# Isotropic And Anisotropic Materials Pdf

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*There are two eigentensors for a linear isotropic elastic material; one is the deviatoric second-rank tensor, and the other is a second-rank tensor proportional to the unit tensor and often called the spherical or hydrostatic part of the tensor. The eigentensors of isotropic elasticity have many properties of physical and mathe-matical significance.*

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- Classification and characterization of electromagnetic materials

*Several applications wood, plants, muscles require modeling the directional dependence of the material elastic properties in three orthogonal directions. We investigate linear orthotropic materials, a special class of linear anisotropic materials where the shear stresses are decoupled from normal stresses, as well as general linear non-orthotropic anisotropic materials.*

Another condition that can fit the anisotropic definition is the presence of different properties in different directions. A different chemical bonding in all directions is also a condition for anisotropy. A mineral can be considered as anisotropic if it allows some light to pass through it. The velocity of light is also different, and there is double refraction which means that light is split in two directions.

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In this paper, we present an efficient method to classify complex electromagnetic materials. This method is based on the directional interaction of incident circularly polarized waves with the materials being tested. The presented method relies on an algorithm that classifies the test materials to one of the following categories: isotropic, chiral, bi-isotropic, symmetric anisotropic or general bianisotropic.

Both analytical and numerical solutions are used to compute fields of the circularly polarized waves from the arbitrary complex material slab. The complex materials are discriminated accordingly and then classified under an appropriate category.

Recent advances in the interaction of electromagnetic fields with complex composite media suggest the feasibility of creating novel materials with unusual electromagnetic properties and the possibility of constructing new electromagnetic devices using such materials. These improved materials can be engineered to possess several unique electromagnetic properties that make them suitable candidates for numerous applications in modern technology systems.

Applications related to the fields of photonics 1 , optoelectronics 2 , 3 , radar cross-section reduction 4 , 5 , 6 , 7 , gigahertz devices 8 , antenna reconfiguration 9 , terahertz plasmonic filters 10 , and biosensors 11 are the leading beneficiaries of such developments.

These complex electromagnetic materials are often further classified to isotropic, bi-isotropic, anisotropic and bianisotropic based on their macroscopic electromagnetic properties 12 , 13 , 14 , which provide the description of certain materials through their constitutive relations.

Isotropic materials like unstressed glass and plastic, water and air, and fluids at rest, behave precisely in the same manner regardless of the direction of the wave propagation axis, because their permittivities and permeabilities are identical in all directions.

In contrast, in anisotropic materials like wood, the electromagnetic properties are different in different directions. Materials with crystalline structure are biaxial anisotropic materials.

For example, yttrium orthosilicate YSO is a dielectric material with biaxial anisotropy at optical frequencies. This well-known rare-earth host material has shown promising performance in quantum-engineered optical devices development Yttrium aluminum perovskite YAP crystal material, with orthorhombic symmetry, is designated as a symmetric anisotropic material Gyrotropic magnetized ferrites are asymmetric anisotropic materials that have been used in microwave engineering for years because of their non-reciprocal behavior that makes them very useful in the design of microwave devices like isolators, polarizers, and circulators Graphene in the static magnetic field is a gyrotropic and uniaxial anisotropic material in the absence of the external magnetic field More recently, materials with optical activity chirality have been considered for application in microwave and infrared regions.

Chiral materials are a particular case of the more general bi-isotropic materials. An artificial composite metamaterial composed of split ring resonators with rods 20 can be modeled as bianisotropic material. Efficient methods for the classification and characterization of such materials based on determining their macroscopic electromagnetic properties are of growing importance but still in evolution 12 , 13 , The traditional characterization techniques used to measure the electromagnetic properties of scalar and tensorial permittivity and permeability of complex materials have received significant efforts in recent years 1 , 3 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , These techniques are, essentially, based on measuring the scattering parameters from the investigated materials in free space 22 , 28 using single or multiple normal 32 and oblique 33 linearly polarized wave incidence.

The characterization procedures start from deducing the effective refractive index and effective wave impedance from the transmitted and reflected fields. However, these methods are liable to obtaining multiple branching ambiguity, which limits using them at high frequencies, especially for thick material slabs.

In addition, the methods that use optimization schemes are usually accompanied by heavy computation costs and often yield multiple solutions. Efforts have been introduced to overcome these discontinuities based on the Kramers—Kronig K—K relation 22 , 34 , and phase correction techniques 35 , 36 , However, the K—K method is saturated at high frequencies, which limits its performance especially for thick structures.

The phase correction techniques are sensitive to simple errors in phase data and they, also, are susceptible to slip to error solutions at zero refractive index values or if they were initialized at an arbitrary starting frequency This paper presents a novel method to classify complex electromagnetic materials based on their behavioral directional interaction with incident circularly polarized waves. The method is based on a classification algorithm of the unknown materials to one of the following categories: isotropic, chiral, bi-isotropic, symmetric anisotropic, or general bianisotropic.

The proposed scheme is a simple and direct classification process without demanding a complete intricate extraction process for the tensor elements that are usually accompanied by multiple solutions that need carful processing. Beside that, after classification, the characterization process extraction of the tensor elements becomes much easier when the investigated material model is predetarmined. Solutions from an analytical method based on the transmission matrix method TMM and numerical results from a full wave simulator are used to compute the transmitted and reflected fields of the circularly polarized waves from the arbitrary complex material slab.

The different complex materials are discriminated accordingly and then classified under an appropriate category. Both analytical and numerical solutions are used to obtain the RCP and LCP transmitted and reflected electromagnetic fields from the investigated materials. The analytical solutions use the TMM 38 , and the numerical solution uses a full wave simulator. If the reflection or transmission coefficients of the incident RCP or LCP waves in these axes are the same, then the investigated material belongs to the general bi-isotropic structure.

