Symmetry Properties of the Elastic Energy of a Woven Fabric with Bending and Twisting Resistance

A woven fabric can be described as a surface made of two families of fibers: in this work we study how the geometry of the weave pattern affects the symmetry properties of the elastic energy of the surface. Four basic symmetry classes of weave patterns are possible, depending on the angle between the fibers and their material properties. The properties of the pattern determine the material symmetry group of the network, under which the elastic energy is invariant. We derive representations for the energy of a woven fabric that are invariant under the symmetry group of the network, and discuss the relation of these invariants with the curvature and twist of the fibers.      

INTEGRAL INVARIANT IN NONCONSERVATIVE SYSTEMS AND ITS APPLICATION IN MODERN PHYSICS

In this paper,the Poincaré and Poincaré-Cartan integral invariants in nonconservativesystems are established.According to the integral invariant,he non-linear oscillation ofparticles in 3-folded symmetry spiral sector cyclotron is investigated.It turns out that themethod is successful.     

Universal hierarchical symmetry for turbulence and general multi-scale fluctuation systems

Scaling is an important measure of multi-scale fluctuation systems. Turbulence as the most remarkable multi-scale system possesses scaling over a wide range of scales. She-Leveque （SL） hierarchical symmetry, since its publication in 1994, has received wide attention. A number of experimental, numerical and theoretical work have been devoted to its verification, extension, and modification. Application to the understanding of magnetohydrodynamic turbulence, motions of cosmic baryon fluids, cosmological supersonic turbulence, natural image, spiral turbulent patterns, DNA anomalous composition, human heart variability are just a few among the most successful examples. A number of modified scaling laws have been derived in the framework of the hierarchical symmetry, and the SL model parameters are found to reveal both the organizational order of the whole system and the properties of the most significant fluctuation structures. A partial set of work related to these studies are reviewed. Particular emphasis is placed on the nature of the hierarchical symmetry. It is suggested that the SL hierarchical symmetry is a new form of the self-organization principle for multi-scale fluctuation systems, and can be employed as a standard analysis tool in the general multi-scale methodology. It is further suggested that the SL hierarchical symmetry implies the existence of a turbulence ensemble. It is speculated that the search for defining the turbulence ensemble might open a new way for deriving statistical closure equations for turbulence and other multi-scale fluctuation systems.     

Optimal Bounds on Texture Coefficients

Let w be the orientation distribution function of a polycrystalline aggregate of crystallites with symmetry G _cr and with group of texture symmetry G _tex. In this paper we obtain a “recipe” on how to derive optimal bounds on the texture coefficients W _lmn associated with w. In particular, we find explicit bounds in the case in which G _tex is a group with orthorhombic symmetry and G _cr is either a group with cubic symmetry or a group with hexagonal symmetry. This revised version was published online in July 2006 with corrections to the Cover Date.     

可降解性表面活性剂体系下水平管内气液两相螺旋流实验研究

通过气液两相螺旋流实验仪器,研究具有可降解性的天然椰子油新型添加剂对于气液两相螺旋流流型影响以及流型的转变规律,并与表面活性剂十二烷基苯磺酸钠(SDBS)进行对比研究。实验工况设定为:实验介质为空气和水,含气率10%~90%,气相折算速度0.01~4.0m/s,液相折算速度0.01~4.0m/s,表面活性剂采用从植物提取的可降解性椰子油和SDBS,起旋装置为叶轮。实验观察到天然椰子油对于螺旋轴状流、螺旋团状流、螺旋弥散流转换特性的影响与SDBS的效果相类似,该三种流型发生条件相比于以往都有所提前,且存在范围被拓宽。浓度为500ppm时椰子油体系下的主要流型为螺旋弥散流,而SDBS体系下则以螺旋团状流为主。     

Dynamics of spiral waves driven by a dichotomous periodic signal

We study the effects of a dichotomous periodic force on meandering and rigidly rotating spiral waves. For a meandering state, the periodic forcing induces more modulating frequencies according to the rules of frequency-locked relations and linear combinations. It can also generate some unique closed tip orbits. On the modulating period T-axis, there exist all kinds of resonant entrainment bands. Arnold tongues exist in the period-amplitude space. The width of entrainment bands is affected by the symmetry of positive and negative parts in each signal unit. In addition, appropriate choices of T-value can be used to eliminate spiral waves. For a rigidly rotating state, the periodic forcing can induce a transition toward meandering spiral waves via generating a transitive bidirectional spiral wave. It is very interesting that, after the transition, the meandering spiral wave has the same primary rotating period as the free meandering states.     

