Modeling Phononic Crystals via the Weighted Relaxed Micromorphic Model with Free and Gradient Micro-Inertia

In this paper the relaxed micromorphic continuum model with weighted free and gradient micro-inertia is used to describe the dynamical behavior of a real two-dimensional phononic crystal for a wide range of wavelengths. In particular, a periodic structure with specific micro-structural topology and mechanical properties, capable of opening a phononic band-gap, is chosen with the criterion of showing a low degree of anisotropy (the band-gap is almost independent of the direction of propagation of the traveling wave). A Bloch wave analysis is performed to obtain the dispersion curves and the corresponding vibrational modes of the periodic structure. A linear-elastic, isotropic, relaxed micromorphic model including both a free micro-inertia (related to free vibrations of the microstructures) and a gradient micro-inertia (related to the motions of the microstructure which are coupled to the macro-deformation of the unit cell) is introduced and particularized to the case of plane wave propagation. The parameters of the relaxed model, which are independent of frequency, are then calibrated on the dispersion curves of the phononic crystal showing an excellent agreement in terms of both dispersion curves and vibrational modes. Almost all the homogenized elastic parameters of the relaxed micromorphic model result to be determined. This opens the way to the design of morphologically complex meta-structures which make use of the chosen phononic material as the basic building block and which preserve its ability of “stopping” elastic wave propagation at the scale of the structure.     

Micromorphic continuum Part I: Strain and stress tensors and their associated rates

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modeling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including non-linear isotropic and non-linear kinematic hardening, have been formulated, even for non-linear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Parts I, II, and III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way.In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the non-linear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively, micromorphic curvature tensors) and stress tensors (respectively, double stress tensors), as well as associated rates, independently of the particular constitutive properties.     

Micromorphic continua: non-redundant formulations

The kinematics of generalized continua is investigated and key points concerning the definition of overall tangent strain measure are put into evidence. It is shown that classical measures adopted in the literature for micromorphic continua do not obey a constraint qualification requirement, to be fulfilled for well-posedness in optimization theory, and are therefore termed redundant. Redundancy of continua with latent microstructure and of constrained Cosserat continua is also assessed. A simplest, non-redundant, kinematic model of micromorphic continua, is proposed by dropping the microcurvature field. The equilibrium conditions and the related variational linear elastostatic problem are formulated and briefly discussed. The simplest model involves a reduced number of state variables and of elastic constitutive coefficients, when compared with other models of micromorphic continua, being still capable of enriching the Cauchy continuum model in a significant way.     

Homogenization of interlocking masonry structures using a generalized differential expansion technique

In this paper a micromorphic continuum is derived for the homogenization of masonry structures with interlocking blocks. This is done by constructing a continuum which maps exactly the kinematics of the corresponding discrete masonry structure and has the same internal and kinetic energy for any ‘virtual’ translational- and rotational-field. The obtained continuum is an anisotropic micromorphic continuum of second order. The enriched kinematics of micromorphic continua allows to model microelement systems undergoing both translations and rotations. The homogenization technique applied here excludes averaging and keeps all the necessary information of the discrete structure. Therefore, all the dispersion curves of the discrete system are reproduced in the continuum model.     

BASIC THEOREMS IN LINEAR MICROMORPHIC THERMOELECTROELASTICITY AND THEIR PRIMARY APPLICATION

As a natural extension of the micromorphic continuum theory, the linear theory of micromorphic thermoelectroelasticity is developed to characterize the nano-micro scale behavior of thermoelectroelastic materials with remarkable microstructures. After the basic governing equations are given and the reciprocal theorem is deduced, both the generalized variational principle and the generalized Hamilton principle for mixed boundary-initial value problems of micromorphic thermoelectroelastodynamics in convolution form are established. Finally, as a primary application, steady state responses of an unbounded homogeneous isotropic micromorphic thermoelectroelastic body to external concentrated loads with mechanical, electric, and thermal origins are analyzed.     

New Principles of Power and Energy Rate of Incremental Rate Type for Generalized Continuum Field Theories

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Variational principles of time-dependent fluid flows through porous media with discontinuous reservoir properties

The use of variational principles as the initial basis for constructing continuum models was investigated by Sedov and his disciples. In this study the variational formalism is developed for calculating time-dependent fluid flows through porous and fractured-porous media with inhomogeneous, discontinuous, and, in particular, piecewise-constant properties. It is proved that, in the case of a medium with discontinuous properties, from the basic variational relation W = 0 there follows not only the differential equations of the flow models but also the conditions on the surfaces of discontinuity of the reservoir properties. This clears the way for the generalization and effective use of direct variational methods for calculating flow fields in complex-structure reservoirs. The methods proposed are illustrated by particular examples.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 115–123.Original Russian Text Copyright © 2004 by Volnitskaya.     

梯度增强Cosserat连续体的广义Hill定理

基于经典Cauchy连续体的Hill定理,在平均场理论的框架下导出了梯度增强Cosserat连续体细、宏观均匀化方法的广义Hill定理。在梯度增强Cosserat连续体中,不仅宏观样条点上的应变和应力张量,而且它们的梯度均作用于与该样条点相关联的细观表征元（RVE）。依据此广义Hill定理,对梯度增强Cosserat连...     

An asymptotic Reissner–Mindlin plate model

A mathematical study via variational convergence of a periodic distribution of classical linearly elastic thin plates softly abutted together shows that it is not necessary to use a different continuum model nor to make constitutive symmetry hypothesis as starting points to deduce the Reissner–Mindlin plate model.     

