Average Tensor Research Articles

Transversely isotropic higher-order averaged structure tensors

For composites or biological tissues reinforced by statistically oriented fibres, a probability distribution function is often used to describe the orientation of the fibres. The overall effect of the fibres on the material response is accounted for by evaluating averaging integrals over all possible directions in space. The directional average of the structure tensor (tensor product of the unit vector describing the fibre direction by itself) is of high significance. Higher-order averaged structure tensors feature in several models and carry similarly important information. However, their evaluation has a quite high computational cost. This work proposes to introduce mathematical techniques to minimise the computational cost associated with the evaluation of higher-order averaged structure tensors, for the case of a transversely isotropic probability distribution of orientation. A component expression is first introduced, using which a general tensor expression is obtained, in terms of an orthonormal basis in which one of the vectors coincides with the axis of symmetry of transverse isotropy. Then, a higher-order transversely isotropic averaged structure tensor is written in an appropriate basis, constructed starting from the basis of the space of second-order transversely isotropic tensors, which is constituted by the structure tensor and its complement to the identity.

Stochastic representations of seismic anisotropy: transversely isotropic effective media models

We apply Jordan's self-consistent, second-order Born theory to compute the effective stiffness tensor for spatially stationary, stochastic models of 3-D elastic heterogeneity. The effects of local anisotropy can be separated from spatially extended geometric anisotropy by factoring the covariance of the moduli into a one-point variance tensor and a two-point correlation function. The latter is incorporated into the rescaled Kneer tensor, which is contracted against the one-point variance tensor to yield a second-order perturbation to the Voigt average. The theory can handle heterogeneity with orthotropic stochastic symmetry, but the calculations presented here are restricted to media with transversely isotropic (TI) statistics. We thoroughly investigate TI stochastic media that are locally isotropic. If the heterogeneity aspect ratio η is unity, the effective medium is isotropic, and the main effect of the scattering is to reduce the moduli. The two limiting regimes are a 2-D vertical stochastic bundle (η → 0), where the P and S anisotropy ratios are negative, and a 1-D horizontal stochastic laminate (η → ∞), where they are positive. The effective-medium equations for the latter yield the second-order approximation to Backus's exact solution, demonstrating the connection between Backus theory and self-consistent effective-media theory. Comparisons of the exact and second-order results for non-Gaussian laminates indicate that the approximation should be adequate for moduli heterogeneities less than about 30 per cent and thus valid for most seismological purposes. We apply the locally isotropic theory to data from the Los Angeles Basin to illustrate how it can be used to explain shallow seismic anisotropy. To assess the relative contributions of geometric and local anisotropy to the effective anisotropy, we consider a rotational model for stochastic anisotropic variability proposed by Jordan. In this model, the axis of a hexagonally symmetric stiffness tensor is locally rotated to align it with a 3-D stochastic Gaussian vector field that has TI statistics and depends on a horizontal-to-vertical orientation ratio ξ. If ξ = 1, the Voigt average and one-point variance tensor are isotropic, and the effective anisotropy depends only on η. In the vertical-alignment regime (ξ → 0), the medium is homogeneous. In the horizontal-alignment regime (ξ → ∞), the hexagonal symmetry axes are uniformly dispersed in the horizontal plane, which maximizes the scattering. In the rotational model, the effects of the local anisotropic alignment are at least an order of magnitude larger than the effects of geometric anisotropy. We apply the rotational model to Earth's inner core. Normal-mode and P-wave estimates of the average anisotropy in the outer half of the inner core can be well fit by a rotational model in which the local anisotropy is specified by ab initio estimate of hexagonal close-packed iron (hcp-Fe) by Martorell et al. The best match to the normal-mode average is given by an orientation ratio ξ = 0.52 with a conditional 95 per cent confidence interval of 0.32–0.85, corresponding to a moderate alignment of the hcp-Fe crystals along Earth's rotation axis.

