Tensor Variables Research Articles

Hydrodynamics beyond local equilibrium: Application to electron gas

Generalized hydrodynamic theory, which does not rest on the requirement of a local equilibrium, is derived in the long-wave limit of a kinetic equation. The theory bridges the whole frequency range between the quasistatic (Navier-Stokes) hydrodynamics and the high frequency (Vlasov) collisionless limit. In addition to pressure and velocity the theory includes new macroscopic tensor variables. In a linear approximation these variables describe an effective shear stress of a liquid and the generalized hydrodynamics recovers the Maxwellian theory of highly viscous fluids - the media behaving as solids on a short time scale, but as viscous fluids on long time intervals. It is shown that the generalized hydrodynamics can be applied to the Landau theory of Fermi liquid. Illustrative results for collective modes in confined systems are given, which show that nonequilibrium effects qualitatively change the collective dynamics in comparison with the predictions of the heuristic Bloch's hydrodynamics. Earlier improvements of the Bloch theory are critically reconsidered.

Further Results on Generating Sets for Isotropic and Anisotropic Functions of Vectors and 2nd-Order Tensors

It has been proved in a previous work that irreducible generating sets for arbitrary-order tensor-valued isotropic or anisotropic functions of any finite number of vector and 2nd order tensor variables may be formed by a union of those for seven types of isotropic or onisotropic functions of three variables. Here we show that, for many kinds of material symmetry groups, some and even all of the latter may further be reduced to cases of two variables and even single variables.

An explicit damage model for the design of composites structures

A behavioural damage model is presented which describes the principal phenomena observed in composites, namely: initial anisotropy, (i.e. the effects of damage on the elastic behaviour) defect growth law, and closure effects in relation to residual strains. We have developed a formulation which allows the use of damage variables related both to microcracks oriented by the constituents and microcracks oriented by the loading. It is now accepted that defects can either be oriented by the microstructure, and therefore expressed by a family of scalar variables, or by the loading, which leads to the use of tensorial variables. We show an application to the case of ceramic-matrix composites ( SiC SiC ) which exhibits two damage modes and requires the use of both scalar and tensorial variables. We emphasise the explicit feature of this model which simplifies parameter identification and the implementation in finite element codes.

48 The impact on prevalence of dementia in the oldest age groups of differential mortality patterns

The comprehension and the quantitative description of the uni- and bidirectional ratchet phenomena remains one of the final aspects to be correctly modelized by the phenomenological approaches employing a model with internal scalar and tensorial variables. Toward this end, the first step consists in creating a set of experimental basis of unidirectional and bidirectional ratchet. This has been done for an austenitic stainless steel at 600°C. With the aid of alternating tension and torsion tests, the unidirectional ratchet can be quantified as a function of the maximal and average stresses. It is shown that the progressive strain only exists when the maximum stress is greater than 210 MPa and has a maximum for an average stress around 25 MPa and fixed maximum stress of 300 MPa. The tension-torsion ratchet is examined in a detailed fashion, and the influence of both primary (axial direction) and secondary (shear direction) loading parameters on the progressive strain rate is demonstrated. To be able to integrate, during the modelization, the nonradiality effects present in this type of loading, several cyclic out-of-phase tension-torsion tests are performed (Φ = 90°). At ambient temperature, several axial-internal pressure ratchet tests agree with the results obtained from tension-torsion tests. However, if ratchet tests were to be performed with two cyclic components (in or out of phase cyclic tension-torsion plus a static stress due to internal pressure), it can be shown that the rate of diametral ratchet is an increasing function of the phase angle between the cyclic components. This set of tests constitutes the experimental basis necessary for the modelization of the ratchet phenomena. It is then shown that is possible to reasonably describe this set of experimental results after taking into account a few modifications in the definition of the evolutionary laws for the tensorial variables of kinematic hardening. The nature of the modifications introduced in the kinematic hardening variables depends on the type of ratchet to be modelized. For uniaxial loadings the progeessive strain is governed by average stress effects, whereas for multiaxial loadings it is essentially governed by directional flow effects.

A new formulation of continuum damage mechanics (CDM) for composite materials

The damaged elastic properties of composite materials are modeled here in the framework of Continuum Damage Mechanics. The original feature of this model is that it combines different types of variables in describing the damage state of the material. It is now admitted that defects, depending on their reinforcement/matrix properties, can be oriented either by the microstructure, and can then be expressed by a family of scalar variables, or by the loading, in which case tensorial variables must be used. This new model can be used to describe all the essential phenomena observed in composites, namely: initial anisotropy; induced anisotropy, i.e. the effects of the damage on the elastic properties; defaults evolution laws; and closure effects in relation with residual strains. After presenting the general framework of the model, we propose a particular application to the case of ceramic matrix composites.

