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Finite Element Analysis for Incipient Flow of Bulk Solid in a Diamondback Hopper

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PAGE 1

FINITE ELEMENT ANALYSIS FOR INCIPI ENT FLOW OF BULK SOLID IN A DIAMONDBACK HOPPER By OSAMA SULEIMAN SAADA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Osama Suleiman Saada

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This document is dedicated to: my father Su leiman Saada, my mother Nema Aude and the rest of my family members for without their support this res earch would not have been completed.

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iv ACKNOWLEDGMENTS I would like to thank the following, because without their help this research would not have been possible: my committee chairman, Dr. Nicolaie Cristescu, for his continuous help; Dr. Kerry Johanson, who was very kind to advise and guide me throughout my graduate research; Dr. Ray Buckli n for his support, not only in scientific decisions but also in academic and personal ones; the rest of my committee members Dr. Frank Townsend and Dr. Bhavani Sankar. I would also like to extend my deepest gratitude to another member of my comm ittee, Dr. Olesya Zhupanska, who gave me detailed help on the modeling part of the rese arch. I would also like to thank the Particle Engineering Center for funding this project. Special thanks go to the PERC faculty and staff for their assistance. Last but not least I would like to extend special thanks to my father, Suleiman Saada, and my mother, Nema Aude, for thei r patience and continuous moral and financial support for the many years of my college career.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii ABSTRACT......................................................................................................................x ii CHAPTER 1 INTRODUCTION........................................................................................................1 2 BACKGROUND INFORMATION.............................................................................4 Bulk Solid Classification..............................................................................................4 Flow Patterns in Bins/Hoppers.....................................................................................4 The Diamondback HopperTM......................................................................................10 Discharge Aids............................................................................................................11 Janssen or Slice Model Analysis................................................................................12 Silo Vertical Section.................................................................................................12 Converging Hopper.............................................................................................14 Diamondback HopperTM......................................................................................15 Direct Shear Testers Jenike Test...............................................................................18 Jenike Wall Friction Test............................................................................................21 Schulze Test................................................................................................................22 Indirect Shear Tester Triaxial Test............................................................................23 The Diamondback HopperTM Measurements.............................................................28 3 FINITE ELEMENT....................................................................................................33 Background in Finite Element Modeling...................................................................33 FEM using ABAQUS on Diamondback.....................................................................36 Explicit Time Integration....................................................................................37 Geometry, Meshing, and Loading.......................................................................43 Plasticity Models: General Discussion................................................................49

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vi 4 CONSTITUTIVE MATERIAL LAWS.....................................................................55 Capped Drucker-Prager Model.................................................................................58 3D General Elastic/ Viscoplastic Model....................................................................59 Numerical Integration of the Elastic/Viscoplastic Equation.....................................63 5 EXPERIMENTAL RESULTS AND DISCUSSION.................................................67 Parameter Determination-Shear Tests......................................................................67 Diamondback hopper measurements-Tekscan Pads.................................................73 Conclusions and Discussion of the Te st results and Testing Procedure...................77 6 FEM RESULTS AND DISCUSSIONS.....................................................................79 Capped Drucker-Prager Model.................................................................................79 Drucker Prager Model Verification..........................................................................81 Drucker Prager Model Hopper FEM Results............................................................82 Conclusions And Discussion of the Pr edictive Capabilities of the Capped Drucker-Prager Model..............................................................................................91 Viscoplastic Model Parameter Determination..........................................................93 Time Effects..............................................................................................................98 Model Validation....................................................................................................101 Viscoplastic Model Hopper FEM Results..............................................................103 Conclusions and Discussion of the Pr edictive Capabilities of the 3D Elastic/Viscoplastic Model.....................................................................................108 7 CONCLUSIONS......................................................................................................110 APPENDIX A NUMERICAL INTEGRATION SCHE ME FOR TRANSIENT CREEP................113 B TESTING PROCEDURE.........................................................................................118 Sample Preparation...................................................................................................118 Membrane Correction...............................................................................................120 Piston Friction Error.................................................................................................121 C TEKSCAN DATA....................................................................................................122 D DRUCKER-PRAGER FEM DATA.........................................................................124 E VISCO-PLASTIC MODEL FLOW CHART...........................................................129 LIST OF REFERENCES.................................................................................................130 BIOGRAPHICAL SKETCH...........................................................................................135

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vii LIST OF TABLES Table page 2.1 Classification of powders...........................................................................................4 2.2 Comparison of mass flow and funnel flow of particulate materials..........................6 2.3 Pad specifications.....................................................................................................30 3.1 Numerical simulation research projects...................................................................34 5.1 Summary of Schulze test results..............................................................................68 5.2 Belt velocity vs. flow rate........................................................................................73 6.1 Capped Drucker-Prager parameters determined from shear tests............................81 6.2 Coefficients of the yield function.............................................................................95 6.3 Coefficients of the viscoplastic potential.................................................................97

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viii LIST OF FIGURES Figure page 2.1 Types of flow obstruction..........................................................................................5 2.2 Patterns of flow..........................................................................................................6 2.3 Types of hoppers........................................................................................................7 2.4 Problems in silos........................................................................................................8 2.5 Trajectories of major principle stress.........................................................................9 2.6 Schematic of the Diamondback Hopper™ showing position of pad.......................10 2.7 Forces acting on differential slice of fill in bin........................................................13 2.8 Forces acting in converging hopper.........................................................................14 2.9 Stresses acting on differential sl ice element in diamondback hopperTM..................16 2.10 Flow wall stresses at an axial positi on Z=3.8 cm below the hopper transition........17 2.10 Yield loci of different bulk solids............................................................................18 2.13 Yield loci................................................................................................................ ..20 2.14 Jenike wall friction test............................................................................................21 2.15 Schulze rotational tester...........................................................................................22 2.16 Experimental set-up..................................................................................................23 2.17 Triaxial chamber on an axial loading device...........................................................24 2.18 Filling procedure......................................................................................................25 2.19 Stress-strain curves..................................................................................................27 2.20 Layers of discs dilating as they are sheared.............................................................28 2.21 TekScanTM Pads.......................................................................................................29

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ix 2.22 TekScan Pads...........................................................................................................30 2.23 Location of TekscanTM pads in the diamondback hopper........................................31 3.1 Flowchart showing steps followed to complete an analysis in ABAQUS...............37 3.2 Summary of the explic it dynamics algorithm..........................................................42 3.3 Geometry and meshing of hopper wall....................................................................43 3.4 Filling procedure for powder....................................................................................44 3.5 Geometry and meshing of bulk solid ......................................................................46 3.6 Frictional behavior...................................................................................................48 4.1 Tresca and Mohr-Coulomb yield surfaces...............................................................55 4.2 Von Mises and Drucker-Prager yield surfaces.........................................................56 4.3 Closed yield surface.................................................................................................57 4.4 The linear Drucker-Prager cap model......................................................................58 4.5 Domains of compressibility and dilatancy...............................................................62 5.1 Output data from Schul ze test on Silica at 8Kg.......................................................67 5.2 Schulze test results for to determ ine the linear Drucker-Prager surface..................68 5.3 Jenike wall friction results to determine boundary conditions.................................69 5.4 Hydrostatic triaxial testing on Silic a Powder (5 confining pressures).....................70 5.5 Axial deformation of 4 deviatoric triaxial tests on Silica Powder and a rate of 0.1 of mm/min................................................................................................................71 5.6 Volumetric deformation of 4 deviatoric triaxial tests on Silica Powder and a rate of 0.1mm/min...........................................................................................................71 5.7 Axial deformation of 4 deviatoric tests on Silica Powder at 13.8KPa and 34.5Kpa....................................................................................................................72 5.8 Discharge mechanism and belt.................................................................................73 5.9 Pad location and output example.............................................................................74 5.10 Static Wall stress data at a di stance 2.8cm below hopper transition........................75

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x 5.11 Wall stress data at a distance 2.8cm be low hopper transition at various speeds and static conditions.................................................................................................75 5.12 Wall stress data at a distance 5.6cm be low hopper transition at various speeds and static conditions.................................................................................................76 5.13 Fourier series analysis on the wall pressure measurements.....................................77 6.1 The Linear Drucker-Prager Cap model....................................................................79 6.2 FEM simulation of a triaxial test using the Drucker-Prager Model.........................81 6.3 Axial stress-strain curves for experi ment and FEM using the Drucker-Prager model........................................................................................................................82 6.4 FEM calculated contact stresses at various filling steps..........................................83 6.5 FEM calculated contact stresses at various discharge steps.....................................84 6.6 FEM vs. TekScan area of interest...........................................................................84 6.7 FEM calculated static wall stresses vs. measured wall stresses at seven locations..................................................................................................................86 6.8 FEM calculated contact stresses vs. m easured wall stresses at seven locations during flow...............................................................................................................87 6.9 FEM calculated contact stresses vs. m easured wall stresses at two locations during flow at two different times............................................................................88 6.10 FEM calculated contact stresses vs. m easured wall stresses at two locations during flow angles of friction values........................................................................89 6.11 FEM calculated contact stresses vs. m easured wall stresses at two locations during flow with various cohesion values................................................................90 6.12 The irreversible volumetric stress work (data points) and function HH(solid line).............................................................................................94 6.13 The irreversible volumetric stress work (data points) and function HD(solid line).............................................................................................94 6.14 The viscoplastic potential derivatives......................................................................97 6.15 History dependence of the stabilization boundary...................................................98 6.16 The irreversible volumetric stress wo rk at two rates 0.1mm/min and 1mm/min.....99 6.17 Deviatoric creep test at a confining pressure of 34.5 Kpa......................................100

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xi 6.18 Variation in time of axial creep rate at a confining pressure of 34.5 Kpa..............100 6.19 Variation in time of volumetric creep rate at a confining pressure of 34.5Kpa..............................................................................................................101 6.20 Theoretically predicted stress-strain cu rves (solid lines) vs. experimentally determined curves at 34.5 Kpa confining pressure................................................103 6.21 FEM calculated static contact stresses vs. measured wall stresses at seven locations below the hopper transition.....................................................................104 6.23 FEM calculated flow contact stresses vs. measured wall stresses at two locations below the hopper transition at two wall friction values.........................................107 6.24 FEM calculated flow contact stresses vs. measured wall stresses at two locations below the hopper transition using adaptive meshing.............................................107

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xii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FINITE ELEMENT ANALYSIS FOR INCIPI ENT FLOW OF BULK SOLID IN A DIAMONDBACK HOPPER By Osama Suleiman Saada December 2005 Chair: Nicolaie Cristescu Major Department: Mechanic al and Aerospace Engineering Storing of bulk materials is essential in a large number of industries. Powders are often stored in containers such as a bin, hopper or silo and discharged through an opening at the bottom of the containe r under the influence of gravit y. This research tackles the frequent problems encountered in handling bulk solids such as flow obstructions and discontinuous flow resulting in doming and piping in hoppers. This problem is of interest to a variety of industries such as chemi cal processing, food, detergents, ceramics and pharmaceuticals. An objective of this work is to use finite element analysis to produce results that could easily be incorporated into the industry and be used by the design engineers. The study includes both a numerical approach, with an appropriate material response model, and an experimental pro cedure implemented on the bulk material of interest. Results from the analysis are used to re late the slopes of th e channels and the size of the outlets. Results from the FEM anal ysis are verified using measured stresses from a DiamondbackTM hopper.

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1 CHAPTER 1 INTRODUCTION Almost every industry handles powders/bulk solids, either as raw materials or as final products. Examples include chemicals a nd chemical processing, foods, detergents, ceramics, minerals, and pharmaceuticals. Powder s are often stored in a container such as a bin, hopper or silo and removed through an opening in the bottom of that container under the influence of gravity. The reliabil ity of the process involved depends on the flowability of these powders. Most powders ar e cohesive, that is, have the tendency to agglomerate or stick together over time. For th e material to flow out of a storage facility, bridging, arching or doming must be prevented. For a stable arch to form, the bulk solid must gain enough strength to support itself with in the constraints of the container. The strength is a function of th e degree of the compaction of the material and stress is a function of spatial position in a piece of process equipment. Thus knowing the stress allows us to compute the strength of th e bulk solid. Stresses are also effective in producing yield or failure of the bulk solid. Blockage occurs when the strength exceeds the stresses needed to fail the material. Since the advent of more powerful com puters, numerical methods have become very useful in research on flow of bulk solids. Numerical methods are very economical and lend themselves to comprehensive parame tric studies. This study presents a finite element approach to solve for displacements, velocities and stresses of a cohesive powder. The domain is discretized into small elements and displacements and loads are

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2 approximated. The commercially available AB AQUS software is used. Both the DruckerPrager yield criterion and a 3-D visco-plastic model are used to describe the material response. To verify the numerical results wall pressure measurements inside a diamondback hopperTM are taken. Measurement of wall loads in hoppers has been difficult due to the expense of constructing multiple loads cell measurement units in a hopper. Recent advances in measuremen t techniques have allowed significant improvements in the measurement of multiple point normal stresses in hoppers. This dissertation presents measurements of wall stre sses using pressure sensitive pads made by TekScanTM. Reliable and complete data on the deformation, failure and flow behavior will allow a fundamental understanding of powder fl ow initiation and flow. An objective of this work is to produce results that could eas ily be incorporated into the industry and be used by the design engineers. Both the fin ite element approach, with an appropriate material response model, and the technique for measurements of wall stresses on a diamondback hopperTM using pressure sensit ive pads are tools towards achieving that objective. Results from the analysis are used to relate the slopes of the channels and the size of the outlets necessary to maintain the flow of a solid of given flowability on walls of given frictional properties. Using the correct material response models, the right boundary conditions combined with the corr ect testing procedure under the correct loading conditions, continuum models with improved predictive capabilities can be developed. The understanding of powder behavior as well as the amount of information on material properties is limited. Generally, th e material properties are determined based on

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3 shear test measurements, yet in order to predict the bulk mechanical response during storage and transport, experimental data corresponding to a variety of deformation, loading conditions, and shear rates are neede d. To fully understand the material behavior of bulk solids, experiments ar e required where all possible stress or strain cases are induced in the material specimen. However, this would be an impossible task and simplified tests are instead utilized. In th ese tests only some stress or strain state components vary independently. In pow der mechanics and technology the most commonly used testing devices are shear testers, such as the Jenike shear cell. This type of device has some limitations su ch as the fact that the loca tion of the shear plane in the sample is determined by the shape of the tester and that the bulk solid is assumed to obey a rigid-hardening/softening pl astic behavior of Mohr-Coul omb type. A combination of direct shear testers (the Jenike tester and the Schulze tester), and the indirect shear tester (such as the triaxial tester) with finite element techniques will improve the analysis techniques of powder flow. The specifi c objectives of this research are: Present measurements of wall stresses on a diamondback hopperTM using pressure sensitive pads made by TekScanTM Present a finite element approach to solve for displacements, velocities and stresses of a bulk material based on a visco-plastic model Verify the method by comparing FEM re sults with measured stresses on a diamondback hopperTM Experimentally determine the material properties needed to use in the constitutive relationships Use the commercially available FEM soft ware ABAQUS with built in models for material response such as the Drucker-Prager model Write a user defined subroutine in FORTRAN to be used with ABAQUS to describe the visco-plastic behavior.

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4 CHAPTER 2 BACKGROUND INFORMATION Bulk Solid Classification A particulate system containing particles of about 100 m and smaller is called a powder. If the majority of the particles in the system are larger in size, it will be called a granular material. A classificat ion of particulate solids based on particle size is given in table 2.1(Brown and Richards, 1970 and Ne dderman, 1992). Granular materials are generally free flowing. To initiate flow in granular material it is sufficient to overcome their friction resistance. Most fine powders are prone to a gglomerate and stick together, causing significant storage and handling problems Table 2.1: Classification of powders Classification Particle size range( m) Ultra-fine powder <1 Superfine powder 1-10 Granular powder 10-100 Granular solid 100-3000 Broken solid >3000 Source: Brown and Richards, 1970 and Nedderman, 1992 Flow Patterns in Bins/Hoppers Before proceeding with a de scription of the flow pattern s in storage facilities the definition of terms that are used throughout this study is necessary. “Hoppers” have inclined sidewalls while “Bins” is the term used to describe a combination of a hopper and a vertical section. Silos is also a term us ed to describe a tall ve rtical container where

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5 the height to diameter ratio is bigger than 1.5. In European literature this term is used to describe a vertical sectio n on top of a hopper section. These containers vary in their usage in industrial applications. In agricultural, mining, cement and refractory applications th ey are used as “primary” storage facilities and have big capacities (up to 1000 tons). In ot her industrial applicati ons these containers are used only for short time storage, the bulk material being subsequently transferred to other containers or mixers. Thus, they are generally much smaller than the “primary hoppers.” In bins, most powders can experience obstruction to flow as shown in figure 2.1. Figure 2.1: Types of flow obstruction For an arch to form, the solid needs to have developed enough strength to support the weight of the obstruction. Some of the met hods used to reinitiate flow are vibration of the bin and manual movement of the powder. These methods have proven to be costly, inefficient and dangerous. There are two major kinds of flow patterns in a contai ner: (1) mass flow, and (2) funnel flow. In mass flow, the entire material flows througho ut the whole vessel during discharge. Figure 2.2a illustrate s such flow. In funnel flow, th ere is a stagnant layer of material at the wall of the vessel. This stagna nt region extends all th e way up to the top of the container. Flow from a funnel flow bi n occurs by first emptying the center flow Piping Doming

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6 channel and then, provided the material is sufficiently free flowi ng, material sloughs off the top surface. If the material is cohesive a stable rathole can form. Flow patterns are illustrated in Figure 2.2b and 2.2c. Figure 2.2: Patterns of flow For most applications, the ideal flow situ ation is mass flow and it occurs when the hopper walls are steep and smooth enough and when there are no abrupt transitions between the bin section and th e hopper section. When the wa lls are rough and the slope angle is too large the flow regime tends to be a funnel flow. Table 2.2 lists the advantages and disadvantages of both mass and funnel flow. Table 2.2: Comparison of mass flow and funnel flow of particulate materials Mass flow Funnel flow Characteristics No stagnant zones Stagnant zone formation Uses full cross-section of vessel Flow occurs within a portion of vessel cross-section First-in, first-out flow First-in, last-out flow Advantages Often minimizes segregation, agglomeration Sm all stresses on vessel walls during flow due to of materials during discharge the ‘buffer effect’ of stagnant zones. Very low particle velocities close to vessel walls; reduced particle attrition and wall wear Disadvantage s Large stresses on vessel walls during flow Promotes segregation and agglomeration during flow Attrition of particles and erosion and wear Di scharge rate less predictable as flow region boundary Of vessel wall surface due to high particle can alter with time Velocities Small storage volume/vessel height ratio Source: Brown and Richards, 1970 a) Mass flow b ) Semi-mass flow c) Funnel flow

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7 Storage containers can have axisymmetric or planar geometry as seen in figure 2.3. The shape, number and position of the discha rge outlet can vary depending upon the silo geometry, bulk solid properties and process requirements. The geometry of the container Figure 2.3: Types of hoppers and properties of the powder determine the flow pattern during filling and discharge. In many industrial applications the same c ontainers are used fo r storing different types of bulk solid. These materials have di fferent mechanical and physical properties such as particle size, particle distribution, pa rticle shape, and bulk density and frictional properties. This results in varying flow regimes for different materials. A hopper designed to provide mass flow for one material may not provide mass flow for a different material. Figure 2.4 shows the relationships among bul k solid material properties, the silo filling process, the flow pattern s during discharge, the wall pr essures, the stress induced in the structure and the conditions that might cause failure of the structure (Rotter 1998). Funnel-flow hoppers Mass-flow hoppers

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8 The flow pattern is strongly affected by th e distribution of densities and orientation of particles which are determined by the mechan ical characteristics of the solid and the Figure 2.4: Problems in silos way it is filled. These characteristics also de termine whether arching or ratholing occurs at the outlet. The flow pattern during discharge also determin es if segrega tion occurs and influences the pressure on the walls. Hence in order to predict the non uniform and unsymmetrical wall pressure it is essentia l that the flow pattern be predicted. This influences directly the wall stress condtions that could induce failure and collapse of the structure. In regards to bulk solids handling in storage facilities, 3 phases can be distinguished: (1)filling, (2)storage and (3)discharge. In the filling stage, with the outlet closed, the material is supplied slowly at th e top of the container. As more and more Solid properties Fillin g Methods Flow pattern Pressure on silo walls Stresses in silo structure Failure conditions for the structure COLLAPSE SEGREGATION ARCHING/RATHOLING Loss of Function Aspect of silo behavior

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9 material is poured in, stresses, due to th e weight of layers, st art developing throughout the container. In the storage stage no material is flowing in or out of the container. For cohesive (fine) powders this may lead to an increase of interpar ticle bonding. Due to its own weight the stored material compacts and its strength increases with time. Also, the slow escape of the air in por es results in stronger surf ace contact and an increase in frictional forces. In the third and final stage the material is discharged through an opening at the bottom of the container. Upon opening the outlet, a dilation zone neighboring the outlet is formed. This zone will spread to the upper layers of the material and cause it to flow. In hoppers with steep and smooth sides, this zone of dilated material will cover the whole cross-sectional area of the containe r resulting in mass-flow, while for rough walls the dilation zone is confined only to the center, resulti ng in funnel flow. he state of stress throughout the containe r, in the discharge stage, drastically differs from the stress state in the filling/st orage phases. The corresponding stress states are called active and passive, respectively. In the filling and storage stages the direction of the major principal stress 1 is vertical with slight curv ature close to the walls. In the discharge stage the principal st ress is horizontal with slight curvature at the walls (Figure 2.5). Figure 2.5: Trajectories of major principle stress 1 a) Filling/storage Active b) discharge Passive

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10 The Diamondback HopperTM The Diamondback Hopper™ consists of a round-to-oval hopper with diverging end walls. This hopper section is positioned a bove an oval-to-round hopper that necks down to a circular outlet. Pressu re sensitive pads (from TekSca n™) were used to measure the normal wall pressure (Johanson, 2001). These pads consist of two plastic sheets with conductive pressure sensitive material printe d in rows on one sheet of plastic and columns on the other sheet of pl astic. When these sheets contact each other they form conductive junctions where contact resistance varies with th e normal stress applied. The effective area of each junction is 1 cm2 and there are over 2000 independent normal stress measurements possible that can be recorded at cycle times up to eight per second. The voltage drops across these juncti ons were sequentially measured and scaled to give real engineering force units using the data ac quisition software. The TekScan™ pad was glued to the inside surface of the round-to-oval hopper with one edge of the pad along the center of the flat plate section (see Figure 2.6). Figure 2.6: Schematic of the Diamondback Hopper™ showing position of pad 61 cm 28 cm Z Pad Pad locations 2064 sensors

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11 Discharge Aids Poor design or incorrect use of a hopper leads to the use of devices that reduce flow problems. The use of the wrong device, however may have the reverse effects and create more problems than it solves. These devices are used if it proves that there are constraints in the overall system that pr event the unaided gravity flow of the material. These aids developed from practices such as beating th e hopper with a blunt instrument and poking the material with some sort of rod. The th ree major types of aids are (1) pneumaticrelying on the applicat ion of air to the pr oduct; (2) vibrational relying on mechanical vibration of the hopper or th e material; (3) mechanical physically extracting the product from the hopper. In the pneumatic aids, aeration devices are used to introduce air at the time that the material is discharged so as to “fluidize” th e material in the region of the outlet opening and to reduce the friction between the mate rial and the hopper wall and the second is to introduce a “trickle of flow” of air during the whole period that the product is stored to prevent the gain of strength in the material. Air is also introduced into inflatable pads that act mechanically against the stored materi al. When using vibrational methods, devices could be used to vibrate the hopper or bin wa lls or to vibrate th e material directly. Vibration should not be applied when the outle t is closed, as this could result in the strengthening of any arch formation. In mech anical methods powered dis-lodgers such as vertical or horizontal stirrers are used to manually move the material

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12 Janssen or Slice Model Analysis Silo Vertical Section Janssen (1895) developed a method of predic ting vertical silo wall pressures over a century ago that is still widely used in th e industry although most of the assumptions he used in the derivations have been shown to be incorrect. The simplest slice for the analysis is th e planar finite boundary slice that has sides that are perpendicular to the walls of the bin. The forces acting on an element slice of stored powder, of thickness dz, at a depth h from the top surface in a deep bin with an overall height h, cross-sectional area A a nd circumference C are shown in figure 2.7. Let the vertical stress at the upper surface, depth h, be v and at the lower surface, depth h+dz, be v+d v. w and w are the normal and tangential (s hear) stresses at the wall due to friction between the walls and the bulk materi al. The weight of the powder in the slice is Adz g where g is the acceleration due to gravity and is the material bulk density which is assumed to remain constant over th e entire depth of the powder. Equating the forces in the vertical direction we get ()0vvvwAAdzgAdCdz (2.1) It is assumed that the ratio of horizontal to vertical pressure is constant everywhere in the element: w v K (2.2) For fully mobilized friction we have tanw w w (2.3)

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13 Figure 2.7: Forces acting on diffe rential slice of fill in bin Expanding equation 2.1 and simplifying we get the first order differential equation: Separating variables and integrating with the boundary condition v=0 at h=0 (at the level of the free surface), and v= v at h=h gives 1kC h A vgA e KC (2.4) and 1kC h A wvgA Ke C (2.5) For large depth of fill the maximum values of vertical and normal stresses are maxvgA KC (2.6) and maxwgA C (2.7) vA v+d v)A A dzg wCdz dz h

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14 while near the top surface: gh topv (2.8) The ratio A/C varies for different silo geometries (e.g., rectangular, circular). As mentioned above recent studies have challenged the accuracy of Janssen’s assumptions of constant K values, angle of wall friction and bulk density. The value of K has a considerable influence on the stress dist ribution in the material and on the wall. Researchers have differing views of the K va lue. Jenike (1973), for example, assumed K=0.4 for most granular material s while others claim it is a f unction of internal angle of friction. Converging Hopper The above analysis is appli cable to deep silos and flat bottom bins ignoring any end effects. For the hopper section a linear hydros tatic pressure grad ient is proposed vo vgh g (2.9) where vo is the vertical pressure at the top of the hopper calculated from equation 2.5 as shown in figure 2.8. Figure 2.8: Forces acting in converging hopper Walker (1966) assumed a constant value of K equal to: v vo w w h

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15 tan tantanwK (2.10) While Jenike (1961) assumed a constant value of: sin2cos sin2sinw wwK (2.11) Diamondback HopperTM A force balance can be done on a small differential slice of material in the Diamondback Hopper™ as shown in Figur e 2.9 (Johanson and Bucklin, 2004). The forces acting on the material slice element are due to the bulk material slice weight, wall frictional forces on the flat plate section a nd the round end walls, vertical stresses, and normal stress conditions at the bin wall. The definition of K-value and the columbic friction condition are used to relate the vert ical pressure acting on the material to the stresses normal and tangent to the bin wall (see Equations 2.12 through 2.15). The resulting differential equation is found in Equation 2.16. The cro ss-sectional area (A) and hopper dimensions (L) and (D) are functions of the axial coordinate as indicated in Equations 2.16 through 2.21. v L LK (2.12) ) tan(w v L LK (2.13) v eK ) ( (2.14) ) tan( ) (w v eK (2.15)

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16 dz L L Wt v ( d ) v v Figure 2.9: Stresses acting on differentia l slice element in diamondback hopperTM 2 02() 1 (tan()tan()) 4 ()tan()tan(())L wD v v ewKLD dA AdzA d g dz D Kd A (2.16) D D L D A ) ( 42 (2.17) ) tan( 2 ) tan( 2 2 2L DD L D dz dA (2.18) ) tan( 2L Tz L L (2.19) ) tan( 2D Tz D D (2.20) minmaxmax11sin1n cKKKK (2.21) Where values of parameters K1min, K1max, and n were chosen to minimize the deviation between the observed wall stress profiles and the calculated wall stresses over the entire axial hopper depth.

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17 The hopper wall slope angle changes around the pe rimeter. The slope angle along the flat plate section equals the hopper angle ( D) while the slope angle along the hopper end wall equals the hopper angle ( L). The variation in th e hopper angle, around the perimeter, is given by Equation 2.23, where is the hopper section angle as defined in Figure 2.9. ) tan( ) sin( ) tan( ) sin( 1 tan ) (L Da (2.22) Figure 2.10 shows Janssen computed wall stresses compared with the actual TekscanTM measurements for the section of the hopper shown (Johanson, 2003). 0 1 2 3 4 5 6 7 0102030405060708090 Hopper Section Angle, (deg)Wall Stress (KPa) Computed Janssen Stress Fully Developed Flow Figure 2.10: Flow wall stresses at an axial position Z=3.8 cm below the hopper transition The Janssen slice model can provide a first approximation to the loads in diamondback hoppers. However, there is sign ificant variation from the simple slice model approach used here. More complex constitutive models will be required to increase the agreement between experime ntal and theoretical approaches.

