Ellipsoidal Inclusions Research Articles

Эффективная диэлектрическая проницаемость композита с эллипсоидальными включениями в матричной оболочке

A composite material based on ellipsoidal inclusions distributed in a bonding medium is considered. A formula is obtained for calculating the effective dielectric constant of such a material, taking into account the presence of a layer of a binder on the inclusions, which prevents their direct contact. It is shown that the calculated values of the effective dielectric constant correspond to the experimental values for a composite based on sections of conductive fibers.

Sheath fold development around deformable inclusions: Integration of field-analysis (Cima Lunga unit, Central Alps) and 3D numerical models

We investigated complex tectonic structures within a km-scale shear zone (Cima Lunga unit), which is traditionally interpreted as generated by multiple, distinct deformation phases, despite showing unique schistosity and lineation. Based on structural analyses we discovered sheath folds developed in relatively weak gneissic/schistose rocks, enveloping inclusions of stronger ultramafics. The internal layering of inclusions experienced superimposed folding, boudinage and folding, attesting to layer-parallel shortening followed by stretching and further again shortening. Using 3D numerical modelling, we explored the structure evolution within and around deformable viscous inclusions under far-field simple shear. The numerical results showed that the internal deformation of ellipsoidal inclusions and the fold development around the inclusions are both dependent on the viscosity ratio, shear strain and the inclusion aspect ratio. The Cima Lunga structural patterns were reproduced for finite strains exceeding 7.5 and viscosity ratio between 2.8 and 9. Inclusions are characterized by persistent rotation of the internal layering, resulting in super-simple shear regime, with kinematic vorticity number >1. An important corollary is that ultramafics and host rocks experienced coupled deformation since the prograde metamorphic evolution. Finally, we emphasise that progressive deformation in shear zones may offer sufficient explanation for complex structural patterns, without invoking unjustified polyphase deformation.

Determination of effective properties of piezoelectric composites containing short fibers

The present paper studies the effects of local field fluctuations on the effective electromechanical coupling coefficients of piezoelectric composites containing short fibers. The finite-element method (FEM) is applied to study the overall behavior of such composites. Geometric models containing spheroid inclusions of two fiber arrangements, simple cubic and body-centered cubic, are modeled with FEM to predict the effective coefficients. The results thus obtained are validated with the results obtained from the analytical method based on Eshelby’s solution to the ellipsoidal inclusion problem. The analytical solution obtained through the Eshelby method is generalized for spheroid inclusions to compare the results with those of the FEM study. All finite-element studies are carried out through the finite-element software package Ansys 18.0. A detailed analysis of results is presented to understand better the behavior of such composites and their effective property variation with change in fiber volume fraction.

Direct determination of phase stress evolution in duplex steel using synchrotron diffraction

The present work deals with the investigation of the micromechanical behaviour of a duplex steel, which consists of two phases (austenite and ferrite) exhibiting significantly different mechanical properties. The stresses in both phases were experimentally determined in the elastic and plastic ranges of deformation using synchrotron radiation diffraction experiments. The used methodology enabled to determine the values of initial stresses and to study the evolutions of the principal phase stresses and the second order stresses during the elastic, as well as the plastic range of deformation. With the help of a self-consistent model, the critical resolved shear stresses and the work hardening parameters for slip systems, active in each phase, were also estimated. Comparison of the measured phase stresses and lattice strains evolutions with the model results showed a good agreement between prediction and experiment when the initial stress state in the sample and shape of grains approximated by ellipsoidal inclusions were taken into account.The overall outcome of the work is the determination of stress partitioning between the two phases of a polycrystalline material for all deformation stages, determined directly from diffraction experiment. The results enabled analyses of von Mises, hydrostatic stresses as well as second order stresses evolutions in both phases during tensile deformation. Finally, the experimental data were successfully compared with predictions of the self-consistent deformation model.

