A homogenisation strategy for micromorphic continua based on particle mechanics

Materials with a granular microstructure frequently fail in narrow zones due to strain localisation. Examplarily, one may look at the shear-zone development in dry sand during bi- and triaxial loading, where grains in the shear-zone exhibit large displacements and rotations. Furthermore, localisation is also observed in materials, where the microstructure consists of grains and a binding material, such as for example metal-casting moulds. Here, sand grains are bound together via a polyurethan-based material and macroscopic material failure originates from the deformation and breakage of the binder material. Within a continuum-based modelling approach, these microstructural effects can be accounted for by the consideration of an additional microcontinuum at each material point of the macroscopic body. These extended continuum theories, such as the micromorphic continua and its micropolar and microstrain sub-formulations, assume a characteristic microcontinuum deformation on a lower scale and have been successfully applied in the field of granular media. Exemplarily, in the framework of a micropolar continua, it is possible to contact forces to stresses and couple stresses via an appropriate homogenisation technique. This method includes the introduction of a Representative Elementary Volume (REV) on the mesoscale situated between the particle and the continuum scale. In this contribution, a homogenisation strategy based on a particle-centre-based REV definition is presented that is generally valid for micromorphic and micropolar continua. Therefore, a grain-binder microstructure is investigated, where particle rotations contribute to the micropolar part, while binder deformations yield the additional macromorphic character. Numerical examples are given, where results from discrete-element simulations are locally averaged and show the individual activation of the microcontinuum characteristics in the localised zones. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

广义连续统场论中新的增率型功率和能率原理

目的是建立广义连续统场论的增率型功率和能率原理.通过组合具有交叉项的增率型虚速度和虚角度原理以及虚应力和虚偶应力原理提出了微极连续统场论中具有交叉项的增率型功率和能率原理,并借助广义Piola定理同时而且无需其它附加要求地推导出微极和非局部微极连续统场论的所有增率型运动方程和边界条件以及能率方程.类似地可以推导出微态连续统的相应结果.文中给出的结果是新的,并可作为建立广义连续统力学相关的增率型有限元方法的理论基础.     

Relaxed loading conditions for higher order homogenisation approaches

The present paper deals with the formulation of minimal loading conditions for the application of numerical homogenisation techniques, namely the FE² methodology. Based on the set of volume averaging rules connecting the heterogeneous micro and the homogeneous macro scale, the minimal constraints on the deformation of a micro volume are derived for a classical Cauchy continuum as well as for a micromorphic continuum theory. For both cases, numerical studies are included highlighting the main aspects of the proposed procedure within the context of small deformations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From particle mechanics to microcontinuum theories

Microcontinuum theories enable the consideration of particle-based microstructures within a continuum mechanical framework. Several classes of microcontinua, such as the micromorphic, the micropolar, the microstrain or the microstrech formulation, have been successfully applied to engineering applications, although a clear physical determination and interpretation of the kinematical extensions and the resulting higher-order stresses within the formulation is frequently missing. In this regard, the present contribution focuses on establishing the physical link between discrete contact forces, stresses and deformation of particle-based microstructures and the characteristic stress states of microcontinuum theories. Representative Elementary Volumes (REVs) are therefore constructed on the mesoscale as ensembles of deformable particles from the mircoscale. Establishing the REV balance relations justifies the common generalisation of the angular momentum balance commonly applied in microcontinuum theories. It furthermore leads to the identification of the continuum stresses based on micro-quantities and enables the application of homogenisation techniques by exploitation of the equilibrium conditions of a REV. In order to investigate the hereby established link from the micro- to the macroscale, granular materials are simulated using the Discrete-Element Method (DEM). In particular, localisation phenomena in granulates, e. g. in biaxial compression tests or during ground-failure processes are studied. This implies the formulation of the contact between particles in an appropriate constitutive manner in accordance to the envisaged granular material behaviour, e. g. whether loose material, such as sand, or bonded multi-component material, such as polyurethan-sand compounds for metal casting applications are of interest. With the full solution of a particle-based initial-boundary-value problem, the homogenisation formalism is applied and enables the study of the extended continuum field quantities, essentially demonstrating the applicability of microcontinuum theories in the field of granular material. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

重建极性连续统理论的基本定律和原理(Ⅴ)——极性热力连续统

在对传统的微极热弹性理论和热压电弹性理论已进行过再研究的基础上重建极性热力连续统的较为完整的基本均衡方程和边界条件.从较为完整的虚功率原理推导出微极热弹性理论的运动方程和局部能率均衡方程.从较为完整的Hamilton原理通过全变分自然地推导出运动方程,熵均衡方程以及所有边界条件.给出的新的动量均衡方程和局部能率均衡方程与现有理论的结果存在本质的差异.通过过渡和归结可从微极热弹性理论分别得到微态热弹性理论的和偶应力热弹性动力学的结果.最后,按照上述思路直接给出微极热压电弹性理论的结果.     

