Ellipsoidal Inclusions Research Articles

Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a, b, and c) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied.

Image-based simulation of dispersed composites and porous rocks

The article presents the results of numerical simulation of the effective elastic properties and stress concentration of the dispersed composite and porous rocks as well as the elastoplastic diagrams of tensile stress of the B4C/2024Al composite using asymptotic averaging method. 2D and 3D simulations of uniaxial tension was carried out using the image-based models of the real structure obtained by X-ray tomography. We also used model structures of the B4C/2024Al composite, consisting of an aluminum matrix with boron carbide inclusions in the form of ellipsoids with a random orientation. It turned out that in the elastic region 3D real structure and 3D ellipsoidal structure lead to the same effective moduli. On contrary, the stress concentrations are much higher in the real structure model than in the model of ellipsoidal inclusions. Also, two types of the models behave completely differently in the plastic region. Comparison of 2D and 3D calculations with experimental data shows that 2D models can be also used for determining the effective elastic moduli of B4C/2024Al composite, but the use of 3D models is preferable for porous geomaterials and dispersed composites.

A mechanistic damage model for solid oxide fuel cell ceramic materials - Part I: Constitutive modeling

A multiscale mechanistic damage model for solid oxide fuel cell (SOFC) ceramics that combines micromechanics of stiffness reduction due to material porosity change and microcracking with a continuum damage mechanics (CDM) description for the evolution of microcracks up to fracture is developed in this work. The model also accounts for volumetric swelling of the anode due to redox cycling. Porosity change and swelling during redox combined with normal operating thermomechanical loads could increase the stress, strain and damage distributions in SOFC stacks leading them to failure. The proposed model involves three governing parameters: (i) the porosity in terms of the pore or void volume fraction, (ii) material swelling magnitude prescribed at a given loading step, and (iii) the damage variable that describes microcracking caused by thermomechanical loads and swelling. Pores and microcracks in the ceramics are modeled as randomly distributed ellipsoidal inclusions with negligible stiffness by an Eshelby-Mori-Tanaka formulation combined with an inclusion orientation distribution method. Microcracking damage evolution is described by a CDM formulation. Volumetric swelling is treated in a similar way to thermal expansion in the constitutive relations. After validation and numerical checks, the damage model is used to analyze a simplified SOFC positive-electrode/electrolyte/negative-electrode (PEN) structure subjected to a redox cycle in addition to thermomechanical loads experienced during normal operation of a SOFC stack.

Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions

The Eshelby formalism for an inclusion in a solid is of significant theoretical and practical implications in mechanics and other fields of heterogeneous media. Eshelby’s finding that a uniform eigenstrain prescribed in a solitary ellipsoidal inclusion in an infinite isotropic medium results in a uniform elastic strain field in the inclusion leads to the conjecture that the ellipsoid is the only inclusion that possesses the so-called Eshelby uniformity property. Previously, only the weak version of the conjecture has been proved for the isotropic medium, whereas the general validity of the conjecture for anisotropic media in three dimensions is yet to be explored. In this work, firstly, we present proofs of the weak version of the generalized Eshelby conjecture for anisotropic media that possess cubic, transversely isotropic, orthotropic, and monoclinic symmetries, which substantiates that only the ellipsoidal shape can transform all uniform eigenstrains into uniform elastic strain fields in a solitary inclusion in infinite media possessing these symmetries. Secondly, we prove that in these anisotropic media, there exist non-ellipsoidal inclusions that can transform particular polynomial eigenstrains of even degrees into polynomial elastic strain fields of the same even degrees in them. These results constitute counter-examples, in the strong sense, to the generalized high-order Eshelby conjecture (inverse problem of Eshelby’s polynomial conservation theorem) for polynomial eigenstrains in both anisotropic media and the isotropic medium (quadratic eigenstrain only). In addition, we also show that there are counter-examples to the strong version of the generalized Eshelby conjecture for uniform eigenstrains in these anisotropic media. A sufficient condition for the existence of those counter-example inclusions is provided. These findings reveal striking richness of the uniformity between the eigenstrains and the correspondingly induced elastic strains in inclusions in anisotropic media beyond the canonical ellipsoidal inclusions. Since the strain fields in embedded and inherently anisotropic quantum dot crystals are effective tuning knobs of the quality of the emitted photons by the quantum dots, the results may have implications in the technology of quantum information, in addition to in mechanics and materials science.

