Micropolar Continuum Research Articles

Localization and Bifurcation Analysis of Granular Materials in Micropolar Continuum

Localization and Bifurcation Analysis of Granular Materials in Micropolar Continuum

Milling matter in a crusher: modeling based on extended micropolar theory

This paper presents a new aspect in generalized continuum theory, namely micropolar media showing structural change. Initially, the necessary theoretical framework for a micropolar continuum is presented. To this end, the standard macroscopic equations for mass and linear and angular momentum are complemented by a recently proposed kinetic equation for the moment of inertia tensor containing a production term. An example for this term is studied: A continuous stream of matter through a crusher is considered. The matter is milled, and consequently, the total number of particles will change. This structural change is the reason for the production of microinertia. The matter is modeled as a Hookean as well as a linear viscous material. The equations are solved numerically based on a finite difference technique.

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

AbstractWe consider the problem to determine the optimal rotations which minimize urn:x-wiley:00442267:media:zamm201800120:zamm201800120-math-0002for a given diagonal matrix with positive entries . The objective function W is the reduced form of the Cosserat shear‐stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy , determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block‐diagonal matrix with blocks of size at most two.

Three‐dimensional discrete element method parallel computation of Cauchy stress distribution over granular materials

SummaryThe paper presents Cauchy stress tensor computation over parallel grids of message passing interface (MPI) parallel three‐dimensional (3D) discrete element method (DEM) simulations of granular materials, considering spherical and nonspherical particles. The stress tensor computation is studied for quasi‐static and dynamic conditions, and its resulting symmetry or asymmetry is discussed within the context of classical continuum mechanics (CCM), granular materials mechanics (GMM), and micropolar continuum mechanics (MCM). The average Cauchy stress tensor computation follows Bagi's and Nicot's formulations and is verified within MPI parallel 3D DEM simulations involving dynamically adaptive compute grids. These grids allow calculation of temporal and spatial distributions of stress across granular materials under static and dynamic conditions. The vertical stress component in gravitationally deposited particle assemblies exhibits nonuniform spatial distributions under static equilibrium, and its zone of maximum value changes during the process of gravitational pluviation and collapse. These phenomena reveal a microstructural effect on stress distribution within granular materials that is attributed to their discrete particulate nature (particle size, shape, gradation, boundary conditions, etc).

On the micro-macro relation for the microdeformation in the homogenization towards micromorphic and micropolar continua

The modeling of size effects requires to employ generalized continuum theories. For instance, the micromorphic theory introduces the microdeformation χ̰ as additional kinematic degree of freedom. Such generalized theories require additional, i. e. non-classical, constitutive relations. The experimental determination of the corresponding non-classical constitutive parameters is cumbersome, even for isotropic and linear-elastic material. Homogenization approaches aim in providing the macroscopic constitutive relations from the behavior of the microscopic constituents of a material. For that purpose, micro-macro relations are required for all kinematic quantities. Though, different micro-macro relations have been used in literature for the microdeformation χ̰.The present contribution investigates the effect of the chosen micro-macro relation for χ̰ on the predicted macroscopic behavior. In particular, a material with pores or inclusions at the microscale is considered for the special case of plane micropolar (Cosserat) elasticity. It is discussed that the coupling modulus should vanish if the material is homogeneous at the micro-scale in order to avoid the artificial prediction of size effects. This goal is reached if the deformation of microscopic heterogeneities is used for the micro-macro relation of the microdeformation. Classical and non-classical micropolar moduli of a material with pores or inclusions are derived in closed form.

Scale Effects in Orthotropic Composite Assemblies as Micropolar Continua: A Comparison between Weak- and Strong-Form Finite Element Solutions.

The aim of the present work was to investigate the mechanical behavior of orthotropic composites, such as masonry assemblies, subjected to localized loads described as micropolar materials. Micropolar models are known to be effective in modeling the actual behavior of microstructured solids in the presence of localized loads or geometrical discontinuities. This is due to the introduction of an additional degree of freedom (the micro-rotation) in the kinematic model, if compared to the classical continuum and the related strain and stress measures. In particular, it was shown in the literature that brick/block masonry can be satisfactorily modeled as a micropolar continuum, and here it is assumed as a reference orthotropic composite material. The in-plane elastic response of panels made of orthotropic arrangements of bricks of different sizes is analyzed herein. Numerical simulations are provided by comparing weak and strong finite element formulations. The scale effect is investigated, as well as the significant role played by the relative rotation, which is a peculiar strain measure of micropolar continua related to the non-symmetry of strain and work-conjugated stress. In particular, the anisotropic effects accounting for the micropolar moduli, related to the variation of microstructure internal sizes, are highlighted.