重建极性连续统理论的基本定律和原理(Ⅸ)——热力学

对现有的微极连续统场论的基本定律进行了再研究,并指出了它们的不完整性．建立起新的微极连续统热静力学和热动力学的第一和第二基本定律．从这些定律可以很自然地和同时推导出热静力学的所有平衡方程和熵不等式以及热动力学的所有均衡方程和熵率不等式．随时对这里得到的新结果与现有微极连续力学专著和教科书中的相应结果进行了比较．着重指出的是,为什么从现有的微极连续统热动力学基本定律不能推导出局部能量均衡方程和局部熵不等式问题已经得到阐明．     

重建极性连续统理论的基本定律和原理(Ⅺ)——兼容性问题

对现有连续统场论中存在若干有关兼容性问题进行了简略的评论．根据重建的极性的连续统基本定律阐述三类兼容性问题．第一类是讨论各基本定律间的兼容性问题．第二类是讨论微极连续统基本定律兼容一般连续统基本定律的问题．第三类是讨论微极连续统基本定律兼容刚体动力学基本方程的问题．推导出的结果有助于深刻理解各种连续统理论的基本定律体系的结构和它们之间的联系．明确地指出,在传统的连续统场论的基本均衡定律的框架下,相容性问题是不可能自然地得到解决．     

Finite element methods for micropolar models of granular materials

We present a thermodynamically based finite element scheme for rate-independent materials and demonstrate its application in modelling the rheological behaviour of granular materials. Starting from the laws of thermodynamics, we have recently developed a new class of micropolar-type constitutive relations for two-dimensional densely packed granular media. This class of constitutive laws is expressed in terms of particle-scale properties, thus providing a direct link between observed macroscopic behaviour and the underlying particle–particle interactions. Here, we demonstrate how the connection to the underlying physics can be maintained and carried through to the finite element implementation phase of the modelling process via the same thermodynamical principles used to construct the constitutive laws. Notably, the study indicates that while the traditional Galerkin-FEM method admits a range of weighting functions, the proposed formulation provides an additional constraint that narrows the choice of admissible weighting functions via the second law of thermodynamics. Additionally, this paper presents insights into the finite element implementation of micropolar models deemed to be appropriate for modelling several classes of heterogeneous media (e.g. granular materials, cellular composites and biological materials). As the kinematics and kinetics of micropolar continua are enriched by the addition of rotational degrees of freedom to each material point, the equations governing boundary value problems for such materials differ from those of other continuum models both from the viewpoint of the constitutive law and the governing conservation laws. Analysis of elastoplastic deformation of micropolar continua is presented.     

Three systems of balance equations and jump conditions for nonlocal polar thermomechanical continua

Three systems of balance equations and jump conditions as well as generalized Clausius-Duhem inequalities for nonlocal polar thermomechanical continua are naturally and systematically derived under the consideration of Euler angles as angular coordinates and the negligence of conservation law of microinertia as well as the introduction of some new definitions. These results are more general than those balance equations and jump conditions as well as generalized Clausius-Duhem inequalities proposed by Eringen for nonlocal micropolar thermomechanical continua and more suitable to treat the problems of finite deformations. Project supported by the National Natural Science Foundation of China and the Research Foundation of Liaoning Province.     

Second order homogenization method based on higher order finite elements

Modeling materials with lattice-like microstructures like open-cell foams requires an extended continuum mechanical setting on the macroscopic scale, e. g. a micropolar or micromorphic theory. In order to avoid the formulation of constitutive equations a higher order numerical homogenization scheme (FE²) is proposed. Therefore, each integration point possesses its own microstructure which, in the present case, consists of beam-like elements representing the cell walls. In this paper, the microstructures are discretized by continuum-based higher order locking free finite elements with high aspect ratios, leading to a numerically efficient treatment of a local displacement-driven boundary value problem according to the macroscopic strain and curvature. The resulting stress distributions in the microstructures are homogenized to macroscopic stresses and couple stresses. The approach is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

隐含的和多重相互作用的非局部微极连续统的本构理论

本文把A.C.Eringen建立的非局部微极连续统的本构理论推广到包括具有隐含的和多重相互作用的非局部性的微极连续统的情形.这里以隐含的和多重相互作用的非局部微极热弹性固体为例说明建立各自本构理论的过程并给出两个相应的有关本构理论的定理.     

