Constitutive Tensors Research Articles

Accurate numerical calculation method for liquid crystal phase shifter loaded with metallic NaM

ABSTRACT Metallic Nanowire Filled Membrane (NaM) exhibit immense potential in enhancing liquid crystal phase shifters (LCPSs) performance. However, conventional numerical calculation methods are inadequate for accurately simulating LCPSs loaded with NaM due to the significant dimensional disparities between the nanoscale NaM and the millimetre or even centimetre-scale LCPS. In this paper, a novel calculation technology is proposed to accurately simulate the electromagnetic (EM) response of LCPSs integrated with NaM, including phase shift and figure of merit (FoM). The effective constitutive tensors of the NaM are extracted and utilised to construct the EM model for characterising the NaM properties. Three NaM-LCPSs are simulated using the proposed EM model and compared with measurement results. The findings reveal that the proposed calculation method exhibits an error of less than 20%, while previously reported numerical calculation method demonstrated error as high as 60%. Furthermore, leveraging the proposed calculation method, the performance of a NaM-LCPS is optimised by systematically varying its dimensional parameters. The results demonstrate a substantial improvement in the FoM of the NaM-LCPS, increasing from 67°/dB to 120°/dB, significantly expanding the potential applications of NaM-LCPSs.

A Moore-Gibson-Thompson heat conduction problem with second gradient

In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.

Overall constitutive properties of stratified lattices with alternating chirality.

The present research focuses on a continuum description of stratified metamaterials achieved through the superposition of layers with alternating chirality. Each layer is constructed as a periodic assembly of centre-symmetric periodic cells, formed by a recurring arrangement of rigid circular discs connected by elastic ligaments. The layers are interconnected through elastic pins passing through the centres of aligned discs, allowing for either restrained or free relative rotation. A micropolar continuum model is used to describe each individual layer. The overall response of the metamaterials to in-plane forces is derived using a multi-field non-local model, expressed in terms of the average and difference of displacement and rotational fields. The overall micropolar and standard (Cauchy) constitutive tensors have been determined in closed form. The validity of the equivalent generalized micropolar model has been confirmed through comparison with discrete Lagrangian solutions of representative examples. In addition, a detailed analysis of a pseudo-indentation test has been carried out. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

A multiscale bifurcation analysis using micromechanical-based constitutive tensor for granular material

The current study presents a multiscale approach that investigates material instability and localization phenomena in plastic granular materials and the discrete-continuum duality. A bifurcation and stability analysis in continuum mechanics usually requires the material’s tangent (stiffness) operator, the computation of which in a micromechanical approach such as Discrete Element Modeling (DEM) requires specific treatment. To bridge the discrete and continuum worlds, a new computational approach incorporating strain probing is proposed to reconstruct the elastoplastic constitutive tensor and its spectral characteristics from DEM simulations. The probing technique permits the computation of the tangent operator that inherits microstructural information from the discrete world to analyze bifurcation in elastoplasticity at the macro level. An incrementally linear constitutive tensor is computed, distinct for each of the ensemble of probing directions belonging to a particular tensorial zone or sector of incremental stress or strain space, thus making it directionally non-linear. Following such an approach, material instability can be evaluated from the spectral characteristics of the tangent constitutive tensor deduced from DEM probing calculations belonging to an identified tensorial zone. A meso-scale analysis is finally offered to detect shear band localization through the well-known Rice’s criterion as a continuum-based concept extended to a micromechanical discrete modeling framework. These new numerical results show that the multiscale proposed approach, which allows access to microstructural information, is consistent with a continuum one such as when predicting the localization angle during shear banding in a granular specimen in DEM.

Continuum-molecular modeling of planar micropolar media: Anisotropy, chiral properties and length-scale effects

This paper presents a continuum-molecular formulation for bi-dimensional micropolar media within the mathematical formalism of a revised peridynamic theory with oriented material points. A variational procedure is considered to derive the fundamental governing equations of the model, which postulates that material points interact through pair potentials allowing generalized pairwise actions to be derived as energy conjugates to properly defined pairwise deformation measures. While continuity of mass is assumed, constitutive laws are produced for long-range micro-interactions to reproduce the effective behavior of the material at the macro-scale. The definition of proper micromoduli functions respecting material symmetries and invariance properties allows then to obtain a non-local micropolar continuum based on pseudo-discrete kinematics, while providing a mechanism-based description of material anisotropy and coupling behaviors uncovered by classical elasticity. An analytic micro-macro moduli correspondence procedure is also established, based on the formal analogy with the constitutive tensor of centrosymmetric planar micropolar continua. As an important result of this research, we show that distinctive chiral effects of micropolar elasticity can be reproduced by introducing a directional independent pseudo-scalar pair potential, which turns out to be analytically vanishing when the rotationally invariant part of the corresponding continuum elastic tensor is invariant to mirror reflections as well. Moreover, the proposed formulation demonstrates sensitivity to elastic bending size-effect as specific property of structured materials homogenized as micropolar continua, and related to the characteristic length of their underlying microstructure. The resulting constitutive non-locality, which plays an important role in fracture problems, is conceptually separated with respect to the intrinsic non-local character of the model related to the integral nature of its governing equations, and then does not vanish when the horizon reduces to zero. The theoretical findings and the effectiveness of the model are successfully verified through illustrative examples referred to representative cases of structured materials homogenized as micropolar continua, including length-scale dependent quasi-static crack propagation as well as the mechanism-based description of the coupling between bulk strain and micro-rotation in elastic bi-dimensional homogenized chiral lattices.