Acoustic waveguide filters made up of rigid stacked materials with elastic joints

The acoustic dispersion properties of monodimensional waveguide filters can be assessed by means of the simple prototypical mechanical system made of an infinite stack of periodic massive blocks, connected to each other by elastic joints. The linear undamped dynamics of the periodic cell is governed by a two degree-of-freedom Lagrangian model. The eigenproblem governing the free propagation of shear and moment waves is solved analytically and the two dispersion relations are obtained in a suited closed form fashion. Therefore, the pass and stop bandwidths are conveniently determined in the minimal space of the independent mechanical parameters. Stop bands in the ultra-low frequency range are achieved by coupling the stacked material with an elastic half-space modelled as a Winkler support. A convenient fine approximation of the dispersion relations is pursued by formulating homogenised micropolar continuum models. An enhanced continualization approach, employing a proper Maclaurin approximation of pseudo-differential operators, is adopted to successfully approximate the acoustic and optical branches of the dispersion spectrum of the Lagrangian models, both in the absence and in the presence of the elastic support.

Generalized micropolar continualization of 1D beam lattices

The enhanced continualization approach proposed in this paper is aimed to overcome some drawbacks observed in the homogenization of beam lattices. To this end an enhanced homogenization technique is proposed and formulated to obtain consistent micropolar continuum models of the beam lattices and able to simulate with good approximation the boundary layer effects and the Floquet–Bloch spectrum of the Lagrangian model. The continualization technique here proposed is based on a transformation of the difference equation of motion of the discrete system via a proper down-scaling law into a pseudo-differential problem; a further McLaurin approximation is applied to obtain a higher order differential problem. The formulation is carried out for simple one-dimensional beam lattices that are, nevertheless, characterized by a rather wide variety of static and dynamic behaviors: the rod lattice, the beam lattice with node rotations and a 1D beam lattice model with generalized displacements. Higher order models may be obtained which are characterized by differential problems involving non-local inertia terms together with spatial high gradient terms. Moreover, the homogenized models obtained by the proposed enhanced continualization technique turn out to be energetically consistent and provide a good simulation of both the static response and of the acoustic spectrum of the original discrete models. The proposed homogenization procedure is first presented for the simple case of monoatomic axial chains. The beam lattice with node rotations and displacement prevented exhibits, in the static regime, decaying oscillations of the nodal rotation in the boundary layer which is well simulated by the homogenized model obtained by the proposed approach. Similar good results are obtained in the simulation of the optical spectrum. It is worth to note that in this case the homogenized model obtained via Padé approximation turns out to be energetically non-consistent. The analysis of the homogenized model derived from the beam lattice with transverse displacement and rotation of the nodes with elastic supports has shown that both the static and the dynamic response are strongly variable on the parameters of the Lagrangian model. Finally, several different cases have been considered and good simulations have been obtained both in describing the static response to prescribed displacements at the end nodes and in representing the Floquet–Bloch spectrum and the polarization vectors.

Numerical simulation of the effect of grain fragmentation on the evolution of microstructure quantities

In this paper the effect of grain fragmentation of a cohesionless granular material on the change of microstructure quantities is investigated using a micropolar continuum model. To this end the change of the grain size distribution is related in a simplified manner to a reduction of the mean grain diameter, which enters the constitutive equation as the internal length. The additional densification of the grain skeleton is modelled by a reduction of the incremental stiffness, which is related to the so-called solid hardness defined in the sense of a continuum description. The reduction of the mean grain diameter and the solid hardness caused by grain fragmentation is described by corresponding evolution equations, which depend on an increasing mean pressure and an increasing deviatoric stress. The focus of the numerical investigations is on the evolution of microstructure quantities like the void ratio, mean grain diameter and couple stresses under monotonic compression. It is shown that for an initially inhomogeneous distribution of the void ratio polar quantities like microrotations and couple stresses can develop even outside of shear bands. With increasing compression the polar quantities can increase, while the range of fluctuation of the void ratio decreases.

Micropolar Modeling of Auxetic Chiral Lattices With Tunable Internal Rotation

Based on micropolar continuum theory, the closed-form stiffness tensor of auxetic chiral lattices with V-shaped wings and rotational joints were derived. Representative volume element (RVE) of the chiral lattice was decomposed into V-shape wings with fourfold symmetry. A unified V-beam finite element was developed to reduce the nodal degrees of freedoms of the RVE to enable closed-form analytical solutions. The elasticity constants were derived as functions of the angle of the V-shaped wings, nondimensional in-plane thickness of the ribs, and the stiffness of the rotational joints. The influences of these parameters on the coupled chiral and auxetic effects were systematically explored. The results show that the elastic moduli were significantly influenced by all three parameters, while Poisson's ratio was barely influenced by the in-plane thickness of the ribs but is sensitive to the angle of the V-shaped wings and the stiffness of the rotational springs. There is a transition region out of which the spring stiffness does not considerably affect the auxeticity and the overall lattice stiffness.

