Ellipsoidal Inclusions Research Articles

Sensing of Ellipsoidal Inclusions in a Biomedical Ti–6Al–4V Alloy by Magnetic Means

In this work, we conducted an experimental research to verify a developed analytical model based on the magnetic sensing of thermoelectric currents produced by ellipsoidal inclusions embedded in an inhomogeneous matrix under the influence of an external temperature gradient. By changing the aspect ratios b/a designated by e of the ellipsoidal inclusions, a wide range of real situations can be simulated. The aspect ratio of the ellipsoidal inclusions varied from 0.33 to 4. An external 1.8°C/cm temperature gradient in the bio-metallic specimen produced magnetic flux densities ranging from 2 to 150 μT at 1.5 mm lift-off distance between the tip of the magnetometer probe and the bio-metallic specimen. Due to the difference in the thermoelectric properties of the host (matrix) and the inclusion (imperfection), the temperature and electric field distributions will be distorted accordingly to the geometry of the inclusion; therefore, the magnetic signal produced by an inclusion will depend not only on its material properties and size, but also on its shape. The experimental magnetic field distribution illustrated the potential for the proposed thermoelectric coupling technique to detect and characterize metallic inclusions of different geometries based on their magnetic signature.

Grain Shape Effect and the Viscoplastic Parameter on the Evolution of Isotropic and Kinematic Hardening of Metallic Materials under Cyclic Loading

The main objective of this work is to study the grain shape effect (aspect ratio α = a / b) and the viscoplastic parameter γ on the evolution of the kinematic and isotropic hardening of FCC type metallic materials, under uniaxial cyclic Tension-Compression ‘‘TC’ and to interpret these results. These parameters of shape and viscoplastic were developed and introduced by Abdul-Latif and Radi, indeed in this study we use their model. Expressed within the framework of a self-consistent approach, the rate-dependent inelastic strain is examined at the crystallographic slip system level describing a constitutive model for FCC metallic polycrystals, whereas the elastic strain is determined at the granular level. Based on the Eshelby’s tensor, the elastic behavior is assumed to be compressible. For a polycrystalline structure, the grains deform plastically by crystallographic slip located at the most favorably oriented systems supporting a high resolved shear stress . The approach considers that the inclusion (grain) form is ellipsoidal of half axes defining by a, b and c such as a ≠b= c. Several numerical tests are carried out highlighting the role of shape and viscoplastic parameter on the evolution of kinematic and isotropic hardening. A general comparison between the and effect on the overall hardening of the polycrystal shows that this work hardening is more sensitive to the parameter (for given ) compared to (for given). Keywords: Grain shape effect, Ellipsoidal inclusion, Viscoplastic parameter effect, Kinematic and isotropic hardening, Uniaxial cyclic ‘‘TC‘‘, Self-consistent model.

Estimates for the Thermoelastic Properties of a Composite with Ellipsoidal Anisotropic Inclusions

A mathematical model describing the thermoelastic characteristics of a composite reinforced by anisotropic ellipsoidal inclusions is proposed. The model is applied to estimating the thermoelastic properties in the case of strengthening the material with single-walled carbon nanotubes. The estimates obtained by the self-consistent method and dual variational formulation of a thermoelasticity problem for an inhomogeneous solid body are compared.

Strength enhancement induced by grain boundary solute segregations in ultrafine-grained alloys

A model is proposed explaining enhanced strength of ultrafine-grained alloys that contain grain boundary (GB) solute segregations. In the framework of the proposed model these segregations are treated as homogeneous ellipsoidal inclusions and act as the sources of elastic stresses affecting the emission of lattice dislocations from GBs. These segregations pin the ends of lattice dislocation segments at the initial stage of dislocation propagation along GBs, and the unpinning requires a load increase, leading to the enhanced yield strength. We calculate the contribution of GB segregations to the yield strength for the ultrafine-grained 1570 Al alloy. We demonstrate that the maximum yield strength of this alloy is achieved in the case of clustered, nearly spherical Mg segregations with a high Mg concentration and a diameter to thickness ratio of 1.0–1.4, depending on the Mg concentration inside segregations. We also briefly discuss the possible role of GB dislocations in the formation of such concentrated solute segregations as well as the influence of GB segregations on the strengthening of alloys containing nanoscale twins. The results of the calculations agree well with experimental data.

