On approximate symmetries of the elastic properties and elliptic orthotropy

The theory of anisotropic elasticity was originally motivated by applications to crystals, where geometric symmetries hold with high precision. In contrast, symmetries of the effective elastic responses of heterogeneous materials are usually approximate due to various imperfections of microgeometry. A related issue is that available data on the elastic constants may be incomplete or imprecise; it may be appropriate to select the highest possible elastic symmetry that fits the data reasonably well. Some of these problems have been discussed in literature in the context of specific applications, primarily in geomechanics. The present work provides a systematic discussion of the related issues, illustrated by examples on the effective elastic properties of heterogeneous materials. We also discuss a special type of orthotropy typical for a variety of heterogeneous materials - elliptic orthotropy - when the fourth-rank tensor of elastic constants can be represented in terms of a certain symmetric second-rank tensor. This representation leads, in particular, to reduced number of independent elastic constants.     

Frontiers and symmetries of dynamical systems

The frontiers of boundedness ? _b of the orbits of dynamical systems X defined on ? ⁿ are studied. When X is completely integrable some topological properties of ? _b are found and, in certain cases, ? _b is localized with the help of symmetries of X. Several examples in dimensions 2 and 3 are provided. In case the number of known first integrals of the vector field X is less than n ? 1, an interesting connection of ? _b with the frontier of boundedness of the level-sets of the first integrals of X is proved. This result also applies to Hamiltonian systems.     

On the stability of the equilibrium state and small oscillations of non-holonomic systems

The stability of small oscillations of non -holonomic systems is discussed. In contrast to Neimark and Fufaev, we have shown here that a non-holonomic system does indeed possess an isolated equilibrium state under certain conditions. We provide conditions under which there exists asymptotic stability of small oscillations of a non-holonomic system about its equilibrium position. The connection between a holonomic and non-holonomic system is also discussed. The results obtained here can be useful for the control of a non-holonomic system     

Lie symmetries and conserved quantities of constrained mechanical systems

Summary The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.     

Spatio-temporal symmetries in SN- and SN × 2-equivariant systems

This paper investigates the spatio-temporal symmetries of periodic trajectories in dynamical systems with S_N and S_N × ₂ symmetry. It turns out that trajectories in S_N-equivariant systems cannot exhibit spatio-temporal symmetries beyond the trivial symmetry of all periodic orbits. More complex symmetries in the trajectories require additional constraints on the dynamics. The possibilities offered by S_N × ₂ symmetric systems are considered and a specific S₃ × ₂-equivariant system is investigated numerically.     

System response matrix calculation using symmetries for dual‐head PET scanners

Positron emission tomography (PET) scanner with dual‐head geometry offers better spatial resolution and higher sensitivity for dedicated application when compared with conventional full‐ring PET scanners. However, this configuration suffers from limited‐angle projection and depth‐of‐interaction (DOI) effects. Accurate modeling of the system response matrix (SRM) and its incorporation into iterative methods can help to obtain images with better quality. In this paper, we proposed a line‐of‐response (LOR) based symmetry approach to calculate the SRM of PET scanners with dual‐head geometry. Both Monte Carlo (MC) and analytically computed SRMs were obtained and named MC SRM and geometrical SRM respectively, with their performances been compared using simulated phantom studies. The point source study shows that the resolution in directions parallel to the detector is rather uniform, with a full width at half maximum (FWHM) of 0.6～0.7 mm and 0.6～0.8 mm using MC and geometrical SRMs respectively. While the spatial resolution in the direction perpendicular to the detector is much worse, when moving towards the detector panel from the field‐of‐view (FOV) center, the FWHM changes from 1.5 mm to 1.8 mm and 2.4 mm to 3.1 mm using MC and geometrical SRMs, which is caused by the missing of projection views. Images generated using MC SRM also show better stochastic quality and quantitative performance. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 205–214, 2013     

Solitons and symmetries

This article serves as an introduction to the special edition of Solitons and Symmetries in the Journal of Engineering Mathematics. Solitons, by their mathematical nature, are deeply connected to underlying symmetries of nonlinear equations. Described in this introduction is an historical and applications overview, with special emphasis directed at the technologically interesting field of soliton communications in fibre optics. A brief discussion of the papers contained in this special edition is included.     