Nonlocal theory: modeling of thermomechanical processes in a composite with inclusions in the form of ellipsoids of revolution

An approach to the construction of a mathematical model of a thermomechanical process in a composite material with inclusions in the form of ellipsoids of revolution is considered. This approach is based on models of generalized continuum mechanics and Eringen’s nonlocal medium. This approach makes it possible to take into account the processes that occur at the micro- and nanolevel. Physical processes are described using internal state parameters and integral terms, which describe the influence of ellipsoidal inclusions on the propagation of heat and stresses using the “influence function” in the corresponding area. Integro-differential relations are obtained to describe the process of propagation of heat and temperature stresses in a composite material with inclusions in the form of ellipsoids of revolution.

Modeling of geometric configuration and fiber interactions in short fiber reinforced composites via new modified Eshelby tensors and enhanced mean-field homogenization

Determination of the effective elastic properties of unidirectional short fiber reinforced composites (SFRCs) is a key fundamental requirement which is usually estimated by mean-field homogenization (MFH) methods. Available classical MFH methods only account for volume fraction and aspect ratio of the fibers and do not consider the physical cylindrical geometry of the fibers. Moreover, classical MFHs are not designed for predicting the effective properties of SFRCs with distinct packing configurations and order of fibers. In this work, new modified Eshelby tensors associated with an enhanced MFH are developed which resolve the mentioned limitations of the available MFH methods and preserve their computational efficiency. While classical Eshelby tensors used in MFHs depend only on the aspect ratio of the fictitious ellipsoidal inclusion and material properties of the matrix, the introduced modified Eshelby tensors additionally account for physical cylindrical geometry as well as interactions of the short fibers with different packing configurations. Moreover, non-uniform homogenizing eigenstrains required for accurate homogenization of the short fibers are also incorporated by the proposed analytical treatment. Predictions of the developed enhanced MFH for various packing configurations are compared with those obtained from Finite Element Method implementing periodic boundary conditions and very good agreements are observed. It is shown that packing configuration and geometrical specifications of short fibers can significantly affect the effective properties of the SFRC.

Erratum to: Ellipsoidal Inclusion with a Shell in an Anisotropic Medium Subjected to a Uniform Electric Field

Erratum to: Ellipsoidal Inclusion with a Shell in an Anisotropic Medium Subjected to a Uniform Electric Field

A Study on Strain Energy of Ellipsoidal Inclusion in Half-space

An inclusion refers to localized eigenstrains appearing in such processes as thermal expansion and plastic deformation. In view of micromechanics, the existence of inclusions may significantly influence the mechanical properties of the engineering materials. A micromechanical model is proposed to determine the variation of the strain energies in the presence of the near-surface inclusions. The corresponding inclusion problem in a half-space is usually difficult to be solved analytically. In this work, the strain energy is evaluated numerically via the method of images, which superposes the counterpart solutions in full-space and eliminates the tractions on the boundary surface of the half-space. The validity of the present work is confirmed by comparing with the published results and the finite element method (FEM).

Well conditioned mathematical modeling for homogenization of thermo-electro-mechanical behaviors of piezoelectric composites

In the present paper, a mathematical modeling of effective properties of fully coupled thermo-electro-mechanical heterogeneous materials is elaborated. Constitutive equations combined with equilibrium equations lead to a coupled partial differential system. A methodological procedure is elaborated based on thermo-electro-elastic Green’s functions, integral equations, and a micro-macro approach. Localization tensors are derived using the Mori–Tanaka mean-field assumptions and the fully coupled effective behavior is obtained through averaging techniques. Due to the large dispersion between elastic, dielectric, and piezoelectric coefficients ill-conditioned localization tensors are resulted. Block matrix decomposition is elaborated to overcome this drawback and the so-regularized localization tensor’s inverses are used to get a well conditioned problem. Numerical computations are done in the general case of ellipsoidal inclusions and anisotropic behavior. Effective thermal conductivity and heat capacity coefficients are obtained with and without the heat strain effect for various volume fractions and types of inclusions. The global properties of thermo-electro-elastic reinforced composites are presented with respect to shape and volume fractions of inclusions.