Mechanical Aspects of Deformation-Induced Surface Roughening in the Presence of Inclusions in a Subsurface Layer. Numerical Modeling

The mechanical aspects of deformation-induced surface roughening inherent in microstructural inhomogeneity are studied numerically using single inclusion models. Three-dimensional finite-element calculations of uniaxial tension are performed for a set of single inclusion models where a cubic-shaped inclusion is embedded into a homogeneous matrix. The inclusion-to-surface distance, tilt angle about the axis of tension, and the ratio between the matrix and inclusion elastic-plastic properties are varied in different combinations to study the effects which these parameters have on the development of out-of-plane surface displacements under uniaxial tension. It has been shown that all stress and strain tensor components in the vicinity of inclusions take on nonzero values, including those directed across the load axis. Thus, the free surface becomes rough under the action of internal forces originated from the inhomogeneous stress-strain fields. Some illustrative examples of surface roughening under uniaxial tension are shown for multiple ellipsoidal inclusions periodically arranged in a subsurface layer of an elastic-plastic material.

An efficient elasto-visco-plastic self-consistent formulation: Application to steel subjected to loading path changes

A novel elasto-visco-plastic self-consistent (EVPSC) formulation based on the scheme of the homogeneous effective medium is presented. The constitutive behavior of a polycrystal is described as that of an elasto-visco-plastic effective medium interacting with grains treated as elasto-visco-plastic ellipsoidal inclusions. The formulation is based on the definition of a unique elasto-visco-plastic compliance, so avoiding the inconsistency arising from assuming superimposed elastic and visco-plastic interaction laws, as made in similar elasto-visco-plastic models. In addition, the elasto-visco-plastic constitutive equation of crystal and aggregate is formulated in terms of stress increments, which leads naturally to a semi implicit solution scheme. The superior numerical stability and computational efficiency of the new incremental EVPSC model (denoted as ΔEVPSC) are demonstrated by applying the model to a 316L austenitic stainless steel and comparing against other elasto-plastic models. The modeling capability for predicting texture, stress-strain response, and Bauschinger effect are demonstrated using a dislocation-density based hardening law applied to low carbon (LC) steel subjected to deformation histories that involve strain-path changes.

New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials

The present study deals with a new micromechanical modeling of the thermal conductivity of multi-coated inclusion-reinforced composites. The proposed approach has been developed in the general frame of anisotropic thermal behavior per phase and arbitrary ellipsoidal inclusions. Based on the Green's function technique, a new formulation of the problem of multi-coated inclusion is proposed. This formulation consists in constructing a system of integral equations, each associated to the thermal conductivity of each coating and the reference medium. Thanks to the concept of interior- and exterior-point Eshelby's conduction tensors, the exact solution of the problem of multicoated inclusion is obtained. Analytical expressions of the intensity in each phase and the effective thermal conductivity of the composite, through homogenizations schemes such as Generalized self-consistent and Mori-Tanaka models are provided. Results of the present model are successfully compared with those issued from both analytical models and finite elements methods for composites with doubly coated inclusions. Moreover, the developed micromechanical model has been applied to a three phase composite materials in order to analyze combined effects of the aspect ratio and the volume fraction of the ellipsoidal inclusions, the anisotropy of the thermal conductivity of interphase, the thermal conductivity contrast between local phases on the predicted effective thermal conductivity.

