Bifurcation in granular materials: An attempt for a unified framework

Sudden collapse mechanisms strictly inside the Mohr–Coulomb plastic limit condition have been observed in granular materials in the laboratory as well as in the field. The purpose of this paper is to show that the theoretical framework of loss of sustainability is convenient to describe such mechanisms. In this context, the notions of both loading and response variables, which characterize the loading path applied to the specimen considered and its response path, are fundamental. Moreover, by investigating the relation between loading and response parameters, it is established that this framework also embeds the notions of loss of constitutive uniqueness and loss of controllability. Therefore, a unified approach is attempted. Finally, by highlighting the basic role played by the loading parameters, the vanishing of the second-order work is shown to be a proper criterion to detect the occurrence of a bifurcation from a quasi-static regime to a dynamic regime leading to the collapse of the material.     

Some micromechanical aspects of failure in granular materials based on second-order work

This paper discusses the notion of failure in a granular assembly by examining the key microstructural mechanisms which are most likely to trigger the nucleation and propagation of instabilities within a granular material. For this purpose, the key variable to predict the occurrence of failure, known as second-order work, is expressed from variables on the grain scale. The local behaviour incidents (where contacts may open or slide), compared to the global response of the assembly, are analysed by two approaches. First, numerical computations made by a discrete element model confirm the microscopic definition of the second-order work. Secondly, a micromechanical model, based on a homogenization procedure, relating the macroscopic behaviour to microscopic ingredients, namely contact planes, points to a close link between the occurrence of failure on the macroscopic scale as well as on the contact planes.     

An energy-based damage model of geomaterials—I. Formulation and numerical results

In this paper, an anisotropic damage model is established in strain space to describe the behaviour of geomaterials under compression-dominated stress fields. The research work focuses on rate-independent and small-deformation behaviour during isothermal processes. It is emphasized that the damage variables should be defined microstructurally rather than phenomenologically for geomaterials, and a second-order fabric tensor is chosen as the damage variable. Starting from it, a one-parameter damage-dependent elasticity tensor is deduced based on tensorial algebra and thermodynamic requirements ; a fourth-order damage characteristic tensor, which determines anisotropic damaging, is deduced within the framework of Rice, 1971 normality structure in Part II of this paper. An equivalent state is developed to exclude the macroscopic stress⧹strain explicitly from the relevant constitutive equations. Finally, some numerical results are worked out to illustrate the mechanical behaviour of this model.     

Micro-mechanical investigation of material instability in granular assemblies

It is now well established that for non-associated materials such as geomaterials, there exists a wide domain strictly within the plastic limit where different failure modes can coexist. In particular, material instability in the sense given by Hill, related to the vanishing of the second-order work, can potentially occur. In this paper we examine the occurrence of such instabilities from the simulation of drained triaxial paths, followed by the computation of two-dimensional, then fully three-dimensional Gudehus response-envelopes using the micro-directional model. This model can be seen in the continuity of Hill’s multi-slip theory, because it accounts for the association of a large number of elementary elasto-plastic bodies. Each body is linked to a contact direction in physical space and therefore takes into account the behavior of the contacts oriented along that direction. Simulations confirmed that some loading directions led to the vanishing of the second-order work. In line with the research initiated by Mandel, a micro-mechanical analysis of the origin of these potential instabilities revealed that this macro-scale phenomenon could be directly related to the constitutive nature of the local contact model between neighboring particles. Finally, this investigation provides a clear physical understanding of Hill’s material stability condition for frictional materials.     

Strain gradient plasticity by internal-variable approach with normality structure

This paper presents a constitutive formulation for materials with strain gradient effects by internal-variable approach with normality structure. Specific micro-structural rearrangements are assumed to account for the inelasticity deformations for this class of materials, and enter the constitutive formulations in form of internal variables. It is further assumed that the kinetic evolution of any specific micro-structural rearrangement may be fully determined by the thermodynamic forces associated with that micro-structural rearrangement, by normality relations via a flow potential. Macroscopic gradient-enhanced inelastic behaviours may then be predicted in terms of the microscopic internal variables and their conjugate forces, and thus a micro–macro bridging formulation is available for strain-gradient-characterised materials. The obtained formulations are first applied to crystallographic materials, and a crystal gradient plasticity model is developed to account for the influence of microscopic slip rearrangements on the macroscopic gradient-dependent mechanical behaviour for this class of materials. Micro-cracked geomaterials are also treated with these formulations and a gradient-enhanced damage constitutive model is developed to address the impacts of the evolutions of micro-cracks on the macroscopic inelastic deformations with strain gradient effects for these materials. The available formulations are further compared with other thermodynamic approaches of constitutive developing.     

