Anisotropic Inclusions Research Articles

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.

Advective Transport Through Three-Dimensional Anisotropic Formations of Bimodal Hydraulic Conductivity

Transport of a non-reactive solute in anisotropic bimodal porous formations is investigated through the evaluation of the breakthrough curve. The hydraulic conductivity $$K$$ structure is modelled as an ensemble of anisotropic and non-overlapping inclusions, with given hydraulic conductivity $$K_{i}$$ , randomly placed in an homogeneous matrix with hydraulic conductivity $$K_{0}$$ . The conductivity contrast $$\kappa =K_{i}/K_{0}$$ between the inclusions and the matrix represents the main source of heterogeneity. The resulting $$K$$ structure is a theoretical representation of natural porous formations where inclusions and matrix play the role of two distinct hydrofacies. Semi-analytical solutions for flow and transport are carried out in both 3D and 2D domains following a Lagrangian approach by adopting the dilute limit approximation. The procedure, formally valid for media with low inclusions–matrix relative volume fraction, is extended to dense media by a simplified formulation of the self-consistent approach. Natural porous formations are intrinsically anisotropic due to the depositional process; hence, particular emphasis is addressed to the quantification of the effects of the anisotropy on solute transport. The solution of our physically based model depends on three structural parameters: the hydraulic conductivity contrast $$\kappa $$ , the relative volume fraction $$n$$ , and the statistical anisotropy ratio $$e$$ . The procedure allows for the physical representation of high-velocity channels and stagnation zones, which can be encountered in bimodal media, and it is particular advantageous for a quick estimation of the solute breakthrough curve. Our analysis shows different transport dynamics, as function of the conductivity contrast $$\kappa =K_{i}/K_{0}$$ , leading to fast preferential flow ( $$\kappa >1$$ ) or mass retention ( $$\kappa <1$$ ), as a consequence of the lack of symmetry of the flow and transport parameters (velocity and travel time residuals). High dispersion is typically found when $$\kappa <1$$ , i.e. for low conductive inclusions, where the latter typically trap the solute particles, determining large travel times. Conversely, when $$\kappa >1$$ flow in the inclusions is constrained by the outer flow, limiting the solute velocity inside the inclusions, which results is a rather limited dispersion. Anisotropy typically enhances the solute spreading for $$\kappa <1,$$ as a consequence of the slower velocity predicted for more anisotropic inclusions, while its impact on formations with high $$\kappa $$ is rather limited.

An iterative FE–BE method and rectangular cell model for effective elastic properties of doubly periodic anisotropic inclusion composites

Based on the rectangular cell model cut from doubly periodic anisotropic inclusion medium, this paper presents the effective elastic properties of the mentioned problems by an iterative FE–BE coupling method. This method is easy to be numerically implemented and especially suitable for the analysis of anisotropic inclusions embedded in an isotropic matrix, allows a wide range of microgeometries of the composite and determination of the complete set of effective elastic properties. Besides, the adopted method also avoids using the fundamental solutions of anisotropic materials and overcomes the difficulties of solving the inclusions with irregular shapes in the BEM. In calculation, the anisotropic inclusion is discretized into finite elements, whereas the boundary of the rectangular cell and the inclusion-matrix interface are meshed into a series of boundary elements. Some numerical examples are used to validate the applicability and reliability of the present scheme.

Multiple Scattering Using Parallel Volume Integral Equation Method: Interaction of SH Waves with Multiple Multilayered Anisotropic Elliptical Inclusions

The parallel volume integral equation method (PVIEM) is applied for the analysis of elastic wave scattering problems in an unbounded isotropic solid containing multiple multilayered anisotropic elliptical inclusions. This recently developed numerical method does not require the use of Green’s function for the multilayered anisotropic inclusions; only Green’s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general two- and three-dimensional elastodynamic problems involving inhomogeneous and/or multilayered anisotropic inclusions whose shape and number are arbitrary. A detailed analysis of the SH wave scattering is presented for multiple triple-layered orthotropic elliptical inclusions. Numerical results are presented for the displacement fields at the interfaces for square and hexagonal packing arrays of triple-layered elliptical inclusions in a broad frequency range of practical interest. It is necessary to use standard parallel programming, such as MPI (message passing interface), to speed up computation in the volume integral equation method (VIEM). Parallel volume integral equation method as a pioneer of numerical analysis enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), multilayer’s shape and geometry, isotropy/anisotropy, and softness/hardness of the multiple multilayered anisotropic elliptical inclusions on displacements at the interfaces of the inclusions.

