Showing papers on "Isotropy published in 2010"

Mean square displacement analysis of single-particle trajectories with localization error: Brownian motion in an isotropic medium.

Abstract: We examine the capability of mean square displacement (MSD) analysis to extract reliable values of the diffusion coefficient D of a single particle undergoing Brownian motion in an isotropic medium in the presence of localization uncertainty. The theoretical results, supported by simulations, show that a simple unweighted least-squares fit of the MSD curve can provide the best estimate of D provided an optimal number of MSD points are used for the fit. We discuss the practical implications of these results for data analysis in single-particle tracking experiments.

Effects of strain on electronic properties of graphene

Abstract: We present first-principles calculations of electronic properties of graphene under uniaxial and isotropic strains, respectively. The semimetallic nature is shown to persist up to a very large uniaxial strain of 30% except a very narrow strain range where a tiny energy gap opens. As the uniaxial strain increases along a certain direction, the Fermi velocity parallel to it decreases quickly and vanishes eventually, whereas the Fermi velocity perpendicular to it increases by as much as 25%. Thus, the low energy properties with small uniaxial strains can be described by the generalized Weyl's equation while massless and massive electrons coexist with large ones. The work function is also predicted to increase substantially as both the uniaxial and isotropic strain increases. Hence, the homogeneous strain in graphene can be regarded as the effective electronic scalar potential.

Limit Analysis and Concrete Plasticity

Abstract: Introduction The Theory of Plasticity Constitutive Equations Extremum Principles for Rigid-Plastic Materials The Solution of Plasticity Problems Reinforced Concrete Structures Yield Conditions Concrete Yield Conditions for Reinforced Disks Yield Conditions for Slabs Reinforcement Design The Theory of Plain Concrete Statical Conditions Geometrical Conditions Virtual Work Constitutive Equations The Theory of Plane Strain for Coulomb Materials Applications Disks Statical Conditions Geometrical Conditions Virtual Work Constitutive Equations Exact Solutions for Isotropic Disks The Effective Compressive Strength of Reinforced Disks General Theory of Lower Bound Solutions Strut and Tie Models Shear Walls Homogenous Reinforcement Solutions Design According to the Elastic Theory Beams Beams in Bending Beams in Shear Beams in Torsion Combined Bending, Shear, and Torsion Slabs Statical Conditions Geometrical Conditions Virtual Work, Boundary Conditions Constitutive Equations Exact Solutions for Isotropic Slabs Upper Bound Solutions for Isotropic Slabs Lower Bound Solutions Orthotropic Slabs Analytical Optimum Reinforcement Solutions Numerical Methods Membrane Action Punching Shear of Slabs Introduction Internal Loads or Columns Edge and Corner Loads Concluding Remarks Shear in Joints Introduction Analysis of Joints by Plastic Theory Strength of Different Types of Joints The Bond Strength of Reinforcing Bars Introduction The Local Failure Mechanism Failure Mechanisms Analysis of Failure Mechanisms Assessment of Anchor and Splice Strength Effect of Transverse Pressure and Support Reaction Effect of Transverse Reinforcement Concluding Remarks

Polarization and Moment Tensors: With Applications to Inverse Problems and Effective Medium Theory

Abstract: Introduction- Layer Potentials and Transmission Problems- Uniqueness for Inverse Conductivity Problems- Generalized Isotropic and Anisotropic Polarization Tensors- Full Asymptotic Formula for the Potentials- Near-Boundary Conductivity Inclusions- Impedance Imaging of Conductivity Inclusions- Effective Properties of Electrical Composites- Transmission Problem for Elastostatics- Elastic Moment Tensor- Full Asymptotic Expansion of the Displacement Field- Imaging of Elastic Inclusions- Effective Properties of Elastic Composites- Appendices- References- Index

Refined beam theories based on a unified formulation

Abstract: This paper proposes several axiomatic refined theories for the linear static analysis of beams made of isotropic materials. A hierarchical scheme is obtained by extending plates and shells Carrera's Unified Formulation (CUF) to beam structures. An N-order approximation via Mac Laurin's polynomials is assumed on the cross-section for the displacement unknown variables. N is a free parameter of the formulation. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. According to CUF, the governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. The governing differential equations are solved via the Navier type, closed form solution. Rectangular and I-shaped cross-sections are accounted for. Beams undergo bending and torsional loadings. Several values of the span-to-height ratio are considered. Slender as well as deep beams are analysed. Comparisons with reference solutions and three-dimensional FEM models are given. The numerical investigation has shown that the proposed unified formulation yields the complete three-dimensional displacement and stress fields for each cross-section as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and loading conditions.