三时间尺度下非光滑电路中的簇发振荡及机理

实际工程应用中存在着诸如冲击、干摩擦、切换等非光滑因素,以此建立的动力学模型是包含非光滑项的系统. 目前针对非光滑动力系统的研究大多基于单一尺度或者两尺度, 而含有更多尺度的非光滑动力系统可能会存在更复杂的动力学现象. 本论文旨在探讨非光滑动力系统中的多尺度效应及其分岔机制.基于典型的非光滑蔡氏电路, 引入一个与系统固有频率存在量级差的周期变化的激励项, 同时通过选取适当的参数值,建立了一个三时间尺度耦合下的、含有两个分界面的四维分段线性电路系统模型, 研究了该系统存在的簇发振荡行为及其分岔机制. 首先,将对应快尺度与中间尺度的变量合并作为快变量, 将对应慢尺度的变量看作慢变量, 重新划分了快慢子系统,从而将三时间尺度耦合问题转化为两时间尺度耦合问题去分析. 然后根据双参数下的Hopf分岔情况, 对应于慢子流形的不同稳定性,给出了不同参数下系统存在的两种典型的簇发振荡行为. 最后, 基于快慢分析法, 结合转换相图以及慢子流形在非光滑分界面上的非光滑动力学行为的详细讨论, 分析了不同簇发振荡相互转化的分岔机制, 发现了一个新的簇发振荡的演化路径, 即由破坏性的擦边分岔诱导的簇发振荡.      

Enhancing the Electro-Mechanical Response of Stacked Dielectric Actuators

Dielectric polymer films subjected to an electric field reduce in thickness and expand in area. A pile up configuration of such films, also known as a stacked dielectric actuator, is capable of exhibiting contractive deformations while subjected to external tensile forces. This work analyzes the capabilities of the stacked actuator according to a new microscopically motivated approach which suggests that the macroscopic response is determined by four microscopic factors—the length of the polymer chains, the local behavior of the monomers, the intensity of the local dipole and the chain-density. With the aim of enhancing the actuators performance, a specific local behavior is assumed and the influence of the remaining three quantities is studied. It is shown that the actuation can be significantly improved with appropriate micro-structural changes. Interestingly, this work demonstrates that these micro-structural alterations depend on the envisaged application.     

Generalized macroscopic traffic model with time delay

The effect of delay or reaction time on traffic flow dynamics has been investigated widely in the literature using microscopic traffic models. Recent studies using second-order Payne-type models have shown analytically that, on a macroscopic scale, time delay does not contribute to whether traffic instabilities occur. This paper will attempt to show that it all depends on the (macroscopic) model used for the analysis that delay does have effect on traffic instabilities or not. To this end, we will formulate a generalized (linear) stability condition for a second-order macroscopic model with delay and investigate analytically the effect of such delay on traffic instabilities in some specific macroscopic models. It is found that the choice of the equilibrium speed function in a (second order) macroscopic model will determine how delay affects such (linear) stability condition     

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.     

A two-scale damage model with material length

The Note presents the formulation of a class of two-scale damage models involving a micro-structural length. A homogenization method based on asymptotic developments is employed to deduce the macroscopic damage equations. The damage model completely results from energy-based micro-crack propagation laws, without supplementary phenomenological assumptions.We show that the resulting two-scale model has the property of capturing micro-structural lengths. When damage evolves, the micro-structural length is given by the ratio of the surface density of energy dissipated during the micro-crack growth and the macroscopic damage energy release rate per unit volume of the material.The use of fracture criteria based on resistance curves or power laws for sub-critical growth of micro-cracks leads to quasi-brittle and, respectively, time-dependent damage models. To cite this article: C. Dascalu, C. R. Mecanique 337 (2009).     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

A Test Method for Characterizing Clear Wood Using a Single Specimen

The constitutive models used for wood and timber have been formulated in different scales. The term clear wood corresponds to the macroscopic scale, where the orthotropic elastic model is commonly used. This model, which is typically applied to local modeling (e.g. connections), is defined by a compliance matrix composed of nine parameters, the elongations and angular distortions of which are uncoupled. The current experimental standard methods for obtaining these nine parameters are time consuming, as different specimen configurations and careful preparation for specific grain orientation are needed. As different specimens are used and the material properties of wood can vary greatly at a local level, this would have a notable influence on the outcome, as this method is not robust. Optical digital measurements (ODM) are changing and improving these traditional procedures. This work presents a new procedure for determining the compliance matrix of clear wood using just one rectangular prism as a specimen, with an orientation of the grain that does not coincide with the axes defined by the edges of the specimen. The specimen is loaded in the three mutually perpendicular directions defined by the prism, in a single axis load set up using a 3D ODM. The method was initially validated using ARAMIS® 3D, with 5 M pixels, on ash (Fraxinus excelsior L). Reasonable results were obtained, according to the resolution used; however, a higher resolution is needed to attain proper accuracy.     