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Since Eshelby’s (1957) result (Eshelby, JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc London A 1957; 421: 379–396) that ellipsoids in an infinite matrix have uniform localization tensors, all attempts to find other finite domain shapes sharing that same property have failed and valuable proofs were provided that none could exist. Since that “uniformity property” also applies to infinite cylinders and layers as limits of prolate and oblate spheroids, we examine the cases of hyperboloidal domains of which infinite cylinders and platelets also are the limits. As members of the quadric surface family, hyperboloids expectably also have uniform Eshelby tensor and Green operator when embedded in infinite media, with specific features expectable too from unboundedness and not convex curvatures. Using the Radon transform method applied by the author to various inclusion shapes, as well as to finite and infinite patterns, since the uniformity of a shape function (inverse Radon transform of the domain indicator function) implies the uniformity of the related Green operator and Eshelby tensor, we examine the shape functions of axially symmetric hyperboloids. We establish that those of the two-sheet types are uniform and those of the one-sheet types are not, an additional neck-related term carrying the non-uniformity. The Green operators are next examined in considering an isotropic embedding medium with elastic- (including dielectric-) like properties. The results regarding the operator (non) uniformity correspond to those concerning the shape functions. The established operator uniformity characteristics imply validity of all Eshelby-derived ellipsoid properties. Yet, determining the Green operator solution calls for overcoming the issue of the infinite hyperbolic planar sections (the operator finiteness), with also attention being paid to positive definiteness. Options are compared from which an obtained satisfying solution with regard to both issues raises questioning theoretical and practical points on mathematical and mechanical grounds. While further studies are in progress, some application tracks are indicated.

Analytical solution for the displacement of a polygonal inclusion with a special application to the case of linear eigenstrain

Previous studies on a polygonal inclusion have been concerned primarily with the stress/strain solutions of uniform eigenstrains, while the analyses of the displacements are less elaborated in literature. By employing the method of Green's function, the displacement solution may be formulated in an area integral, which is then converted to a contour integral around the boundary of the inclusion. This work presents a straightforward and closed-form evaluation of the displacements when the boundary of the inclusion is composed of line elements. The proposed method of solution can not only deal with the widely investigated problem of uniform eigenstrains, but is also effective for handling linearly distributed eigenstrains. The strain Eshelby tensor may be derived in a straightforward manner from the displacement solution. Unlike the classical Eshelby ellipsoidal inclusion, where the solutions for the exterior field are complicated, the displacements for both interior and exterior points of a polygonal inclusion can be written in a unified algebraic form of elementary functions. It is well-known that the stress and strain fields have logarithmic singularities at the corners of a polygonal inclusion. In contrast, the present work shows that the displacement is continuous and finite at the vertices of the polygon.

Application of micro-CT to Mori-Tanaka method for non-randomly oriented pores in air-entrained cement pastes

The homogenization technique has played a key role in elucidating the mechanical, thermal, and chemical behaviors of composite materials. Specifically, the cement-based material is a representatively complex composite material consisting of various inclusions, such as pores, fibers, and aggregates. Recent studies have mathematically introduced the Mori-Tanaka method to express non-randomly oriented ellipsoidal inclusions. Therefore, the present study aimed to practically apply pore information obtained from micro-computed tomography (micro-CT) to the Mori-Tanaka method. Cement-paste samples were prepared with 0, 1, 2, 5, and 10 wt% of an air-entraining (AE) agent. The representative ellipsoidal shape and orientation distribution function (ODF) of the pores were obtained from the micro-CT, and this information was incorporated into the Mori-Tanaka method. The computation results revealed good agreement between the results of the Mori-Tanaka method and the finite-element method (FEM). Additional sensitivity studies using the Mori-Tanaka model allowed for quantifying the anisotropic degree of the pores in the AE-agent added cement pastes.

An explicit analytical solution for temperature rise in a half-space induced by ellipsoidal inclusions and its application for problems related to friction heating

Green functions for temperature rise in a semi-infinite space containing an ellipsoidal inclusion are obtained in the present study. Explicit expression for disturbed temperature rise generated by eigen-temperature gradients uniformly distributed within a domain is derived. Further, the proposed analytical solution method is utilized to deal with temperature rise in heterogeneous half-space subjected to friction heating via applying the equivalent inclusion method (EIM), whose results are proven to be in good agreements with those of the benchmarks. Influences of heat load velocity, spatial orientation and aspect ratio of ellipsoidal inhomogeneity on temperature rise in a semi-infinite space are discussed. Finally, a model of semi-infinite medium with embedded dispersed ellipsoidal inhomogeneities of arbitrary spatial orientation is adopted to explore the application scope of the proposed solution method.

