output. The mechanical properties determined experimentally by indentation are compared with corresponding results from standard mechanical tests, and the results are encouraging considering the fact that non-linear effects are also influencing the outcome of the experiments.     

Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

Uniqueness and stability of steady states of continuous chemical reactors

We examine the uniqueness and stability of the solutions to the problem of steady-state operation of a continuous chemical reactor in which longitudinal diffusion and heat conduction are taken into account. We investigate an adiabatic reactor in which the concentration and temperature distributions are similar (the thermal diffusivity and diffusion coeffecient are equal) and an isothermic reactor. These two cases are considered together because the mathematical formulations of the problem are equivalent.The question of the existence and number of steady states was analyzed in [1, 2], where references were made to earlier investigations. The results obtained in [1, 2] are now extended. The stability of the steady states is investigated by the small-perturbation method.