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18 Direct Shear TestersJenike Test For the past 30 years the assessment of flow properties of bulk solids has been done through shear testing (Jenike, 1961). The bulk solid is assumed to obey a rigidhardening/softening plastic behavior of M ohr-Coulomb type (Figure 2.10) for coarse cohesion-less, particles (granular particle s), the yield locus is approximated by: tan while for fine powders, the yield locus is tan c (2.24) Figure 2.10: Yield loci of different bulk solids As shown in figure 2.11, the cel l consists of a base, an upper ring and a cover. The cover has a fixture that can hold a normal force N. The motor causes the loading stem to push against the shear ring and displace it horizont ally. The shear force is measured and is divided by the cross-secti onal area of the ring to provide the shear stress( ). c c b) Yield locus of a cohesive nonCoulomb solid c) Yield locus of a freeflowing sand a) Yield locus of a cohesive Coulomb solid

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19 Figure 2.11: Jenike shear test The test begins when the sample of bulk solid is poured into th e shear cell and is pre-consolidated manually by placing a special top on the cell, applying a normal load and twisting the top to consolidate the material. Then a normal stress n is applied and the specimen is sheared. The shear force con tinuously increases with time until constant shear is obtained. Corresponding to the change in shear is a change in bulk density. After a period of time, the bulk density reac hes a constant value for a value of n. At that point, the deformation is known as steady state deform ation. The sample is first pre-sheared under a constant n and The shearing is stopped when steady-state deformation is reached. Then the sample is sheared under the normal stress n that is smaller than the shear stress. To insure the same initial bulk density the sample is then pre-sheared under the same normal stress n and sheared at a lower n than the first test. This process is repeated several times.

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20 The yield locus, usually called the effectiv e yield locus is used to determine the flow parameters. As seen from figure 2.12 it is a straight line passing through the origin and tangential to the Mohrs circ le at steady state deformation (the largest circle). This line has the inclination e or the effective angle of internal friction (angle with the axis). At lower normal stresses the yield loci is curved instead of straight. Figure 2.12: Yield loci Therefore a linearized yield lo cus is used to define the internal friction angle ( i). Using a different n for pre-shearing the sample will re sult in different consolidation and bulk density values. This will produce several yi eld loci and several Mohr circles as seen in Figure 2.13 Figure 2.13: Yield loci c c Yield locus Linearized Yield locus e i

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21 Jenike Wall Friction Test The Jenike tester (1961) can also be utilized to determ ine the wall friction angle. The test is modified so that the base of the tester is repl aced by a flat plate of the wall material that the hopper is made of (Figure 2.14). Figure. 2.14: Jenike wa ll friction test The procedure is similar to the description above in the sample preparation stage. A maximum normal force is applied at the top co ver and is decreased in a series of steps while the maximum shearing force is being m easured. As explained above the wall yield locus can be plotted with vs and the coulomb equation is: arctan (2.25) The coefficient of friction is expressed as the angle of wall friction given by: (2.26)

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22 Schulze Test The Jenike test is limited to a small disp lacement of about 6mm and is suitable for fine particles only. Rotationa l shear testers such as the annular Schulze (1994) tester (figure 2.15) have unlimited st rains. The powder is containe d in the cell and loaded from Figure 2.15: Schulze rotational tester the top with a normal force N through the lid. After calibrating and filling the cell with powder a predetermined consolidation weight was added to a hanger, which is connected to the top of the cell. The m achine was then turned on and a llowed to run until it reaches a steady state. The normal weight was adjusted to about 70% of the steady state value and the sample was sheared. During the test the shea r cell rotates slowly in the direction of while the cover is prevented from rotation by two tie rods. This causes the powder to shear. The forces F1 and F2 and the normal forces are recorded. This procedure was F1 F2

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23 repeated several times with decreasing weight. Between each test the sample was preconsolidated to the steady state value. The da ta reduction procedure is same as the Jenike test and gives the yield loci and the internal angle of friction. Indirect Shear TesterTriaxial Test To investigate the stress-strain behavior of powders, a triaxial testing apparatus is used. Its capabilities are wider th an those of uniaxial testers, and it can be described as an axially symmetric device having 2 degrees of freedom. A schematic diagram and picture of the triaxial apparatus are shown in Figure 2.16 and 2.17. Figure 2.16: Experimental set-up In an ideal triaxial test, all three major principal stresses would be independently controlled. However the independent control would lead to mechanical difficulties that limit the conduction of such tests to special a pplications. The commonly used triaxial test refers to the axisymmetric compression test. Th is test has been used since the beginning 13 Pressure Source Water Rubber membrane Porous stone Volume Change Device

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24 of the 19th century in testing the strength of soils. The soil sample is pre-compacted, saturated with water, and compressed both hydrostatically and deviatorically and the water discharged from the sample is used to measure the volume change and hence the density change. In recent years, the test has been used to test other bu lk solids: grains and fine powders. The testing procedure was modifi ed so as to allow testing of dry powders at low stresses. The behavior in this regime is essential for the applications of interest. Figure 2.17: Triaxial chambe r on an axial loading device A latex membrane is stretched out in to a cylindrical shape using a 2-piece cylindrical mold and vacuum pressure. The powder is poured into the membrane dispersed and compressed using a predetermi ned stress. Figure 2.18 shows the steps followed in the filling procedure. The value of H was determined using Janssen’s equation. To insure repeatable initial conditions, both the mass and volume of the

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25 material have to be measured. The membrane is sealed by rubber O-rings to a pedestal at the bottom, and to a cap at the top. Vacuum grease is used the membrane and the pistons so as to reduce the chances of leaks. Any test consists of two phases: (1) a hydrostatic phase followed by (2) a deviatoric phase. In the hydrostatic test the assembly is placed in a chamber that is filled with water. The water is pressurized to the desired confining pressure. This hydrostatic pressure is increased up to a desired value 3 using a pressurizing device. The sample can support itself due to the filling procedure and thus the mould is removed. Under the applied hydrostatic pressure the powder, compacts and the specimen takes the form of a cylinder. The change in volume of the speci men is recorded using a data acquisition system that measures the volume of the water displaced. The initial volume of the sample is determined by multiplying the height by the arithmetic mean of the diameters at 3 locations in the vicinity of the middle of the sample. Figure 2.18: Filling procedure In the deviatoric phase, the assembly is placed on an axial loading device. An additional axial stress is applied by means of a piston passing through a frictionless H

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26 bushing at the top of the chamber. The test is carried out under stra in control conditions in which a predetermined rate of axial deformation is imposed and the axial load required to maintain the rate of deformation is measured via a load cell. As the sample deforms, water is displaced from the chamber. The quantities measured during the deviatoric phase are: the confining pressure 3 (which provides an all-ar ound pressure on the lateral surface of the sample), the axial force applied to the piston, the change in length of the sample and the change in the volume of the sample which is determined by the change in the volume of water existing in the chamber. Thus, in the test, the change in volume of the sample we can be monitored continuously as it is sheared. Th ese tests will produce the following stress-strain curves: the axial strain 1 versus the deviatoric stress 13 ( 1 is the axial stress) and the volumetric strain V versus the deviatoric stress 13 respectively. Here and throughout the text comp ressive stresses and strains are considered to be positive. The axial strain is defined as: 0 0 1l l l wherel0 is the initial length of the sample and lis the current length ; the volumetric strain is VVV V 0 0, V0and V being the initial and current volume, respectively.Any change in the volumetric strain reflects change of the relative positions of the powder particles. Failure corresponds to an observed instability in the specimen when one pa rt slides with respect to the other along a plane inclined at about 45 with respect to the vertical axis of the specimen. Failure shows up in the stress strain curve as a maximum deviatoric pressure.

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27 Figure 2.19 Stress-strain curves Figure 2.19 shows the stress-strain curves obtained in a typical test (Saada, 1999). This is a test carried out on a powder at a c onfining pressure of 40. 56kPa. The axial strain curve shows that the powder undergoes elas tic deformation up to about 50% of its strength, followed by work hardening and fa ilure. The stress-volumetric strain curve shows two distinct regimes of behavi or: first, the powder compacts (i.e. v increases) very slightly, then starts to expands (i.e. v decreases). The volume increase, which is observed on the last portion of the curve ( 13 ) -v is called dilatancy. The powder expands because of the interlocking of the particles: deformation can proceed only if some particles are able to ride up or rotate over other particles. This phenomena can simplistically be visualized with the aid of Figure 2.20, which shows two layers of discs, one on top of the other. If a shear stress is ap plied to the upper layer, then each disc in this layer has to rise (increasing the volume occupied by the particulate material) for the sample to undergo any shear deformation. 0 50 100 150 200 250 300 -55000-35000-15000500025000450006500085000105000Srain(E-6)(kPa) e1(test 3) ev(test 3) Powder Test Date 3(kPa) i(g/cm^3) Poly#1 3 12/10 40.56 0.8594

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28 Figure 2.20: Layers of discs dilating as they are sheared The Diamondback HopperTM Measurements The core of the TekscanTM pressure measurement syst em (1987) consists of an extremely thin 0.004 in (0.1 mm), flexible t actile force sensor. Sensors come in both gridbased and single load cell configurations, and are available in a wide range of shapes, sizes and spatial resolutions (sensor spacing). These sensors are capable of measuring pressures ranging from 0-15 kPa to 0-175 MPa. Each application requires an optimal match between the dimensional characteristics of the object(s) to be tested and the spatial resolution and pressure range provided by Tekscan's sensor technology. Sensing locations within a matrix can be as small as .0009 square inches (.140 mm2); therefore, a one square centimeter area can contain an array of 170 of these locations. Teksan’s Virtual System Architecture (VSA) allows the user to integrate several sensors into a uniform whole. The standard sensor consists of two th in, flexible polyester sheets which have electrically conductive electr odes deposited in varying patte rns as seen in Figure 2.21 (Tekscan Documentation, 1987). In a simplifie d example below, the inside surface of one sheet forms a row pattern while the inner surf ace of the other employs a column pattern. The spacing between the rows and columns va ries according to sensor application and can be as small as ~0.5 mm.

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29 Figure 2.21: TekScanTM Pads Before assembly, a patented, thin semi-c onductive coating (ink) is applied as an intermediate layer between the electrical contac ts (rows and columns). This ink, unique to Tekscan sensors, provides the electrical re sistance change at each of the intersecting points. When the two polyester sheets are pl aced on top of each other, a grid pattern is formed, creating a sensing location at each in tersection. By measuring the changes in current flow at each intersection point, the applied force distribution pattern can be measured and displayed on the computer screen. With the TekscanTM system, force measurements can be made either statically or dynamically and the information can be seen as graphically informative 2-D or 3-D displays. In use, the sensor is installed between two mating surfaces. Tekscan's matrix-based systems provide an array of force sensitive cells that enable measurement of the pressure distribution between the two su rfaces (figure 2.22). The 2-D and 3-D displays show the location and magnitude of the forces exerted on the surface of the sensor at each sensing

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30 location. Force and pressure changes can be observed, measured, recorded, and analyzed throughout the test, providing a powerful engineering tool. Figure 2.22: TekScan Pads Table 2.3: Pad specifications General Dimensions Sensing Region Dimensions Summary Overall Overall Tab Length Width Length L W A Matrix Matrix Width Height Columns Rows MW MH CW CS Qty RW RS Qty # of Sensels Sensels Density (mm) (mm) (mm) 622 530 130 (mm) (mm) 488 427 (mm) (mm) 6.35 10.2 48 (mm) (mm) 6.35 10. 2 42 ( per cm2) 2016 0.97 Several pleats or folds were made in the upper portion of the pad to make it conform to the upper cylinder wall (figure 2.23) A thin retaining ring was placed at the transition between the hopper and cylinder. This held the pad close to the hopper wall

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31 Figure 2.23: Location of TekscanTM pads in the diamondback hopper and helped maintain the plea ts in the cylinder. Contact paper was placed on the hopper and cylinder surface to protect the pad and produce a consistent wall friction angle (of about 17 degrees) in the bin. The pad lead was fed through the hopper wall well above the transition and connected to the data ac quisition system. Spot calibration checks were made on groups of four load sensors. Th is arrangement produced a load measurement system capable of measuring normal wall loads to within + 10 %. The bin level was maintained by a choke fe d standpipe located at the bin centerline and at an axial position about 60 cm above th e hopper transition. The material used was fine 50 micron silica. During normal operation, the feed system ma intains a repose angle at the bottom of the standpipe and produces a relatively constant level as material discharges from the hopper. Flow from the hopper was controlled by means of a belt feeder below the Diamondback Hopper™. There was a gate valve between the Diamondback Hopper™ and the belt. Initially, this gate was closed and the lower hopper was charged with a small quantity of materi al. The amount of material in the hopper Pad locations 2064 sensors

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32 initially was enough to fill only the lower ovalto-round section of the bin. The gate was then opened, allowing material to fill the voi d region between the gate and the belt. This was done to avoid surges onto the belt during the initial filling rou tine. The remaining hopper was then filled slowly, using the existing conveying system until the diamondback test hopperTM and a surge bin above this test hopper were filled. Material level was maintained in the surge hopper by re cycling any material leaving the belt into the surge hopper feeding the Diamondback Hopper™ bin. Once the bin was full, material was allowed to stand at rest for about 10 minutes as material deaerated. The speed of the belt was preset and flow was init iated. The wall load data acquisition system recorded changes in wall stress as flow wa s initiated. The initial stress condition shows a large peak pressure concentrating at the top point of the triangular flat plate in the hopper. This peak is focused at the tip of the plate. During flow these loads decrease and delocalize producing the highest peak stresses at along th e edge of the plate.

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33 CHAPTER 3 FINITE ELEMENT Background in Finite Element Modeling Great effort has been made in recent d ecades to understand flow phenomena of bulk solids. Since the 1970’s, a large number of research teams have worked on the application of finite element analysis to hopper problems. Models and programs were hampered due to limited capacity of com puters and the high cost of equipment. Nowadays, it is possible to use computers whose capacity and speed is continually increasing and a large number of programs exis t that manipulate, analyze and present the results. In principle there are no rest rictions with regards to th e hopper geometry and to the bulk material in FEM analysis. However stud ies have to account for complex geometry of the container, complex material behavi or and interaction w ith the wall. Hopper geometry is not always axisymmetric and ofte n has shapes that require three-dimensional numerical modeling, as is the case with th e diamondback hopper. Bulk solids exhibit complex mechanical behavior such as anisotr opy, plasticity, dilatanc y and so on. Most of these properties are developed during discha rge due to the large strains that are experienced. Some properties, such as anisotropy are devel oped during the filling process. It is very essential that deta iled mathematical models, accompanied with a proper experimental procedure, are used to describe these features. The appropriate simulation has to be used to model the intera ction between the solid and the wall. For the

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34 case of smooth wall, a constant friction coe fficient can be satisfactory while for rough walls it might be required to use more sophisticated modeling. In a book edited by C.J. Brown and J. Niel sen (1998) several FEM research on silo flow are presented. This book is a compilati on of reports brought about by a research project on silos called the CA-Silo Project. It was funded by the European commission and completed in 1997. The first chairman of the group, G. Rombach (1998), assembled data for comparison of existing programs. Ta ble 3.1 lists several projects regarding granular solid behavior simulation. It incl udes both finite element and discrete element simulations. Table 3.1: Numerical simulation research projects Author/user Name of program Static/ Dyna mic 2D/3D Cohesive/ Non Cohesive 1. Aubry (Ecole Central Paris) 2. Eibl (Karlsruhe U, Germany) 3. Eibl (Karlsruhe U, Germany) 4. Eibl (Karlsruhe U, Germany) 5. Klisinski (Lulea U, Sweden) 6. Klisinski (Lulea U, Sweden) 7. Martinez (INSA, Rennes, France) 8. Martinez (INSA, Rennes, France) 9. Schwedes (Braunschweig U, Germany) 10. Schwedes (Braunschweig U, Germany) 11. Tuzun (University of Surrey, UK) 12. Ooi, Carter (Edinburgh U, Scotland) 13. Ooi (Edinburgh U, Scotland) 14. Rong (Edinburgh U, Scotland) 15. Thompson (Edinburgh U, Scotland) 16. Cundall (INSA, Rennes, France) GEFDYN SILO ABAQUS SILO BULKFEM AMG/PLAXIS MODACSIL ABAQUS HAUFWERK HOPFLO AFENA ABAQUS DEM.F SIMULEX TRUBALL,PFC S/D S/D S/D S/D S/D S/D S/D S S/D S S/D S S/D S/D S/D S/D 2D/3D 2D/3D 2D/3D 2D/3D 2D 2D 2D 2D/3D 2D/3D 2D 2D/3D 2D 2D/3D 2D 2D 2D/3D C/N C/N C/N C/N N N N C/N C/N N N C/N C/N N N N Also listed in the table are: the name of the program used, whether it analyzed static or dynamic material, the geometry used (2D or 3D) and whether it an alyzed cohesive or non-cohesive material. The material models used include the wellknown elastic-plastic models and yield criteria su ch as Mohr-Coulomb, Tresca a nd Drucker-Prager. Other law

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35 include: Lade (1977), Kolymbas (1998), Boyce (1980), Wilde (1979) and critical state. These laws have been modified for cohe sive bulk solids and dynamic calculations. Aside from this project there has not been extensive work published on finite element modeling flow of cohesive materi al in hoppers. Haussler and Eibl (1984) developed a model using 18 governing e quations: 3 for dynamic equilibrium, 6 constitutive relations and 9 kinematic relatio ns to describe flow behavior. They used FEM, with triangular elements, to determine the spatial response of sand flow from a hopper. They concluded that the velocity in the bin section was constant, while in the hopper section it was maximum at the center li ne. Schmidt and Wu (1989) used a similar FEM procedure except that they used a modifi ed version of the viscosity coefficient and used rectangular elements. Runesson a nd Nilsson (1986) used the same dynamic equations and modeled the flow of granular material as a viscous-plastic fluid with a Newtonian part and a plastic part. The Drucke r-Prager (1952) yield criteria was used to represent purely frictional flow. Link and Elwi (1990) used an elastic-pe rfectly plastic mode l with wall interface elements to describe incipient flow of cohesi on-less material. Several steps were used to simulate the filling process in layers while in cipient release as opposed to full release was simulated for discharge. Flow was detected by a sudden increase in displacements at the outlet followed by failure of the FEM to c onverge to a solution. The material was simulated using eight-node isoparametric elements while the in terface used six-node isoperimetric elements. The wall pressure re sults were close to the results obtained by Jenike while the maximum outlet pressure was lower than the Jenike analysis. The results, however were not verifi ed by experimental procedures.

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36 A FEM analysis, with a new secant constitutive relationship, was used, by Meng and Jofriet (1992), to simulate cohessionless gr anular material (soy bean) flow. The outlet velocities were comparable with preliminary experimental results. The solution converged quickly and was stable throughout the simulation. Another model, developed by Diez and Godoy (1992), used viscoplastic behavior and incompressible flow to describe cohesive materials. This model us ed the Drucker-Prager (1952) yield theory and was applied to conical and wedge shaped hoppe rs. Results obtained compared well with other published work for the conical hopper fo r cohessionless materials except at the bottom of the hopper. Kamath and Puri (1999) used the modified Cam-clay equations in a FEM code to predict incipient flow behavior of wheat flour in a mass flow hopper bin. Incipient flow was characterized by the characterization of th e first dynamic arch. This arch represented the transition from the static to the dynami c state during discharge. Incipient flow is assumed to occur when the mesh at the outle t of the hopper displays 7% or more axial strain. The bin was discretized using rect angular 4-node elements while the hopper is discritized using quadrilateral elements. A thin wall interface element was the for the powder wall interaction. The FEM results were verified experimentally using a transparent plastic laboratory size mass flow hopper bin. The FEM results were within 95% of the measured values. FEM using ABAQUS on Diamondback The numerical model is based on a consis tent continuum mechanics approach. In this project the commercially available softwa re ABAQUS (Inc. 2003) is being used. It is a general-purpose analysis pr oduct that can solve a wide range of linear and nonlinear problems involving the static, dynamic, thermal, and electrical response of components.

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37 It has several built in models to describe material response. Such materials include: metals, plastics, concrete, sand and other fric tional material. It also allows for a user subroutine to be written that allows any ma terial to be implemented. Some of these plastic models are listed in figure 3.1 (A BAQUS 2003). The figure al so shows the steps involved in an analysis. The geometry and meshing and material pr operties are created in the preprocessor (ABAQUS/CAE), which creates an input f ile that is submitted to the processor (ABAQUS/Standard) or (ABAQUS/Explicit). The results of the simulation can be viewed as a data file or as visualization in ABAQUS/Viewer. Figure 3.1: Flowchart showing steps followe d to complete an analysis in ABAQUS Explicit Time Integration Exact solutions of problems where findi ng stresses and deformations on bodies subjected to loading, requires that both fo rce and moment equilibrium be maintained at all times over any arbitrary volume of the body. The finite element method is based on Plasticity User Materials • User Subroutines allows any m model to be implemented • Creep • Volumetric Swelling • Two-layer viscoplasticity • Extended Drucker-Prager • Capped Drucker-Prager • Cam-Clay • Mohr-Coulomb • Crushable Foam • Strain-rate-dependent Plasticity Preprocessing ABAQUS/CAE or other software Input file: job.inp Simulation ABAQUS/Standard Or ABAQUS/Explicit Output files: job.obd, job.dat job.res, job.fil Postprocessing ABAQUS/CAE or other software

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38 approximating this equilibrium requirement by replacing it with a weaker requirement, that equilibrium must be maintained in an average sense over a finite number of divisions of the volume of the body. Starting from the force equilibr ium equation for the volume: 0SVtdsfdV (3.1) where: V is the volume occupied by a part of the body in the current configuration, S is the surface bounding this volume. t is the force per unit of current f is the body force per unit volume and : tn where is the Cauchy stress matrix and n is the unit outward normal. From the principles of continuum mechanic s the deferential form of the equilibrium equation of motion is derived as: .0 f (3.2) The displacement-interpolation finite el ement model is developed by writing the equilibrium equations in “weak form”. This produces the virtual work equation in the classical form: :VSVDdVvtdSvfdV (3.3) where: D is virtual strain rate (virtual rate of deformation) v is a virtual velocity field This equation tells us that the rate of work done by the external forces subjected to any virtual velocity field is equal to the rate of work done by th e equilibrating stresses on the rate of deformation of the same virtual velocity field. The principle of virtual work is the

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39 “weak form” of the equilibrium equations and is used as the basic equilibrium statement for the finite element formulation. ABAQUS can solve problems explicitly or implicitly (using ABAQUS/Standard) For both the explicit and the implicit time integration procedures, the equilibrium is M uPI (3.4) where: P are the external applied forces I are the internal element forces M is the mass matrix u are the nodal accelerations Both procedures solve for nodal accelerations and use the same element calculations to determine the internal element fo rces. In the implicit procedure a set of linear equations is solved by a direct solution method to obt ain the nodal accelerations. This method however becomes expensive and time consuming compared to using the explicit method. The implicit scheme uses the full Newton’s iterative solution method to satisfy dynamic equilibrium at the end of the increment at time tt and compute displacements at the same time. The time increment, t is relatively large compared to that used in the explicit method because the implicit scheme is unconditionally stable. For a nonlinear problem each increment typically requires se veral iterations to obtain a solution within the prescribed tolerances. The iterations continue until several quantities—force residual, displacement correction, etc.—are within the pr escribed tolerances. Since the simulation of flow in a diamondback hopper contains highly discontinuous processes, such as contact and frictional sliding, quadratic convergence may be lost and a large number of iterations may be required. Explicit schemes are often very efficient in solving certain classes of pr oblems that are essentially static and involve complex contact such as forging, rolling, and flow. Flow problems are characterized by

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40 large distortions, and contact in teraction with the hoppe r wall. It also has the advantage of requiring much less disk space and memory than implicit simulation. To get accurate solutions for a bulk solid in a hopper a relativel y dense mesh with thousands of elements needs to be used. And since the explicit method shows great cost savings over the implicit method it is used in this research. Hence using an explicit scheme where the solution is determin ed without iterating by explicitly advancing the kinematic state from th e previous increment is preferred. A central difference rule, to integrate the equations of motion e xplicitly through time, using the kinematic conditions at one incremen t to calculate the kinematic conditions at the next increment, is used. The accelerations at the beginning of the current increment (time ) are calculated as: 1 t tuMPI (3.5) Since the explicit procedure always uses a diagonal, or lumped, mass matrix, solving for the accelerations is trivial; there are no simultaneous equations to solve. The acceleration of any node is determined comple tely by its mass and the net force acting on it, making the nodal calculat ions very inexpensive. The accelerations are integrated through time using the central difference rule, which calculates the change in velocity assu ming that the acceleration is constant. This change in velocity is added to the velocity from the middle of the previous increment to determine the velocities at the middle of the current increment:

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41 222ttt tt tttttuuu (3.6) The velocities are integrated through tim e and added to the displacements at the beginning of the increment to determine the di splacements at the end of the increment: 2 t tttttttuuu (3.7) Thus, satisfying dynamic equilibrium at the beginning of the increment provides the accelerations. Knowing the acceler ations, the velocities and displacements are advanced “explicitly” through time. The term “explicit” refers to the fact that th e state at the end of the increment is based solely on the displa cements, velocities, and accelerations at the beginning of the increment. This method inte grates constant accelerations exactly. For the method to produce accurate results, the time increments must be quite small so that the accelerations are nearly constant during an increment. Since the time increments must be small, analyses typically require many thousands of increments. Fortunately, each increment is inexpensive because there are no simultaneous equations to solve. Most of the computational expense lies in the element calculations to determine the internal forces of the elements acting on the nodes. The element calculations include determining element strains and applying material constitutive relationships (the element stiffness) to determine element stresses and, consequently, internal forces. Here is a summary of the explicit dynamics algorithm:

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42 Figure 3.2: Summary of the explicit dynamics algorithm Contact conditions and other extremely discontinuous events are readily formulated in the explicit method and can be enforced on a node-by-node basis w ithout iteration. The nodal accelerations can be adjusted to balan ce the external and in ternal forces during contact. The most striking feature of the e xplicit method is the l ack of a global tangent stiffness matrix, which is required with imp licit methods. Since the state of the model is advanced explicitly, iterations and tolerances are not required. Nodal calculations. Dynamic equilibrium. 1 t tuMPI Integrate explicitly through time. 222ttt tt tttttuuu 2 t tttttttuuu Element calculations. Compute element strain increments, d from the strain rate, Compute stresses, from constitutive equations ,ttt f d Assemble nodal forces, ttI Set tt to t

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43 Geometry, Meshing, and Loading The advantage of FEM is that a variety of problems such as the hopper geometry, the size and locations of the openings and th e interaction between the material and the walls can be investigated. The diamondback hopper geometry is relatively complicated relative to other hopper geometri es. Because of this complexity generating the mesh with the appropriate density and correct elements is not a trivial task. The steel wall is simulated as a rigid body as shown in figure 3.3 (from different view points). Since this research focused on the behavior of the bulk material in the hopper, no finite element analysis was conducted on the wall. Because of the complexity of the geometry however, it was required to discritize the wall. It was meshed into 6872 triangular elements (R3D3) as seen in figure 3.3. The mesh had to be of particular density so as to simulate contact analysis correctly. Figure 3.3: Geometry and meshing of hopper wall

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44 It was decided to simulate the process of filling the bulk solid in the hopper in several steps. The solid was simulated as a deformable body and divided into 11 different sets. A uniform load is applied to each set to simulate gravity (9.81m/s2) in 11 different steps (as seen in figure 3.4). Discharge was simulated in a 12th step by applying a small displacement to the bottom surface of the powde r. By default, all previously defined loads are propagated to the curr ent step. The starting condition for each general step is the ending condition of the previous general step Thus, the model's response evolves during a sequence of general steps in a simulation. Figure 3.4: Filling procedure for powder It was decided to use second order elements as opposed to first order elements. This is because of the standard first-order elements are essentially constant strain elements and the solutions they give are ge nerally not accurate and, thus of little value. The secondorder elements are capable of representing all possible linear strain fields and much higher solution accuracy per degree of freedom is usually available with the higher-order elements. However, in plasticity problems discontinuities occur in the solution if the 1 2 1 1 2 3 11 10 9 8 7 6 5 4

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45 finite element solution is to exhibit accuracy, th ese discontinuities in the gradient field of the solution should be reasonably well modeled. With a fixed mesh that does not use special elements that admit discontinuities in their formulation, this suggests that the first-order elements are likely to be the mo st successful, because, for a given number of nodes, they provide the most locations at wh ich some component of the gradient of the solution can be discontinuous (the element edge s). Therefore first-order elements tend to be preferred for plastic cases. Tetrahedral elements are geometrically versatile and are used in many automatic meshing algorithms. It is very convenient to mesh a complex shape with tetrahedra, and the second-order and modified tetrahedral elements (C3D10, C3D10M,) in ABAQUS are suitable for gene ral usage. However, tetrahedra are less sensitive to initial element shape. The elemen ts become much less accurate when they are initially distorted. A family of modifi ed 6-node triangular and 10-node tetrahedral elements is available that provides improve d performance over the first-order tetrahedral elements and that avoids some of the problems that exist for regular second-order triangular and tetrahedral elements, mainly re lated to their use in contact problems. The modified tetrahedron elements use a speci al consistent interp olation scheme for displacement. Displacement de grees of freedom are active at all user-defined nodes. These elements are used in contact simula tions because of their excellent contact properties. In ABAQUS/Explicit these modified triangular and tetrahedral elements are the only second-order elements available. In addition, the regular elements may exhibit “volumetric locking” when incompressibility is approached, such as in problems with a large amount of plastic deformation. The modified elements are more expensive computationally than lower-order quadrilate rals and hexahedron and sometimes require a

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46 more refined mesh for the same level of accuracy. However, in ABAQUS/Explicit they are provided as an attractive alternative to the lower-order tetrahedron to take advantage of automatic tetrahedral mesh generators. In the diamondback the powder body is meshed into quadratic tetrahedron (C3D10M) elements and the wall into discrete rigid body. The deformation of the wall is negligible compared to the deformation of the powder. The mesh density and element type are two factors that have a major influe nce on the accuracy of results and computer time. Hence in this research the appropriat e mesh had to be chosen. Figure 3.5 shows 4 different meshes that were used for analys is. As the project progressed, one mesh was used with the various models. Figure 3.5: Geometry and meshing of bulk solid 1266 elements 5815 elements 10129 elements 11325 elements

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47 The wall friction coefficient is appl ied between the wa ll and the body using Coulomb criteria. The basic c oncept of the Coulomb fricti on model is to relate the maximum allowable frictional (s hear) stress across an interface to the contact pressure between the contacting bodies. In the basic form of the Coulomb friction model, two contacting surfaces can carry shear stresses up to a certain magnitude across their interface before they start slidi ng relative to one another; this state is known as sticking. The Coulomb friction model define s this critical shear stress,crit at which sliding of the surfaces starts as a fraction of the contact pressure, p between the surfaces (crit p ). The stick/slip calculations determine when a point transitions from sticking to slipping or from slipping to sticking. The fraction, is known as the coef ficient of friction. The basic friction model assumes that is the same in all directions (isotropic friction). For a three-dimensional simula tion there are two or thogonal components of shear stress, 1 and 2 along the interface between the two bodies. These components act in the slip directions for the contact surfaces or contact elements. ABAQUS combines the two shear stress components into an “equivalent shear stress,” for the stick/slip calculations, where 22 12 In addition, ABAQUS combines the two slip velocity components into an equivalent slip rate, 22 11eq The stick/slip calculations define a surface. The friction coefficient is defined as a function of the equi valent slip rate and contact pressure: ,eq p (3.8) where eq is the equivalent slip rate, p is the contact

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48 The solid line in Figure 3.6 summarizes the behavior of the Coulomb friction model: there is zero relative motion (slip) of the surfaces when they ar e sticking (the shear stresses are below ). Figure 3.6: Frictional behavior Simulating ideal friction behavior can be very difficult; therefore, by default in most cases, ABAQUS uses a penalty friction formulation with an allowable “elastic slip,” shown by the dotted line in Figure 12–5. The “el astic slip” is the sm all amount of relative motion between the surfaces that occurs when the surfaces should be sticking. ABAQUS automatically chooses the penalty stiffness (t he slope of the dotted line) so that this allowable “elastic slip” is a very small frac tion of the characteristic element length. The penalty friction formulation works well for most problems, including most metal forming applications. In those problems where the ideal stick-sl ip frictional behavior must be included, the “Lagrange” friction formulation can be us ed in ABAQUS/Standard and the kinematic friction formulation can be used in ABAQUS/Explicit. The “Lagrange” friction formulation is more expensive in terms of the computer resources used because ABAQUS/Standard uses additional variable s for each surface node with frictional contact. In addition, the solution converges mo re slowly so that a dditional iterations are usually required. This friction formula tion is not discussed in this guide.