Evolution and control of nonmetallic inclusions in FeSiB alloy

Inclusions have a great influence on the quality of FeSiB amorphous ribbon. The evolution behavior of inclusions was analyzed. The results show that the spherical and elliptical inclusions, consisting of Al2O3 and SiO2, mainly come from industrial pure iron. Ellipsoidal inclusions of 20–30 μm in pure iron were removed by floatation during the smelting process, and spherical inclusions of 1–3 μm combined with deoxidized products in FeSiB melt form Al2O3·SiO2 and CaO·Al2O3·SiO2 inclusions. Some inclusions accumulated on the inner wall of the nozzle during the spraying process, and others flowed out of the nozzle and remained in FeSiB amorphous ribbon. By controlling the total oxygen content in industrial pure iron to 31 × 10−6, clogging on the inner wall of the nozzle can be reduced and the free surface smoothness of the amorphous ribbon can be improved.

Crack‐Induced Shear‐Wave Orthorhombic Anisotropy: Modeling and Inversion of Shear‐Wave Borehole Measurements

AbstractRocks in horizontally layered sediments often exhibit transversely isotropic (TI) characteristics and contain aligned fractures/cracks due to tectonic movement and stress. We model the effective elastic properties of the TI medium containing vertically aligned cracks using the sphere‐equivalency of the effective scattering method. Results show that the corresponding anisotropy of the VTI background medium containing vertically fluid‐filled ellipsoidal inclusions can be characterized by the orthorhombic anisotropy model. The shear‐wave anisotropy of the model is governed by three anisotropy parameters, corresponding to the horizontal plane, and the two vertical planes that are normal and parallel to the crack plane, respectively. Effects on anisotropy parameters due to cracks are examined in detail, showing that one can use shear‐wave measurements to evaluate both the TI and crack parameters of the cracked TI‐rock. Based on the theoretical analysis, we develop an inversion method to estimate shear anisotropy parameters from the four‐component shear‐wave logging measurement in a vertical borehole penetrating the orthorhombic rock. The inversion method is validated with synthetic data and applied to interpret acoustic anisotropy from borehole measurements acquired in a fractured shale formation, where theoretical predictions and measurements are found in good agreement.

A numerical application of the Eshelby theory for geobarometry of non-ideal host-inclusion systems

In the complex geodynamic processes occurring at convergent plate margins, rocks can be subducted at depth into the Earth experiencing metamorphism. A mineral inhomogeneity entrapped into another mineral, after exhumation to the Earth surface, will exhibit stress and strain fields different from those of the host because of the different thermoelastic properties. In the present paper, we propose a finite-element-based approach to determine the Eshelby and the relaxations tensors for any morphology of the inhomogeneity and for any crystallographic symmetry of the host. The proposed procedure can be directly applied in the framework of elastic geobarometry to estimate, on the basis of the Eshelby theory, the entrapment conditions (pressure and temperature) from the residual strain field measured in the inhomogeneity. This aspect represents a step forward to currently available models for geobarometry allowing the investigation of complex morphologies of the inhomogeneity in systems with general material anisotropy. We validate the proposed approach versus Eshelby analytical solutions available for spherical and ellipsoidal inclusions and we show the application to a real geological case of high pressure metamorphic rocks.

Effects of friction heating on a half space involving ellipsoidal inclusions with non-uniform eigentemperature gradients

Performance of material containing inhomogeneities/inclusions is severely affected by frictional heat resulting from contact behaviors between two mechanical components. An effective method for dealing with thermal field of a half space with distributed non-uniform ellipsoidal inclusions subjected to frictional heating load was proposed in the current investigation, which can be simplified to handle problems of uniform ellipsoidal inclusion and extended to deal with inhomogeneity problems. Two categories of non-uniform inclusions with Gaussian and exponential eigentemperature gradients are modeled. Temperature rise and thermal stress are analogously solved via the equivalent inclusion method (EIM). Results obtained by the proposed method totally correspond to those of the benchmarks. Parametric studies on eigentermperature gradient distribution and thermal field of a half space involving a non-uniform spherical inclusion are carried out. Surface convection effect is further taken into consideration to extend the application scope of the proposed method, and the influence of surface heat-transfer coefficients on the thermal field of a semi-infinite medium is explored.