Visco-electro-elastic models of fiber-distributed active tissues

We present a constitutive model for stochastically distributed fiber reinforced visco-active tissues, where the behavior of the reinforcement depends on the relative orientation of the electric field. Following our previous works, for the passive behaviors we adopt a second order approximation of the strain energy density associated to the parameters of the fiber distribution. Consistently, we also assume that the active behavior accounts for the stochastic distribution of the fibers. The ensuing mechanical quantities result to be dependent on two average structure tensors. We introduce an extended Helmholtz free energy density characterized by the inclusion of a directional active potential, dependent on a stochastic anisotropic permittivity tensor. The permittivity tensor is expanded in Taylor series up to the second order, allowing to obtain an approximated active potential with the same structure of the passive Helmholtz free energy density. In particular, the explicit expression of active stress and stiffness are dependent on the two average structure tensors that characterize the passive response. Anisotropy follows from the fiber distribution and inherits its stochastic nature through statistics parameters. The active fiber distributed model is extended here to viscous materials by including the contribution of a dual dissipation potential in the variational formulation of the constitutive updates. Additionally, we present a computational example of application of the electro-viscous-mechanical material model by simulating peristaltic contractions on a portion of human intestine.

Critical investigation of calculation methods for the elastic velocities in anisotropic ice polycrystals

Abstract. Crystallographic texture (or fabric) evolution with depth along ice cores can be evaluated using borehole sonic logging measurements. These measurements provide the velocities of elastic waves that depend on the ice polycrystal anisotropy, and they can further be related to the ice texture. To do so, elastic velocities need to be inverted from a modeling approach that relate elastic velocities to ice texture. So far, two different approaches can be found. A classical model is based on the effective medium theory; the velocities are derived from elastic wave propagation in a homogeneous medium characterized by an average elasticity tensor. Alternatively, a velocity averaging approach was used in the glaciology community that averages the velocities from a given population of single crystals with different orientations. In this paper, we show that the velocity averaging method is erroneous in the present context. This is demonstrated for the case of waves propagating along the clustering direction of a highly textured polycrystal, characterized by crystallographic c axes oriented along a single maximum (cluster). In this case, two different shear wave velocities are obtained while a unique velocity is theoretically expected. While making use of this velocity averaging method, reference work by Bennett (1968) does not end with such an unphysical result. We show that this is due to the use of erroneous expressions for the shear wave velocities in a single crystal, as the starting point of the averaging process. Because of the weak elastic anisotropy of ice single crystal, the inversion of the measured velocities requires accurate modeling approaches. We demonstrate here that the inversion method based on the effective medium theory provides physically based results and should therefore be favored.

Patterns of stress drop in earthquakes of the Northern Tien Shan

Source parameters of local earthquakes that occurred between 1998 and 2012 within the Kyrgyz seismological network (KNET) are used to study the patterns of stress drop in the Northern Tien Shan. The dynamic parameters of source radius r, the seismic moment M0, and the stress drop Ao are estimated for eighty seven M = 3.0-5.5 (K = 9.5-13.7) events, which make up 86% of earthquakes with this magnitude (99) for the period of observations. Earthquake mechanisms are determined either from P-wave arrivals (66 events) or by inversion (14 events). The distribution of earthquake mechanisms with the maximum (14 events) and minimum (14 events) stress drop shows that the amount of released stress correlates with slip geometry and revere-slip events contribute the most to the stress drop in the area. The spatial distribution of events with Ao > 0.9 MPa is compared with maps of kinematic parameters (Lode-Nadai coefficient or stress ratio, qo, and strain angle, o, based on average strain tensors according to the complete earthquake catalog for the Northern Tien Shan (1056 events for 1998-2012). The two sets of source parameters (r, M0, Ao and jio, o) correlate within the eastern Kyrgyz Range and the Karamoinok and Syndyk Ranges. © 2016, V.S. Sobolev IGM, Siberian Branch of the RAS. Published by Elsevier B.V. All rights reserved.

Magnetic-field-induced stepwise director reorientation and untwisting of a planar cholesteric structure with finite anchoring energy.