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49 Kinematic enforcement of the frictional constraints in ABAQUS/Explicit is based on a predictor/corrector algorithm. The force re quired to maintain a node's position on the opposite surface in the predicted configurati on is calculated using the mass associated with the node, the distance th e node has slipped, and the time increment. If the shear stress at the node calculated usi ng this force is greater than the surfaces are slipping, and the force corresponding to is applied. In either case the forces result in acceleration corrections tangentia l to the surface at the slav e node and the nodes of the master surface facet that it contacts. Often the friction coefficient at the initia tion of slipping from a sticking condition is different from the friction coe fficient during established slidi ng. The former is typically referred to as the static friction coefficient, and the latter is referred to as the kinetic friction coefficient. Plasticity Models: General Discussion The elastic-plastic response models in ABAQUS have the same general form. They are written as rate-ind ependent models or as rate-depe ndent. A rate-independent model is one in which the constitutive response does not depend on the rate of deformation as opposed to a rate-dependent models where tim e effect such as creep are considered. A basic assumption of elastic-plastic models is that the deformation can be divided into an elastic part and a plasti c part. In its most general form this statement is written as: elpl F FF (3.9) where: Fis the total deformation gradient, elFis the fully recoverable part of the deformation plFis the plastic deformation

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50 The rigid body rotation at the point can be included in the definition of either elFor plFor can be considered separately before or after either part of the decomposition. This decomposition can be reduced to an additive strain rate decomposition, if the elastic strains are assumed to be infinitesimal: elpl (3.10) where: is the total strain rate el is the elastic strain rate pl is the plastic strain rate The strain rate is the rate of deformation: v sym x (3.11) The above decomposition implies that the elastic response must always be small in problems in which these models are used. In practice this is the case: plasticity models are provided for metals, soils, polymers, crus hable foams, and concrete; and in each of these materials it is very unlikely that the el astic strain would ever be larger than a few percent. The elastic part of the re sponse derived from an elastic strain energy density potential, so the stress is defined by: elU (3.12) where: Uis the strain energy density potential The stress tensor is defined as the Cauchy stress tensor. For several of the plasticity models provide d in ABAQUS the elasticity is linear, so the strain energy density potential has a ve ry simple form. For the soils model the

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51 volumetric elastic strain is propor tional to the logarithm of th e equivalent pressure stress. The rate-independent plasticity models have a region of purely elastic response. The yield function, f defines the limit to this region of purely elastic response and, for purely elastic response is written so that: ,,0 fH (3.13) where: is the temperature H are a set of hardening parameters(the subscript is introduced simply to indicate that there may be several hardening parameters) In the simplest plasticity model (perfect plasticity) the yield surface acts as a limit surface and there are no hardening parameters at all: no part of the model evolves during the deformation. Complex plasticity models usually include a large number of hardening parameters. Only one is used in the isotropi c hardening metal model and in the Cam-clay model; six are used in the simple kinematic hardening model. When the material is flowing inelastically the inelastic part of the deformation is defined by the flow rule, which we can write as: pl i i ig dd (3.14) where pldis plastic strain increase id proportionality factor for the ith system ,,,iigH plastic potential for the ith system Cauchy stress state In an “associated flow” plasticity model th e direction of flow is the same as the direction of the outward norm al to the yield surface:

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52 ii igf c (3.15) where: ic is a scalar. These flow models are useful for materi als in which dislo cation motion provides the fundamental mechanisms of plastic flow when there are no sudden changes in the direction of the plastic strain rate at a poin t. They are generally not accurate for materials in which the inelastic deformation is primarily caused by frictional mechanisms. The metal plasticity models in ABAQUS (excep t cast iron) and the Cam-clay soil model use associated flow. The cast iron, granular/polymer, crushable foam, Mohr-Coulomb, Drucker-Prager/Cap, and jointed material models provide nonassociated flow with respect to volumetric straining and equivalent pressure stress. Since the flow rule and the hardening evolution rules are written in rate form, they must be integrated. The only rate equations are the evolutionary rule for the hardening, the flow rule, and the strain rate decomposition. The simplest operator that provides unconditional stability for inte gration of rate equations is the backward Euler method: applying this method to the flow gives pl i i ig (3.16) and applying it to the hardenin g evolution equations gives ,, iii H h (3.17) where ,ih is the hardening law for ,iH The strain rate decomposition is integrated over a time increment as:

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53 elpl (3.18) where is defined by the central difference operator: 1 2t x xsymx (3.19) The total values of each strain measure as the sum of the va lue of that strain at the start of the increment, rotated to account for rigid body motion during the increment, and the strain increment, are inte grated. This integration allows the strain rate decomposition to be integrated into: elpl (3.20) From a computational viewpoi nt the problem is now al gebraic: the integrated equations of the constitutive model for the state at the end of the increment, must be solved. The set of equations that define the algebraic problem are the strain decomposition, the elasticity, the integrated fl ow rule, the integrated hardening laws, and for rate independent models the yield constraints: 0if (3.21) For some plasticity models the algebraic problem can be solved in closed form. For other models it is possible to reduce the prob lem to a one variable or a two variable problem that can then be solved to give th e entire solution. For example, the Mises yield surface—which is generally used for isotropi c metals, together with linear, isotropic elasticity—is a case for which th e integrated problem can be solved exactly or in one variable.

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54 For other rate-independent models with a single yield system th e algebraic problem is considered to be a problem in the components of pl Once these have been found— the elasticity—together with the integrated strain rate decomposition—define the stress. The flow rule then defines and the hardening laws define the increments in the hardening variables.

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55 CHAPTER 4 CONSTITUTIVE MATERIAL LAWS For bulk materials a constitutive highly nonlin ear material model needs to be used that describes the solid-like behavior of the material dur ing small deformation rates as well as the fluid-like behavior during flow conditions. Most of the models describing bulk solids were developed in ci vil engineering to describe sa nd, clay and rock behavior. Due to difference in stress magnitudes, loadi ng and flow, only some of these models can be applied to bulk solid applications. Th ey are based on a consistent continuum mechanics approach. The simplest model to use would be an elastic model based on Hooke’s law. This model was used by Ooi and Rotter (1990). Other elastic concepts such as non-linear elasticity (Bishara, Ayoub and Mahdy, 1983) and hypo-elasticity (Weidner, 1990) have also been used. However these models cannot predict phenomena such as cohesion, dilatancy, plasticity and other effects that are essential for the model. These effects need to be described by elastic-plastic models that can describe irreversible deformations. One of the first plastic mode ls used is the Moh r-Coulomb model (1773). Figure 4.1: Tresca and M ohr-Coulomb yield surfaces 1 2 3 1= 2=3 Tresca Mohr-Coulomb

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56 It is derived from the Tresca criteria for meta l plasticity and states that a material can sustain a maximum shear stress for a normal load in a plane. If the shear load is more than that maximum stress then the material starts to flow. Figur e 4.1 shows the yield surfaces for both Mohr-Coulomb and Tresca in the principle stress space. Another yield surface that is numerically ea sier to handle is the Drucke r-Prager (1952) surface because it does not have the edges of the Mohr-Coulom b. Because of this, researchers have a preference to use this model. It is derived from the von-Mise s criteria and is shown in Figure 4.2 Figure 4.2: Von Mises and Dr ucker-Prager yield surfaces One disadvantage to the above models is that under hydrostatic loading plastic deformation can not be predicted. That is because during isotropic loading (on the line 1=2=3), the stress lies within the yield surface. However, it is known from experiments that plastic defo rmation does occur during isotr opic loading. Therefore a cap is introduced to create a closed yield surface that limits elastic deformation in hydrostatic loading. So, as can be seen from figure 4.3, the model consists of a yield cone and a yield cap. 3 2 1 Drucker-Prager Von Mises 1= 2=3

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57 p ij ijG dd Figure 4.3: Closed yield surface In the above elastic-plastic models, both the elastic behavior inside the yield surface and the plastic flow rule have to be determined. The flow rule can either be associated, where the shape of the yield surf ace determines that flow direction or nonassociated, where an additional surface, the flow potential, is defined. The general form of the flow rule is usuall y assumed to be potential where the strain increments are related to the stress in crements by the following relationship: (4.1) where: plastic strain increase proportionality factor plastic potential Cauchy stress state The flow rule is associated if the potential G is equal to the yield f unction F. If they are not equal then it is non-associated. p ijdG ij d 3 1 2 1= 2= 3

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58 Capped Drucker-Prager Model (1952) The capped Drucker-Prager model is used to model cohesive mate rials that exhibit pressure-dependent yield. It is based on the addition of a cap yield surface to the DruckerPrager plasticity model, which provides an inelastic hardening mechanism to account for plastic compaction and helps to control volume dilatancy when the material yields in shear. It can be used in conjunction with an el astic material model and allows the material to harden or soften isotropical ly. It is shown in Figure 4.4. Figure 4.4: The linear Druc ker-Prager cap model The Drucker-Prager failure surface is written as: (4.2) where and d represent the angle of fricti on of the material and its cohesion, respectively. 1 3 Ptrace ---------------is the equivalent pressure stress 3 : 2 q SS ----------------------is the misses equivalent stress 1 39 .: 2 r SSS ----------------is the third stress invariant P IS-----------------------------i s the deviatoric stress Hardening Hardening Shear Failure, Fs Transition surface, Ft Cap, Fc 131 2 3 P 13q Elastic Plastic tan0sFqpd

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59 The cap yield surface has an elliptical shape with constant eccentr icity. The cap surface hardens or softens as a function of the volum etric inelastic strain. It is defined by the following equation: (4.3) where, R is a material parameter that controls the shape of the cap and is a small parameter that controls the transition yield surface so that the model provides a smooth intersection between the cap and failure surfaces: (4.4) As can be seen from the equations above the model itself needs 7 parameters or measured material properties for calibration. In addi tion to describe the elastic behavior two parameters are needed (EYoung’s modulus, and -poisson’s ratio). The coefficient of wall friction also needs to be determined. 3D General Elastic/Viscoplastic Model In order to be able to pred ict the mechanical behavior of cohesive powders in silos or hoppers, a physically adequate constitutiv e model has been developed (Cristescu, 1987). In general, the constitutive model cap tures all major features of material mechanical response and accurately describe s the evolution of deformation and volume change. Dilatancy and the rela ted effects, such as microc rack formation damage, and creep failure can be predicted. 2 2tan0 1 coscaaRp FppRdp 2 21tantan0 costaaaFpppdpdp

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60 The three-dimensional general elastic/vi scoplastic constitutive equation with nonassociated flow rule (Cristescu, 1987) is 11() 1 232(,)I TWtF k GKGH 1 (4.5) = rate of deformation tensor (,)H = yield function = Cauchy stress tensor ()IWt= irreversible stress work per unit volume, = mean stress F = viscoplastic potential, K, G = shear and bulk moduli Tk = viscosity coefficient for transient creep 1is the identity matrix. The function () 1 (,)IWt H is chosen to represent mechanical behavior of the material due to transient creep and symbol is known as Macaulay bracket: 1 2AAA The last term in equation 4.5 describes the mechanical behavior of the elastic/viscoplas tic material that exhibits vi scous properties in the plastic region only. In the elastic/viscoplastic form ulations stress and strain are time-dependent variables, and time is considered as an independent variab le. In this problem th e phenomena of creep and plasticity cannot be treated separately as only the s uperposition effect. It is worthy to note that the model initially included a term for steady-state creep given by the equation: I SSS k (4.6) where S is the viscoplastic potentia l for steady-state creep and Sk is a viscosity coefficient. This term howev er describes behavior over long periods of time such as

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61 creep in tunnels. However, transient creep can describe the behavior of powders in hoppers at relatively sma ller stresses and hence th e above term is ignored. The material is assumed to be isotropic and homogeneous. Hence the constitutive equation depends on the stress invari ants; primarily the mean stress: 1231 3 (4.7) and the equivalent stress or the octahedral shear stress : 222 12312132322 39 (4.8) or in terms of an arbitrary (1,2,3) coordinate system: 222 222 1122113322331223131 666 3 (4.9) The strain rate consists of elastic and irreversible parts: E I (4.10) The elastic part of the strain rate tensor is 11 232EGKG 1 (4.11) and the irreversible part is () 1 (,)I I TWtF k H (4.12) Equation 4.11 describes an instantaneous re sponse of the material to loading. The irreversible part, (equation 4.12), depends on the time history as well as on the path history of the stress. One of the time e ffects that the model describes is creep.

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62 Deformation due to transient creep stops when the stabilization boundary is reached. The equation of this boundary (the locus of the stre ss states at the end of transient creep) is (,)()I H Wt (4.13) The irreversible stress work per unit volume ()IWt describes the irreversible isotropic hardening: id t W : (4.14) One of the concepts describe d by the model is whether the bulk solid is compressing or dilating. Figure 4.5 shows these domains. Tr iaxial experiments on cohesive materials show that powders show co mpressibility at lower stresses followed by dilatancy. Between the two domains we have the comp ressibility/dilatancy bounda ry. In the case of a triaxial test the boundary is determined by determining the where the slopes of the plots of the stress vs. volumetri c strain are vertical. Figure 4.5: Domains of compressibility a nd dilatancy The viscoplastic flow occurs when: (4.15) () 10 (,)IWt H FAILURE COMPRESSIBLE INCOMPRESSIBLE DILATANT

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63 This model will enable us to capture all major features of powder mechanical response and accurately describe the evoluti on of deformation and volume change. It can predict the dilatancy and the related effects, such as creep failure. It is calibrated by numerous hydrostatic and devi atoric triaxial testing. Numerical Integration of the El astic/Viscoplastic Equation The main difficulties in implementation of the elastic/viscoplastic constitutive model in the finite element code is the numerical integration of the irreversible (viscoplastic) part of the strain tensor. Suppose that at the time moment t all values of stress and strains in 4.5 are known. Consider the moment of time tt First, we represent the increment of the irreversible strain rate in the form of the truncated Ta ylor series with respect to the stress and irreversible strain increments: II Itt (4.16) where () 1 (,)IWtF H (4.17) and ()().IIIIWttWttttt (4.18) The total strain increment consists from elasti c and viscoplastic parts. The elastic part is EI (4.19)

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64 and from the Hooke’s law C I (4.20) where C is constant stiffness matrix de pendent on the elastic constants. The increment of irreversible strain is IIIttt (4.21) and 1IIItttt (4.22) If 0 the explicit scheme (Euler forward sc heme) for the integration of viscoplastic strains results. On the other hand, if 1 the fully implicit scheme (Euler backward scheme) for the integration is obtained. The case 1 2 results in the so-called "implicit trapezoidal" scheme. Substituting (4.22) in (4.21): ()I II Ittt t (4.23) or () 1I I I Ittttttt tt (4.24) Returning to formula (4.20) the expre ssion for stress tensor at the moment tt gives:

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65 1 1 1 C C C C II I Itttttttttttt tttt (4.25) The important issue to be addressed is stability of the numerical integration scheme. Note that the Euler forward me thod does not give an accurate numerical solution. For 1 2a the integration process can proceed only for values of t less than some critical value (this critical value should be determined in some way). The accuracy of the numerical integration scheme with respect to stability and convergence has been estimated by comparing the theoretically pr edicted values for irreversible stress work and irreversible st rain rate (obtained an alytically by direct integration of the constitutive equation) w ith those obtained nume rically during creep. There is good correspondence of results as fo r the irreversible stress work as for the irreversible strain rate. The elastic component is cal culated from Hooke’s law: 2ijijij (4.26a) or C (4.26b) where 1,, 0,.ijij ij (4.27) and are Lame constants. In another form

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66 2000 2000 2000 000200 000020 000002 x x xx yy yy z z zz x y xy x z xz yz yz (4.28) Inverse matrix 1C may be calculated using using the C matrix above. 1C (4.29) Relations of Lame’s constants the bulk and the shear moduli are 2 3 K (4.30) (32) E (4.31) G (4.32)

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67 CHAPTER 5 EXPERIMENTAL RESULTS AND DISCUSSION Parameter Determination-Shear Tests Both direct shear testers and indirect shear testers are used to determine the parameters that are needed for the model per the procedures described in Chapter 2 (Saada, 2005). Silica powder with particle size of around 50 microns was used. To determine the internal angle of friction, and the cohesion, d the linear failure surface was determined using several Schulze tests. The Schulze test (1994) was chosen over the Jenike test (1961) because of the simplicity of the sample preparation and the test procedure. The consolidation weights ranged from 2kg to 25kg. The specimen was sheared at values between 20% and 80% of Figure 5.1: Output data from Schulze test on Silica at 8Kg the steady-state value. Figure 5.1 shows the resu lts of a consolidation weight of 8Kg. The slope of the yield locus (strai ght red line) is the internal angle of friction and the Yintercept is the cohesion. Plots of cohesion and internal angle of fric tion are plotted vs the

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68 principal stress 1 in Figure 5.2. Table 5.1 summarizes the results of 16 tests (Saada, 2005). The tests were also repeated to check for consistency. Table 5.1: Summary of Schulze test results Weight (g) (kPa) (internal friction) d (cohesion) 2000 1.57 28.71 0.359 2000 1.55 27.87 0.364 3000 2.29 29.00 0.43 3000 2.36 28.80 0.5 4000 3.05 31.29 0.47 4000 3.13 30.37 0.566 6000 4.55 34.86 0.46 6000 4.64 33.18 0.639 8000 6.02 31.47 0.75 8000 6.50 30.07 0.849 12000 9.16 32.33 1.04 12000 9.08 33.83 0.935 16000 11.97 34.09 1.11 16000 11.98 33.65 1.16 22000 15.99 32.18 1.38 25000 18.24 32.05 1.56 The cohesion ranged from about 0.36KPa to about 1.56Kpa at a 1 value of 12Kpa. The data can be fit with a se cond order polynomial as seen in Figure 5.2 d (Cohesion)0 0.2 0.4 0.6 0.8 1 1.2 1.4 0510151 (Kpa)d (Kpa) Figure 5.2: Schulze test results for to determine the linear Drucker-Prager surface (internal friction)0 10 20 30 40 50 051015 1(Kpa) deg

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69 The internal angle of fricti on is relatively constant ar ound the 32 deg value except for 1 values that are less than 3Kpa where the angle drops slightly (Figure 5.2). To determine the wall friction coefficien t, Jenike tests are preformed between 304 stainless steel slab, 120mm x 120mm (with 2B finish) and silica (S aada, 2005). The test was conducted using the proce dure described above with weights decreasing from 16kg down to 2kg. To check for repeat ability the test was repeated twice. Results are shown in Figure 5.3: Jenike wall friction re sults to determine boundary conditions Figure 5.3. It can be seen that the values of wall friction angles ar e slightly higher under the 5 kpa, but decrease in value and reach a c onstant value of about 22 deg as the normal stress is increased. This provides a value of =0.4 to be used in th e coulomb criteria. In the stress area of interest the test are repeatable and the ma ximum error is less than 5%. Detailed studies of wall friction are outside th e scope of this research but are currently undertaken by other research groups. Future wo rk in the area of simulation involves the study of more complex frictional models. 20 21 22 23 24 25 26 27 28 29 30 0.05.010.015.020.025.0 Normal Stress (kPa)Wall Frction Angle (deg) Test 1 Test 2

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70 Triaxial tests are required to determine th e parameters for the cap in the model and the elastic parameters. A hydrostatic test where the sample is pressuri zed in all directions is preformed and the pressure and volume cha nges are recorded. Several deviatoric tests covering the range of confining pr essures of interest are also needed. In these tests a fixed confining pressure is applied while the sample is given axial deformation. The stress, axial and volumetric strains are measured. Figure 5.4 shows results for five hydrostatic tests up to 69.0 kPa (Saada, 2005). The confining pressure is increased at a constant rate up to five values: 6.9KPa, 34.5KPa, 41.4KPa, 55. 2KPa, and 69.2KPa. It is important to 0 10 20 30 40 50 60 70 00.010.020.030.040.050.06v(kpa) 6.9 Kpa 34.5 Kpa 41.4 Kpa 55.2 Kpa 69.0 Kpa Figure 5.4: Hydrostatic triaxial testing on Silica Powder (5 confining pressures) note that the specimen had an initial axial st ress of 3.7KPa prior to the initiation of hydrostatic testing due to the filling procedure described in Appendix A. This is reflected in Figure 5.4 where the plots of the stress -strain curves show a clear jump at the beginning of the hydrostatic testing. The hydrosta tic tests are followed by deviatoric tests where the confining pressure is kept consta nt and the axial deformation is increasing.

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71 Figures 5.5 and 5.6 show the axial strain and volumetric strain respectively of five deviatoric tests. 0 50 100 150 200 250 -0.0050.0450.0950.1450.1950.245 1(kpa) 34.5Kpa 13.8Kpa 46.9Kpa 4.8Kpa Figure 5.5: Axial deformation of 4 de viatoric triaxial tests on Silica Powder and a rate of 0.1 mm/min 0 50 100 150 200 250 00.0050.010.0150.020.0250.030.0350.04v(kpa) 34.5Kpa 13.8Kpa 4.8Kpa 46.9Kpa Figure 5.6: Volumetric deforma tion of 4 deviatoric triaxial tests on Silica Powder and a rate 0.1mm/min As expected the graphs show an increase in measured axial stress with increasing confining pressure. Each test shows an increa se in strength until a plateau is reached. The

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72 volumetric strain graphs show a compacti on region followed by a dilation region. The data has been corrected for the confining effects of the latex rubber membrane using Hooke’s law and the elastic properties of the rubber. To check for repeatability several tests were repeated more than once. Figure 5.7 shows the axial strain results of two deviatoric tests at 13.8KPa and 34.5Kpa. 0 50 100 150 200 250 -0.0050.0450.0950.1450.1950.2451(kpa) 34.5Kpa 13.8Kpa 34.5Kpa 13.8Kpa Figure 5.7: Axial deformation of 4 deviatoric tests on Si lica Powder at 13.8KPa and 34.5Kpa It can be seen that the error in the shear st ress is maximum at the higher deformations and has is less than 5%. Because of the low pressures that are being investigated, the plot s show results that require the ‘modification’ of the testing procedure. This is because of the apparent oscillations in the graphs and the not very distinct value of failure for the specimen. Another issue to re solve is the sharp jumps and drops in stresses under 10kpa (shown in the circle), which are due to the loading piston friction while it is lining up with the center of the specimen. These os cillations are described and explained in

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73 Appendix A under piston fricti on error. In spite of these challenges these results do produce values that are satisfa ctory for use in the model. Diamondback hopper measurements-Tekscan Pads The experimental procedure for the meas urements of wall stresses in the hopper was described in chapter 2. Measurements were taken when the powder was static and at belt speeds of 40, 60, 70, and 100 (Johanson, 2003). Figure 5.8 shows the geometry of the mechanism at the outlet of the hopper Figure 5.8: Discharge mechanism and belt Based on the dimensions of all the parts and the outlet, the ve locity of the belt, table 5.2 was created to relate velocity and flow rate of the powder in the chute. Table 5.2: Belt veloci ty vs. flow rate Belt velocity Flow rate (cm3/sec) 40 35.6 60 52.7 70 60.9 100 80.6 Figures 5.9 shows the location of the pad in the hopper and an outpu t from a test. The area that is covered by the rectangles re presents the line in the hopper where the transition between the vertical and the converging parts of the hopper. The left side of the pad starts midway through the flat sec tion and loops around the curved section. Powder on belt Belt Powder in chute

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74 Figure 5.9: Pad locatio n and output example Rectangles are used to obtain wall stress as opposed to lines so as to obtain an average and not localized stresses. The data were averaged along eight paths shown by the white lines in figure 5. 10 (Saada, 2005). They are 2.8c m apart and span from the middle of the flat section of the hopper to halfway around the curved section (90 deg). Figure 5.10 shows the contact pressure measur ed while the powder was static at a depth of 2.8cm from the hopper tran sition. These stresses gene rally decrease around the perimeter and are smaller at greater material depths. The results show a peak at the flat section of the hopper and oscillations going around the curved section of the hopper.

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75 Wall Stress (@2.8 cm) Static0 2 4 6 8 10 12 14 020406080 Angle (deg)Stress (kPa) Initial Figure 5.10: Static Wall stress data at a distance 2.8cm below hopper transition Figure 5.11 shows wall stress measurements were along the same path of depth 2.8cm conducted at various speeds 60, 70 and 100 in a ddition to the static state (Saada, 2005). The plots show agreement in the pattern and magnitu de of wall stress at th e various belt speeds. 0 2 4 6 8 10 12 14 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Figure 5.11: Wall stress data at a distance 2.8c m below hopper transition at various speeds and static conditions 2.8cm

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76 Figure 5.12 shows the same data at the same speeds at a depth of 5.6cm. Appendix A shows the results at the rest of the six paths shown in Figure 4.10. 0 1 2 3 4 5 6 7 8 9 10 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Figure 5.12: Wall stress data at a distan ce 5.6cm below hopper transition at various speeds and static conditions Figure 5.13 shows the results of a Fourier series analysis c onducted at the results from two different regions (Johans on, 2005). Region numbered #1 is in the flat section of the converging hopper. In this region the fluctuatio ns in the stress have a wide range of frequencies. Region #7 is in the round section. In this region the fluctuations have a wide range of frequencies. It is hypothesized th at the converging dive rging nature of the Diamondback hopper causes a shock zone form ation across the hopper that propagates at points along the edge of the fl at plate section. Here the radius of curvature changes causing a stress discontinuity. This should be a region of steep stress gradients. This is validated by experiment it will be compared to FEM calculation.

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77 Figure 5.13: Fourier series analysis on the wall pressure measurements. Conclusions and Discussion of the Test results and Testing Procedure Both simple direct shear tests and more complex triaxial tests are needed to determine the parameters for the models that are used in this research. Sixteen Schulze tests were conducted to determine the internal an gle of friction,, and the cohesion, c (or d). The tests were conducted at various consolidation stresses and were repeated at least twice to check for repeatability. The intern al angle of friction was determined to be constant at a value of 32o and the cohesion was increasing qua draticaly. Jenike tests were conducted to determine the wall friction angle. That angle was determined to be 22o. Both hydrostastic tests and devi atoric triaxial tests were pe rformed for the calibration of the viscoplastic model. The deviatoric tests were conducted at a confining pressure of up to 47Kpa. They were repeatable and showed the expected increas e in strength with increasing confining pressure and the showed the powder going from compressibility to dilatancy. 101001 1030 0.01 0.02 0.03 0.04 0.05 0.06 Period of Signal (sec)Amplitude (P7) 050010001500200 0 0 2 4 Time (sec)Pressure 7 (KPa) Radius of curvature changes causing a stress discontinuity. This is a region of steep stress gradients. 101001 1030 0.01 0.02 0.03 0.04 0.05 0.06 Period of Signal (sec)Amplitude (P1) 0500100015002000 2 4 6 8 Time (sec)Pressure 1 (KPa)

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78 Performing the tests at this lower confin ing pressure (less than 50Kpa) created issues that could introduce some error to the resu lts. This is partially seen by the fact that the material does not show an apparent failure point. The two main issues are: 1) the influence of the rubber membrane on the de formation of the specimen, and 2) the influence on the confining pressure of the water height. This 2nd issue is a problem due to the fact that the stress (due to the water pressu re) at the top of the sample is close to zero while at the bottom it is around 2KPa which c ould present some erro r especially at the test with 4.7Kpa confining pressure. For the 1st issue, Appendix B provides a procedure to correct for membrane pressure based on hoop’s law and the rubber properties. To minimize the influence of the membrane it is suggested that a much thinner membrane is used. This, however, may cause some complicat ions in the ‘current sample preparation procedure’, since extra care has to be taken so as not to cause any damage to the membrane. It is also suggested that plasti c wrap be used instead of rubber membrane, since it is thinner and because of plasticity, it would be easier to correct for it. The ideal procedure would be to conduct the test without a membrane but since this is impossible with the current setup it is suggested to use another test er such the Cubi cal Biaxial test.