On a micropolar continuum approach to some problems of thermo- and electrodynamics

A new nonlinear model of a micropolar continuum is suggested. The peculiarity of the model is that the constitutive equations depend only on the strain measures associated with rotational degrees of freedom, and at the same time, the stress tensor turns out to be different from zero. This mathematical model has been created with the view of its use for modeling various processes, including processes at the micro-scale level. Following the terminology of nineteenth-century scientists, we call our model the ether model, though in its mathematical content, it differs from the nineteenth-century ether models very significantly. There may be different points of view concerning the physical meaning of our model. On the one hand, one can suppose the same meaning that nineteenth-century scientists implied in their ether models. On the other hand, one can imagine a continuum consisting of quasi- or virtual particles. The choice of one of the aforementioned physical interpretations is not important for constructing the mathematical model. Our method of modeling thermo- and electrodynamic processes is as follows. In the framework of our model, we introduce mechanical analogies of physical quantities such as temperature, entropy, the electric field vector, the magnetic induction vector. We show that under certain simplifying assumptions the equations of our model coincide with well-known equations, in particular, with Maxwell’s equations. We explore the properties of our mathematical model in its most general form, investigate what processes can be described in the framework of our model, and suggest a possible interpretation of these processes.

Ultrastrong nanocomposites with interphases: Nonlocal deformation and damage behavior

The focus of this work is on matrix-inclusion nanocomposites and their mechanical behavior, particularly taking into account inelasticity and damage. We propose a nonlocal damage model based on a micropolar continuum theory which captures the heterogeneous interphases in the nanostructure. The model is valid for a large range of matrix-inclusion nanocomposites. The effects of the interaction between nanoparticles, the thickness of the interphase and the size of nanoparticles on the overall mechanical properties of the nanocomposite are analyzed to optimize the microstructure of the nanocomposites. 3D micromechanical simulations are conducted for the particular example of a tailored, ultrastrong nanocomposite. It has extraordinary mechanical properties, and to date, is the strongest synthetic inorganic-organic nanocomposite. The simulation results are compared to experimental data from microcantilever beams for different process temperatures.

On plane thermoelastic waves in hemitropic micropolar continua

Рассматриваются связанные термические и динамические уравнения гемитропной термоупругой микрополярной среды относительно подлежащих определению полей перемещений, микровращений и температурного инкремента. Механизм теплопроводности предполагается термодиффузионным. Определяющие постоянные гемитропного термоупругого тела редуцированы к минимальному набору, обеспечивающему его термоупругую полуизотропность. Изучаются решения связанных уравнений в форме распространяющихся плоских волн. Определены их пространственные поляризации. Получено алгебраическое бикубическое уравнение для определения волновых чисел и установлено, что для связанной волны в действительности существуют ровно три нормальных комплексных волновых числа. Исследуется также холодная атермическая волна. Пространственные поляризации в этом случае образуют (вместе с волновым вектором) триэдр взаимно ортогональных направлений. Для атермической волны находятся (в зависимости от случая) либо два вещественных нормальных волновых числа, либо одно.

Asymmetric tensor representations in micropolar continuum mechanics theories

Получены новые представления трехмерного асимметричного тензора напряжений и соответствующие им формы дифференциальных уравнений равновесия. Асимметричные теории механики деформируемого твердого тела по прежнему привлекают пристальное внимание в связи с необходимостью математического моделирования механического поведения современных материалов (например, ауксетиков с помощью теорий гемитропной микрополярной упругости). Исследование ограничивается только такими асимметричными тензорами второго ранга, для которых удается сохранить понятие о вещественных собственных значениях, но отказаться от взаимной ортогональности направлений главного триэдра. Обсуждается точная алгебраическая формулировка указанных условий асимметричности. В статье обобщаются тензорные представления симметричного тензора напряжений, основанные на естественном репере асимптотических направлений. Полученные результаты являются ярким свидетельством в пользу алгебраической «гиперболичности» симметричных и асимметричных тензоров второго ранга в трехмерном пространстве.

Micromorphic and micropolar continua: A comparative study based on homogenisation of particle simulations

AbstractExtended continuum‐mechanical theories, such as the micromorphic, the micropolar or the microstrain continuum, are suitable formulations when additional characteristic microstructural deformation modes should be included into macroscopic modelling techniques. This is especially the case when the microscopic view of the considered material reveals a granular or particulate structure. Then, by means of a particle‐centre‐based homogenisation technique, it is possible to link the microstructural information that can be gained, for example, from Discrete‐Element (DE) simulations, to the extended deformation and stress states of the above mentioned extended continua. In this context the present contribution focuses on two main aspects: at first, different granular microstructures, in particular bonded and unbonded granular systems, are investigated using a discrete‐element formulation with the emphasis on the evolution of strain localisation phenomena. Secondly, the obtained results of these initial‐boundary‐value problems are homogenised towards micropolar and micromorphic stress measures. Depending on the microstructural characteristics, the activation and evolution of the extended stress states are compared and evaluated with special emphasis on the size of the chosen homogenisation domain.

