Motion decomposition,frame-indifferent derivatives,and constitutive relations at large displacement gradients from the viewpoint of multilevel modeling

Widespread approaches to generalizing geometrically linear constitutive relations to the case of large displacement gradients have been considered. These approaches are based on the replacement of the material derivatives of stress and strain tensors by frame-indifferent corotational or convective derivatives. The correctness of choosing the indifferent derivatives is analyzed from a more general viewpoint of motion decomposition into rigid and strain-induced motion. It is shown that the use of the Zaremba-Jaumann derivative in constitutive relations corresponds to motion decomposition by the Cauchy-Helmholtz theorem according to which instantaneous rigid rotation of a material particle with small neighborhood is described by the vorticity tensor. The relations derived with the use of the so-called "logarithmic spin" are analyzed. It is noted that the spin tensors entering into these relations are not associated with the material fibers (in particular with the symmetry axes of anisotropic materials) during the entire studied process of deformation. Hence these spins do not describe the rotation of the reference frame (crystallographic one for metals) in which the material property tensor is defined. A new method of motion decomposition is proposed on the basis of a two-level (macro and meso) approach for single and polycrystalline metals. The mesoscopic spin is determined by the rotation rate of the corotational coordinate system associated with the crystallographic direction and crystallographic plane. Mesoscopic constitutive relations are formulated using the proposed spin. The spin of a representative macrovolume is determined by averaging the spins of the crystallites contained in this volume. This spin is used to formulate rate-type elastic constitutive equations. Examples are given to illustrate the stress state determination for loading along closed strain paths and two-segment paths for isotropic and anisotropic (with cubic symmetry, hcp) elastic materials, and an elastoviscoplastic fcc crystallite. The determination is carried out by using the corotational derivatives in the constitutive relations which are obtained by different motion decomposition methods.     

Material frame representation of equivalent stress tensor for discrete solids

In this paper, we derive expressions for equivalent Cauchy and Piola stress tensors that can be applied to discrete solids and are exact for the case of homogeneous deformation. The main principles used for this derivation are material frame formulation, long wave approximation and decomposition of particle motion into continuum and thermal parts. Equivalent Cauchy and Piola stress tensors for discrete solids are expressed in terms of averaged interparticle distances and forces. No assumptions about interparticle forces are used in the derivation, thereby ensuring our expressions are valid irrespective of the choice of interatomic potential used to model the discrete solid. The derived expressions are used for calculation of the local Cauchy stress in several test problems. The results are compared with prediction of the classical continuum definition (force per unit area) as well as existing discrete formulations (Hardy, Lucy, and Heinz-Paul-Binder stress tensors). It is shown that in the case of homogeneous deformations and finite temperatures the proposed expression leads to the same values of stresses as classical continuum definition. Hardy and Lucy stress tensors give the same result only if the stress is averaged over a sufficiently large volume. Thus, given the lack of sensitivity to averaging volume size, the derived expressions can be used as benchmarks for calculation of stresses in discrete solids.     

Comparing,Modeling, and Assigning Chemical-Shift Tensors in the Cartesian,Irreducible Spherical,and Icosahedral Representations

A measure of the difference between two chemical-shift tensors is developed by defining the scalar distance between them. Chemical-shift tensors are treated as functions whose domain is the surface of a sphere and the mathematical definition of the quadratic distance between two functions is invoked. Expressions for the distance between two chemical-shift tensors are developed in the Cartesian and irreducible spherical representations and in a new icosahedral representation. A representation wherein the chemical-shift tensor is specified by the shifts when the magnetic field is along six directions defined by the vertices of an isosahedron is developed and its properties are discussed. The expression for the distance between two tensors is found to be particularly attractive and useful in this icosahedral representation. The distance between tensors computed in the icosahedral representation is useful in fitting linear models to tensor data. It is shown how such fitting can contribute to the assignment of tensors obtained from single-crystal studies. A quantitative figure of merit useful for comparing multiple assignment possibilities is developed. The results derived are applicable to any physical phenomenon described by real zero-rank and second-rank tensors.     