格栅夹层梁热弯曲的等效微极热弹性分析

将格栅夹层梁热弯曲等效为微极热弹性梁的受热变形,利用平面微极热弹性理论建立了微极梁受热变形的控制方程组,给出了温度载荷下微极梁的位移表达式.通过胞元能量等效的方法,得到了研究的格栅夹层梁等效微极热弹性梁材料参数.对比了等效微极梁模型和ANSYS有限元软件计算得到的温度载荷下悬臂格栅夹层梁受热弯曲变形的数值结果,两种方法得到的结果非常接近,证明了微极热弹性梁是一种简单有效的模拟格栅夹层梁热变形的等效模型.     

重建极性连续统理论的基本定律和原理(Ⅳ)——表面守恒定律

从普遍均衡定律的平移和转动的不变性出发来重新建立较为完整的微极热力连续统的表面守恒定律,提出了广义的能量动量和能量动量矩张量.给出了Piola型、Cauchy型和Kirchhoff型微极热力连续统的表面守恒定律的具体形式.现有的结果都可以当做是特殊情形从该结果自然地推导出来,并可从归结过程中清楚地看出现有理论的不完整性程度.非局部微极热力连续统的表面守恒定律可通过局部化得到.     

Dispersion of elastic waves in a micropolar metamaterial plate with periodical arranged resonators

The wave propagation in a micropolar elastic metamaterial is investigated in this paper. The elastic metamaterial is composed of the micropolar elastic host material and the periodically arranged local resonators. Compared with the classical elastic metamaterial, the micropolar elastic metamaterial has more material parameters that can be elaborately designed to manipulate the elastic wave propagation. By introducing additional displacement fields, a multi-displacement continuum model of the micropolar elastic metamaterial is presented to characterize the resonance behavior of the resonators and the microstructure effects of the unit cell. According to this continuum model, two independent wave systems exist: one is a longitudinal system and the other is a shear and rotation coupled transversal system. The dispersive curves and band gaps of the longitudinal and transversal systems are numerically discussed and the influences of the resonators are mainly considered.     

重建极性连续统理论的基本定律和原理(Ⅱ)——微态连续统理论和偶应力理论

重建微态连续统理论和偶应力理论的动量和动量矩均衡定律以及能量守恒定律,并由这些定律自然地推导出相应的局部和非局部均衡方程。这些结果可由耦合型微极连续统理论过渡和归结而得到。把推导出的结果和传统的质量和微惯性守恒定律以及熵不等式结合在一起就构成微态连续统理论和偶应力理论的基本均衡定律和方程体系。还弄清了以前的各种连续统理论的不完整性层次。最后,给出了几种特殊情形。     

Localisation in granular media: Particle approach,homogenisation and continuum modelling

The individual motion of grains in granular material has a strong influence on the macroscopic material behaviour, which is in particular the case for the phenomena of strain localisation in shear zones and justifies the need for techniques that incorporate a micro-macro transition. In this contribution, granular media are investigated in three steps. Firstly, a microscopic particle-based modelling is set up, where individual grains are considered as rigid uncrushable particles while their motion is obtained through Newton's equations of state. The inter-particle contact forces are thereby determined via constitutive contact-force formulations, which have to account for the envisaged material behaviour. The second step is the homogenisation of the obtained particle's displacements and contact forces through a particle-centre-based strategy towards continuum quantities. Therefore, Representative Elementary Volumes (REV) are introduced on the mesoscale and the specific construction of the REV boundary leads to the understanding of granular media as a micropolar continuum. Finally, in order to verify the homogenisation strategy, a continuum based micropolar model is applied to model localisation phenomena and a comparative study of the results is carried out in a qualitative way. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Cosserat material parameters obtained by homogenization of a Cauchy continuum

If a higher order continuum theory is used in a numerical simulation, the material parameters have to be derived. The experimental determination is laborious and sometimes impossible. Alternatively homogenization methods can be used for the numerical identification of overall material parameters. A short introduction to the micropolar continuum and the homogenization approach is followed by the discussion of the identified properties. Therefore some parameter studies are presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigations on an elastic micropolar continuum model for large deformations

The mechanical behaviour of cellular materials is dominated by the influence of the microtopology. If the characteristic microstructural length scale becomes comparable to the macroscopical length scale, classical homogenisation strategies fail and have to be replaced by extended approaches. In the present contribution, we present a micropolar continuum model for large deformations introducing the microrotation as an additional degree of freedom. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical homogenisation strategy for micromorphic continua

In this contribution, we apply a numerical homogenisation scheme for micromorphic continua replacing constitutive equations on the macroscale by a microscopic boundary value problem. The aim of this procedure is to describe the influence of the microtopology on the effective behaviour of microstructured materials such as biological tissues as well as polymer or metal foams. On the one hand, that allows for avoiding the numerically expensive calculation of a fully resolved microstructure. On the other hand there is no need to identify additional material parameters which are in general hard to interpret from the physical point of view. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The relaxed micromorphic continuum model

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. We prove that the new relaxed micromorphic model supports well-posedness results. Relations with other theories are also pointed out. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)