Thermal properties of composite two-layer systems with a fractal inclusion structure

In this work, we study the thermal transport properties of platelike composite two-layer samples made of polyester resin and magnetite inclusions. By means of photoacoustic spectroscopy and thermal relaxation, their effective thermal diffusivity and conductivity were experimentally measured. The composite layers were prepared under the action of a static magnetic field, resulting in anisotropic (fractal) inclusion structures with the formation of chain-like magnetite aggregates parallel to the faces of the layers. In one kind of the bilayers, a composite layer was formed on top of a resin layer while their relative thickness was varied. These samples can be described by known models. In contrast, bilayers with the same concentration of inclusions and the same thickness on both sides, where only the angle between their inclusion structures was systematically varied, show a nontrivial behaviour of their thermal conductivity as a function of this angle. Through a multifractal and lacunarity analysis, we explain the observed thermal response in terms of the complexity of the interface between the layers.

Mica Inclusions inside Host Mica Grains from the Sutlej Section of the Higher Himalayan Crystallines, India-Morphology and Constrains in Genesis

Biotite and muscovite inclusions inside mica host minerals from the Sutlej section of the Higher Himalayan Crystalline were studied under an optical microscope. These inclusions formed possibly by local recrystallization of mica grains during regional prograde metamorphism, with some affected by top-to-SW shear leading to parallelogram shapes. Recrystallization may have been assisted by solution transfer along the cleavage planes of the host grains. The relative competency of deformed phyllosilicate inclusions with the same or different composition to the host depends on the size and orientation of (001) cleavage planes of the inclusions relative to the host. Shearing of mica inclusions led to their parallelogram geometries within the contained mica inclusions. Some of the sheared inclusions deflect cleavage planes in the host minerals and define flanking microstructures. Trapezoid-shaped inclusions are a new finding that deserves more attention for their genesis. These structurally anisotropic inclusions did not originate from sub-grains, secondary infillings or retrogression. These inclusions are also not related to pseudomorphism, isomorphism, folding of the bulk rock etc. Some of the inclusions formed by recrystallization of the host mineral during top-to-SW ductile shear.

A new internal variables homogenization scheme for linear viscoelastic materials based on an exact Eshelby interaction law

A new time-incremental internal variables homogenization scheme for Maxwellian linear viscoelastic heterogeneous materials is proposed. This scheme is based on the exact solution of the ellipsoidal Eshelby inclusion problem obtained in the time domain. In contrast with current existing methods, the effective behavior as well as the evolution laws of the averaged stresses per phase are solved incrementally in the time domain without need to analytical or numerical inverse Laplace–Carson transforms. This is made through a time-differential equation in addition to the more classic strain rate concentration equation. In addition, the new derived interaction law for the Eshelby inclusion problem is provided in a compact matrix form. It is proved that this is an exact formulation for an arbitrary anisotropic ellipsoidal Maxwellian inclusion embedded in an isotropic Maxwellian matrix. In order to show the interest of the present approach, the results are reported and discussed with a Mori–Tanaka homogenization scheme for two-phase composites in comparisons with other exact or approximate methods.

Two-sided Estimates of Effective Thermal Conductivity of the Composite with Anisotropic Lamellar Inclusions

Two-sided Estimates of Effective Thermal Conductivity of the Composite with Anisotropic Lamellar Inclusions

Dispersion properties of liquid crystal core photonic crystal fibers calculated by a multipole method modified for anisotropic inclusions

Temperature dependence of the dispersion properties of liquid crystal core photonic crystal fibers with large air fraction in clads between 300 and 2000nm for different core diameters have been calculated by a multipole method, which is modified to treat anisotropic inclusions rigorously. In calculations, air holes are assumed to be arranged in a regular hexagonal array in fused silica and a central hole is filled with liquid crystal to create a core. Below the clearing point temperature, the liquid crystal molecules are assumed to be oriented parallel to the cylindrical axis of the holes, where the liquid crystal is highly anisotropic. The large changes of the dispersion properties are found at the clearing point temperature, where the liquid crystal becomes isotropic.