A Study of Uniaxial Compressive Strength of Shale Based on Homogenization Method

Uniaxial compressive strength of shale is one of primary input parameters during shale gas development. In order to analyze uniaxial compression failure mechanism and obtain strength formula, the present study regards shale as heterogeneous material composed of pores, brittle minerals and clays. Pores are distributed as oblate ellipsoidal inclusion in clays, while brittle minerals are distributed as spheroidal inclusion in the equivalent clay matrix formed by pores and clays. With the help of the basic solution to Eshelby’s matrix–inclusion problem and the homogenization method, a two-step homogenization framework for shale was established. The influences of pore fluid on the rock mechanical behavior were also discussed by categories. Based on the composite mechanics, the failure strength of shale under uniaxial compression was deduced at last. The results show that the shale strength is affected by the mechanical properties of diagenetic minerals and enhanced with the increase of brittle mineral content. Since pore water promotes the initiation and propagation of cracks, shale strength decreases non-linearly with increasing water saturation.

Micro and mesomechanics of fibre reinforced composites using mean field homogenization formulations: A review

Efficient deployment of composites necessitates robust modelling approaches for homogenization and micromechanics calculations. Mean field homogenization (MFH) methods are a class of versatile and accurate methods used for analysis of different types of composites varying from unidirectional composites to textile composites to short fibre reinforced composites to nano-tube reinforced composites. Micro and meso-mechanics of composites using MFH methods involves transformation of fibre architecture to assembly of ellipsoid inclusions, homogenization of the transformed RVE by making suitable simplifications about interactions between inclusions, and localization of stresses leading to damage/defect modelling. An in-depth review of the three steps for composites with varied reinforcement architecture including continuous straight, discontinuous short, discontinuous curved or wavy (including nano-tubes), textile and fibres with imperfect interphases reinforcements is presented.

Effective electrokinetic parameters of porous media with ellipsoidal inclusions. The differential effective medium approximation

Many porous media such as natural rocks have mesoscale inhomogeneities. The characteristic sizes of such inhomogeneities are much larger than the pore sizes but much less than the characteristic scale of the problem, such as the length of the sample on which the measurements taken. In this paper, we have solved the one-particle problem for depolarization of an ellipsoidal particle located in a porous medium with electrokinetic effect. To calculate the effective physical properties of a porous medium with many interacted ellipsoidal inclusions we have applied the homogenization scheme named differential effective medium method. This method leads to some system of differential equations for the effective constant determination. General theory is illustrated by calculations of these constants for media containing spherical and ellipsoidal inclusions.

Analysis of three-dimensional ellipsoidal inclusions in thermoelectric solids

ABSTRACT Thermoelectric (TE) materials as energy functional semiconductors can directly convert heat into electrical power. Due to the brittle nature of semiconductors, voids or inclusions are easy to be produced in the TE materials. This paper studies the three-dimensional ellipsoidal inclusion problem in thermoelectric materials. By introducing two auxiliary functions, we successfully simplify the non-linear coupled governing equations into linear un-coupled equations and the linearization is validated by two practical cases. Based on the Green's function method, the analytical solutions of this problem are derived. We find that the thermoelectric field inside the ellipsoidal inclusion is always uniform when subjected to far-field uniform loads. Furthermore, we derive the effective material properties of the matrix-inclusion system in closed-form. We find that it is possible to enhance the thermoelectric properties as well as the figure of merit by inserting specific inclusions. Moreover, among the different shapes of inclusions, the elliptical cylinder fiber that lies along the loading direction has the most significant improvements to the material properties. This paper is the first that derives the analytical solutions of three-dimensional inclusion problems in TE materials.

