Anisotropic Inclusions Research Articles

Elastic Waves in Spatially Periodic Anisotropic Media: Diffraction by Periodic Arrays of Voids or Anisotropic Inclusions

Abstract A two-scale asymptotic analysis coupled with the spatially periodic fundamental solutions are used for analyzing diffraction of elastic bulk waves propagating in anisotropic media containing periodic inclusions or voids. Explicit equations are derived for the scattering cross sections and velocities of bulk waves propagating in spatially periodic media with arbitrary elastic anisotropy.

Elastic Analysis of the Infinite Plate with Anisotropic Inclusion

Elastic Analysis of the Infinite Plate with Anisotropic Inclusion

Anisotropic ellipsoidal inclusion with an anisotropic shell in an isotropic medium subjected to a uniform electric field

An electrostatic problem has been solved for a dielectric inclusion that consists of an anisotropic core and an anisotropic shell. The inclusion is immersed in a uniform isotropic medium (matrix) subjected to a uniform electric field. It is assumed that the outer boundaries of the core and shell are ellipsoidal and become confocal after a linear nonorthogonal transformation that removes the anisotropy of the dielectric properties of the shell. Analytical expressions have been derived for the potential and strength of the electric field in the matrix and also in the shell and core of the inclusion, and an expression for the polarizability tensor of the inclusion has been deduced. It has been shown that the results agree with the well-known solutions in partial (limiting) cases.

On the critical Casimir interaction between anisotropic inclusions on a membrane.

Using a lattice model and a versatile thermodynamic integration scheme, we study the critical Casimir interactions between inclusions embedded in a two-dimensional critical binary mixtures. For single-domain inclusions we demonstrate that the interactions are very long range, and their magnitudes strongly depend on the affinity of the inclusions with the species in the binary mixtures, ranging from repulsive when two inclusions have opposing affinities to attractive when they have the same affinities. When one of the inclusions has no preference for either of the species, we find negligible critical Casimir interactions. For multiple-domain inclusions, mimicking the observations that membrane proteins often have several domains with varying affinities to the surrounding lipid species, the presence of domains with opposing affinities does not cancel the interactions altogether. Instead we can observe both attractive and repulsive interactions depending on their relative orientations. With increasing number of domains per inclusion, the range and magnitude of the effective interactions decrease in a similar fashion to those of electrostatic multipoles. Finally, clusters formed by multiple-domain inclusions can result in an effective affinity patterning due to the anisotropic character of the Casimir interactions between the building blocks.

Determination of Stresses and Ultimate Loads for Composite Plates with Elastic Inclusions

In the article, the algorithm for determination of stresses in anisotropic plates with elastic inclusions of another anisotropic material was developed on the basis of complex singular integral equations. The solving of integral equations has been carried out numerically using the method of mechanical quadratures. The strength analysis (calculation of strength) of composite plates with inclusions has been performed using the Hoffman criterion.

New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces

In this work, a new micromechanical model to determine the effective elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces is developed. By using the Green's functions technique, the integral equation of the linear elastic medium is reformulated in the presence of imperfect interfaces. The integral equation is then solved in the case of an ellipsoidal inclusion embedded in an infinite matrix with imperfect interface. By introducing the concept of interior and exterior-point Eshelby tensors, the solution of the inclusion problem is obtained in the case of displacement discontinuities and continuous traction across the interface between inclusion and matrix. The concentration equations are then exactly expressed in the general case of anisotropic elastic behavior and ellipsoidal inclusions with imperfect interface. In the case of spring interface model, the explicit concentration equations are obtained. In the case of isotropic elasticity, we retrieved the analytical strain concentration tensors from the literature for spherical inclusion morphology. The Mori-Tanaka homogenization scheme is used to determine the local strain and stress fields in each phase and effective elastic moduli of composite materials. In the case of fibrous composites under plane strain loading, the effective elastic moduli are successfully compared to those obtained by complex variable methods. The combined influence of shape of inclusions and interface parameters is analyzed on the effective elastic behavior.

Equivalent-inclusion approach for the conductivity of isotropic matrix composites with anisotropic inclusions

Many effective medium approximations for effective conductivity are elaborated for matrix composites made from isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres, ... In this paper, we focus specially on the effective conductivity of the isotropic composites containing the disorderly oriented anisotropic inclusions. We aim to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified conductivities. Available simple approximations for the equivalent circular (spherical)-inclusion media then can be used to estimate the effective conductivity of the original composite. The equivalent-inclusion approach agrees well with numerical extended finite elements results.