Inverse design method of thermal devices with thermal Hall effect

Efficient thermal management is important for both performance and efficiency improvements of thermal devices. For designing reasonable materials and structures of such devices, various design methods were proposed where the material thermal conductivity tensors were positive definite and symmetric based on the physical requirements. Here, we propose an inverse design method for thermal devices considering the thermal Hall effect, which makes the material thermal conductivity tensor asymmetric. Enlarging the design space consisting of the symmetric constitutive tensors to that of the asymmetric ones, there is a possibility of improving the theoretical performance limit of thermal devices. We formulate an inverse problem based on the free material optimization formalism, parameterizing the design space so that the physically available property could be naturally satisfied. Several numerical experiments are provided to show the validity and the utility of the proposed method.

A progressive damage model for 3D woven composites integrating kinking-buckling competition

In this work, a progressive damage model (Kinking-Buckling PDM) considering fiber micro-buckling failure, fiber kinking failure, transverse inter-fiber failure of yarns, matrix failure and interface debonding, is proposed to predict the compressive mechanical behavior of 3D woven composites (3DWC). Fiber kinking and fiber micro-buckling failure are adopted in the form of competition to characterize the longitudinal failure of fiber yarns. A bilinear damage evolution model is established to control the damage accumulation of each constituent. Meanwhile, a material tangent constitutive tensor characterizing the constitutive behavior of 3DWC is derived. The complex failure process of 3DWC under compression is displayed through Kinking-Buckling PDM. As the fiber initial misalignment angle increases, the compressive strength of 3DWC declines significantly, meanwhile, the main failure mode of 3DWC will be transformed from fiber micro-buckling to fiber kinking in warp yarns. The relevant experimental results are used to confirm the accuracy of the PDM.

Two-Dimensional Nye Figures for Some Micropolar Elastic Solids

The paper is devoted to problems related to two-dimensional Nye figures for micropolar continua. The representation technique known from studies on crystallography for 4th, 3rd and 2nd rank tensors is employed by two-dimensional matrices, supplied by relationships between their elements. Such representations are commonly used to simplify a script of the equations of athermic micropolar elastic solids. This method allows us to graphically represent micropolar constitutive tensors and pseudotensors in the form of specific two-dimensional blocks. The Nye figures for the micropolar elastic solids are obtained. The Nye figures enables us to discriminate anisotropic solids and figure out general anisotropic, hemitropic, isotropic, and ultraisotropic continua.

Dynamic continualization of masonry-like structured materials

Block-lattice materials featuring periodic planar running-bond tessellation of regular rigid blocks and linear elastic homogeneous isotropic interfaces are considered. The governing equations of the discrete masonry-like Lagrangian model are properly manipulated via the novel enhanced continualization scheme, in such a way as to obtain equivalent integral type non-local continua, whose band structure turns out to be coincident with that of the corresponding discrete models. The formal Taylor series expansion of the integral kernels allows deriving homogeneous generalized micropolar higher-order continuum models, characterized by non-local constitutive and inertial terms. The enhanced continualization exhibits thermodynamic consistency in the definition of the overall non-local constitutive tensors, as well as qualitative agreement and quantitative convergent matching of the complex frequency band structure in the regime of both homogeneous and non-homogeneous Bloch waves. The theoretical findings are effectively validated by studying the dispersion relations and the spatial attenuation properties, as referred to realistic representative cases of masonry-like block-lattice micro-structures.

