Showing papers on "Cauchy stress tensor published in 1970"

The stress system in a suspension of force-free particles

Abstract: The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).

Stress tensor for a nematic liquid crystal

Abstract: Constitutive equations for nematic liquid crystals were first established by Ericksen and Leslie. They used a stress tensor with six viscosity coefficients αi (i = 1...6). It is shown in this paper that the Onsager reciprocal relations lead to the relation α2 + α 3 = α6 - α5 . Hence there are only five independent viscosity coefficients for a nematic liquid crystal.

Dynamics of Extended Bodies in General Relativity. II. Moments of the Charge-Current Vector

Abstract: The problem is considered of defining multipole moments for a tensor field given on a curved spacetime, with the aim of applying this to the energy-momentum tensor and charge-current vector of an extended body. Consequently, it is assumed that the support of the tensor field is bounded in spacelike directions. A definition is proposed for 'a set of multipole moments' of such a tensor field relative to an arbitrary bitensor propagator. This definition is not fully determinate, but any such set of moments completely determines the original tensor field. By imposing additional conditions on the moments in two different ways, two uniquely determined sets of moments are obtained for a vector field J$^{\alpha}$. The first set, the complete moments, always exists and agrees with moments defined less explicitly by Mathisson. If $ abla $$\_{\alpha}$J $^{\alpha}$ = 0, as is the case for the charge-current vector, these moments are interrelated by an infinite set of corresponding restrictions. The second set, the reduced moments, exists if and only if $ abla $$\_{\alpha}$J $^{\alpha}$ = 0. These avoid such an infinite set of interrelations, there being instead only one such restriction, the constancy of the total charge of the body. The energy-momentum tensor will be treated in a subsequent paper.

Lagrangian methods in plasma dynamics. I. General theory of the method of the averaged Lagrangian

Abstract: Whitham's averaged Lagrangian method is used to yield a relativistically covariant formalism for wave packets in a weakly inhomogeneous (and time dependent) medium. Provided the physics of the medium can be based on a Lagrangian density, a procedure of expansion and averaging is available which gives separate Lagrangians for the dynamics of the background and of the waves. The Euler—Lagrange equations then give immediately the equations of motion for the background, including non-linear reaction of the waves, and the dispersion relation, equations for ray tracing, conservation of wave action and non-linear coupling coefficients, for the waves. Many of these results can be interpreted in an illuminating way by considering the corresponding expansion of the canonical four-dimensional stress tensor.

Transition conditions: the yield condition

Abstract: The transformation of a continuum from a state A through a state T into a state B may be represented as a mapping. T is asymptotic in character, and hence is characterised by a contraint on the invariants of the strain or stress tensor of the field, giving rise to a functional relation between the otherwise independent invariants, I 1 , I 2 , I 3 . At present ad-hoc forms are used for this relation. It should be obtained from the condition that the modulus of transformation from A to B tends to become zero or infinite. As a consequence the corresponding reciprocal deformation ellipsoid degenerates into a cylinder, a plane or a point. For an isotropic medium the normalised yield condition is found to be; 3I′ 2 + 2(27I 3 + 3 2 I 1 I′ 2 − 3 I ) = 2, I′ 2 = 2I 2 1 − 6I 2 . This includes the classical yield conditions and also the Bauchinger effect. A similar condition is obtained for orthotropic bodies.

Self-Consistent Theory of Second-Order Elastic Constants with an Application to Noble-Gas Crystals

Abstract: Formal expressions for the first two strain derivatives of the first-order self-consistent free-energy density are rederived, and presented in a form suitable for numerical computation. The first strain derivative is the first-order self-consistent stress tensor and the second derivatives are the corresponding elastic constants. Because of the self-consistency condition, these elastic constants contain thermally averaged third-and fourth-order force constants. Special reference is made to an approximation first introduced by Horner in 1967. The expressions are applied to solid Ne, Ar, Kr, and Xe using a (12-6) Mie-Lennard-Jones potential. Calculations are carried out for the temperature range 0\ifmmode^\circ\else\textdegree\fi{}K to their respective melting points at zero pressure. The calculations are presented for the 0\ifmmode^\circ\else\textdegree\fi{}K volume, the experimental volume at zero pressure, and the volume produced by first-order self-consistent theory (SC). The volume effect is often large. However, at the same volume, the bulk moduli derived from ${F}_{\mathrm{ISC}}$ and ${F}_{\mathrm{SC}}$ differ by at most a few percent. This is taken to indicate the probable accuracy of our results.

