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.     

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.     

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

A thermodynamically consistent continuum theory for single-phase, single-constituent cohesionless granular materials is presented. The theory is motivated by dimensional inconsistencies of the original Goodman-Cowin theory [1–3]; it is constructed by removing these inconsistencies through the introduction of an internal length ℓ. Four constitutive models are proposed and discussed in which ℓ is (i) a material constant (Model I), (ii) an independent constitutive variable (Model II), (iii) an independent dynamic field quantity (Model III) and (iv) an independent kinematic field quantity (Model IV). Expressions of the constitutive variables emerging in the systems of the balance equations in these four models in thermodynamic equilibrium are deduced by use of a thermodynamic analysis based on the Müller-Liu entropy principle. Comments on the validity of these four models are given and discussed; the results presented in the current study show a more general formulation for the constitutive quantities and can be used as a basis for further continuum-based theoretical investigations on the behaviour of flowing granular materials. Numerical results regarding simple plane shear flows will be discussed and compared in Part II of this work.     

A generalized Mohr–Coulomb criterion implied by the revised Goodman–Cowin theory

In the present study, the constitutive relations derived in the revised Goodman–Cowin theory for granular matter are shown to imply a generalized Mohr–Coulomb criterion for impending flows. Due to the concept of microcontinuum and the incorporation of the internal friction into the expression of the Cauchy stress tensor, a constrained equilibrium stress state characterized by the Mohr–Coulomb criterion is yet obtained under uniform distributions of the grains.     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

考虑破碎影响的颗粒材料亚塑性模型及应变局部化模拟

在基于2ndP-K应力率的亚塑性模型基础上,通过引入一个能够考虑颗粒破碎影响的孔隙比-平均压力临界状态方程,形成了一个能够模拟颗粒破碎影响的颗粒材料亚塑性模型,数值算例考查了颗粒破碎对应变局部化模式及位移-承载曲线的影响,结果表明,所建议模型具有模拟破碎对颗粒材料应变局部影响的良好性能。     

Global–local analysis of granular media in quasi-static equilibrium

A method of global–local analysis is developed for quasi-static equilibrium problems for granular media. The two-scale modeling based on mathematical homogenization theory enables us to formulate two separate boundary value problems in terms of macro- and microscales. The macroscale problem governs the equilibrium of a global structure composed of granular assemblies, while the microscale one is posed for the particulate nature of a local structure with the friction-contact mechanism between particles. The local structure is identified with a periodic representative volume element, or equivalently, a unit cell, over which averaging is performed. The mechanical behavior of unit cells is analyzed by a discrete numerical model, in which spring and friction devices connect rigid particles, whereas the continuum-based finite element method is used for the macroscopic one. Representative numerical examples are presented to demonstrate the capability of the proposed two-scale analysis method for granular materials.     

Probability density functions of contact forces for cohesionless frictional granular materials

A theory is developed for the probability density functions of contact forces for cohesionless, frictional granular materials in quasi-static equilibrium. This theory is based on a maximum information entropy principle, with an expression for information entropy that is appropriate for granular materials. Entropy is maximized under the constraints of a prescribed stress and that the normal component of the contact force is compressive and that the tangential component of the contact force is limited by Coulomb friction. The theory results in a dependence of the probability density function for the tangential contact forces on the friction coefficient. The theoretical predictions are compared with results from discrete element simulations on isotropic, two-dimensional assemblies under hydrostatic stress. Good qualitative agreement is found for means and standard deviations of contact forces and the shape of the probability density functions, while the quantitative agreement is fairly good. Discrepancies between theory and simulations, such as the difference in shape of the probability density function for the normal force and the observed dependence on elastic properties of the exponential decay rate of tangential forces, are attributed to the fact that the method does not take into account any kinematics, which are essential in relation to elastic effects.     