A time-incremental homogenization method for elasto-viscoplastic particulate composites based on a modified secant formulation

Starting from the exact solution of the heterogeneous linear viscoelastic Eshelby ellipsoidal inclusion problem obtained in the time domain for an ellipsoidal inclusion embedded in an isotropic matrix (Berbenni et al., 2015), a direct time-incremental homogenization Mori–Tanaka scheme for two-phase non linear elasto-viscoplastic materials is proposed. The extension to non-linear behavior is based on a “modified secant” formulation to obtain the linear comparison composite (LCC) properties. In contrast with more classic homogenization method based on the first moment of stresses (i.e. classic secant formulation), the present approach is based on the second-order moment of stresses making use of the Hill’s lemma. Hence, a modified secant viscoplastic compliance is considered for the matrix phase. The effective behavior as well as the evolution laws of the averaged strains and stresses per phase are directly solved in the time domain, without the need of Laplace-Carson transforms. The estimates provided by the present homogenization model are compared to full-field calculations from the literature for two-phase non linear particulate composites constituted of an isotropic elasto-viscoplastic matrix phase and isotropic elastic particles, and, for radial and non-radial loadings.

FFT and Equivalent-inclusion approach for effective conductivity of three-phase composites with random coated-ellipse inclusion

Unlike sphere inclusion, every ellipsoid inclusion is anisotropic, but compound-inclusion composite material with random distribution of coated-ellipse inclusion can be considered as isotropic. The paper introduces the Fast Fourier transformation method (FFT) and equivalent inclusion approach to calculate the effective conductivity of the transverse isotropic unidirectional coated-fiber composite with coated-ellipse inclusion. In the equivalent inclusion approximation, the two-phase of coated-ellipse inclusions are substituted by the equivalent homogeneous one of the same size and corresponding mechanical properties, from here we can build a simple approximation formula (EI) to determine effective conductivity of three-phase composites. Beside that, the FFT simulation can be combined with the equivalent inclusion approach to obtain the results FFT-EI.

Thermal metadevices with geometrically anisotropic heterogeneous composites

The classical effective medium theory (EMT) provides a powerful tool in designing heterogeneous-composite functional thermal metadevices. But it is usually appropriate for spatially symmetric or geometrically isotropic systems, which limits the flexibility or adjustability in control. Here, we develop a generalized EMT to predict effective thermal conductivities of heterogeneous systems with elliptical (or ellipsoidal) inclusions. With the generalized EMT, omnidirectional thermal invisibility and directional Janus thermal illusion are designed and experimentally demonstrated just by tuning orientations of shape-anisotropy particles. In particular, we try to clarify the respective scopes of the generalized Maxwell-Garnett and Bruggeman theory based on both theoretical interpretations and numerical simulations. Our work may offer a promising fundamental framework and application prototype in elaborating thermal metadevices with asymmetry, irregularity, or anisotropy in configuration.

Analytical solution for residual stress and strain preserved in anisotropic inclusion entrapped in an isotropic host

Abstract. Raman elastic thermobarometry has recently been applied in many petrological studies to recover the pressure and temperature (P–T) conditions of mineral inclusion entrapment. Existing modelling methods in petrology either adopt an assumption of a spherical, isotropic inclusion embedded in an isotropic, infinite host or use numerical techniques such as the finite-element method to simulate the residual stress and strain state preserved in the non-spherical anisotropic inclusions. Here, we use the Eshelby solution to develop an analytical framework for calculating the residual stress and strain state of an elastically anisotropic, ellipsoidal inclusion in an infinite, isotropic host. The analytical solution is applicable to any class of inclusion symmetry and an arbitrary inclusion aspect ratio. Explicit expressions are derived for some symmetry classes, including tetragonal, hexagonal, and trigonal. The effect of changing the aspect ratio on residual stress is investigated, including quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Quartz is demonstrated to be the least affected, while rutile is the most affected. For prolate quartz inclusion (c axis longer than a axis), the effect of varying the aspect ratio on Raman shift is demonstrated to be insignificant. When c/a=5, only ca. 0.3 cm−1 wavenumber variation is induced as compared to the spherical inclusion shape. For oblate quartz inclusions, the effect is more significant, when c/a=0.5, ca. 0.8 cm−1 wavenumber variation for the 464 cm−1 band is induced compared to the reference spherical inclusion case. We also show that it is possible to fit an effective ellipsoid to obtain a proxy for the averaged residual stress or strain within a faceted inclusion. The difference between the volumetrically averaged stress of a faceted inclusion and the analytically calculated stress from the best-fitted effective ellipsoid is calculated to obtain the root-mean-square deviation (RMSD) for quartz, zircon, rutile, apatite, and diamond inclusions in garnet host. Based on the results of 500 randomly generated (a wide range of aspect ratio and random crystallographic orientation) faceted inclusions, we show that the volumetrically averaged stress serves as an excellent stress measure and the associated RMSD is less than 2 %, except for diamond, which has a systematically higher RMSD (ca. 8 %). This expands the applicability of the analytical solution for any arbitrary inclusion shape in practical Raman measurements.

