Cosserat Materials Research Articles

Odd Cosserat elasticity in active materials.

Stress-strain constitutive relations in solids with an internal angular degree of freedom can be modeled using Cosserat (also called micropolar) elasticity. In this paper, we explore Cosserat materials that include chiral active components and hence odd elasticity. We calculate static elastic properties and show that the static response to rotational stresses leads to strains that depend on both Cosserat and odd elasticity. We compute the dispersion relations in odd Cosserat materials in the overdamped regime and find the presence of exceptional points. These exceptional points create a sharp boundary between a Cosserat-dominated regime of complete wave attenuation and an odd-elasticity-dominated regime of propagating waves. We conclude by showing the effect of Cosserat and odd-elasticity terms on the polarization of Rayleigh surface waves.

Weakened interfaces in Cosserat bi-materials with constrained rotation

A continuum mechanics–based model is proposed to describe the mechanical behavior of a weakened interface lying between materials with microstructure. The weakened interface is simulated by a surface elastic medium adhering on either side with bulk elastic continua exhibiting behavior of Cosserat materials with constrained rotation. Under general frame invariance considerations, it is shown that the deformation state of the surface continuum is completely specified by an extension and a rotation vector with the latter being tangential to the surface continuum. Conjugate to these deformation quantities are a surface force and a surface couple being in local equilibrium with the reduced force traction and the reduced couple traction of the bulk continua acting on their connected surfaces. The constitutive equations describing the mechanical behavior of the surface continuum within the context of linear elasticity of small deformation are five relationships that linearly relate each component of the surface force and couple to the corresponding component of the extension and rotation vector. It can be said that the surface continuum represents a spring-type interface since its behavior can been seen as that of linear and torsional springs of vanishing thickness distributed continuously over its surface. Then, the proposed model is used to investigate the effect of a weakened interface on the stress concentration around a spherical inhomogeneity embedded in an unbounded matrix both of which consist of Cosserat materials with constrained rotation.

Existence and uniqueness of Rayleigh waves in isotropic elastic Cosserat materials and algorithmic aspects

We discuss the propagation of surface waves in an isotropic half space modelled with the linear Cosserat theory of isotropic elastic materials. To this aim we use a method based on the algebraic analysis of the surface impedance matrix and on the algebraic Riccati equation, and which is independent of the common Stroh formalism. Due to this method, a new algorithm which determines the amplitudes and the wave speed in the theory of isotropic elastic Cosserat materials is described. Moreover, this method allows us to prove the existence and uniqueness of a subsonic solution of the secular equation, a problem which remains unsolved in almost all generalized linear theories of elastic materials. Since the results are suitable to be used for numerical implementations, we propose two numerical algorithms which are viable for any elastic material. Explicit numerical calculations are made for aluminium-epoxy in the context of the Cosserat model. Since the novel form of the secular equation for isotropic elastic material has not been explicitly derived elsewhere, we establish it in this paper, too.

Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples

In this work, the derivation of the effective properties for heterogeneous micropolar media with periodic structure using the two-scale asymptotic homogenization method (AHM) is reported. Analytical expressions for the local problems and the effective coefficients are explicitly described. As a particular case, periodic laminated composites are also analyzed, focusing on centro-symmetric Cosserat composites with isotropic and cubic constituents. Also, closed-form formulae of the effective properties are obtained for both constituent symmetries, and numerical values are reported and discussed. The resulting composite belongs to the orthotropic symmetry under rotations of 90°about the unitary vector e3, i.e., it has eighteen effective independent properties: nine stiffness and nine torques. As a limit case, a comparison between classical and Cosserat effective elastic properties is shown for a laminated composite with isotropic constituents. Finally, the engineering moduli of centro-symmetric laminated Cosserat materials with isotropic and cubic constituents are reported, and the numerical values are analyzed.

