Determination of the scattered fields of an SH-wave by an eccentric coating-fiber ensemble using DEIM

To date, the existing theories pertinent to the determination of the scattered fields of an inhomogeneity have been limited to certain topological symmetries for which the method of wave-function expansion is widely used. In the literature the wave-function expansion method has also been employed to the case involving concentric coated fiber. An alternative approach is the dynamic equivalent inclusion method (DEIM) proposed by Fu and Mura [L.S. Fu, T. Mura, The determination of elastodynamic fields of an ellipsoidal inhomogeneity. ASME J. Appl. Mech. 50 (1983) 390-396.] who found the scattered field of a single spheroidal inhomogeneity. The pioneering work of Eshelby [J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. London, Ser. A A241 (1957) 376-396.] on elastostatic EIM is based on polynomial form of eigenstrains which holds certain useful properties and subsequently its application is only effective for certain relevant situations and not necessarily efficient for other problems. Nevertheless, Fu and Mura’s analysis is also based on polynomial eigenstrains. It will be shown that taking the dynamic homogenizing eigenstrains in the form of the series expansion whose general term is products of functions of r and trigonometric functions of θ, is more rigorous and attractive for the problem under consideration. This natural form of solution gives very accurate result with just the first few terms of the series. Moreover, this work aims to extend the DEIM to the case of coated fiber obstacle with the rather complex topology where the coating-fiber phases are not concentric. The effect of variableness of the coating thickness on the elastodynamic fields is examined. Comparison with other analytical solutions, whenever available, establishes the remarkable accuracy and robustness of the proposed theory.     

The electro-elastic scattered fields of an SH-wave by an eccentric two-phase circular piezoelectric sensor in an unbounded piezoelectric medium

The dynamic equivalent inclusion method (DEIM) which was first proposed by Fu and Mura (1983), in its original context has some shortcomings, which were pointed out and remedied by Shodja and Delfani (2009) who introduced the new consistency conditions along with the related micromechanically substantiated notion of eigenstress and eigenbody-force fields. However, these theories are bound to elastic media with isotropic phases. The present work extends the idea of the above-mentioned new DEIM to the dynamic electro-mechanical equivalent inclusion method (DEMEIM) for the treatment of the scattering of SH-waves by a two-phase circular piezoelectric obstacle bonded to a third phase piezoelectric matrix. All the three transversely isotropic media have the same rotational axis of symmetry and the same poling direction which are parallel to the axis of the coated fiber, but perpendicular to the direction of propagation of the incident SH-wave. In general, the nested circular media are considered to be eccentric, i.e., the core fiber has a coating with variable thickness. Realization of the nature of the behavior of the field quantities a priori and its appropriate implementation in to the new extended consistency conditions is a critical step to insure a rigorous mathematical framework. As it will be shown, the expansion of the Green’s function and the eigenelectric, eigenstress, and eigenbody-force fields in terms of the eigenfunctions of the pertinent field equations rather than the commonly considered polynomials in the traditional equivalent inclusion method (EIM) leads to an accurate solution with high convergence rate.The exact analytical expression for the total scattering cross-section which is influenced by the piezoelectric couplings is derived. The effects of the piezoelectric couplings and the properties of the fiber, coating, and the matrix as well as the wave number on the electro-mechanical scattered fields are examined.     

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby’s language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions.     

A general treatment of piezoelectric double-inhomogeneities and their associated interaction problems

The present paper addresses an analytical method to determine the electroelastic fields over a double-phase piezoelectric reinforcement interacting with an ellipsoidal single-inhomogeneity. The approach is based on the extension of the electro-mechanical equivalent inclusion method (EMEIM) to the piezoelectric double-inhomogeneity system. Accordingly, the double-inhomogeneity is replaced by an electroelastic double-inclusion problem with proper polynomial eigenstrains-electric fields. The long- and short-range interaction effects are intrinsically incorporated by the homogenizing eigenfields. The equivalent double-inclusion is subsequently decomposed to the single-inclusion problems by means of a superposition scheme. The methodology is further extended to the piezoelectric multi-inhomogeneity, where the particle core is surrounded by many layers of coatings of ellipsoidal shapes. Through consideration of various examples, including (1) 2D and 3D interaction problems of a coated piezoelectric reinforcement near a lamellar inhomogeneity and (2) a two-phase spherical particle with thick coating of variable thickness, the validity and robustness of the present theory are thoroughly demonstrated.     

