A thermo-mechanical continuum theory with internal length for cohesionless granular materials

This article continues Part I. Here the non-equilibrium responses of the constitutive variables t (Cauchy stress tensor), q (heat flux vector), h (equilibrated stress vector), Γ (flux term associated with the internal length ℓ), Π (production term associated with ℓ) and f (equilibrated intrinsic body force) as well as the Helmholtz free energy Ψ are postulated by use of a quasi-linear theory for three of four models deduced in Part I. In so doing, together with the equilibrium responses gained in Part I, a complete set of constitutive equations for the constitutive quantities for each model is obtained. The implemented models are applied to investigate typical isothermal steady granular shearing flows with incompressible grains, namely, simple plane shear flow, inclined gravity-driven flow and vertical channel-flow. The emphasis is on the models in which ℓ is considered a material constant (Model I) and an independent dynamic field quantity (Model III). Numerical results show that Model III is more appropriate than Model I since in the former model the effect of the motion of an individual grain can better be taken into account. Such a result is in particular significant for avalanches, since it verifies the existence of a thin layer immediately above the base of an avalanche, in which the grains are colliding strongly with one another, and provides a quantitative means to measure such a thin layer.     

Constitutive models for granular materials including quasi-static frictional behaviour: Toward a thermodynamic theory of plasticity

This work deals with the thermodynamic formulation of constitutive models for materials whose quasi-static behaviour is governed by internal friction, e.g., dry granular materials. The process of internal friction is represented here phenomenologically with the help of a second-order, symmetric-tensor-valued internal variable. A general class of models for the evolution of this variable is considered, including as special cases a hypoelastic-like form for this relation as well as the hypoplastic form of Kolymbas (1991). The thermodynamic formulation is carried out in the context of the Müller-Liu entropy principle. Among other things, it is shown that for the hypoelastic-type models, a true equilibrium inelastic Cauchy stress exists. On the other hand, such a stress does not exist for the hypoplastic model due to its rate-independence and incremental non-linearity. With the help of a slight generalization of the notion of thermodynamic equilibrium, i.e., to thermodynamic “quasi-equilibrium,” however, such a Cauchy stress can be formulated for the hypoplastic model. As it turns out, this quasi-equilibrium for the Cauchy stress represents a thermodynamic generalization of the so-called quasi-static stress postulated for example by Goddard (1986) in the context of his viscoplastic model for a frictional-dissipative, and in particular for granular, materials. Received January 22, 1999     

A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material

In the present study an evolution equation for the Cauchy stress tensor is proposed for an isotropic elasto-visco-plastic continuum. The proposed stress model takes effects of elasticity, viscosity and plasticity of the material simultaneously into account. It is ascribed with some scalar coefficient functions and, in particular, with an unspecified tensor-valued function N, which is handled as an independent constitutive quantity. It is demonstrated that by varying the values and the specific functional forms of these coefficients and N, different known models in non-Newtonian rheology can be reproduced. A thermodynamic analysis, based on the Müller–Liu entropy principle, is performed. The results show that these coefficients and N are not allowed to vary arbitrarily, but should satisfy certain restrictions. Simple postulates are made to further simplify the deduced general results of the thermodynamic analysis. They yield justification and thermodynamic consistency of the existing models for a class of materials embracing thermoelasticity, hypoelasticity and in particular hypoplasticity, of which the thermodynamic foundation is established successively for the first time in literature. The study points at the wide applicability and practical usefulness of the present model in different fields from non-Newtonian fluid to solid mechanics. In this paper the thermodynamic analysis of the proposed evolution-type stress model is discussed, its applications are reported later.      

A Variational Principle for the Revised Goodman–Cowin Theory with Internal Length

In the present study a variational principle is proposed for the revised Goodman–Cowin theory with internal length for cohesionless granular materials (Fang et al. in Continuum Mech Thermodyn in press). The balance equations of the internal variables employed in the theory in equilibrium states, the equilibrium expressions of the constitutive variables and the corresponding natural boundary conditions are derived by use of the proposed variational principle for both cases of compressible and incompressible grains. It is demonstrated that the derived results coincide with those obtained by use of the thermodynamic analysis. The current work serves as a supplementary variational verification of the constitutive theory proposed in Fang et al. (in Continuum Mech Thermodyn in press).     

