Constitutive Tensors Research Articles

Perfect electromagnetic conductor medium

This article presents a review of a novel concept in electromagnetics, the Perfect Electromagnetic Conductor (PEMC). In the Minkowskian representation of the material response to electromagnetic fields, PEMC corresponds to the axion part of the constitutive tensor. From the electrical engineering point of view, PEMC is a generalization of the perfect electric conductor (PEC) and perfect magnetic conductor (PMC) materials which are useful concepts as ideal boundaries in the modeling of electromagnetic problems. PEMC is characterized by a pseudoscalar, admittance-type parameter M, with limits 1/M (PEC) and M (PMC). This paper discusses how the PECM medium as a boundary or in particulate form generalizes earlier known electromagnetic problems (reflection, scattering, materials modeling, waveguides, resonators, image principles). The main effect of PEMC boundary is to rotate the polarization plane of the reflected electric field vector. Also practical realization aspects of a PEMC medium are discussed in the paper.

Finite element analysis of viscoelastic composite structures based on a micromechanical material model

The stress and creep analysis of structures made of micro-heterogeneous composite materials is treated as a two-scale problem, defined as a mechanical investigation on different length scales. Reinforced composites show by definition a heterogeneous texture on the microlevel, determined by the constitutive behaviour of the matrix material and the embedded fibres as well as the characteristics of the bonding properties in the interphase. All these heterogeneities are neglected by the finite element analysis of structural elements on the macroscale, since a ficticious and homogeneous continuum with averaged properties is assumed. Therefore, the constitutive equations of the substitute material should well reflect the mechanical behaviour of the existing micro-heterogeneous composite in an average sense.The paper at hand starts with the brief outline of a micromechanical model, named generalized method of cells (GMC), which provides the macrostress responses due to macrostrain processes as well as the homogenised constitutive tensor of the substitute material. The macroscopic stresses and strains are obtained as volume averages of the corresponding microfields within a representative volume element. The effective material tensor constitutes the mapping between the macro-strains and the macro-stresses. The cells method is used for the homogenisation of the unidirectionally reinforced single layers of laminates made of viscoelastic resins and flexibly embedded elastic fibres. The algorithm for the homogenisation of the constitutive properties runs simultaneously to the finite element analysis at each point of numerical integration and provides the macro-stresses and the homogenised constitutive properties. The validity of the proposed two-scale simulation is investigated by solving boundary value problems and comparing the numerical results for the structures to the experimental data of creep and relaxation tests or analytical solutions.

Metamaterial Blueprints for Reflectionless Waveguide Bends

A derivation of metamaterial blueprints for the reflectionless (perfectly matched) guiding of electromagnetic waves through waveguide bends is performed. The sensitivity of the response with respect to small perturbations in the associated constitutive tensors is examined.

Conformal Perfectly Matched Layer for the Mixed Finite Element Time-Domain Method

We introduce a conformal perfectly matched layer (PML) for the finite-element time-domain (FETD) solution of transient Maxwell equations in open domains. The conformal PML is implemented in a mixed FETD setting based on a direct discretization of the first-order coupled Maxwell curl equations (as opposed to the second-order vector wave equation) that employs edge elements (Whitney 1-form) to expand the electric field and face elements (Whitney 2-form) to expand the magnetic field. We show that the conformal PML can be easily incorporated into the mixed FETD algorithm by utilizing PML constitutive tensors whose discretization is naturally decoupled from that of Maxwell curl equations (spatial derivatives). Compared to the conventional (rectangular) PML, a conformal PML allows for a considerable reduction on the amount of buffer space in the computational domain around the scatterer(s).

Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure

We report a polarization beam splitter (PBS) utilizing the anomalous reflection and transmission of an anisotropic metamaterial slab. By properly design of the constitutive tensors of the slab, PBS is achieved with little dependence on incident angle and slab thickness. The realization of the proposed PBS in the optical range is achieved by layered metal-dielectric nanostructured system, which could be modeled as an effective anisotropic metamaterial with the desired constitutive tensors. The full-wave electromagnetic simulation has confirmed the optical polarization splitting with an extinction ratio of over 20dB by a slab of an alternating silver and silicon carbide multilayer.