Bianchi type III models with anisotropic dark energy

Abstract: The general form of the anisotropy parameter of the expansion for Bianchi type-III metric is obtained in the presence of a single diagonal imperfect fluid with a dynamically anisotropic equation of state parameter and a dynamical energy density in general relativity. A special law is assumed for the anisotropy of the fluid which reduces the anisotropy parameter of the expansion to a simple form ( $${\Delta\propto H^{-2}V^{-2}}$$ , where Δ is the anisotropy parameter, H is the mean Hubble parameter and V is the volume of the universe). The exact solutions of the Einstein field equations, under the assumption on the anisotropy of the fluid, are obtained for exponential and power-law volumetric expansions. The isotropy of the fluid, space and expansion are examined. It is observed that the universe can approach to isotropy monotonically even in the presence of an anisotropic fluid. The anisotropy of the fluid also isotropizes at later times for accelerating models and evolves into the well-known cosmological constant in the model for exponential volumetric expansion.

A constitutive model for plastically anisotropic solids with non-spherical voids

Abstract: Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.

Generalized Hooke's law for isotropic second gradient materials

Abstract: In the spirit of Germain the most general objective stored elastic energy for a second gradient material is deduced using a literature result of Fortun\'e & Vall\'ee. Linear isotropic constitutive relations for stress and hyperstress in terms of strain and strain-gradient are then obtained proving that these materials are characterized by seven elastic moduli and generalizing previous studies by Toupin, Mindlin and Sokolowski. Using a suitable decomposition of the strain-gradient, it is found a necessary and sufficient condition, to be verified by the elastic moduli, assuring positive definiteness of the stored elastic energy. The problem of warping in linear torsion of a prismatic second gradient cylinder is formulated, thus obtaining a possible measurement procedure for one of the second gradient elastic moduli.

Brownian motion of a self-propelled particle

Abstract: Overdamped Brownian motion of a self-propelled particle is studied by solving the Langevin equation analytically. On top of translational and rotational diffusion, in the context of the presented model, the "active" particle is driven along its internal orientation axis. We calculate the first four moments of the probability distribution function for displacements as a function of time for a spherical particle with isotropic translational diffusion as well as for an anisotropic ellipsoidal particle. In both cases the translational and rotational motion is either unconfined or confined to one or two dimensions. A significant non-Gaussian behavior at finite times t is signalled by a non-vanishing kurtosis. To delimit the super-diffusive regime, which occurs at intermediate times, two time scales are identified. For certain model situations a characteristic t^3 behavior of the mean square displacement is observed. Comparing the dynamics of real and artificial microswimmers like bacteria or catalytically driven Janus particles to our analytical expressions reveals whether their motion is Brownian or not.

Conformal anisotropic relativistic charged fluid spheres with a linear equation of state

Abstract: We obtain two new families of compact solutions for a spherically symmetric distribution of matter consisting of an electrically charged anisotropic fluid sphere joined to the Reissner–Nordstrom static solution through a zero pressure surface. The static inner region also admits a one parameter group of conformal motions. First, to study the effect of the anisotropy in the sense of the pressures of the charged fluid, besides assuming a linear equation of state to hold for the fluid, we consider the tangential pressure p ⊥ to be proportional to the radial pressure p r , the proportionality factor C measuring the grade of anisotropy. We analyze the resulting charge distribution and the features of the obtained family of solutions. These families of solutions reproduce for the value C=1, the conformal isotropic solution for quark stars, previously obtained by Mak and Harko. The second family of solutions is obtained assuming the electrical charge inside the sphere to be a known function of the radial coordinate. The allowed values of the parameters pertained to these solutions are constrained by the physical conditions imposed. We study the effect of anisotropy in the allowed compactness ratios and in the values of the charge. The Glazer’s pulsation equation for isotropic charged spheres is extended to the case of anisotropic and charged fluid spheres in order to study the behavior of the solutions under linear adiabatic radial oscillations. These solutions could model some stage of the evolution of strange quark matter fluid stars.

