Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.     

Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model

Macro-scale deformation of granular solids comprising large number of grains (>10⁶) are most efficiently described within the framework of continuum mechanics. It is notable, however that the micro-scale deformations in these materials are concentrated at the grain-boundaries or grain-contacts. Thus, the deformation energies in these systems must be modeled by considering the deformations concentrated in the neighborhood of the grain-boundaries or grain-contacts. To address this issue, grain-interactions has been widely described in the Hertzian sense by considering the relative movement of points on either side of a grain boundary or contact treated as an imperfect interface. This communication introduces the relevant kinematic variables given in the terms of the grain displacements, spins and size that can be used to estimate the relative movement of a grain boundary or contact. The macro-scale kinematic variables useful for continuum modeling are then identified with the grain-scale kinematic variables. The deformation energy density of the granular solid can thus be expressed both in terms of the grain-scale as well as the macro-scale kinematic variables providing the necessary pathway for micro-macro identification which can lead to non-classical micromorphic continuum models that incorporate grain-scale representation.     

Evolving internal length scales in plastic strain localization for granular materials

A numerical model in the Cosserat continuum for strain localization phenomena in granular materials is developed and proposed in this paper. The model assumes a constant internal length scale that is used to describe the shear band thickness. However, it is observed that the internal length scales need to change to accommodate the possible change in the contact surface between the particles, damage of the particles or/and any change in the local void ratio within the domain, which will change the shear band thickness. The mathematical formulations used in the present numerical model were equipped with evolution equations for the length scales through the Micropolar theory, those formulations are proposed and discussed in this paper. The evolution equations of the internal length scales describe any possible change in the contact surface between the particles, damage of the particles if exists and/or any change in the local void ratio within the domain. Hence, the strain localization described by the enhanced model with evolving internal length scales is more accurate and closer to the real solution. The solution for the shear bands thickness shows more accurate correlation with the experimental results and less dependency on the mesh size when such evolution equations are used. Moreover, the shear band thickness and inclination evolve during the deformation process.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

A micro-mechanical investigation of bifurcation in granular materials

In this paper, the notion of loss of sustainability of a mechanical state in a granular assembly is investigated. The vanishing of the second-order work, defined on the macroscopic scale from tensorial variables, is shown to play a fundamental role in detecting the occurrence of this type of bifurcation. Then a link is established between the macroscopic second-order work on the specimen scale and a discrete local expression that introduces microscopic variables defined on each contact scale. This relation opens up a micro-mechanical interpretation allowing one to examine which micro-structural features are responsible for the vanishing of the macroscopic second-order work. Finally, it is established that both geometrical and material micro-structural origins may combine to induce the occurrence of bifurcation on a specimen scale.     

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.     

Modeling shear localization along granular soil–structure interfaces using elasto-plastic Cosserat continuum

The current study presents finite element simulations of shear localization along the interface between cohesionless granular soil and bounding structure under large shearing movement. Micro-polar (Cosserat) continuum approach is applied in the framework of elasto-plasticity in order to overcome the numerical problems of localization modeling seen in the conventional continuum mechanics. The effects of different micro-polar kinematic boundary conditions, along the interface, on the evolution and location of shear band are shown by the numerical results. Furthermore, shear band thickness is also investigated for its dependence on the initial void ratio, vertical pressure and mean grain size. Here, the distribution and evolution of static and kinematic quantities are the main focuses regarding infinite layer of micro-polar material during plane shearing, especially with advanced large movement of bounding structure. The influence of such movement has not been investigated yet in the literature. Based on the results obtained from this study, shear localization appears parallel to the direction of shearing. It occurs either in the middle of granular layer or near boundaries, regarding the assumed micro-polar kinematic boundary conditions at the bottom and top surfaces of granular soil layer. Narrower shear band is observed in lower rotation resistance of soil particles along the interface. It is emphasized that the displacement magnitude of bounding structure has significant effect on the distribution and evolution of state variables and polar quantities in the granular soil layer. However, continuous displacement has no meaningful effect on the thickness of shear band. Here, smooth distributions of void ratio and shear stress components are obtained within the shear band, what the other previous numerical investigations did not receive. Despite indirect linking of Lade’s model to the critical state soil mechanics, state variables tend towards asymptotical stationary condition in large shear deformation.     

Framework using functional forms of hardening internal state variables in modeling elasto-plastic-damage behavior

This work gives the thermodynamically consistent theoretical formulations and the numerical implementation of a plasticity model fully coupled with damage. The formulation of the elasto-plastic-damage behavior of materials is introduced here within a framework that uses functional forms of hardening internal state variables in both damage and plasticity. The damage is introduced through a damage mechanics framework and utilizes an anisotropic damage measure to quantify the reduction of the material stiffness. In deriving the constitutive model, a local yield surface is used to determine the occurrence of plasticity and a local damage surface is used to determine the occurrence of damage. Isotropic hardening and kinematic hardening are incorporated as state variables to describe the change of the yield surface. Additionally, a damage isotropic hardening is incorporated as a state variable to describe the change of the damage surface. The hardening conjugate forces (stress-like terms) are general nonlinear functions of their corresponding hardening state variables (strain-like terms) and can be defined based on the desired material behavior. Various exponential and power law functional forms are studied in this formulation. The paper discusses the general concept of using such functional forms. however, it does not address the relevant appropriateness of certain forms to solve different problems. The proposed work introduces a strong coupling between damage and plasticity by utilizing damage and plasticity flow rules that are dependent on both the plastic and damage potentials. However, in addition to that the coupling is further enhanced through the use of the functional forms of the hardening variables introduced in this formulation.The use of this formulation in solving boundary value problems will be presented in future work. The fully implicit backward Euler scheme is developed for this model to be solved in a Newton–Raphson solution procedure.     