Editorial: Ann. Phys. 1/2008

The aim of this paper is to provide an algebraic decomposition of the principal part of the electromagnetic constitutive tensor χμνρσ (a pre-metric structure), in a sum of terms of the form where the Z(i) are invariant scalars and gμν(i)unimodular conformal metrics. If the space-time is also equipped with a volume measure, such a decomposition is equivalent to a similar decomposition of the associated area metric, where the gμν(i) are ordinary metrics. It has been recently argued [8] that, if such a particular decomposition should be available, it would then be possible to find a compatible connection and derived tensors.

Algebraic decomposition of the electromagnetic constitutive tensor. A step towards a pre‐metric based gravitation?

AbstractThe aim of this paper is to provide an algebraic decomposition of the principal part of the electromagnetic constitutive tensor χμνρσ (a pre‐metric structure), in a sum of terms of the form where the Z(i) are invariant scalars and gμν(i)unimodular conformal metrics. If the space‐time is also equipped with a volume measure, such a decomposition is equivalent to a similar decomposition of the associated area metric, where the gμν(i) are ordinary metrics. It has been recently argued [8] that, if such a particular decomposition should be available, it would then be possible to find a compatible connection and derived tensors.

Algebraic decomposition of the electromagnetic constitutive tensor. A step towards a pre-metric based gravitation?

It is known that the electromagnetic constitutive tensor can be algebraically decomposed in three parts: the so-called principal part, the axion part and the skewon part (see gr-qc/0506042). The aim of this paper is to provide a deeper decomposition of the principal part.

Simulation of asymmetric effects for shape memory alloys

AbstractExperimental results of shape memory alloys show a pronounced asymmetric behaviour between tension, compression and shear. For simulation of these effects in the constitutive equations different transformation strain tensors are introduced. These are related to the different variants for the multi‐variant‐ and detwinned‐martensite as a consequence of different stress states. In the framework of plasticity the concept of stress mode dependent weighting functions is applied in order to characterize the different stress states. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behaviour of shape memory alloys with different hardening characteristics in tension, compression and shear. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Diffraction by an imperfect half plane in a bianisotropic medium

A general theory to study the electromagnetic diffraction by imperfect half planes immersed in linear homogeneous bianisotropic media is presented. The problem is formulated in terms of Wiener‐Hopf equations by deriving explicit spectral domain expressions for the characteristic impedances of bianisotropic media, which allow one to exploit their analytical properties. In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener‐Hopf equations involve matrices of order 2, which can be factorized in closed form if the constitutive tensors of the bianisotropic material are of special form. Four of these special cases are discussed in detail. In order to deal with the more general problem, a technique to numerically factorize the Wiener‐Hopf matrix kernels is presented. Our numerical approach is discussed on one example, by considering the previously unsolved problem of a perfect electric conducting half plane in a gyrotropic medium. The reported numerical results show that the diffracted field contribution is obtained by use of the saddle point integration method.

Two-frequency radiative transfer: Maxwell equations in random dielectrics

The paper addresses the space-frequency correlations of electromagnetic waves in general random, bianisotropic media whose constitutive tensors are complex Hermitian matrices. The two-frequency Wigner distribution (2f-WD) for polarized waves is introduced to describe the space-frequency correlations, and the closed form Wigner-Moyal equation is derived from the Maxwell equations. Two-frequency radiative transfer (2f-RT) equations are then derived from the Wigner-Moyal equation by using the multiple-scale expansion. For the simplest isotropic medium, the result coincides with Chandrasekhar's transfer equation. In birefringent media, the 2f-RT equations take the scalar form due to the absence of depolarization. A number of birefringent media such as chiral, uniaxial, and gyrotropic media are examined. For the unpolarized wave in an isotropic medium the 2f-RT equations reduces to the 2f-RT equation previously derived in part I of this research [J. Opt. Soc. Am. A24, 2248 (2007)]. A similar Fokker-Planck-type equation is derived from the scalar 2f-RT equation for the birefringent media.

Mutual coherence of polarized light in disordered media: two-frequency method extended

The paper addresses the two-point correlations of electromagnetic waves in general random, bi-anisotropic media whose constitutive tensors are complex Hermitian, positive- or negative-definite matrices. A simplified version of the two-frequency Wigner distribution (2f-WD) for polarized waves is introduced and the closed form Wigner–Moyal equation is derived from the Maxwell equations. In the weak-disorder regime with an arbitrarily varying background the two-frequency radiative transfer (2f-RT) equations for the associated 2 × 2 coherence matrices are derived from the Wigner–Moyal equation by using the multiple-scale expansion. In birefringent media, the coherence matrix becomes a scalar and the 2f-RT equations take the scalar form due to the absence of depolarization. A paraxial approximation is developed for spatially anisotropic media. Examples of isotropic, chiral, uniaxial and gyrotropic media are discussed.

