Discrete element method for modelling solid and particulate materials

The collaborative structural analysis (CSA) system is capable of performing highly sophisticated structural analyses utilizing the beneficial features of existing individual structural analysis programmes. It requires a time consuming static condensation procedure if adopting an implicit integration scheme. The operator splitting (OS) method, which does not require the tangential stiffness, can be used to improve the system efficiency. Furthermore, the conventional OS method is not able to provide enough numerical stability, particularly for the analyses considering geometrical non‐linearity. Thus, improvement is needed. To this end, a modified OS method is proposed, which treats unbalanced forces in the current step as pseudo‐external forces in the immediate following step. In this paper, first, the conventional OS method is reformulated in an incremental form, and a CSA scheme based on it is proposed. Second, a modified OS method is developed to improve the numerical stability. Third, a fixed‐base steel column with a lumped mass assigned at its top is analysed using the CSA system as a numerical example. It is found that the OS methods are effective for CSA, and the modified OS method exhibits better numerical stability than the conventional one for the analysis considering geometrical non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.     

A general finite element approach for contact stress analysis

This paper presents a general finite element approach for the treatment of contact stress problems. Stanctard shape function routines are used for the detection of contact between previously separate meshes and for the application of displacement constraints where contact has been identified. The mesh contact routines are installed in an incremental approach whereby the contact constraints are imposed by using either penalty functions or Lagrange multipliers.     

A discrete element analysis of elastic properties of granular materials

The elastic properties of a regular packing of spheres with different tolerances were evaluated using the discrete element method to elucidate the mechanisms behind the discrepancies between laboratory experiments and theoretical predictions of the classic Hertz-Mindlin contact law. The simulations indicate that the elastic modulus of the packing is highly dependent on the coordination number and the magnitude and distribution of contact normal forces, and this dependence is macroscopically reflected as the influence of confining pressure and void ratio. The increase of coordination number and the uniformity of contact normal forces distribution with increasing confining pressure results in the stress exponent $n$ for elastic modulus being higher than 1/3 as predicted by the Hertz-Mindlin law. Furthermore, the simulations show that Poisson’s ratio of a granular packing is not a constant as commonly assumed, but rather it decreases as confining pressure increases. The variation of Poisson’s ratio appears to be a consequence of the increase of the coordination number rather than the increase of contact normal forces with confining pressure.     

A simplified updated Lagrangian approach for combining discrete and finite element methods

This paper proposes the use of a specific combination of discrete and finite element methods for the simulation of systems of deformable bodies in order to reduce the computational cost, when certain assumptions can be made. In particular, the Updated Lagrangian finite element formulation and the central difference time integration method are employed together with certain simplifying assumptions in order to linearize this highly nonlinear contact problem and obtain solutions with realistic computational cost and sufficiently good accuracy. Furthermore, the paper discusses software implementation issues and the advantages that the Java technologies can offer in the development of such engineering applications.     

An implicit finite element material model for energetic particulate composite materials

The predisposition of energetic particulate composite materials, or high explosives (HE), to ignition by bulk heating (cookoff) poses serious safety problems. Because unexpected initiation of HE must be a major consideration in any activity involving employment of the material, its behaviour under a variety of conditions is of much interest. The formulation of a numerical constitutive model that can be employed in an implicit finite element code to predict the mechanical and ignition behaviour of HE is presented. The capability of the developed material model is then demonstrated through its implementation in the ABAQUS finite element code to simulate the response of HE test configurations. The simulated response is found to compare favourably with the physical test results, in the cases where test data exist. Copyright © 2000 John Wiley & Sons, Ltd.     

A general two-dimensional,graphical finite element preprocessor utilizing discrete transfinite mappings

Finite element preprocessor programs have been developed in recent years in order to expedite the task of data preparation. A key decision in the design of these programs is the choice of a method for automated mesh generation. Three popular methods (Laplacian, isoparametric and transfinite mappings) are compared. Transfinite mappings with boundary information represented in discrete form are found to possess distinct advantages for the task of mesh generation. A two-dimensional preprocessor program utilizing the discrete transfinite mappings is described. Interactive computer graphics techniques are used extensively to facilitate data preparation and display. The geometry-generating routines are general, and may be used in any finite element application. A specialized graphical ‘attribute editor’ for structural mechanics problems is also described. This editor provides an efficient method of specifying boundary conditions, material properties, loads, etc.     

A discrete element method-based approach to predict the breakage of coal

Pulverization is an essential pre-combustion technique employed for solid fuels, such as coal, to reduce particle sizes. Smaller particles ensure rapid and complete combustion, leading to low carbon emissions. Traditionally, the resulting particle size distributions from pulverizers have been determined by empirical or semi-empirical approaches that rely on extensive data gathered over several decades during operations or experiments, with limited predictive capabilities for new coals and processes. This work presents a Discrete Element Method (DEM)-based computational approach to model coal particle breakage with experimentally characterized coal physical properties. The effect of select operating parameters on the breakage behavior of coal particles is also examined.     

