Fundamentals in generalized elasticity and dislocation theory of quasicrystals: Green tensor,dislocation key-formulas and dislocation loops

The present work provides fundamental quantities in generalized elasticity and dislocation theory of quasicrystals. In a clear and straightforward manner, the three-dimensional Green tensor of generalized elasticity theory and the extended displacement vector for an arbitrary extended force are derived. Next, in the framework of dislocation theory of quasicrystals, the solutions of the field equations for the extended displacement vector and the extended elastic distortion tensor are given; that is, the generalized Burgers equation for arbitrary sources and the generalized Mura–Willis formula, respectively. Moreover, important quantities of the theory of dislocations as the Eshelby stress tensor, Peach–Koehler force, stress function tensor and the interaction energy are derived for general dislocations. The application to dislocation loops gives rise to the generalized Burgers equation, where the displacement vector can be written as a sum of a line integral plus a purely geometric part. Finally, using the Green tensor, all other dislocation key-formulas for loops, known from the theory of anisotropic elasticity, like the Peach–Koehler stress formula, Mura–Willis equation, Volterra equation, stress function tensor and the interaction energy are derived for quasicrystals.     

A note on Killing tensors

The connection between symmetric and skew-symmetric Killing tensors is studied. Some theorems on skew-symmetric Killing tensors are generalized, and it is shown that in all type-D vacuum metrics admitting a symmetric Killing tensor, this Killing tensor can be given in terms of a skew-symmetric Killing tensor.     

Bargmann–Wigner方程与高自旋场方程

在坐标表象中由Bargmann-Wigner方程导出了便于求解的高自旋场方程,并给出了相应的拉氏函数.     

Stress functions and internal stresses in linear three-dimensional anisotropic elasticity

A stress function method is presented in order to give a general solution of the incompatibility problem of three-dimensional linear anisotropic elasticity theory. A relation between the internal stress tensor in terms of derivatives of a sixth-order stress function tensor will be derived. Using this formulation the so far open problem of the representation of the second-order stress function tensor by the fourth-order stress function tensor in anisotropic elasticity theory is solved in general.     

On microcontinuum field theories: the Eshelby stress tensor and incompatibility conditions

We investigate linear theories of incompatible micromorphic elasticity, incompatible microstretch elasticity, incompatible micropolar elasticity and the incompatible dilatation theory of elasticity (elasticity with voids). The incompatibility conditions and Bianchi identities are derived and discussed. The Eshelby stress tensor (static energy momentum) is calculated for such inhomogeneous media with microstructure. Its divergence gives the driving forces for dislocations, disclinations, point defects and inhomogeneities which are called configurational forces.     

On potentials for several classes of spinor and tensor fields in curved spacetimes

Already known results with respect to the existence of a vector potential for the Maxwell field tensor and a tensor potential for Weyl's conformal curvature tensor in four-dimensional spacetimes are generalized. It is shown that there exists a spinor potential of type (n–1,1) for any totally symmetric spinor field of rankn. From this theorem we deduce a series of corollaries, for example, that every antisymmetric tensor of second rank admits a linear representation in terms of the first derivatives of two vector fields. Further, some investigations are made on the existence of potentials for arbitrary symmetric spinors of type (n, m).     

Measuring the Dielectric Permittivity of Sapphire at Temperatures 93–343 K

We consider a method for measuring the elements of the dielectric-permittivity tensor of a slightly lossy uniaxial anisotropic dielectric using the spectrum of resonance frequencies of a metal-dielectric resonator in the form of an axially anisotropic cylindrical specimen with endface metal mirrors. Measurements of both elements of the dielectric-permittivity tensor of sapphire at temperatures 93–343 K are performed using the spectrum of symmetric E modes. To avoid the influence of the residual gaps between the specimen and the mirrors, the endface surfaces of the specimen are metallized. We performed independent measurements of the axial and transverse elements of the permittivity tensor at room temperature from the spectra of azimuthal E modes and symmetric H modes. The values of the axial and transverse elements of the dielectric-permittivity tensor of sapphire in the range 93–343 K are presented with a spacing of 10 K.     

A natural one-form for the Schouten concomitant

The Poisson bracket in classical mechanics arises from the existence of a natural one-form on a cotangent bundle. The Schouten concomitant of two symmetric contravariant tensor fields is closely related to the Poisson bracket. We show that it arises in an analogous way from a natural onecochain, where the chains are chains of derivations from the module of symmetric contravariant tensor fields into itself.     

Optical metrics and birefringence of anisotropic media

The material tensor of linear response in electrodynamics is constructed out of products of two symmetric second rank tensor fields which in the approximation of geometrical optics and for uniaxial symmetry reduce to “optical” metrics, describing the phenomenon of birefringence. This representation is interpreted in the context of an underlying internal geometrical structure according to which the symmetric tensor fields are vectorial elements of an associated two-dimensional space.     

Zero-frequency elastic moduli of uniform fluids

A statistical mechanical treatment of equilibrium elasticity of a uniform fluid phase based on density functional theory is presented. Bulk expressions for the stress tensor and the zero-frequency elastic moduli tensor involving the direct correlation function are found.     