Distance to a constitutive tensor isotropy stratum by the Lasserre polynomial optimization method

Distance to a constitutive tensor isotropy stratum by the Lasserre polynomial optimization method

Hilbert’s energy–momentum tensor extended

In classical field models, Hilbert’s energy–momentum tensor for a matter field is defined by a variational derivative of a Lagrangian with respect to the metric tensor. Constitutive relationships between fundamental field variables are a subject that is dealt with by a wide range of models in solid-state physics. In this case, a higher-order constitutive tensor replaces the second-order metric tensor. A similar premetric description with a linear constitutive relation has recently been provided for classical field models of gravity and electrodynamics. In this paper, we analyze the extension of the Hilbert definition of the energy–momentum tensor to models with general linear constitutive law. In order to treat integrals on a differential manifold covariantly, we need the differential forms description. It follows that the Lagrangian, electromagnetic current, and energy–momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. The constitutive relation then appears as a linear map on forms. We derive a commutative variation identity for an arbitrary linear map, enabling direct variation processes without having to deal with the individual components. We present evidence for the close relationship between the explicit form of the energy–momentum current of Hilbert’s type and the commutative-variation identity. A variety of field models with a general linear constitutive law are subject to this construction.

Toupin–Mindlin first strain‐gradient elasticity for cubic and isotropic materials at small scales

AbstractThe Toupin–Mindlin anisotropic first strain‐gradient elasticity is a generalized continuum field theory valid at small scales. The field theoretical framework, the constitutive tensors, and the material parameters are given for anisotropic, cubic, and isotropic materials in first strain gradient elasticity. Since the characteristic lengths are in the unit‐range of Ångström, first strain‐gradient elasticity leads to Ångström mechanics, in a straightforward manner.

Tensors with Constant Components in the Constitutive Equations of a Hemitropic Micropolar Solids

The present paper is devoted to elastic potentials and the constitutive equations of mechanics of anisotropic micropolar solids, the kinematics of which can be specified by two independent vector fields: a contravariant field of translational displacements and a contravariant pseudovector field of microrotations of weight +1. The quadratic stress potential is represented by three constitutive tensors of the fourth rank, two of which are pseudotensor in nature and can be assigned weights –2 and –1. Such a solid is completely specified by the 171st micropolar elastic modulus. The main attention is focused on the model of a hemitropic (half-isotropic, demitropic) micropolar elastic solid characterized by nine constitutive constants. The components of the constitutive pseudo-tensor of weight ‒1 turn out to be sensitive to mirror reflection transformations in three-dimensional space. A peculiar algebraic structure of the constitutive tensors of a hemitropic solid, more precisely, their absolute analogues obtained by multiplying by integer powers of a pseudoscalar unity, is studied. It is shown that these tensors can always be constructed from isomers (isomer) of a tensor with constant components (generally insensitive to any transformations of the coordinate system) and one additional fourth-rank tensor constructed, in turn, from the components of the metric tensor.

Self-bridging metamaterials surpassing the theoretical limit of Poisson’s ratios

A hallmark of mechanical metamaterials has been the realization of negative Poisson’s ratios, associated with auxeticity. However, natural and engineered Poisson’s ratios obey fundamental bounds determined by stability, linearity and thermodynamics. Overcoming these limits may substantially extend the range of Poisson’s ratios realizable in mechanical systems, of great interest for medical stents and soft robots. Here, we demonstrate freeform self-bridging metamaterials that synthesize multi-mode microscale levers, realizing Poisson’s ratios surpassing the values allowed by thermodynamics in linear materials. Bridging slits between microstructures via self-contacts yields multiple rotation behaviors of microscale levers, which break the symmetry and invariance of the constitutive tensors under different load scenarios, enabling inaccessible deformation patterns. Based on these features, we unveil a bulk mode that breaks static reciprocity, providing an explicit and programmable way to manipulate the non-reciprocal transmission of displacement fields in static mechanics. Besides non-reciprocal Poisson’s ratios, we also realize ultra-large and step-like values, which make metamaterials exhibit orthogonally bidirectional displacement amplification, and expansion under both tension and compression, respectively.

Computation of minimal covariants bases for 2D coupled constitutive laws

We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and — possibly coupled — laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any n-uplet of bidimensional constitutive tensors.

Analysis of a strain-gradient problem arising in MGT thermoelasticity

In this work, we study a dynamic problem arising in the MGT-thermoelasti-city. Under some assumptions on the constitutive tensors, the basic equations of the model are derived, leading to a coupled system written in terms of the displacements and the thermal displacements. A MGT dissipation mechanism is considered for the heat equation. An existence and uniqueness result is proved in the three-dimensional case, and the one-dimensional version is analyzed in the case that the coefficients of the constitutive functions are assumed constants. We prove that at least a polynomial decay is found and that, in general, an exponential decay cannot be expected. A couple of remarks concerning particular situations are also proposed. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme, proving an a priori error estimates result which, under some additional regularity, leads to the linear convergence of the approximation. Finally, we perform some one-dimensional numerical simulations which show the accuracy and the behavior of the solution.

