Future continuum models for granular materials in penetration analyses

This paper presents an investigation on future continuum models for granular materials in penetration analyses. A two-dimensional discrete element method has been used to numerically simulate penetration tests on a granular ground. The stress paths of soil elements in the ground have been studied, and then used to highlight the main features of granular materials based on most-advanced knowledge in soil mechanics. The study shows that the penetration makes the soil near the penetrometer undergo a significant changes of stresses in both magnitude and direction. The soil of large deformation rate may arrive at a stress state slightly over the strength envelope obtained from conventional tests. As a result, shear dilatancy, rate-dependency, non-coaxiality and particle crushing are the four main features that future continuum models should capture for granular materials in penetration analyses.     

Influence of relative density on granular materials behavior: DEM simulations of triaxial tests

The rheological behavior of non-cohesive soils results from the arrangement and complex geometry of the grains. Numerical models based on discrete element modeling provides an opportunity to understand these phenomena while considering the discrete elements with a similar shape to that of the grains the soil is composed of. However, dealing with realistic shapes would lead to a prohibitive calculation cost. In a macroscopic modeling approach, simplification of the discrete elements’ shape can be done as long as the model can predict experimental results. Since the intrinsic non-convex geometry property of real grains seems to play a major role on the response of the granular medium, it is thus possible to keep this geometrical feature by using cluster of spherical discrete elements, which has the advantage to reduce dramatically the computation cost. Since the porosities found experimentally could not always be obtained with the numerical model—owing to the huge difference in shape, the notion of relative density, which requires a search for minimum and maximum porosities for the model, was chosen to compare the experimental and numerical results. Comparing the numerical simulations with the experimental triaxial tests conducted with relative densities and different confining pressures shows that the model is able to predict the experimental results.     

A phenomenological law for complex granular materials from Mohr-Coulomb theory

Manipulating powders still entails some clumsy and risky operations even now in the middle of the fourth industrial revolution. This is because there is a lack of well-understood theory about granular matter due to its ravelled complexity. However, granular matter is the second most handled material by man after water and is thus ubiquitous in daily life and industry only after water. Since the eighteenth century, mechanical and chemical engineers have been striving to manage the many difficulties of grain handling, most of which are related to flow problems. Many continuum models for dense granular flow have been proposed. Herein, we investigated Mohr–Coulomb failure analysis as it has been the cornerstone of stress distribution studies in industrial applications for decades. This research gathers over 130 granular materials from several industrial sectors, as varied as cement and flour, including raw materials, food, pharmaceuticals, and cosmetics. A phenomenological law derived from the yield locus and governed exclusively by one dimensionless number from adhesive interactions has been found. Surprisingly, and in contrast to the common perception, flow in the quasi-static regime is actually independent of the friction, the packing fraction and any other grains/bulk intrinsic properties. The simplicity and accuracy of the model are remarkable in light of the complex constitutive properties of granular matter.     

An experimental and numerical study of the effects of length scale and strain state on the necking and fracture behaviours in sheet metals

Within sheet metal forming, crashworthiness analysis in the automotive industry and ship research on collision and grounding, modelling of the material failure/fracture, including the behaviour at large plastic deformations, is critical for accurate failure predictions. In order to validate existing failure models used in finite element (FE) simulations in terms of dependence on length scale and strain state, tests recorded with the optical strain measuring system ARAMIS have been conducted. With this system, the stress–strain behaviour of uniaxial tensile tests was examined locally, and from this information true stress–strain relations were calculated on different length scales across the necking region. Forming limit tests were conducted to study the multiaxial failure behaviour of the material in terms of necking and fracture. The failure criteria that were verified against the tests were chosen among those available in the FE software Abaqus and the Bressan–Williams–Hill (BWH) criterion proposed by Alsos et al, 2008. The experimental and numerical results from the tensile tests confirmed that Barba's relation is valid for handling stress–strain dependence on the length scale used for strain evaluation after necking. Also, the evolution of damage in the FE simulations was related to the processes ultimately leading to initiation and propagation of a macroscopic crack in the final phase of the tensile tests. Furthermore, numerical simulations using the BWH criterion for prediction of instability at the necking point showed good agreement with the forming limit test results. The effect of pre-straining in the forming limit tests and the FE simulations of them is discussed.     

Construction and characterization of a sputter deposition system for coating granular materials

In many modern ceramic or metal matrix composites the interface between the matrix and the reinforcements (particles or fibres) plays an important role. Either no or only weak mechanical bonding is observed or severe reactions between the matrix and the filler during the manufacturing process take place. A method to promote adhesion or to avoid severe reactions is to use coated reinforcements. A uniform film can act as an adhesion promoter, a compliant layer, a reaction inhibitor or a promoter of thermal transport across the interface.The aim of this work was to construct a particle coating system based on magnetron sputter deposition which allows to keep the particles or the granular material in motion during the deposition process to guarantee a homogenous coating on every single particle. As particles to be coated diamond granulates and carbon fibres were investigated. For transparent diamond particles the uniformity of a metallic coating could be evaluated by transmission optical methods and was found to be quite high. Carbon fibres, on the other hand, could only partially be coated due to agglomeration and shadowing effects. The system presented here can be considered as suitable for coating spherical or close to spherical granular matter.     