Evolution of Strain Energy During Recrystallization of Plated Cu Films

Microstructural evolution within thin films is dictated by energy minimization that can arise due to different mechanisms. Conventional treatments that estimate the driving force associated with elastic strain energy often employ a fiber-textured microstructure to arrive at an analytical solution. By approximating the case of a recrystallized grain in a randomly oriented film as an elastically anisotropic inclusion in an elastically isotropic matrix, we can apply Eshelby's inclusion method to calculate both the interaction strains and the strain energy density generated by this elastic incompatibility. A comparison of these two approaches for grains with cubic symmetry reveals that a lower amount of elastic strain energy is generated in the case of an elastically anisotropic grain in an isotropic matrix, suggesting that it is less energetically favorable for Cu (111) grains to recrystallize in films possessing strong (111) texture than in randomly textured films. High-resolution X-ray diffraction measurements are used to extract the difference in the elastic response between recrystallized grains and the untransformed matrix, providing experimental quantification of the induced interaction strains. In addition, the corresponding elastic strain energy values are compared with the appearance of recrystallized (100) grains as a function of film thickness, allowing for a threshold energy density to be extracted. This threshold energy density can be correlated to the difference in surface energy values between (100)- and (111)-oriented grains.

Metamaterial ground cloak with anisotropic solid inclusions

There has been much research on the use of metamaterials within ground cloaks to hide objects on a rigid surface. Within the literature, ground cloaks made with metamaterials based on unit cells with solid inclusions have required the use of at least two distinct materials within the unit cell to achieve the necessary anisotropic material properties calculated from a transformation method. Here, we present a method to calculate the material properties of an anisotropic solid inclusion for use in a metamaterial ground cloak. The generated materials properties are then shown to be realized through an additively manufactured anisotropic solid inclusion. Finally, numerical results of cloaking performance of the metamaterial ground cloak are presented.

On the dynamic generalization of the anisotropic Eshelby ellipsoidal inclusion and the dynamically expanding inhomogeneities with transformation strain

The self-similarly dynamically (subsonically) expanding anisotropic ellipsoidal Eshelby inclusion is shown to exhibit the constant stress “Eshelby property” in the interior domain of the expanding inclusion on the basis of dimensional analysis, analytic properties and the proof for the static inclusion alone. As an example of this property and the application of the dynamic Eshelby tensor (constant in the interior domain), it is shown that the Eshelby equivalent inclusion method always allows for the determination of the equivalent transformation strain for a self-similarly dynamically expanding inhomogeneous spherical inclusion when the Poisson's ratio is in the real range (positive definiteness of the strain energy). Thus, the solution of dynamically self-similarly expanding inhomogeneities (chemical phase change) with transformation strain can be obtained, as well as the driving force per unit area of the expanding inhomogeneity.

The “rigorous” Maxwell homogenization scheme in 2D elasticity: Effective stiffness tensor of composite with elliptic inhomogeneities

Maxwell’s homogenization scheme is formulated in terms of the induced dipole moments of the representative volume element (RVE) of actual composite and properly defined equivalent inclusion. Taking interactions between the inhomogeneities into account makes this scheme rigorous in the sense that the effective properties converge to their exact values with increasing RVE size. To demonstrate potential of the “rigorous” Maxwell’s scheme, the 2D elasticity problem for a composite with elliptical inhomogeneities is considered. The complete series solutions for a finite cluster of inhomogeneities and equivalent anisotropic inclusion have been obtained by the method of complex potentials. The effective elastic moduli of composite have been found by equating the dipole strengths of RVE and equivalent inclusion. The convergence of solution is verified and an effect of structural parameters on the effective stiffness tensor of the composite has been evaluated. Numerical study shows that the proposed version of Maxwell’s scheme enables evaluation of the effective properties of both periodic and random structure composites with accuracy, comparable with that of Rayleigh’s method.

Automatic three-dimensional geometry and mesh generation of periodic representative volume elements for matrix-inclusion composites

This paper introduces an efficient method to automatically generate and mesh a periodic three-dimensional microstructure for matrix-inclusion composites. Such models are of major importance in the field of computational micromechanics for homogenization purposes utilizing unit cell models. The main focus of this contribution is on the creation of cubic representative volume elements (RVEs) featuring a periodic geometry and a periodic mesh topology suitable for the application of periodic boundary conditions in the framework of finite element simulations. Our method systematically combines various meshing tools in an extremely efficient and robust algorithm. The RVE generation itself follows a straightforward random sequential absorption approach resulting in a randomized periodic microstructure. Special emphasis is placed on the discretization procedure to maintain a high quality mesh with as few elements as possible, thus, manageable for computer simulations applicable to high volume concentrations, high number of inclusions and complex inclusion geometries. Examples elucidate the ability of the proposed approach to efficiently generate large RVEs with a high number of anisotropic inclusions incorporating extreme aspect ratios but still maintaining a high quality mesh and a low number of elements.