Transverse intersection of invariant manifolds in perturbed multi-symplectic systems

A multi-symplectic system is a PDE with a Hamiltonian structure in both temporal and spatial variables. This article considers spatially periodic perturbations of symmetric multi-symplectic systems. Due to their structure, unperturbed multi-symplectic systems often have families of solitary waves or front solutions, which together with the additional symmetries lead to large invariant manifolds. Periodic perturbations break the translational symmetry in space and might break some of the other symmetries as well. In this article, periodic perturbations of a translation invariant PDE with a one-dimensional symmetry group are considered. It is assumed that the unperturbed PDE has a three-dimensional invariant manifold associated with a solitary wave or front connection of multi-symplectic relative equilibria. Using the momentum associated with the symmetry group, sufficient conditions for the persistence of invariant manifolds and their transversal intersection are derived. In the equivariant case, invariance of the momentum under the perturbation gives the persistence of the full three-dimensional manifold. In this case, there is also a weaker condition for the persistence of a two-dimensional submanifold with a selected value of the momentum. In the non-equivariant case, the condition leads to the persistence of a one-dimensional submanifold with a seleceted value of the momentum and a selected action of the symmetry group. These results are applicable to general Hamiltonian systems with double zero eigenvalue in the linearization due to continuous symmetry. The conditions are illustrated on the example of the defocussing non-linear Schrödinger equations with perturbations which illustrate the three cases. The perturbations are: an equivariant Hamiltonian perturbation which keeps the momentum level sets invariant; an equivariant damped, driven perturbation; and a perturbation which breaks the rotational symmetry.     

弹性力学辛体系广义动量矩定理

类似于动力学中动量矩的概念,提出了弹性力学辛体系中广义动量矩的概念,给出了平面直角坐标辛体系和极坐标辛体系中广义动量、广义动量矩的统一定义,对广义动量矩的相关理论进行了研究。由Hamilton对偶方程导出了广义动量矩定理,得到了广义动量矩的守恒律,给出了守恒条件,分析了广义动量矩定理和守恒律的物理意义,经典问题验证了广义动量矩定理和守恒律。这一定理进一步丰富了弹性力学辛体系理论的内容,它和Hamilton函数、广义动量等相关理论一起,形成了辛体系中类似于动力学基本定理的一个理论体系。     

Sommerfeld effect in rotationally symmetric planar dynamical systems

Sommerfeld effect concerns the non-linear jump phenomena induced due to the influence of the unbalance response on a non-ideal drive around the critical speed of the excited structure. In this work, we study the influence of external and internal dampings and gyroscopic forces on the Sommerfeld effect in rotationally symmetric planar dynamical systems. The rotational symmetry assumption allows us to obtain neat analytical results for the steady state dynamics. We show that the rotating material or internal damping and the gyroscopic forces influence the spin rate of the non-ideal system and the former changes the system dynamics in an unexpected manner. In particular, we show that the stability threshold may restrict the jump phenomena due to the Sommerfeld effect for larger values of internal damping. Moreover, it is also shown that the Sommerfeld effect would cease to exist under certain conditions. A stability condition for various steady-state equilibriums (branches of steady-state solutions) is derived. A rotor dynamics problem and a structural dynamics problem where the systems interact with a non-ideal source are considered as illustrative examples. A few numerical results are given to validate the analytical solutions.     

Radiation pressure and the photon momentum in dielectrics

We review the magnitudes of the photon momenta in free space, derived by Maxwell and Einstein, and those in homogeneous, dispersionless and lossless dielectrics, associated with Abraham and Minkowski. These momenta determine the forces exerted by light beams on material objects, as measured in radiation pressure experiments. It is shown that conservation conditions for components of the electromagnetic energy–momentum tensors derived by Abraham and Minkowski, also by Einstein and Laub, are equivalent relations simply derived from Maxwell's equations. The challenge for theory is the reliable interpretation of experimental momentum transfers from light to matter, which requires extensions of the basic theory to include material dispersion and surface effects. The main experiments are reviewed, together with details of a Lorentz-force theory that accounts for them. It is shown that the Abraham kinetic momentum is associated with the overall motion of a dielectric sample, while the Minkowski canonical momentum applies to the motion of bodies embedded in the dielectric.     