Lattice-vibrational properties of chromium in the harmonic approximation on the basis of the tensor model

Abstract: Chromium lattice vibrational properties based on fourth-nearest-neighbor tensor force model, obtaining agreement with inelastic neutron diffraction data and elastic constants

Effect of gyro-viscosity on rayleigh--taylor instability of a plasma.

Abstract: The effect of finite ion-Larmor radius is considered on the Rayleigh-Taylor instability of a plasma. The macroscopic equations of motion are used where the finite ion-Larmor radius effects are incorporated in the stress tensor. The finite ion-Larmor radius stabilization of the interchange mode is demonstrated. For disturbances propagating along the magnetic field, it is found that the inclusion of gyro-viscosity has a stabilizing effect; it increases the critical wavenumber (compared to its value in the absence of gyro-viscosity) beyond which all modes are unstable.

Stress‐Tensor Commutators in Nonlinear Electrodynamics

Abstract: Equal‐time stress‐tensor commutators are shown to have a simple form for general systems of spin zero and one Electrodynamics with arbitrary self‐coupling is treated in detail Ordering and other problems of a purely quantum nature are not discussed

Spin--two mesons, the stress tensor, and a field--source identity. i.

Abstract: It is suggested that the hypothesis that the matrix elements of the stress tensor are dominated by the neutral isoscalar spin-two mesons may be expressed by a field-source identity. This states that, to the lowest order in the gravitational constant $\ensuremath{\kappa}$, the traceless part of the complete stress-energy tensor is proportional to a linear combination of the renormalized field operators of the neutral, isoscalar spin-two mesons. It is shown how this identity may be realized in a Lagrangian field theory, in a manner analogous to the work of Kroll, Lee, and zumino for the vector-meson dominance of the electromagnetic current. The conditions imposed by Lorentz covariance are discussed. The field-source identity determines the parts of the singular terms in the stress-tensor commutation relations which are the most singular in the coupling strength.

A new systematic approach to non-equilibrium thermodynamics

Abstract: The theory presented here is based on the following definitions and postulates: (i) A process is defined as a set of functions p(x, t), x(x, t), (x, t) representing the density, the motion and the (empirical) temperature. (ii) Constitutive equations are formulated for the stress tensor t, .the internal energy e and the heat flux q, such that a set t, , q belongs to every process. (iii) A thermodynamic process is defined as a process which is a solution of the equations of balance for the mass, the momentum and the energy. (iv) (Entropy Principle) In a body there exists a scalar extensive quantity which cannot decrease in any thermodynamic process, if its flux through the surface of vanishes, and whose density and flux i are given by constitutive relations. This quantity is called entropy. The ideas (i)—(iv) can be put to use to derive restrictive conditions on the constitutive relations for t, s, q, ij and , and these restrictions are valid for arbitrary non-equilibrium processes. In order to obtain restrictions on t, and q alone one needs further information about thermodynamic processes in a body and essentially the only processes about which useful information exists are uniform equilibrium processes. We know from experience that: (v) There exists a uniform equilibrium in a body on whose boundary the fluxes of mass and energy vanish. (vi) If such a body 9T consists of two subbodies £i (cx 1, 2) of different materials, there exists a uniform equilibrium in each of the subbodies and their temperatures are equal. (vii) If ET, HT and VT are internal energy, entropy and volume of T' and E, W and V the internal energies, entropies and volumes of , then the following relations hold: (oET /LE2 j) = and = \ / IITV2 \ / ill \ /HTV' \ /H2 From (i)—(vii) the existence of an absolute temperature can be proved for uniform equilibrium processes, but not more generally. The theory based on (i)—(vii) avoids any specific assumptions on the entropy and the entropy flux, in particular it avoids the customary assumptions that the flux and supply of entropy are equal to the quotients of flux and supply of internal energy and absolute temperature. If the principles (i)—(vii) are applied to a simple heat-conducting fluid, one may obtain, apart from all the familiar results for the material, a hyperbolic

