Showing papers on "Isotropy published in 2003"

Body Force Equivalents for Seismic Dislocations

Abstract: An explicit expression is derived for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation. This equivalent force depends only upon the source and the elastic properties of the medium in the immediate vicinity of the source and not upon the proximity of any reflecting surfaces. The theory is developed for dislocations in an anisotropic inhomogeneous medium; in the examples isotropy is assumed. For displacement dislocation faults, the double couple is an exact equivalent body force.

On nonuniqueness for Calderón’s inverse problem

Abstract: We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.

Linear dynamics of double-porosity dual-permeability materials. I. Governing equations and acoustic attenuation

Abstract: The equations governing the linear acoustics of composites with two isotropic porous constituents are derived from first principles using volume-averaging arguments. The theory is designed for modeling acoustic propagation through heterogeneous porous structures. The only restriction placed on the geometry of the two porous phases is that the overall composite remains isotropic. The theory determines the macroscopic fluid response in each porous phase in addition to the combined bulk response of the grains and fluid in the composite. The complex frequency-dependent macroscopic compressibility laws that are obtained allow for fluid transfer between the porous constituents. Such mesoscopic fluid transport between constituents within each averaging volume provides a distinct attenuation mechanism from the losses associated with the net Darcy flux within individual constituents as is quantified in the examples.

Transient elastography in anisotropic medium: Application to the measurement of slow and fast shear wave speeds in muscles

Abstract: From the measurement of a low frequency (50–150 Hz) shear wave speed, transient elastography evaluates the Young’s modulus in isotropic soft tissues. In this paper, it is shown that a rod source can generate a low frequency polarized shear strain waves. Consequently this technique allows to study anisotropic medium such as muscle. The evidence of the polarization of low frequency shear strain waves is supported by both numeric simulations and experiments. The numeric simulations are based on theoretical Green’s functions in isotropic and anisotropic media (hexagonal system). The experiments in vitro led on beef muscle proves the pertinent of this simple anisotropic pattern. Results in vivo on man biceps shows the existence of slow and fast shear waves as predicted by theory.

Magnetoelastic modelling of elastomers

Abstract: In this paper we first summarize the equations governing the deformation of magneto-sensitive (MS) elastic solids with particular reference to elastomers whose mechanical properties may be changed rapidly by the application of a magnetic field. These 'smart materials' typically consist of micron-sized ferrous particles dispersed within an elastomeric matrix. Constitutive relations for isotropic MS-elastic solids are examined. The equations are then applied to a representative geometry appropriate for applications, that in which the material is confined to a circular cylindrical tube in the presence of a radial magnetic field. The material is then subject to an axial shear deformation. Results are illustrated for two specific material models, for each of which the shear stiffness of the material increases with the magnetic field strength, as observed in experiments on MS elastomers.