基于深度学习和细观力学的颗粒材料本构关系研究

颗粒材料的本构关系对岩土工程等众多领域至关重要. 不同于传统的唯象本构理论, 本文基于机器学习模型探索了一种细观力学理论指导下的数据驱动型颗粒材料本构关系预测方法. 根据Vogit均质化假设, 建立了小应变条件下颗粒材料应力?应变解析关系, 此关系唯一地确定了一组与颗粒材料本构行为相关的细观组构变量. 这些变量与反应颗粒材料宏观性质的主应力和主应变信息通过一系列离散元三轴压缩数值试验获得. 考虑到细观组构变量为内变量, 不能直接作为本构模型的输入. 本文基于有向图方法将颗粒材料微观结构信息隐式地包含在应力?应变的预测当中, 并采用门控循环单元(GRU)循环神经网络作为基础深度学习模型描述有向图中结点之间的映射关系. 通过将有向图从目标节点沿源节点展开, 整个应力?应变预测模型可由两个神经网络分别训练并组装而成. 将训练后的深度学习模型在全新的数据集上进行测试, 结果表明该训练策略能有效捕捉到颗粒材料在常规三轴任意加卸载, 等中主应力系数b的真三轴加载, 和等平均有效应力p的真三轴加卸载等复杂多轴加载工况下的应力?应变响应关系, 模型具有良好的内插和外推预测能力. 考虑到深度学习模型捕捉颗粒材料力学响应的能力及其开放式学习的特点, 充分结合数据驱动方法和理论本构模型可能是颗粒材料本构研究的一个重要方向.      

On the equilibrium response of different modeling approaches for the porosity variation in dry granular matter

A selection of models for the variation in porosity in dry granular flows is investigated and compared on the basis of thermodynamic consistency to illustrate their performance and limitations in equilibrium situations. To this end, the thermodynamic analysis, based on the Müller–Liu entropy principle, is employed to deduce the ultimate constitutive equations at equilibrium. Results show that while all the models deliver appropriate equilibrium expressions of the Cauchy stress tensor for compressible grains, the model in which the variation in porosity is treated kinematically yields a spherical stress tensor for incompressible grains. Only the model in which the variation in porosity is modeled by a dynamic equation can give rise to a non-spherical stress tensor at equilibrium. The present study illuminates the validity and thermodynamic justification of the two modeling approaches for the porosity variation in dry granular matter.     

Micromechanical definition of an entropy for quasi-static deformation of granular materials

A micromechanical theory is formulated for quasi-static deformation of granular materials, which is based on information theory. A reasoning is presented that leads to the definition of an information entropy that is appropriate for quasi-static deformation of granular materials. This definition is based on the hypothesis that relative displacements at contacts with similar orientations are independent realisations of a random variable. This hypothesis is made plausible based on the results of Discrete Element simulations. The developed theory is then used to predict the elastic behaviour of granular materials in terms of micromechanical quantities. The case considered is that of two-dimensional assemblies consisting of non-rotating particles with an elastic contact constitutive relation. Applications of this case are the initial elastic (small-strain) deformation of granular materials. Theoretical results for the elastic moduli, relative displacements, energy distribution and probability density functions are compared with results obtained from the Discrete Element simulations for isotropic assemblies with various average numbers of contacts per particle and various ratios of tangential to normal contact stiffness. This comparison shows that the developed information theory is valid for loose systems, while a theory based on the uniform-strain assumption is appropriate for dense systems.     