A modelling and verification approach for soybean seed particles using the discrete element method

To build a discrete element method (DEM) model of soybean seed particles, the shape and size of soybean seed particles were measured and analysed. The results showed that the shape of a soybean seed particle could be approximated to an ellipsoid and that the dispersity in size could be approximated by a normal distribution. Additionally, a certain functional relationship between the primary dimension and secondary dimensions was determined. On this basis, an approach for modelling soybean seed particles based on the multi-sphere (MS) method was proposed. The soybean seed particle was simplified to an ellipsoid with the averaged size of one hundred randomly selected soybean seeds. The model of a single soybean seed particle was built by filling spheres within the ellipsoid. For modelling soybean seed assembly, the primary dimension was generated according to the normal distribution, and the other secondary dimensions were calculated based on their relationships with the primary dimension. In this way, the model of soybean seed assembly with different sizes and distributions was built. In this paper, four varieties of soybean seed were used. By comparing the simulated results and experimental results both in piling tests and “self-flow screening” tests, when the number of filling spheres was five, the simulated results were close to those obtained experimentally. Therefore, the feasibility and validity of the modelling method for soybean seed particles that we proposed were verified. Finally, an application case was employed to show how to use the soybean seed particle model and the discrete element method to analyse the discharging process of a silo.     

A predictive model for particulate filled composite materials

Our predictive model for particulate filled composite materials is applied to epoxy resin toughened by rubber spheres. Good agreement is found between predicted values of stiffness and experimental measurements. The variability of yield stress with volume fraction is explained. The observed fracture processes, including rubber cavitation, are explored; the importance of thermal stresses is highlighted. The fracture behaviour of these materials is discussed in the light of these predictions.     

Analysis of a triangulation based approach for specimen generation for discrete element simulations

Discrete element methods are emerging as useful numerical analysis tools for engineers concerned with granular materials such as soil, food grains, or pharmaceutical powders. Obviously, the first step in a discrete element simulation is the generation of the geometry of the system of interest. The system geometry is defined by the boundary conditions as well as the shape characteristics (including size) and initial coordinates of the particles in the system. While a variety of specimen generation methods for particulate materials have been developed, there is no uniform agreement on the optimum specimen generation approach. This paper proposes a new triangulation based approach that can easily be implemented in two or three dimensions. The concept of this approach (in two dimensions) is to triangulate a system of points within the domain of interest, creating a mesh of triangles. Then the particles are inserted as the incircles of these triangles. Extension to three dimensions using a mesh of tetrahedra and inserting the inspheres is relatively trivial. The major advantages of this approach include the relative simplicity of the algorithm and the small computational cost associated with the preparation of an initial particle assembly. The sensitivity of the characteristics of the particulate material that is generated to the topology of the triangular mesh used is explored. The approach is compared with other currently used methods in both two and three dimensions. These comparisons indicate that while this approach can successfully generate relatively dense two-dimensional particle assemblies, the three- dimensional implementation is less effective at generating dense systems than other available approaches. The research presented in this paper made use of software developed by other researchers. For the two-dimensional study the program Triangle developed by Jonathan Shewchuk was used. The three-dimensional analysis used the Geompack++ program developed by Barry Joe as well as an implementation of the Jodrey and Tory (1985) algorithm by Monika Bargiel and Jacek Moscinski called NSCP3D.     

Absorption-coefficient-determination method for particulate materials

A method is presented for determining the optical absorption coefficient, or the imaginary refractive index, of particulate material that has been collected from aerosols or hydrosols by means of filtration. The method, based on the Kubelka-Munk theory of diffuse reflectance, is nondestructive and requires no other knowledge of the sample than the amount present, the specific gravity, and an estimate of the real index of refraction. The theoretical development of the method is discussed along with an analysis of photometric and gravimetric errors. We test the method by comparing results obtained for powdered didymium glass with measurements made before the glass was crushed. An example of the method's application to the determination of the absorption coefficient of atmospheric dust at UV, visible, and near-IR wavelengths is also presented.     

A hybrid approach for modeling of breakable granular materials using combined finite-discrete element method

It is well known that particle breakage plays a critical role in the mechanical behavior of granular materials and has been a topic subject to intensive studies. This paper presents a three dimensional fracture model in the context of combined finite-discrete element method (FDEM) to simulate the breakage of irregular shaped granular materials, e.g., sands, gravels, and rockfills. In this method, each particle is discretized into a finite element mesh. The potential fracture paths are represented by pre-inserted non-thickness cohesive interface elements with a progressive damage model. The Mohr–Coulomb model with tension cut-off is employed as the damage initiation criterion to rupture the predominant failure mode at the particle scale. The particle breakage modeling using combined FDEM is validated by the qualitative agreement between the results of simulated single particle crushing tests and those obtained from laboratory tests and prior DEM simulations. A comprehensive numerical triaxial tests are carried out on both the unbreakable and breakable particle assemblies with varied confining pressure and particle crushability. The simulated stress–strain–dilation responses of breakable granular assembly are qualitatively in good agreement with the experimental observations. The effects of particle breakage on the compressibility, shear strength, volumetric response of the fairly dense breakable granular assembly are thoroughly investigated through a variety of mechanism demonstrations and micromechanical analysis. This paper also reports the energy input and dissipation behavior and its relation to the mechanical response.     