Comparative FE-studies of interface behavior of granular Cosserat materials under constant pressure and constant volume conditions

This article shows the outcomes of a systematic series of finite element (FE) calculations relevant to the shear behavior of a particulate-continuum interface system under different normal boundary conditions. In this respect, shearing of a thin and long granular Cosserat layer in the vicinity of a rigid moving wall with varied surface roughness values is analyzed under constant normal pressure and constant volume conditions. The material behavior is defined with a special elasto-plastic Cosserat model, taking into account micro-rotation, micro-curvature, couple stress, and mean particle size. The interaction between the layer of boundary particles and the surface roughness of the adjoining bottom wall is modeled by the rotation resistance of particles along the wall surface. Herein, the coupled effects of normal confining constraints imposed on the layer and the surface roughness of the bottom wall, are considered on the response of granular material under shearing. The influences of pressure level and initial void ratio are explored as well. Numerical results demonstrate that the dilatancy constraint prescribed to the interface plane in the normal direction, and the wall roughness have visible influences on the interface shear resistance as well as the deformation field formed within the layer. After large shearing, the width of the localized zone along the wall does not necessarily depend on the normal confining constraint and the applied pressure level. However, the localized zone characteristics and the interface shear response are mainly affected by the initial void ratio of the material. In addition to FE analyses, DEM-based simulations are also performed to investigate the micro-mechanical response of granular medium adjacent to a wall under shearing. FE predictions are qualitatively compared with DEM results, and reasonable agreement is observed.

A variational discrete element method for the computation of Cosserat elasticity

The variational discrete element method developed in [28] for dynamic elasto-plastic computations is adapted to compute the deformation of elastic Cosserat materials. In addition to cellwise displacement degrees of freedom (dofs), cellwise rotational dofs are added. A reconstruction is devised to obtain P 1 non-conforming polynomials in each cell and thus constant strains and stresses in each cell. The method requires only the usual macroscopic parameters of a Cosserat material and no microscopic parameter. Numerical examples show the robustness of the method for both static and dynamic computations in two and three dimensions.

Theoretical and experimental study of micropolar elastic materials using acoustic waves in air

This study focuses on the development of a theoretical and experimental acoustic method and setup for studying the micropolar behavior of materials. Two types of low density closed-cell foams made from polystyrene (expanded and extruded) materials that were previously reported as materials exhibiting the non-classical elastodynamic behavior were investigated in order to reveal the particular comportment using acoustic waves. Two different types of transducers were employed to provide a large frequency band to study the influence of the Cosserat parameters. The frequency band from 1–40 kHz was covered using a tweeter loudspeaker whereas that from 30–120 kHz, by an air coupled ultrasonic transducer. The materials’ micropolar elastic behavior were revealed through the comparison between the theoretical and experimental transmission coefficients. At the cut-off frequency where the theoretical phase velocity of the transverse acoustic mode was established to be infinite, the acoustic wave transmission by the panels was found to be very small. This makes the Cosserat materials appealing to harmonic noise reduction.

A higher order geometrically nonlinear Cosserat‐shell model with initial curvature effects

AbstractUsing a geometrically motivated 8‐parameter ansatz through the thickness, we reduce a three‐dimensional shell‐like geometrically nonlinear Cosserat material to a fully two‐dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed‐form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order O(h5) in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending‐curvature tensor, with coefficients depending on the initial curvature of the shell.

Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending–curvature tensor, with coefficients depending on the initial curvature of the shell.

Mathematical analysis of a solution method for finite-strain holonomic plasticity of Cosserat materials

This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler–Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energy-minimizing solutions based on the limited memory Broyden–Fletcher–Goldstein–Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

The dynamics of folding instability in a constrained Cosserat medium.