Within the continuum approach we study the equilibrium configurations of a cholesteric liquid crystal confined between two parallel plates, when a magnetic field is applied perpendicularly to the plates. We analyze the role of soft anchoring boundary conditions on magnetic-field-induced cholesteric-nematic transitions in a finite thickness cholesteric cell, treated to induce soft planar alignment. We study the stepwise behavior of cholesteric pitch as a function of the anchoring energy, the thickness of a layer, and the field strength. We analyze some kinds of soft anchoring potentials, including the case of degeneration of the easy axes. We show that the variation of the thickness or intrinsic pitch induces the the stepwise behavior of a pitch of the planar cholesteric structure, and the stepwise variations of the average tensor of diamagnetic susceptibility. The values of these jumps are determined by the anchoring energy. We find the values of critical parameters for the transitions between planar and confocal cholesteric states, and homeotropic nematic state. We show that the variation of the anchoring energy leads to change of the phase transition character; the conditions for hysteresis behavior are obtained. We show that for rather soft anchoring the confocal state is metastable, and the increase of a magnetic field leads to the direct transition between the planar cholesteric and homeotropic nematic phases. We also give a detailed derivation of the threshold and saturation properties of planar cholesteric to homeotropic nematic transition.

Combined near- and far-field high-energy diffraction microscopy dataset for Ti-7Al tensile specimen elastically loaded in situ

High-energy diffraction microscopy (HEDM) constitutes a suite of combined X-ray characterization methods, which hold the unique advantage of illuminating the microstructure and micromechanical state of a material during concurrent in situ mechanical deformation. The data generated from HEDM experiments provides a heretofore unrealized opportunity to validate meso-scale modeling techniques, such as crystal plasticity finite element modeling (CPFEM), by explicitly testing the accuracy of these models at the length scales where the models predict their response. Combining HEDM methods with in situ loading under known and controlled boundary conditions represents a significant challenge, inspiring the recent development of a new high-precision rotation and axial motion system for simultaneously rotating and axially loading a sample. In this paper, we describe the initial HEDM dataset collected using this hardware on an alpha-titanium alloy (Ti-7Al) under in situ tensile deformation at the Advanced Photon Source, Argonne National Laboratory. We present both near-field HEDM data that maps out the grain morphology and intragranular crystallographic orientations and far-field HEDM data that provides the grain centroid, grain average crystallographic orientation, and grain average elastic strain tensor for each grain. Finally, we provide a finite element mesh that can be utilized to simulate deformation in the volume of this Ti-7Al specimen. The dataset supporting this article is available in the National Institute of Standards and Technology (NIST) repository ( https://doi.org/hdl.handle.net/11256/599 ).

STATISTICAL DECOUPLING OF A LAGRANGIAN FLUID PARCEL IN NEWTONIAN COSMOLOGY

ABSTRACT The Lagrangian dynamics of a single fluid element within a self-gravitational matter field is intrinsically non-local due to the presence of the tidal force. This complicates the theoretical investigation of the nonlinear evolution of various cosmic objects, e.g., dark matter halos, in the context of Lagrangian fluid dynamics, since fluid parcels with given initial density and shape may evolve differently depending on their environments. In this paper, we provide a statistical solution that could decouple this environmental dependence. After deriving the evolution equation for the probability distribution of the matter field, our method produces a set of closed ordinary differential equations whose solution is uniquely determined by the initial condition of the fluid element. Mathematically, it corresponds to the projected characteristic curve of the transport equation of the density-weighted probability density function (ρPDF). Consequently it is guaranteed that the one-point ρPDF would be preserved by evolving these local, yet nonlinear, curves with the same set of initial data as the real system. Physically, these trajectories describe the mean evolution averaged over all environments by substituting the tidal tensor with its conditional average. For Gaussian distributed dynamical variables, this mean tidal tensor is simply proportional to the velocity shear tensor, and the dynamical system would recover the prediction of the Zel’dovich approximation (ZA) with the further assumption of the linearized continuity equation. For a weakly non-Gaussian field, the averaged tidal tensor could be expanded perturbatively as a function of all relevant dynamical variables whose coefficients are determined by the statistics of the field.

