Deviatoric Tensor Research Articles

Three-Dimensional Characterization and Simulation of Anisotropic Soil Fabric

A procedure for constructing resin impregnated sand specimens for fabric (microstructure) analysis is presented. The impregnation device provides excellent control of the impregnation rate and confinement for different types of soils. Such control is necessary to minimize disturbance to the soil fabric during impregnation. The proposed technique also allows monitoring of the uniformity of impregnation throughout the process. Image analysis techniques are used to examine the variation of porosity at the microscopic and macroscopic levels within specimens. The analysis shows that about twice the deviation in local porosity from average porosity is obtained when microscopic measurements are used, as compared with the macroscopic measurements. Soil fabric anisotropy is characterized using two directional functions. Each of these functions has two components: an average value that describes the isotropic condition and a deviatoric tensor that represents the deviation from the average. They are shown to capture the relative distribution of air voids and their connectivity in different directions. The functions are also used to simulate the three-dimensional anisotropic fabric of soils in a computer.

Irreducible decompositions of physical tensors of high orders

In this paper we use the term 'physical tensor' to stand for a tensor that belongs to a tensor subspace. Based on the relationship among the characters of rotation representation, some techniques are developed in order to give the numbers of independent deviatoric tensors contained in the irreducible decompositions of physical tensors, even prior to the constructions of the irreducible decompositions. A number of examples are shown and many of them are new results.

The Tertiary dynamics of the northern Eastern Alps (Austria): changing palaeostresses in a collisional plate boundary

Six Tertiary deviatoric palaeostress tensor groups from 165 stations in the Calcareous Alps describe the upper crustal dynamics of the leading edge of the Adriatic plate during protracted continental collision with the European lower plate. Palaeostress changes are correlated to the kinematic evolution of the plate boundary and to independently derived plate kinematic data of the Alpine-Carpathian-Pannonian area. Each palaeostress direction is defined by 32 to 89 individual tensors which constrain: (1) Late Eocene to ?Oligocene NW-directed compression during thrusting and dextral shearing on WNW-striking faults; NW-directed compression along the northern margin of the Adriatic plate is correlated to its N-directed translation combined with a counterclockwise rotation. (2a) ?Late Eocene to Early Miocene N-directed compression during N-directed thrusting of Penninic and Helvetic units and the Molasse; (2b) Early to Middle Miocene N-directed strike-slip compression during the onset of eastward lateral extrusion; the change from NW- to N-directed compression is interpreted to result from strain partitioning at the Periadriatic fault which has formed the new Adriatic plate boundary since the Oligocene. Dextral shearing on this fault accommodated continued anticlockwise rotation and decoupled the northern Eastern Alps from the rotational component. N-directed compression in the Calcareous Alps paralleled the N-directed translation of the Adriatic plate. (3) Middle Miocene NE-directed compression of thrust- and strike-slip type during lateral extrusion; NE-directed compression in the Calcareous Alps resulted from the drag of the eastward extruding central Eastern Alps. Sinistral shear stress was transmitted across the ENE-trending Salzach-Ennstal fault. NE-oriented σ 1-axes include 45° with the E-W-striking zone of distributed sinistral shear in the Calcareous Alps. (4) Middle Miocene E-directed extension associated with orogen-parallel normal faulting of the central Eastern Alps; E-directed extension paralleled the direction of mass transfer towards the Pannonian Basin during lateral extrusion. Normal faulting was induced by reduced lateral confinement due to removal of material east of the Alps. On the large scale, eastward motion was enabled by the E-directed retreat of the Eastern Carpathian subduction zone. (5) Late Miocene E-directed compression; orogen parallel compression obstructed lateral extrusion. The stress change from extension to E-directed compression was caused by the end of subduction retreat in the Eastern Carpathians due to dynamic changes in the subduction zone. (6) N-directed extension; Late- to post-Miocene extension paralleled recent topographic slopes in the foothill of the Eastern Alps. Palaeostresses changes correspond to distinct phases in the kinematic evolution of the northern Eastern Alps and to the systematic formation and reactivation of major fault zones. WNW-striking faults changed deformation styles from dextral shear to dextral transpression, reverse faulting and finally to sinistral shear. The Early to Middle Miocene Salzach-Ennstal fault which formed the northern boundary of the extruded central Eastern Alps changed from sinistral transpression to sinistral shear, sinistral transtension and finally to dextral shear.