On flows of granular materials

This article reviews the behavior of materials made up of a large assemblage of solid particles under rapid and quasi static deformations. The focus is on flows at relatively high concentrations and for conditions when the interstitial fluid plays an insignificant role. The momentum and energy exchange processes are then primarily governed by interparticle collisions and Coulomb-type frictional contact. We first discuss some physical behavior —dilatancy, internal friction, fluidization and particle segregation — that are typical to the understanding of granular flows. Bagnold's seminal Couette flow experiments and his simple stress analysis are then used to motivate the first constitutive theories that use a microstructural variable — the fluctuation energy or granular temperature — governing the subscale fluctuating motion. The kinetic theories formalize the derivation of the field equations of bulk mass, momentum and energy, and permit derivation of constitutive relations for stress, flux of fluctuation energy and its dissipation rate for simple particle assemblages and when frictional rubbing contact can be ignored. These statistical considerations also show that formulation of boundary conditions needs special attention. The frictional-collisional constitutive behavior in which both Coulomb-type rubbing contact and collisional encounters are significant are discussed. There is as yet no rigorous formulation. We finally present a phenomenological approach that describes rapid flows of granular materials under simultaneous transport of heat and close with a summary of stability analyses of the basic flow down an inclined plane.Dedicated to Professor Dr.-Ing. Franz Gustav Kollmann on the occasion of his sixtieth brithday     

On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation

The purpose of this work is the unified formulation and generalization of selected models for extended, gradient, or “higher-order” crystal plasticity via the application of a recently developed rate variational approach to the formulation of continuum thermodynamic models for history-dependent, inelastic systems. The investigation here includes models which were not originally formulated in a thermodynamic or “work-conjugate” fashion. The approach is based on the formulation of rate potentials for each model whose form is determined by (i) energetic processes via the free energy, (ii) kinetic processes via the dissipation potential, and (iii) the form of the evolution relations for the internal-variable-like quantities upon which the free energy and dissipation potential depend. For the case of extended crystal plasticity, these latter quantities include for example the inelastic local deformation, or dislocation densities. The stationarity conditions of the corresponding rate functional then yield volumetric and surficial balance-like field relations determining in the current context for example the form of momentum balance or that of the generalized glide-system flow rule. With the help of this approach, we derive thermodynamically consistent forms of specific models for extended crystal plasticity. Since most of these were formulated for small deformation, we also investigate their generalization to large deformation with the help of, e.g., form invariance. Among other things, the current rate variational approach implies that, beyond the form of the free energy itself, it is form of the evolution relations for the dislocation densities which is important in determining whether or not higher-order model quantities like the glide-system back stress can be formulated in a thermodynamic fashion.     

Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations

The purpose of this work is the formulation of constitutive models for the inelastic material behaviour of single crystals and polycrystals in which geometrically necessary dislocations (GNDs) may develop and influence this behaviour. To this end, we focus on the dependence of the development of such dislocations on the inhomogeneity of the inelastic deformation in the material. More precisely, in the crystal plasticity context, this is a relation between the density of GNDs and the inhomogeneity of inelastic deformation in glide systems. In this work, two models for GND density and its evolution, i.e., a glide-system-based model, and a continuum model, are formulated and investigated. As it turns out, the former of these is consistent with the original two-dimensional GND model of Ashby (Philos. Mag. 21 (1970) 399), and the latter with the more recent model of Dai and Parks (Proceedings of Plasticity ’97, Neat Press, 1997, p. 17). Since both models involve a dependence of the inelastic state of a material point on the (history of the) inhomogeneity of the glide-system inelastic deformation, their incorporation into crystal plasticity modelling necessarily implies a corresponding non-local generalization of this modelling. As it turns out, a natural quantity on which to base such a non-local continuum thermodynamic generalization, i.e., in the context of crystal plasticity, is the glide-system (scalar) slip deformation. In particular, this is accomplished here by treating each such slip deformation as either (1), a generalized “gradient” internal variable, or (2), as a scalar internal degree-of-freedom. Both of these approaches yield a corresponding generalized Ginzburg-Landau- or Cahn-Allen-type field relation for this scalar deformation determined in part by the dependence of the free energy on the dislocation state in the material. In the last part of the work, attention is focused on specific models for the free energy and its dependence on this state. After summarizing and briefly discussing the initial-boundary-value problem resulting from the current approach as well as its algorithmic form suitable for numerical implementation, the work ends with a discussion of additional aspects of the formulation, and in particular the connection of the approach to GND modelling taken here with other approaches.     

Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example

In the context of a special problem, this paper investigates the possibility of modeling dissipative mechanical response in solids on the basis of the equilibrium theory of finite elasticity for materials that may lose ellipticity at large strains. Quasi-static motions for such materials are in general dissipative if the associated equilibrium fields involve discontinuous displacement gradients. For the problem treated, consideration of such deformations is shown to lead naturally to an internal variable formalism similar to those used to describe macroscopic plastic behavior arising from microstructural effects. For quasi-static motions which are maximally dissipative in a specified sense, this formalism leads to a mechanical response which resembles that associated with the pseudo-elastic effect in shape-memory alloys.     