Different from Cauchy elastic materials, generalized continua, and in particular constrained Cosserat materials, can be designed to possess extreme (near a failure of ellipticity) orthotropy properties and in this way to model folding in a three-dimensional solid. Following this approach, folding, which is a narrow zone of highly localized bending, spontaneously emerges as a deformation pattern occurring in a strongly anisotropic solid. How this peculiar pattern interacts with wave propagation in the time-harmonic domain is revealed through the derivation of an antiplane, infinite-body Green’s function, which opens the way to integral techniques for anisotropic constrained Cosserat continua. Viewed as a perturbing agent, the Green’s function shows that folding, emerging near a steadily pulsating source in the limit of failure of ellipticity, is transformed into a disturbance with wavefronts parallel to the folding itself. The results of the presented study introduce the possibility of exploiting constrained Cosserat solids for propagating waves in materials displaying origami patterns of deformation.This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

An energy residual-based approach to gradient effects within the mechanics of generalized continua

AbstractGeneralized continua exhibiting gradient effects are addressed through a method grounded on the energy residual (ER)-based gradient theory by the first author and coworkers. A main tool of this theory is the Clausius-Duhem inequality cast in a form differing from the classical one only by a nonstandard extra term, the (nonlocality) ER, required to satisfy the insulation condition (its global value has to vanish or to take a known value). The ER carries in the nonlocality features of the mechanical problem through a strain-like rate field, being the specific nonlocality source, and a concomitant higher-order long-range stress (or microstress) field. The thermodynamic restrictions on the constitutive equations are determined by the latter inequality with no need for microstress equilibrium equations, whereas the principle of virtual power (PVP) is left in a standard format. The derived state equations include a set of partial differential equations involving the nonlocality-source strain-like quantity and the related long-range stress, as well as the associated higher-order boundary conditions determined by the insulation condition. Second-grade materials within gradient elasticity, gradient plasticity and crystal plasticity, as well as materials with microstructure (micromorphic and Cosserat materials) are considered to derive the pertinent constitutive equations. The proposed ER-based approach to gradient effects is shown to constitute a more straightforward and “economic” way to formulate the relevant constitutive equations than the PVP-based one.

Enhanced mathematical model for Cosserat plate bending

The purpose of this paper is to present a bending plate model, which assumes physically and mathematically motivated approximations over the plate thickness for asymmetric stress, couple stress, displacement, and microrotation. The approximations are consistent with the three-dimensional Cosserat elasticity equilibrium equations, boundary conditions and the constitutive relationships. Based on the generalized Hellinger–Prange–Reissner variational principle for Cosserat materials and the proposed strain–displacement relation, we formulate the Cosserat bending plate variational principle. This principle allows us to obtain the equilibrium equations, constitutive relations and optimal value for the minimization of the elastic energy with respect to the splitting parameter in the approximation of the σ33 stress component. The obtained solution is proved to be unique. The preliminary computations for a plate bending produce a relative error of the order 1% in comparison with the exact three-dimensional solution. The results are compatible with the precision of the well-known Reissner model used for bending of simple elastic plates. The low relative error is stable with respect to the standard range of the plate thickness.

Finite element analysis of 3D elastic–plastic frictional contact problem for Cosserat materials

The objective of this paper is to develop a finite element model for 3D elastic---plastic frictional contact problem of Cosserat materials. Because 3D elastic---plastic frictional contact problems belong to the unspecified boundary problems with nonlinearities in both material and geometric forms, a large number of calculations are needed to obtain numerical results with high accuracy. Based on the parametric variational principle and the corresponding quadratic programming method for numerical simulation of frictional contact problems, a finite element model is developed for 3D elastic---plastic frictional contact analysis of Cosserat materials. The problems are finally reduced to linear complementarity problems (LCP). Numerical examples show the feasibility and importance of the developed model for analyzing the contact problems of structures with materials which have micro-polar characteristics.

A finite element model for 3D frictional contact analysis of Cosserat materials

The formulation and numerical framework for 3D frictional contact analysis of Cosserat materials have been constructed. Based on the parametric variational principle and the corresponding quadratic programming method for numerical simulation of frictional contact problems, a finite element model is developed for 3D frictional contact analysis of Cosserat materials. The contact conditions are expressed in the form of a non-associated plastic flow problem, and the penalty factors introduced into the algorithm for contact problem are eliminated by using a definite numerical technique. The calculations of numerical examples show the feasibility and importance of the proposed algorithm. The developed contact model provides a fundamental understanding of contact problems of structures with materials which have micro-polar characteristics.