An Explicit Behavioural Damage Model for the Design of Components in Ceramic Matrix Composites

We suggest a behavioural damage model. The model describes the principal phenomena observed in Ceramic Matrix Composites, namely : initial anisotropy, induced anisotropy, in other words the effects of damage on the elastic behaviour, default evolution law, closure effects in relation with residual strains. We have developed a formulation which allows to use damage variables both related to microcracks oriented by the constituents and microcracks oriented by the loading. It is now admitted that defects can either be oriented by the microstructure, and therefore expressed by a family of scalar variables, or by the loading, which leads to use tensorial variables. We show an application to the case of ceramic matrix composites (SiC/SiC) which presents two damage modes and requires to use both scalar and tensorial variables. We emphasise on the explicit feature of this model which simplifies the parameter identification and the implementation in finite element codes.

One-point modelling of rapidly deformed homogeneous turbulence

One-point turbulence models are important tools for engineering analysis. A good model should have a viscoelastic character, predicting turbulent stresses proportional to the mean strain rate for slow deformations and stresses determined by the amount of strain for rapid distortions. Our goal is to build a one-point turbulence model with this character, and this requires a one-point model for rapid distortions. Here it is shown that the turbulent stresses introduced by Osborne Reynolds do not, by themselves, provide an adequate tensorial base for one-point modelling of rapidly distorted turbulence because they do not carry critical information about the turbulence structure. The deficiency is shown to be most pronounced in flows subjected to strong mean rotation. Additional one-point tensors that do carry the missing information are introduced, and the complexities of a model that would have an adequate tensorial base are assessed. A new type of one-point structure-based turbulence model that overcomes the basic deficiency of Reynolds-stress transport models, but without the excessive complexity of multiple tensor variables, is then described. The ideas behind the rapid distortion version of this new model are presented, along with results for some special cases.

Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on an austenitic stainless steel

Abstract The comprehension and the quantitative description of the uni- and bidirectional ratchet phenomena remains one of the final aspects to be correctly modelized by the phenomenological approaches employing a model with internal scalar and tensorial variables. Toward this end, the first step consists in creating a set of experimental basis of unidirectional and bidirectional ratchet. This has been done for an austenitic stainless steel at 600°C. With the aid of alternating tension and torsion tests, the unidirectional ratchet can be quantified as a function of the maximal and average stresses. It is shown that the progressive strain only exists when the maximum stress is greater than 210 MPa and has a maximum for an average stress around 25 MPa and fixed maximum stress of 300 MPa. The tension-torsion ratchet is examined in a detailed fashion, and the influence of both primary (axial direction) and secondary (shear direction) loading parameters on the progressive strain rate is demonstrated. To be able to integrate, during the modelization, the nonradiality effects present in this type of loading, several cyclic out-of-phase tension-torsion tests are performed ( Φ = 90°). At ambient temperature, several axial-internal pressure ratchet tests agree with the results obtained from tension-torsion tests. However, if ratchet tests were to be performed with two cyclic components (in or out of phase cyclic tension-torsion plus a static stress due to internal pressure), it can be shown that the rate of diametral ratchet is an increasing function of the phase angle between the cyclic components. This set of tests constitutes the experimental basis necessary for the modelization of the ratchet phenomena. It is then shown that is possible to reasonably describe this set of experimental results after taking into account a few modifications in the definition of the evolutionary laws for the tensorial variables of kinematic hardening. The nature of the modifications introduced in the kinematic hardening variables depends on the type of ratchet to be modelized. For uniaxial loadings the progeessive strain is governed by average stress effects, whereas for multiaxial loadings it is essentially governed by directional flow effects.