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79 tan0sFqpd CHAPTER 6 FEM RESULTS AND DISCUSSIONS Capped Drucker-Prager Model The capped Drucker-Prager model is used to model cohesive mate rials that exhibit pressure-dependent yield. It is based on the addition of a cap yield surface to the DruckerPrager plasticity model, which provides an inelastic hardening mechanism to account for plastic compaction and helps to control volume dilatancy when the material yields in shear. It can be used in conjunction with an el astic material model and allows the material to harden or soften isotropical ly. It is shown in Figure 6.1. Figure 6.1: The Linear Dr ucker-Prager Cap model The Drucker-Prager failure surface is written as: (6.1) where and d represent the angle of fricti on of the material and its cohesion, respectively. Hardening Hardening Shear Failure, Fs Transition surface, Ft Ca p Fc 131 2 3 P 13q Elastic Plastic

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80 1 3 Ptrace ---------------is the equivalent pressure stress 3 : 2 q SS ----------------------is the misses equivalent stress 1 39 .: 2 r SSS ----------------is the third stress invariant P IS-----------------------------i s the deviatoric stress The cap yield surface has an elliptical shap e with constant eccentricity. The cap surface hardens or softens as a function of the volumetric inelastic stra in. It is defined by the following equation: (6.2) where, R is a material parameter that controls the shape of the cap and is a small parameter that controls the transition yield surface so that the model provides a smooth intersection between the cap and failure surfaces (6.3) As can be seen from the equations above the model itself needs 7 parameters for calibration. In addition to describe the elas tic behavior two parameters are needed (EYoung’s modulus, and -poisson’s ratio). The coefficient of wall friction also needs to be determined. Results from the shear testers listed in Ch apter 5 are used to determine the required parameters and are listed in Table 6.1. 2 2tan0 1 coscaaRp FppRdp 2 21tantan0 costaaaFpppdpdp

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81 Table 6.1: Capped Drucker-Prager parame ters determined from shear tests Drucker Prager Model Verification The Drucker-Prager model was used to pe rform FEM simulations on the results of a triaxial test. The results are used to co mpare the way perfect plasticity approximates powder behavior. The specimen was simula ted as a deformable body (height=15cm, diameter=7cm) and discritized into 1063 tetrah edral elements as seen in Figure 6.2 a). A confining pressure of 10KPa was applied around the specimen and the specimen was deformed axially by 3cm (up to 20% strain) as seen in Figure 6.2 b). Figure 6.2: FEM simulation of a triaxial test using the Drucker-Prager Model E c (or d)R K v0 Pav Run 1 4.6x107 0.24 33.71.16 0.330.041 0.0024 4.4 10 20 30 0 0.0086 0.021 0.0321 Run 2 4.6x107 0.24 28.70.359 0.330.041 0.0001 0.5 10 20 30 0 0.0086 0.021 0.0321 a)Undeformedspecimenb)Deformedspecimen

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82 The parameters for the model where determined from the shear tests and are shown in Table 6.1. The axial stress was calculated fr om the average of the stress at the top of the specimen. The axial stress-s train curves from the simulation were compared with the results that were obtai ned experimentally as seen in figure 6.3. 0 5 10 15 20 25 30 35 40 00.020.040.060.080.1 Strain (kpa) Experimental FEM Using Drucker-PragerSilica @1mm/min Figure 6.3: Axial stress-strai n curves for experiment and FEM using Drucker-Prager the model Drucker Prager Model Hopper FEM Results The results from the shear tests above were used to run an ABAQUS FEM simulation on the diamondback hopper in the static case ( during storage). As de scribed in section 3.2, the powder is simulated as a deformab le body with quadratic tetrahedron (C3D10M) elements. The powder was divide d into 11 different sets and the gravitational force was applied to each set to simulate filing. The st eel wall is simulated as a discrete rigid body that required discritization. The boundary conditions where the tangential Coulomb friction of =0.4 on the walls and a vertical disp lacement of zero at the outlet. The loading is a gravitational constant of 9.81m/s2. Figure 6.4 shows a side view of the

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83 calculated contact stresses at the end of each filling step (Saada, 2005). They show the increase in stress at the bottom of the hoppe r and moving up as the force is applied. Figure 6.4: FEM calculated contact stresses at various filling steps It is important to note that the stress concen trations in the vertical section of the hopper (above the transition to a conve rging section), are not a pow der phenomena but are rather due to FEM error in the contact analysis proced ure. This is due to the fact that some of the elements in the powder penetrate the wa ll and cause stress concentrations. Also the non-symmetry in the simulation could be due the fact that the me shing is not 100% symmetrical (automatic meshi ng techniques were used). Anot her explanation is probably due to the fact that the simulation of filling is not completely static and there is movement of powder nodes on wall nodes causing non-symmetry. To simulate discharge the surface at the bottom of the powder was moved by a distance of 5cm. Figure 6.5 shows the contact stresses at 4 different intervals (Saada, 2005). Note

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84 Figure 6.5: FEM calculated contact st resses at various discharge steps the stress change close to the outlet and in th e converging section. The comments above explain the non-symmetry of th e simulations. The area where the stress measurements are being compared and analyzed is shown in Figure 6.6. The FEM section shown shows the nodes where contact stresses ar e recorded. They ar e then compared with the Tekscan measurements along the corresponding paths shown as white lines in the plot. Figure 6.6: FEM vs. TekScan area of interest FEM TekScan

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85 It is important to note that the reasons the whit e lines are curved is due to the fact that the pad is curved inside the hopper a nd is presented a flat section in the figure. Also due to the complexity of the contact analysis si mulation, a balance between mesh density, element types and computer simulation time an appropriate mesh had to be chosen. The mesh chosen had 14 nodes across each of the pa ths that were analysed These are seen as the black dots in the FEM contour plot. Figure 6.7 shows the comparisons of the static wall stresses at seven locations (from 2.8cm to 19.6cm below the hopper transiti on). The plots here show a similar trend. They capture the general magnitude of the wa ll stresses that are hi gher at the middle of the flat section but are decr easing as we go round in the curved section. The simulations show some oscillations, but fail to calculate the same oscillations that are recorded by the tekscan pads. The magnitudes of the stresses are better approximated at the flat section (between 0 and 40 deg) but are slightly higher at the range of 40 deg to 90 deg. It is possible that the oscillations in the pad readings are due to th at fact that the powder is not completely static and there is some movement while the powder is settling and the measurement are being made. As explained earlier the si mulation of flow was done by applying a displacement of 5 cm to the bottom surface of the powder. A nother method attempted was the removal of the boundary conditions at the surface this how ever produced large deformations of the elements and caused the simulation to abort. The data analysis was conducted at various locations in the flow simulation. Figure 6. 8 shows the comparisons at two seven paths

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86 0 2 4 6 8 10 12 14 04590 Angle (deg)Stress (kPa) Measured Static Wall Stress FEM Static Wall Stress (@8.4 cm) 0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @11.2 cm 0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @14 cm 0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa) @16.8 cm 0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @19.6 cm 0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa) Figure 6.7: FEM calculated st atic wall stresses vs. measured wall stresses at seven locations 0 1 2 3 4 5 6 7 8 9 10 04590 Angle (deg)Stress (kPa) Measured Static Wall Stress FEM Static Wall Stress

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87 Figure 6.8: FEM calculated contact stresses vs. measured wall stresses at seven locations during flow 0 1 2 3 4 5 6 7 8 9 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm 0 1 2 3 4 5 6 7 8 9 10 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm @11.2 cm 0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @19.6 cm 0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) @14 cm 0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa) @16.8 cm 0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) @8.4 cm 0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa)

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88 and at two simulation displacements (0.25cm and 5cm). All the results show that the Drucker-Prager model approximates the wall stresses during flow better than during storage. The stresses follow the same pattern of decreasing as the angle is increased. The simulation captures a good estim ate of magnitude and some of the oscillatory pattern. We see that the magnitude of the oscillations incr eases as the displacement is increased. This is probably due to the increas e in contact wall friction crit eria between the nodes at the wall and the nodes at the surface of the powder. Several other parameters that affect the m odel were also investig ated. The effect of flow rate is shown in Figure 6.9. The 5 cm displacement was applied at times of 1 second and 0.1 seconds. The figure shows the stresse s at 5.6 cm and 8.4 cm transitions. Figure 6.9: FEM calculated contact stresses vs. measured wall stresses at two locations during flow at two different times 0 1 2 3 4 5 6 7 8 9 020406080 Angle (deg)Stress (kPa) MEASURED Flow Wall Stress FEM at Discharge time=0.1 sec FEM at Discharge time=1 sec 0 1 2 3 4 5 6 7 020406080 Angle (deg)Stress (kPa) MEASURED Flow Wall Stress FEM at Discharge time=0.1 sec FEM at Discharge time=1 sec a) At 5.6cm from hopper transition b) At 8.4cm from hopper transition

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89 The plots show that there are no major diffe rences in stress values between the two rates. The discharge at a higher rate does show stresses that are slightly higher at certain locations in the hopper. They do however follo w a similar path as the slower rate. Table 6.1 shows there are 7 parameters th at control the functions of the plastic Drucker-Prager model. Several simulations we re conduced to analyze the effects of the angle of internal friction and the cohesion, bot h parameters that determine the linear line in the model. Figure 6.10 shows results of st resses at two locations (2.8cm and 5.6cm). Figure 6.10: FEM calculated contact st resses vs. measured wall stresses at two locations during flow angles of friction values The angle was changed in the model from 25o,29o and 33o. Increasing the angle resulted in changes of stress at certain locat ions in the hopper while others locations had the same stress. Stresses at locations that we re in the flat section of the hopper increased with increasing angle while the locations towa rds the curved sections that did not have 0 2 4 6 8 10 12 14 16 020406080 Angle (deg)Stress (kPa) MEASURED WALL STRESS FEM Flow Wall Stress Angle=25 deg FEM Flow Wall Stress Angle=29 deg FEM Flow Wall Stress Angle=33 deg 0 1 2 3 4 5 6 7 8 9 020406080 Angle (deg)Stress (kPa) MEASURED(Initial) FEM Flow Wall Stress Angle=25 deg FEM Flow Wall Stress Angle=29 deg FEM Flow Wall Stress Angle=33 deg a) At 2.8cm from hopper transition b) At 5.6cm from hopper transition

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90 any major oscillations did not show any major change in stress with a change in angle. The reason for this increase is that the incr ease in the angle increas es the slope of the failure plane hence increasing the elastic domain in the powder. The cohesion value was decreased fro m a value of 1160 Pa, to 500pa, and to 0.0001pa (a value that simulates zero cohesi on since the value zero is not accepted by the model). The results at two locations are s hown in figure 6.11. Again an increase in Figure 6.11: FEM calculated contact st resses vs. measured wall stresses at two locations during flow with various cohesion values cohesion causes an increase in wall stress at locations on the flat section and no increase in the curved section. There is actually a reduc tion in stress at the highe r angles in plot of the path at 5.6cm. The overall in crease in stresses results from the fact that th e increase of the cohesion value causes the linear failure plane to move up and increases the elastic domain in the powder. 0 2 4 6 8 10 12 14 16 020406080 Angle (deg)Stress (kPa) MEASURED(Initial) FEM Flow Wall Stress-Cohesion=500Kpa FEM Flow Wall Stress-Cohesion=1160Kpa FEM Flow Wall Stress-Cohesion=0.0001Kpa 0 1 2 3 4 5 6 7 8 9 020406080 Angle (deg)Stress (kPa) MEASURED(Initial) FEM Flow Wall Stress-Cohesion=500Kp FEM Flow Wall Stress-Cohesion=1160 K FEM Flow Wall Stress-Cohesion=0.0001 a) At 2.8cm from hopper transition b) At 5.6cm from hopper transition

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91 Conclusions And Discussion of the Predic tive Capabilities of the Capped DruckerPrager Model Several researchers have used FEA in c onjunction with the Drucker-Prager model in the predictions of wall stresses in silo s and hoppers. The application of the cap model to the complicated geometry of the diam ondback hopper is unique to this study. The previous section presented a comparison be tween the wall stresses calculated from the model and experimentally measured stresses. Out of the 8 parameters used to calibrate the model, 5 were measured using shear testers and 3 were fit parameters. The Young’s modulus, E and Poisson’s ratio, were measured using the triaxial tester. The tester was also, hydrostatically used to calculate the hardening parameters. The angle of internal friction, and the cohesion, c were measured using the Schulze tester. The transition parameter,, and the radius of curvature of the cap, R, were fit parameters because of problems faced with the triaxial testing setup at the low confining regime. The ch allenges in using this tester are discussed at the end of chapter 5. A simulation of the triaxial test using a perfectly plastic model, where the yield surface is fixed in stress space, showed the weakness of using the Drucker-Prager model. By definition, the stress-strain curve for th e model is represented by a linear elastic line followed by the a straight constant stress line when yield is re ached. Figure 6.3 shows that the material shows a gra dually increasing curve, that re quires a more complex model that takes into account the time effect s and the volumetric deformations. In spite of the inherit flaws in the model the FEA simulations provided flow wall stress predictions that are more accurate than stresses predicted using the Janssen equation. In Figure 6.8, the seven locations where the simulation results were compared

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92 with the model experimental measurements ther e was relative agreement in values. In all the locations analyzed the simulations mana ged to capture some of the oscillatory movement of the wall stress going across the paths from 0o to 90o. The simulations also captured the decreasing magnitude of the stress es with increasing angle. The simulations were overestimated at an gles higher than around 60o (in the curved section) while they were underestimated in the flat section. It is important to note that FEM related parameters such as geometry and loading procedure also influence results. For exam ple, changing the amount of displacement of the bottom surface of the powder during simula ted discharge causes the predicted wall stresses to shift. Increasing the value of the displacement increases th e oscillations in the stresses causing the accuracy to increase. Because of the comput er power required in some of these calculations much more time is required in determining the full influence of all the parameters. Depending on the co mputer memory, processor speed and the model parameters used, the simulation might take up to 4 days to complete.

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93 Viscoplastic Model Parameter Determination The viscoplastic model is described in detail in Chapter 3. The Elastic parameters for the Silica powder were determined from cy clic triaxial tests performed in previous research. The yield function is determined from the equation of the stabilization boundary, IHW The isotropic hardening and the changes of the yi eld surfaces during deformation is considered to be influenced by the work hardening and not the strain rate. The volumetric and deviatoric part s of the irreversible work (I H VHW ,I D DHW ) are used to estimate the yield function following the equation: '' 00()()().() TT IIII VDVWWWttdtttdt (6.4) where, V is the irreversible volumetric strain rate and I is the rate of deviatoric deformation tensor, 'is deviator of the Cauchy stress tensor, and T is the actual time. The hydrostatic triaxial tests were used to determine the values of irreversible work. Figure 6.12 (Saada, 2005) shows the experiment al data and the yield function that was approximated with: 2 I HVoHWk (6.5) where ko=0.0006 Kpa. Here and below: **, 11 KpaKpa are the dimensionless mean st ress and equivalent stress.

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94 0.0 0.2 0.4 0.6 0.8 1.0 1.2 01020304050 (kPa)WI v (kPa) 6.89 kPa 13.79 kPa 20.69 kPa 41.37 kPa Figure 6.12: The irreversible volumetric stress work (data points) and function HH(solid line) A similar procedure is used to determine I D DHW from the deviatoric tests. Figure 6.13 shows the results of four tests at four confining pressures. 0 5 10 15 20 25 30 35 050100150200250 (Kpa)WD I (Kpa) 4.8KPa 13.8Kpa 34.5Kpa 46.9Kpa H Fit Figure 6.13: The irreversible volumetric stress work (data points) and function HD(solid line The experimental data is a pproximated with the function:

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95 ** 3 *3***, 3ob DoooHaecd (6.6) Table 6.2: Coefficients of the yiel d function Parameter value ao 1.403x10-5 bo -0.03626 co 5.798x10-4 do 6.706x10-3 From both the hydrostatic and deviatoric appr oximations the yield f unction is obtained: ** 2 3 **3***, 3ob ooooHkaecd (6.7) The surfaces ,H =const satisfies the differential equation: dH dH (6.8) The yield function is not uni que and any equation that satisfies the above equation can be used as the yield function. The only restrictions are that H has to be greater than zero everywhere in the constitutive domain except boundary 0 where H can be equal to zero. The next step is to determine the viscopl astic potential. The yield function and experimental data is used to determine th e derivatives of the viscoplastic potential: () 1 (,)I V TF k Wt H (6.9a)

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96 132 3 () 1 (,)II TF k Wt H (6.9b) As can be seen from the data, Silica powde r exhibits both compr essible and dilatant volumetric behavior. The derivativ e of the visoplastic potential TF k must posses the following properties: 0TF k (or I V >0) for compressibility 0TF k (or I V =0) for compressibility/dilatancy boundary 0TF k (or I V <0) for compressibility The experimental data for ()Wt,12 II I V and the yield function determined above are used to determine the following expressions: ** *3**3*** 2222ln 33TF kabcd (6.10a) * 1* 3 *3*** 1113b TF kaecd (6.10b) The experimental data and the model fits are shown in figure 6.14.

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97 -0.003 -0.001 0.001 0.003 0.005 0.007 0.009 0.011 0.013050100150200(KPa)Viscoplastic potential derivativeskTdf/d kTdf/d Figure 6.14: The viscoplastic potential derivatives. Integration of both derivativ es gives a closed-form expr ession for the viscoplastic potential: * 1***2 32 3 ******2 1 11111 4 1 ****2* *3****3*** 22223 ,954162 292 ln 33323b Ta kFebbbcd b abcd Table 6.3: Coefficients of the viscoplastic potential Parameter value Parameter Value a1 5.277x10-9 a2 3.164x10-10 b1 -0.03823 b2 -1.442x10-9 c1 2.495x10-7 c2 -2.023x10-8 d1 -5.442x10-7 d2 5.112x10-6

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98 Time Effects It is important to determine if the Silica powder possesses time-dependent properties such as creep and relaxation. When the stress is kept constant during creep, W(t) is increasing in time and W(t) approaches H(,). Thus equilibrium is reached when W(t)= H(,). The ideal way to determine the parameters would involve determining the stabilization boundary. Experimentally this w ould be determined by loading the material incrementally and holding the stress constant until stabilization of the transient creep (I V =0 ) is obtained as shown in figure 6.15. Figure 6.15: History dependence of the stabilization boundary. But since tests of long durations are not avai lable, any triaxial test will suffice, but the model will underestimate the magnitude of th e transient strain, but will still describe the main features: dilatancy, co mpressibility, creep failure etc… For the case of the Silica powder the deviat oric tests were run at two rates: 0.1 mm/min and 1mm/min to determine rate effects. The irreversible work plots are shown in figure 6.16 1 Stabilization boundary Creep

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99 0 5 10 15 20 25 30 35 050100150200250 (Kpa)WD I (Kpa) 0.1 mm/min 1 mm/min Figure 6.16: The irreversible volumetric stress work at two rates: 0.1mm/min and 1mm/min It can be seen from the plot that at th e lower confining pressures and at the rate difference of one order of magnitude (10 time s faster), there are no major differences in work and hence the yield functions and the viscoplastic potentials are very similar. To determine the influence of creep a de viatoric test where the stress was held constant for five hours was conducted. The st ress and axial test results are shown in figure 6.17. The creep state is followed by an unloading and loading cycle to check for hysteresis.

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100 0 50 100 150 200 250 00.050.10.150.20.250.31 (Kpa) Figure 6.17: Deviatoric creep test at a c onfining pressure of 34.5 Kpa The creep data was used to determine the axial strain rate (Figure 6.18). We see from the plots that after around 30 minutes de formation practically stops (stabilization boundary). This time is well within the 5 hours it takes for the deviat oric tests at 0.1 mm/min. -5.E-05 0.E+00 5.E-05 1.E-04 2.E-04 2.E-04 05000100001500020000time (seconds) I 1 Figure 6.18: Variation in time of axial creep rate at a confining pressure of 34.5 Kpa

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101 The creep test was also repeat ed in the hydrostatic state (Fig ure 6.19). Again we see that the strain rate is reached after about 30min. -2.E-06 0.E+00 2.E-06 4.E-06 6.E-06 8.E-06 1.E-050200040006000800010000 time (seconds)I v Figure 6.19: Variation in time of volumetric creep rate at a confining pr essure of 34.5Kpa Model Validation The model is tested by comparing the theore tically predicted values of irreversible stress with the experimental data. The e quation for the irreversible strain rate I is used and the formula for calculating the irreversible stress work is obtai ned by integrating the following equation:

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102 0() 1. (,)II i TWt F Wtk H (6.12) This gives the irreversible stress work during creep as: 00. (,)()(,)(()(,))IITk F tt HWtHWtHe (6.13) where t0 is the time when the material starts to creep. To check the validity of the model the stress-strain curves resulting from the model are compared with the experimental curves. To obtain the strain, both the elastic and inelastic parts of the constitutive equation have to be integrated. Assuming that in each step, the loading increases instantaneously a nd then remains constant as in creep, the following formula for strain at each i-step is obtained: 00 (,)() 1 (,) 11 1 1 232 (,)T ii k F tt HWt F H t F GKG He 1 (6.14) Where 0 itt and 0 itis the time when the material starts to creep. The correct loading history of the experimental data has to be followed for the comparison since the strain tensor is very sensit ive to changes in the stress tensor. Figure 6.20 shows the comparison betw een the model predicted and the experimentally obtained stress-strain curves for a deviatoric test at a confining pressure of 34.5 Kpa. It is shown from the plots that the chosen functions for the yield and viscoplastic potentials can reproduce the axia l, transversal, and volumetric strains.

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103 0 20 40 60 80 100 120 140 160 180 200 -0.2-0.100.10.20.3strain (KPa)1v3 Figure 6.20: Theoretically predicted stress-strain curves (solid lines) vs. experimentally determined curves at 34.5 Kpa confining pressure Viscoplastic Model Hopper FEM Results Just as in the Drucker-Prager model, the parameters obtained are used to run an ABAQUS FEM simulation on the diamondback hopper in the static case (during storage) and during flow. As described in section 3.2, the powder is simulated as a deformable body with quadratic tetrahedron (C3D10M) el ements. The powder was divided into 11 different sets and the gravitational force was applied to each set to simulate filing. The steel wall is simulated as a discrete rigid body that requir ed discritization. The boundary conditions where the tangen tial coulomb friction of =0.4 on the walls and a vertical displacement of zero at the ou tlet. The loading is a gravitational constant of 9.81m/s2. The results of the static simulations along the seven locations below the hopper transition are presented in Figure 6.21. As can be seen the simulations provide results

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104 @2.8 cm0 2 4 6 8 10 12 14 04590 Angle (deg)Stress (kPa) Measured Static Wall Stress FEM Static Wall Stress @5.6 cm0 1 2 3 4 5 6 7 8 9 10 04590 Angle (deg)Stress (kPa) Measured Static Wall Stress FEM Static Wall Stress @8.4 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @11.2 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @14 cm0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa) @16.8 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @19.6 cm0 1 2 3 4 5 6 7 04590 Angle (deg)Stress (kPa) Figure 6.21: FEM calculated static contact st resses vs. measured wall stresses at seven locations below the hopper transition

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105 oscillations but underestimate the wall stresses by about half in certain locations (2.8cm, 5.6cm, 8.4cm and 11.2cm below hopper tran sition). At locations 14cm, 16.8cm and 19.6cm below hopper transition there is good agreement between experimental and calculated results. Figure 6.22 show the results of the model simulations during flow at the same seven locations below hopper transition. The st ress predictions are more accurate in the discharge stage. For locations 2.8cm, 5.6cm, and 8.4cm below hopper transition the magnitude and pattern are predicted with re lative accuracy. For locations 11.2cm, 14cm, 16.8cm and 19.6cm below hopper transitions the predictions are accurate above 30o but are underestimated at angles lower th an that (flat section of the hopper). This could be attributed to the experimental problems that were described in the triaxial tests. The triaxial tests was de signed to provide results at a pressure range that is above the range of intere st in the hopper. Other parameters that are FEM related were investigated. Figure 6.23 shows the effects of increasing the wall friction coefficient from 0.42 to 0.52. We see that there is no major change in the stress values. The wall stresses at =0.52 do show an increase in oscillations however. Figure 6.24 shows the results of using adap tive meshing in the simulation. Adaptive meshing is used when th ere are major deformations in the mesh due to loading. The body is re-meshed during the simulation to control the deformations. We see some slight changes in stress during the re-meshing. With adaptive meshing the magnitude of stresses are about the same but there are increase s in oscillations.

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106 @2.8 cm 0 2 4 6 8 10 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress @5.6 cm0 2 4 6 8 10 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress @8.4 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @11.2 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) @14 cm 0 1 2 3 4 5 6 7 8 9 04590 Angle (deg)Stress (kPa) @16.8 cm0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) @19.6 cm0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) Figure 6.22: FEM calculated flow contact stresses vs. measured wall stresses at seven locations below the hopper transition

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107 @5.6 cm0 1 2 3 4 5 6 7 8 9 10 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction Coefficient=0.42 Friction coefficient=0.52 @8.4 cm0 1 2 3 4 5 6 7 8 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction Coefficient=0.42 Friction Coefficient=0.52 Figure 6.23: FEM calculated flow contact st resses vs. measured wall stresses at two locations below the hopper transiti on at two wall friction values @16.8 cm 0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM With Adaptive Mesh @19.6 cm 0 2 4 6 8 10 12 04590 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM with Adaptive Mesh Figure 6.24: FEM calculated flow contact st resses vs. measured wall stresses at two locations below the hopper transition using adaptive meshing

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108 Conclusions and Discussion of the Predictive Capabilities of the 3D Elastic/Viscoplastic Model The complexity of powder flow behavior in hoppers prompted the investigation of more complex material response models. In the preceding section a Viscoplastic model is used to determine wall stresses in a diamondback hopper. It has the capability to captures all major features of material mechanical re sponse and accurately describes the evolution of deformation and volume change of frictional materials. It is integrated numerically and implemented into a code used in FEM to describe the wall stresses in a diamondback hopper during flow. Triaxial compression tests were used to determine the 14 parameters used in the simulations. Four hydrostatic test s and four deviatoric tests (p resented in Chapter 5) were used to determine the yield function and the viscoplastic potential. It is shown that the chosen functions for the yield and viscopl astic potentials can reproduce the axial, transversal, and volumetric strains. To determine the time effects including creep the deviatoric test s were repeated at rates of 0.1mm/min and 1mm/min. It was dete rmined from the experimental data at the confining pressure used that there are no signi ficant differences in the determination of the parameters. Creep tests showed that the strain rate stabilized around 30 minutes ( very close to zero). This means that the deviat oric tests conducted at 0.1mm/min already contain the effects of creep since they are conducted over a period of 5 hours. The wall stresses predicted during storage in the diamondback were closer to the measured results than the predictions of the perfectly plastic model. Three of the locations analyzed provided stresses that are co mparable to the experimental results while in the Drucker-Prager model six of the locati ons analyzed show stre sses that are higher

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109 than the experimental measurements. None of the models, however, showed wall stresses that have magnitudes of oscillations equivalent to the measured ones. The model predicted flow wall stresses more accurately than the static wall stresses. The simulated discharge stresses show more oscillations a nd smaller variations from the measured stresses. As discussed earlier, the parameters for th e model are determined using a triaxial tester. The challenges and disadvantages of using the triaxial test er at low confining pressure are discussed at the end of chapter 5. An improvement in the testing procedure is recommended to improve the pr ediction capabilities of the model.