A wrinkling model based on material modification for isotropic and orthotropic membranes

The intrinsic inability to withstand compression is a challenge in numerical simulation of a membrane with the finite element method. Basically there are two possibilities to deal with this problem. On the one hand wrinkles would be detected in detail by a refined mesh of shell elements once a bending formulation is used. On the other hand this problem is computed on a coarser mesh of membrane elements augmented with a wrinkling model, which considers effects of unresolved deformation patterns. This contribution focuses on the material modification approach, which is considered as a member of the latter. Within this category, works in [P. Contri, B.A. Schrefler, A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model, Commun. Appl. Numer. Methods 4 (1988) 5–15; X. Liu, C.H. Jenkins, W.W. Schur, Large deflection analysis of pneumatic envelopes using a penalty parameter modified material model, Finite Elements Anal. Des. 37 (2001) 233–251; X. Liu, Fine scale analysis of wrinkled membrane, Int. J. Numer. Comput. Engrg. Sci. 1 (2) (2000) 1017–1038; R. Rossi, M. Lazzari, R. Vitaliani, E. Onate, Simulation of light-weight membrane structures by wrinkling model, Int. J. Numer, Methods Engrg. 62 (15) (2005) 2127–2153; R. Rossi, E. Onate, Convergence of the modified material model for wrinkling simulation of light-weight membrane structures, in: E. Onate, B. Kröpin (Eds.), Textile Composites and Inflatable Structures (Structural Membranes 2003), CIMNE, 2003; R. Rossi, Light Weight Structures: Structural Analysis and Coupling Issues, Ph.D. Diss., The university of Bologna, 2005] are attractive because their wrinkling models on a coarse grid are simple and intuitive. The basic idea of this model is as follows: whenever compression occurs in a membrane, the corresponding components of the constitutive tensor in the direction of the compressive stress are penalized to weaken the compressive stiffness of the membrane. However, such abrupt change causes inevitably an unphysical oscillation of stress redistribution. Therefore, an algorithm to stabilize this oscillation is required. The first objective of this paper is to give a systematic verification of a wrinkling model based on the material modification approach by means of an analogy between wrinkling and plasticity. The second one is to include this model in a static analysis for both isotropic and orthotropic membranes. This model holds following advantages: it is computationally inexpensive and virtually able to reproduce the exact stress field, caused by wrinkling, on a macroscopic scale. To the author’s knowledge applications of a material modification approach with orthotropic materials are scarcely available. This paper demonstrates the potential of the model by means of numerical examples, compared to ones in literature.

Simulation of asymmetric effects for shape memory alloys using the concept of stress mode dependent weighting functions

Experimental results of shape memory alloys show a pronounced asymmetric behaviour between tension, compression and shear. For simulation of these effects in the constitutive equations different transformation strain tensors are introduced. These are related to the different variants for the multi-variant- and detwinned-martensite as a consequence of different stress states. In the framework of plasticity the concept of stress mode dependent weighting functions is applied in order to characterize the different stress states. Verification of the proposed methodology is succeeded for simulation of the pseudoelastic behaviour of shape memory alloys with different hardening characteristics in tension and shear.

New perspectives on the relevance of gravitation for the covariant description of electromagnetically polarizable media

By recognizing that stress–energy–momentum tensors are fundamentally related to gravitation in spacetime it is argued that the classical electromagnetic properties of a simple polarizable medium may be parameterized in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial (accelerating) frames and in the presence of weak but variable gravitational fields. After establishing some geometric notation, discussion is given to basic concepts of stress, energy and momentum in the vacuum where the useful notion of a drive form is introduced in order to associate the conservation of currents involving the flux of energy, momentum and angular momentum with spacetime isometries. The definition of the stress–energy–momentum tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. General constitutive properties of material continua are formulated in terms of spacetime tensors including those that describe magneto-electric phenomena in moving media. This leads to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric bulk matter in arbitrary motion. The question of an invariant characterization of intrinsically magneto-electric media is explored. An action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate stress–energy–momentum tensors is discussed in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.