2213 マイクロポーラ弾性体を対象としたトポロジー最適化(OS6-3 設計と最適化III)

Micropolar elasticity represents mechanical behavior of materials with microstructures. Some real materials exhibit complex nonlocal behavior strongly influenced by their microstructure. In order to analyze these behavior, complex continua, such as Cosserat materials and micropolar elasticity, have been proposed recently. In this paper, the topology optimization problem for micropolar elasticity is dealt with. Topology optimization leads to optimal configurations characterized by the inclusion of holes and multiply-connected material regions. Topology optimization methods have been extensively applied to a variety of structural optimization problems. To mitigate numerical instabilities, such as mesh dependency, checkerboard patterns and grayscales, this paper proposes a level set-based topology optimization method. This method can control the geometrical complexity of obtained optimal configurations, using a fictitious interface energy based on the concept of the phase field model. Optimal material distributions are obtained for minimum mean compliance problems and compliant mechanism design problems. The influence of the micropolar constants on the optimal configuration is discussed.

A finite element model for 2D elastic–plastic contact analysis of multiple Cosserat materials

In this paper, a finite element model is reported for 2D elastic–plastic contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed to solve the elastic–plastic contact problem of Cosserat materials. Both the elastic–plastic problem and the contact problem with friction between two Cosserat materials are studied by using the parametric variational principle, which can greatly simplify the solution process of the original double nonlinear problem. The penalty factors, which are introduced into the algorithm for contact problem, are eliminated by using a definite numerical technique, resulting in high accuracy of the numerical results. Numerical examples are performed to demonstrate the accuracy and feasibility of the proposed algorithm. It can be found that sometimes the contact forces are strongly dependent on the material parameters of Cosserat materials.

Homogenisation of periodic Cosserat media

The present work deals with the development of homogenisation procedures for Cosserat materials with periodic micro-structure. The procedure developed by Forest et al. (2001) for continuum heterogeneous linear elastic Cosserat media is recalled and extended to discrete Cosserat media. The nature of the homogeneous equivalent medium depends on the hierarchy of three characteristic lengths: the size l of the heterogeneities, the Cosserat intrinsic length lc and the typical size L of the considered structure. When l and lc are comparable and much smaller than L, the effective medium is a Cauchy one, whereas the case lc ∼ L leads to a Cosserat effective medium. Applications to square, hexagonal, cubic and tetrakaidecahedral cell micro-structures are provided. We critically address the other existing homogenisation procedures.

Analysis of Cosserat materials with Voronoi cell finite element method and parametric variational principle

The elastic–plastic analysis of heterogeneous Cosserat materials is carried out with the Voronoi cell finite element method (VCFEM) under small deformation. The parametric complementary energy principle of the Cosserat theory is developed. By using the parametric variational principle, the quadratic programming model and finite element formulation for the VCFEM are established. Compared with the conventional finite element method (FEM), the newly developed method requires no tedious iterative procedures, and has no convergence problems. Based on the developed method, the influence of microscopic heterogeneities on the overall mechanical responses of heterogeneous materials is studied. Numerical examples are given to demonstrate the validity and efficiency of the method developed.

A finite element model for contact analysis of multiple Cosserat bodies

The objective of this paper is to develop a finite element model for multi-body contact analysis of Cosserat materials. Based on the parametric virtual work principle, a quadratic programming method is developed for finite element analysis of contact problems. The contact problem with friction between two Cosserat bodies is treated in the same way as in plastic analysis. The penalty factors, that are normally introduced into the algorithm for contact analysis, have a direct influence on accuracy of solution. There is no available rule for choosing a reasonable value of these factors for simulation of contact problems of Cosserat materials, and they are therefore cancelled through a special technique so that the numerical results can be of high accuracy. Compared with the conventional work on Cosserat elasticity, the newly developed model is on the contact analysis of the Cosserat materials and is seldom found in the existing literatures. Four examples are computed to illustrate the validity and importance of the model developed.