Thermoelasto-viscous Materials

A thermoelasto-viscous material is defined by a set of constitutive laws in which the stress, entropy, heat flux and free energy are functions of the present configuration, temperature, temperature gradient and the rate of change of all three of these. Here these materials are presented within the framework of Walter Noll’s new theory of simple materials, so that the constitutive laws are specified without the use of a frame of reference. The Coleman-Noll procedure is carried out, and the symmetry group of the material is also discussed, both without using a frame of reference. It is then shown what form the constitutive laws of a fluid thermoelasto-viscous material take when a frame of reference is considered. Finally, the governing equations for these materials are explicitly obtained and discussed. The results in this paper may serve as a foundation for new and better mathematical models to deal with phenomena such as heat transfer, heat exchanges and thermophoresis.     

Mechanical behavior of amorphous polymers in shear

Based on the non-equilibrium thermodynamic theory, a new thermo-viscoelastic constitutive model for an incompressible material is proposed. This model can be considered as a kind of generalization of the non-Gaussian network theory in rubber elasticity to include the viscous and the thermal effects. A set of second rank tensorial internal variables was introduced, and in order to adequately describe the evolution of these internal variables, a new expression of the Helmholtz free energy was suggested. The mechanical behavior of the thermo-viscoelastic material under simple shear deformation was studied, and the “ viscous dissipation induced“ anisotropy due to the change of orientation distribution of molecular chains was examined. Influences of strain rate and thermal softening produced by the viscous dissipation on the shear stress were also discussed. Finally, the model predictions were compared with the experimental results performed by G‘ Sell et al. , thus the validity of the proposed model is verified.     

非均匀颗粒材料的类固-液相变行为及本构方程

以非均匀颗粒介质为研究对象,采用三维离散元方法对其在不同密集度和剪切速率下的动 力过程进行了数值模拟,分析了其在由瞬时接触的快速流动向持续接触的准静态流动的转变 过程及其行为特点. 通过对不同材料性质下相变过渡区内颗粒材料的宏观应力、接触时间数、 配位数、团聚颗粒数量、有效摩擦系数等参量的计算,更加全面地描述了非均匀颗粒材料在 类固-液相变过程中的基本特征. 基于以上数值计算结果,建立了一个适用于颗粒材料 类固态、类液态以及其相变过程的本构方程,并通过剪切室实验结果验证了它的合理性.     

基于深度强化学习算法的颗粒材料应力?应变关系数据驱动模拟研究

颗粒材料的宏观力学行为受颗粒组分等材料参数, 孔隙率、配位数等状态参数的影响, 同时又具备复杂的加载路径和加载历史相关性, 建立包含多个内变量以及各变量间相互关联的颗粒材料本构模型是一个重要的科学难题. 不同于传统的基于屈服面、流动法则和硬化函数框架下的唯象本构模型, 本文基于颗粒物质力学的研究基础, 以颗粒材料平均孔隙率、细观组构参数和弹性刚度参数作为内变量, 结合深度学习方法建立以有向图表征的数据本构模型. 有向图中以不同的链接网络表示不同的内变量信息流动方向, 各个内变量间的映射关系采用循环神经网络来建立, 将各个神经网络相互组合, 形成包含不同内变量且具有不同预测能力的本构模型. 该本构模型的建立过程等价于在众多可能的内变量链接关系空间中寻找最能描述实际材料宏观应力应变行为的优化问题. 因此, 可将有向图本构模型的建立过程看作“马尔可夫决策过程”, 采用深度强化学习算法构建有向图的内变量链接组合优化过程, 具体采用AlphaGo Zero算法自动寻找最优的颗粒材料数据驱动本构模型建模路径. 研究结果表明, 采用有向图和深度强化学习算法可建立起完全依靠“数据驱动”的颗粒材料应力?应变关系. 此外, 本方法提供了一种将不同理论模型从数据角度统一起来, 且基于人工智能算法发展更优模型的研究思路, 可为相似问题的研究提供借鉴.      