Handbook of elasticity solutions

Abstract: 1: Basic Equations of Elasticity. 1.1. Cartesian Coordinates. 1.2. Cylindrical Coordinates. 1.3. Spherical Coordinates. 1.4. Hooke's Law for Anisotropic Materials. 2: Point Forces and Systems of Point Forces in Three-Dimensional Space and Half-Space. 2.1. Point Force in an Infinite Isotropic Solid. 2.2. Systems of Forces Distributed in a Small Volume of an Infinite Isotropic Solid. 2.3. Dynamic Problems of a Suddenly Introduced Point Forces Couples and Dipoles in an Infinite Isotropic Solid. 2.4. Point Force in the Isotropic Half-Space (Mindlin's Problem). 2.5. Point Force Applied at the Boundary of the Isotropic Half-Space. 2.6. Point Force of an Infinite Transverse Isotropic Solid. 2.7. Point Force Applied at the Boundary of the Transversely Isotropic Half-Space. 2.8. Two Joined Isotropic Half-Spaces with Different Moduli: Solution for a Point Force. 3: Selected Two-Dimensional Problems. 3.1. Introductory Material. 3.2. Infinite 2-D Solid. Isotropic and Orthotropic Materials. 3.3. 2-D Isotropic Half-Plane. 3.4. Stress Concentrations near Holes and Inclusions. 3.5. Equilibrium of an Elastic Wedge. 3.6. Circular Ring Loaded by External and Internal Pressures. 4: Three-Dimensional Crack Problems for the Isotropic or Transversely Isotropic Infinite Solid. 4.1. Circular (Penny-Shaped) Crack. 4.2. Half-Plane Crack. 4.3. External Circular Crack. 4.4. Elliptical Crack. 5: A Crack in an Infinite Isotropic Two-Dimensional Solid. 5.1. A Pair of Equal and Opposite Point Forces Applied at an Arbitrary Pointof the Crack. 5.2. Uniform Loading at Crack Faces. 5.3. Crack Tip Fields. 5.4. Far Field Asymptotics. 6: A Crack in an Infinite Anisotropic Two-Dimensional Solid. 6.1. Notations and General Representations for a 2-D Anisotropic Elastic Solid. 6.2. A Pair of Equal and Opposite Point Forces Applied at an Arbitrary Point of the Crack. 6.3. Uniform Loading at Crack Faces. 6.4. Crack Tip Fields. 6.5. Far Field Asymptotics. 6.6. Crack Compliance Tensor. 6.7. Appendix. 7: Thermoelasticity. 7.1. Basic Equations. 7.2. Stationary 3-D Problems. 7.3. Non-Stationary 3-D Problems. 7.4. Stationary 2-D Problems. 7.5. Non-Stationary 2-D Problems. 7.6. Thermal Stresses in Heated Infinite Solid Containing an Inhomogeneity or a Cavity. 8: Contact Problems. 8.1. 2-D Problems for a Rigid Punch on the Isotropic and Anisotropic Elastic Half-Plane. 8.2. 3-D Problems for a Rigid Punch on the Isotropic and Transversely Isotropic Elastic Half-Space. 8.3. Contact of Two Elastic Bodies (Hertz' Problem). 9: Eshelby's Problem and Related Results. 9.1. Inclusion Problem. 9.2. Ellipsoidal Inhomogeneity. 9.3. Eshelby's Tensor for Various Ellipsoidal Shapes. 9.4. Alternative Form of Solution for Ellipsoidal Inhomogeneity. 9.5. Expressions for Tensors P, Q, A and GBPIiGBP. 9.6. Quantities Relevant for Calculation of the Effective Elastic Properties. 10: Elastic Space Containing a Rigid Ellipsoidal Inclusion Subjected to Translation and Rotation. 10.1.

The electric field induced in the brain by magnetic stimulation: a 3-D finite-element analysis of the effect of tissue heterogeneity and anisotropy

Abstract: We investigate the effect of tissue heterogeneity and anisotropy on the electric field and current density distribution induced in the brain during magnetic stimulation. Validation of the finite-element (FE) calculations in a homogeneous isotropic sphere showed that the magnitude of the total electric field can be calculated to within an error of approximately 5% in the region of interest, even in the presence of a significant surface charge contribution. We used a high conductivity inclusion within a sphere of lower conductivity to simulate a lesion due to an infarct. Its effect is to increase the electric field induced in the surrounding low conductivity region. This boost is greatest in the vicinity of interfaces that lie perpendicular to the current flow. For physiological values of the conductivity distribution, it can reach a factor of 1.6 and extend many millimeters from the interface. We also show that anisotropy can significantly alter the electric field and current density distributions. Either heterogeneity or anisotropy can introduce a radial electric field component, not present in a homogeneous isotropic conductor. Heterogeneity and anisotropy are predicted to significantly affect the distribution of the electric field induced in the brain. It is, therefore, expected that anatomically faithful FE models of individual brains which incorporate conductivity tensor data derived from diffusion tensor measurements, will provide a better understanding of the location of possible stimulation sites in the brain.

Fusion of Multiple Imaging Planes for Isotropic Imaging in MRI and Quantitative Image Analysis using Isotropic or Near-isotropic Imaging

Abstract: In accordance with the present invention there is provided methods for generating an isotropic or near-isotropic three-dimensional images from two-dimensional images. In accordance with the present invention the method includes, obtaining a first image of a body part in a first plane, wherein the first image generates a first image data volume; obtaining a second image of the body part in a second plane, wherein the second image generates a second image data volume; and combining the first and second image data volumes to form a resultant image data volume, wherein the resultant image data volume is isotropic or near-isotropic.