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110 CHAPTER 7 CONCLUSIONS The problem that is tackled in this research is of intere st to various industries that handle bulk solids. Solving th e problem of flow obstructi on or discontinuous flow can save a numerous amount of manpower and can make the discharge process from hoppers, under gravity loading, more cost-efficient. Th e determination of stresses and velocities in the hopper provides data that can provid e help in the design of the hoppers. This research provides an experimental and numerical approach to determine the wall stresses in a diamondback hopper. This hopper was chosen because of its complex geometry that helps powder flow. An experi mental procedure usi ng pressure-sensitive pads was successfully developed to m easure the wall stresses in the hopper. Measurements were obtained for silica powder during storage and during flow at various speeds. The experimental data proved to be repeatable. It was proved that pressure sensitive pads are a useful t ool in examining the spatial variation in hopper wall loads The Janssen slice model can provide a first approximation to the loads in diamondback hoppers. However, a significant variation from the simple slice model approach was used here. This method was able to capture the magnitude of the stresses but not the pattern. Finite element analysis with more comple x constitutive models are used to increase the agreement between experime ntal and theoretical approaches. The two models used are the perfectly plastic Drucker-Prager model and a 3D general visco-plastic model. The calibration of the models required modificati on of standard testing procedures in the

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111 triaxial shear tests because of the lower stre sses that are encountered inside the hopper. It was found that there was a different between th e parameters determined using the direct shear tests and the indirect sh ear tests. This was attributed to the rubber membrane that holds the powder together dur ing testing. It is suggested that a new procedure that reduces the effect of the membrane be developed. Using the measured parameters and fit parameters from the shear tests FEM simulations were completed to calculate stresses on the hopper wall and compare them with measured stresses. Various model parame ters and FEM parameters were varied to investigate their effect. The Drucker-Prager m odel obtained values for the wall stress that captured the magnitude of stre sses and decreasing pattern al ong a path inside the hopper. It also managed to capture some of the osci llatory pattern captured in the experimental data. The simulations provided flow wall stre ss predictions that are more accurate than stresses predicted using the Janssen equation. In the seven locations where the simulation results were compared with the model expe rimental measurements there was relative agreement in values. In all the locations analyzed the simulations managed to capture some of the oscillatory movement of th e wall stress going across the paths from 0o to 90o. The simulations also captured the decreasing magnitude of the stresses with increasing angle. The simulations were overestim ated at angles higher than around 60o (in the curved section) while they were und erestimated in the flat section. The visco-plastic model also captured some of the oscillatory pattern during flow. The wall stresses predicted during storage in the diamondback were closer to the measured results than the predictions of the perfectly plastic model. Three of the locations analyzed provided stresses that are co mparable to the experimental results while

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112 in the Drucker-Prager model six of the locati ons analyzed show stre sses that are higher than the experimental measurements. None of the models, however, showed wall stresses that have magnitudes of oscillations equivalent to the measured ones. The model predicted flow wall stresses more accurately than the static wall stresses. The simulated discharge stresses show more oscillations a nd smaller variations from the measured stresses.

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113 APPENDIX A NUMERICAL INTEGRATION SCHE ME FOR TRANSIENT CREEP 1 1 1II I Itttttttttttt tttt C C C C With: () 1 (,)IWtF H 2 2 2()1()() 1 (,)(,)(,)IIIWtHWtFWtF t HHH 1() (,)I IIWtF t H where ()IWt It, and ()I IWt t 112233122331IIIIIIIt 112233122331t To be determined based on stability requirements To be determined based on accuracy requirements ()IIIIWttWttttt t

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114 The following equations are for dry sand: 2 1..oo TooFXYZW kYXgg where: 2 1 2 1 2312 2 3 ,1 23 22ooo oo oo of f X f ff ,ln21 3ooooYf 12 2oo ooof Z f 2 1 32 ,212121 333 2ooooooof Wff f 2 1 1 23122 4 3 1 23 22oo oo oo ooo o of X f f ff 21 3 21 3oo oo oof Y f 1, 2 22oo oooo ooZ f ff 1 3 2, 2 2121 33 2 21241 33oo oo oo ooo oooW f ff f f

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115 22 22222 2 11 2222222.2.2oooo T oooFXXYYZW kYXggg where: 2 222 1 22 2222 2 2 1 3 21212 33 2 22 4 1 3 2 o oo oooooo o ooo of X f ff f f 2 22 2 22 2 221 21 3 3 21 21 3 3oo oo oo oo oof f Y f f 2 222 1 2222, 2 2 22oo oooooo ooZ f ff 2 22 1 3 2 22 22 22, 2 212121 333 2 2122 3oo oooo oo oooo ooW f fff f f 2 2 241 3ooo o 6 2 112(,) 3 3oo oooooo o oHaabcc 66 65 1 21 12 675611 65 66 33 2 3333oooo oo ooo oo ooo oo oooo ooooaa aa H bcc

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116 The following equations are for Microcrystalline Cellulose: 0.81.20.20.20.290.20.00000050.0000030.0000030.000013oooo TooooooF k 2 2 71.81.270.80.270.81.2 2 2 2 70.20.860.20.270.71 24101210510 6103108.710 ooooo Toooooo ooooo oooooF kxxx xxx 22 60.2960.850.2 22 +3102.610+1.310oooo oooxxx 2 2(,) 7 3o o o oHab 2 21 2 3 2 7 7 3 3oo o o o o o o o o ob b H a Parameters: 1122331 3o ………………………………………is the mean stress o=3.0x10-3 1 =5.3x10-6 Kpa-1 =0.984 f=0.696 go=0.003 g1=1.7x10-8 ao=3x10-4 a1=2.1x10-5 bo=3.4x10-3 c1=-1.5x10-6 Kpa-1 c2=1.8x10-3

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117 3 2oo …………………………………………………..is equivalent shear stress 222 211221133223312132322 33D oIssssss …is the octahedral shear stress where: 1 3ijijijijs or in terms of an arbitrary (1,2,3) coordinate system: 222 222 1122113322331223131 666 3o We get: 111213 212223 1323331 00 3 1 00 3 1 00 3ooo oooo ooo 2 2 o 0 112233 1213 111213 221133 1223 212223 332211 1323 1323332 33 2 2 1 33 2 2 33 2ooo oooo o ooo 222 11223313 12 222 111213 111213 2222 221133 12 222 2 12 212223 222 222 13233323 3 133 2 2 13 31 2ooo ooo ooo ooooo ooo ooo 23 2223 1323332211 1323333 3 332 331 2oo o ooo ooo

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118 APPENDIX B TESTING PROCEDURE Sample Preparation The initial stress state of th e powder proved very influenc ial in the results of the triaxial tests. Therefore it was very importa nt the specimen be consistent in uniformity and initial density. The preparation was done using the procedure shown in the schematic below. Jansen’s equation provided the H needed to get uniform distribution of the powder. In the first part measured amount is poured into the membra ne inside the mould and is evenly distributed across area using a special brush. Then a weight is used to compress the layer of powder to a stress lower than any hydrostatic pressure that the test is to be conducted at. This step is repeat ed several times until the required height is reached. H a) Pour powder into mould b) Compre ss to a predetermined c) Repeat until mould isfull Membrane Brush Consolidating weight

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119 The required H was calculated using Jansen’s Equation: 4tan/4tan/1 4tanvvvoKHDKHDGD Kee The following constant values were used: 1.1o Kv=0.4, 40o, D=Diameter= 0.072m

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120 Membrane Correction The results of the triaxial tests were corrected for the influence of the additional confining pressure added by the rubber membrane that is containing the powder. The procedure is provided in the American Society For Test ing And materials ( ASTM: D 4767-95). The following equation is used: 3 134Et D Where: 13 =correction to be subtracted from th e measured principal stress difference D= diameter of specimen E= Young’s modulus for the membrane material t= thickness of membrane 3 =axial strain This equation is derived from Hoop’s law for a thin walled pressure vessel 12Prt where: P=Pressure r= radius t=thickness And Hooke’s Law: 33121 E The Young’s modulus of he rubber is determined using a thin strip of the membrane and measuring the force per unit strain ob tained by stretching the membrane. 1 P

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121 Piston Friction Error The experimental setup for the triaxial te st caused some slight error at the lower stresses. Several of the deviatoric test plot s showed oscillations at the beginning of the test. This is due to the fact that the piston applying the axial deformation was not perfectly lined up with the center of the piston at the beginning of the test. The friction between the two metal pieces cau sed these oscillations. The pl ot below illustrates this phenomenon 0 10 20 30 40 50 60 00.050.10.15 1(kpa) 6.9KPa (test 2)

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122 APPENDIX C TEKSCAN DATA Wall Stress (@0 cm)0 5 10 15 20 25 30 35 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Wall Stress (@8.4 cm)0 1 2 3 4 5 6 7 8 9 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Wall Stress (@11.2 cm)0 1 2 3 4 5 6 7 8 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100

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123 Wall Stress (@14 cm)0 2 4 6 8 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Wall Stress (@16.8 cm)0 2 4 6 8 10 12020406080Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100 Wall Stress (@19.6 cm)0 2 4 6 8 10 12 020406080 Angle (deg)Stress (kPa) Initial speed 60 speed 70 speed 100

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124 APPENDIX D DRUCKER-PRAGER FEM DATA Wall Stress (@8.4 cm)0 1 2 3 4 5 6 7 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Sress at 0.25cm FEM Flow Wall Stress at 5 cm Wall Stress (@5.6 cm) 0 1 2 3 4 5 6 7 8 9020406080Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm Wall Stress (@11.2 cm)0 2 4 6 8 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm

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125 Wall Stress (@14 cm) 0 1 2 3 4 5 6 7 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm Wall Stress (@19.6 cm)0 2 4 6 8 10 12 020406080Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm Wall Stress (@16.8 cm)0 2 4 6 8 10 12 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM Flow Wall Stress at 0.25 cm FEM Flow Wall Stress at 5 cm

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126 VISCO-PLASTIC FEM DATA Wall Stress (@11.2 cm)0 2 4 6 8020406080Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction coefficient=0.42 Friction Coefficient=0.52 Wall Stress (@14 cm) 0 2 4 6 8 10 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction Coefficient=0.42 Friction Coefficient=0.52 Wall Stress (@16.8 cm) 0 2 4 6 8 10 12 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction Coefficient=0.42 Friction Coefficient=0.52

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127 Wall Stress (@19.6 cm) 0 2 4 6 8 10 12 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress Friction Coefficient=0.42 Friction Coefficient=0.52 Wall Stress (@5.6 cm) 0 2 4 6 8 10 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM with Adaptive Meshing Wall Stress (@8.4 cm) 0 1 2 3 4 5 6 7 8 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM with Adaptive Mesh

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128 Wall Stress (@11.2 cm) 0 1 2 3 4 5 6 7 8 0102030405060708090 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM with Adaptive Mesh Wall Stress (@14 cm) 0 2 4 6 8 10 12 020406080 Angle (deg)Stress (kPa) Measured Flow Wall Stress FEM without Adaptive Mesh FEM with Adaptive Mesh

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129 APPENDIX E VISCO-PLASTIC MODEL FLOW CHART User Subroutine VUMAT 1-Pass on Parameters from ABAQUS to subroutine (a,b,G,K,). 2-Determine C and C-1 matrix 3-Calculate elastic stress E for first increment. 4-Determine total strain increment, and elastic strain increment, E 5-Define mean stress,o octahedral shear stress,o and equivalent shear stress, o 6-Define derivatives: o 2 2o o 2 2o o 2 2o 7Define 1st and 2nd derivatives of visco-plastic potential: TF k 2 2TF k 8Define yield functi on and it’s derivative:(,)H H 9Determine irreversible strain at beginning and end of increment: I i I i+1 10Determine irreversible work hardening: WI(t). 11Define: t It 12Calculate: V(t), L(t), M(t) and Q(t). 13Calculate: total stress tt 14Go to step 4 and repeat process.

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130 LIST OF REFERENCES Abdel-Hadi A., 2003, Experimental And Theoretical Formulation of Constitutive Laws for Particulate Materials, PhD Dissertation, University of Florida. ABAQUS version 6.4 Documentation, ABA QUS, Inc. 2003, Provide nce, RI, U.S.A. ASTM, 2000, Standard Test Method for Shear Testing of Bulk Bolids using the Jenike Shear Cell, in: Annual book of ASTM sta ndards, 04.09 no D6128-00, ASTM, Philadelphia, USA. ASTM, 2002, Standard Shear Test Method for Bulk Solids Using the Schulze Ring Shear Tester, in: Annual book of ASTM standards, 04.09 no D6773-02, ASTM, Philadelphia, USA. Bhushan B., 2002, Introduction to Tribology, John Wiley & Sons, New York, USA. Bishop A.W., Henkel D. J., 1962, The Measurement of Soil Properties in the Triaxial Test, pp. 228, Edward Arnold, 2nd edition, London. Bishara A. G., Ayoub S. F., Mahdy A. S., 1983, Static Pressures in Concrete Circular Silos Storing Granular Materials, ACI Journal, pp. 210-216. Boyce H. R., 1980, A Non-Linear Model for Elastic Behaviour of Granular Material under Repeated Loading, International Symposium on So ils under Cyclic and Transient Loading, Swansea, UK., pp. 103. Brown C. J., Nielsen J., 1998, Silos, Fundamentals of Theory, Behaviour and Design, Headington Hill Hall, Oxford. Brown R. L., Richards J. C., 1970, Principles of Powder Mechanics-Essays on the Packing and Flow of Powders and Solids, Pergamon Press Ltd., Headington Hill Hall, Oxford. Cazacu O., Jin J., Cristescu N. D., 1997, A New Constitutive Model for Alumina Powder Compaction, Kona 15, pp. 103-112. Cristescu N. D.,1987, Elastic/Viscoplastic Consti tutive Equations for Rocks, Int. J. Plasticity, Min. Sci. & Geomech Abstract. 24, pp. 271-282.

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131 Cristescu N. D., 1991, Nonassociated Elastic/Viscoplastic Constitutive Equations for Sand, Int. J. Plasticity 7, pp. 41-64. Cristescu N. D., Hunsche U., 1998, Time Effects in Rock Mechanics, John Wiley & Sons Ltd, West Sussex, England. Diez M. A., Godoy L. A., 1992, International Journal of Mechanical Sciences 34, pp.395 Drescher A.,1991, Analytical Methods in Bin-Load Analysis, Elsevier Science Publishers, Amsterdam, The Netherlands. Drucker D. C., Prager W., 1952, Soil Mechanics and Plastic Analysis or Limit Design, Quarterly of Applied Mathematics 10(2), 157-165. EFCE 1989, Standard Shear Testing Technique for Pa rticulate Solids using the Jenike Shear Cell, Working Party on the Mechanics of Particulate Solids, The Institution of Chemical Engineers, Rugby, UK. Genovese C., 2003, Mechanical Behavior of Particul ate Systems: Experimental and Modeling, PhD Dissertation, University of Florida. Genovese C., Cazacu O., Bucklin R., 2002, Experimental and Theoretical Investigation of the Behavior of Silica Powder under Compression, in: 4th World Congress On Particle Technology, Sidney, Australia, pp. 1 Guaita M., Couto A., Ayuga F., 2003, Numerical Simulation of Wall Pressure during Discharge of Granular Material from Cy lindrical Silos with Eccentric Hoppers, Biosystems Engineering 85 (1), pp.101-109 Guaita M., Aguado P., Ayuga F., 2001, Static and Dynamic Silo Loads using Finite Element Models, J. Agric. Engng Res. 78(3), pp. 299-308. Haussler U., Eibl J., 1984, J ournal of Engineering Mechanics 100, pp. 957-963. Holtz R. D., Kovacs W. D., An Introduction To Geotechnical Engineering, 1981, Prentice-Hall Inc., Englewood Cliffs, N.J. Janssen H. A., 1895 Versuche ber Getreidedruck in Silozellen, Z. Ver.Dt. Ing. 39, pp. 1045-1049. Janssen R. J., Verwijs M. J., Scarlett B., 2005, Measuring Flow Functions with the Flexible Wall Biaxial Tester, Powder Technology 158(1-3), pp. 34-44. Jenike A. W., 1961, Gravity Flow of Bulk Solids, Bulletin 108, Utah Engineering Experimental Station,University of Utah, Salt Lake City, USA.

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132 Jenike A. W., 1964, Storage and Flow of Solids, Bulletin 123, Utah Engineering Experimental Station, University of Utah, Salt Lake City, USA. Johanson K., Bucklin R., 2004, Measurement of K-values in Diamondback Hoppers using Pressure Sensitive Pads, Powder Technology 140(1-2), pp. 122-130. Kamath S., Puri V. M., 1999, Finite Element Model Development and Validation for Incipient Flow Analysis of C ohesive Powders from Hopper Bins, Powder Technology 102, pp. 184-193, Pennsylvania State University, USA. Karlsson T., Klisinski M., Runesson K., 1999, Finite Element Simulation of Granular Material Flow in Plane Silo s with Complicated Geometry, Powder Technology 99, pp. 29-39. Kolymbas D., 1989, Stress Strain Behavior of Granular Media, Proc, Third Int. Conf. Bulk Material Storage, Handling and Trans potation, I.E. Aust.,Newcastle, June, pp. 141149. Lade P. V., 1977, Elasto-Plastic Stress-Strain Theory for Cohesionless Soil with Curved Yield Surface, International J ournal of Solids and Structures 13, pp. 1014-1035. Lambe T. W., Whitman R. V., 1969, Soil Mechanics, Massachusetts Institute of Technology, John Wiley & Sons, New York. Link R. A., Elwi A. E., 1990, Incipient Flow in Silo-Hopper Configurations, Journal of Engineering Mechanics 116(1), pp.172-188. Li F., Puri V. M., 1996, Measurement of Anisotropic Behavior of Dry Cohesive and Cohesionless Powders using a Cubical Triaxial Tester, Powder Technology 89(3), pp. 197-207. Lu Z., Negi C., Jofriet J. C., 1997, A Numerical Model for Flow of Granular Material in Silos. Part 1: Model Development, J. Agric. Egng Res. 68, pp. 223-229. Meng Q., Jofriet J., 1992, ASAE paper No 924016, ASAE, St. Joseph, MI, USA, pp. 1423 Meng Q., Jofriet J., Negi S., 1997, Finite Elemen t Analysis of Bulk Solids Flow, J. Agric. Egng Res. 67, pp. 141-150. Nedderman R. M., 1992, Statics and Kinematics of Granular Materials, Cambridge University Press, Camridge, England. Ooi J.Y., She K.M., 1997, Finite Element Analysis of Wall Pressure in Imperfect Silos, Int. J. Solids Structures 34(16), pp.2061-2072.

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133 Ooi J. Y., Rotter, J. M., 1990, Wall Pressures in Squat Steel Silos from Finite Element Analysis, Computers and Structures 37(4), pp. 361-374. Rotter J. M., 1998, Challenges for the Future in Numerical Simulation, SILOS, Fundamentals of Theory, Behavior and Design, Routledge, London, pp.585-604. Runesson K., Nilsson L., 1986, Internationa l Journal of Bulk Solids and Handling 9, pp. 877. Saada O., 1999, Assessment of the Flowab ility of Particulate Systems using Triaxial Testing, M.S. Thesis, University of Florida. Saada O., 2005, Finite Element Analysis for Inci pient Flow of Bulk Solid in a Diamondback Hopper using Perfect-Plasticity, in Preparation, University of Florida. Saada O., 2005, Finite Element Analysis for Inci pient Flow of Bulk Solid in a Diamondback Hopper using Visco-Plasticity, in Preparation, University of Florida. Schwedes J., 1996, Measurement of Flow properties of Bulk Solids, Powder Technology 88, pp.285-290. Shamlou P. A., 1988, Handling of Bulk Solids, Theory and Practice, Butterworth & Co. Ltd, London, England Schmidt L.C., Wu Y. H., 1989, Internati onal Journal of Bulk Solids and Handling 9, pp.333-340. TekscanTM Documentation, Inc. 1987, Boston, MA, U.S.A. Van Der Kraan M., 1996, Techniques for the Measuremen t of the Flow Properties of Cohesive Powders, PhD Dissertation, Technishe Universiteit, Delft. Verwijs M. J., Abdel-Hadi A. I., Cristescu N. D., Scarlett B., 2003, Comparison of a Cylindrical and Cubical Biaxial Powder Tester, Proceedings 4th Conference for Conveying and Handling Particulate Mate rial, Budapest, Hungary, pp. 18-25. Vidal P., Guaita M., Ayuga F., 2004, Simulation of Discharging Processes in Metallic Silos, ASAE/CSAE Presentation, Paper Numb er 044151, Ontario, Canada, pp. 1-10. Walker D. M., 1966, An Approximate Theory for Pressures and Arching in Hoppers, Chm. Engng. Sci. 21, pp. 975-997. Weidner J., 1990, Vergleich von Stoffgesetzen Granularer Schuttguter zur Silodruckermkttlung, PhD Dissertation, University of Karlsruhe.

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134 Wilde, 1979, Principes Mathematiques et Physiques des Modeles Elastoplastiques des sols Pulverulents, Compte Rendu du Colloque Franco-Pol onais de Paris, LCPC, France. Woodcock C. R., Mason, J.S, 1987, Bulk Solids Handling, Blackie & Sons Limited, London, U.K. Zhupanska O. I., Verwijs M. J., Scarlett B., 2003, Anisotropy in Powders: From Microto Macroscale, Proceedings annual AIChE meeting, San Francisco. Zhupanska O. I, Abdel-Hadi A. I, Cristescu N. D., 2002, Mechanical Properties of Microcrystalline Cellulose Part II: Constitutive Model, Mechanics of Materials 34, pp.391-399.

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135 BIOGRAPHICAL SKETCH Osama Saada graduated with a Bachel or of Science degree in aerospace engineering from the University of Florid a in 1996. He continued his education and obtained a Master of Science in engineering mechanics from the same university in 1999. He then decided to join Daimle rChrysler in Michigan where he worked as a test engineer at their proving grounds. In 2001 he decided to continue his higher education and returned to the Mechanical and Aerospace Engi neering Department at the University of Florida to pursue a PhD.


Permanent Link: http://ufdc.ufl.edu/UFE0013054/00001

Material Information

Title: Finite Element Analysis for Incipient Flow of Bulk Solid in a Diamondback Hopper
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013054:00001

Permanent Link: http://ufdc.ufl.edu/UFE0013054/00001

Material Information

Title: Finite Element Analysis for Incipient Flow of Bulk Solid in a Diamondback Hopper
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0013054:00001


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FINITE ELEMENT ANALYSIS FOR INCIPIENT FLOW OF BULK SOLID IN A
DIAMONDBACK HOPPER















By

OSAMA SULEIMAN SAADA


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

Osama Suleiman Saada

































This document is dedicated to: my father Suleiman Saada, my mother Nema Aude and
the rest of my family members for without their support this research would not have
been completed.















ACKNOWLEDGMENTS

I would like to thank the following, because without their help this research would

not have been possible: my committee chairman, Dr. Nicolaie Cristescu, for his

continuous help; Dr. Kerry Johanson, who was very kind to advise and guide me

throughout my graduate research; Dr. Ray Bucklin for his support, not only in scientific

decisions but also in academic and personal ones; the rest of my committee members Dr.

Frank Townsend and Dr. Bhavani Sankar. I would also like to extend my deepest

gratitude to another member of my committee, Dr. Olesya Zhupanska, who gave me

detailed help on the modeling part of the research. I would also like to thank the Particle

Engineering Center for funding this project. Special thanks go to the PERC faculty and

staff for their assistance.

Last but not least I would like to extend special thanks to my father, Suleiman

Saada, and my mother, Nema Aude, for their patience and continuous moral and financial

support for the many years of my college career.
















TABLE OF CONTENTS



ACKNOW LEDGM ENTS ........................................ iv

LIST OF TABLES ...... ......... ..... .... ........................... vii

LIST OF FIGU RE S ........................................... ................................... viii

ABSTRACT.................. .................. xii

CHAPTER

1 INTRODUCTION ................... ............................ ......... .. .......... 1

2 BACKGROUND INFORM ATION ........................................................................4

B ulk Solid C classification .........................................................................4
Flow Patterns in Bins/Hoppers ..........................................4
The Diam ondback H opper ............................................... ........ 10
Discharge Aids......................................................... .... ......... ...............1
Janssen or Slice M odel A analysis ........................................ ................. 12
Silo -Vertical Section......................................... .........12
Converging H opper .......................................................... .......... ....... ........14
Diam ondback HopperTM ....................................................... 15
Direct Shear Testers-Jenike Test ................................ ............ ............18
Jenike W all Friction Test....................................................... 21
Schulze Test....................................................................22
Indirect Shear Tester-Triaxial Test.............. ................................ ....... 23
The Diamondback HopperTM Measurements .................................. .....28

3 FINITE ELEM EN T ................................. .....................................33

Background in Finite Elem ent M odeling ..................................... ......... ......33
FEM using ABAQUS on Diamondback................................. ..........................36
Explicit Time Integration ................. ...............................37
Geom etry, M eshing, and Loading ....................... ........................... ...............43
Plasticity M odels: General D iscussion..............................................................49






v









4 CONSTITUTIVE MATERIAL LAWS ..........................................55

Capped Drucker-Prager M odel ................................. .................... .... ........... 58
3D General Elastic/Viscoplastic Model................... ............................................59
Numerical Integration of the Elastic/Viscoplastic Equation...............................63

5 EXPERIMENTAL RESULTS AND DISCUSSION...............................................67

Param eter D term ination-Shear Tests ...................................................... 67
Diamondback hopper measurements-Tekscan Pads.......................73
Conclusions and Discussion of the Test results and Testing Procedure...................77

6 FEM RESULTS AND DISCUSSIONS ....................................................79

Capped Drucker-Prager M odel ................................. .................... .... ........... 79
Drucker Prager M odel Verification .............................. ............... 81
Drucker Prager Model Hopper FEM Results...........................82
Conclusions And Discussion of the Predictive Capabilities of the Capped
D rucker-Prager M odel .................................................. ............... 91
Viscoplastic Model Parameter Determination.................................................93
Time Effects ...................................... ....... ........... .98
M odel V alidation ..................................... ......................... ..... .............. 101
Viscoplastic Model Hopper FEM Results ...................... .....103
Conclusions and Discussion of the Predictive Capabilities of the 3D
Elastic/Viscoplastic Model ..................... ......... ..................108

7 CONCLUSIONS ................ .... ............. ............... ...10

APPENDIX

A NUMERICAL INTEGRATION SCHEME FOR TRANSIENT CREEP................113

B TE ST IN G PR O C E D U R E .................................................................................... 118

Sam ple Preparation ................... ............................. ......... .... ..... ................. 118
M embrane Correction............................................. 120
Piston Friction Error .................. .............................................. .. ........ 121

C TEK SC A N D A TA ................................................................................. 122

D DRUCKER-PRAGER FEM DATA............................ ......... 124

E VISCO-PLASTIC M ODEL FLOW CHART...........................................................129

L IST O F R E F E R E N C E S ............................................................................................ 130

BIOGRAPH ICAL SKETCH .............. ................................................................. .....135
















LIST OF TABLES

Table page

2.1 Classification of powders ......................... ...... ........4

2.2 Comparison of mass flow and funnel flow of particulate materials .......................6

2.3 Pad specifications ................ ........ .. ............. ........ .... .. .... .. 30

3.1 Numerical simulation research projects ..................... ............ 34

5.1 Summary of Schulze test results ................................. ............... 68

5.2 B elt velocity vs. flow rate ............................................... ............... 73

6.1 Capped Drucker-Prager parameters determined from shear tests............................81

6.2 Coefficients of the yield function.................................................................. 95

6.3 Coefficients of the viscoplastic potential .............................................................. 97
















LIST OF FIGURES

Figure page

2.1 Types of flow obstruction ................................................. ...............5

2.2 Patterns offlow ........................................... .........6

2.3 Types of hoppers ................ ..... ... ....... .. .... ........ .............7

2.4 Problem s in silos ............................................... .......... 8

2.5 Trajectories of m ajor principle stress ...................... ........... ................................9

2.6 Schematic of the Diamondback HopperTM showing position of pad .....................10

2.7 Forces acting on differential slice of fill in bin ....................................................13

2.8 Forces acting in converging hopper .......................................................................14

2.9 Stresses acting on differential slice element in diamondback hopperTM...............16

2.10 Flow wall stresses at an axial position Z=3.8 cm below the hopper transition........ 17

2.10 Yield loci of different bulk solids .................................. ...........18

2.13 Yield loci .......................................................20

2.14 Jenike w all friction test ............................................................... 21

2.15 Schulze rotational tester .............................................................................22

2.16 E xperim ental set-up..................................................23

2.17 Triaxial chamber on an axial loading device ................. ................. ...........24

2.18 Filling procedure ....................................................... ...............25

2.19 Stress-strain curves .............................................. ..... ...27

2.20 Layers of discs dilating as they are sheared .................................. .....28

2.21 TekScanTM Pads ................. ..............................29









2 .22 T ek Scan P ads ....................................................... 30

2.23 Location of TekscanTM pads in the diamondback hopper.................... ...........31

3.1 Flowchart showing steps followed to complete an analysis in ABAQUS............37

3.2 Summary of the explicit dynamics algorithm ...................................................42

3.3 Geometry and meshing of hopper wall ..............................43

3.4 Filling procedure for powder...................... ........ ...............44

3.5 Geometry and meshing of bulk solid .......................................... 46

3.6 Frictional behavior ......................................... ...... .... .....48

4.1 Tresca and Mohr-Coulomb yield surfaces .........................................55

4.2 Von M ises and Drucker-Prager yield surfaces.............................. ........... ....56

4.3 Closed yield surface ...................... .......... ........ ........ 57

4.4 The linear Drucker-Prager cap model ........................................... 58

4.5 Domains of compressibility and dilatancy ..............................................62

5.1 Output data from Schulze test on Silica at 8Kg ........................................... 67

5.2 Schulze test results for to determine the linear Drucker-Prager surface..................68

5.3 Jenike wall friction results to determine boundary conditions...............................69

5.4 Hydrostatic triaxial testing on Silica Powder (5 confining pressures)...................70

5.5 Axial deformation of 4 deviatoric triaxial tests on Silica Powder and a rate of 0.1
of mm/min ............... ............... ................. 71

5.6 Volumetric deformation of 4 deviatoric triaxial tests on Silica Powder and a rate
of 0. 1mm/min ............... .................................... ..71

5.7 Axial deformation of 4 deviatoric tests on Silica Powder at 13.8KPa and
3 4 5 K p a ...................... .. .. ......... .. .. ....................... 7 2

5.8 D ischarge m echanism and belt.................................... .................. 73

5.9 Pad location and output example ................................. ............... 74

5.10 Static Wall stress data at a distance 2.8cm below hopper transition........................75









5.11 Wall stress data at a distance 2.8cm below hopper transition at various speeds
and static conditions ................... ........................................................................ ..........75

5.12 Wall stress data at a distance 5.6cm below hopper transition at various speeds
and static conditions ................... ........................................................................ ..........76

5.13 Fourier series analysis on the wall pressure measurements. .............................77

6.1 The Linear Drucker-Prager Cap model........................................ 79

6.2 FEM simulation of a triaxial test using the Drucker-Prager Model.........................81

6.3 Axial stress-strain curves for experiment and FEM using the Drucker-Prager
m odel ................... ......... .......................... .......................... 82

6.4 FEM calculated contact stresses at various filling steps .......................................83

6.5 FEM calculated contact stresses at various discharge steps .....................................84

6.6 FEM vs. TekScan area of interest ............................... ............... 84

6.7 FEM calculated static wall stresses vs. measured wall stresses at seven
locations ........................................................86

6.8 FEM calculated contact stresses vs. measured wall stresses at seven locations
during flow ...................................... ................................. ......... 87

6.9 FEM calculated contact stresses vs. measured wall stresses at two locations
during flow at two different times........................ ........ ............................ 88

6.10 FEM calculated contact stresses vs. measured wall stresses at two locations
during flow angles of friction values.................................................89

6.11 FEM calculated contact stresses vs. measured wall stresses at two locations
during flow with various cohesion values...............................................................90

6.12 The irreversible volumetric stress work (data points) and
function HH(solid line) ................... ............ .. ....................94

6.13 The irreversible volumetric stress work (data points) and
function HD(solid line) ................... ............ .. ....................94

6.14 The viscoplastic potential derivatives. ....................................... 97

6.15 History dependence of the stabilization boundary. ...............................................98

6.16 The irreversible volumetric stress work at two rates 0. 1mm/min and 1mm/min.....99

6.17 Deviatoric creep test at a confining pressure of 34.5 Kpa .....................................100









6.18 Variation in time of axial creep rate at a confining pressure of 34.5 Kpa..............100

6.19 Variation in time of volumetric creep rate at a confining pressure
of 34.5Kpa ................... ... ............................ ......... 101

6.20 Theoretically predicted stress-strain curves (solid lines) vs. experimentally
determined curves at 34.5 Kpa confining pressure .................................... 103

6.21 FEM calculated static contact stresses vs. measured wall stresses at seven
locations below the hopper transition............... .................................... 104

6.23 FEM calculated flow contact stresses vs. measured wall stresses at two locations
below the hopper transition at two wall friction values ......................................107

6.24 FEM calculated flow contact stresses vs. measured wall stresses at two locations
below the hopper transition using adaptive meshing ..........................................107
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FINITE ELEMENT ANALYSIS FOR INCIPIENT FLOW OF BULK SOLID IN A
DIAMONDBACK HOPPER
By

Osama Suleiman Saada

December 2005

Chair: Nicolaie Cristescu
Major Department: Mechanical and Aerospace Engineering

Storing of bulk materials is essential in a large number of industries. Powders are

often stored in containers such as a bin, hopper or silo and discharged through an opening

at the bottom of the container under the influence of gravity. This research tackles the

frequent problems encountered in handling bulk solids such as flow obstructions and

discontinuous flow resulting in doing and piping in hoppers. This problem is of interest

to a variety of industries such as chemical processing, food, detergents, ceramics and

pharmaceuticals. An objective of this work is to use finite element analysis to produce

results that could easily be incorporated into the industry and be used by the design

engineers. The study includes both a numerical approach, with an appropriate material

response model, and an experimental procedure implemented on the bulk material of

interest. Results from the analysis are used to relate the slopes of the channels and the

size of the outlets. Results from the FEM analysis are verified using measured stresses

from a DiamondbackTM hopper.