The study of relations between loading history and yield surfaces in powder materials using discrete finite element simulations

Only a few studies in the literature have applied the finite-element method to analyse assemblies of meshed particles. These studies illustrated the relevance of this method for granular materials. Here, the compaction of ductile metal powders was studied through a numerical assembly of elastic–plastic and rate-independent spherical particles. The aim of this paper was to understand the evolution of yield surfaces with complex loading paths up to high relative density at the macroscopic scale and at the granular scale. Simulation results revealed that yield surfaces evolved with both isotropic and kinematic hardening mechanisms, depending on the compaction stage. An analysis of the sample microstructure was proposed, and a detailed study of contacts between particles revealed some of the mechanisms that led to the observed evolution of yield surfaces.     

Original ArticleKinematic hardening rules in finite plasticity Part II: Some examples

We consider finite plasticity laws which are represented by means of dual variables and satisfy the intrinsic dissipation inequality. Within two families of dual variables, two different formulations for a kinematic hardening rule, referred to as Model 1 and Model 2, are proposed. In order to discuss basic properties of the two models, the predicted response for some simple deformations is calculated. It turns out that e.g. with respect to the simple shear and simple torsion the two models differ mainly in the effects of second order. Received June 14, 1995     

A note on formulations of static shakedown analysis with bounded kinematic hardening

Two different formulations for the two-surface model of bounded kinematic hardening can be found in literature on shakedown analysis with the von Mises yield criterion. This short paper explains that these two formulations are not equivalent, although there exists literature asserting that they are equivalent. More specifically, the formulation using the constraint on the stress is not a sufficient condition for the two-surface model. Consequently, the static shakedown analysis using this formulation over-evaluates the shakedown factor in general.     

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.     

Suitability of yield functions for the approximation of subsequent yield surfaces

Many consitutive models in the plasticity of metals are based on the existence of a yield function, which is not only used to mark the elastic limit but also as a potential function for the plastic strain rate. The construction of this function therefore deserves the utmost interest. Measurements of the elastic limit show the essential features of how the geometry of the yield function contour lines should change with further straining. Against these typical geometric forms of the yield surface existing proposals for invariant formulations of the yield function taking into account isotropic, kinematic and formative hardening are tested. Even if no evolution equations for the constitutive variables contained in the yield functions are specified, best approximations of measured yield surfaces can be computed by optimisation of a quality function. It can be shown that most of the representations are not even able to describe the experimental results qualitatively. The numeric results show further that the yield function is essentially of grade three in the deviatoric stresses. The evolution of internal variables can be deduced from the approximations of the measurements.     

A three-invariant cap model with isotropic–kinematic hardening rule and associated plasticity for granular materials

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticity of granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associated flow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decomposition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. The model assessment and procedure for determination of material parameters are described. Finally, the applicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pressure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.     

New constitutive equations for the rapid flow of granular materials—II

This paper compliments a previous paper, (J. Non-Newtonian Fluid Mech.,9(1981) 147), which discussed new constitutive equations for rapid collisionally maintained flow of granular materials as a non-Newtonian microfluid in which the gradient of microrotation played an important role as a kinematic variable. In this article we discuss another variation of such constitutive equations; one for which we do not consider the effects of gradients of the mean microrotation of grains. Derived are expressions for the dispersive normal and shear stresses for a plane shear rapid flow of a granular material in the presence of intergranular slip and friction. These expressions reduce to classical results if friction is set equal to zero. A graph of the variation of stresses versus the friction factor is also presented which reveals a kind of choking phenomenon at larger values of μ_k.     

On Limit Design of Frames Using Linear Programming

Abstract

This paper deals with a broad class of optimum frame design problems amenable to the mathematical model of linear programming: allowance is made for self-weight (design dependent loads) and technological constraints “assigned minimum for yield moments, prescribed variation laws of yield moments along members”. Two alternative “static” formulations and the corresponding dual “kinematic” formulations are discussed and compared to each other. The main limit design theorems, generalized to the present broader context, are derived on the basis of duality theory of linear programming. Numerical examples, worked out by means of standard LP computer codes, are given.     

A divide and conquer algorithm for constrained multibody system dynamics based on augmented Lagrangian method with projections-based error correction

This paper presents a?new parallel algorithm for dynamics simulation of general multibody systems. The developed formulations are iterative and possess divide and conquer structure. The constraints equations are imposed at the acceleration level. Augmented Lagrangian methods with mass-orthogonal projections are used to prevent from constraint violation errors. The proposed approaches treat tree topology mechanisms or multibody systems which contain kinematic closed loops in a?uniform manner and can handle problems with rank deficient Jacobian matrices. Test case results indicate good accuracy performance dependent on the expense put in the iterative correction of constraint equations. Good numerical properties and robustness of the algorithms are observed when handling systems with single and coupled kinematic loops, redundant constraints, which may repeatedly enter singular configurations.