Comparaison de deux modèles de comportement viscoplastique à variables internes

The aim of this paper is about the comparison between two unified models with internal variables which have been established with 17-12 Mo SPH austenitic stainless steel experimental results. One is developed at the National Office of Aerospatial Research and Studies, the other, at the Applied Mechanical Laboratory of Besancon. The study proved their validity when applicated to a well known experimental loadings at high temperature, 500-600 o C. The two models report correctly the phenomena corresponding to classical loadings like monotonic traction, creep and cyclic hardening. However, there are important differences about transient creep and cyclic hardening under stress control. In the present state of the models, the progressive strain under uni or bidirectional loading (1D and 2D ratchet) is strongly overestimated. However, it is shown that it is possible to correctly describe the two types of progressive strain after taking into account a few modifications in the definition of the evolutionary laws for the tensorial variables of kinematical hardenings. Finally, the comparison does not allow to prefer one of the two models

Quantum symmetries from quantum phases, fermions from bosons, A [formula omitted]2 anomally and Galilean invariance

In the presence of a topologically nontrivial Wess-Zumino term, the wave functions are not functions on the configuration space Q of the system; instead they are functions on a U(1) bundle Q̂ over this configuration space. The space Q̂ is obtained by associating circles to points of Q in a suitable way. If a group G operates on Q and is a symmetry of the action, then it can happen that G does not operate on Q̂ and on wave functions. Instead, the “quantum mechanical” symmetry group G QM which does act on Q̂, and thereby wave functions, is in general a central extension of G; it may not even contain G as a subgroup. In this paper, we give an explicit and elementary construction of G QM. The construction is applied to the charge-monopole system and the Skyrme solitons, and used to explain why the states of these systems may be spinorial in nature even though the action contains only tensorial (integral spin) variables. The well known Z 2 anomaly of certain SU(2) gauge theories is also established using a modification of this construction. When a classical system with Galilei invariance is quantized, the quantum theoretic Galilei group G QM is different from the classical Galilei group G. Again the former is a central extension of the latter even though there is no Wess-Zumino term in this case. We present a novel explanation of this fact as well by constructing the lift of G from (galilean) spacetime Q to the appropriate U(1) bundle Q̂.

Irreducible invariants of fourth-order tensors

In solid mechanics of isotropic and anisotropic materials representing scalar-valued tensor functions or symmetric second-order tensor-valued tensor functions is of major concern, For instance, the plastic potential is scalar-valued, whereas constitutive equations are tensor-valued. In this paper scalar-valued functions have been represented. Anisotropic effects have been characterized by material tensors of rank two and four. For instance, the effect of damage or the behaviour of oriented solids have been characterized by second-order tensors, and with respect to more general cases inborn anisotropy has been described by using a rank four. In representing scalar-valued tensor functions, a set of irreducible invariant involving the above mentioned tensor variables has been constructed. The central problem is: to find an integrity basis for the argument tensors. Together with the invariants of the single argument tensors the system of simultaneous or joint invariants is considered. Such invariants are given by traces of outer products formed from different argument tensors considered. In finding irreducible invariants of a fourth-order tensor the characteristic polynomial for a fourth-order tensor is derived by the definition of the eigenvektor . Furthermore, Hamilton-Cayley's theorem is applied to a fourth-order tensor. Thus irreducible invariants of a fourth-order tensor can be expressed through sums of principal minors.

Large strain inelastic state variable theory

A precise format for the construction of three-dimensional, large strain inelastic state variable models through the continuum modeling of micromechanism is proposed. This general theory is based on a concept of state wherein each material element in its present configuration is regarded as a “black box” which converts possible future stimuli (deformations) into future response (stress and internal energy). Fundamental considerations suggest that state is determined by the instantaneous spatial distribution of the material bonds. The parameterization of the accessible states with N-tuples of tensor state variables is considered and the explicit assumptions which make it possible to reduce causal functional forms to response and incremental evolution equations are enumerated. The specific restrictions associated with material frame invariance and material symmetry are specified at each step of this development. The presentation concludes by considering some important specialized theories which contain virtually all successful models in the field of inelasticity.

A unified approach to constitutive equations of inelasticity based on tensor function representations

The applicability of the tensor function theory to the development of elaborate and rational constitutive equations of inelasticity is discussed to facilitate the description of the complicated inelastic response of structural components. After a brief review of the representation theory of scalar-valued as well as tensor-valued functions of a set of vector and tensor variables, previous results in this line of approach are first presented. Then, the formulation of some non-classical constitutive equations of inelasticity is discussed in some detail to elucidate the utility and the practical procedures of this theory; a yield criterion and a non-associated flow law of prestrained plastic materials, those of transversely isotropic materials, isotropic and anisotropic hardening theories of creep, and a constitutive equation of anisotropic creep damage in metals. It is shown that the present approach provides the general framework to materialize accurate constitutive equations of inelasticity.

Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case

The problem on mapping between two Lagrangian descriptions (using a commuting c-number spinor ψα or anticommuting pseudovector ξμ and pseudoscalar ξ5 variables) of the spin degrees of freedom of a color spinning massive particle interacting with background non-Abelian gauge field, is considered. A general analysis of the mapping between a pair of Majorana spinors (ψα,θα) (θα is some auxiliary anticommuting spinor) and a real anticommuting tensor aggregate (S,Vμ,Tμν⁎,Aμ,P), is presented. A complete system of bilinear relations between the tensor quantities, is obtained. The analysis we have given is used for the above problem of the equivalence of two different ways of describing the spin degrees of freedom of the relativistic particle. The mapping of the kinetic term (iħ/2)(θ¯θ)(ψ¯˙ψ−ψ¯ψ˙), the term (1/e)(θ¯θ)x˙μ(ψ¯γμψ) that provides a couple of the spinning variable ψ and the particle velocity x˙μ, and the interaction term ħ(θ¯θ)QaFμνa(ψ¯σμνψ) with an external non-Abelian gauge field, are considered in detail. In the former case a corresponding system of bilinear identities including both the tensor variables and their derivatives (S˙,V˙μ,T˙μν⁎,A˙μ,P˙), is defined. A detailed analysis of the local bosonic symmetry of the Lagrangian with the commuting spinor ψα, is carried out. A connection of this symmetry with the local SUSY transformation of the Lagrangian containing anticommuting pseudovector and pseudoscalar variables, is considered. The approach of obtaining a supersymmetric Lagrangian in terms of the even ψα and odd θα spinors, is offered.

Quasi-elastic light scattering study of the depolarized lines from supercooled benzyl benzoate

The depolarized scattered light intensity from supercooled benzyl benzoate has been analysed in the whole spectral range for a 90° scattering angle with a Fabry-Perot and an intensity correlation technique. In particular, a detailed analysis of the central part of the spectrum of both polarized and depolarized intensities, is carried out. Our results are consistent with the local order theory and allow the experimental determination of the relaxation times and coupling coefficients of the tensorial variables introduced in that theory.

Null-coordinate gravidynamics and the spin coefficients

By choosing a very natural tetrad it is shown that the null-energy density, found previously by two of us from a first-order action principle, is four times the modulus squared of the Newman-Penrose coefficient tau. Moreover, a computation is made in terms of the usual tensorial field variables of gravity of the more important three tetradic scalars of the conformal Weyl tensor. Consequently, this enables writing down the mass formulas in terms of the field variables. Also given is a null-energy formula in terms of line and surface integrals. All the results are given for fields which are asymptotically flat and asymptotically ''uniformly smooth.'' (AIP)

Classical effects from a factorized spinor representation of Maxwell's equations

It was demonstrated in earlier work that the vector representation of electromagnetic theory can be factorized into a pair of two-component spinor field equations (Sachs & Schwebel, 1962). The latter is a generalization of the usual formalism, in the sense that in addition to predicting all of the effects that are implied by the vector theory, it predicts additional observable effects that are out of the domain of prediction of the Maxwell formalism. The latter extra predictions were derived in previous publications (Sachs & Schwebel, 1961, 1963; Sachs, 1968a, b). In this paper, the spinor formalism is applied to effects that are expected to agree with the predictions of the standard formalism—the Coulomb force between point charges and the measured speed of a charged particle which moves in an electric potential. While there are no vector or tensor variables involved in this formalism, the results are found, as expected, to be in agreement with the conventional representation of electromagnetic theory. The analysis serves the role of demonstrating that in the appropriate limiting case, the factorized spinor formulation of electromagnetism does predict the explicit classical effects that are also predicted by Maxwell's field equations. The paper also presents a derivation of the general form of the solutions of the spinor field equations.