CHAPTER 1
INTRODUCTION

Almost every industry handles powders/bulk solids, either as raw materials or as

final products. Examples include chemicals and chemical processing, foods, detergents,

ceramics, minerals, and pharmaceuticals. Powders are often stored in a container such as

a bin, hopper or silo and removed through an opening in the bottom of that container

under the influence of gravity. The reliability of the process involved depends on the

flowability of these powders. Most powders are cohesive, that is, have the tendency to

agglomerate or stick together over time. For the material to flow out of a storage facility,

bridging, arching or doing must be prevented. For a stable arch to form, the bulk solid

must gain enough strength to support itself within the constraints of the container. The

strength is a function of the degree of the compaction of the material and stress is a

function of spatial position in a piece of process equipment. Thus knowing the stress

allows us to compute the strength of the bulk solid. Stresses are also effective in

producing yield or failure of the bulk solid. Blockage occurs when the strength exceeds

the stresses needed to fail the material.

Since the advent of more powerful computers, numerical methods have become

very useful in research on flow of bulk solids. Numerical methods are very economical

and lend themselves to comprehensive parametric studies. This study presents a finite

element approach to solve for displacements, velocities and stresses of a cohesive

powder. The domain is discretized into small elements and displacements and loads are









approximated. The commercially available ABAQUS software is used. Both the Drucker-

Prager yield criterion and a 3-D visco-plastic model are used to describe the material

response. To verify the numerical results, wall pressure measurements inside a

diamondback hopperTM are taken. Measurement of wall loads in hoppers has been

difficult due to the expense of constructing multiple loads cell measurement units in a

hopper. Recent advances in measurement techniques have allowed significant

improvements in the measurement of multiple point normal stresses in hoppers. This

dissertation presents measurements of wall stresses using pressure sensitive pads made by

TekScanTM

Reliable and complete data on the deformation, failure and flow behavior will

allow a fundamental understanding of powder flow initiation and flow. An objective of

this work is to produce results that could easily be incorporated into the industry and be

used by the design engineers. Both the finite element approach, with an appropriate

material response model, and the technique for measurements of wall stresses on a

diamondback hopperTM using pressure sensitive pads are tools towards achieving that

objective. Results from the analysis are used to relate the slopes of the channels and the

size of the outlets necessary to maintain the flow of a solid of given flowability on walls

of given frictional properties. Using the correct material response models, the right

boundary conditions combined with the correct testing procedure under the correct

loading conditions, continuum models with improved predictive capabilities can be

developed.

The understanding of powder behavior as well as the amount of information on

material properties is limited. Generally, the material properties are determined based on









shear test measurements, yet in order to predict the bulk mechanical response during

storage and transport, experimental data corresponding to a variety of deformation,

loading conditions, and shear rates are needed. To fully understand the material behavior

of bulk solids, experiments are required where all possible stress or strain cases are

induced in the material specimen. However, this would be an impossible task and

simplified tests are instead utilized. In these tests only some stress or strain state

components vary independently. In powder mechanics and technology the most

commonly used testing devices are shear testers, such as the Jenike shear cell. This type

of device has some limitations such as the fact that the location of the shear plane in the

sample is determined by the shape of the tester and that the bulk solid is assumed to obey

a rigid-hardening/softening plastic behavior of Mohr-Coulomb type. A combination of

direct shear testers (the Jenike tester and the Schulze tester), and the indirect shear tester

(such as the triaxial tester) with finite element techniques will improve the analysis

techniques of powder flow. The specific objectives of this research are:

* Present measurements of wall stresses on a diamondback hopperTM using pressure
sensitive pads made by TekScanTM

* Present a finite element approach to solve for displacements, velocities and stresses
of a bulk material based on a visco-plastic model

* Verify the method by comparing FEM results with measured stresses on a
diamondback hopperTM

* Experimentally determine the material properties needed to use in the constitutive
relationships

* Use the commercially available FEM software ABAQUS with built in models for
material response such as the Drucker-Prager model

* Write a user defined subroutine in FORTRAN to be used with ABAQUS to
describe the visco-plastic behavior.















CHAPTER 2
BACKGROUND INFORMATION

Bulk Solid Classification

A particulate system containing particles of about 100[lm and smaller is called a

powder. If the majority of the particles in the system are larger in size, it will be called a

granular material. A classification of particulate solids based on particle size is given in

table 2.1(Brown and Richards, 1970 and Nedderman, 1992). Granular materials are

generally free flowing. To initiate flow in granular material it is sufficient to overcome

their friction resistance. Most fine powders are prone to agglomerate and stick together,

causing significant storage and handling problems

Table 2.1: Classification of powders
Classification Particle size range(pm)
Ultra-fine powder <1
Superfine powder 1-10
Granular powder 10-100
Granular solid 100-3000
Broken solid >3000
Source: Brown and Richards, 1970 and Nedderman, 1992

Flow Patterns in Bins/Hoppers

Before proceeding with a description of the flow patterns in storage facilities the

definition of terms that are used throughout this study is necessary. "Hoppers" have

inclined sidewalls while "Bins" is the term used to describe a combination of a hopper

and a vertical section. Silos is also a term used to describe a tall vertical container where









the height to diameter ratio is bigger than 1.5. In European literature this term is used to

describe a vertical section on top of a hopper section.

These containers vary in their usage in industrial applications. In agricultural,

mining, cement and refractory applications they are used as "primary" storage facilities

and have big capacities (up to 1000 tons). In other industrial applications these containers

are used only for short time storage, the bulk material being subsequently transferred to

other containers or mixers. Thus, they are generally much smaller than the "primary

hoppers."

In bins, most powders can experience obstruction to flow as shown in figure 2.1.










Doming
Doing Piping

Figure 2.1: Types of flow obstruction

For an arch to form, the solid needs to have developed enough strength to support

the weight of the obstruction. Some of the methods used to reinitiate flow are vibration of

the bin and manual movement of the powder. These methods have proven to be costly,

inefficient and dangerous.

There are two major kinds of flow patterns in a container: (1) mass flow, and (2)

funnel flow. In mass flow, the entire material flows throughout the whole vessel during

discharge. Figure 2.2a illustrates such flow. In funnel flow, there is a stagnant layer of

material at the wall of the vessel. This stagnant region extends all the way up to the top of

the container. Flow from a funnel flow bin occurs by first emptying the center flow









channel and then, provided the material is sufficiently free flowing, material sloughs off

the top surface. If the material is cohesive a stable rathole can form. Flow patterns are

illustrated in Figure 2.2b and 2.2c.








a) Mass flow b) Semi-mass flow c) Funnel flow

Figure 2.2: Patterns of flow

For most applications, the ideal flow situation is mass flow and it occurs when the

hopper walls are steep and smooth enough and when there are no abrupt transitions

between the bin section and the hopper section. When the walls are rough and the slope

angle is too large the flow regime tends to be a funnel flow. Table 2.2 lists the advantages

and disadvantages of both mass and funnel flow.

Table 2.2: Comparison of mass flow and funnel flow of particulate materials
Mass flow Funnel flow

Characteristics
No stagnant zones Stagnant zone formation
Uses full cross-section of vessel Flow occurs within a portion of vessel cross-section
First-in, first-out flow First-in, last-out flow

Advantages
Often minimizes segregation, agglomeration Small stresses on vessel walls during flow due to
of materials during discharge the 'buffer effect' of stagnant zones. Very low
particle velocities close to vessel walls; reduced
particle attrition and wall wear

Disadvantages
Large stresses on vessel walls during flow Promotes segregation and agglomeration during flow
Attrition of particles and erosion and wear Discharge rate less predictable as flow region boundary
Of vessel wall surface due to high particle can alter with time
Velocities
Small storage volume/vessel height ratio
Source: Brown and Richards, 1970











Storage containers can have axisymmetric or planar geometry as seen in figure 2.3.


The shape, number and position of the discharge outlet can vary depending upon the silo


geometry, bulk solid properties and process requirements. The geometry of the container


(C) CYLINDRICAL
FLAT-BOTTOMED
SLOT OPENING


r Mass-flow


hoppers


E)WEDGE PYRAMID


Funnel-flow
hoppers



(D) CYLINDRICAL
FLAT-BOTTOMED
CIRCULAR OPENIF


Figure 2.3: Types of hoppers

and properties of the powder determine the flow pattern during filling and discharge.


In many industrial applications the same containers are used for storing different


types of bulk solid. These materials have different mechanical and physical properties


such as particle size, particle distribution, particle shape, and bulk density and frictional


properties. This results in varying flow regimes for different materials. A hopper


designed to provide mass flow for one material may not provide mass flow for a different


material.


Figure 2.4 shows the relationships among bulk solid material properties, the silo


filling process, the flow patterns during discharge, the wall pressures, the stress induced


in the structure and the conditions that might cause failure of the structure (Rotter 1998).


(B) SQUARE OPENING (C) CHISEL


A) CONICAL HOPPER


F AMID.
-i i -RE OPENING


BQ
(B) CONICAL









The flow pattern is strongly affected by the distribution of densities and orientation

of particles which are determined by the mechanical characteristics of the solid and the

Aspect of silo behavior Loss of Function


Solid properties
ARCHING/RATHOL

Filling Methods


'' SEGREGATION
Flow pattern



Pressure on silo walls



Stresses in silo structure



Failure conditions for the structure -- COLLAPSE


Figure 2.4: Problems in silos

way it is filled. These characteristics also determine whether arching or ratholing occurs

at the outlet. The flow pattern during discharge also determines if segregation occurs and

influences the pressure on the walls. Hence in order to predict the non-uniform and

unsymmetrical wall pressure it is essential that the flow pattern be predicted. This

influences directly the wall stress conditions that could induce failure and collapse of the

structure.

In regards to bulk solids handling in storage facilities, 3 phases can be

distinguished: (1)filling, (2)storage and (3)discharge. In the filling stage, with the outlet

closed, the material is supplied slowly at the top of the container. As more and more









material is poured in, stresses, due to the weight of layers, start developing throughout

the container. In the storage stage no material is flowing in or out of the container. For

cohesive (fine) powders this may lead to an increase of interparticle bonding. Due to its

own weight the stored material compacts and its strength increases with time. Also, the

slow escape of the air in pores results in stronger surface contact and an increase in

frictional forces. In the third and final stage the material is discharged through an opening

at the bottom of the container. Upon opening the outlet, a dilation zone neighboring the

outlet is formed. This zone will spread to the upper layers of the material and cause it to

flow. In hoppers with steep and smooth sides, this zone of dilated material will cover the

whole cross-sectional area of the container resulting in mass-flow, while for rough walls

the dilation zone is confined only to the center, resulting in funnel flow.

The state of stress throughout the container, in the discharge stage, drastically

differs from the stress state in the filling/storage phases. The corresponding stress states

are called active and passive, respectively. In the filling and storage stages the direction

of the major principal stress c0I is vertical with slight curvature close to the walls. In the

discharge stage the principal stress is horizontal with slight curvature at the walls (Figure

2.5).
Active Passive










a) Filling/storage b) discharge


Figure 2.5: Trajectories of major principle stress









The Diamondback HopperTM

The Diamondback HopperTM consists of a round-to-oval hopper with diverging end

walls. This hopper section is positioned above an oval-to-round hopper that necks down

to a circular outlet. Pressure sensitive pads (from TekScanTM) were used to measure the

normal wall pressure (Johanson, 2001). These pads consist of two plastic sheets with

conductive pressure sensitive material printed in rows on one sheet of plastic and

columns on the other sheet of plastic. When these sheets contact each other they form

conductive junctions where contact resistance varies with the normal stress applied. The

effective area of each junction is 1 cm2 and there are over 2000 independent normal stress

measurements possible that can be recorded at cycle times up to eight per second. The

voltage drops across these junctions were sequentially measured and scaled to give real

engineering force units using the data acquisition software. The TekScanTM pad was

glued to the inside surface of the round-to-oval hopper with one edge of the pad along the

center of the flat plate section (see Figure 2.6).












Pad 61 cm




Pad locations
2064 sensors


Figure 2.6: Schematic of the Diamondback HopperTM showing position of pad











Discharge Aids

Poor design or incorrect use of a hopper leads to the use of devices that reduce flow

problems. The use of the wrong device, however, may have the reverse effects and create

more problems than it solves. These devices are used if it proves that there are constraints

in the overall system that prevent the unaided gravity flow of the material. These aids

developed from practices such as beating the hopper with a blunt instrument and poking

the material with some sort of rod. The three major types of aids are (1) pneumatic-

relying on the application of air to the product; (2) vibrational-relying on mechanical

vibration of the hopper or the material; (3) mechanical-physically extracting the product

from the hopper.

In the pneumatic aids, aeration devices are used to introduce air at the time that the

material is discharged so as to "fluidize" the material in the region of the outlet opening

and to reduce the friction between the material and the hopper wall and the second is to

introduce a "trickle of flow" of air during the whole period that the product is stored to

prevent the gain of strength in the material. Air is also introduced into inflatable pads that

act mechanically against the stored material. When using vibrational methods, devices

could be used to vibrate the hopper or bin walls or to vibrate the material directly.

Vibration should not be applied when the outlet is closed, as this could result in the

strengthening of any arch formation. In mechanical methods powered dis-lodgers such as

vertical or horizontal stirrers are used to manually move the material .









Janssen or Slice Model Analysis

Silo -Vertical Section

Janssen (1895) developed a method of predicting vertical silo wall pressures over a

century ago that is still widely used in the industry although most of the assumptions he

used in the derivations have been shown to be incorrect.

The simplest slice for the analysis is the planar finite boundary slice that has sides

that are perpendicular to the walls of the bin. The forces acting on an element slice of

stored powder, of thickness dz, at a depth h from the top surface in a deep bin with an

overall height h, cross-sectional area A and circumference C are shown in figure 2.7. Let

the vertical stress at the upper surface, depth h, be ov and at the lower surface, depth

h+dz, be ov+dov. ow and Tw are the normal and tangential (shear) stresses at the wall due

to friction between the walls and the bulk material. The weight of the powder in the slice

is Adzpg where g is the acceleration due to gravity and p is the material bulk density

which is assumed to remain constant over the entire depth of the powder. Equating the

forces in the vertical direction we get


UA + Adzpg A(a, + do,) + r,Cdz = 0 (2.1)

It is assumed that the ratio of horizontal to vertical pressure is constant everywhere

in the element:


U = K (2.2)
0-v

For fully mobilized friction we have



/=tan W(2.3)













OyA h



rCdz ___ dz


(oC+dov)A

Adzpg




Figure 2.7: Forces acting on differential slice of fill in bin

Expanding equation 2.1 and simplifying we get the first order differential equation:

Separating variables and integrating with the boundary condition ov=0 at h=0 (at the level

of the free surface), and ov=ov at h=h gives

gA kPC h
ua =KC e )A (2.4)


and


pgA k h]
a = Ka 1- e (2.5)


For large depth of fill the maximum values of vertical and normal stresses are


oa (max) =pgA (2.6)
puKC

and


a, (max) =gA (2.7)
tiC









while near the top surface:

- (top) pgh (2.8)

The ratio A/C varies for different silo geometries (e.g., rectangular, circular).

As mentioned above recent studies have challenged the accuracy of Janssen's

assumptions of constant K values, angle of wall friction and bulk density. The value of K

has a considerable influence on the stress distribution in the material and on the wall.

Researchers have differing views of the K value. Jenike (1973), for example, assumed

K=0.4 for most granular materials while others claim it is a function of internal angle of

friction.

Converging Hopper

The above analysis is applicable to deep silos and flat bottom bins ignoring any end

effects. For the hopper section a linear hydrostatic pressure gradient is proposed


9, = pg[ o +h] (2.9)
P9

where avo is the vertical pressure at the top of the hopper calculated from equation 2.5 as

shown in figure 2.8.


\

zcyw I







Figure 2.8: Forces acting in converging hopper

Walker (1966) assumed a constant value of K equal to:










K = tan (2.10)
(tan p, + tan a)

While Jenike (1961) assumed a constant value of:

sin (2a cos) (2.11)
[sin (, + 2a) + sin 9]

Diamondback HopperT

A force balance can be done on a small differential slice of material in the

Diamondback HopperTM as shown in Figure 2.9 (Johanson and Bucklin, 2004). The

forces acting on the material slice element are due to the bulk material slice weight, wall

frictional forces on the flat plate section and the round end walls, vertical stresses, and

normal stress conditions at the bin wall. The definition of K-value and the columbic

friction condition are used to relate the vertical pressure acting on the material to the

stresses normal and tangent to the bin wall (see Equations 2.12 through 2.15). The

resulting differential equation is found in Equation 2.16. The cross-sectional area (A)

and hopper dimensions (L) and (D) are functions of the axial coordinate as indicated in

Equations 2.16 through 2.21.


L = KL .-v (2.12)

rL = KL .a,tan(w) (2.13)

a = K, (a) c, (2.14)

z, = Ke (a) a, tan(O) (2.15)













dz Lr

-I-^It'^


a


Figure 2.9: Stresses acting on differential slice element in diamondback hopperTM

1 dA 2-K, -(L-D))
dA + 2.K (L D)(tan(O) + tan(D))+
di- A dz A
=-Yg- 4.D (2.16)
A K, (a) (tan() + tan(O(a))) da
0

A=-D +(L-D)-D (2.17)


d= -2F [ -D+L Ltan(OD)-2-D-tan(OL) (2.18)

L L -2-.z-tan(OL) (2.19)

D= D -2- z tan(OD) (2.20)

K, (a) = (K1m, Klmax) (sin(a))" +K1ma (2.21)

Where values of parameters Kimin, Kimax, and n were chosen to minimize the deviation

between the observed wall stress profiles and the calculated wall stresses over the entire

axial hopper depth.










The hopper wall slope angle changes around the perimeter. The slope angle along the flat

plate section equals the hopper angle (OD) while the slope angle along the hopper end

wall equals the hopper angle (OL). The variation in the hopper angle, around the

perimeter, is given by Equation 2.23, where. qa is the hopper section angle as defined in

Figure 2.9.

0(a) = a tan[( sin(a)). tan(OD) + sin(a) tan(OL)] (2.22)

Figure 2.10 shows Janssen computed wall stresses compared with the actual TekscanTM

measurements for the section of the hopper shown (Johanson, 2003).


7






-4
Hopper Section Ange, (deg)













-Computed Janssen Stress A Fully Developed Flow
Figure 2.10: Flow wall stresses at an axial position Z=3.8 cm below the hopper transition
1 :A


0 10 20 30 40 50 60 70 80 90
Hopper Section Angle, a (deg)
-Computed Janssen Stress A Fully Developed Flow
Figure 2. 10: Flow wall stresses at an axial position Z=3.8 cm below the hopper transition

The Janssen slice model can provide a first approximation to the loads in

diamondback hoppers. However, there is significant variation from the simple slice

model approach used here. More complex constitutive models will be required to

increase the agreement between experimental and theoretical approaches.









Direct Shear Testers-Jenike Test

For the past 30 years the assessment of flow properties of bulk solids has been done

through shear testing (Jenike, 1961). The bulk solid is assumed to obey a rigid-

hardening/softening plastic behavior of Mohr-Coulomb type (Figure 2.10) for coarse

cohesion-less, particles (granular particles), the yield locus is approximated by:

cT=atan4 (2.23)

while for fine powders, the yield locus is

c=atan4 + c (2.24)














a) Yield locus of a b) Yield locus of a c) Yield locus of a free-
cohesive Coulomb cohesive non-
solid Coulomb solid

Figure 2.10: Yield loci of different bulk solids

As shown in figure 2.11, the cell consists of a base, an upper ring and a cover. The cover

has a fixture that can hold a normal force N. The motor causes the loading stem to push

against the shear ring and displace it horizontally. The shear force is measured and is

divided by the cross-sectional area of the ring to provide the shear stress(c).


















SHEAR FORCE

X HEAR FORCE




POWDER SAMPLE
Figure 2.11: Jenike shear test

The test begins when the sample of bulk solid is poured into the shear cell and is

pre-consolidated manually by placing a special top on the cell, applying a normal load

and twisting the top to consolidate the material. Then a normal stress on is applied and

the specimen is sheared. The shear force continuously increases with time until constant

shear is obtained. Corresponding to the change in shear is a change in bulk density. After

a period of time, the bulk density reaches a constant value for a value of on. At that point,

the deformation is known as steady state deformation. The sample is first pre-sheared

under a constant on and T. The shearing is stopped when steady-state deformation is

reached. Then the sample is sheared under the normal stress on that is smaller than the

shear stress.

To insure the same initial bulk density the sample is then pre-sheared under the

same normal stress on and sheared at a lower on than the first test. This process is

repeated several times.









The yield locus, usually called the effective yield locus is used to determine the

flow parameters. As seen from figure 2.12 it is a straight line passing through the origin

and tangential to the Mohrs circle at steady state deformation (the largest circle). This

line has the inclination 4e or the effective angle of internal friction (angle with the o-

axis). At lower normal stresses the yield loci is curved instead of straight.


Linearized Yield locus


Yield locus .,* .





a e


Figure 2.12: Yield loci

Therefore a linearized yield locus is used to define the internal friction angle (4i).

Using a different on for pre-shearing the sample will result in different consolidation and

bulk density values. This will produce several yield loci and several Mohr circles as seen

in Figure 2.13










FC
Fc 21 Y

Figure 2.13: Yield loci









Jenike Wall Friction Test

The Jenike tester (1961) can also be utilized to determine the wall friction angle.

The test is modified so that the base of the tester is replaced by a flat plate of the wall

material that the hopper is made of (Figure 2.14).


APPLIED WEIGHT
'.NORMAL FORCE)


SHEAR FORCE


POWDER SAMPLE


Figure. 2.14: Jenike wall friction test

The procedure is similar to the description above in the sample preparation stage. A

maximum normal force is applied at the top cover and is decreased in a series of steps

while the maximum shearing force is being measured. As explained above the wall yield

locus can be plotted with c vs o and the coulomb equation is:


= arctan (u) (2.25)


The coefficient of friction is expressed as the angle of wall friction given by:

T = Pu(


(2.26)


BRACKET-


SHEAR FORCE









Schulze Test

The Jenike test is limited to a small displacement of about 6mm and is suitable for

fine particles only. Rotational shear testers such as the annular Schulze (1994) tester

(figure 2.15) have unlimited strains. The powder is contained in the cell and loaded from

F, a






















F2
Figure 2.15: Schulze rotational tester

the top with a normal force N through the lid. After calibrating and filling the cell with

powder a predetermined consolidation weight was added to a hanger, which is connected

to the top of the cell. The machine was then turned on and allowed to run until it reaches

a steady state. The normal weight was adjusted to about 70% of the steady state value and

the sample was sheared. During the test the shear cell rotates slowly in the direction of

o while the cover is prevented from rotation by two tie rods. This causes the powder to

shear. The forces Fi and F2 and the normal forces are recorded. This procedure was









repeated several times with decreasing weight. Between each test the sample was pre-

consolidated to the steady state value. The data reduction procedure is same as the Jenike

test and gives the yield loci and the internal angle of friction.


Indirect Shear Tester-Triaxial Test

To investigate the stress-strain behavior of powders, a triaxial testing apparatus is

used. Its capabilities are wider than those of uniaxial testers, and it can be described as an

axially symmetric device having 2 degrees of freedom. A schematic diagram and picture

of the triaxial apparatus are shown in Figure 2.16 and 2.17.




Water

Rubber membrane
Volume Change Device





Porous stone *


Figure 2.16: Experimental set-up

In an ideal triaxial test, all three major principal stresses would be independently

controlled. However the independent control would lead to mechanical difficulties that

limit the conduction of such tests to special applications. The commonly used triaxial test

refers to the axisymmetric compression test. This test has been used since the beginning









of the 19th century in testing the strength of soils. The soil sample is pre-compacted,

saturated with water, and compressed both hydrostatically and deviatorically and the

water discharged from the sample is used to measure the volume change and hence the

density change. In recent years, the test has been used to test other bulk solids: grains and

fine powders. The testing procedure was modified so as to allow testing of dry powders

at low stresses. The behavior in this regime is essential for the applications of interest.


Figure 2.17: Triaxial chamber on an axial loading device

A latex membrane is stretched out into a cylindrical shape using a 2-piece

cylindrical mold and vacuum pressure. The powder is poured into the membrane

dispersed and compressed using a predetermined stress. Figure 2.18 shows the steps

followed in the filling procedure. The value of AH was determined using Janssen's

equation. To insure repeatable initial conditions, both the mass and volume of the






material have to be measured. The membrane is sealed by rubber O-rings to a pedestal at
the bottom, and to a cap at the top. Vacuum grease is used the membrane and the pistons
so as to reduce the chances of leaks. Any test consists of two phases: (1) a hydrostatic
phase followed by (2) a deviatoric phase.
In the hydrostatic test the assembly is placed in a chamber that is filled with
water. The water is pressurized to the desired confining pressure. This hydrostatic
pressure is increased up to a desired value ca using a pressurizing device. The sample
can support itself due to the filling procedure and thus the mould is removed. Under the
applied hydrostatic pressure the powder, compacts and the specimen takes the form of a
cylinder. The change in volume of the specimen is recorded using a data acquisition
system that measures the volume of the water displaced. The initial volume of the sample
is determined by multiplying the height by the arithmetic mean of the diameters at 3
locations in the vicinity of the middle of the sample.


Figure 2.18: Filling procedure
In the deviatoric phase, the assembly is placed on an axial loading device. An
additional axial stress is applied by means of a piston passing through a frictionless


I~









bushing at the top of the chamber. The test is carried out under strain control conditions

in which a predetermined rate of axial deformation is imposed and the axial load required

to maintain the rate of deformation is measured via a load cell. As the sample deforms,

water is displaced from the chamber. The quantities measured during the deviatoric phase

are: the confining pressure 03 (which provides an all-around pressure on the lateral

surface of the sample), the axial force applied to the piston, the change in length of the

sample and the change in the volume of the sample which is determined by the change in

the volume of water existing in the chamber. Thus, in the test, the change in volume of

the sample we can be monitored continuously as it is sheared. These tests will produce

the following stress-strain curves: the axial strain E, versus the deviatoric stress a, Ca

(o, is the axial stress) and the volumetric strain Es versus the deviatoric stress a7 1 3,

respectively. Here and throughout the text compressive stresses and strains are considered

to be positive.