Light propagation in statistically homogeneous and isotropic universes with general matter content

Abstract: We derive the relationship of the redshift and the angular diameter distance to the average expansion rate for universes which are statistically homogeneous and isotropic and where the distribution evolves slowly, but which have otherwise arbitrary geometry and matter content. The relevant average expansion rate is selected by the observable redshift and the assumed symmetry properties of the spacetime. We show why light deflection and shear remain small. We write down the evolution equations for the average expansion rate and discuss the validity of the dust approximation.

Non-Gaussianity and Statistical Anisotropy from Vector Field Populated Inflationary Models

Abstract: We present a review of vector field models of inflation and, in particular, of the statistical anisotropy and non-Gaussianity predictions of models with SU(2) vector multiplets. Non-Abelian gauge groups introduce a richer amount of predictions compared to the Abelian ones, mostly because of the presence of vector fields self-interactions. Primordial vector fields can violate isotropy leaving their imprint in the comoving curvature fluctuations 𝜁 at late times. We provide the analytic expressions of the correlation functions of 𝜁 up to fourth order and an analysis of their amplitudes and shapes. The statistical anisotropy signatures expected in these models are important and, potentially, the anisotropic contributions to the bispectrum and the trispectrum can overcome the isotropic parts.

The bending of single layer graphene sheets: the lattice versus continuum approach

Abstract: The out-of-plane bending behaviour of single layer graphene sheets (SLGSs) is investigated using a special equivalent atomistic-continuum model, where the C-C bonds are represented by deep shear bending and axial stretching beams and the graphene properties by a homogenization approach. SLGS models represented by circular and rectangular plates are subjected to linear and nonlinear geometric point loading, similar to what is induced by an atomic force microscope (AFM) tip. The graphene models are developed using both a lattice and a continuum finite element discretization of the partial differential equations describing the mechanics of the graphene. The minimization of the potential energy allows us to identify the thickness, elastic parameters and force/displacement histories of the plates, in good agreement with other molecular dynamic (MD) and experimental results. We note a substantial equivalence of the linear elastic mechanical properties exhibited by circular and rectangular sheets, while some differences in the nonlinear geometric elastic regime for the two geometrical configurations are observed. Enhanced flexibility of SLGSs is observed by comparing the nondimensional force versus displacement relations derived in this work and the analogous ones related to equivalent plates with conventional isotropic materials.

Bending Analysis of Functionally Graded Sandwich Plates Under the Effect of Mechanical and Thermal Loads

Abstract: The bending response of sandwich plates subjected to thermo-mechanical loads is studied. The sandwich plate faces are assumed to have isotropic, two-constituent (metal-ceramic) material distribution through the thickness, and the modulus of elasticity, Poisson's ratio, and thermal expansion coefficient of the faces are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used, taking into account the symmetry of the plate and the thickness of each layer. Field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory are derived. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Exact solutions for functionally ...

Guidelines and Recommendations to Construct Theories for Metallic and Composite Plates

Abstract: This work has evaluated the refinement of some classical theories, such as the Kirchhoff and Reissner-Mindlin theories, adding generalized displacement variables (up to fourth-order) to the Taylor-type expansion in the thickness plate direction. Isotropic, orthotropic, and laminated plates have been analyzed, varying the thickness ratio, orthotropic ratio, and stacking sequence of the layout. Higher-order theories have been implemented according to the compact scheme known as the Carrera unified formulation. The results have been restricted to simply-supported orthotropic plates subjected to harmonic distributions of transverse pressure for which closed-form solutions are available. For a given plate problem (isotropic, orthotropic, or laminated), the effectiveness of each employed generalized displacement variable has been established comparing the error obtained accounting for and removing the variable in the plate governing equations. A number of theories have therefore been constructed imposing a given error with respect to the available best results. Guidelines and recommendations that are focused on the proper selection of the displacement variables that have to be retained in refined plate theories are then furnished. It has been found that the terms that have to be used according to a given error vary from problem to problem, but they also vary when the variable that has to be evaluated (displacement, stress components) is changed. Diagrams (errors in terms of geometrical and orthotopic ratios) and graphical schemes have been built to establish the appropriate theories with respect to the data of the problem under consideration.