Lagrangian Formulation of Statics of Shells

Abstract: A Lagrangian formulation of the equations of equilibrium of nonlinear thin elastic shell theory referred to nonorthogonal midsurface coordinates is presented. In analogy with the definition of the Lagrangian stress tensor of nonlinear continuum mechanics, stress and moment resultants are introduced and the equations of equilibrium with reference to the undeformed state derived. All results are “exact” within the Kirchhoff-Love hypothesis and are stated both in tensorial as well as in physical component form.

4TH-Order Tensor Invariants and Geometric Representation

Abstract: : First and second order invariants of 4th- order tensor are derived. Geometric representations analogous to Mohr's circle are presented to aid the visualization and operations of 4th-order tensors such as transformation, determination of principal direction and optimization procedures. The newly derived second order invariants may be considered as additional intrinsic material properties and may be used to simplify optimization of the physical properties of laminated composites.

Power Flow and Energy Storage in Piezoelectric Semiconductor Devices

Abstract: Traditionally the phenomenological constitutive relations for piezoelectric materials explicitly relate the electric displacement D/spl ovbr/ the electric intensity E/spl ovbr/, the stress tensor, and the strain tensor. This paper presents a new formulation for the theory of coupled wave interactions in a class of important hexagonal piezoelectric devices; here an equivalent dielectric description explicitly involving only D/spl ovbr/ and E/spl ovbr/ replaces (without approximation) the traditional formulation. The new formulation supplies the foundation for a new determination of power flow and energy storage on a basis broad enough to include the effects of diffusion and collisions on multivelocity multispecies carrier streams. The results, when specialized to a single-velocity single-species carrier stream, differ significantly with others recently proposed for those circumstances. The general results display a considerable degree of compactness and simplicity and are "electrically invariant" in that they hold for insulating, photoconducting, and semiconducting piezoelectric materials without any change in basic form.

On the rational foundations of the non-equilibrium thermodynamics of ordinary fluid mixtures

Abstract: Despite the arbitrary and ambiguous character of certain fundamental parts of the usual theory of non-equilibrium thermodynamics of mixtures, it has been successful in describing a variety of experimental situations. The goals of the research partially reported in this paper have been (1) to strengthen the foundations of the usual, practical theory and (2) to extend its usefulness to experimental situations hitherto regarded as too complicated to analyse. The principles and methods of rational mechanics are used to deduce balance equations which take full account of the kinetic energy of each component and of the partial stress tensor of each component Similar equations have been obtained previously by similar techniques. The new results reported here stem from a particular choice of independent variables; namely, for a v-component, non-reacting mixture of fluids, the independent variables are the v component densities and their gradients, the temperature and its gradient, the v — 1 independent diffusive velocities, and the v symmetric and v — 1 antisymmetric parts of the gradients of component velocity. For the special but very important case of an ordinary fluid mixture, whose constitutive relations are linear in the diffusion velocities and the independent gradients, the source term in the entropy balance equation is a bi-linear form in the diffusion velocities and the independent gradients. This bi-linear form for the entropy source term, which has eluded previous workers, has two immediate consequences: (1) Coupled with the second law, it leads to unambiguous conditions for equilibrium; namely, for equilibrium with respect to all processes in a v-component, non-reacting fluid mixture, it is necessary and sufficient that the following shall all vanish: the temperature gradient, the diffusion velocities, and the symmetric and antisymmetric parts of the gradients of component velocity. (2) It permits full comparison with the practical theory, wherein the entropy source plays a central role. It also opens new experimental avenues since it contains terms which arise from component kinetic energy and component stress tensor.