Neutral spherical inhomogeneities in a twisted circular cylindrical bar

We consider the torsion problem of a circular cylindrical bar which is filled up with composite spherical inclusions. The composite inclusions consist of a core and coating both of which are spherically orthotropic with the volume fractions of the core being the same in every composite inclusion. The center points of the spherical inhomogeneities are on the axis of revolution of the circular cylinder. The neutral inhomogeneity in the considered problem of elastic equilibrium is defined as a foreign body (inclusion) which can be introduced in a host body without disturbing the elastic field (displacements, stresses) in it. The conditions of the neutral inhomogeneity for the twisted circular cylindrical bar are derived, and some special cases of inhomogeneity are analyzed. The present paper gives a new example for neutral inhomogeneity in the field of elasticity.     

Effects of interfacial excess energy on the elastic field of a nano-inhomogeneity

A semi-analytical approach based on a variational framework is developed to obtain the three-dimensional solution for a nano-scale inhomogeneity with arbitrary eigenstrains embedded in a matrix of infinite extent. Both the inhomogeneity and the matrix can be elastically anisotropic. The Gurtin–Murdoch surface/interface model is used to describe the elastic behavior of the inhomogeneity/matrix interface. The displacement fields in the inhomogeneity and the matrix are represented, respectively, by two sets of polynomials. Coefficients of these polynomials are determined by solving a system of linear algebraic equations that are derived from minimizing the total potential energy of the system. In the case of an isotropic spherical inhomogeneity with dilatational eigenstrain in an isotropic matrix, our solution shows an excellent agreement with the corresponding analytical solution available in the literature. To demonstrate the capabilities of the method developed here and to investigate the effect of interfacial excess energy, numerical examples are also presented when the inhomogeneity and matrix are both elastically anisotropic. Both dilatational and pure shear eigenstrains are considered in these examples.     

Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials

Most effective medium approximations for isotropic inhomogeneous materials are based on dilute solutions of some typical inclusions in an infinite matrix medium, while the simplest approximations are those for the composites with spherical and circular inclusions. Practical particulate composites often involve inhomogeneities of more complicated geometry than that of the spherical (or circular) one. In our approach, those inhomogeneities are supposed to be substituted by simple equivalent spherical (circular) inclusions from a comparison of their respective dilute solution results. Then the available simple approximations for the equivalent spherical (circular) inclusion material can be used to estimate the effective conductivity of the original composite. Numerical illustrations of the approach are performed on some 2D and 3D geometries involving elliptical and ellipsoidal inclusions.     

Computation of electromagnetic fields scattered by a cylindrical inhomogeneity in a homogeneous medium of infinite extent

Summary A method is presented for computing time harmonic electromagnetic fields scattered by a cilindrical inhomogeneity in a homogeneous medium of infinite extent. Geometrically, the homogeneity is a cylinder of arbitrary cross-section. Outside a circular cylinder that completely surrounds the inhomogeneity, the electromagnetic field is expanded in terms of wave functions of the circular cylinder. Inside this cylinder, the electromagnetic field equations are transformed into a system of ordinary differential equations in the radial direction. The relevant system behaves numerically unstable and is therefore transformed into a stable one through a specific transformation scheme. To elucidate the validity and the versatility of the method, numerical results are presented for fields scattered by a number of different cylindrical inhomogeneities.     

Periodic distributions of inclusions

Elastic fields caused by periodically distributed spherical inclusions in an infinite isotropic medium are investigated when eigenstrains in the inclusions are homogeneous polynomials of degree $\text{l}$ of the local coordinates taken at the centers of the inclusions. The stress field can be expressed as a convergent sum of the solutions of individual inclusions when $\text{l}$ is an odd number. The stress field must be modified when $\text{l}$ is an even number. If a distribution of inclusions is on a plane or a line, the stress field can be expressed as a convergent sum of the individual inclusions for all $\text{l}$. The analysis is extended to general ellipsoidal inclusions with arbitrary eigenstrains in anisotropic media.     

Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites

A novel computational model is presented using the eigenstrain formulation of the boundary integral equations for modeling the particle-reinforced composites. The model and the solution procedure are both resulted intimately from the concepts of the equivalent inclusion of Eshelby with eigenstrains to be determined in an iterative way for each inhomogeneity embedded in the matrix. The eigenstrains of inhomogeneity are determined with the aid of the Eshelby tensors, which can be readily obtained beforehand through either analytical or numerical means. The solution scale of the inhomogeneity problem with the present model is greatly reduced since the unknowns appear only on the boundary of the solution domain. The overall elastic properties are solved using the newly developed boundary point method for particle-reinforced inhomogeneous materials over a representative volume element with the present model. The effects of a variety of factors related to inhomogeneities on the overall properties of composites as well as on the convergence behaviors of the algorithm are studied numerically including the properties and shapes and orientations and distributions and the total number of particles, showing the validity and the effectiveness of the proposed computational model.     

Interacting circular inhomogeneities in plane elastostatics

Summary This paper provides a general series solution to the problem of interacting circular inhomogeneities in plane elastostatics. The analysis is based upon the use of the complex stress potentials of Muskhelishvili and the Laurent series expansion method. The general forms of the complex potentials are derived explicitly for the circular inhomogeneity problem under arbitrary plane loading. Using the superposition principle, these general expressions were subsequently employed to treat the problem of an infinitely extended matrix containing any number ofarbitrarily located inhomogeneities. The above procedure reduces the problem to a set of linear algebraic equations which are solved with the aid of a perturbation technique. The current method is shown to be capable of yielding approximate closed-form solutions for multiple inhomogeneities, thus providing the explicit dependence of the solution upon the partinent parameters.     

The problem of thermal conduction for two ellipsoidal inhomogeneities in an anisotropic medium and its relevance to composite materials

Summary We consider the problem of thermal conduction for an unbounded medium containing two ellipsoidal inhomogeneities subjected to a remote homogeneous boundary condition of temperature. The constituents are anisotropic and the ellipsoids could be at arbitrary orientations. In the formulation we first introduce some appropriate transformations into the heterogeneous medium and transform the problem into an isotropic matrix consisting of two analogous ellipsoidal inhomogeneities. Next, we replace the effect of inhomogeneities by some polynomial types of equivalent eigen-intensities by the concept of equivalent inclusion. These procedures allow us to write the local fields in terms of harmonic potentials and their derivatives. Numerical results show that linear approximations of eigen-fields yield accurate results in comparison with existing solutions by Honein et al. [2] for moderately separated inhomogeneities. Solutions of this type are used to estimate the overall thermal conductivity of composites with periodic microstructure. Finally, we present results for composites consisting of spherical inclusions with body-centered cubic, face-centered cubic, body-centered orthorhombic, and face-centered orthorhombic arrays.     

Thermal and moisture expansion of material elements and macroscopic bodies with geometrical symmetries—wood logs and sawn goods as practical examples

Thermal and moisture expansion states available to material elements with particular symmetries, as well as macroscopic bodies with particular structural symmetries, are investigated in terms of Group Theory and Curie’s principle. It is found that in the case of one-dimensional material symmetry, Eigen shear strain of the material element must be zero within any plane perpendicular to the reflection plane. The same applies to the most simple two-dimensional rotation symmetry, within any plane including the rotation axis. The corresponding reflection-rotation symmetry requires all the Eigen shear strains to be zero. Greater 2-d rotation symmetries require, in addition, normal eigenstrains perpendicular to the rotation axis to be equal, which means transverse isotropy of the eigenstrain tensor. Such a strain state complies with three-dimensional rotation symmetries, as long as not more than one of the symmetries is of order greater than 2. In such case all the normal eigenstrains must be equal, corresponding to isotropy of the eigenstrain tensor. Unlike two-dimensional symmetries, any three-dimensional reflection–rotation symmetry does not place more restrictions on the eigenstrain state than the corresponding rotation symmetry. There is, however, a three-dimensional symmetry which places no restriction on the eigenstrain state. Possible non-uniform eigenstrain states of macroscopic bodies are discussed. Eigendeformations of wood logs are illustrated as examples. It is found that one abstract symmetry group may correspond to several geometrical symmetry groups, and the geometrical symmetry group of a body may depend on the choice of co-ordinate system. Log symmetries translate into sawn goods provided the sawing pattern complies with the log symmetries.     