A method for reconstructing the variance of a 3D physical field from 2D observations: application to turbulence in the interstellar medium

Abstract: We introduce and test an expression for calculating the variance of a physical field in three dimensions using only information contained in the two-dimensional projection of the field. The method is general but assumes statistical isotropy. To test the method we apply it to numerical simulations of hydrodynamic and magnetohydrodynamic turbulence in molecular clouds, and demonstrate that it can recover the three-dimensional (3D) normalized density variance with ∼10 per cent accuracy if the assumption of isotropy is valid. We show that the assumption of isotropy breaks down at low sonic Mach number if the turbulence is sub-Alfvenic. Theoretical predictions suggest that the 3D density variance should increase proportionally to the square of the Mach number of the turbulence. Application of our method will allow this prediction to be tested observationally and therefore constrain a large body of analytic models of star formation that rely on it.

A molecular based anisotropic shell model for single-walled carbon nanotubes

Abstract: Single-walled carbon nanotubes (SWCNTs) are frequently modeled as isotropic elastic shells. However, there are obvious evidences showing that SWCNTs exhibit remarkable chirality induced anisotropy that should not be neglected in some cases. In this paper, we derive the closed-form expressions for the anisotropic elastic properties of SWCNTs using a molecular mechanics model. Based on these anisotropic elastic properties, we develop a molecular based anisotropic shell model (MBASM) for predicting the mechanical behavior of SWCNTs. The explicit expressions for the coupling of axial, circumferential, and torsional strains, the radial breathing mode frequency, and the longitudinal and torsional wave speeds are obtained. We show that the MBASM is capable of predicting the effects of size and chirality on these quantities. The efficiency and accuracy of the MBASM are validated by comparisons of the present results with the existing results.

Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries

Abstract: Recently, there are emerging demands for isotropic material parameters, arising from the broadband requirement of the functional devices. Since inverse Laplace’s equation with sliding boundary condition will determine a quasi-conformal mapping, and a quasi-conformal mapping will minimize the transformation material anisotropy, so in this work, the inverse Laplace’s equation with sliding boundary condition is proposed for quasi-isotropic transformation material design. Examples of quasi-isotropic arbitrary carpet cloak and waveguide with arbitrary cross sections are provided to validate the proposed method. The proposed method is very simple compared with other quasi-conformal methods based on grid generation tools.

Thermal buckling analysis of functionally graded material beams

Abstract: Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.

Transversely isotropic nonlinear magneto-active elastomers

Abstract: Magneto-active elastomers are smart materials composed of a rubber-like matrix material containing a distribution of magneto active particles. The large elastic deformations possible in the rubber-like matrix allow the mechanical properties of magneto-active elastomers to be changed significantly by the application of external magnetic fields. In this paper, we provide a theoretical basis for the description of the nonlinear properties of a particular class of these materials, namely transversely isotropic magneto-active elastomers. The transversely isotropic character of these materials is produced by the application of a magnetic field during the curing process, when the magneto active particles are distributed within the rubber. As a result the particles are aligned in chains that generated a preferred direction in the material. Available experimental data suggest that this enhances the stiffness of the material in the presence of an external magnetic field by comparison with the situation in which no external field is applied during curing, which leads to an essentially random (isotropic) distribution of particles. Herein, we develop a general form of the constitutive law for such magnetoelastic solids. This is then used in the solution of two simple problems involving homogeneous deformations, namely simple shear of a slab and simple tension of a cylinder. Using these results and the experimental available data we develop a prototype constitutive equation, which is used in order to solve two boundary-value problems involving non-homogeneous deformations—the extension and inflation of a circular cylindrical tube and the extension and torsion of a solid circular cylinder.