Dyakonov surface waves at the interface of nanocomposites with spherical and ellipsoidal inclusions

The dispersion properties of different types of surface electromagnetic waves which are supported by the interface of two nanocomposite materials with inclusions of different shape are studied theoretically and numerically. Both nanocomposites are made of doped n-type semiconductor inclusions which are evenly distributed in a transparent dielectric matrix. First nanocomposite contains semiconductor inclusions of spherical shape and can be represented as isotropic material with scalar effective dielectric permittivity. Second nanocomposite contains semiconductor inclusions of ellipsoidal shape and can be modeled as a uniaxial anisotropic crystal with two different principal effective permittivity components. Influence of physical and geometrical parameters of the nanocomposites on dispersion properties of surface waves is studied. It is shown that when dissipations in the semiconductor inclusions are taken into account the angular spectrum of Dyakonov surface wave existence becomes wider, and several dispersion curves of surface modes can simultaneously exist.

A simple model to predict effective conductivity of multicomponent matrix-based composite materials with high volume concentration of particles

For high-contrast-component-property composites, most popular effective medium approximations that are based on the dilute analytical solution for an ellipsoidal inclusion embedded in an infinite matrix diverge significantly from each other and from experimental and numerical data. This paper discusses the challenges in this research area and proposes some homogenization strategies to solve the problems. In this work, a simple effective medium approximation is constructed to predict the effective conductivity of multicomponent matrix-based composites containing high concentrations of particles in d-dimensional space (d=2,3). The maximum volume fraction that depends on particle size distribution and the geometric approximation are accounted for in the micromechanical model to obtain new closed-form solutions. A flexible version of the model containing a free parameter is proposed to make the predictions more accurate at very high-volume fraction of inclusions. Applications are illustrated by comparing the theoretical predictions with the available experimental data or finite element simulations for various types of material to show the agreeable results close to the maximal packing points of the interactions.

A new energy-based effective strain theory for mechanical properties of multiphase composites

In this study, the energy-based effective strain method is proposed to better exhibit the energy stored in the three-phase composites by replacing average strain utilized in most analytical methods with an effective strain assessment. The overall composite materials can be treated statistically as a transversely isotropic solid for the case of aligned ellipsoidal inclusions, or as an isotropic solid for the case of randomly oriented inclusions. The paper aims to predict mechanical properties of spheroidal particle-reinforced composites using a new model considering effect of the interphase between particles and matrix. Either hard or soft interphases surrounding the fillers are taken into account for analyzing the effective properties of composites. The current methodology is generally extended to the case of composites containing multiple types of inclusions. The effects of the particle shape/size and the interphase thickness on effective elastic moduli are analyzed. The obtained solutions are shown to agree well with the experimental data.

Stress State of an Orthotropic Piezoelectric Body with a Triaxial Ellipsoidal Inclusion Subject to Tension

The problem of the stress state in an orthotropic piezoelectric body with a triaxial ellipsoidal inclusion under homogeneous force and electric loads is considered. The problem is solved by the Eshelby method of equivalent inclusion generalized to the case of a piezoelectric orthotropic space. The approach is validated against the example of a spheroidal cavity in a transversely isotropic material (the axis of revolution coincides with the symmetry axis) for which the exact solution is known. The stress distribution over the surface of the ellipsoidal cavity subject to tension is analyzed numerically.

Testing popular rock-physics models

In the spirit of classic rock physics, and as an ideal foundation for conventional quantitative interpretation workflows, we consider several popular models relating elastic rock properties to their composition, microstructure, and effective stress on the background of a worldwide log data set, incorporating sands and shales characterized by the maximum dynamic impedance range. We demonstrate that the patchy cement model, ellipsoidal inclusion model, and siliciclastic diagenesis model may be calibrated successfully against the world data set and used in seismic rock property log restoration/editing. We also demonstrate that some of these models present obvious challenges in terms of the information derived from quantitative seismic interpretation. Notably, the key input parameters used in these rock-physics models may show little resemblance to the rock parameters actually observed in geologic studies. Replacing the true rock parameters with the effective ones may do disservice to the science of rock physics in general.