零维物质体和微态物质体

在这篇文章里,R.G Muncaster的零维弹性体被推广到一般的高阶零维物质体。从非极连续介质力学理论出发,我们推出了零维物质体的所有平衡方程和热力学不等式。再从这些方程和不等式推出微态物质体的相应平衡方程和热力学不等式。这样,我们在零维物质体理论和微态物质体理论之间建立了一个类似于刚性质点力学和古典非极连续介质力学之间关系的关系。     

Stability of surfaces of strong discontinuity in gases

The existence of solutions with surfaces of strong discontinuity is one of the principal features of the continua whose motions are described by systems of differential equations of hyperbolic type. Shock waves in gas dynamics, magnetohydrodynamics and in solids, detonation waves and combustion fronts, contact discontinuities, etc. are well-known examples of these surfaces. The discontinuities are usually investigated in accordance with the following scheme: 1) derivation of the boundary conditions on the discontinuity from the input system of differential equations in integral form; 2) verification of the fulfilment of the evolution conditions; 3) solution of the problem of the discontinuity structure and, when the occasion requires, obtaining supplementary boundary conditions; 4) investigation of the stability of the discontinuity. Only after obtaining positive results in all fours stages can we assert that the existence of the discontinuity is theoretically justified and that it can be used for constructing the solutions of particular boundary value problems. In the present paper attention will be concentrated on the problem of the stability of discontinuities, all the material, with the exception of the general results of Sec.1, being concerned with gas media and relating to discontinuities on whose surface the normal mass flow is nonzero. Having no way of exploring all the aspects of the problem of the stability of discontinuities in the same detail within the limited context of this paper, the authors hope to demonstrate the most general ideas and approaches which could subsequently be used to investigate the stability of discontinuities in various particular models of continua.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–22, March–April, 1996.     

Micro-macro homogenization of gradient-enhanced Cosserat media

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro-macro homogenization modeling of heterogeneous gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients at each macroscopic sampling point are attributed to associated microstructural representative volume element (RVE). The admissible boundary conditions required to prescribe on the RVE for the modeling are extracted as a corollary of the presented generalized Hill’s lemma and discussed to ensure the satisfaction of the enhanced Hill–Mandel energy condition and the average-field theory.     

A directed continuum model of micro- and nano-scale thin films

As is well known, classical continuum theories cease to adequately model a material’s behavior as long-range loads or interactions begin to have a greater effect on the overall behavior of the material, i.e., as the material no longer conforms to the locality requirements of classical continuum theories. It is suggested that certain structures to be analyzed in this work, e.g., columnar thin films, are influenced by non-local phenomena. Directed continuum theories, which are often used to capture non-local behavior in the context of a continuum theory, will therefore be used. The analysis in this work begins by establishing the kinematics relationships for a discrete model based on the physical structure of a columnar thin film. The strain energy density of the system is calculated and used to formulate a directed continuum theory, in particular a micromorphic theory, involving deformations of the film substrate and deformations of the columnar structure. The resulting boundary value problem is solved analytically to obtain the deformation of the film in response to applied end displacements.     

Micropolar nematic model for polarized liquid crystals

A micromorphic electroelastic model for polarized liquid crystals is proposed on the basis of a representation of electric multipoles in terms of microdeformation. Nematic liquid crystals are modeled as micropolar continua endowed with intrinsic electric dipole and quadrupole. A nonlinear dimensionless problem for a homogeneous nematic layer is formulated and solved numerically. The existence of a threshold electric potential is discussed, and the corresponding linearized system is also obtained to compare results on small values of deformation and electric field. Differently from common results of the classical continuum approach, asymmetric deformations and electric potentials within the layer are obtained due to the occurrence of non-null intrinsic quadrupole.     

A nonlinear generalized continuum approach for electro-elasticity including scale effects

Materials characterized by an electro-mechanically coupled behaviour fall into the category of so-called smart materials. In particular, electro-active polymers (EAP) recently attracted much interest, because, upon electrical loading, EAP exhibit a large amount of deformation while sustaining large forces. This property can be utilized for actuators in electro-mechanical systems, artificial muscles and so forth. When it comes to smaller structures, it is a well-known fact that the mechanical response deviates from the prediction of classical mechanics theory. These scale effects are due to the fact that the size of the microscopic material constituents of such structures cannot be considered to be negligible small anymore compared to the structure's overall dimensions. In this context so-called generalized continuum formulations have been proven to account for the micro-structural influence to the macroscopic material response. Here, we want to adopt a strain gradient approach based on a generalized continuum framework [Sansour, C., 1998. A unified concept of elastic-viscoplastic Cosserat and micromorphic continua. J. Phys. IV Proc. 8, 341-348; Sansour, C., Skatulla, S., 2007. A higher gradient formulation and meshfree-based computation for elastic rock. Geomech. Geoeng. 2, 3-15] and extend it to also encompass the electro-mechanically coupled behaviour of EAP. The approach introduces new strain and stress measures which lead to the formulation of a corresponding generalized variational principle. The theory is completed by Dirichlet boundary conditions for the displacement field and its derivatives normal to the boundary as well as the electric potential. The basic idea behind this generalized continuum theory is the consideration of a micro- and a macro-space which together span the generalized space. As all quantities are defined in this generalized space, also the constitutive law, which is in this work conventional electro-mechanically coupled nonlinear hyperelasticity, is embedded in the generalized continuum. In this way material information of the micro-space, which are here only the geometrical specifications of the micro-continuum, can naturally enter the constitutive law. Several applications with moving least square-based approximations (MLS) demonstrate the potential of the proposed method. This particular meshfree method is chosen, as it has been proven to be highly flexible with regard to continuity and consistency required by this generalized approach.