Coupled electro-mechanical models of fiber-distributed active tissues

We discuss a constitutive model for stochastically distributed fiber reinforced tissues, where the active behavior of the fibers depends on the relative orientation of the electric field. Unlike other popular approaches, based on numerical integration over the unit sphere, or on the use of second order structure tensors, for the passive behavior we adopt a second order approximation of the strain energy density of the distribution. The purely mechanical quantities result to be dependent on two (second and fourth order, respectively) averaged structure tensors. In line with the approximation used for the passive behavior, we model the active behavior accounting for the statistical fiber distribution. We extend the Helmholtz free energy density by introducing a directional active potential, dependent on a stochastic permittivity tensor associated to a particular direction, and approximate the total active potential through a second order Taylor expansion of the permittivity tensor. The approximation allows us to derive explicitly the active stress and the active constitutive tensors, which result to be dependent on the same two averaged structure tensors that characterize the passive response. Active anisotropy follows from the distribution of the fibers and inherits its stochastic parameters. Examples of passive and active behaviors predicted by the model in terms of response to biaxial testing are presented, and comparisons with passive experimental data are provided.

Micromechanics prediction of effective modulus for asphalt mastic considering inter-particle interaction

This paper presents the development and validation of a new micromechanical model, Ju–Chen (J–C) model, to predict the effective viscoelastic modulus of asphalt mastic by considering the inter-particle interaction. Based on the approximate solutions for two-particle interaction problem, the radial distribution function is integrated into the ensemble-volume averaged eigenstrain tensor so as to consider the inter-particle interaction, and the solution could be extended and simplified to predict the effective complex modulus of asphalt mastic with different filler volume fractions. It is found that the inter-particle interaction increases with the filler volume fraction and decreases with the test frequency, and the predictions agree well with the experiment data at low and moderate filler volume fractions, not exceeding 50%. Compared with two commonly used micromechanical models (M–T model and DSEM model), the J–C model relatively gives the best estimation of the effective modulus of mastics, and the applicability of J–C model could be further improved.

Propagation of elastic waves through textured polycrystals: application to ice.

The propagation of elastic waves in polycrystals is revisited, with an emphasis on configurations relevant to the study of ice. Randomly oriented hexagonal single crystals are considered with specific, non-uniform, probability distributions for their major axis. Three typical textures or fabrics (i.e. preferred grain orientations) are studied in detail: one cluster fabric and two girdle fabrics, as found in ice recovered from deep ice cores. After computing the averaged elasticity tensor for the considered textures, wave propagation is studied using a wave equation with elastic constants c=〈c〉+δc that are equal to an average plus deviations, presumed small, from that average. This allows for the use of the Voigt average in the wave equation, and velocities are obtained solving the appropriate Christoffel equation. The velocity for vertical propagation, as appropriate to interpret sonic logging measurements, is analysed in more details. Our formulae are shown to be accurate at the 0.5% level and they provide a rationale for previous empirical fits to wave propagation velocities with a quantitative agreement at the 0.07–0.7% level. We conclude that, within the formalism presented here, it is appropriate to use, with confidence, velocity measurements to characterize ice fabrics.

Fluctuations of an exposed π-helix involved in lipoxygenase substrate recognition.

The second helix in lipoxygenases adapts to permit substrate access to the active site, but details of this process are varied and poorly understood. We therefore examined the dynamics of helix 2 in solutions of spin-labeled soybean lipoxygenase-1 and spin relaxation at 60 K of the spin-labels by catalytic iron. Helix 2 in soybean lipoxygenase structures is surface-exposed and contains one turn of π-helix, centrally located. A site-directed spin-label scan of 18 of the 21 helix 2 residues, and electron paramagnetic resonance, showed that the π-helical segment became unusually mobile, on a nanosecond time scale, under conditions favoring substrate binding (pH 9 and lipid addition), while segments before and after had relatively unchanged dynamics. Backbone dynamics of residues in the π-helical segment appeared to be correlated, at pH 9. Samples also were frozen to examine the polarity and proticity of the local environments, the effect of the local environment on intrinsic relaxation, and dipolar relaxation by two symmetries of catalytic iron. The average hyperfine tensor component, Azz, of four π-helix residues decreased by 1.75 G, with an increase in pH from 7 to 9, while it remained unaffected for nearby buried residues. Power saturation data suggested the change in polarity specific to the π-helix altered the intrinsic relaxation rates. Different symmetries of iron contributed to distance-dependent magnetic relaxation. We interpret these data to mean that a π-helix in the second helix of plant lipoxygenases is highly dynamic and is the site where lipid chains penetrate to inner helices that outline the substrate pocket.