Bifurcations in granular media: macro- and micro-mechanics approaches

Various failure modes related to different kinds of bifurcations occur in nonassociated elastoplastic materials such as geomaterials. After presenting experimental evidence, we study this question by means of phenomenological constitutive relations and direct numerical simulations based on the discrete element method. The second-order work criterion related to diffuse failure modes is particularly considered within the framework of continuum and discrete mechanics. The equations of the bifurcation domain boundary and unstable stress direction cones are established. Diffuse failure is simulated numerically by perturbing bifurcation states. To cite this article: F. Darve et al., C. R. Mecanique 335 (2007).     

Computational homogenization of cellular materials

In this work we propose to study the behavior of cellular materials using a second-order multi-scale computational homogenization approach. During the macroscopic loading, micro-buckling of thin components, such as cell walls or cell struts, can occur. Even if the behavior of the materials of which the micro-structure is made remains elliptic, the homogenized behavior can lose its ellipticity. In that case, a localization band is formed and propagates at the macro-scale. When the localization occurs, the assumption of local action in the standard approach, for which the stress state on a material point depends only on the strain state at that point, is no-longer suitable, which motivates the use of the second-order multi-scale computational homogenization scheme. At the macro-scale of this scheme, the discontinuous Galerkin method is chosen to solve the Mindlin strain gradient continuum. At the microscopic scale, the classical finite element resolutions of representative volume elements are considered. Since the meshes generated from cellular materials exhibit voids on the boundaries and are not conforming in general, the periodic boundary conditions are reformulated and are enforced by a polynomial interpolation method. With the presence of instability phenomena at both scales, the arc-length path following technique is adopted to solve both macroscopic and microscopic problems.     

含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

Micro-mechanical bases of some salient constitutive features of granular materials

Although phenomenological constitutive models have the ability to exhibit most salient macroscopic features of granular materials, they generally do not provide any convincing interpretation of them: their basic origin remains hidden. It is now well established that the micro-structure of granular materials plays a significant role in their overall constitutive behavior. In the past few years, a great deal of theoretical and experimental research has been devoted to this domain, giving rise to efficient micro-mechanical approaches. First, this paper reviews the micro-directional model, which is a micro-mechanically based constitutive relation. Then an extension is proposed to describe the possible collapse of force chains. This micro-structural adjunction is shown to be sufficient to simulate work-hardening and softening mechanisms. A granular assembly, containing a multitude of frictional contacts whose orientation is generally anisotropically distributed, exhibits various other typical features such as the nonassociated character of plastic strains. The micro-structural origin of this feature is investigated, and further conclusions related to the existence of a regular or a singular flow rule are drawn.     

Experimental Evidence of Thermal Effects in Multiphase Ceramic Specimens Subjected to Cyclic Loading

The aim of this work is to observe and to analyze various phenomena that exist in a multiphase ceramic material subjected to cyclic compressive loads. An infrared camera is used for this purpose. The material under study is an andalusite-based low-cement castable, which exhibits a pre-existing diffused damage (microcracks and debonded interfaces) before mechanical testing. The temperature variation in the specimen during the tests is investigated both at a macroscopic scale and a mesoscopic scale. In the first case, the material compaction, the thermoelastic coupling and the temperature increase due to mechanical dissipation are clearly evidenced. In the second case, local temperature variations related to microcracks are observed. The technique used and the results obtained are described and discussed in the paper.     

The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach

After recalling the constitutive equations of finite strain poroelasticity formulated at the macroscopic level, we adopt a microscopic point of view which consists of describing the fluid-saturated porous medium at a space scale on which the fluid and solid phases are geometrically distinct. The constitutive equations of poroelasticity are recovered from the analysis conducted on a representative elementary volume of porous material open to fluid mass exchange. The procedure relies upon the solution of a boundary value problem defined on the solid domain of the representative volume undergoing large elastic strains. The macroscopic potential, computed as the integral of the free energy density over the solid domain, is shown to depend on the macroscopic deformation gradient and the porous space volume as relevant variables. The corresponding stress-type variables obtained through the differentiation of this potential turn out to be the macroscopic Boussinesq stress tensor and the pore pressure. Furthermore, such a procedure makes it possible to establish the necessary and sufficient conditions to ensure the validity of an ‘effective stress’ formulation of the constitutive equations of finite strain poroelasticity. Such conditions are notably satisfied in the important case of an incompressible solid matrix.