The structure of the linear anisotropic elastic symmetries

An insightful, structurally appealing and potentially utilitarian formulation of the anisotropic form of the linear Hooke's law due to Lord Kelvin was independently rediscovered by R ychlewski (1984, Prikl. Mat. Mekh. 48, 303) and M ehrabadi and C owin (1990, Q. J. Mech. appl. Math. 43, 14). The eigenvectors of the three-dimensional fourth-rank anisotropic elasticity tensor, considered as a second-rank tensor in six-dimensional space, are called eigentensors when projected back into three-dimensional space. The maximum number of eigentensors for any elastic symmetry is therefore six. The concept of an eigentensor was introduced by K elvin (1856, Phil. Trans. R. Soc. 166, 481) who called eigentensors “the principal types of stress or of strain”. Kelvin determined the eigentensors for many elastic symmetries and gave a concise summary of his results in the 9th edition of the Encyclopaedia Britannica (1878). The eigentensors for a linear isotropic elastic material are familiar. They are the deviatoric second-rank tensor and a tensor proportional to the unit tensor, the spherical, hydrostatic or dilatational part of the tensor. M ehrabadi and C owin (1990, Q. J. Mech. appl. Math. 43, 14) give explicit forms of the eigentensors for all of the linear elastic symmetries except monoclinic and triclinic symmetry. We discuss two approaches for the determination of eigentensors and illustrate these approaches by partially determining the eigentensors for monoclinic symmetry. With the nature of the eigentensors for monoclinic symmetry known, a rather complete table of the structural properties of all linear elastic symmetries can be constructed. The purpose of this communication is to give the most specifically detailed presentation of the eigenvalues and eigentensors of the Kelvin formulation to date.

An adaptive mesh refinement technique for the analysis of shear bands in plane strain compression of a thermoviscoplastic solid

We have developed an adaptive mesh refinement technique that generates elements such that the integral of the second invariant of the deviatoric strain-rate tensor over an element is nearly the same for all elements in the mesh. It is shown that the finite element meshes so generated are effective in resolving shear bands, which are narrow regions of intense plastic deformation that form in high strain-rate deformation of thermally softening viscoplastic materials. Here we assume that the body is deformed in plane strain compression at a nominal strain-rate of 5000 sec-1, and model a material defect by introducing a temperature perturbation at the center of the block.

Evolution of stress and strain during the consolidation of a fluid-saturated porous elastic sphere

The evolution of stress and strain during the consolidation of a fluid-saturated porous elastic sphere subjected to a uniform normal surface traction is investigated. The solution of Cryer [Q. J. Mech. Appl. Math. 16, 401 (1963)] for the pore fluid pressure is extended by deriving and fully analyzing new analytical solutions for the strain tensor of the elastic skeleton, the total stress tensor, the trace-free stress deviator tensor, and the effective stress tensor. Asymptotic expansions for small values of time of the stress and strain components, which determine the behavior of these components during the initial stages of the consolidation, are derived. Computer-generated graphs of the exact analytical solutions are presented, which provide insights regarding the redistribution of stress and strain throughout the consolidation. As fluid is expelled from the outermost layers of the sphere, fluid discharge builds up quickly to a peak rate and then subsides slowly as successively deeper layers are drained. During this initial consolidation, the stresses, strains, deviator stresses, and effective stresses within the outermost layers all change very rapidly. The nature of the Mandel–Cryer effect (pore fluid pressure increase within the sphere during initial fluid expulsion) is illuminated. Two other quantities, the transverse component of the total stress tensor and the total mean normal pressure, also show changes of opposite sign in the near surface and deeper regions, respectively: diminution in the outermost layers and simultaneous increase in the deeper portions. All of these effects are derived from Biot’s theory of poroelasticity, in which the dilatation satisfies the diffusion equation.

The Proportional Anisotropic Elastic Invariants

There are two proportional invariants for a linear isotropic material, the hydrostatic invariant, and the deviatoric invariant. The former is proportional to the trace of the tensor and the latter is proportional to the trace of the square of the associated deviatoric tensor. The hydrostatic stress and strain and the von Mises stress and strain are directly related to the hydrostatic and deviatoric proportional invariants, respectively, for an isotropic, linear elastic material. For each anisotropic linear elastic material symmetry there are up to six proportional invariants. In this paper we illustrate the six proportional invariants of an orthotropic elastic material using the elastic constants for spruce as the numerical example. The proportional elastic invariants play a role in anisotropic linear elasticity similar to the roles played by the hydrostatic stress and strain and the von Mises stress and strain in isotropic elasticity. They are the unique parameters whose contours represent both the stress and the strain distributions. They also have potential for representing failure or fracture criteria.