Gravity-driven dry granular slow flows down an inclined moving plane: a comparative study between two concepts of the evolution of porosity

Two concepts in modeling the effects of the evolution of porosity in dry granular flows are investigated to illuminate their performance and limitations. To this end, the thermodynamic analysis, based on the Müller-Liu entropy principle, and the quasi-linear theory, are employed to deduce the ultimate constitutive models and the restrictions on their thermodynamic consistencies. The models are employed to study an isothermal dry granular slow flow down an inclined moving plane, of which the results are compared with the experimental outcomes. Results show that, while the two models deliver appropriate equilibrium expressions of the Cauchy stress tensor for compressible grains, the model in which the evolution of porosity is treated kinematically yields a spherical stress tensor for incompressible grains. Only the model with a dynamic evolution of porosity can give rise to a non-spherical stress tensor at equilibrium. Moreover, whilst the former model can better capture the characteristics of flows with slow to moderate speeds, the latter model is more able to describe the features of very rapid flows like avalanches. The present study illustrates the essential difference between the two concepts in modeling the effects of the evolution of porosity, and can be extended for further studies on other microstructural effects in granular flows.     

Numerical simulations of particle migration in suspension flows: Frame-invariant formulation of curvature-induced migration

The objective of this work is to investigate what mechanisms should be employed to qualitatively/quantitatively predict particle migration in a suspension flow. Based on the diffusive flux model originally proposed by Phillips et al. [R.J. Phillips, R.C. Armstrong, R.A. Brown, A.L. Graham, A constitutive equation for concentrated suspensions that account for shear-induced particle migration, Phys. Fluids A 4 (1992) 30–40], we survey the accuracy of three models including original Phillips model (Model I), modified Phillips model with curvature-induced migration mechanism (Model II), and finally the modified Model II with volume-fraction-dependent parameters (Model III). The empirical parameters which appear in the three models are determined by fitting to independent concentric Couette experiments. The accuracy of three models in concentric Couette problem is comparable except that Model III shows more improved predictions near the inner cylinder. However, the predictions of the three models are entirely different on a qualitative level for parallel plate problems and the existence and direction of particle migration are severely model-dependent. Models II and III predict no migration or very slight migration at high volume fraction, which is in good agreement with the previous experiments, whereas Model I predicts inward migration. We show that Model III accurately predicts a solid-free region near the center at low volume fraction, which was experimentally observed.In addition to a survey of migration mechanisms, we developed a frame-invariant curvature-induced migration model applicable to multi-dimensional flows. A transient 2D mixed-order finite element method (FEM) code was implemented to compare the predictions of the three models in a 2D problem. In this work, we considered the eccentric Couette problem, which is often used as a benchmarking problem. Though there is not much difference among the three models, Model III predicts that the particle migration is slightly retarded at high shear rate regions.     

A Simple Nonlinear Thermodynamic Theory of Arbitrary Elastic Beams

A ‘classical’ theory of beams (i.e., a theory in which the basic kinetic variables are a stress resultant and a stress couple) undergoing elastic, thermodynamic processes is developed by first deriving exact beamlike (one-dimensional) equations of motion and a beamlike Second Law (Clausius–Duhem inequality) by descent from three-dimensions. Then what may be considered as the three basic assumptions of a classical theory are introduced: an assumed form of the First Law (conservation of energy), a relaxed form of the Second Law, and a general form of the constitutive relations. Throughout, detailed specification of geometry, kinematics, or constitution is minimized. It is shown how the kinematic Kirchhoff hypothesis may be avoided by first introducing a mixed-energy density and then imposing a logically more satisfying constitutive Kirchhoff hypothesis. Mathematics Subject Classifications (2000) 74A15, 74B20, 74K10     

A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II – Validation and localization analysis

We study the mechanical failure of cemented granular materials (e.g., sandstones) using a constitutive model based on breakage mechanics for grain crushing and damage mechanics for cement fracture. The theoretical aspects of this model are presented in Part I: Tengattini et al. (2014), A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables, Part I – Theory (Journal of the Mechanics and Physics of Solids, 10.1016/j.jmps.2014.05.021). In this Part II we investigate the constitutive and structural responses of cemented granular materials through analyses of Boundary Value Problems (BVPs).The multiple failure mechanisms captured by the proposed model enable the behavior of cemented granular rocks to be well reproduced for a wide range of confining pressures. Furthermore, through comparison of the model predictions and experimental data, the micromechanical basis of the model provides improved understanding of failure mechanisms of cemented granular materials. In particular, we show that grain crushing is the predominant inelastic deformation mechanism under high pressures while cement failure is the relevant mechanism at low pressures. Over an intermediate pressure regime a mixed mode of failure mechanisms is observed. Furthermore, the micromechanical roots of the model allow the effects on localized deformation modes of various initial microstructures to be studied. The results obtained from both the constitutive responses and BVP solutions indicate that the proposed approach and model provide a promising basis for future theoretical studies on cemented granular materials.     

