Mechanical behaviour of a granular solid and its contacting deformable structure under uni-axial compression – Part I: Joint DEM–FEM modelling and experimental validation
Introduction
Many of the materials handled by industry are of a granular or particulate character, such as pharmaceutical powders, chemical pellets, agricultural grains, sands, gravels, coals and other minerals. These granular materials constitute a large proportion of the feedstock. Especially in chemical industry, approximately one-half of the products and at least three-quarters of the raw materials are in the form of granular solids (Nedderman, 1992). There are many handling processes in industry including filling, conveying, transporting, vibrating and storing, to name a few. During these handling processes, the granular materials undergo a variety of loading and deformation regimes. In the meantime, the contacting structures or machines also experience some stress state, and are even worn and damaged at some occasions. For example, the eccentric discharge of stored granular solids from fairly slender metal silos has caused many dramatic and catastrophic buckling failures (Rotter, 2001b, Song and Teng, 2003, Sadowski and Rotter, 2010). For another example in the alloy powders of hydrogen storage (Nasako et al., 1998, Qin et al., 2008, Charlas et al., 2012), large volume expansion of the stored metal hydride during hydriding/dehydriding cycles induces high wall stresses, hence leading to bulging and damage of the storage vessels.
In the above systems, the particulate material (discrete nature) and the containing structure or machine component (continuum nature) interact in a very complex manner, which is still inadequately understood even by specialists in shell structures. A better modelling for such systems should shed new light on the failure mechanism and lead to improved methods for the handling processes. Finite Element Method (FEM) (Zienkiewicz and Taylor, 2000) is an excellent approach for modelling continuum media, while Discrete Element Method (DEM) (Cundall and Strack, 1979) is a well-established method for modelling granular assemblies. However, modelling of the above problems using DEM or FEM alone has been either cumbersome or with strong limitation in complex deformation scenarios. In order to achieve a better analysis of both discrete and continuum components of the systems, there is hence a need to develop a joint modelling technique to solve these particulate solid–structure interaction problems.
The research for evaluating stresses of a deformable body subjected to a variety of loading conditions resulting from granular solids is less reported. The first author and a co-worker have developed a joint DEM–FEM modelling technique to evaluate the loading of particulate solids on a contacting structure or machine (Chung and Ooi, 2012). This Joint DEM–FEM modelling strategy was developed with the aim that it can be applied in the modelling of handling systems, such as rotating drums, vibrating beds, shear cells and silos for evaluating the loading of granular materials on a contacting machine. In the authors׳ ongoing research, this proposed modelling technique has been applied to investigating high wall stress problems of hydrogen storage vessels containing metal hydride powders, which are subjected to hydriding/dehydriding cycles. In addition, DEM has been increasingly used to model rock fall and landslide problems in geo-technique engineering (Campbell et al., 1995, Lo et al., 2011). The impact forces induced by debris flows or falling rocks often cause the damage of neighbouring houses and buildings. The safety analysis of protective structures and buildings can also be assessed by the proposed modelling technique.
Theoretically, the proposed Joint DEM–FEM modelling technique can be applied to any granules–structure interaction scenario, provided that the scale of the discrete phase is within the capacity of the DEM and so is the continuum phase of the FEM, primarily the former. Modelling large-scale industrial problems involving large number of particles has been formidable in the past due to extensive computer resource requirement. Nevertheless, there are significant advances and potentials with the adoption of various computational strategies to minimise the computational costs. For example, density scaling method can be employed to reduce the computational time for quasi-static problems (Thornton and Antony, 2000, O׳Sullivan, 2011). For certain classes of problems, particle stiffness may not have a significant influence on the studied physical quantities and may hence be reduced to gain computational advantage (Chung and Ooi, 2008b). Alternatively, the development of parallel DEM codes (parallel computing) allows the DEM simulations involving millions of particles to be computationally feasible (Cleary and Sawley, 2002). These computational strategies are useful to the proposed Joint DEM–FEM modelling technique and worthy of further study.