On delaminated thin Timoshenko inclusions inside elastic bodies

In the paper, we consider equilibrium problems for 2D elastic bodies with thin inclusions modeled in the frame of Timoshenko beam theory. It is assumed that a delamination of the inclusion takes place thus providing a presence of cracks between the inclusion and the elastic body. Nonlinear boundary conditions at the crack faces are imposed to prevent a mutual penetration between the faces. Different problem formulations are analyzed: variational and differential. Dependence on physical parameters characterizing the mechanical properties of the inclusion is investigated. The paper provides a rigorous asymptotic analysis of the model with respect to such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non‐penetration conditions, respectively. Also anisotropic inclusions with parameters are analyzed when parameters tend to zero and infinity. In particular, in the limit, we obtain the so called semi‐rigid inclusions. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Shear-Wave Birefringence Measurements in Faulted Near-Surface Sediment: Fluorspar Area Fault Complex, Central United States

Abstract The location and geometry of seismogenic faults are fundamental to understanding the regional seismic hazard; however, surface evidence for active faults in the seismically active central United States is rare, thus contributing to the uncertainty in the hazard estimation. Multicomponent S ‐wave reflection energy was acquired near the northern boundary of the New Madrid Seismic Zone across a well‐constrained N61°E‐oriented fault that extends above the top of Paleozoic bedrock and into near‐surface Quaternary sediment in order to evaluate the effectiveness of shear‐wave splitting measurements for determining azimuthal orientation of fault‐deformed unlithified sediment that lack a surface manifestation. The data were analyzed within the shear‐wave window (i.e., without wide‐angle distortion) using orthogonal source–receiver orientations. Normalized reflection energy spectra of the main and cross‐diagonal elements were rotated in 5° increments to determine fast (S1) and slow (S2) S ‐wave directions associated with the azimuthally anisotropic inclusion. The resultant natural coordinate system was N60°E and N30°W for S1 and S2, respectively. The S1 orientation correlated with the established N61°E fault strike. In addition, there was an average dynamic mistie of 11.3 ms that suggests 3.6% azimuthal anisotropy between the top of bedrock and ground surface. These results suggest that shear‐wave birefringence can be a useful tool for defining azimuthal orientation of neotectonic inclusions that lack surface or geomorphic expression, as well as for areas that may have limited accessibility (i.e., urban environments) for deploying multiple widespread geophysical surveys.

Thermal properties of composite materials with a complex fractal structure

In this work, we report the thermal characterization of platelike composite samples made of polyester resin and magnetite inclusions. By means of photoacoustic spectroscopy and thermal relaxation, the thermal diffusivity, conductivity and volumetric heat capacity of the samples were experimentally measured. The volume fraction of the inclusions was systematically varied in order to study the changes in the effective thermal conductivity of the composites. For some samples, a static magnetic field was applied during the polymerization process, resulting in anisotropic inclusion distributions. Our results show a decrease in the thermal conductivity of some of the anisotropic samples, compared to the isotropic randomly distributed ones. Our analysis indicates that the development of elongated inclusion structures leads to the formation of magnetite and resin domains, causing this effect. We correlate the complexity of the inclusion structure with the observed thermal response through a multifractal and lacunarity analysis. All the experimental data are contrasted with the well known Maxwell–Garnett effective media approximation for composite materials.

Modeling and characterizing anisotropic inclusion orientation in heterogeneous material via directional cluster functions and stochastic microstructure reconstruction

We present a framework to model and characterize the microstructure of heterogeneous materials with anisotropic inclusions of secondary phases based on the directional correlation functions of the inclusions. Specifically, we have devised an efficient method to incorporate both directional two-point correlation functions S2 and directional two-point cluster functions C2 that contain non-trivial topological connectedness information into the simulated annealing microstructure reconstruction procedure. Our framework is applied to model an anisotropic aluminum alloy and the accuracy of the reconstructed structural models is assessed by quantitative comparison with the actual microstructure obtained via x-ray tomography. We show that incorporation of directional clustering information via C2 significantly improves the accuracy of the reconstruction. In addition, a set of analytical “basis” correlation functions are introduced to approximate the actual S2 and C2 of the material. With the proper choice of basis functions, the anisotropic microstructure can be represented by a handful of parameters including the effective linear sizes of the iron-rich and silicon-rich inclusions along three orthogonal directions. This provides a general and efficient means for heterogeneous material modeling that enables one to significantly reduce the data set required to characterize the anisotropic microstructure.