Material Tensors and Pseudotensors of Weakly-Textured Polycrystals with Orientation Distribution Function Defined on the Orthogonal Group

A representation theorem on material tensors of weakly-textured polycrystals was recently established by Man and Huang, which delineates quantitatively the effect of crystallographic texture on the material tensor in question. Man and Huang’s theorem is based on the classical assumption in texture analysis that ODFs are defined on the rotation group SO(3), which strictly speaking makes it applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. In this paper we let ODFs be defined on the orthogonal group O(3) and extend the representation theorem of Man and Huang to cover pseudotensors and polycrystals with crystallite symmetry defined by any of the 21 improper point groups. In SO(3)-based texture analysis, as a result of the inherent limitation imposed by the restricted definition of the ODF, each improper group of crystallite symmetry in question is routinely treated as if it were its peer proper group in the same Laue class. In light of the extended representation theorem, we examine the conditions under which this ad hoc practice will still work as far as effects of texture on material tensors and pseudotensors are concerned and the circumstances under which it won’t.     

Stabilization of coordinated motion for underwater vehicles

This paper presents a coordinating and stabilizing control law for a group of underwater vehicles with unstable dynamics. The coordinating law is derived from a potential that only depends on the relative configuration of the underwater vehicles. Being coordinated,the group behaves like one mechanical system with symmetry,and we focus on stabilizing a family of coordinated motions,called relative equilibria. The stabilizing law is derived using energy shaping to stabilize the relative equilibria which involve each vehicle translating along its longest(unstable) axis without spinning,while maintaining a relative configuration within the group. The proposed control law is physically motivated and avoids the linearization or cancellation of nonlinearities.     

Integration of third order ordinary differential equations possessing two-parameter symmetry group by Lie’s method

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: (1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). (2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.     

Complete group classification and exact solutions to the extended short pulse equation

In this paper, the complete group classification is performed on the extended short pulse equation (ESPE), which including many important non-linear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary parameters of the equation. Furthermore, the symmetry reductions and exact solutions to the short pulse types of equations are investigated, and the physical significance of the solutions are considered from the transformation group point of view.     

When is a symmetric pin-jointed framework isostatic?

Maxwell’s rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework.     

Fish and Plankton Interplay Determines Both Plankton Spatio-Temporal Pattern Formation and Fish School Walks: A Theoretical Study

The fascinating variety of spatio-temporal patterns in aquatic ecosystems and the understanding of the governing mechanisms of its generation and further dynamics requires ongoing experimental and theoretical studies. After introducing a certain hybrid mathematical model, this paper makes an attempt to demonstrate that the predation of a mobile planktivorous fish school on zooplankton can initiate both plankton pattern formation and fish school walks. Nonlinear interactions in the model of a fish-zooplankton-algae trophic chain prevent a simple intuitive understanding of the system dynamics. It is shown that the fish school predation and motion can give rise to plankton spiral waves. In the course of the spiral wave formation, the amplitudes of the spatially averaged plankton density oscillations are decreasing dramatically. Fish school walks are shown to resemble a fractional Brownian motions with a Hurst exponent depending on the fish predation rate.     

The structure of symmetric attractors

We consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems. We show, for example, that the symmetry of an attractor cannot, in general, be an arbitrary subgroup of the group of symmetries. In addition, there are group-theoretic restrictions on the symmetry of connected components of a symmetric attractor. The symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time-average of a chaotic dynamical system.Our methods are topological in nature and exploit connectedness properties of the ambient space. In particular, we prove a general lemma about connected components of the complement of preimage sets and how they are permuted by the mapping.These methods do not themselves depend on equivariance. For example, we use them to prove that the presence of periodic points in the dynamics limits the number of connected components of an attractor, and, for one-dimensional mappings, to prove results on sensitive dependence and the density of periodic points.     