The axial strain is defined as: ei=- where 41 is the initial length of the sample
10


and / is the current length ; the volumetric strain is E, = Vo Vo and V being the
0V

initial and current volume, respectively. Any change in the volumetric strain reflects

change of the relative positions of the powder particles. Failure corresponds to an

observed instability in the specimen when one part slides with respect to the other along a

plane inclined at about 450 with respect to the vertical axis of the specimen. Failure

shows up in the stress strain curve as a maximum deviatoric pressure.












300

4*
S250 p




150 elv(test 3)


100 *


50 Powder Test Date u3(kPa) pl(g/cm^3)
Poly#1 3 12/10 4056 08594


-55000 -35000 -15000 5000 25000 45000 65000 85000 105000
Srain(E-6)

Figure 2.19 Stress-strain curves

Figure 2.19 shows the stress-strain curves obtained in a typical test (Saada, 1999).

This is a test carried out on a powder at a confining pressure of 40.56kPa. The axial strain


curve shows that the powder undergoes elastic deformation up to about 50% of its

strength, followed by work hardening and failure. The stress-volumetric strain curve


shows two distinct regimes of behavior: first, the powder compacts (i.e. Sv increases) very


slightly, then starts to expands (i.e. S; decreases). The volume increase, which is observed


on the last portion of the curve (o, c3) Sv is called dilatancy. The powder expands


because of the interlocking of the particles: deformation can proceed only if some

particles are able to ride up or rotate over other particles. This phenomena can

simplistically be visualized with the aid of Figure 2.20, which shows two layers of discs,


one on top of the other. If a shear stress is applied to the upper layer, then each disc in

this layer has to rise (increasing the volume occupied by the particulate material) for the


sample to undergo any shear deformation.
















Figure 2.20: Layers of discs dilating as they are sheared

The Diamondback HopperTM Measurements

The core of the TekscanTM pressure measurement system (1987) consists of an

extremely thin 0.004 in (0.1 mm), flexible tactile force sensor. Sensors come in both grid-

based and single load cell configurations, and are available in a wide range of shapes,

sizes and spatial resolutions (sensor spacing). These sensors are capable of measuring

pressures ranging from 0-15 kPa to 0-175 MPa. Each application requires an optimal

match between the dimensional characteristics of the objects) to be tested and the spatial

resolution and pressure range provided by Tekscan's sensor technology. Sensing locations

within a matrix can be as small as .0009 square inches (.140 mm2); therefore, a one

square centimeter area can contain an array of 170 of these locations. Teksan's Virtual

System Architecture (VSA) allows the user to integrate several sensors into a uniform

whole.

The standard sensor consists of two thin, flexible polyester sheets which have

electrically conductive electrodes deposited in varying patterns as seen in Figure 2.21

(Tekscan Documentation, 1987). In a simplified example below, the inside surface of one

sheet forms a row pattern while the inner surface of the other employs a column pattern.

The spacing between the rows and columns varies according to sensor application and

can be as small as -0.5 mm.































Figure 2.21: TekScanTM Pads

Before assembly, a patented, thin semi-conductive coating (ink) is applied as an

intermediate layer between the electrical contacts (rows and columns). This ink, unique to

Tekscan sensors, provides the electrical resistance change at each of the intersecting

points. When the two polyester sheets are placed on top of each other, a grid pattern is

formed, creating a sensing location at each intersection. By measuring the changes in

current flow at each intersection point, the applied force distribution pattern can be

measured and displayed on the computer screen. With the TekscanTM system, force

measurements can be made either statically or dynamically and the information can be

seen as graphically informative 2-D or 3-D displays.

In use, the sensor is installed between two mating surfaces. Tekscan's matrix-based

systems provide an array of force sensitive cells that enable measurement of the pressure

distribution between the two surfaces (figure 2.22). The 2-D and 3-D displays show the

location and magnitude of the forces exerted on the surface of the sensor at each sensing











location. Force and pressure changes can be observed, measured, recorded, and analyzed


throughout the test, providing a powerful engineering tool.


Overall Width (W)
-I Matrix Width (MW)


* Tab Length (A)

U ____


Matrix Height .IH,


SRow Width (rw)
Column Width (cw)

Su- Row Spacing (rs)



Column Spacing (cs)
Sensell
Exploded View


Figure 2.22: TekScan Pads

Table 2.3: Pad specifications
General Dimensions Sensing Region Dimensions Summary
Overall Overall Tab Matrix Matrix
Length Width Length Width Height Columns Rows # of Sensels
L W A MW MH CW CS Qty RW RS Qty Sensels Density
(mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (per cm2)
622 530 130 488 427 6.35 10.2 48 6.35 10. 2 42 2016 0.97



Several pleats or folds were made in the upper portion of the pad to make it


conform to the upper cylinder wall (figure 2.23). A thin retaining ring was placed at the


transition between the hopper and cylinder. This held the pad close to the hopper wall


Overall Length (L)









Pad locations
2064 sensors
















Figure 2.23: Location of TekscanTM pads in the diamondback hopper

and helped maintain the pleats in the cylinder. Contact paper was placed on the hopper

and cylinder surface to protect the pad and produce a consistent wall friction angle (of

about 17 degrees) in the bin. The pad lead was fed through the hopper wall well above

the transition and connected to the data acquisition system. Spot calibration checks were

made on groups of four load sensors. This arrangement produced a load measurement

system capable of measuring normal wall loads to within +10 %.

The bin level was maintained by a choke fed standpipe located at the bin centerline

and at an axial position about 60 cm above the hopper transition. The material used was

fine 50 micron silica. During normal operation, the feed system maintains a repose angle

at the bottom of the standpipe and produces a relatively constant level as material

discharges from the hopper. Flow from the hopper was controlled by means of a belt

feeder below the Diamondback HopperTM. There was a gate valve between the

Diamondback HopperTM and the belt. Initially, this gate was closed and the lower hopper

was charged with a small quantity of material. The amount of material in the hopper









initially was enough to fill only the lower oval-to-round section of the bin. The gate was

then opened, allowing material to fill the void region between the gate and the belt. This

was done to avoid surges onto the belt during the initial filling routine. The remaining

hopper was then filled slowly, using the existing conveying system until the

diamondback test hopperTM and a surge bin above this test hopper were filled. Material

level was maintained in the surge hopper by recycling any material leaving the belt into

the surge hopper feeding the Diamondback HopperTM bin. Once the bin was full,

material was allowed to stand at rest for about 10 minutes as material deaerated. The

speed of the belt was preset and flow was initiated. The wall load data acquisition system

recorded changes in wall stress as flow was initiated. The initial stress condition shows a

large peak pressure concentrating at the top point of the triangular flat plate in the hopper.

This peak is focused at the tip of the plate. During flow these loads decrease and de-

localize producing the highest peak stresses at along the edge of the plate.















CHAPTER 3
FINITE ELEMENT

Background in Finite Element Modeling

Great effort has been made in recent decades to understand flow phenomena of

bulk solids. Since the 1970's, a large number of research teams have worked on the

application of finite element analysis to hopper problems. Models and programs were

hampered due to limited capacity of computers and the high cost of equipment.

Nowadays, it is possible to use computers whose capacity and speed is continually

increasing and a large number of programs exist that manipulate, analyze and present the

results.

In principle there are no restrictions with regards to the hopper geometry and to the

bulk material in FEM analysis. However studies have to account for complex geometry

of the container, complex material behavior and interaction with the wall. Hopper

geometry is not always axisymmetric and often has shapes that require three-dimensional

numerical modeling, as is the case with the diamondback hopper. Bulk solids exhibit

complex mechanical behavior such as anisotropy, plasticity, dilatancy and so on. Most of

these properties are developed during discharge due to the large strains that are

experienced. Some properties, such as anisotropy are developed during the filling

process. It is very essential that detailed mathematical models, accompanied with a

proper experimental procedure, are used to describe these features. The appropriate

simulation has to be used to model the interaction between the solid and the wall. For the









case of smooth wall, a constant friction coefficient can be satisfactory while for rough

walls it might be required to use more sophisticated modeling.

In a book edited by C.J. Brown and J. Nielsen (1998) several FEM research on silo

flow are presented. This book is a compilation of reports brought about by a research

project on silos called the CA-Silo Project. It was funded by the European commission

and completed in 1997. The first chairman of the group, G. Rombach (1998), assembled

data for comparison of existing programs. Table 3.1 lists several projects regarding

granular solid behavior simulation. It includes both finite element and discrete element

simulations.

Table 3.1: Numerical simulation research projects
Name of Static/ 2D/3D Cohesive/
Author/user program Dyna Non
mic Cohesive
1. Aubry (Ecole Central Paris) GEFDYN S/D 2D/3D C/N
2. Eibl (Karlsruhe U, Germany) SILO S/D 2D/3D C/N
3. Eibl (Karlsruhe U, Germany) S/D 2D/3D C/N
4. Eibl (Karlsruhe U, Germany) ABAQUS S/D 2D/3D C/N
5. Klisinski (Lulea U, Sweden) SILO S/D 2D N
6. Klisinski (Lulea U, Sweden) BULKFEM S/D 2D N
7. Martinez (INSA, Rennes, France) AMG/PLAXIS S/D 2D N
8. Martinez (INSA, Rennes, France) MODACSIL S 2D/3D C/N
9. Schwedes (Braunschweig U, Germany) ABAQUS S/D 2D/3D C/N
10. Schwedes (Braunschweig U, Germany) HAUFWERK S 2D N
11. Tuzun (University of Surrey, UK) HOPFLO S/D 2D/3D N
12. Ooi, Carter (Edinburgh U, Scotland) AFENA S 2D C/N
13. Ooi (Edinburgh U, Scotland) ABAQUS S/D 2D/3D C/N
14. Rong (Edinburgh U, Scotland) DEM.F S/D 2D N
15. Thompson (Edinburgh U, Scotland) SIMULEX S/D 2D N
16. Cundall (INSA, Rennes, France) TRUBALL,PFC S/D 2D/3D N



Also listed in the table are: the name of the program used, whether it analyzed static

or dynamic material, the geometry used (2D or 3D) and whether it analyzed cohesive or

non-cohesive material. The material models used include the well-known elastic-plastic

models and yield criteria such as Mohr-Coulomb, Tresca and Drucker-Prager. Other law









include: Lade (1977), Kolymbas (1998), Boyce (1980), Wilde (1979) and critical state.

These laws have been modified for cohesive bulk solids and dynamic calculations.

Aside from this project there has not been extensive work published on finite

element modeling flow of cohesive material in hoppers. Haussler and Eibl (1984)

developed a model using 18 governing equations: 3 for dynamic equilibrium, 6

constitutive relations and 9 kinematic relations to describe flow behavior. They used

FEM, with triangular elements, to determine the spatial response of sand flow from a

hopper. They concluded that the velocity in the bin section was constant, while in the

hopper section it was maximum at the center line. Schmidt and Wu (1989) used a similar

FEM procedure except that they used a modified version of the viscosity coefficient and

used rectangular elements. Runesson and Nilsson (1986) used the same dynamic

equations and modeled the flow of granular material as a viscous-plastic fluid with a

Newtonian part and a plastic part. The Drucker-Prager (1952) yield criteria was used to

represent purely frictional flow.

Link and Elwi (1990) used an elastic-perfectly plastic model with wall interface

elements to describe incipient flow of cohesion-less material. Several steps were used to

simulate the filling process in layers while incipient release as opposed to full release was

simulated for discharge. Flow was detected by a sudden increase in displacements at the

outlet followed by failure of the FEM to converge to a solution. The material was

simulated using eight-node isoparametric elements while the interface used six-node

isoperimetric elements. The wall pressure results were close to the results obtained by

Jenike while the maximum outlet pressure was lower than the Jenike analysis. The

results, however were not verified by experimental procedures.









A FEM analysis, with a new secant constitutive relationship, was used, by Meng

and Jofriet (1992), to simulate cohessionless granular material (soy bean) flow. The outlet

velocities were comparable with preliminary experimental results. The solution

converged quickly and was stable throughout the simulation. Another model, developed

by Diez and Godoy (1992), used viscoplastic behavior and incompressible flow to

describe cohesive materials. This model used the Drucker-Prager (1952) yield theory and

was applied to conical and wedge shaped hoppers. Results obtained compared well with

other published work for the conical hopper for cohessionless materials except at the

bottom of the hopper.

Kamath and Puri (1999) used the modified Cam-clay equations in a FEM code to

predict incipient flow behavior of wheat flour in a mass flow hopper bin. Incipient flow

was characterized by the characterization of the first dynamic arch. This arch represented

the transition from the static to the dynamic state during discharge. Incipient flow is

assumed to occur when the mesh at the outlet of the hopper displays 7% or more axial

strain. The bin was discretized using rectangular 4-node elements while the hopper is

discritized using quadrilateral elements. A thin wall interface element was the for the

powder wall interaction. The FEM results were verified experimentally using a

transparent plastic laboratory size mass flow hopper bin. The FEM results were within

95% of the measured values.

FEM using ABAQUS on Diamondback

The numerical model is based on a consistent continuum mechanics approach. In

this project the commercially available software ABAQUS (Inc. 2003) is being used. It is

a general-purpose analysis product that can solve a wide range of linear and nonlinear

problems involving the static, dynamic, thermal, and electrical response of components.










It has several built in models to describe material response. Such materials include:

metals, plastics, concrete, sand and other frictional material. It also allows for a user

subroutine to be written that allows any material to be implemented. Some of these

plastic models are listed in figure 3.1 (ABAQUS 2003). The figure also shows the steps

involved in an analysis.

The geometry and meshing and material properties are created in the preprocessor

(ABAQUS/CAE), which creates an input file that is submitted to the processor

(ABAQUS/Standard) or (ABAQUS/Explicit). The results of the simulation can be

viewed as a data file or as visualization in ABAQUS/Viewer.

Plasticity
Preprocessing User Materials
ABAQUS/CAE or other software
User Subroutines allows any n
Input file: model to be implemented
Input file:
job.inp Creep

Volumetric Swelling
Simulation Two-layer viscoplasticity
ABAQUS/Standard
Or ABAQUS/Explicit Extended Drucker-Prager
Capped Drucker-Prager

Output files: cam-(
job.obd, job.dat Mohr-Coulomb
job.res, job.fil
res, oCrushable Foam


Postprocessing Strain-rate-dependent Plasticity
ABAQUS/CAE or other software

Figure 3.1: Flowchart showing steps followed to complete an analysis in ABAQUS

Explicit Time Integration

Exact solutions of problems where finding stresses and deformations on bodies

subjected to loading, requires that both force and moment equilibrium be maintained at

all times over any arbitrary volume of the body. The finite element method is based on









approximating this equilibrium requirement by replacing it with a weaker requirement,

that equilibrium must be maintained in an average sense over a finite number of divisions

of the volume of the body.

Starting from the force equilibrium equation for the volume:

Stds + fdV = 0 (3.1)
S V

where: V is the volume occupied by a part of the body in the current configuration,
S is the surface bounding this volume.
t is the force per unit of current
f is the body force per unit volume
and t = n : (T where c is the Cauchy stress matrix and n is the unit outward normal.

From the principles of continuum mechanics the deferential form of the equilibrium

equation of motion is derived as:

V .07 + f = 0 (3.2)

The displacement-interpolation finite element model is developed by writing the

equilibrium equations in "weak form". This produces the virtual work equation in the

classical form:


fu :SDdV = fSv-tdS+ fv- fdV (3.3)
V S V

where:

9D is virtual strain rate (virtual rate of deformation)
9v is a virtual velocity field

This equation tells us that the rate of work done by the external forces subjected to any

virtual velocity field is equal to the rate of work done by the equilibrating stresses on the

rate of deformation of the same virtual velocity field. The principle of virtual work is the









"weak form" of the equilibrium equations and is used as the basic equilibrium statement

for the finite element formulation.

ABAQUS can solve problems explicitly or implicitly (using ABAQUS/Standard)

For both the explicit and the implicit time integration procedures, the equilibrium is

M ii = P I (34)

where: P are the external applied forces
I are the internal element forces
M is the mass matrix
ii are the nodal accelerations

Both procedures solve for nodal accelerations and use the same element calculations to

determine the internal element forces. In the implicit procedure a set of linear equations is

solved by a direct solution method to obtain the nodal accelerations. This method

however becomes expensive and time consuming compared to using the explicit method.

The implicit scheme uses the full Newton's iterative solution method to satisfy dynamic

equilibrium at the end of the increment at time t + At and compute displacements at the

same time. The time increment, At, is relatively large compared to that used in the

explicit method because the implicit scheme is unconditionally stable. For a nonlinear

problem each increment typically requires several iterations to obtain a solution within

the prescribed tolerances. The iterations continue until several quantities-force residual,

displacement correction, etc.-are within the prescribed tolerances.

Since the simulation of flow in a diamondback hopper contains highly

discontinuous processes, such as contact and frictional sliding, quadratic convergence

may be lost and a large number of iterations may be required. Explicit schemes are often

very efficient in solving certain classes of problems that are essentially static and involve

complex contact such as forging, rolling, and flow. Flow problems are characterized by









large distortions, and contact interaction with the hopper wall. It also has the advantage of

requiring much less disk space and memory than implicit simulation. To get accurate

solutions for a bulk solid in a hopper a relatively dense mesh with thousands of elements

needs to be used. And since the explicit method shows great cost savings over the

implicit method it is used in this research.

Hence using an explicit scheme where the solution is determined without iterating

by explicitly advancing the kinematic state from the previous increment is preferred.

A central difference rule, to integrate the equations of motion explicitly through time,

using the kinematic conditions at one increment to calculate the kinematic conditions at

the next increment, is used.

The accelerations at the beginning of the current increment (time t) are calculated

as:


t) =M (P I) (3.5)


Since the explicit procedure always uses a diagonal, or lumped, mass matrix,

solving for the accelerations is trivial; there are no simultaneous equations to solve. The

acceleration of any node is determined completely by its mass and the net force acting on

it, making the nodal calculations very inexpensive.

The accelerations are integrated through time using the central difference rule,

which calculates the change in velocity assuming that the acceleration is constant. This

change in velocity is added to the velocity from the middle of the previous increment to

determine the velocities at the middle of the current increment:









Atl + A1l))
t+ At ( I t) (3.6)

The velocities are integrated through time and added to the displacements at the

beginning of the increment to determine the displacements at the end of the increment:

u i .\ = u ( \ + A-t u, st 1 A/
U (t+At) UI(t) + A (t+At) t+- (3.7)

Thus, satisfying dynamic equilibrium at the beginning of the increment provides the

accelerations. Knowing the accelerations, the velocities and displacements are advanced

"explicitly" through time. The term "explicit" refers to the fact that the state at the end of

the increment is based solely on the displacements, velocities, and accelerations at the

beginning of the increment. This method integrates constant accelerations exactly. For

the method to produce accurate results, the time increments must be quite small so that

the accelerations are nearly constant during an increment. Since the time increments must

be small, analyses typically require many thousands of increments. Fortunately, each

increment is inexpensive because there are no simultaneous equations to solve. Most of

the computational expense lies in the element calculations to determine the internal forces

of the elements acting on the nodes. The element calculations include determining

element strains and applying material constitutive relationships (the element stiffness) to

determine element stresses and, consequently, internal forces.

Here is a summary of the explicit dynamics algorithm:











4


Integrate explicitly through time.
(AtI(A) + AtI()

uI(i = u 2 t u --

u(t+At) u (t) + A (t+At) t+-


Element calculations.
Compute element strain increments, de, from the strain rate, E
Compute stresses, a, from constitutive equations
A (tet)= f( o(rcs)
Assemble nodal forces, (tAt)


Set t + At to t


Figure 3.2: Summary of the explicit dynamics algorithm

Contact conditions and other extremely discontinuous events are readily formulated

in the explicit method and can be enforced on a node-by-node basis without iteration. The

nodal accelerations can be adjusted to balance the external and internal forces during

contact. The most striking feature of the explicit method is the lack of a global tangent

stiffness matrix, which is required with implicit methods. Since the state of the model is

advanced explicitly, iterations and tolerances are not required.


Nodal calculations.
Dynamic equilibrium.
id )= M 1 (P I )









Geometry, Meshing, and Loading

The advantage of FEM is that a variety of problems such as the hopper geometry,

the size and locations of the openings and the interaction between the material and the

walls can be investigated. The diamondback hopper geometry is relatively complicated

relative to other hopper geometries. Because of this complexity generating the mesh with

the appropriate density and correct elements is not a trivial task.

The steel wall is simulated as a rigid body as shown in figure 3.3 (from different

view points). Since this research focused on the behavior of the bulk material in the

hopper, no finite element analysis was conducted on the wall. Because of the complexity

of the geometry however, it was required to discritize the wall. It was meshed into 6872

triangular elements (R3D3) as seen in figure 3.3. The mesh had to be of particular

density so as to simulate contact analysis correctly.


Figure 3.3: Geometry and meshing of hopper wall









It was decided to simulate the process of filling the bulk solid in the hopper in

several steps. The solid was simulated as a deformable body and divided into 11 different

sets. A uniform load is applied to each set to simulate gravity (9.81m/s2) in 11 different

steps (as seen in figure 3.4). Discharge was simulated in a 12th step by applying a small

displacement to the bottom surface of the powder. By default, all previously defined

loads are propagated to the current step. The starting condition for each general step is the

ending condition of the previous general step. Thus, the model's response evolves during

a sequence of general steps in a simulation.




















Figure 3.4: Filling procedure for powder

It was decided to use second order elements as opposed to first order elements. This

is because of the standard first-order elements are essentially constant strain elements and

the solutions they give are generally not accurate and, thus, of little value. The second-

order elements are capable of representing all possible linear strain fields and much

higher solution accuracy per degree of freedom is usually available with the higher-order

elements. However, in plasticity problems discontinuities occur in the solution if the









finite element solution is to exhibit accuracy, these discontinuities in the gradient field of

the solution should be reasonably well modeled. With a fixed mesh that does not use

special elements that admit discontinuities in their formulation, this suggests that the

first-order elements are likely to be the most successful, because, for a given number of

nodes, they provide the most locations at which some component of the gradient of the

solution can be discontinuous (the element edges). Therefore first-order elements tend to

be preferred for plastic cases. Tetrahedral elements are geometrically versatile and are

used in many automatic meshing algorithms. It is very convenient to mesh a complex

shape with tetrahedra, and the second-order and modified tetrahedral elements (C3D10,

C3D1OM,) in ABAQUS are suitable for general usage. However, tetrahedra are less

sensitive to initial element shape. The elements become much less accurate when they are

initially distorted. A family of modified 6-node triangular and 10-node tetrahedral

elements is available that provides improved performance over the first-order tetrahedral

elements and that avoids some of the problems that exist for regular second-order

triangular and tetrahedral elements, mainly related to their use in contact problems. The

modified tetrahedron elements use a special consistent interpolation scheme for

displacement. Displacement degrees of freedom are active at all user-defined nodes.

These elements are used in contact simulations because of their excellent contact

properties. In ABAQUS/Explicit these modified triangular and tetrahedral elements are

the only second-order elements available. In addition, the regular elements may exhibit

volumetricc locking" when incompressibility is approached, such as in problems with a

large amount of plastic deformation. The modified elements are more expensive

computationally than lower-order quadrilaterals and hexahedron and sometimes require a









more refined mesh for the same level of accuracy. However, in ABAQUS/Explicit they

are provided as an attractive alternative to the lower-order tetrahedron to take advantage

of automatic tetrahedral mesh generators.

In the diamondback the powder body is meshed into quadratic tetrahedron

(C3D10OM) elements and the wall into discrete rigid body. The deformation of the wall is

negligible compared to the deformation of the powder. The mesh density and element

type are two factors that have a major influence on the accuracy of results and computer

time. Hence in this research the appropriate mesh had to be chosen. Figure 3.5 shows 4

different meshes that were used for analysis. As the project progressed, one mesh was

used with the various models.


Figure 3.5: Geometry and meshing of bulk solid









The wall friction coefficient is applied between the wall and the body using

Coulomb criteria. The basic concept of the Coulomb friction model is to relate the

maximum allowable frictional (shear) stress across an interface to the contact pressure

between the contacting bodies. In the basic form of the Coulomb friction model, two

contacting surfaces can carry shear stresses up to a certain magnitude across their

interface before they start sliding relative to one another; this state is known as sticking.

The Coulomb friction model defines this critical shear stress, r,,, at which sliding of the

surfaces starts as a fraction of the contact pressure, p between the surfaces ( T, = ,up).

The stick/slip calculations determine when a point transitions from sticking to slipping or

from slipping to sticking. The fraction, ji, is known as the coefficient of friction.

The basic friction model assumes that u is the same in all directions isotropicc

friction). For a three-dimensional simulation there are two orthogonal components of

shear stress, r, and rz, along the interface between the two bodies. These components act

in the slip directions for the contact surfaces or contact elements. ABAQUS combines the

two shear stress components into an "equivalent shear stress," 7, for the stick/slip

calculations, where T = r 2 + r, In addition, ABAQUS combines the two slip velocity


components into an equivalent slip rate, Pe 712 +2 The stick/slip calculations define

a surface. The friction coefficient is defined as a function of the equivalent slip rate and

contact pressure:


P = {(eqp ) (3.8)


where y,, is the equivalent slip rate, p is the contact.










The solid line in Figure 3.6 summarizes the behavior of the Coulomb friction model:

there is zero relative motion (slip) of the surfaces when they are sticking (the shear

stresses are below IIP)).


i (shearstress) slipping

sticking
Y (slip)



Figure 3.6: Frictional behavior

Simulating ideal friction behavior can be very difficult; therefore, by default in

most cases, ABAQUS uses a penalty friction formulation with an allowable "elastic slip,"

shown by the dotted line in Figure 12-5. The "elastic slip" is the small amount of relative

motion between the surfaces that occurs when the surfaces should be sticking. ABAQUS

automatically chooses the penalty stiffness (the slope of the dotted line) so that this

allowable "elastic slip" is a very small fraction of the characteristic element length. The

penalty friction formulation works well for most problems, including most metal forming

applications.

In those problems where the ideal stick-slip frictional behavior must be included,

the "Lagrange" friction formulation can be used in ABAQUS/Standard and the kinematic

friction formulation can be used in ABAQUS/Explicit. The "Lagrange" friction

formulation is more expensive in terms of the computer resources used because

ABAQUS/Standard uses additional variables for each surface node with frictional

contact. In addition, the solution converges more slowly so that additional iterations are

usually required. This friction formulation is not discussed in this guide.









Kinematic enforcement of the frictional constraints in ABAQUS/Explicit is based

on a predictor/corrector algorithm. The force required to maintain a node's position on the

opposite surface in the predicted configuration is calculated using the mass associated

with the node, the distance the node has slipped, and the time increment. If the shear

stress at the node calculated using this force is greater than 7Trit, the surfaces are slipping,

and the force corresponding to Tritis applied. In either case the forces result in

acceleration corrections tangential to the surface at the slave node and the nodes of the

master surface facet that it contacts.

Often the friction coefficient at the initiation of slipping from a sticking condition is

different from the friction coefficient during established sliding. The former is typically

referred to as the static friction coefficient, and the latter is referred to as the kinetic

friction coefficient.

Plasticity Models: General Discussion

The elastic-plastic response models in ABAQUS have the same general form. They

are written as rate-independent models or as rate-dependent. A rate-independent model is

one in which the constitutive response does not depend on the rate of deformation as

opposed to a rate-dependent models where time effect such as creep are considered.

A basic assumption of elastic-plastic models is that the deformation can be divided

into an elastic part and a plastic part. In its most general form this statement is written as:


F=Fe-.FPl (3.9)

where: F is the total deformation gradient,
F"1 is the fully recoverable part of the deformation
FPl is the plastic deformation









The rigid body rotation at the point can be included in the definition of either F"1 or

F"1 or can be considered separately before or after either part of the decomposition. This

decomposition can be reduced to an additive strain rate decomposition, if the elastic

strains are assumed to be infinitesimal:


8 = 8 el + pl (3.10)

where: E is the total strain rate
s"1 is the elastic strain rate
P'1 is the plastic strain rate

The strain rate is the rate of deformation:

~Sv~
S= sym (3.11)


The above decomposition implies that the elastic response must always be small in

problems in which these models are used. In practice this is the case: plasticity models

are provided for metals, soils, polymers, crushable foams, and concrete; and in each of

these materials it is very unlikely that the elastic strain would ever be larger than a few

percent.

The elastic part of the response derived from an elastic strain energy density

potential, so the stress is defined by:


a = (3.12)


where: U is the strain energy density potential

The stress tensor a is defined as the Cauchy stress tensor.

For several of the plasticity models provided in ABAQUS the elasticity is linear, so

the strain energy density potential has a very simple form. For the soils model the









volumetric elastic strain is proportional to the logarithm of the equivalent pressure stress.