Relativistic symmetry of position-dependent mass particle in Coulomb field including tensor interaction

The spatially-dependent mass Dirac equation is solved exactly for attractive scalar and repulsive vector Coulomb potentials including a tensor interaction under the spin and pseudospin symmetric limit. Closed forms of the energy eigenvalue equation and wave functions are obtained for arbitrary spin-orbit quantum number κ. Some numerical results are given too. The effect of the tensor interaction on the bound states is presented. It is shown that the tensor interaction removes the degeneracy between two states in the spin doublets. We also investigate the effects of the spatially-dependent mass on the bound states under the conditions of the spin symmetric limit in the absence of tensor interaction.     

On mechanical characteristics of nanocrystals

The dependence of the elastic moduli of a nanocrystal on its size is investigated theoretically with reference to a two-dimensional single-crystal strip. It is shown that the uncertainty (of a fundamental nature) in the size of a nanocrystal causes the determination of many of its mechanical characteristics to be ambiguous. It is found that the Cauchy-Green relations are modified and the elastic-constant tensor ceases to be symmetric; the size and shape of a nanocrystal render its mechanical properties more anisotropic. For a single-crystal strip, the Poisson ratio decreases and the Young modulus increases with decreasing thickness of the strip; in the case of a very thin crystal film (two atomic layers thick), these elastic moduli can differ from their macroscopic values by a factor of two. The size effects which make the continuum elasticity theory inapplicable to nanocrystals are estimated. The size effects that occur when the molecular dynamics method is applied for modeling macroscopic objects are also discussed.     

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.     

Eshelby's tensor fields and effective conductivity of composites made of anisotropic phases with Kapitza's interface thermal resistance

Eshelby's results and formalism for an elastic circular or spherical inhomogeneity embedded in an elastic infinite matrix are extended to the thermal conduction phenomenon with a Kapitza interface thermal resistance between matrix and inclusions. Closed-form expressions are derived for the generalized Eshelby interior and exterior conduction tensor fields and localization tensor fields in the case where the matrix and inclusion phases have the most general anisotropy. Unlike the relevant results in elasticity, the generalized Eshelby conduction tensor fields and localization tensor fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of Kapitza's interface thermal resistance. With the help of these results, the size-dependent overall thermal conduction properties of composites are estimated by using the dilute, Mori–Tanaka, self-consistent and generalized self-consistent models. The analytical estimates are finally compared with numerical results delivered by the finite element method. The approach elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media, and to the mathematically identical phenomenon of anti-plane elasticity.     

A generalization of the relativistic equilibrium equations for a non-rotating star

The problem of elastomechanical equilibrium for a static, spherically symmetric star composed of an elastic material is analyzed. A suitable formulation of relativistic elasticity theory is used, and the second order equilibrium equations are found. It is shown that the equilibrium conditions with anisotropic pressure introducedad hoc by some authors are in fact the dynamical conditions for a relativistic elastic material. The corresponding first order equations for the components of the metric and of the energy-momentum tensor reduce to the Tolman-Oppenheimer-Volkhoff equations if the material exhibits no shape-rigidity. Two interesting classes of solutions are discussed.     

An invariant theory of the constitutive parameters of a plasma and a ferrite

Under the influence of a constant magnetic field, the electric property of a plasma and the magnetic property of a ferrite are anisotropic. In this paper, the general coordinatefree invariant forms of the dielectric tensor of a plasma and the permeability tensor of a ferrite are obtained. The tensors are expressed explicitly as a sum of three tensors: a unit tensor, a symmetric tensor and an antisymmetric tensor, each of which is weighted by different constants. The symmetric and antisymmetric tensors are related to the unit vector of the constant magnetic field. The invariant forms in terms of the sum of the projectors of the tensors are also derived. When a Cartesian coordinate system is introduced, the invariant forms are easily reduced to the commonly used matrix representations. The invariant forms clearly show the effects of the constant magnetic field on the anisotropies of the media. Moreover, they effectuate and simplify the deduction of the general solutions of problems involving wave propagation and excitation in plasma and ferrite and thus facilitate interpretations of the final results.     

Second-Order Covariant Tensor Decomposition in Curved Spacetime

Local four-dimensional tensor decomposition formulae for generic vectors and 2-tensors in spacetime, in terms of scalar and antisymmetric covariant tensor potentials, are studied within the framework of tensor distributions. Earlier first-order decompositions are extended to include the case of four-dimensional symmetric 2-tensors and new second-order decompositions are introduced.     

On the evolution equations for ideal magnetohydrodynamics in curved spacetime

We examine the problem of the construction of a first order symmetric hyperbolic evolution system for the Einstein-Maxwell-Euler system. Our analysis is based on a 1?+?3 tetrad formalism which makes use of the components of the Weyl tensor as one of the unknowns. In order to ensure the symmetric hyperbolicity of the evolution equations implied by the Bianchi identity, we introduce a tensor of rank 3 corresponding to the covariant derivative of the Faraday tensor. Our analysis includes the case of a perfect fluid with infinite conductivity (ideal magnetohydrodynamics) as a particular subcase.     

Conformally symmetric semi-Riemannian manifolds

In this paper we introduce the concept of conformal curvature-like tensor on a semi-Riemannian manifold, which is weaker than the notion of conformal curvature tensor defined on a Riemannian manifold. By such kind of conformal curvature-like tensor we give a complete classification of conformally symmetric semi-Riemannian manifolds with generalized non-null stress energy tensor.