Viscoplastic self-consistent polycrystal modeling of texture evolution of ultra-low carbon steel during cold rolling

The viscoplastic self-consistent (VPSC) polycrystal model, where each grain is regarded as an ellipsoidal inclusion interacting with a homogeneous effective medium represented by the polycrystal has long been successful in explaining the mechanical response and crystallographic texture development during plastic deformation. VPSC has especially been used to model low-symmetry materials such as Mg-alloys, uranium, etc, which could not be addressed by the conventional Taylor model. Although there are several reports on the modeling of steel that successfully reproduce its measured crystallographic texture at specific reductions, few studies have been carried out to reproduce the evolution of the intensity of texture fibers at various rolling reductions. In this study the texture evolution of body-centered-cubic ultra-low carbon steel during cold rolling was calculated using the VPSC model and compared with textures measured at 40%, 60%, 70%, 80% and 90% reduction. The deformation by cold rolling was accommodated by and slip systems interacting via a Voce hardening law. Addressing a wide range of rolling reductions poses a more severe test on the modeling. It is found that accounting for grain shape, grain interaction with the surroundings, and their evolution with strain, is critical for explaining the experimentally observed evolution of texture intensities. Specifically, the mechanisms of grain-neighbor co-rotation and the role played by the relative plastic stiffness of grain and surrounding medium were explored in this work. The predicted evolution of the main texture components, namely, α fiber (RD//) and γ fiber (ND//) was compared with experimental data measured by x-ray Schultz method, resulting in that the calculated texture evolution using the above models captures well the experimental results which show a gentle ODF intensity increase in the γ fiber and a strong ODF increase near components in the α fiber during cold rolling.

Volumetric response of an ellipsoidal liquid inclusion: implications for cell mechanobiology

Elastic composites containing liquid inclusions exist widely in nature and in engineered systems. The volumetric response of liquid inclusions is important in many cases, such as an isolated cell embedded in an extracellular matrix or an oil pocket embedded within shale. In this study, we developed a model for describing the volumetric response of an ellipsoidal liquid inclusion. Specifically, we investigated the volumetric response of an ellipsoidal liquid inclusion embedded in a three-dimensional (3D) matrix through an analytical expression of the volumetric response. We performed parametric analysis and found that loading along the shortest axis can induce the most volume change, while loading along the longest axis can induce the least volume change. We also found that the volumetric response decreases with increasing Poisson ratio of the matrix. These results could be used to understand some cell behavior in a 3D matrix, for example, cell alignment under mechanical load.

Effective electrical conductivity of transversely isotropic rocks with arbitrarily oriented ellipsoidal inclusions

This paper addresses the problem of the electrical conductivity tensor calculation for a transversely isotropic material that contains inhomogeneities of arbitrary orientation. For this goal, we first construct the electrical conductivity contribution tensor for an arbitrarily oriented isolated ellipsoidal anisotropic inhomogeneity embedded in a transversely isotropic matrix. The general case of an orthotropic ellipsoidal inhomogeneity unaligned in an anisotropic matrix with different classes of symmetry can be considered. This solution is used as the basic building block of various homogenization techniques: the Mori–Tanaka–Benveniste scheme, Maxwell scheme, and differential scheme. The approach is illustrated by an application to a transversely isotropic mudstone rock, composed of a clay matrix containing inhomogeneities of calcite and quartz. We analyse the origins of the extent of anisotropy of the effective conductivity tensor, distinguishing among the shape, orientation distribution, and anisotropy of the inhomogeneities on the one hand and the anisotropy of the matrix on the other hand. Numerical results show that the orientation distribution of the inhomogeneities significantly affects the overall anisotropy in the case of inhomogeneities with low aspect ratio(s). Limiting cases of aligned and randomly oriented inhomogeneities provide bounds of the extent of anisotropy for the overall conductivity tensor.