Generalized Formula for Effective Dielectric Permeability of the Medium with Ellipsoidal Inclusions

Generalized Formula for Effective Dielectric Permeability of the Medium with Ellipsoidal Inclusions

On the computational modelling of nonlinear electro-elasticity in heterogeneous bodies at finite deformations

Dielectric elastomer actuators (DEA) have been demonstrated to exhibit a quasi-immediate electro-mechanical actuation response with relatively large deformation capability. The properties of DEA make them suitable to be used in the form of major active components within soft robotics and biomimetic artificial muscles. However, some of the electro-active material properties impose limitations on its applications. Therefore, researchers attempt to modify the structure of the homogeneous DEA material by the incorporation of fillers that possess distinct electro-mechanical properties. This modification of the material’s structure leads to a fabricated inhomogeneous composite. From the point of mathematical material modelling and numerical simulation, we propose a material model and a computational framework using the finite element method, which is capable of emulating nonlinear electro-elastic interactions. We consider a coupled electro-mechanical description with the electric and the electro-mechanical properties of the material assumed to be nonlinearly dependent on the deformation. Furthermore, we demonstrate a coupled ansatz that expresses the electric response as dielectrically quasi-linear with only density-dependent electric permittivity. We couple the electro-mechanical models to the extended tube model, which is a suitable approach for the realistic emulation of the hyperelastic response of rubber-like materials. Thereafter, we demonstrate analytical and numerical solutions of a homogeneous electro-elastic body with the Neo-Hookean material model and the extended tube model to express the hyperelastic response. Finally, we use the finite element method to investigate several heterogeneous configurations consisting of soft DEA matrix filled with spherical stiff inclusions with changing volume fraction and ellipsoidal inclusions with varying aspect ratio.

“Volume collapse” instabilities in deep-focus earthquakes: A shear source nucleated and driven by pressure

Symmetry-breaking instabilities in high-pressure phase transformation produce the counterintuitive phenomenon of “volume collapse” producing only shear radiation, with little, or no, volumetric component, even under conditions of full isotropy, and explain the mystery of the long-standing observations in deep-focus earthquakes (400–700 km). Due to instability, at a critical “nucleation pressure”, an arbitrarily small densified region in the shape of a “pancake-like” flattened ellipsoidal Eshelby inclusion, grows self-similarly as a “lacuna” (zero particle velocity) with the phase transformation occurring under conditions of equilibrium in uniform strain/stress and at constant potential energy (at the vanishing of the M integral, when the radius-expanding driving force pεkk∗ overcomes the radius-shrinking self-force). The symmetry-breaking flattened shape favors minimization of the energy needed for the boundary to grow large, while for the accommodation of the large collapsing volume in the very thin inclusion deviatoric stresses are developed to avoid openings and overlaps. It is shown that, if an arbitrarily small flattened densified region is generated planarly, and the pressure exceeds the critical nucleation value, then it will necessarily produce a shear seismic source, with little or no, volumetric component, nucleated and driven to propagate by the pressure. The ellipsoid of phase change forms in the direction that minimizes the interaction energy with the pre-stress field and will be close to the direction of max shear pre-stress in the mantle. The obtained stress/deformation fields of a densified 2D flattened elliptical inclusion constitute a new defect that models the “anticrack” in geophysics and densified shear bands. The instability analysis can be extended to the nucleation and growth of the phase transition from water to a solid ice phase under high pressure, with the discovered instabilities providing insight to other phenomena of dynamic phase transformations, such as failure waves, amorphization, planetary impacts, etc.

