Terms Of Invariant Quantities Research Articles

Invariant-based formulation of a triangular finite element for geometrically nonlinear thermal analysis of composite shells

The paper describes an invariant-based formulation of a triangular finite element for geometrically nonlinear analysis of shear flexible composite shells subjected to thermal loads. Transverse shear deformation is taken into account using the first order shear deformation theory. The focus is on the representation of the strain energy of the shell in terms of invariant quantities which depend on the components of the strain tensor and elastic constants of the material. Based on the invariant expression for the strain energy density, algorithmic relations are derived for computing the stiffness matrix of the shell finite element. The finite element formulation is used to study stability of equilibrium configurations in the region of large thermal displacements. A positive definite second variation of the total energy is used as a sufficient criterion for stability of equilibrium configurations. A series of numerical examples are given to estimate performance of the finite element in solving nonlinear problems of composite plates and shells under uniform temperature rise. Solution of some classical problems of laminated plates and shells shows that there exist equilibrium configurations not previously reported in the literature.

Sensitivity of the DVCS Cross Section to the Compton form Factors in Scalar QED

Using deeply virtual Compton scattering as a tool to study the structure of hadrons in an exclusive process, one expresses the amplitudes in terms of invariant quantities: the Compton form factors. In this paper the sensitivity of the hadronic part of the cross section to the Compton form factors is determined.

Extrema of Young’s modulus in hexagonal materials

The optimum values of Young’s modulus for hexagonal materials are obtained by using the representation of the elasticity tensor by Ahmad and Khan [F. Ahmad, R.A. Khan, Eigenvectors of a rotation matrix, Q. J. Mech. Appl. Math. 62 (2009) 297–310]. The expression of Young’s modulus E(n) for a hexagonal material is written in terms of n3 only such that it reveals an axis of rotational symmetry in the direction x3, which is perpendicular to the transverse isotropy plane, that is x1x2-plane: indeed the components n1,n2 of the unit vector n have no influence on the value of Young’s modulus. Moreover Young’s modulus is expressed in terms of invariant quantities, i.e. eigenvalues rather than components of the compliance tensor. The problem is solved in a simple manner and is applied to some real materials.

INVARIANT REPRESENTATION OF PROPAGATION PROPERTIES FOR BI-COUPLED PERIODIC STRUCTURES

General bi-coupled periodic systems are dealt with by means of transfer matrices of single units. The solutions of the associated characteristic equation are discussed in terms of invariant quantities by exploiting the well-known reversibility of its coefficients. An exhaustive description of the free wave propagation patterns is given on the invariant plane where propagation domains with qualitatively different character are identified. The asymptotic behavior of the roots of the characteristic equation when the invariants tend to infinity is analyzed. The contour plot of the real part of the propagation constants, responsible for the amount of attenuation of the characteristic waves, is illustrated on the invariants' plane. Next, several models of bi-coupled periodic structures made up of beams resting on elastic supports are considered. A non-linear mapping from the invariants' plane to the physical parameters plane provides a concise representation of the pattern of the propagation domains. A mechanical interpretation associated with the boundaries of these regions is given. Finally, the proper'selection of the physical parameters governing the propagation modes is discussed.

Viewpoint--invariance of contour-curvature discrimination over extended performance levels.

A traditional approach to the problem of human object recognition is to assume that the visual system represents an object in terms of invariant quantities. This study considers a generalization of this approach to the problem of visually recognizing differences in shape, specifically between contour markings on a planar surface, as the position of the surface varies in space. For a given 'shape cue', perceived differences in contour may be quantified by threshold values--Weber fractions--at a particular criterion level of performance. A necessary theoretical condition for the Weber fraction to be constant with the relative viewpoint of the observer is that the cue should be a relative invariant under the natural spatial transformations of the image. Some data are reviewed showing how statistically efficient some cues are in explaining the observed discriminability of symmetric curved contours at a fixed criterion performance level of 75%. A new, fuller analysis is presented showing that the efficiency of the most efficient cue satisfying the invariance condition is maintained over a wide range of criterion performance levels, from 53% to 92% correct, corresponding to discrimination index values of 0.1 to 2.0. Over these levels, Weber's law was found empirically to hold almost exactly.

On the curvature description of gravitational fields

The local geometry of a Riemannian manifold is described completely by the curvature tensor and a finite number of its covariant derivatives. The authors derive necessary and sufficient conditions for a line element to exist that gives rise to these quantities. The resulting system of equations can be written as a set of integrable differential equations along with a set of algebraic equations. This gives a technique for searching for solutions to Einstein's equations with special properties. It also makes it possible to perform perturbative calculations entirely in terms of invariant quantities. They illustrate the methods on homogeneous rotationally symmetric spacetimes.

Some invariant properties of the polarization scattering matrix

A set of parameters can be based on properties of the polarization scattering matrix which are invariant to change in the axial ratio of the polarization ellipse for the measuring antenna, or to rotation of the antenna about the line of sight. These invariant quantities have the advantage that they are free of the effects of Faraday rotation and of certain errors in antenna polarization, yet depend upon the nature of the scattering body and as such can be used to classify certain characteristics of the body. Some of the parameters of special representations for the scattering matrix, such as the null polarization and eigenpolarization, are simply related to the invariant quantities. However, not all of the significant characteristics of the scattering matrix can be specified in terms of invariant quantities, since two degrees of freedom are necessary, one to account for the axial ratio of measuring antenna and one to account for the relative orientation of the antenna and scatterer about the line of sight. The decomposition of these effects into the product of a rotation operator and an ellipticity operator offers a convenient method for critical examination of polarization dependent characteristics.

The geometry of Riemannian spaces

The primary purpose of this paper is to expose, in as simple and clear a form as is possible, the fundamentals of the geometric structure of a Riemannian space. It is a general truth that the methods which pierce most deeply into the heart of a geometric theory are invariant methods, that is, methods which are independent of the choice of the coordinates in terms of which the theory is expressed analytically. In the case of Riemannian geometry, these are the methods of tensor analysis. As important, perhaps, as the use of invariant methods is the expression of the analytic theory, so far as possible, in terms of invariant quantities alone. For it is in this form that the theory becomes most illuminating and suggestive. But, in ordinary tensor analysis, the components of a tensor are not invariants. A first step toward our goal will be, then, to introduce for Riemannian geometry an intrinsic tensor analysis, that is, a form of tensor analysis in which the components of all tensors are invariants. Any theory of the geometry of a Riemannian space presupposes that the space is referred to a certain ennuple of congruences of curves. In the ordinary theory, this ennuple consists of the parametric curves. In the intrinsic theory, it is an ennuple E, whose choice, as will presently be evident, is entirely arbitrary. The ordinary components of a tensor, that is, the components in the ordinary theory, are referred to the differentials of the coordinates xi pertaining to the ennuple of the parametric curves. The intrinsic components are referred to the differentials of arc, dsi, of the curves of the ennuple E. There is no need in the intrinsic theory of actual coordinates pertaining to the ennuple E; the differentials of arc dsi suffice. Accordingly, E can be chosen arbitrarily; it does not have to be a parametric ennuple, that is, an ennuple with which coordinates can be associated. It may be, and we shall ordinarily take it to be, an ennuple of general type, and hence, of course, not necessarily orthogonal. Intrinsic components, referred to E, of the covariant derivative of a tensor are made possible by the introduction of invariant Christoffel symbols in