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

Abstract: A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress–strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading–reloading cycles are presented.

Anisotropic Stars II: Stability

Abstract: We investigate the stability of self-gravitating spherically symmetric anisotropic spheres under radial perturbations. We consider both the Newtonian and the full general-relativistic perturbation treatment. In the general-relativistic case, we extend the variational formalism for spheres with isotropic pressure developed by Chandrasekhar. We find that, in general, when the tangential pressure is greater than the radial pressure, the stability of the anisotropic sphere is enhanced when compared to isotropic configurations. In particular, anisotropic spheres are found to be stable for smaller values of the adiabatic index γ.

Instabilities of isotropic solutions of active polar filaments.

Abstract: We study the dynamics of an isotropic solution of polar filaments coupled by molecular motors which generate relative motion of the filaments in two and three dimensions. We investigate the stability of the homogeneous state for constant motor concentration taking into account excluded volume and an estimate of entanglement. At low filament density the system develops a density instability, while at high density entanglement drives the instability of orientational fluctuations.

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

Abstract: The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh–Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.

Optical application and measurement of torque on microparticles of isotropic nonabsorbing material

Abstract: We show how it is possible to controllably rotate or align microscopic particles of isotropic nonabsorbing material in a TEM00 Gaussian beam trap, with simultaneous measurement of the applied torque using purely optical means. This is a simple and general method of rotation, requiring only that the particle is elongated along one direction. Thus, this method can be used to rotate or align a wide range of naturally occurring particles. The ability to measure the applied torque enables the use of this method as a quantitative tool - the rotational equivalent of optical tweezers based force measurement. As well as being of particular value for the rotation of biological specimens, this method is also suitable for the development of optically-driven micromachines.

Nonlinear Elasticity, Anisotropy, Material Stability and Residual Stresses in Soft Tissue

Abstract: In this chapter the basic equations of nonlinear elasticity theory needed for the analysis of the elastic behaviour of soft tissues are summarized. Particular attention is paid to characterizing the material symmetries associated with the anisotropy that arises in soft tissue from its fibrous constituents (collagens) that endow the material with preferred directions. The importance of the issue of convexity in the construction of constitutive laws (strain-energy functions) for soft tissues is emphasized with reference to material stability. The problem of extension and inflation of a thick-walled circular cylindrical tube is used throughout as an example that is closely associated with arterial wall mechanics. This is discussed first for isotropic materials, then for cylindrically orthotropic materials. Since residual stresses have a very important role in, in particular, arterial wall mechanics these are examined in some detail. Specifically, for the tube extension/inflation problem the residual stresses arising from the assumption that the circumferential stress is uniform under typical physiological conditions are calculated for a representative constitutive law and compared with those calculated using the ‘opening angle’ method.

Linearly Forced Isotropic Turbulence

Abstract: Stationary isotropic turbulence is often studied numerically by adding a forcing term to the Navier-Stokes equation. This is usually done for the purpose of achieving higher Reynolds number and longer statistics than is possible for isotropic decaying turbulence. It is generally accepted that forcing the Navier-Stokes equation at low wave number does not influence the small scale statistics of the flow provided that there is wide separation between the largest and smallest scales. It will be shown, however, that the spectral width of the forcing has a noticeable effect on inertial range statistics. A case will be made here for using a broader form of forcing in order to compare computed isotropic stationary turbulence with (decaying) grid turbulence. It is shown that using a forcing function which is directly proportional to the velocity has physical meaning and gives results which are closer to both homogeneous and non-homogeneous turbulence. Section 1 presents a four part series of motivations for linear forcing. Section 2 puts linear forcing to a numerical test with a pseudospectral computation.