Rotation-reversal symmetries in crystals and handed structures

Symmetry is a powerful framework to perceive and predict the physical world. The structure of materials is described by a combination of rotations, rotation-inversions and translational symmetries. By recognizing the reversal of static structural rotations between clockwise and counterclockwise directions as a distinct symmetry operation, here we show that there are many more structural symmetries than are currently recognized in right- or left-handed helices, spirals, and in antidistorted structures composed equally of rotations of both handedness. For example, we show that many antidistorted perovskites possess twice the number of symmetry elements as conventionally identified. These new 'roto' symmetries predict new forms for 'roto' properties that relate to static rotations, such as rotoelectricity, piezorotation, and rotomagnetism. They enable a symmetry-based search for new phenomena, such as multiferroicity involving a coupling of spins, electric polarization and static rotations. This work is relevant to structure-property relationships in all materials and structures with static rotations.     

Bifurcations in dynamical systems with interior symmetry

We propose a definition of interior symmetry in the context of general dynamical systems. This concept appeared originally in the theory of coupled cell networks, as a generalization of the idea of symmetry of a network. The notion of interior symmetry introduced here can be seen as a special form of forced symmetry breaking of an equivariant system of differential equations. Indeed, we show that a dynamical system with interior symmetry can be written as the sum of an equivariant system and a ‘perturbation term’ which completely breaks the symmetry. Nonetheless, the resulting dynamical system still retains an important feature common to systems with symmetry, namely, the existence of flow-invariant subspaces. We define interior symmetry breaking bifurcations in analogy with the definition of symmetry breaking bifurcation from equivariant bifurcation theory and study the codimension one steady-state and Hopf bifurcations. Our main result is the full analogues of the well-known Equivariant Branching Lemma and the Equivariant Hopf Theorem from the bifurcation theory of equivariant dynamical systems in the context of interior symmetry breaking bifurcations.     

Hidden symmetries and pattern formation in Lapwood convection

We study Lapwood convection (convection of a fluid in a porous medium) on a two-dimensional rectangular domain. The linearized eigenmodes are symmetric pxq cellular patterns, which we call (p, q) modes. Numerical calculations of the branching structure near mode interaction points have derived bifurcation diagrams for the (3, 1)/(1, 1) and (3, l)/(2, 2) mode interactions which are non-generic, even when the rectangular symmetry of the domain is taken into account. This has raised questions about the accuracy of the numerical method used, a finite-element Galerkin approximation implemented using Harwell's ENTWIFE code. We show that this apparent lack of genericity is partly a consequence of 'hidden' translational symmetries, which arise when the problem is extended to one with periodic boundary conditions. This extension procedure has become standard for partial differential equations (PDEs) with Neumann or Dirichlet boundary conditions, and it reveals restrictions on the Liapunov-Schmidt reduced bifurcation equations and the resulting singularity-theoretic normal forms. Its application to Lapwood convection is unusual in that the PDE involves a mixture of both Neumann and Dirichlet boundary conditions. Specifically, on the vertical sidewalls the stream function satisfies Dirichlet boundary conditions (is zero), but the temperature satisfies Neumann (no-flux) boundary onditions. Nevertheless, we show that for abstract group-theoretical reasons the same symmetry constraints that occur for purely Neumann boundary conditions are imposed on the Liapunov-Schmidt reduced bifurcation equations, and therefore the same list of normal forms is valid. The hidden symmetries force certain terms in the reduced bifurcation equations to be zero and change the generic branching geometry. With the aid of MACSYMA, we determine a small number of low-order coefficients of the reduced bifurcation equations which are needed to find the correct normal form. We show that in some cases the normal form is more degenerate than might be anticipated, but that when these degeneracies are taken into account the resulting branching geometry reproduces that found in the earlier numerical approach. In particular, we obtain an analytic vindication of the numerical method     