A novel approach to examining double-shearing type models for granular materials

A novel method, designated as the Rotation of Principal Axes Method (RPAM), capable of examining the double-shearing type kinematic models for granular materials is presented herein. A planar velocity field, which is proposed to represent a continuous rotation of principal strain rate axes, is applied to each model to analyse the rotation of principal stress axes. The proposed approach was proven to show main features of the double-shearing model, the double-sliding free-rotating model, and the revised double-shearing model, in a simple way interesting to geo-researchers. Furthermore, the RPAM was efficient in investigating the choice of a Cosserat rotation rate in kinematic theories and determining a key model parameter in the revised double-shearing model.     

A large deformation breakage model of granular materials including porosity and inelastic distortional deformation rate

A general constitutive model of crushable granular materials is developed within the context of large deformations. The time evolution equations for breakage, inelastic porous compaction and dilation, and distortional deformations are coupled by a yield surface and restrictions are imposed to ensure that these inelastic processes are dissipative. Some of the most salient mechanisms of such materials are described, including: (1) stiffness dependent on the breakage (a variable index of grading), porosity, and pressure; (2) critical comminution pressure and isotropic hardening, also dependent on the breakage and porosity; (3) jamming transition between solid and gaseous states; (4) a dilation law that embodies competition between porous compaction (due to the rate of breakage) and bulking (porous dilation at positive pressure due to the rate of inelastic distortional deformation); and finally, (5) the non-unique critical state relation between stress and porosity, in terms of the loading history and grading changes.     

FE-investigations of a deterministic and statistical size effect in granular bodies within a micro-polar hypoplasticity

The paper deals with numerical investigations of a deterministic and statistical size effect in granular bodies during shearing of an infinite layer under plane strain conditions and free dilatancy. For a simulation of the mechanical behavior of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. The proposed model captures the essential mechanical features of granular bodies in a wide range of densities and pressures with a single set of constants. To describe a deterministic size effect, the calculations were carried out with an uniform distribution of the initial void ratio for four different heights of the granular layer: 5, 50, 500 and 2,000 mm. To investigate a statistical size effect, the distribution of the initial void ratio in infinite granular layers was assumed to be spatially correlated. As only primary stochastic calculations were performed, single examples of different random fields of the initial void ratio were generated. For this purpose a conditional rejection method was used.     

Influence of particle breakage on the dynamic compression responses of brittle granular materials

The dynamic compression responses of dry quartz sand are tested with a modified spilt Hopkinson pressure bar (MSHPB), and the quasi-static compression responses are tested for comparison with a material testing system. In the experiments, the axial stress–strain responses and the confining pressure of the jacket are both measured. Comparison of the dynamic and the quasi-static axial stress–strain curves indicate that dry quartz sand exhibits obvious strain-rate effects. The grain size distributions of the samples after dynamic and quasi-static loading are obtained with the laser diffractometry technique to interpret the rate effects. Quantitative analyses of the grain size distributions show that at the same stress level, the particle breakage extent under quasi-static loading is larger than that under dynamic loading. Moreover, the experimental and the theoretical relationships of the particle breakage extent versus the plastic work show that the energy efficiency in particle breakage is higher under quasi-static loading, which is the intrinsic cause of the strain-rate effects of brittle granular materials. Using the discrete element method (DEM), the energy distributions in the brittle granular material under confined compression are discussed. It is observed that the input work is mainly transformed into the frictional dissipation, and the frictional dissipation under dynamic loading is higher than that under quasi-static loading corresponding to the same breakage extent. The reason is that more fragmentation debris is produced during dynamic breakage of single grains, which promotes particle rearrangement and the corresponding frictional dissipation significantly.     

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.     

A probabilistic model for cleavage fracture with a length scale - Parameter estimation and predictions of growing crack experiments

A probabilistic model for the cumulative probability of failure by cleavage fracture was applied to experimental results where cleavage fracture was preceded by ductile crack growth. The model, introduced by Kroon and Faleskog [Kroon M, Faleskog J. A probabilistic model for cleavage fracture with a length scale - influence of material parameters and constraint. Int J Fract 2002;118:99-118], includes a non-local stress with an associated material related length scale, and it also includes a strain measure to account for the number of nucleated cleavage initiation sites. The experiments were performed on single edge cracked bend test specimens with three different crack lengths at the temperature 85 °C, which is in the upper transition region for the steel in question. The ductile rupture process is modelled using the cell model for nonlinear fracture mechanics. The original cleavage fracture model had to be modified in order to account for the substantial number of cleavage initiators being consumed by the ductile process. With this modification, the model was able to accurately capture the experimental failure probability distribution.     