The aim of this paper is to validate the proposed numerical methodology against physical experiments. The uni-axial compression test of granular solids in a cylindrical container is a typical and fundamental compression system. Many researchers have studied this complex system by means of diverse experimental and numerical methods (Martin et al., 2003, Foo et al., 2004, Samimi et al., 2005, Wu et al., 2005, Tien et al., 2007, O׳Sullivan and Cui, 2009). However, even now the knowledge and understanding of compression mechanism are not adequate, especially in the complex interaction between the granular materials and the containing structure. In addition, the tractions along the solid height which the particles exert on the containing structure, and the stress state within the granular solid have seldom been measured in the literature.
In view of this, a high-quality uni-axial compression test setup was designed and built up to investigate the mechanical response of a granular material under vertical loading and the load transfer to the containing wall. In this study, confined compression tests were conducted on spherical polystyrene beads and the corresponding joint DEM–FEM simulations were performed for comparison. To achieve a high level of quantitative validation, the key grain properties required for DEM simulations, such as friction coefficient and coefficient of restitution, were not just given but measured from testers developed in house. The system characteristics during compression, including load–displacement response, force transmission to boundary surfaces, lateral pressure ratio, bulk wall friction and stress distribution in the cylindrical shell, were analyzed. The comparison between the numerical simulation and the experimental observation was made and discussed in this paper. This paper presents the first part of our series of work on such a complex particulate solid–structure interaction system and mainly focuses on the macro behaviour of the granular assemblies. The Joint DEM–FEM model successfully matched most of the observed features of the physical tests. Subsequently, the DEM data are used to further explore the meso and micro behaviour of the compressed granular assembly (meso-properties: solid fraction, stress in bulk solid; micro-properties: force chain, coordination number and friction mobilized factor). These internal properties will be reported on our companion paper PART II.
Section snippets
Experimental setup and procedures
A confined compression test was designed and set up to investigate the mechanical response of a granular solid under vertical loading and the load transfer to the contacting wall, as shown in Fig. 1. The apparatus was modified from the K0 tester (Masroor et al., 1987) and is more exquisite than that proposed by the European Standard to measure the lateral pressure ratio for silo design (EN 1991-4, 2004). The tester comprises an instrumented thin-walled circular cylinder, two platens, a
Mathematical formulation for linking DEM with FEM
Our research rationale is as follows: (1) DEM is first used to simulate the granular solid under confined compression; (2) the contact forces between particles and surface boundaries are then extracted from the DEM output and serve as the input for a FEM analysis; (3) FEM is finally adopted to analyze the cylindrical wall (continuum structure). A methodology for linking DEM with FEM has previously been developed (Chung and Ooi, 2012). The proposed linking methodology is briefly reviewed here.
In
DEM modelling for polystyrene beads
This study adopted the DEM with soft contact approach to model the particulate system. Detailed description for DEM is available elsewhere (Cundall and Strack, 1979, Tsuji et al., 1992, O׳Sullivan, 2011) and will not be elaborated here. There are nevertheless several key modelling features, such as the contact force model, DEM input parameters, boundary mesh representation, computational time step and particle generation, which are summarized below.
The simplified Hertz-Mindlin contact force
Results and discussion
Five tests were repeated under the same test conditions to investigate the variation of the confined compression tests. In each test, a funnel with a 50 mm orifice was placed centrally in the cylinder, filled with polystyrene beads and then pulled up slowly. Although the five tests followed the same “central” filling method, the settled packing structures for these five samples may be still different but probably with a closer packing configuration. In earlier trials of the experiment, it was
Conclusions
Physical calibration experiment of confined compression has been established and conducted to validate a previously developed numerical methodology, Joint DEM–FEM approach. The corresponding DEM and FEM simulations have been performed and comparisons between numerical and experimental results have also been made. The majority of compared physical quantities showed reasonable to good agreement, providing a not only qualitative but also quantitative validation for the proposed modelling
Acknowledgment
The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Grant no. NSC 100-2221-E-008-123-MY2 and Grant no. MOST 103-2221-E-008-015-MY3. The first author is very grateful to Professor J.Y. Ooi at the University of Edinburgh for his inspiration on this research topic. The authors greatly thank Dr. Jun Ai, INTECSEA (UK) Ltd., for his valuable advice and discussion.
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