A finite strain thermomechanically-coupled constitutive model for phase transformation and (transformation-induced) plastic deformation in NiTi single crystals

A 3D finite-strain constitutive model for the deformation response of NiTi at the single crystal level is proposed. The model accounts for reversible phase transformation from austenite to martensite habit plane variants, plastic deformation in the austenite phase, and rate effects induced from latent heat. It is developed within the formalism of irreversible thermodynamics with internal state variables based on the Eulerian logarithmic strain and its corrotational objective rate. The inelastic deformation is defined as an average over a representative volume element as classical in the micromechanics-based modeling approach. Transformation-induced plastic deformation is viewed as a mechanism for accommodation of the local deformation incompatibility at the austenite–martensite interface. It is accounted for by introducing an interaction term in the free energy, which is described through the Eshelby tensor by regarding the habit plane variants as ellipsoidal inclusions embedded in the austenite matrix, in order to accurately reflect the internal stress states that contribute to dislocation slipping. The numerical implementation of the model in an efficient scheme and its calibration are described in detail. The proposed model is validated by comparing simulations with available experimental data in single NiTi crystals. Numerical simulations of polycrystals are performed to obtain an insight into the interaction between phase transformation and plastic deformation induced by intergrannular constraints. The efficiency of the numerical implementation of the model is verified by simulations of indentation tests.

Research and application of seismic porosity inversion method for carbonate reservoir based on Gassmann’s equation

Abstract Carbonate rocks frequently exhibit less predictable seismic attribute–porosity relationships because of complex and heterogeneous pore geometry. Pore geometry plays an important role in carbonate reservoir interpretation, as it influences acoustic and elastic characters. So in porosity prediction of carbonate reservoirs, pore geometry should be considered as a factor. Thus, based on Gassmann’s equation and Eshelby–Walsh ellipsoidal inclusion theory, we introduced a parameter C to stand by pore geometry and then deduced a porosity calculating expression from compressional expression of Gassmann’s equation. In this article, we present a porosity working flow as well as calculate methods of every parameter needed in the porosity inverting equation. From well testing and field application, it proves that the high-accuracy method is suitable for carbonate reservoirs.

Need the Uniform Stress Field Inside Multiple Interacting Inclusions Be Hydrostatic?

Since the pioneering work of Eshelby on a single ellipsoidal inclusion embedded in an infinite space, much attention has been devoted in the literature to the question of the uniformity of the stress field inside inclusions surrounded by an elastic matrix. Over the last decade or so, researchers have established the existence of multiple (interacting) inclusions enclosing uniform internal stress distributions when embedded in an infinite elastic matrix subjected to a uniform far-field loading and identified a variety of shapes of such inclusions. In the design of multiple inclusions with uniform internal stresses, it is customary to assume that the uniform stress field inside each inclusion (each with different shear modulus distinct from that of the matrix) is hydrostatic. In this paper, we examine whether this assumption is actually necessary to ensure the required existence of multiple inclusions enclosing uniform stresses. By establishing several theorems in the theory of functions of a complex variable, we prove rigorously that for any collection of multiple inclusions enclosing uniform stresses in an infinite isotropic plane subjected to uniform remote in-plane loading, the internal uniform stress field must indeed be hydrostatic if the corresponding inclusion’s shear modulus is distinct from that of the matrix.

An ellipsoidal inclusion and an elliptical crack in the elastic media with ellipsoidal anisotropy

An isolated ellipsoidal inclusion and an elliptical crack in a homogeneous elastic medium with ellipsoidal anisotropy are considered. For constant external stress (strain) fields applied to the medium, the solutions of these problems are reduced to calculation of 2D-regular integrals. For materials with ellipsoidal anisotropy, the integrand functions in these integrals can be presented in explicit analytical forms that simplifies substantially construction of the numerical solutions of the problems. The stress intensity factors at the crack edge are expressed in the forms of 1D-integrals which integrand functions have also explicit analytical forms for media with ellipsoidal anisotropy.