EIGENTENSORS OF LINEAR ANISOTROPIC ELASTIC MATERIALS

There are two eigentensors for a linear isotropic elastic material; one is the deviatoric second-rank tensor, and the other is a second-rank tensor proportional to the unit tensor and often called the spherical or hydrostatic part of the tensor. The eigentensors of isotropic elasticity have many properties of physical and mathe-matical significance. In this paper a method of construction of the eigentensors for the anisotropic elastic-material symmetries is presented and applied to determine the eigentensors of each anisotropic elastic symmetry. The eigentensors for the anisotropic symmetries are shown to have the same important properties as those possessed by the eigentensors of isotropic elasticity.

Constitutive Equations for Concrete in Failure State

For concrete, two simple three‐dimensional failure conditions depending on the first invariant of the stress tensor as well as on the second and third invariant of the stress deviator tensor with two and four material parameters and a nonassociated flow rule are proposed in this paper. First, the basic equations like stress deviator tensor and invariants of the stress tensor and a stress deviator tensor are introduced. Then the proposed failure conditions are presented in a geometrically interpretable form. In the next step, the material parameters are determined by fitting the failure conditions to several test results. Finally, a nonassociated flow rule is introduced. An example illustrates the usefulness of the proposed constitutive equations in the failure state.

Description of flow of sand using state parameters

The concept of “Critical State” as originally envisaged [1] is extended to include the influence of the third invariant of the stress deviation tensor. The concept of “Steady State of Flow” as proposed by Poulos and Castro [2,3] is also examined in some detail and it is concluded that the two concepts are substantially equivalent. It is pointed out that both nomenclatures are unfortunate: a more appropriate terminology is the “Ultimate State” as all samples when distorted far enough would ultimately tend to approach this state. The framework for a unified approach to the formulation of the constitutive law for cohesionless granular media utilizing the state parameters is discussed. Within this framework a number of observations such as the curvature of the yield surface and the state boundary surface and the laws of similarity under the so called 1-g testing conditions [4] are accounted for.

Planar deformation of cohesionless granular media

A theory for planar deformation of cohesionless granular media is presented. Although this study follows closely the methodology used by Mandel (1947), Spencer (1964) and Mehrabadi and Cowin (1978) it differs in many respects from the contents of these papers. Both the angle of dilatancy and the orientation of the family of the velocity characteristics (slip lines) with respect to the direction of the principal axes of stress are assumed variant and dependent on the loading history of the element. The velocity and stress characteristics are assumed not to be coincidental (necessarily) and it is reasoned that deformation may proceed along one family of the slip lines only. A hypothesis regarding the symmetry of orientation of slip lines with respect to the principal stress directions and a second one regarding the minimal value for a so called ‘energy correction factor’ involving the ratio of the first invariant of the strain rate tensor to the second invariant strain rate deviation tensor is stated. It is shown that the adaptation of these hypotheses leads directly to St. Venant's principle: the coaxiality of the stress and the strain rate tensors. It should be stressed that coaxiality conclusion reached in this paper is in variance with the conclusions reached by the above authors (see also Spencer, 1981) and the conclusions of Christoffersen et al. (1981) who approached the problem more fundamentally using micromechanical modeling.

Flow law for polycrystalline ice in glaciers: Comparison of theoretical predictions, laboratory data, and field measurements

Theoretical considerations, laboratory experiments, and limited field data support a value of 3 for the exponent n in the commonly used empirical flow law ϵ = (τ/B)n relating stress and strain rate in polycrystalline ice. If this value is accepted, the viscosity parameter B can be determined for a wide variety of experiments. In a plot of log B against reciprocal temperature, points scatter about a line defined by an empirical equation of the form B = B0 exp {(T0/T) ‐ [C/(Tr ‐ T)k]}, where T is the temperature in kelvins and B0, C, T0, Tr, and k are empirically determined constants. For laboratory data the scatter is equivalent to approximately a factor of 5 variation in strain rate for a given stress and temperature. The cause of this variation is unclear, but because results from any single laboratory are generally internally consistent, sample preparation procedures should be studied. Field experiments yield values of B that are systematically higher than laboratory results. Thus natural ice appears stronger than laboratory ice, despite the coarser texture and the presence of anisotropic fabrics in the natural ice, both of which should tend to soften it. In addition, natural ice in glaciers appears stronger than natural ice deformed in the laboratory. These observations suggest either that the stress is systematically overestimated in field studies or that a flow law based on the von Mises yield criterion (or the second invariant of the stress deviator tensor) does not provide an adequate description of the deformation of ice in multiaxial stress fields.