A complementarity-based rolling friction model for rigid contacts

In this work (also, preprint ANL/MCS-P3020-0812, Argonne National Laboratory) we introduce a complementarity-based rolling friction model to characterize dissipative phenomena at the interface between moving parts. Since the formulation is based on differential inclusions, the model fits well in the context of nonsmooth dynamics, and it does not require short integration timesteps. The method encompasses a rolling resistance limit for static cases, similar to what happens for sliding friction; this is a simple yet efficient approach to problems involving transitions from rolling to resting, and vice-versa. We propose a convex relaxation of the formulation in order to achieve algorithmic robustness and stability; moreover, we show the side effects of the convexification. A natural application of the model is the dynamics of granular materials, because of the high computational efficiency and the need for only a small set of parameters. In particular, when used as a micromechanical model for rolling resistance between granular particles, the model can provide an alternative way to capture the effect of irregular shapes. Other applications can be related to real-time simulations of rolling parts in bearing and guideways, as shown in examples.     

On the modelling of anisotropic elastic and inelastic material behaviour at large deformation

The purpose of this work is the formulation and discussion of an approach to the modelling of anisotropic elastic and inelastic material behaviour at large deformation. This is done in the framework of a thermodynamic, internal-variable-based formulation for such a behaviour. In particular, the formulation pursued here is based on a model for plastic or inelastic deformation as a transformation of local reference configuration for each material element. This represents a slight generalization of its modelling as an elastic material isomorphism pursued in earlier work, allowing one in particular to incorporate the effects of isotropic continuum damage directly into the formulation. As for the remaining deformation- and stress-like internal variables of the formulation, these are modelled in a fashion formally analogous to so-called structure tensors. On this basis, it is shown in particular that, while neither the Mandel nor back stress is generally so, the stress measure thermodynamically conjugate to the plastic “velocity gradient”, containing the difference of these two stress measures, is always symmetric with respect to the Euclidean metric, i.e., even in the case of classical or induced anisotropic elastic or inelastic material behaviour. Further, in the context of the assumption that the intermediate configuration is materially uniform, it is shown that the stress measure thermodynamically conjugate to the plastic velocity gradient is directly related to the Eshelby stress. Finally, the approach is applied to the formulation of metal plasticity with isotropic kinematic hardening.     

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.     

Hypoplasticity theory for granular materials—II: Three-dimensional axially symmetric exact solutions

In part I of this paper, we consider the governing equations of hypoplasticity theory for two-dimensional steady quasi-static plane strain compressible gravity flow and determine some exact analytical solutions applying for certain special cases. Similarly, for the three-dimensional situation considered here in part II, we undertake a similar mathematical investigation to determine some simple solutions of the governing equations for three-dimensional steady quasi-static axially symmetric compressible gravity flow for hypoplastic granular materials. We again find that for certain special cases, we are able to determine some exact solutions for the stress and velocity profiles. We comment that hypoplasticity theory generally gives rise to complicated systems of coupled non-linear differential equations, for which the determination of any analytical solutions is not a trivial matter, and that the solutions determined here might be exploited as benchmarks for full numerical schemes.     

Boundary effects on behaviour of granular material during plane strain compression

The paper investigates the boundary effect on the behaviour of granular materials during plane strain compression using finite element method. A micro-polar hypoplastic constitutive model was used. The numerical calculations were carried out with different initial densities and boundary conditions. The behaviour of initially dense, medium dense and loose sand specimen with very smooth or very rough horizontal boundary was investigated. The formation of shear zones gave rise to different global and local stress and strain. Comparisons of the mobilized internal friction, dilatancy and non-coaxiality between global and local quantities were made.     

Evolution Equations for Non-Simple Viscoelastic Solids

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.     

Interface models coupling adhesion and friction

Interface models coupling friction and adhesion, where adhesion is regarded as interface damage, are briefly reviewed. The most widely used cohesive zone models are presented and discussed. A general framework for these laws, recently developed by Del Piero and Raous in the form of a unified model, is outlined. As an example, it is here established that the RCCM (Raous–Cangémi–Cocou–Monerie) model is a specific case in this general framework. The variational formulation and some associated solvers are briefly recalled in the context of non-smooth mechanics in the cases of both quasi-static and dynamic problems. A few examples in various fields of application are given. Lastly, some open problems and ongoing researches in this field are presented and discussed.