Evaluation of the Linear Thermal Expansion Coefficient of a Composite with Disperse Anisotropic Inclusions by the Self- Consistency Method

The self-consistency method is used to deduce a relation between the linear thermal expansion coefficient of a composite reinforced with anisotropic spherical inclusions and the thermoelastic characteristics of the matrix and inclusions and their volume concentration. This method considers the thermomechanical interaction of a separate inclusion with a homogeneous isotropic medium having the required value of the coefficient mentioned. On the basis of the dual variation formulation of a problem of thermoelasticity for a solid heterogeneous body, bilateral limits of possible values of this coefficient allowing one to check the reliability of the dependences obtained and to estimate the limit value of the error of calculation results are constructed. The relations presented can be used to predict the linear thermal expansion coefficient of a composite with disperse anisotropic inclusions, nanostructural ones among them.

Estimations and bounds of the effective conductivity of composites with anisotropic inclusions and general imperfect interfaces

The present work aims to determine the effective thermal conductivity of composites made of an isotropic matrix phase in which circular or spherical inhomogeneities are embedded. The inhomogeneity phases can be anisotropic and the interface between the inhomogeneity and matrix phase can be modeled by a general thermal imperfect interface model across which both the temperature and normal heat flux across can suffer a discontinuity. To achieve this objective, we derive first a unified and exact solution for the thermal fields of the inhomogeneity problem consisting of a spherical or circular anisotropic inhomogeneity inserted via a general thermal imperfect interface into an infinity isotropic matrix medium subjected to a remote uniform loading at its external surface. Unlike the relevant results in elasticity, the intensity and heat flux fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of the general thermal imperfect interface and anisotropy of inhomogeneity. Next, with the help of the foregoing solution results for the heterogeneity problem, the differential scheme is extended to predicting the effective thermal conductivity of composites with taking into account the imperfect interfaces between the constituent phases. Finally, the minimum potential and complementary energy principles and the morphologically representative pattern approach based on the Hashin–Shtrikman variational principles and the variational polarization principles are applied to such inhomogeneous materials and to bracketing their effective thermal properties. By constructing trial appropriate intensity and heat flux fields, the first- and second-order upper and lower bounds are obtained for the effective thermal conductivity of multiphase materials consisting of spherical or circular inhomogeneities embedded in a matrix. The estimations obtained by the differential scheme for the effective conductivity are shown to comply with the first- and second-order upper and lower bounds. Numerical results are provided to illustrate the dependence of the effective conductivity on the sizes of inhomogeneities and to compare the estimations with the relevant upper and lower bounds.

Planar auxeticity from elliptic inclusions

Abstract Composites with elliptic inclusions of long semi-axis a and short semi-axis b are studied by the Finite Element method. The centres of ellipses form a square lattice of the unit lattice constant. The neighbouring ellipses are perpendicular to each other and their axes are parallel to the lattice axes. The influence of geometry and material characteristics on the effective mechanical properties of these anisotropic composites is investigated for deformations applied along lattice axes. It is found that for anisotropic inclusions of low Young's modulus, when a + b → 1 the effective Poisson's ratio tends to −1, while the effective Young's modulus takes very low values. In this case the structure performs the rotating rigid body mechanism. In the limit of large values of Young's modulus of inclusions, both effective Poisson's ratio and effective Young's modulus saturate to values which do not depend on Poisson's ratio of inclusions but depend on geometry of the composite and the matrix Poisson's ratio. For highly anisotropic inclusions of very large Young's modulus, the effective Poisson's ratio of the composite can be negative for nonauxetic both matrix and inclusions. This is a very simple example of an auxetic structure being not only entirely continuous, but with very high Young's modulus. A severe qualitative change in the composite behaviour is observed as a/b reaches the limit of 1, i.e. inclusions are isotropic. The observed changes in both Poisson's ratio and Young's modulus are complex functions of parameters defining the composite. The latter allows one to tailor a material of practically arbitrary elastic parameters.