Closed-Form Metamaterial Blueprints for Electromagnetic Masking of Arbitrarily Shaped Convex PEC Objects

In this letter, we derive closed-form expressions of constitutive tensors (blueprints) for masking of arbitrarily shaped convex PEC objects. The derivation is based on the metric invariance of Maxwell equations, as recently explored for derivation of metamaterial blueprints for electromagnetic cloaking. The present masking layer can be seen as a partial cloaking layer, whose effectiveness depends on the geometry of the masked object

Wave Dispersion Decoupling in Micropolar Thermoelasticity

AbstractFor a system of field equations of micropolar thermoelasticity we derive a propagation condition for thermoelastic disturbance in a form of monochromatic plane wave in deformation, micro rotation and temperature. The corresponding dispersion relation is given in an explicit form, together with dependence of characteristic coefficients on principal material constants forming the constitutive tensor of isothermal macro and micro elasticity, phenomenological heat conductivity and coupled macroscopic thermoelasticity. It is shown that due to the centro–symmetric nature of microelasticity and particular form of temperature coupling in a free energy function, the separation between the optical and acoustical branch of dispersion relation is inherent. For such systems dispersion relations due to the micropolar fields on one side and macroscopic thermoelastic fields on the other side are completely independent without any cross–coupling. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite element analysis of partly wrinkled reinforced prestressed membranes

Wrinkling is a well known phenomenon experimented by tension membranes in Civil Engineering applications. This paper will present an efficient numerical technique for the computational simulation of such wrinkles in a prestressed membrane. In particular, the relaxed energy approach (Pipkin in IMA J Appl Math 36:85–99, 1986) is particularized for prestressed membranes (Gil in Textile composites and inflatable structures, CIMNE, 2003) undergoing moderate strains. Wrinkling conditions in terms of the Euler-Lagrange finite deformation tensor along principal directions will be obtained. This will provide a framework to describe properly the initial instant when wrinkles start to be encountered in a prestressed Saint Venant–Kirchhoff hyperelastic membrane. Subsequently, a modified Helmholtz’s free energy functional will be introduced with the purpose of describing the modified constitutive behaviour of the continuum after the onset of wrinkling. Consistent derivations of the stress tensor as well as the constitutive tensor will de depicted. The results will be particularized for membranes and cables in a Finite Element discretization basis. Some numerical examples will prove the accuracy and robustness of the described algorithm.

Study on Constitutive Relation and Permeable Coefficient Tensor of Rocks under Tensile Stress Loading with Micromechanics

In the present paper, a micromechanical based damage model and corresponding permeable coefficient tensor are developed for rocks under tensile stress and hydraulic pressure loading based on the concept of the domain of microcrack growth (DMG) . After choosing an appropriate fracture criterion for microcracks, we obtain the equations from which the DMG under a monotonically increasing proportional plane stress and an invariable hydraulic pressure loading is obtained. Then the overall effective stress-strain relations and the permeable coefficient tensor of damaged rocks are calculated. The theory is verified by test.

Transient analysis of spectrally asymmetric magnetic photonic crystals with ferromagnetic losses

We analyze transient electromagnetic pulse propagation in spectrally asymmetric magnetic photonic crystals (MPCs) with ferromagnetic losses. MPCs are dispersion-engineered materials consisting of a periodic arrangement of misaligned anisotropic dielectric and ferromagnetic layers that exhibit a stationary inflection point in the (asymmetric) dispersion diagram and unidirectional frozen modes. The analysis is performed via a late-time stable finite-difference time-domain method (FDTD) implemented with perfectly matched layer (PML) absorbing boundary conditions, and extended to handle (simultaneously) dispersive and anisotropic media. The proposed PML-FDTD algorithm is based on a $\mathbf{D}$-$\mathbf{H}$ and $\mathbf{B}$-$\mathbf{E}$ combined field approach that naturally decouples the FDTD update into two steps, one involving the (anisotropic and dispersive) constitutive material tensors and the other involving Maxwell's equations in a complex coordinate space (to incorporate the PML). For ferromagnetic layers, a fully dispersive modeling of the permeability tensor is implemented to include magnetic losses in a consistent fashion. The numerical results illustrate some striking properties of MPCs, such as wave slowdown (frozen modes), amplitude increase (pulse compression), and unidirectional characteristics. The numerical model is also used to investigate the sensitivity of the MPC response against excitation (frequency and bandwidth), material (ferromagnetic losses), and geometric (layer misalignment and thickness) parameter variations.