The use of strain tensor to estimate thoracic tumors deformation

Respiration-induced kinematics of thoracic tumors suggests a simple analogy with elasticity, where a strain tensor is used to characterize the volume of interests. The application of the biomechanical framework allows for the objective determination of tumor characteristics. Four-dimensional computed tomography provides the snapshots of the patient's anatomy at the end of inspiration and expiration. Image registration was used to obtain the displacement vector fields and deformation fields, which allows one for the determination of the strain tensor. Its departure from the identity matrix gauges the departure of the medium from rigidity. The tensorial characteristic of each GTV voxel was determined and averaged. To this end, the standard Euclidean matrix norm as well as the Log-Euclidean norm were employed. Tensorial anisotropy was gauged with the fractional anisotropy measure which is based on the normalized variance of the tensors eigenvalues. Anisotropy was also evaluated with the geodesic distance in the Log-Euclidean framework of a given strain tensor to its closest isotropic counterpart. The averaged strain tensor was determined for each of the 15 retrospectively analyzed thoracic GTVs. The amplitude of GTV motion varied from 0.64 to 4.21 with the average of 1.20 cm. The GTV size ranged from 5.16 to 149.99 cc with the average of 43.19 cc. The tensorial analysis shows that deformation is inconsiderable and that the tensorial anisotropy is small. The Log-Euclidean distance of averaged strain tensors from the identity matrix ranged from 0.06 to 0.31 with the average of 0.19. The Frobenius distance from the identity matrix is similar and ranged from 0.06 to 0.35 with the average of 0.21. Their fractional anisotropy ranged from 0.02 to 0.12 with the average of 0.07. Their geodesic anisotropy ranged from 0.03 to 0.16 with the average of 0.09. These values also indicate insignificant deformation. The tensorial framework allows for direct measurements of tissue deformation. It goes beyond the evaluation of deformation via comparison of shapes. It is an independent and objective determination of tissue properties. This methodology can be used to determine possible changes in lung properties due to radiation therapy and possible toxicities.

Short-term seismic crustal deformation of Iran

<p>In this paper the short-term seismic deformation of Iran is determined by the earthquake moment tensor summation. The study areas include the Alborz, Kopeh-Dagh, eastern Iran, Makran and Zagros orogenic belts. The spatial distribution and focal mechanisms of the earthquakes delineate the deformation zones. The mean directions of the P and T axes are determined by the equal area projection of the seismic moment tensors. The orientations of the P-axes are dominantly correlated with the NE crustal motion of Iran relative to Eurasia. The average strain rates are calculated in all of the regions. The maximum shear strain and dilatation rates are defined by the eigenvalues of the average strain rate tensors. The dilatation rate indicates that not only the dominant compression but also the subsidiary tension affects the Alborz and Makran orogenic belts. The velocity tensor components discriminate the vertical thickening and thinning of the crust in some regions of Iran. The seismic deformation rates, which are determined by the velocity tensors, are smaller than the geodetic deformation rates. In the high seismic deformation zones, such as the eastern Iran and Alborz, the geodetic deformation rate is comparable with the seismic deformation rate. Our results indicate that the NW Zagros and Kopeh-Dagh have the lowest seismic deformation rates. The seismic shortening rate increases from NW to SE in the Zagros orogenic belt. The seismic deformation orientations are different from the P-axes, probably due to the lateral translation. The maximum percentage of the seismic deformation in the study areas is related to the eastern Iran and the minimum one is related to the Makran orgenic belt. The average shape tensors indicate that the focal mechanisms in the Kopeh-Dagh have the highest internal similarity. The eastern Iran has the largest seismic moment rate, while the central Zagros has the lowest one.</p>

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.