Ultrawide-angle generalized Brewster metasurface

A generalized Brewster impedance matching theory is proposed in this paper to overcome the performance degradation of conventional metasurfaces at large incidence angles and achieve ultrawide angular stability. To begin, in the case of transmission, the characteristic impedance of anisotropy metasurface, denoted by incidence angle and constitutive parameter tensor, is obtained by extending the constitutive parameter from the isotropy scalar to the anisotropy tensor. The theoretical formulas used to derive the generalized Brewster impedance matching condition extend the classical Brewster phenomenon to the entire angular domain. The constitutive parameters of the anisotropy metasurface are then extended to the complex domain for the case of absorption, and the complex reflection coefficient of the absorptive metasurface is established in the angular domain. To realize ultrawide-angle absorption, the generalized Brewster complex-impedance matching condition is derived, allowing the extension of ultrawide-angle transmission to absorption. Finally, ultrawide-angle transmissive and absorptive metasurfaces are designed, and their transmission of 0${~}^{\circ}$–80${~}^{\circ}$ and absorption performance of 0${~}^{\circ}$–75${~}^{\circ}$ are demonstrated using simulations and measurements.

AN ALGEBRAIC DECOMPOSITION OF HEMITROPIC PSEUDOTENSORS IN N DIMENSIONS AND APPLICATIONS TO MICROPOLAR CONTINUUM THEORIES

The paper is devoted to problems related to algebraic decompositions and coordinate representations of tensors with constant components, hemitropic tensors and pseudotensors. Such objects of tensor algebra are interesting from the micropolar continuum mechanics viewpoint. Algebraic properties and coordinate representations of tensors and pseudotensors with constant components are discussed. The base examples of tensors with constant components usable in continuum mechanics are given. An algebraic algorithm for coordinate representations of tensors and pseudotensors with constant components proposed by prof. G.B. Gurevich is highlighted and employed.The notions of fully isotropic, conventionally isotropic, unconventionally isotropic, semi-isotropic (demitropic, hemitropic, chiral) fourth rank tensors and pseudotensors are proposed. The coordinate representations of a Cartesian hemitropic fourth rank tensor in three-dimensional space are obtained in terms of the Kronecker delta and metric tensor products. Based on an unconventional definition of a hemitropic fourth rank tensor, general coordinate representations in dimensions in terms of the Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth rank tensor and a tensor with constant components is carried out. A general form of the elastic potential of a linear anisotropic micropolar elastic continuum is obtained by the pseudotensor technique. Obtained anisotropic micropolar potential is reduced to a hemitropic one by proposed coordinate representations of fourth rank tensors. Coordinate representations for constitutive tensors and pseudotensors usable in mathematical modeling of linear hemitropic micropolarelastic continua are obtained as a modification of pseudotensors with constant components in three-dimensional space.

Anisotropic structure of two-dimensional linear Cosserat elasticity

In the present contribution the anisotropic structure of the two-dimensional linear Cosserat elasticity is investigated. The symmetry classes of this model are derived and detailed in a synthetic way. Particular attention is paid to specific features of Cosserat Elasticity which are the sensitivity to non-centrosymmetry as well as to chirality. These aspects are important for the application of this continuum theory to the mechanical modelling of lattices and metamaterials. In order to give a parameterisation to the Cosserat constitutive law, an explicit harmonic decomposition of its constitutive tensors is provided. Finally, using an algorithm introduced in a side paper, a minimal integrity basis, which is the minimal set of polynomial invariants generating the algebra of O(2)-invariant polynomials, is finally reported.

Constitutive tensor in the geometrized Maxwell theory

It is generally accepted that the main obstacle to the application of Riemannian geometrization of Maxwell’s equations is an insufficient number of parameters defining a geometrized medium. In the classical description of the equations of electrodynamics in the medium, a constitutive tensor with 20 components is used. With Riemannian geometrization, the constitutive tensor is constructed from a Riemannian metric tensor having 10 components. It is assumed that this discrepancy prevents the application of Riemannian geometrization of Maxwell’s equations. It is necessary to study the scope of applicability of the Riemannian geometrization of Maxwell’s equations. To determine whether the lack of components is really critical for the application of Riemannian geometrization. To determine the applicability of Riemannian geometrization, the most common variants of electromagnetic media are considered. The structure of the dielectric and magnetic permittivity is written out for them, the number of significant components for these tensors is determined. Practically all the considered types of electromagnetic media require less than ten parameters to describe the constitutive tensor. In the Riemannian geometrization of Maxwell’s equations, the requirement of a single impedance of the medium is critical. It is possible to circumvent this limitation by moving from the complete Maxwell’s equations to the approximation of geometric optics. The Riemannian geometrization of Maxwell’s equations is applicable to a wide variety of media types, but only for approximating geometric optics.