Material symmetries and inhomogeneous waves in piezoelectric media

Summary This study is an experimental investigation on a novel oscillation phenomenon of a water rivulet on a smooth hydrophobic surface. It is found that the water rivulet running down on a smooth Plexiglass plate exhibits all together four distinctive patterns with increasing either the Froude number, the Weber number or the Reynolds number. Once either the Froude number, the Weber number or the Reynolds number exceeds the third critical value, the rivulet on a smooth Plexiglass plate is restabilized, and becomes almost straight. However, in the restable rivulet, several beads of a rosary are formed following immediately downstream of the pipe mouth. This is no more than the novel oscillation phenomenen of the water rivulet on a smooth Plexiglass plate. The oscillatory motion is steady in the hydrodynamical sense, because the phase of the oscillation is always the same at each point in space.It is found that with increasing either the Froude number, the Weber number or the Reynolds number, not only the wave-length and width of the beads in the restable rivulet increase, but also the rivulet itself becomes more straight and stable. On one hand, with increasing distance from the pipe mouth along the central axis of the rivulet, the local maximum width of each of the beads decreases, but its wave-length increases.It is suggested that the oblique surface waves generated at the left and right contact lines of the rivulet play a primary role in the contraction process of the novel oscillation phenomenon of the restable rivulet.     

Broken symmetries and thermodynamic limit in quantum antiferromagnets

We study how the Néel order breaks the symmetries of the hamiltonian in quantum antiferromagnet. The construction of a quantum wave packet that mimics the classical groundstate implies the existence of an extensive number of states: 1) with an energy scale much smaller than the spin-waves excitations (magnons); 2) with well defined symmetries.For the case of the triangular lattice, we find that all necessary properties to obtain a Néel state are well verified, already on periodic samples as small as 21 spins. For this system, this is a new and strong evidence of Néel order in the thermodynamic limit.     

Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries

This paper is devoted to the construction of a class of prior stochastic models for non‐Gaussian positive‐definite matrix‐valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi‐scale approaches where the randomness induced by fine‐scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least‐square optimization problems. An application is finally provided. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient uncertainty quantification of dynamical systems with local nonlinearities and uncertainties

Idealized modeling of most engineering structures yields linear mathematical models, i.e., linear ordinary or partial differential equations. However, features like nonlinear dampers and/or springs can render nonlinear an otherwise linear model. Often, the connectivity of these nonlinear elements is confined to only a few degrees-of-freedom (DOFs) of the structure. In such cases, treating the entire structure as nonlinear results in very computationally expensive solutions. Moreover, if system parameters are uncertain, their stochastic nature can render the analysis even more computationally costly. This paper presents an approach for computing the response of such systems in a very efficient manner. The proposed solution procedure first segregates the DOFs appearing in the nonlinear and/or stochastic terms from those DOFs that involve only linear deterministic operations. Second, the responses of nonlinear/stochastic terms are determined using a non-standard form of a nonlinear Volterra integral equation (NVIE). Finally, the responses of the remaining DOFs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. Three examples are presented to demonstrate the efficacy and accuracy of the proposed method. It is shown that, even for moderately sized systems (∼1000 DOFs), the proposed method is about three orders of magnitude faster than a conventional Monte Carlo sampling method (i.e., solving the system of ODEs repeatedly).     

Single step synthesis of Ce doped CaAl2O4 and CaAl4O7 systems

Both CaAl₂O₄ (CA2) and CaAl₄O₇ (CA4) oxide-systems possess monoclinic crystal structure. Herein, we have prepared CA2 and CA4 systems via single step combustion route. A good correlation is observed between calculated and the standard lattice parameters. Ce³⁺ ions were deliberately doped as extrinsic impurities in order to understand the crystal symmetry effects on the emission characteristics in the as-prepared matrices. Large red-shift was observed in CA4-emission spectrum despite of their same crystal structures. Possible reasons are discussed.