Generalization of Hydrodynamics to Include Single Particle Modes and Fluctuations

Abstract: Two exact hierarchies of hydrodynamic equations which follow from Newton's equations for systems of identical particles interacting with a spherically symmetric potential are presented. The first hierarchy, essentially given by Born and Green in 1947, is shown to be capable of predicting hydrodynamic fluctuations and therefore has important applications near the critical point. Explicit microscopic forms of the higher level stress tensors and heat fluxes are given. Detailed macroscopic approximations for these quantities are obtained using the isotropic symmetry of the fluid, reduction conditions which relate these quantities to the ordinary stress tensor and heat flux, and the known equilibrium limit of the hierarchy (the Yvon‐Born‐Green equation). The second hierarchy is a generalization of Born and Green's hierarchy to include time correlations. At equilibrium it yields an equation for the Van Hove correlation function, G(R, t), which contains both single‐particle and collective modes and is the analog...

Inelastic buckling of axisymmetric shells.

Abstract: : An incremental variational method is presented for the determination of the inelastic load-deformation relationship, as well as the buckling or collapse load for a shell of revolution which is made of a work hardening material and subjected to axisymmetric loadings. A variational principle involving the Kirchhoff stress tensor, the Green strain tensor, and their rates is employed. The isothermal stress-strain relationship based on a modified J2 incremental theory of plasticity is used. A seven sheet sandwich shell idealization and the finite difference method of variational calculus are utilized in a numerical procedure in order to determine discrete velocities and a subsequent deformation process. The buckling loads of a number of elastic and inelastic cylindrical and truncated conical shells as well as inelastic spherical shell caps are determined. The results of the present analysis, obtained numerically by a Univac 1107 digital computer, compare favorably with available experimental data. (Author)

A Contact Problem In Poroelasticity

Abstract: The paper examines the axisymmetric problem of the indentation of poroelastic layer which rests in contact with a rough rigid and impermeable base. The indenting punch has a circular planform and the smooth contact is assumed to be either permeable or impermeable. The paper develops the integral equations governing the problem for the generalized case where the the pore-fluid is considered to be compressible. Numerical results are presented to illustrated the influence of layer thickness, drainage conditions, and the compressibility characteristics of the pore fluid on the degree of consolidation settlement of the indenting punch. INTRODUCTION The classical theory of poroelasticity which is applicable to the study of the mechanics of deformable fluid saturated porous medium was first developed by Biot [5]. This theory has been widely applied to the study of various loading and contact problems associated with halfspace and infinite space regions. A majority of these discussions have focussed on the evaluation of the response of the poroelastic medium which is saturated with an incompressible pore fluid. In this paper, attention is focussed on the problem of the indentation of a thin poroelastic layer by a smooth rigid circular punch. The term "thin" is intended to signify a layer thickness which is smaller than the diameter of the indenting circular punch. The punch is in smooth contact with the poroelastic layer which exhibits either permeable or impermeable pore pressure boundary conditions over the entire surface. The layer is underlain by an impermeable rigid base 1 Currently at The Infrastructure Laboratory, Institute for Research in Construction, National Research Council of Canada, Ottawa, Ontario, Canada K1A OR6 Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 286 Contact Mechanics and the displacement contact conditions are considered to be adhesive (Figure 1). The paper summarizes the integral equations governing the mixed boundary value problems associated with boundary variables in the Laplace transform domain. The paper illustrates the influences of the relative thickness of the consolidating layer and the Poisson's ratios of the pore fluid on the degree of the consolidation of the poroelastic layer. A permeable or impermeable surface A rigid circular smooth indentor t A poroelastic layer saturated with a compressible pore fluid xA rough rigid and impermeable base Figure 1. An axisymmetric contact problem in poroelasticity GOVERNING EQUATIONS In the ensuing we shall present a brief account of the governing equations referred to a Cartesian tensor notation. The constitutive equations governing the quasi-static response of a poroelastic medium, which consists of an isotropic poroelastic soil skeleton saturated with a compressible pore fluid takes the forms (1) 3(1 2r/,) where cr^ is the total stress tensor, p is the pore fluid pressure, Cij are the soil skeleton strains defined by £.. = !(„..+„,.) (2) where Ui are the corresponding displacement components. In the absence of body forces, the quasi-static equations of equilibrium take the forms °ij,i = 0 (3) The equations governing quasi-static fluid flow are defined by Darcy's law which take the form %, = -/W (4) Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 Contact Mechanics 287 and the continuity equation associated with quasi-static fluid flow is The above governing equations are characterized by the five basic material parameters which are represented by the drained and undrained Poisson's ratios v and z/% respectively, the shear modulus //, Skempton's [7] pore pressure coefficient B, and the K (=fc/7 , where k is the coefficient of permeability and 7%, is the unit weight of pore fluid). From the consideration of the positive definiteness of a strain energy potential, it can be shown that the material parameters should satisfy the following thermodynamic constraints: /j > 0, 0 0 (see. e.g., Rice and Cleary [6]). Avoiding details, it can be shown (Yue [3]) that the following sets of solution representations exist for the field variables in a linear, isotropic, poroelastic medium of layer extent saturated with a compressible pore fluid. In either the temporal domain or the Laplace transform domain and in the cylindrical coordinate systems (r, #, z) and (p, y, z), we have % roc r2ir } u(r, 0,z,t) = — I I -TlawK 27T Jo JQ p I roc /-27T ,(r, 0, z, f ) = — / / n, rA 2?r Vo Vo