Scale-dependent constitutive relations and the role of scale on nominal properties

Size effects in strength and fracture energy of heterogeneous materials is considered within a context of scale-dependent constitutive relations. Using tools of wavelet analysis, and considering the failure state of a one-dimensional solid, constitutive relations which include scale as a parameter are derived from a ‘background’ gradient formulation. In the resulting theory, scale is not a fixed quantity independent of deformation, but rather directly dependent on the global deformation field. It is shown that strength or peak nominal stress (maximum point at the engineering stress–strain diagram) decreases with specimen size while toughness or total work to fracture per nominal area (area under the curve in the engineering stress–strain diagram integrated along the length of the considered one-dimensional specimen) increases. This behavior is in agreement with relevant experimental findings on heterogeneous materials where the overall mechanical response is determined by variations in local material properties. The scale-dependent constitutive relations are calibrated from experimental data on concrete specimens.     

Curing effects on the acoustical properties of epoxy polymers

This paper presents an experimental method for measuring the attenuation and the velocity of longitudinal ultrasonic waves propagating through flat epoxy polymer samples. The study takes place in the first phase of epoxy polymer's polymerization, where these materials pass slowly from liquid state to the solid state. For this purpose an experimental setup was introduced, suitable for the accurate evaluation of the acoustic properties Δα andc _ℓ ^e , when the epoxy polymers are in their first phase of polymerization, while they are cured for 24 hours at room temperature (20°C). The ultrasonic method used is the pulse echo-through transmission technique. From the variation ofc _ℓ ^e and Δα during the first phase of epoxy polymers curing, the three characteristic states: liquid, semi-solid and solid, are clearly determined. It is also observed that plasticizer reduces the testability and the semi-solid state shows greater attenuation than either the liquid or the solid state.     

Temperature-rate dependent constitutive theory in thermo-viscoplastic media

In this paper, a temperature-rate dependent constitutive theory in thermo-viscoplastic media is proposed. On the basis of Clausius-Duhem (C-D) entropy inequality and the generalized Helmholtz free energy defined by Green and Lindsay, it is shown under some reasonable assumptions that the constitutive models in classical thermo-plasticity theory can be extended to the temperature-rate dependent (TRD) media provided that the temperature-rate effect is taken into account and the absolute temperatureT taken as an independent constitutive variable is replaced by coldness function T^* introduced by Müller, and discussed in detail by Ru and Duan.     

DISCRETE AND CONTINUUM MODELLING OF GRANULAR FLOW

This paper analyses three popular methods simulating granular flow at different time and length scales： discrete element method （DEM）, averaging method and viscous, elastic-plastic continuum model. The theoretical models of these methods and their applications to hopper flows are discussed. It is shown that DEM is an effective method to study the fundamentals of granular flow at a particle or microscopic scale. By use of the continuum approach, granular flow can also be described at a continuum or macroscopic scale. Macroscopic quantities such as velocity and stress can be obtained by use of such computational method as FEM. However, this approach depends on the constitutive relationship of materials and ignores the effect of microscopic structure of granular flow. The combined approach of DEM and averaging method can overcome this problem. The approach takes into account the discrete nature of granular materials and does not require any global assumption and thus allows a better understanding of the fundamental mechanisms of granular flow. However, it is difficult to adapt this approach to process modelling because of the limited number of particles which can be handled with the present computational capacity, and the difficulty in handling non-spherical particles. Further work is needed to develoo an aoorooriate aooroach to overcome these problems.