Influence of particle shape and angularity on the behaviour of granular materials: a numerical analysis

Abstract: This paper analyses the influence of grain shape and angularity on the behaviour of granular materials from a two-dimensional analysis by means of a discrete element method (Contact Dynamics). Different shapes of grains have been studied (circular, isotropic polygonal and elongated polygonal shapes) as well as different initial states (density) and directions of loading with respect to the initial fabric. Simulations of biaxial tests clearly show that the behaviour of samples with isotropic particles can be dissociated from that of samples with anisotropic particles. Indeed, for isotropic particles, angularity just tends to strengthen the behaviour of samples and slow down either local or global phenomena. One of the main results concerns the existence of a critical state for isotropic grains characterized by an angle of friction at the critical state, a critical void ratio and also a critical anisotropy. This critical state seems meaningless for elongated grains and the behaviour of samples generated with such particles is highly dependent on the direction of loading with respect to the initial fabric. The study of local variables related to fabric and particle orientation gives more information. In particular, the coincidence of the principal axes of the fabric tensor with those of the stress tensor is sudden for isotropic particles. On the contrary, this process is gradually initiated for elongated particles. Copyright © 2003 John Wiley & Sons, Ltd.

Pair dispersion and preferential concentration of particles in isotropic turbulence

Abstract: The objective of the paper is to present a statistical model for predicting pair dispersion and preferential concentration of particles suspended in an isotropic homogeneous turbulent flow field. This model is based on a kinetic equation for the probability density function of the relative velocities of two particles. The model developed is applied to predict the pair relative velocity statistics and the accumulation effect of heavy particles in a steady-state suspension. The effect of particle inertia on the predicted Eulerian two-point particle velocity correlations is demonstrated and compared with known results of numerical simulation.

Plasma, thermal, and elastic waves in semiconductors

Abstract: The system of partially coupled plasma, thermal, and elastic wave equations and conditions for neglecting the coupling between them is analyzed. The treatment considers a semiconductor elastic medium with isotropic and homogeneous thermal and elastic properties. The solution of the coupled system of plasma, thermal and elastic equations is a very complex problem. In most practical cases partially coupled treatment is sufficient. Conditions where it is possible to neglect the coupling between the plasma, thermal, and elastic waves are considered in this work. Special cases for coupled thermal and elastic waves (the thermoelastic problem) and coupled plasma and elastic waves (the electronic deformation) are taken into consideration. An approximate quantitative analysis of the coupling effects is given.

Finite element model predictions of static deformation from dislocation sources in a subduction zone: Sensitivities to homogeneous, isotropic, Poisson-solid, and half-space assumptions

Abstract: [1] Dislocation models can simulate static deformation caused by slip along a fault. These models usually take the form of a dislocation embedded in a homogeneous, isotropic, Poisson-solid half-space (HIPSHS). However, the widely accepted HIPSHS assumptions poorly approximate subduction zone systems of converging oceanic and continental crust. This study uses three-dimensional finite element models (FEMs) that allow for any combination (including none) of the HIPSHS assumptions to compute synthetic Green's functions for displacement. Using the1995 Mw = 8.0 Jalisco-Colima, Mexico, subduction zone earthquake and associated measurements from a nearby GPS array as an example, FEM-generated synthetic Green's functions are combined with standard linear inverse methods to estimate dislocation distributions along the subduction interface. Loading a forward HIPSHS model with dislocation distributions, estimated from FEMs that sequentially relax the HIPSHS assumptions, yields the sensitivity of predicted displacements to each of the HIPSHS assumptions. For the subduction zone models tested and the specific field situation considered, sensitivities to the individual Poisson-solid, isotropy, and homogeneity assumptions can be substantially greater than GPS measurement uncertainties. Forward modeling quantifies stress coupling between the Mw = 8.0 earthquake and a nearby Mw = 6.3 earthquake that occurred 63 days later. Coulomb stress changes predicted from static HIPSHS models cannot account for the 63-day lag time between events. Alternatively, an FEM that includes a poroelastic oceanic crust, which allows for postseismic pore fluid pressure recovery, can account for the lag time. The pore fluid pressure recovery rate puts an upper limit of 10−17 m2 on the bulk permeability of the oceanic crust.

Anisotropic Stars II : Stability

Abstract: We investigate the stability of self-gravitating spherically symmetric anisotropic spheres under radial perturbations. We consider both the Newtonian and the full general-relativistic perturbation treatment. In the general-relativistic case, we extend the variational formalism for spheres with isotropic pressure developed by Chandrasekhar. We find that, in general, when the tangential pressure is greater than the radial pressure, the stability of the anisotropic sphere is enhanced when compared to isotropic configurations. In particular, anisotropic spheres are found to be stable for smaller values of the adiabatic index $\gamma$.