Numerical calculation of permeability of periodic porous materials: Application to periodic arrays of spheres and 3D scaffold microstructures

SummaryIn this paper, an efficient numerical method is proposed to calculate the anisotropic permeability in porous materials characterized by a periodic microstructure. This method is based on pore‐scale fluid dynamic simulations using a static volume of fluid method. Unlike standard solution procedures for this type of problem, we here solve an average constitutive equation over both fluid and solid domain by use of a subgrid model to accurately capture momentum transfer from the fluid to solid interface regions. Using numerical simulations on periodic arrays of spheres, we first demonstrate that, by using the subgrid interface model, more accurate results can be produced, for the velocity and pressure fields, than via more conventional approaches. We then apply numerical upscaling over the unit cell to calculate the full anisotropic permeability from the pore‐scale numerical results. The obtained permeability values for a variety of periodic arrays of spheres in different arrangements and packing orders are in good agreement with semianalytical results reported in literature. This validation allows for the permeability assessment of more complex structures such as isotropic gyroid structures, or anisotropic cases, here modeled in their simplest form, the ellipsoidal inclusion.

Esh3D, an Analytical and Numerical Hybrid Code for Full Space and Half‐Space Eshelby's Inclusion Problems

I present an open source code Esh3D that evaluates the quasi‐static elastic displacement and stress fields perturbed by ellipsoidal inclusions, known as the Eshelby solution. The improvement over my previous work includes obtaining solutions in half‐space by numerically canceling the traction to stage a free surface. The discrete numerical domain allows nonplanar surface. I first briefly introduce the ellipsoidal inclusion problem and solution evaluation. Then, I describe the traction cancellation method in forming the half‐space solutions. Next, I emulate a planar fault with a low aspect ratio ellipsoid and compare the half‐space solution against the well‐known Okada solution for flat and topographic surfaces. Finally, I provide comparisons between the full space and half‐space solutions to illustrate the free surface effects on the displacement and stress fields.

Size-dependent Eshelby’s ellipsoidal inclusion problem based on generalized first strain gradient elasticity theory

An analytical solution of Eshelby’s ellipsoidal inclusion problem is derived in this paper using the generalized first strain gradient theory (GFSGT) of elasticity. The GFSGT employed here is a reformulation of Mindlin’s first-order strain gradient theory, retaining only three independent material length scales. The Eshelby’s tensor for an ellipsoidal inclusion embedded in a strain gradient elastic continuum is obtained from the result for an arbitrary inclusion, which is available in the literature. The fourth-order Eshelby’s tensor depicts size-dependent behaviour, and a position-dependence is noted even within the domain of the inclusion. The volume average of this position-dependent tensor is determined inside and outside the inclusion, for the evaluation of size-dependent homogenized properties of composites. As expected, the results of this study asymptotically approach those of classical elasticity when the geometric dimensions of the inclusion are increased. For inclusions having sizes on the order of the material length scale parameters, these volume averages are strongly influenced by gradient effects. A comparison of the current model based on the GFSGT with that built on the simplified theories of strain gradient elasticity is made to illustrate the complete strain gradient effects. This comparison, along with the suitability of a generalized model involving different length scales for crystals and polymers, illustrates the importance of using GFSGT for determining the effective properties of micro- and nanocomposites.

Numerical evaluation on the effective thermal conductivity of the composites with discontinuous inclusions: Periodic boundary condition and its numerical algorithm

Boundary condition plays an important role in prediction of the effective thermal conductivity of composites. In this research, the periodic boundary condition and the representative volume element (RVE) based finite element (FE) homogenization method are adopted to evaluate the effective thermal conductivities of the composites reinforced by the spherical, ellipsoidal and cylindrical inclusions, and the emphases are on the numerical implementation algorithm and validation of the periodic boundary condition. The heat flux continuity of the node pairs on the opposite surfaces of the RVEs of the composites is analyzed and the effective thermal conductivity of the composites are homogenized. The results show that the heat flux continuity of the node pairs on the opposite surfaces of the RVEs of the composites can be guaranteed by the proposed numerical implementation algorithm for the periodic boundary condition, and that the predicted effective thermal conductivities of the composites agree well with those determined by the Lewis-Nielsen model and the experimental tests. Therefore, the RVE based FE homogenization method with the periodic boundary condition can accurately evaluate the effective thermal conductivity of the composites with discontinuous inclusions.