Goal‐oriented adaptivity for parameter identification in linear micromorphic elasticity

AbstractIt is well‐known that the classical Cauchy‐Boltzmann continuum, in spite of its simplicity, has its limitations, e.g. on the simulation of strain localization phenomena or size effects. For a remedy, one may resort to the so‐called generalized continuum theories, see e.g. [2] and references therein. Amongst others, we consider a class of higher order continua, namely the micromorphic continua, where the kinematics are enriched by means of a microstructure undergoing an affine micro deformation. As a drawback, the enriched kinematics lead to additional constitutive equations, increasing the number of material parameters, and therefore making the parameter identification procedure more involved. Here, the parameter identification is viewed as an inverse problem minimizing a least‐square function, see also [3] for a micropolar continuum. For simplicity, we restrict ourselves to a small strain theory. The increasing number of the degrees of freedom and the material parameters clearly motivates an application of adaptive methods. For an application on the direct micromorphic problems, we refer to [5]. Here for the inverse micromorphic problems, we follow the general ansatz by [1] for goal‐oriented error estimation and adaptivity, with which we control the spatial discretization errors of the finite element method. For illustration, numerical examples are presented.

Coupled Multi-Physics Modelling in Continuum Mechanics

Contemporary models of solids of thermoelasticity requires to include multiphysics coupling and employ non classical e.g. elastic behavior. The permanent generalization of elastic model is the Cosseratt micropolar model. Now this model is to be applied to growing solids, biomaterials, granular media, concrete. The basic concepts of the continuum mechanics is considered in connection with the difference in the dimensions of the continuum and external space. New field variables are introduced representing complex continuum properties. A generalization of the model of the micropolar continuum has been proposed. The action and the action density for the complex continuum model are discussed. A new field-theoretic model of a nonlinearly elastic continuum is developed assuming existence of an isometric immersing into an external plane space.

A micropolar model for elastic properties in functionally graded materials

By considering the description of phase volume fractions, a micromechanics model is presented for predicting the elastic mechanical properties of isotropic two-phase functionally graded materials. The particle size dependence of the overall elasticity of functionally graded materials is not generally considered by classical continuum micromechanics; however, being based on micropolar theory, the presented micromechanics model is able to study such size effects. As the effective material properties vary gradually along the gradation direction, a functionally graded material can be divided into two distinct zones: the particle–matrix zone and the transition zone. In the particle–matrix zone, the matrix material is idealized as a micropolar continuum and Mori–Tanaka’s method is extended to the micropolar medium to evaluate the effective elastic properties. The effective properties across the gradation forms are further derived and the effects of particle size on the effective properties of a functionally graded materials are also studied. The validity and effectiveness of the present model is demonstrated by comparing the model results with other model outputs and experimental data.

A predictive micropolar continuum model for a novel three-dimensional chiral lattice with size effect and tension-twist coupling behavior

This work extends the micropolar theory for the constitutive behaviors of three-dimensional (3D) cubic chiral lattice. A novel 3D chiral lattice is proposed by introducing noncentrosymmetry into microstructure design of artificial materials. The independent micropolar elastic constants of the proposed 3D chiral lattice are deduced and divided into the isotropic and anisotropic types with orthogonal irreducible decomposition of tensor. A homogenization method for the proposed 3D chiral lattice is developed to solve mathematical relations between the elastic constants and structure parameters of the chiral lattice which are admitted by finite element (FE) analysis. Two representative numerical examples are analyzed with the FE model and the continuum model where the size effect and tension-twist coupling behavior of the proposed 3D cubic chiral lattice are accurately predicted. This work establishes a fundamental link between the macroscopic mechanical properties and microstructure of the chiral material, and provides a new channel for 3D chiral microstructure design.

Measurement of higher-order stress-strain effects in granular materials undergoing non-uniform deformation

Discrete element (DEM) simulations demonstrate that granular materials are non-simple, meaning that the incremental stiffness of a granular assembly depends on the gradients of the strain increment as well as on the strain increment itself. In quasi-static simulations, two-dimensional granular assemblies were stiffer when the imposed deformation was non-uniform than for uniform deformation. The contacts between particles were modeled as linear–frictional contacts with no contact moments. The results are interpreted in the context of a higher-order micro-polar continuum, which admits the possibility of higher-order stress and couple-stress. Although the behavior was non-simple, no evidence was found for a couple-stress or an associated stiffness. The experimental results apply consistently to three particle shapes (circles, ovals, and a non-convex cluster shape), to assemblies of three sizes (ranging from 250 to 4000 particles), and at pre-peak and post-peak strains.