The flow rate of granular materials through an orifice

The flow rate of grains through large orifices is known to be dependent on its diameter to a 5/2 power law. This relationship has been checked for big outlet sizes, whereas an empirical fitting parameter is needed to reproduce the behavior for small openings. In this work, we provide experimental data and numerical simulations covering a wide span of outlet sizes, both in three- and two-dimensions. This allows us to show that the laws that are usually employed are satisfactory only if a small range of openings is considered. We propose a new law for the mass flow rate of grains that correctly reproduces the data for all the orifice sizes, including the behaviors for very large and very small outlet sizes.     

Regimes of segregation and mixing in combined size and density granular systems: an experimental study

Granular segregation in a rotating tumbler occurs due to differences in either particle size or density, which are often varied individually while the other is held constant. Both cases present theoretical challenges; even more challenging, however, is the case where density and size segregation may compete or reinforce each other. The number of studies addressing this situation is small. Here we present an experimental study of how the combination of size and density of the granular material affects mixing and segregation. Digital images are obtained of experiments performed in a half-filled quasi-2D circular tumbler using a bi-disperse mixture of equal volumes of different sizes of steel and glass beads. For particle size and density combinations where percolation and buoyancy both contribute to segregation, either radial streaks or a “classical” core can occur, depending on the particle size ratio. For particle combinations where percolation and buoyancy oppose one another, there is a transition between a core composed of denser beads to a core composed of smaller beads. Mixing can be achieved instead of segregation if the denser beads are also bigger and if the ratio of particle size is greater than the ratio of particle density. Temporal evolution of these segregated patterns is quantified in terms of a “segregation index” (based on the area of the segregated pattern) and a “shape index” (based on the area and perimeter of the segregated pattern).     

Mechanistic modelling of tests of unbound granular materials

Various tests are used to characterise the strength and resilience of granular materials used in the subbase of a pavement system, but there is a limited understanding of how particle properties relate to the bulk material response under various test conditions. Here, we use discrete element method (DEM) simulations with a mechanistically based contact model to explore influences of the material properties of the particle on the results of two such tests: the dynamic cone penetrometer (DCP) and the resilient modulus tests. We find that the measured resilient modulus increases linearly with the particle elastic modulus, whereas the DCP test results are relatively insensitive to particle elastic modulus. The DCP test results are also relatively insensitive to inter-particle friction coefficient but strongly dependent on the particle shape. We discuss strengths and weaknesses of our modelling approach and include suggestions for future improvements.     

Granular flows down inclined channels with a strain-rate dependent friction coefficient. Part I: Non-cohesive materials

The flow of a granular material down an incline of finite width with a strain-rate dependent coefficient of friction and a conical yield criterion is semi-analytically obtained using a characteristic method for flows on a deep layer of grains. This analysis leads to a flow field with three distinct zones: a Bagnold-flow zone below the free surface, a dead-zone and a matching zone between the two, linked to slippage at the wall. A good agreement between the computed flow field and experimental data is obtained.     

Granular flows down inclined channels with a strain-rate dependent friction coefficient. Part II: cohesive materials

A strain-rate dependent rheology for granular materials with a constant cohesion, is proposed and applied in the case of a flow down an inclined channel. The results obtained show that a plug flow zone appears below the free surface and show that low cohesion may affect drastically the flow behaviour when the inclination of the channel is close to the repose angle. These results give also a basic understanding of h _stop, the depth of the remaining granular layer on an inclined channel, treating dilatancy like a cohesion-like behaviour.     

Application of decohesive model with mixed damage scale in fracture analysis of composite materials

A decohesive model using a mixed damage scale and using the total fracture energy to simulate the fracture process of composite materials has been developed in this article. The model assumes a bilinear interfacial decohesion function and is incorporated into an interface finite element developed as a user subroutine in the commercial FEA package ABAQUS. In comparison with traditional numerical methods in fracture mechanics, this approach can automatically predict the failure load, crack path and the residual stiffness of bodies undergoing the fracture process. Applications given in this paper are simulation of a typical fracture test with a double cantilever beam (DCB) specimen; modelling a stiffened composite laminated panel under four-point bending and modelling a repaired composite sandwich panel under four-point bending. Good correlation was seen between modelling predictions and experimental results.     

Development of a 4-node finite element for the computation of nano-structured materials

The molecular structure of a material determines its mechanical, thermal and chemical properties. Thus, to better understand characteristic mechanical properties like damping behavior or softening, in principle, one just has to model the interactions of a sufficiently large number of atoms. Various force field approaches have been proposed for that purpose, which are based on molecular-dynamic simulations, or rather quantum-mechanical ab initio calculations. They provide the potential energy of a structure in dependence of sort and number of chemical and physical bonds. In general, the different energy forms can be represented by nonlinear normal, bending and torsional springs which suggests the use of a finite element code. However, standard finite elements like truss, beam or shell elements are not very applicable because of the interaction of many atoms and, considered from a mechanical perspective, the absence of rotational degrees of freedom. For example, a bending of beam elements would lead to unrealistic constraints of neighboring molecular groups. In order to overcome this disadvantage, a new 4-node finite element is introduced, which uses only translational degrees of freedom and therefore is capable of representing the different energy forms exactly.