On the nature of the boundary layers instabilities in a flow between a rotating and a stationary discSur la nature des instabilités de couche limite dans un écoulement entre un disque fixe et un disque tournant

Both theoretical linear stability analysis and direct numerical simulation are performed to study the transition flow between a stationary and a rotating disc. This paper concerns three-dimensional spiral and annular patterns computed with a high-order (spectral) numerical method and related to Bödewadt layer instabilities. The characteristic parameters of these boundary layer patterns are compared to the theoretical results and interpreted in terms of type I and type II generic instabilities. Moreover, the absolute instability regions are also theoretically identified and the critical Reynolds numbers of the convective/absolute transition in both layers are given. To cite this article: E. Tuliska-Sznitko et al., C. R. Mecanique 330 (2002) 91–99.     

Conditions for nonlinear stability of flows of an ideal incompressible liquid

Stability of steady state flows of an ideal incompressible liquid with homogeneous density with some type of symmetry (translational, axial, rotational, or helical) is considered. Two types of sufficient conditions for nonlinear stability are obtained, which can be proven by constructing two types of functionals which have absolute minima at the given steady state solutions. Each of the functionals used is the sum of the kinetic energy and some other integral, specific to the given class of motion. The first type of stability conditions are a generalization to the case of finite perturbations and a new class of flows of the well known Rayleigh criterion [1] for centrifugal stability of rotating flows relative to perturbations with rotational symmetry. In the same sense the second type of stability conditions generalize another result, also originally proposed by Rayleigh, according to which plane-parallel flow of a liquid is stable in the absence of an inflection point in the velocity profile [1]. A nonlinear variant of the latter condition for the class of planar motions was first obtained in [2]. To systematize the results extensive use is made of the analogy between the effects of density stratification and rotation in the form of [3], The results to be presented relate to stability of a wide class of hydrodynamic flows having the required symmetry. For example, they relate to flows in tubes and channels which rotate or are at rest, and flows with concentrated annular or spiral vortices.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 70–78, May–June, 1986.     

Influence of turning radius on wheeled military vehicle induced rut formation

A rut is a depression or groove formed into the ground by the travel of wheels and tracks. Ruts can cause severe influences on soil and vegetation, and reduce vehicle mobility. In this paper, rut depth and rut width were used as the main indicators to quantify a rut. A new indicator, rut index, was proposed, combining rut depth and rut width. A Light Armored Vehicle (LAV) and a High Mobility Multi-purpose Wheeled Vehicle (HMMWV) were used for testing the influence of turning radius on rut depth, rut width and rut index. The LAV and the HMMWV were operated in spiral patterns at different speeds. Differential GPS data for the vehicles were collected every second during the spiral. Rut measurements were manually taken every 4-7 m along each of the spiral tracks. The results of field tests indicate that rut depth, rut width and rut index increase with the decrease of turning radius, especially when turning radius is less than 20 m. Velocity influences rut formation for the LAV but not HMMWV.     

基于拉格朗日乘子法和群论的具有任意位移边界条件的旋转周期对称结构有限元分析

对任意位移边界条件下的旋转周期对称结构，由拉格朗日乘子法建立有限元方程。在对称适应的坐标系下，由结构刚度矩阵的块循环性质，利用群变换给出一种新的求解方法。数值验证给出令人满意的结果。     

Three-dimensional advective flows in a horizontal cylinder of square section with thermally insulated lateral boundaries

Three-dimensional advective flows in a horizontal cylinder of square section are numerically investigated under thermally insulated lateral boundaries and in the presence of a uniform longitudinal temperature gradient. The flow structure is shown to considerably depend on the Grashof number, the channel length, and the Prandtl number; depending on these parameters, the flow symmetrymode and its temporal behavior may be different. It is established that different cases of transition to oscillatory flow regimes are possible, namely, either with a preliminary violation of the flow symmetry (fork bifurcation) or without a change in the type of symmetry. The parameter range on which only the fork bifurcation is observable, while oscillatory flow patterns do not appear, is also determined.     

A twinning study

For some materials the constitutive function may be multi-valued. One such material is a twinned crystal, which consists of two regions related by a symmetry transformation. In this paper, multi-valued functions based on twinning, trilling, etc., are examined. Group theory is used to obtain some new results on the invariance groups of multi-valued constitutive functions. With these results and ideas of material symmetry used in classical elasticity, estimates for the invariance groups are provided. In addition, a theorem is proved which gives a necessary and sufficient condition relating the invariance group of any branch to the number of branches of the constitutive function. Finally, an example is worked out which uses the theory of stress-free joints. In this example, various ways of bonding two regions together so that the resulting crystals forms a twin are shown.