Shear Viscosity and Temperature Inequalities for Multi-Species Plasmas

The first coefficients in the orthogonal expansions of the velocity distribution functions with respect to Grad's tensorial (three-dimensional) Hermite polynomials are shown to be proportional to ther-modynamic fluxes, if the weight functions are local Maxwellians centered at the mean mass velocity and widened with a mean temperature. Balance equations for the stress tensors are established and reduced to linear algebraic systems under certain restrictions. Explicit solutions for the traces and the tracefree parts of the stress tensors are given.     

Mechanical stress tensor for a fluid in a magnetostatic field

Summary Two constraints on the form of the mechanical stress tensor for an uncharged and linear dielectric fluid at rest in a magnetostatic field are determined. Moreover, it is shown that the stress tensor proposed by Helmholtz, that proposed by Einstein-Laub and that proposed by Liu-Müller are unacceptable as mechanical stress tensors for a dielectric fluid in the conditions stated above.     

Quantum fluctuations of vacuum stress tensors and spacetime curvatures

We analyze the quantum fluctuations of vacuum stress tensors and spacetime curvatures, using the framework of linear response theory which connects these fluctuations to dissipation mechanisms arising when stress tensors and spacetime metric are coupled. Vacuum fluctuations of spacetime curvatures are shown to be a sum of two contributions at lowest orders; the first one corresponds to vacuum gravitational waves and is restricted to light-like wavevectors and vanishing Einstein curvature, while the second one arises from gravity of vacuum stress tensors. From these fluctuations, we deduce noise spectra for geodesic deviations registered by probe fields which determine ultimate limits in length or time measurements. In particular, a relation between noise spectra characterizing spacetime fluctuations and the number of massless neutrino fields is obtained.     

Boundary sources in the Doran-Lobo-Crawford spacetime

We take a null hypersurface (causal horizon) generated by a congruence of null geodesics as the boundary of the Doran-Lobo-Crawford spacetime to be the place where the Brown-York quasilocal energy is located. The components of the outer and inner stress tensors are computed and shown to depend on time and on the impact parameter b of the test-particle trajectory. The spacetime is a solution of Einstein’s equations with an anisotropic fluid as source. The surface energy density σ on the boundary is given by the same expression as that obtained previously for the energy stored on a Rindler horizon. For time intervals long compared to b (when the stretched horizon tends to the causal one), the components of the stress tensors become constant.      

Covariant electrodynamics in a medium,II

The energy-momentum and angular momentum emission rates for an arbitrarily moving charge (whose speed is less than that of light in the medium) in a uniform transparent medium are calculated in manifestly covariant form. The calculations are executed for three types of stress tensor: Minkowski, Abraham, and Marx. Among other things it is found that the energy-momentum emission rates for the latter two tensors are equal and differ from that of the former. Further, the angular momentum emission rates for all three tensors are found to be equal. Only for the Marx tensor is this rate independent of the orientation of the associated asymptotic space-like surface.     

Yield stress of nanocrystalline materials: role of grain-boundary dislocations,triple junctions and Coble creep

A theoretical model is suggested which describes the strengthening of nanocrystalline materials due to the effects of triple junctions of grain boundaries as obstacles for grain-boundary sliding. In the framework of the model, a dependence of the yield stress characterizing grain-boundary sliding on grain size and triple-junction angles is revealed. With this dependence we have found that, in as-fabricated nanocrystalline materials, the yield stress depends upon a competition between conventional dislocation slip and grain-boundary sliding. On the other hand, yield stress dependence on grain size in heat-treated nanocrystalline materials is described as that caused by a competition between conventional dislocation slip and Coble creep. Grain-size and triple-junction angle distributions are incorporated into the consideration to account for distributions of grain size and triple-junction angles, occurring in real specimens. The results of the model are compared with experimental data from as-fabricated and heat-treated nanocrystalline materials and shown to be in good agreement.     