Diffuse-interface modeling of composition evolution in the presence of structural defects

A diffuse-interface phase field model is described for modeling the interactions between compositional inhomogeneities and structural defects. The spatial distribution of these structural defects is described by the space-dependent eigenstrains. It takes into account the effect of the coherency elastic energy of a compositional inhomogeneity, and the elastic coupling between the coherency strains and defect strains. The temporal evolution of composition is described by the Cahn–Hilliard equation. Particularly, the solute segregation, and nucleation and growth around dislocation slip bands and crack-like condensed interstitial dislocation loops are discussed. The effect of nucleated coherent precipitates on the stress field around these defects is analyzed     

Superconducting transition widths in applied magnetic fields

The width of superconducting-normal transitions for inhomogeneous ellipsoidal superconductors in an applied magnetic field is calculated. A simple model is used for the variation in transition temperature with position and the range of this variation is assumed to be larger than the Landau-Ginzburg coherence length. It is found that the decrease in transition width with increasing applied magnetic fields at low fields occurs because the boundaries between superconducting and normal regions begin to be determined less by the inhomogeneities and more by the loss of condensation energy between domains, and by sample surface effects. At higher fields, the transition width increases with applied field because of the finite demagnetization coefficient.Contribution of the National Bureau of Standards. Not subject to copyright.     

A new finite element for treating plane thermomechanical heterogeneous solids

This study is concerned with the development and implementation of a new finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids under mechanical and thermal loadings. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The newly proposed eight‐noded plane element can be used to treat quite readily the two‐dimensional steady‐state heat conduction and thermoelastic problems of an elastic circular inclusion embedded in an elastic matrix with different thermomechanical properties. Moreover, the devised element may be applied to deal with arbitrarily and periodically located multiple inhomogeneities under general mechanical and thermal loading conditions using a very limited number of elements. The current element also enables the determination of the local and effective thermoelastic properties of composite materials with relative ease. Three numerical examples are given to demonstrate its versatility, accuracy and efficiency. Copyright © 1999 John Wiley & Sons. Ltd.     

A novel finite element for treating inhomogeneous solids

This study is concerned with the development and implementation of a novel finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The strength of the proposed eight-noded plane element is demonstrated by its ability to treat arbitrarily and periodically located multiple inhomogeneities under general loading conditions using a limited number of elements. Assessment of the accuracy and efficiency of the devised element is obtained by comparing its performance against existing analytical and traditional finite element attempts. The current element enables the determination of the local and effective elastic properties of composite materials with relative ease.     

Multiple 3D inhomogeneous inclusions in a half space under contact loading

A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading.     

On the finite deformation of an inhomogeneity in a viscous liquid

Previous work on the problem of the slow deformation of an ellipsoidal inhomogeneity inside a viscous liquid of different viscosity is extended to derive the equations governing the flow of an initially spherical inhomogeneity when a two-dimensional pure strain rate is applied at infinity. The equations are solved numerically to find the three-dimensional change of shape of the inhomogeneity.     

The scattering of electro-elastic waves by a spherical piezoelectric particle in a polymer matrix

This paper is devoted to the study of scattering of plane harmonic waves by a piezoelectric sphere with spherical isotropy embedded in an unbounded isotropic polymer matrix. The scattered displacement field and the electric potential in the matrix are expressed in terms of spherical vector wave functions and spherical harmonic functions, respectively. For the field points inside the inhomogeneity, new displacement functions are introduced. Expansion of the new displacement functions and the electric potential in terms of spherical harmonic functions, the equations of motion and electrostatic lead to four second order ordinary differential equations (odes), where three of them are coupled. The coupled system of odes is solved by the generalized Frobenius series. This approach is readily used to handle low and high frequencies. Three different types of piezoelectric inhomogeneities, PZT-4, PZT-5H, and BaTiO₃ are considered and the associated piezoelectric effects on the electro-mechanical fields, differential and total scattering cross-sections are addressed.