Anisotropic Inhomogeneous Poroelastic Inclusions: With Application to Underground Energy-Related Problems

Understanding the stress change in a reservoir generated by fluid production/injection is important for field development purposes. In this paper, we provide the Eshelby solution for stress and strain distribution inside and outside of an anisotropic poroelastic inhomogeneity due to pore pressure changes inside the inhomogeneity. The term anisotropic inhomogeneity refers to an inhomogeneity with anisotropic poroelastic constants. Some graphical results for strain and stress ratios for different material properties and geometries are presented as well. Anisotropy in elastic properties has been studied extensively in the last century; however, anisotropy in poroelastic properties, despite its potential significant impact in different engineering problems, has not been explored thoroughly. The results show how neglecting the effect of anisotropic poroelastic properties may result in large differences in calculated stresses. Due to the authors' primary interest in geomechanical problems, the discussions and examples are chosen for applications involving fluid withdrawal/injection into hydrocarbon reservoirs.

Polarizability of radially anisotropic elliptic inclusion

This article discusses a two-dimensional electrostatic scattering problem where an elliptic inclusion is suspended in a homogeneous background and impinged by an electric field which is uniform and static. The novelty of the discussion stems from the inclusion's material parameters. The material of the inclusion is assumed to be axially anisotropic, so that the axis of anisotropy aligns itself with the radial unit vector of the elliptic coordinate system. Similar varieties of anisotropy have been formerly referred to as radial anisotropy, and the same term is employed herein. The radially anisotropic elliptic inclusions are studied with an analytic method. The validation is likewise analytic. The validation method compares the new results with the results for radially anisotropic circles and homogeneous two-dimensional needles. The elliptic inclusion is found to facilitate both cloaking and field concentration.

Numerical analysis of SH wave field calculations for various types of a multilayered anisotropic inclusion

The Parallel Volume Integral Equation Method (PVIEM) is applied for the analysis of elastic wave scattering problems in an unbounded isotropic solid containing various types of a multilayered anisotropic inclusion. It should be noted that this numerical method does not require the use of the Green׳s function for the multilayered anisotropic inclusion-only the Green׳s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general elastodynamic problems involving arbitrary shape and number of inhomogeneous and/or multilayered anisotropic inclusions. A detailed analysis of the SH wave scattering is presented for various types of a multilayered orthotropic inclusion. Numerical results are presented for the displacement and stress fields at the interfaces of the multilayered inclusion in a broad frequency range. Parallel volume integral equation method as a pioneer of numerical analysis enables us to investigate the effects of single/multiple layer(s), multilayer׳s shape and geometry, isotropy/anisotropy, and softness/hardness of the multilayered anisotropic inclusion on displacements and stresses at the interfaces of the inclusion.

Effective elastic properties of a composite containing multiple types of anisotropic ellipsoidal inclusions, with application to the attachment of tendon to bone

Estimates of the effective stiffness of a composite containing multiple types of inclusions are needed for the design and study of a range of material systems in engineering and physiology. While excellent estimates and tight bounds exist for composite systems containing specific classes and distributions of identical inclusions, these are not easily generalized to systems with multiple types of inclusions. The best estimate available for a composite containing multiple classes of inclusions arises from the Kanaun–Jeulin approach. However, this method is analogous to a generalized Benveniste approach, and therefore suffers from the same limitations: while excellent for low volume fractions of inclusions, the Kanaun–Jeullin and Benveniste estimates liebelow three-point bounds at higher volume fractions. Here, we present an estimate for composites containing multiple classes of aligned ellipsoidal inclusions that lies within known three-point bounds at relatively higher volume fractions of inclusions and that is applicable to many engineering and biological composites. The approach involves replacing the averaged strains used in the Kanaun-Jeulin method with an effective strain measure. We demonstrate application of the constitutive model to the graded tissue system at the attachment of tendon to bone.

Modal method for the 2D wave propagation in heterogeneous anisotropic media.

A multimodal method based on a generalization of the admittance matrix is used to analyze wave propagation in heterogeneous two-dimensional anisotropic media. The heterogeneity of the medium can be due to the presence of anisotropic inclusions with arbitrary shapes, to a succession of anisotropic media with complex interfaces between them, or both. Using a modal expansion of the wave field, the problem is reduced to a system of two sets of first-order differential equations for the modal components of the field, similar to the system obtained in the rigorous coupled wave analysis. The system is solved numerically, using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed. The convergence of the method is discussed, considering arrays of anisotropic inclusions with complex shapes, which tend to show that Li's rules are not concerned within our approach. The method is validated by comparison with a subwavelength layered structure presenting an effective anisotropy at the wave scale.