Three-phase anisotropic elliptical inclusions with internal uniform in-plane and anti-plane stresses

We study the stress field of a three-phase composite in which an internal elliptical inclusion is bonded to the surrounding matrix through an interphase layer. The linearly elastic materials occupying both the inclusion and the matrix are generally anisotropic, whereas the interphase layer is made of an isotropic elastic material. The two interfaces of the three-phase composite are confocal ellipses. Two conditions are found that ensure that the internal in-plane and anti-plane stress field is uniform. When these conditions are met, the mean stress within the isotropic interphase layer is also uniform. A real form expression of the internal uniform stress field inside the inclusion is derived. Several examples are presented to demonstrate and validate the obtained results.

Effects of morphology and geometry of inclusions on two-point correlation statistics in two phase composites

A spectral form of the two-point correlation functions for two-phase composites is presented and the effects of secondary phase morphology, geometry and volume fraction on the coefficients of the corresponding series are studied. Initially, a principal component analysis is performed on the coefficients of the series and then, the effects of different morphological and geometrical parameters of the second phase inclusions on two-point correlation functions are examined. A direct linkage between the principal components of the coefficients with the morphological features (inclusion shape, aspect ratio and size) is established and the influence of preferred orientation of the anisotropic inclusions is also investigated. Finally a composite hull is constructed using the principal components of the coefficients in a three-dimensional space, which helps in determining the interrelation of properties and microstructures of any composite containing these types of inclusions.

Electromechanical response of (3–0, 3–1) particulate, fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.

Control of boundary impedance in two-dimensional material-body cloaking by the wave flow method

Control problems are considered for a two-dimensional electromagnetic field model describing electromagnetic wave scattering in a unbounded homogeneous medium containing an anisotropic permeable inclusion with a partially covered (cloaked) boundary. The control is a function involved in the impedance boundary condition on the covered part of the boundary. The solvability of the original mixed transmission problem for the two-dimensional Helmholtz equation and of the control problems is proved. Optimality systems describing necessary extremum conditions are derived. The uniqueness and stability of optimal solutions with respect to certain perturbations of the cost functional and the incident wave are established.

Non‐linear deformation properties of materials with stochastically distributed anisotropic inclusions

AbstractIn the present contribution, the problem of non‐linear deformation of materials with stochastically distributed anisotropic inclusions is considered on the basis of the methods of mechanics of stochastically non‐homogeneous media. The homogenization model of materials of stochastic structure with physically non‐linear components is developed for the case of a matrix which is strengthened by unidirectional ellipsoidal inclusions. It is assumed that the matrix is isotropic, deforms non‐linearly; inclusions are linear‐elastic and have transversally‐isotropic symmetry of physical and mechanical properties. Stochastic differential equations of physically non‐linear elasticity theory form the underlying equations. Transformation of these equations into integral equations by using the Green's function and application of the method of conditional moments allow us to reduce the problem to a system of non‐linear algebraic equations. This system of non‐linear algebraic equations is solved by the Newton‐Raphson method. On the analytical as well as the numerical basis, the algorithm for determination of the non‐linear effective characteristics of such a material is introduced. The non‐linear behavior of such a material is caused by the non‐linear matrix deformations. On the basis of the numerical solution, the dependences of homogenized Poisson's coefficients on macro‐strains and the non‐linear stress‐strain diagrams for a material with randomly distributed unidirectional ellipsoidal pores are predicted and discussed for different volume fractions of pores. (© 2013 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Shear localization in three-dimensional amorphous solids

In this paper we extend the recent theory of shear localization in two-dimensional (2D) amorphous solids to three dimensions. In two dimensions the fundamental instability of shear localization is related to the appearance of a line of displacement quadrupoles that makes an angle of 45^{∘} with the principal stress axis. In three dimensions the fundamental plastic instability is also explained by the formation of a lattice of anisotropic elastic inclusions. In the case of pure external shear stress, we demonstrate that this is a 2D triangular lattice of similar elementary events. It is shown that this lattice is arranged on a plane that, similarly to the 2D case, makes an angle of 45^{∘} with the principal stress axis. This solution is energetically favorable only if the external strain exceeds a yield-strain value that is determined by the strain parameters of the elementary events and the Poisson ratio. The predictions of the theory are compared to numerical simulations and very good agreement is observed.

Thermal conductivity of the textured composite with anisotropic inclusions in the form of ellipsoids of rotation

Thermal conductivity of the textured composite with anisotropic inclusions in the form of ellipsoids of rotation