A Statistical Framework for Generating Microstructures of Two-Phase Random Materials: Application to Fatigue Analysis

Random microstructures of heterogeneous materials play a crucial role in the material macroscopic behavior and in predictions of its effective properties. A common approach to modeling random multiphase materials is to develop so-called surrogate models approximating statistical features of the material. However, the surrogate models used in fatigue analysis usually employ simple microstructure, consisting of ideal geometries such as ellipsoidal inclusions, which generally does not capture complex geometries. In this paper, we introduce a simple but flexible surrogate microstructure model for two-phase materials through a level-cut of a Gaussian random field with covariance of Mat\'ern class. Such parametrization of the covariance function allows for the representation of a few key design parameters while representing the geometry of inclusions in a more general setting for a large class of random heterogeneous two-phase media. In addition to the traditional morphology descriptors such as porosity, size and aspect ratio, it provides control of the regularity of the inclusions interface and sphericity. These parameters are estimated from a small number of real material images using Bayesian inversion. An efficient process of evaluating the samples, based on the Fast Fourier Transform, makes possible the use of Monte-Carlo methods to estimate statistical properties for the quantities of interest in a given material class. We demonstrate the overall framework of the use of the surrogate material model in application to the uncertainty quantification in fatigue analysis, its feasibility and efficiency, and its role in the microstructure design.

Novel theories on magneto-electro-elastic ellipsoidal multi-inclusions and inhomogeneities and associated impotent fields

The exact nature of the induced coupled-fields of anisotropic magneto-electro-elastic ellipsoidal inclusions, multi-inclusions, and inhomogeneities with non-uniform eigenfields under polynomial magneto-electro-elastic far-field loadings is of particular interest. For the sake of prediction of the induced coupled-fields of magneto-electro-elastic multi-inclusions due to piecewise polynomial generalized eigenfields several theorems and corollaries are stated and proved. Some classes of impotent generalized eigenfields associated with encapsulated ellipsoidal multi-inclusion result in vanishing generalized disturbance strains within the innermost ellipsoidal domain. On the other hand, it is found that there are certain classes of impotent generalized eigenfields for which the generalized disturbance stresses vanish everywhere. Furthermore, it is proved that there exists a unique equivalent magneto-electro-elastic inclusion associated with an anisotropic magneto-electro-elastic ellipsoidal inhomogeneity subjected to polynomial far-field loadings. It is shown that associated with every anisotropic magneto-electro-elastic ellipsoidal inhomogeneity, there exists a set of infinite numbers of far-field loadings for which the inhomogeneity does not induce any disturbance generalized stresses anywhere within the entire medium. In other words, under certain far-field loadings the inhomogeneity is not sensed by the surrounding magneto-electro-elastic medium.

A microstructure-based constitutive model accounting for rate dependence of superelastic NiTi alloy under cyclic deformation

In this work, a micromechanical constitutive model based on single crystal is proposed to account for the rate dependence and functional degradation of superelastic NiTi. Correspondence variant pair is treated as ellipsoid inclusion embedded in austenite matrix, and the stress distribution is obtained through Mori–Tanaka homogenization. Two inelastic mechanisms, that is, martensitic transformation and transformation-induced plasticity, are taken into consideration. Slipping is modeled to originate from austenite–martensite interface and is transferred between austenite and martensite. The heat equilibrium equation and thermodynamic driving force are deduced from the first law of thermodynamics and Clausius–Duhem inequality, respectively. Via the introduction of dominant <111> texture and Taylor approximation, the single crystal model is extended to polycrystalline version. The calibration and numerical implementation procedures for the model are systemized. The effects of key prescribed material parameters are discussed. Simulation results are validated against the experimental data. It is found that simulated thermomechanical responses agree with experimental observations reasonably.

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.