Flow stability of a Grad-model liquid layer on an inclined plane

Abstract: A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force. It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector ω, the internal moment stresses are negligibly small, and in the general case the force stress tensor τkj is asymmetric. The model is characterized by the usual Newtonian viscosity η, the Newtonian “rolling” viscosity ηr, and the relaxation time τ=ρ J/4 ηr, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, ρ is the density. It is assumed that the medium is incompressible, the coefficients η, ηr, J are constant [2]. The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid. The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].

Computer Modeling Of Unsteady ProcessingOf Semi-dilute And Dilute Suspensions

Abstract: An alternative approach for solving unsteady flows of fiber suspensions and predicting the implied fiber orientation is discussed. The approach uses the traditional orientation tensor representation for describing suspensions flows but approximates them using finite deformation tensors. Using this approach, the stress tensor given by the theory used by Lipscomb et. al. [I] for dilute suspensions and Dinh-Armstrong theory for large aspect ratio fibers is expressed in terms of the Finger and Cauchy tensors using the Currie approximation. A semiimplicit finite difference scheme is then used to express the Finger and Cauchy tensors in terms of the kinematics of the flow. The proposed method has the advantage that it does not require any closure approximation as in the case of the traditional approach. Moreover, since the stress tensor is expressed in terms of the kinematics of the flow explicit solution for the solution for the components of the orientation tensor are not required. This reduces the computational time and storage requirements.

A FBEM-formulation For Hyperelastic AndElastoplastic Problems At Finite Strains

Abstract: A novel formulation of a Field-Boundary-Element-Method (FBEM) for hyperelasticity and elastoplasticity at finite strains is presented. The proposed formulation uses a Total-Lagrange scheme. Therefore the system matrices have to be generated only once. The constitutive model used is based on the concept of an intermediate configuration. This enables to split the constitutive equations into a time-independent hyperelastic relation relative to the intermediate configuration and into more complex evolution equations for the intermediate configuration and for further internal variables. For the derivation of the boundary integral equation only the time-independent hyperelastic relation for the stress tensor is needed. Consequently the basic boundary integral equation itself is time-independent and does not depend on the explicit structure of the evolution equations. The proposed formulation finally provides a nonlinear set of equations with identical structure, for both boundary and internal nodes, for which a consistent linearisation can be derived. The basic unknowns are the displacement gradients.