Thermodynamic Substate Variables for a Solid

A set of tensorial thermodynamic substate variables ${V}_{\mathrm{ij}}$ has been found such that under hydrostatic pressure the enthalpy $H$ of a solid, defined by $H\ensuremath{\equiv}U\ensuremath{-}{\ensuremath{\tau}}_{\mathrm{ij}}{V}_{\mathrm{ij}}$, where ${\ensuremath{\tau}}_{\mathrm{ij}}\ensuremath{\equiv}{(\frac{\ensuremath{\partial}U}{\ensuremath{\partial}{V}_{\mathrm{ij}}})}_{S}$, reduces to the conventional enthalpy $U+pV$. $U$ is the internal energy per unit mass, $p$ the pressure, $V$ the specific volume, and $S$ the entropy per unit mass. Previously used tensorial variables (e.g., the Lagrangian strain components) lead to a different enthalpy. The new variables are defined by ${V}_{\mathrm{ij}}=(\frac{1}{3\overline{\ensuremath{\rho}}}){{D}^{(\frac{3}{2})}}_{\mathrm{ij}}$, ${D}_{\mathrm{ij}}=(\frac{\ensuremath{\partial}{x}_{k}}{\ensuremath{\partial}{X}_{i}})(\frac{\ensuremath{\partial}{x}_{k}}{\ensuremath{\partial}{X}_{j}})$. Here the ${X}_{i}$ are the Cartesian coordinates of the particles of the body in some arbitrarily stressed reference configuration of density $\overline{\ensuremath{\rho}}$ and stresses ${\overline{T}}_{\mathrm{ij}}$, ${x}_{i}$ are the present coordinates, and ${{D}^{(\frac{3}{2})}}_{\mathrm{ij}}$ is a symbol for the $\mathrm{ij}$ element of the positive real 3/2 power of the tensor ${D}_{\mathrm{ij}}$. Under these definitions, the enthalpy of a solid of arbitrary symmetry has been proved to reduce to $U+pV$ whenever x and X both correspond to states in which the stress in hydrostatic pressure. As a special case, X may of course be an unstressed configuration. The above choice of variables is not unique. In fact, the enthalpy reduces to $U+pV$ if ${V}_{\mathrm{ij}}$ is any matrix function of ${D}_{\mathrm{ij}}$ whose determinant is proportional to the 3/2 power of the determinant of the matrix ${D}_{\mathrm{ij}}$. However, the above choice of ${V}_{\mathrm{ij}}$ has the additional desirable property that when evaluated at x = X, the thermodynamic tensions ${\ensuremath{\tau}}_{\mathrm{ij}}$ equal the stresses ${T}_{\mathrm{ij}}$. In the case of cubic crystals and isotropic media under hydrostatic pressure, the present ${V}_{\mathrm{ij}}$ and ${\ensuremath{\tau}}_{\mathrm{ij}}$ reduce to ${V}_{\mathrm{ij}}=\frac{1}{3}V{\ensuremath{\delta}}_{\mathrm{ij}}$, ${\ensuremath{\tau}}_{\mathrm{ij}}=\ensuremath{-}p{\ensuremath{\delta}}_{\mathrm{ij}}$.

Irreversible Processes in Liquids and the Density Matrix: Monatomic Molecules

A density matrixρ̄ is introduced to describe a macroscopically small subvolume of an infinite, pure, monatomic liquid. It is assumed that the effect of molecular interactions across the wall bounding the subvolume may be represented by a random force which is a functional of the density matrix itself. The form of the functional is determined by expanding bothρ̄ and the random force potential as linear sums of a set of scalar, vector, and tensor variables, for which the time rates of change are calculated from the Von Neumann equation. It is shown that the system should relax in two stages to the local equilibrium whose decay is described by the Navier-Stokes equations and that the inertial effects which are important in the initial stage are described by the parameters in the imaginary part ofρ̄. From the low-frequency limiting forms of the pressure, heat flux, and rate equations, general expressions for the transport coefficients are written down.

On invariant theory under restricted groups

A previous paper, ‘Invariant theory, tensors and group characters’, dealt with invariant theory under the full linear group. In this paper the methods are extended to restricted groups of transformation such as the orthogonal group. A knowledge of the characters of the group is shown to be an essential preliminary to any adequate study of invariants under the group. The characters of the orthogonal and symplectic groups, previously obtained by Schur and Weyl by transcendental methods involving group integration, are here obtained by methods entirely algebraic. Concerning transformation groups with a system of fundamental tensors, a fundamental theorem is proved that every concomitant may be obtained by multiplication and contraction of ground-form tensors, tensor variables, fundamental tensors and the alternating tensor. A characteristic analysis is developed, involving the operation denoted by ®, which enables the numbers and types of the concomitants of any given degree in any system of ground forms to be predicted. The determination of the actual concomitants is also discussed. Application is made for the orthogonal group to the quadratic, the ternary cubic and the quaternary quadratic complex; for the ternary symplectic group, to the quadratic, the linear complex and the quadratic complex. Various applications are also made for intransitive and imprimitive groups of transformation.