Numerical simulation of granular materials by an improved discrete element method

In this paper, we present an improved discrete element method based on the non‐smooth contact dynamics and the bi‐potential concept. The energy dissipated during the collisions is taken into account by means of restitution coefficients. The interaction between particles is modelled by Coulomb unilateral contact law with dry friction which is typically non‐associated: during the contact, the sliding vector is not normal to the friction cone. The main feature of our algorithm is to overcome this difficulty by means of the bi‐potential theory. It leads to an easy implement predictor–corrector scheme involving just an orthogonal projection onto the friction cone. Moreover the convergence test is based on an error estimator in constitutive law using the corner stone inequality of the bipotential. Then we present numerical simulations which show the robustness of our algorithm and the various possibilities of the software ‘MULTICOR’ developed with this approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Finite element approach for modelling fatigue damage in fibre-reinforced composite materials

Today, a lot of research is dedicated to the fatigue behaviour of fibre-reinforced composite materials due to their increasing use in all sorts of applications. These materials have a quite good rating as regards to lifetime in fatigue, but the same does not apply to the number of cycles to initial damage, or to the evolution of damage. Composite materials are inhomogeneous and anisotropic, and their behaviour is more complicated than that of homogeneous and isotropic materials such as metals. A new finite element approach is proposed in order to deal with two conflicting demands: (i) due to the gradual stiffness degradation of a fibre-reinforced composite material under fatigue, stresses are continuously redistributed across the structure and as a consequence, the simulation should follow the complete path of successive damage states; (ii) the finite element simulation should be fast and computationally efficient to meet the economic needs. The authors have adopted a cycle jump approach which allows to calculate a set of fatigue loading cycles at deliberately chosen intervals and to account for the effect of the fatigue loading cycles in between in an accurate manner. The finite element simulations are compared against the results of fatigue experiments on plain woven glass/epoxy specimens with a [#45°]₈ stacking sequence.     

A discrete strong discontinuity approach

In this paper, strong discontinuities are embedded in finite elements to describe fracture in quasi-brittle materials. A new formulation is presented in which global nodes are introduced along the crack path. The displacement jumps are transferred to the element nodes as a rigid body motion. This approach is compared to the discrete approach, in which interface elements are inserted to model discontinuities. The adopted embedded discontinuities and the interface elements share similar kinematics as well as the same numerical integration schemes. Thus, the present formulation is obtained within the framework of a discrete approach and this is why it is called the discrete strong discontinuity approach (DSDA). Numerical tests are considered, namely a shear band, a mode-I and a mixed-mode fracture examples and a failure test of a RC beam externally reinforced with a steel sheet. Results are compared with those obtained from analyses using interface elements and with experimental results. Finally, conclusions are drawn with respect to mesh independence and robustness of the method.     

A variational discrete element method for quasistatic and dynamic elastoplasticity

We propose a new discrete element method supporting general polyhedral meshes. The method can be understood as a lowest-order discontinuous Galerkin method parametrized by the continuous mechanical parameters (Young's modulus and Poisson's ratio). We consider quasistatic and dynamic elastoplasticity, and in the latter situation, a pseudoenergy conserving time-integration method is employed. The computational cost of the time-stepping method is moderate since it is explicit and used with a naturally diagonal mass matrix. Numerical examples are presented to illustrate the robustness and versatility of the method for quasistatic and dynamic elastoplastic evolutions.     

A discrete element method for the simulation of CFRP cutting

Orthogonal machining of unidirectional carbon fiber-reinforced polymer (UD-CFRP) composites is simulated using discrete element method (DEM). The objective of this work is to present a simple numerical model that allows the study the machining of unidirectional composites during orthogonal cutting. To control the physicochemical phenomena that occur during cutting, it is necessary to identify the parameters of contact, very difficult to measure experimentally. The DEM numerical simulation is presented then as an alternative to the problem. This tool has helped to recreate the physical mechanisms identified experimentally and to understand the origin of the abrasive wear of carbide tools. The observation of the chip formation using a high speed video camera made possible to validate qualitatively the results of numerical simulation by discrete elements. This tool can also determine the cutting forces quite close to reality.     

A general quadrilateral plate element

The stiffness matrix and consistent nodal load vector for a general quadrilateral plate element are developed for both in-plane and bending analysis. The basis chosen for the development is the hybrid stress method because

• i it has been shown to produce results of relatively high accuracy.
• ii it employs only primary nodes and
• iii all integration required in the derivations can be achieved in closed form.

Various numerical examples, relating to both plane stress and bending conditions, are considered in order to establish the validity and generality of the derivations.