A Statistical Construction of the Reynolds-Stress Closure Model in Inhomogeneous Turbulence

The paper proposes a statistical construction of the Reynolds-stress closure model which is frequently used, as well as the large-eddy simulation, in the investigation of real turbulence. Production-, redistribution- and diffusion terms in the Reynolds-stress transport equation are evaluated from a general theory for the inhomogeneous turbulence with arbitrary mean flows (A. Yoshizawa: J. Phys. Sac. Jpn. 46 (1979) 669 and 47 (1979) 1665). Using the result, transport equations are derived for the turbulent energy and the squared virtual deviatoric tensor, where the virtual deviatoric tensor is defined by the product of the eddy viscosity and the rate-of-strain tensor of mean velocity. These two equations are combined with the equation for mean flow to lead to a Reynolds-stress closure medel with no empirical information.

Experimental examination of the hypothesis of local determinability in the plastic deformation of metals

Ilyushin's general plasticity theory expressed in a vector space corresponding to the deviatoric tensor space is available in formulating precise stress-strain relations of inelastic deformations by incorporating the deformation history of metals under complex loadings. In this case, the hypothesis of local determinability proposed by L ensky (1960) is useful for determining the explicit form of the stress-strain relation. In the present paper, the hypothesis is discussed by using not only data obtained by an experiment in which the history effect on the succeeding deformation is clearly estimated, but also data described in previous work. As the result, the hypothesis of local determinability has been confirmed to hold with high accuracy in every case.

Free Oscillations of the Laterally Inhomogeneous Earth: Quasi-degenerate Multiplet Coupling

Summary The scheme leading to the computation of the normal mode eigen-frequencies of the laterally inhomogeneous Earth is derived. Rayleigh's variational principle is used to calculate the first-order corrections to the eigenfrequencies of a spherically symmetric, non-rotating and elastic Earth model due to the lateral variations in the real Earth. The Wigner-Eckart theorem is used, whenever applicable, to simplify many complicated angular integrals involving the strain deviator tensor to more basic and manageable integrals involving three spherical harmonics. In particular, near-resonance multiplet coupling for spheroidal-spheroidal, spheroidal-toroidal and toroidal-toroidal multiplets is presented along with the uncoupled case of a single arbitrary spheroidal or toroidal multiplet.

Tresca-type plastic materials in the theory of hypoelasticity II. Optical constitutive equations and birefringence in simple shear

Behavior of a Tresca type plastic dielectric is investigated theoretically from a continuum mechanical point of view. The optical constitutive equations are defined as special cases of a hypo-elastic dielectric of grade two. The singularity condition of the constitutive equations satisfies the Tresca yield criterion. The index deviator tensor is proportional to the stress deviator tensor and, then, the birefringence and the extinction angle are expressed by the stress deviator. Their numerical variations with the angle of shear in simple shear deformation are shown.

Yielding and Flow of Sand in Triaxial Compression: Part I

A theoretical and experimental study of the nature of deformation of a sand sample when tested in the triaxial apparatus is presented. The medium is shown to be elastic-strain hardening plastic but does not conform to certain rules usually adopted in the classical theory of plasticity. Experimental verification of an earlier suggestion by Poorooshasb (1961) leads to a proof of the existence of a potential function, known as the plastic potential, of the form Ψ(σ, e) the parameters σ representing the stress and e the voids ratio. The plastic potential curves defined by Ψ = constant, e = constant, trace a family of geometrically similar curves when plotted in a stress space. The yield loci, on the other hand, are found to be independent of voids ratio and are only functions of the ratio of the second invariant of the stress deviation tensor to the first invariant of the stress tensor. It is noted, therefore, that the plastic potential and yield surface are not coincidental. This defies the normality condition and in this respect the medium's behaviour appears to deviate from that of a classical plastic body. Although the scope of the study is limited, being applicable only to stress conditions provided in a triaxial compression test, extensions are made to incorporate stress and strain in their more general form. This is done to stimulate further research on the subject, although a fair amount of evidence is available to support, qualitatively at least, the validity of the propositions made. The paper is concluded by presenting the flow rule for the medium and discussing the application of the theory presented to the solution of equilibrium and eigenvalue problems involving stress and deformation.

Remarks Concerning the Paper “Conditions for the Derivation of the Stress Deviator Tensor”

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Extension of Remarks on the Derivation of the Stress Deviator Tensor

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Conditions for the Derivation of the Stress Deviator Tensor

The decomposition of the stress tensor into the spherical stress tensor and the stress deviator found in theories of plasticity is shown to depend upon the postulate that the stress tensor is the tensor sum of two physically independent stress constituents. The decomposition is derived by two different methods, the first method based on energy arguments, and the second method based on cause and effect arguments.