If not, the sample is a general bianisotropic material. For general bi-isotropic structures, if the refractive indices related to the RCP and LCP waves are similar, then the material is simply isotropic. If the imaginary part of the refractive indices related to the RCP and LCP are the same, then the material is chiral In this section, the scattering parameters of the selected materials are determined using analytical or numerical solutions. Bi-isotropic media are the most general linear, homogenous and isotropic materials.

The constitutive relations that define the bi-isotropic media are given by. Biaxial and gyrotropic media are used in this study as illustrations for symmetric and asymmetric anisotropic materials, respectively. Gyrotropic media is defined by permittivity and permeability tensors as. Recently, researches focus their efforts on the study of the interaction of electromagnetic fields with complex materials in the terahertz frequency ranges for possible new applications and devices.

Terahertz investigations of structures like graphene-dielectric stacks 42 , metal and graphene hybrid metasurface 43 , organic crystals for THz photonics 44 , terahertz metamaterials 45 , 46 and chiral metamaterials 46 are few examples of such efforts.

The methods presented in this paper are general and can be applied to any frequency band; still it is demonstrated here at the terahertz bands.

Drude and Lorentz Models 22 are used to select parameters of the tested bi-isotropic material as shown in Fig. The permittivity is modelled using the Drude module, whereas the permeability and the magnetoelectric coupling coefficients are modelled using the Lorentz model.

For chiral material, the Tellegen coefficient is set to zero, and the other parameters are similar to those of the Tellegen general bi-isotropic material. The selected permittivities of the symmetric and asymmetric anisotropic materials are based on Drude 22 and Debye 47 models, respectively, as shown in Fig. The permeabilities of the symmetric and asymmetric anisotropic material are considered as unity in this investigation.

Parameters of illustrated a bi-isotropic and chiral materials, b symmetric anisotropic material, and c asymmetric anisotropic material. Three different orthogonal measurements of the RCP and LCP waves transmitted, and reflected electromagnetic fields from the investigated materials are engaged. The dielectric tensor after rotation in the xyz coordinate system is given by.

The scattering parameters of both chiral and bi-isotropic materials are the same for different propagation axes. It is zero for chiral materials and nonzero for bi-isotropic materials as described in the classification algorithms.

Both of them are dependent on the orientation of the propagation axis. This information can help to classify the investigated material as an asymmetric anisotropic material. Absolute difference of refractive indices of RCP and LCP of symmetric and asymmetric anisotropic materials for three measurements of x, y and z propagation axes.

After the classification of complex materials, characterization of electromagnetic materials becomes easier. For bi-isotropic materials, the permittivity and permeability are defined as For the z propagation axis, the parameters of the gyrotropic material are given as 41 , For the x or y propagation axes, the third permittivity element can be determined as. The extracted parameters agree with the correct parameters in the numerical scales of the presented figures.

Extracted parameters of the tested a bi-isotropic material b gyrotropic material. This paper presented an efficient method to classify complex electromagnetic materials based on their directional interaction with incident circularly polarized waves.

The method used an algorithm that classifies unknown materials to one of the following categories: isotropic, chiral, bi-isotropic, symmetric anisotropic or general bianisotropic. Solutions from an analytical method based on the transmission matrix method TMM and numerical results from a full wave simulator were used to compute the fields of the circularly polarized waves from the arbitrary complex material slab.

The complex materials were discriminated accordingly and classified under an appropriate category. Chen, X. Robust method to retrieve the constitutive effective parameters of metamaterials. E 70 , Hua-Qiang, W. Graphene applications in electronic and optoelectronic devices and circuits. B 22 , Fokin, V.

Method for retrieving effective properties of locally resonant acoustic metamaterials. B 76 , Gao, C. Song, J. Broadband and tunable RCS reduction using high-order reflections and salisbury-type absorption mechanisms. Chaudhury, B. Study and optimization of plasma-based radar cross section reduction using three-dimensional computations. IEEE Trans. Plasma Sci. Singh, H. Guerriero, E. Gigahertz integrated graphene ring oscillators. ACS Nano 7 , — Malhat, H.

Moazami, A. High efficiency tunable graphene-based plasmonic filter in the THz frequency range.

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Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. In this paper, we present an efficient method to classify complex electromagnetic materials. This method is based on the directional interaction of incident circularly polarized waves with the materials being tested.

Isotropic and anisotropic are two important terms widely used to explain the material properties in material science and crystal morphology in basic crystallography. In certain materials like crystals, the orientation of atoms is very important as it affects their physical and mechanical properties. Based on the orientation of atoms, materials are broadly divided into two classes namely: isotropic materials and anisotropic materials. The main difference between isotropic and anisotropic is that the properties of isotropic materials are the same in all directions, whereas in anisotropic materials, the properties are direction dependent. What is the difference between Isotropic and Anisotropic. If the properties mechanical, physical, thermal and electrical properties of a material do not change with different crystallographic orientations, or in other words, the properties are direction independent, that material is called isotropic. Isotropic crystals have one refractive index in all directions.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: The following paper presents the results of modelling of the magnetic characteristics of isotropic Ni—Zn ferrite and anisotropic material Fe40Ni38Mo4B18 amorphous alloy in as quenched state , both useful for sensor applications. For the modelling an extended Jiles—Atherton model was used. View via Publisher.

## Classification and characterization of electromagnetic materials

Even W. Hovig, Amin S. Improving the success rate in additive manufacturing and designing highly optimized structures require proper understanding of material behaviour. This study proposes a novel experimental method by which anisotropic mechanical properties of additively manufactured materials can be assessed.

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