Internal geometric current,and the Maxwell equation as a Hamiltonian system on configuration surfaces

Internal tension tensors on Riemannian manifolds with magnetic field are defined by means of a quantization mapping and quantum statistical averaging. The internal geometric density, current, and stress are derived from asymptotic expansions of the corresponding quantum tensors in the quantization parameter. On Riemannian surfaces with magnetic field, the Maxwell-Lorentz equation is interpreted as a Hamiltonian system. The effect of geometric superconductivity is discussed. The research was supported by the Russian Foundation for Basic Research under grant no. 05-01-00918.     

Conservation laws and stress–energy–momentum tensors for systems with background fields

This article attempts to delineate the roles played by non-dynamical background structures and Killing symmetries in the construction of stress–energy–momentum tensors generated from a diffeomorphism invariant action density. An intrinsic coordinate independent approach puts into perspective a number of spurious arguments that have historically lead to the main contenders, viz the Belinfante–Rosenfeld stress–energy–momentum tensor derived from a Noether current and the Einstein–Hilbert stress–energy–momentum tensor derived in the context of Einstein’s theory of general relativity. Emphasis is placed on the role played by non-dynamical background (phenomenological) structures that discriminate between properties of these tensors particularly in the context of electrodynamics in media. These tensors are used to construct conservation laws in the presence of Killing Lie-symmetric background fields.     

Raman tensor of germanium and zincblende-type semiconductors

It is shown that the one-phonon Raman tensor of germanium and zincblende-type materials can be calculated from a model density of states consisting only of the E₀ and the E₀ + Δ₀ parabolic critical points. The calculated Raman tensors and their spectral dependence are compared with experimental results for GaP, GaAs, and ZnSe and with a recent calculation for Ge based on the complete band structure.     

A new coordinate condition,and elementary flatness,for axially symmetric interior solutions in general relativity

Two general results for stationary axially symmetric interior solutions of the Einstein or Einstein-Maxwell equations in cylindrical coordinates are derived.Firstly, a coordinate condition for interior solutions is proposed, corresponding to the Weyl coordinate condition used in the exterior.Secondly, it is shown that elementary flatness in the interior is always ensured by realistic boundary conditions and matter tensors, given elementary flatness in the exterior metric.A physical discussion of the results is given, particularly in reference to solutions which have singular struts in them.     

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.     

Canonical Quantization of Electromagnetic Field in the Presence of Nonlinear Anisotropic Magnetodielectric Medium with Spatial-Temporal Dispersion

Modeling a nonlinear anisotropic magnetodielectric medium with spatial-temporal dispersion by two continuum collections of three dimensional harmonic oscillators, a fully canonical quantization of the electromagnetic field is demonstrated in the presence of such a medium. Some coupling tensors of various ranks are introduced that couple the magnetodielectric medium with the electromagnetic field. The polarization and magnetization fields of the medium are defined in terms of the coupling tensors and the oscillators modeling the medium. The electric and magnetic susceptibility tensors of the medium are obtained in terms of the coupling tensors. It is shown that the electric field satisfy an integral equation in frequency domain. The integral equation is solved by an iteration method and the electric field is found up to an arbitrary accuracy.     

Integrability conditions for a gravitational theory with nonmetric connection

In the framework of a new gravitational theory with nonmetric connection recently introduced by one of us (J. K.), it is shown that the matter stress tensors satisfy a certain identity, which, via the contracted Bianchi identities, turns out to be a formal integrability condition for the gravitational field equations. The conservation law for the Hilbert tensor is also discussed.     

A new gravitational energy tensor

The Bel-Robinson tensor is the most used gravitational energy tensor; however, it has the dimensions of energy squared. How to construct tensors with the dimensions of energy by using Lancoz tensors is shown here. The resulting tensors have a large number of arbitrary parameters, frequently have spacelike currents, and frequently do not reduce to familiar pseudo-energy tensors in the weak field limit. Two particular examples of interest are one with well-behaved currents and one which reduces to an energy pseudo-tensor in the weak field limit.     

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.