Alterations in the optic radiations of very preterm children—Perinatal predictors and relationships with visual outcomes

Children born very preterm (VPT) are at risk for visual impairments, the main risk factors being retinopathy of prematurity and cerebral white matter injury, however these only partially account for visual impairments in VPT children. This study aimed to compare optic radiation microstructure and volume between VPT and term-born children, and to investigate associations between 1) perinatal variables and optic radiations; 2) optic radiations and visual function in VPT children. We hypothesized that optic radiation microstructure would be altered in VPT children, predicted by neonatal cerebral white matter abnormality and retinopathy of prematurity, and associated with visual impairments.142 VPT children and 32 controls underwent diffusion-weighted magnetic resonance imaging at 7 years of age. Optic radiations were delineated using constrained spherical deconvolution tractography. Tract volume and average diffusion tensor values for the whole optic radiations and three sub-regions were compared between the VPT and control groups, and correlated with perinatal variables and 7-year visual outcome data.Total tract volumes and average diffusion values were similar between VPT and control groups. On regional analysis of the optic radiation, mean and radial diffusivity were higher within the middle sub-regions in VPT compared with control children. Neonatal white matter abnormalities and retinopathy of prematurity were associated with optic radiation diffusion values. Lower fractional anisotropy in the anterior sub-regions was associated with poor visual acuity and increased likelihood of other visual defects.This study presents evidence for microstructural alterations in the optic radiations of VPT children, which are largely predicted by white matter abnormality or severe retinopathy of prematurity, and may partially explain the higher rate of visual impairments in VPT children.

Advances in X-ray diffraction contrast tomography: flexibility in the setup geometry and application to multiphase materials

Diffraction contrast tomography is a near-field diffraction-based imaging technique that provides high-resolution grain maps of polycrystalline materials simultaneously with the orientation and average elastic strain tensor components of the individual grains with an accuracy of a few times 10−4. Recent improvements that have been introduced into the data analysis are described. The ability to process data from arbitrary detector positions allows for optimization of the experimental setup for higher spatial or strain resolution, including high Bragg angles (0 < 2θ < 180°). The geometry refinement, grain indexing and strain analysis are based on Friedel pairs of diffraction spots and can handle thousands of grains in single- or multiphase materials. The grain reconstruction is performed with a simultaneous iterative reconstruction technique using three-dimensional oblique angle projections and GPU acceleration. The improvements are demonstrated with the following experimental examples: (1) uranium oxide mapped at high spatial resolution (300 nm voxel size); (2) combined grain mapping and section topography at high Bragg angles of an Al–Li alloy; (3) ferrite and austenite crystals in a dual-phase steel; (4) grain mapping and elastic strains of a commercially pure titanium sample containing 1755 grains.

Two-dimensional finite deformations evaluated from pre- and post-deformation markers: Application to balanced cross sections

A method is proposed for determining the 2D deformation gradient tensor that represents the deformations of pre- and post-deformation markers with arbitrary shapes. The deformation is not necessarily coaxial. The tensor is evaluated in a least-square sense. Therefore, the method can deal with heterogeneous deformations, and calculate their average tensor. The inverse method has the residuals that can be directly converted to the logarithmic strain needed to transform the calculated post-deformation shapes to observed ones. In addition, we propose the measure of heterogeneity for finite deformations. The method was applied to artificial and natural data from balanced cross sections.

Isotropy theorem for cosmological Yang-Mills theories

We consider homogeneous non-abelian vector fields with general potential terms in an expanding universe. We find a mechanical analogy with a system of N interacting particles (with N the dimension of the gauge group) moving in three dimensions under the action of a central potential. In the case of bounded and rapid evolution compared to the rate of expansion, we show by making use of a generalization of the virial theorem that for arbitrary potential and polarization pattern, the average energy-momentum tensor is always diagonal and isotropic despite the intrinsic anisotropic evolution of the vector field. We consider also the case in which a gauge-fixing term is introduced in the action and show that the average equation of state does not depend on such a term. Finally, we extend the results to arbitrary background geometries and show that the average energy-momentum tensor of a rapidly evolving Yang-Mills fields is always isotropic and has the perfect fluid form for any locally inertial observer.