The rate-independent plasticity models have a region of purely elastic response. The yield

function, f defines the limit to this region of purely elastic response and, for purely

elastic response is written so that:


f(O*, H,) <0 (3.13)

where: 0 is the temperature
H, are a set of hardening parameters(the subscript o is introduced simply to
indicate that there may be several hardening parameters)


In the simplest plasticity model (perfect plasticity) the yield surface acts as a limit

surface and there are no hardening parameters at all: no part of the model evolves during

the deformation. Complex plasticity models usually include a large number of hardening

parameters. Only one is used in the isotropic hardening metal model and in the Cam-clay

model; six are used in the simple kinematic hardening model.

When the material is flowing inelastically the inelastic part of the deformation is defined

by the flow rule, which we can write as:


d c = d (3.14)


where ds = is plastic strain increase
dA = proportionality factor for the ith system
g (-, H,,) = plastic potential for the ith system
a = Cauchy stress state

In an "associated flow" plasticity model the direction of flow is the same as the

direction of the outward normal to the yield surface:









Df,
S= i (3.15)

where: c, is a scalar.

These flow models are useful for materials in which dislocation motion provides

the fundamental mechanisms of plastic flow when there are no sudden changes in the

direction of the plastic strain rate at a point. They are generally not accurate for materials

in which the inelastic deformation is primarily caused by frictional mechanisms. The

metal plasticity models in ABAQUS (except cast iron) and the Cam-clay soil model use

associated flow. The cast iron, granular/polymer, crushable foam, Mohr-Coulomb,

Drucker-Prager/Cap, and jointed material models provide nonassociated flow with

respect to volumetric straining and equivalent pressure stress.

Since the flow rule and the hardening evolution rules are written in rate form, they

must be integrated. The only rate equations are the evolutionary rule for the hardening,

the flow rule, and the strain rate decomposition. The simplest operator that provides

unconditional stability for integration of rate equations is the backward Euler method:

applying this method to the flow gives


AcpV= Y AA, agi
A og (3.16)

and applying it to the hardening evolution equations gives


AHa = AA ,hi,a (3.17)

where hi,, is the hardening law for H, .


The strain rate decomposition is integrated over a time increment as:









A = Acel + A (3.18)

where As is defined by the central difference operator:



S= sym Ax (3.19)
a \x, + AAx


The total values of each strain measure as the sum of the value of that strain at the

start of the increment, rotated to account for rigid body motion during the increment, and

the strain increment, are integrated. This integration allows the strain rate decomposition

to be integrated into:

8 = C el+ pl (3.20)

From a computational viewpoint the problem is now algebraic: the integrated

equations of the constitutive model for the state at the end of the increment, must be

solved. The set of equations that define the algebraic problem are the strain

decomposition, the elasticity, the integrated flow rule, the integrated hardening laws, and

for rate independent models, the yield constraints:


fi = 0 (3.21)

For some plasticity models the algebraic problem can be solved in closed form. For

other models it is possible to reduce the problem to a one variable or a two variable

problem that can then be solved to give the entire solution. For example, the Mises yield

surface-which is generally used for isotropic metals, together with linear, isotropic

elasticity-is a case for which the integrated problem can be solved exactly or in one

variable.






54


For other rate-independent models with a single yield system the algebraic problem

is considered to be a problem in the components of AsE'. Once these have been found-

the elasticity-together with the integrated strain rate decomposition-define the stress.

The flow rule then defines AA and the hardening laws define the increments in the

hardening variables.















CHAPTER 4
CONSTITUTIVE MATERIAL LAWS

For bulk materials a constitutive highly nonlinear material model needs to be used

that describes the solid-like behavior of the material during small deformation rates as

well as the fluid-like behavior during flow conditions. Most of the models describing

bulk solids were developed in civil engineering to describe sand, clay and rock behavior.

Due to difference in stress magnitudes, loading and flow, only some of these models can

be applied to bulk solid applications. They are based on a consistent continuum

mechanics approach. The simplest model to use would be an elastic model based on

Hooke's law. This model was used by Ooi and Rotter (1990). Other elastic concepts such

as non-linear elasticity (Bishara, Ayoub and Mahdy, 1983) and hypo-elasticity (Weidner,

1990) have also been used. However these models cannot predict phenomena such as

cohesion, dilatancy, plasticity and other effects that are essential for the model. These

effects need to be described by elastic-plastic models that can describe irreversible

deformations. One of the first plastic models used is the Mohr-Coulomb model (1773).
S1I= C2= C3



Mohr-Coulomb
Tresca


CY


C3 Tr
Figure 4.1: Tresca and Mohr-Coulomb yield surfaces









It is derived from the Tresca criteria for metal plasticity and states that a material can

sustain a maximum shear stress for a normal load in a plane. If the shear load is more

than that maximum stress then the material starts to flow. Figure 4.1 shows the yield

surfaces for both Mohr-Coulomb and Tresca in the principle stress space. Another yield

surface that is numerically easier to handle is the Drucker-Prager (1952) surface because

it does not have the edges of the Mohr-Coulomb. Because of this, researchers have a

preference to use this model. It is derived from the von-Mises criteria and is shown in

Figure 4.2



C 1= C2= 3
C72

Drucker-Prager
\ Von Mises






C03

Figure 4.2: Von Mises and Drucker-Prager yield surfaces

One disadvantage to the above models is that under hydrostatic loading plastic

deformation can not be predicted. That is because during isotropic loading (on the line

CI= 02= 03), the stress lies within the yield surface. However, it is known from

experiments that plastic deformation does occur during isotropic loading. Therefore a cap

is introduced to create a closed yield surface that limits elastic deformation in hydrostatic

loading. So, as can be seen from figure 4.3, the model consists of a yield cone and a yield

cap.










C1- 02- 3
02










C3 1

Figure 4.3: Closed yield surface

In the above elastic-plastic models, both the elastic behavior inside the yield

surface and the plastic flow rule have to be determined. The flow rule can either be

associated, where the shape of the yield surface determines that flow direction or non-

associated, where an additional surface, the flow potential, is defined.


The general form of the flow rule is usually assumed to be potential where the strain

increments are related to the stress increments by the following relationship:


S= d G (a) (4.1)
de" =dA -'


where:

dEs = plastic strain increase

dA= proportionality factor

G = plastic potential

o-, = Cauchy stress state

The flow rule is associated if the potential G is equal to the yield function F. If they are

not equal then it is non-associated.









Capped Drucker-Prager Model (1952)

The capped Drucker-Prager model is used to model cohesive materials that exhibit

pressure-dependent yield. It is based on the addition of a cap yield surface to the Drucker-

Prager plasticity model, which provides an inelastic hardening mechanism to account for

plastic compaction and helps to control volume dilatancy when the material yields in

shear. It can be used in conjunction with an elastic material model and allows the material

to harden or soften isotropically. It is shown in Figure 4.4.
Plastic Transition surface, Ft

Shear Failure, Fs



L-V

Hardening e \ \ Hardening
Elastic \


11


Figure 4.4: The linear Drucker-Prager cap model 3 3 +23

The Drucker-Prager failure surface is written as:

F = q p tan / d = 0 (4.2)

where P and d represent the angle of friction of the material and its cohesion,

respectively.

p -trace (u ) --------------is the equivalent pressure stress
3
q J s -----------------------is the misses equivalent stress

r 9 S3 -----------------is the third stress invariant
----------------------is the deviatoric stress
S = c + PI ------------------------------is the deviatoric stress









The cap yield surface has an elliptical shape with constant eccentricity. The cap surface

hardens or softens as a function of the volumetric inelastic strain. It is defined by the

following equation:

2

F= [P P]+2 + Rp R (d + p, tanp) = 0 (4.3)
P FI+a -a/ 8


where, R is a material parameter that controls the shape of the cap and a is a small

parameter that controls the transition yield surface so that the model provides a smooth

intersection between the cap and failure surfaces:


-2
F [PPa]-2+LpK (d+P ,tani) -a(d+ptan)=0 0 (4.4)



As can be seen from the equations above the model itself needs 7 parameters or measured

material properties for calibration. In addition to describe the elastic behavior two

parameters are needed (E- Young's modulus, and v-poisson's ratio). The coefficient of

wall friction [t also needs to be determined.

3D General Elastic/Viscoplastic Model

In order to be able to predict the mechanical behavior of cohesive powders in silos

or hoppers, a physically adequate constitutive model has been developed (Cristescu,

1987). In general, the constitutive model captures all major features of material

mechanical response and accurately describes the evolution of deformation and volume

change. Dilatancy and the related effects, such as microcrack formation damage, and

creep failure can be predicted.









The three-dimensional general elastic/viscoplastic constitutive equation with non-

associated flow rule (Cristescu, 1987) is

i 1 1 1 W(t) 8F
E= -+ --- l+k, 1- (4.5)
2G 3K 2G) H(, Tr) 8(5)


= rate of deformation tensor H(cr, r) = yield function
a = Cauchy stress tensor W' (t) = irreversible stress work per unit volume,
a = mean stress F = viscoplastic potential,
K, G = shear and bulk moduli k, = viscosity coefficient for transient creep

W1 (t)
1 is the identity matrix. The function 1- W') is chosen to represent mechanical
H(, z-)

behavior of the material due to transient creep and symbol ( ) is known as Macaulay


bracket: (A)=-(A+ JAI). The last term in equation 4.5 describes the mechanical


behavior of the elastic/viscoplastic material that exhibits viscous properties in the plastic

region only.

In the elastic/viscoplastic formulations stress and strain are time-dependent variables, and

time is considered as an independent variable. In this problem the phenomena of creep

and plasticity cannot be treated separately as only the superposition effect.

It is worthy to note that the model initially included a term for steady-state creep

given by the equation:

. C 9S
s = ks (4.6)


where S(oa) is the viscoplastic potential for steady-state creep and ks is a viscosity

coefficient. This term however describes behavior over long periods of time such as









creep in tunnels. However, transient creep can describe the behavior of powders in

hoppers at relatively smaller stresses and hence the above term is ignored.

The material is assumed to be isotropic and homogeneous. Hence the constitutive

equation depends on the stress invariants; primarily the mean stress:

1
( =1 +02 + +03 (47)


and the equivalent stress (T or the octahedral shear stress Z :

-2 2 2 2
T=-U- = _1 +_2 +3 1 2 1 3 2 3 (4.8)


or in terms of an arbitrary (1,2,3) coordinate system:


= (11 22)2 +(11 -33) +(02 -033 2 6 22+ 23 +6132 (4.9)


The strain rate consists of elastic and irreversible parts:


= E + (4.10)

The elastic part of the strain rate tensor is


S= + -- 1 (4.11)
2G 3K 2G) (4.11)

and the irreversible part is


S= k 1 )- O(4. 12)


Equation 4.11 describes an instantaneous response of the material to loading. The

irreversible part, (equation 4.12), depends on the time history as well as on the path

history of the stress. One of the time effects that the model describes is creep.









Deformation due to transient creep stops when the stabilization boundary is reached. The

equation of this boundary (the locus of the stress states at the end of transient creep) is


H(o, ) =W'(t) (4.13)

The irreversible stress work per unit volume W'(t) describes the irreversible isotropic

hardening:

W (t) = r:dE' (4.14)

One of the concepts described by the model is whether the bulk solid is compressing or

dilating. Figure 4.5 shows these domains. Triaxial experiments on cohesive materials

show that powders show compressibility at lower stresses followed by dilatancy.

Between the two domains we have the compressibility/dilatancy boundary. In the case of

a triaxial test the boundary is determined by determining the where the slopes of the plots

of the stress vs. volumetric strain are vertical.


FAILURE


DILATANT



I CO 4P E 1S3B E



COMPRESSIBLE

Figure 4.5: Domains of compressibility and dilatancy G

The viscoplastic flow occurs when:


K> W'(t) > 0 (4.15)









This model will enable us to capture all major features of powder mechanical

response and accurately describe the evolution of deformation and volume change. It can

predict the dilatancy and the related effects, such as creep failure. It is calibrated by

numerous hydrostatic and deviatoric triaxial testing.

Numerical Integration of the Elastic/Viscoplastic Equation

The main difficulties in implementation of the elastic/viscoplastic constitutive

model in the finite element code is the numerical integration of the irreversible

(viscoplastic) part of the strain tensor.

Suppose that at the time moment t all values of stress and strains in 4.5 are known.

Consider the moment of time t + At. First, we represent the increment of the irreversible

strain rate in the form of the truncated Taylor series with respect to the stress and

irreversible strain increments:


Ai' = (t)a + {(t)AE', (4.16)


where


(= W(t) )1-F (4.17)
SH(c, T) /O

and


W' (t + At) = W (t) + (t) ( (t + At) 1 (t)). (4.18)


The total strain increment consists from elastic and viscoplastic parts. The elastic part is


E =E-E (4.19)









and from the Hooke's law


(==C E-. ) (4.20)


where C is constant stiffness matrix dependent on the elastic constants.

The increment of irreversible strain is


As = I (t + At) 7 (t) (4.21)


and


A' =^ ((1-a) e(t)+c (t+)) (4.22)


If a = 0, the explicit scheme (Euler forward scheme) for the integration of viscoplastic

strains results. On the other hand, if a = 1, the fully implicit scheme (Euler backward

scheme) for the integration is obtained. The case a = results in the so-called "implicit
2

trapezoidal" scheme.

Substituting (4.22) in (4.21):


= ij (t)+a -(t)Aa + a (t)AEI (4.23)
At 0G de

or


E (t)At + aAt (t) (a (t + At) -a (t))
Ae' (4.24)
1 aAt (t)e t


Returning to formula (4.20) the expression for stress tensor at the moment t + At gives:










,(t+AO) I {C(loN(t)((t+AO)c't))A 0
1- 2V-(t)+a -(t) a1(4.25)


The important issue to be addressed is stability of the numerical integration

scheme. Note that the Euler forward method does not give an accurate numerical

1
solution. For a < -, the integration process can proceed only for values of At less than


some critical value (this critical value should be determined in some way).

The accuracy of the numerical integration scheme with respect to stability and

convergence has been estimated by comparing the theoretically predicted values for

irreversible stress work and irreversible strain rate (obtained analytically by direct

integration of the constitutive equation) with those obtained numerically during creep.

There is good correspondence of results as for the irreversible stress work as for the

irreversible strain rate.

The elastic component is calculated from Hooke's law:

o-, = A25 + 2/E1 (4.26a)

or

a = C (4.26b)

where

1 l, i = j,
= 1= (4.27)


i and pu are Lame constants.

In another form






66


oa A +2p A A 0 0 0 ~,
Y l A A+2pl A 0 0 0 yy
Cz A A A+ 2p 0 0 0 (4.28)
CXY 0 0 0 2p 0 0 exy
oxz 0 0 0 0 2p 0 E5
-CY 0 0 0 0 0 2p _L s

Inverse matrix C-' may be calculated using using the C matrix above.
8 = C1 ( (4.29)

Relations of Lame's constants the bulk and the shear moduli are

2
K = i + u (4.30)
3

E= (3 ) (4.31)

G = u (4.32)















CHAPTER 5
EXPERIMENTAL RESULTS AND DISCUSSION

Parameter Determination-Shear Tests

Both direct shear testers and indirect shear testers are used to determine the

parameters that are needed for the model per the procedures described in Chapter 2

(Saada, 2005). Silica powder with particle size of around 50 microns was used. To

determine the internal angle of friction, /. and the cohesion, d the linear failure surface

was determined using several Schulze tests. The Schulze test (1994) was chosen over the

Jenike test (1961) because of the simplicity of the sample preparation and the test

procedure. The consolidation weights ranged from 2kg to 25kg. The specimen was

sheared at values between 20% and 80% of

YeldLout















Figure 5.1: Output data from Schulze test on Silica at 8Kg

the steady-state value. Figure 5.1 shows the results of a consolidation weight of 8Kg. The

slope of the yield locus (straight red line) is the internal angle of friction and the Y-

intercept is the cohesion. Plots of cohesion and internal angle of friction are plotted vs the










principal stress o- in Figure 5.2. Table 5.1 summarizes the results of 16 tests (Saada,

2005). The tests were also repeated to check for consistency.

Table 5.1: Summary of Schulze test results
i g ka3 (internal
Weight (g) oa (kPa) intn) d (cohesion)
friction)


2000
2000
3000
3000
4000
4000
6000
6000
8000
8000
12000
12000
16000
16000
22000
25000


1.57
1.55
2.29
2.36
3.05
3.13
4.55
4.64
6.02
6.50
9.16
9.08
11.97
11.98
15.99
18.24


28.71
27.87
29.00
28.80
31.29
30.37
34.86
33.18
31.47
30.07
32.33
33.83
34.09
33.65
32.18
32.05


0.359
0.364
0.43
0.5
0.47
0.566
0.46
0.639
0.75
0.849
1.04
0.935
1.11
1.16
1.38
1.56


The cohesion ranged from about 0.36KPa to about 1.56Kpa at a o0 value of 12Kpa. The

data can be fit with a second order polynomial as seen in Figure 5.2


d (Cohesion)


p(internal friction)


30

,20


0 0
SOo


5 a,(Kpa) 10


ai (Kpa)


Figure 5.2: Schulze test results for to determine the linear Drucker-Prager surface


1.4
1.2
1
f-0.8
_n


* x

x


-a






69


The internal angle of friction is relatively constant around the 32 deg value except foro-,

values that are less than 3Kpa where the angle drops slightly (Figure 5.2).

To determine the wall friction coefficient, Jenike tests are preformed between 304

stainless steel slab, 120mm x 120mm (with 2B finish) and silica (Saada, 2005). The test

was conducted using the procedure described above with weights decreasing from 16kg

down to 2kg. To check for repeatability the test was repeated twice. Results are shown in



30 -
29 -
M 28 -
27 -
0 26 -
-- Test 1
25 ....... .......Test 2
0
24
u- 23 D
22 0. .. ....
21
20
0.0 5.0 10.0 15.0 20.0 25.0
Normal Stress (kPa)
Figure 5.3: Jenike wall friction results to determine boundary conditions

Figure 5.3. It can be seen that the values of wall friction angles are slightly higher under

the 5 kpa, but decrease in value and reach a constant value of about 22 deg as the normal

stress is increased. This provides a value of u =0.4 to be used in the coulomb criteria. In

the stress area of interest the test are repeatable and the maximum error is less than 5%.

Detailed studies of wall friction are outside the scope of this research but are currently

undertaken by other research groups. Future work in the area of simulation involves the

study of more complex frictional models.










Triaxial tests are required to determine the parameters for the cap in the model and

the elastic parameters. A hydrostatic test where the sample is pressurized in all directions

is preformed and the pressure and volume changes are recorded. Several deviatoric tests

covering the range of confining pressures of interest are also needed. In these tests a fixed

confining pressure is applied while the sample is given axial deformation. The stress,

axial and volumetric strains are measured. Figure 5.4 shows results for five hydrostatic

tests up to 69.0 kPa (Saada, 2005). The confining pressure is increased at a constant rate

up to five values: 6.9KPa, 34.5KPa, 41.4KPa, 55.2KPa, and 69.2KPa. It is important to



70

60


0.
40 -4
000 0 o)K 6.9 Kpa
u030 Xj.rb A 34.5 Kpa
)b so@.a0 -41.4 Kpa
20- #0z x 55.2 Kpa
10Y o 69.0 Kpa



0 0.01 0.02 0.03 0.04 0.05 0.06
Ev

Figure 5.4: Hydrostatic triaxial testing on Silica Powder (5 confining pressures)

note that the specimen had an initial axial stress of 3.7KPa prior to the initiation of

hydrostatic testing due to the filling procedure described in Appendix A. This is reflected

in Figure 5.4 where the plots of the stress-strain curves show a clear jump at the

beginning of the hydrostatic testing. The hydrostatic tests are followed by deviatoric tests

where the confining pressure is kept constant and the axial deformation is increasing.







71


Figures 5.5 and 5.6 show the axial strain and volumetric strain respectively of five

deviatoric tests.


250 -
oC-a3(kpa)
200 AAAA


150 -

10 A 00000
A ^ 0000000
100 0, 000000000000

8 0o0o00 x34.5Kpa
50 A- 4 o 13.8Kpa
& 46.9Kpa
Or IU o 4.8Kpa
-0.005 0.045 0.095 0.145 0.195 0.245
8i

Figure 5.5: Axial deformation of 4 deviatoric triaxial tests on Silica
Powder and a rate of 0.1 mm/min


250 -
o1-G3(kpa)

200 A A A A


150 -Xx
o x 34.5Kpa
100 -D 0 Ao-^*13.8Kpa
06, 53 4.8Kpa




0
50 -X o o oY00 A46.9Kpa



0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
ev


Figure 5.6: Volumetric deformation of 4 deviatoric triaxial tests on Silica Powder and a
rate 0.1mm/min

As expected the graphs show an increase in measured axial stress with increasing

confining pressure. Each test shows an increase in strength until a plateau is reached. The










volumetric strain graphs show a compaction region followed by a dilation region. The

data has been corrected for the confining effects of the latex rubber membrane using

Hooke's law and the elastic properties of the rubber. To check for repeatability several

tests were repeated more than once. Figure 5.7 shows the axial strain results of two

deviatoric tests at 13.8KPa and 34.5Kpa.

250 -
oa-o3(kpa)
200


150 -








Figure 5.7: Axial deformation of 4 deviatoric tests on Silica Powder at 13.8KPa and
34.5Kpa50 -34.5Kpa













has is less than 50.
13.8Kpa


-0.005 0.045 0.095 0.145 0.195 0.245



Figure 5.7: Axial deformation of 4 deviatoric tests on Silica Powder at 13.8KPa and
34.5Kpa

It can be seen that the error in the shear stress is maximum at the higher deformations and

has is less than 5%.

Because of the low pressures that are being investigated, the plots show results that

require the 'modification' of the testing procedure. This is because of the apparent

oscillations in the graphs and the not very distinct value of failure for the specimen.

Another issue to resolve is the sharp jumps and drops in stresses under 10kpa

(shown in the circle), which are due to the loading piston friction while it is lining up

with the center of the specimen. These oscillations are described and explained in









Appendix A under piston friction error. In spite of these challenges these results do

produce values that are satisfactory for use in the model.

Diamondback hopper measurements-Tekscan Pads

The experimental procedure for the measurements of wall stresses in the hopper

was described in chapter 2. Measurements were taken when the powder was static and at

belt speeds of 40, 60, 70, and 100 (Johanson, 2003). Figure 5.8 shows the geometry of

the mechanism at the outlet of the hopper .
Powder in chute

Pow der on belt B l

S; Belt

c---r


Figure 5.8: Discharge mechanism and belt

Based on the dimensions of all the parts and the outlet, the velocity of the belt, table 5.2

was created to relate velocity and flow rate of the powder in the chute.

Table 5.2: Belt velocity vs. flow rate
Belt velocity Flow rate
(cm3 /sec)
40 35.6
60 52.7
70 60.9
100 80.6


Figures 5.9 shows the location of the pad in the hopper and an output from a test. The

area that is covered by the rectangles represents the line in the hopper where the

transition between the vertical and the converging parts of the hopper. The left side of the

pad starts midway through the flat section and loops around the curved section.










































Figure 5.9: Pad location and output example

Rectangles are used to obtain wall stress as opposed to lines so as to obtain an

average and not localized stresses. The data were averaged along eight paths shown by

the white lines in figure 5.10 (Saada, 2005). They are 2.8cm apart and span from the

middle of the flat section of the hopper to halfway around the curved section (90 deg).

Figure 5.10 shows the contact pressure measured while the powder was static at a depth

of 2.8cm from the hopper transition. These stresses generally decrease around the

perimeter and are smaller at greater material depths. The results show a peak at the flat

section of the hopper and oscillations going around the curved section of the hopper.


I










Wall Stress (@2.8 cm)
Static


Initial




2 cm '

^ LgA
M~ LI~t


40
Angle (deg)


Figure 5.10: Static Wall stress data at a distance 2.8cm below hopper transition

Figure 5.11 shows wall stress measurements were along the same path of depth 2.8cm

conducted at various speeds 60, 70 and 100 in addition to the static state (Saada, 2005). The

plots show agreement in the pattern and magnitude of wall stress at the various belt speeds.


14-

12

10 \ Initial
M (M -.-speed 60
8 u--speed 70
6 speed 100
U,4

2-


0 20 40 60 80
Angle (deg)

Figure 5.11: Wall stress data at a distance 2.8cm below hopper transition at various
speeds and static conditions









Figure 5.12 shows the same data at the same speeds at a depth of 5.6cm. Appendix A

shows the results at the rest of the six paths shown in Figure 4.10.


10
9 Initial
8 -*-speed 60
S7 \ --speed 70
speed 100



(43



1
0
0 20 40 60 80
Angle (deg)


Figure 5.12: Wall stress data at a distance 5.6cm below hopper transition at various
speeds and static conditions

Figure 5.13 shows the results of a Fourier series analysis conducted at the results from

two different regions (Johanson, 2005). Region numbered #1 is in the flat section of the

converging hopper. In this region the fluctuations in the stress have a wide range of

frequencies. Region #7 is in the round section. In this region the fluctuations have a wide

range of frequencies. It is hypothesized that the converging diverging nature of the

Diamondback hopper causes a shock zone formation across the hopper that propagates at

points along the edge of the flat plate section. Here the radius of curvature changes

causing a stress discontinuity. This should be a region of steep stress gradients. This is

validated by experiment it will be compared to FEM calculation.







77


0.(0 Radius of curvature 0.6
0.5 changes causing a 0.5
stress discontinuity. R
o0.(4 This is a region of O.o 4
steep stress
B 0.0
5. Ogradients. .




10 100 11 10 100 i.
Period of Signal (sec) Period of Signal (sec)








0 500 1000 1500 .. .
.Time (se 0 500 1000 1500 2001
Time (sec)
Time (sec)
Figure 5.13: Fourier series analysis on the wall pressure measurements.

Conclusions and Discussion of the Test results and Testing Procedure

Both simple direct shear tests and more complex triaxial tests are needed to

determine the parameters for the models that are used in this research. Sixteen Schulze

tests were conducted to determine the internal angle of friction, 3, and the cohesion, c (or

d). The tests were conducted at various consolidation stresses and were repeated at least

twice to check for repeatability. The internal angle of friction was determined to be

constant at a value of 320 and the cohesion was increasing quadraticaly. Jenike tests were

conducted to determine the wall friction angle. That angle was determined to be 220.

Both hydrostastic tests and deviatoric triaxial tests were performed for the calibration of

the viscoplastic model. The deviatoric tests were conducted at a confining pressure of up

to 47Kpa. They were repeatable and showed the expected increase in strength with

increasing confining pressure and the showed the powder going from compressibility to

dilatancy.









Performing the tests at this lower confining pressure (less than 50Kpa) created

issues that could introduce some error to the results. This is partially seen by the fact that

the material does not show an apparent failure point. The two main issues are: 1) the

influence of the rubber membrane on the deformation of the specimen, and 2) the

influence on the confining pressure of the water height. This 2nd issue is a problem due to

the fact that the stress (due to the water pressure) at the top of the sample is close to zero

while at the bottom it is around 2KPa which could present some error especially at the

test with 4.7Kpa confining pressure. For the 1st issue, Appendix B provides a procedure

to correct for membrane pressure based on hoop's law and the rubber properties. To

minimize the influence of the membrane it is suggested that a much thinner membrane is

used. This, however, may cause some complications in the 'current sample preparation

procedure', since extra care has to be taken so as not to cause any damage to the

membrane. It is also suggested that plastic wrap be used instead of rubber membrane,

since it is thinner and because of plasticity, it would be easier to correct for it. The ideal

procedure would be to conduct the test without a membrane but since this is impossible

with the current setup it is suggested to use another tester such the Cubical Biaxial test.















CHAPTER 6
FEM RESULTS AND DISCUSSIONS

Capped Drucker-Prager Model

The capped Drucker-Prager model is used to model cohesive materials that exhibit

pressure-dependent yield. It is based on the addition of a cap yield surface to the Drucker-

Prager plasticity model, which provides an inelastic hardening mechanism to account for

plastic compaction and helps to control volume dilatancy when the material yields in

shear. It can be used in conjunction with an elastic material model and allows the material

to harden or soften isotropically. It is shown in Figure 6.1.



q -o3 Plastic Transition surface, Ft

Shear Failure, Fs \

2 \Cap, F


1 \\
Hardening 1 \\ \ Hardening
'-o Elastic \




P (-, + 203)
Figure 6.1: The Linear Drucker-Prager Cap model

The Drucker-Prager failure surface is written as:

F = q p tan 8 d = 0 (6.1)

where 3 and d represent the angle of friction of the material and its cohesion,

respectively.









p L Ltrace (a ) --------------is the equivalent pressure stress
3
q --------------------is the misses equivalent stress


2
S = a + PI ----------------------------is the deviatoric stress

The cap yield surface has an elliptical shape with constant eccentricity. The cap

surface hardens or softens as a function of the volumetric inelastic strain. It is defined by

the following equation:

12

F,= 7[P Pa 2 + Rp R (d + p, tan/ )= 0 (6.2)
S / os 8

where, R is a material parameter that controls the shape of the cap and a is a small

parameter that controls the transition yield surface so that the model provides a smooth

intersection between the cap and failure surfaces

-2
f= [p-p]2+[p-1-- (d+p.tanf) -a(d+p tan/) =0
cosa p1 (6.3)

As can be seen from the equations above the model itself needs 7 parameters for

calibration. In addition to describe the elastic behavior two parameters are needed (E-

Young's modulus, and v-poisson's ratio). The coefficient of wall friction [t also needs to

be determined.

Results from the shear testers listed in Chapter 5 are used to determine the required

parameters and are listed in Table 6.1.










Table 6.1: Capped Drucker-Prager parameters determined from shear tests
E u p c (or d) R a K Evo Pa ev


4.4 0
Run 1 4.6x107 0.24 33.7 1.16 0.33 0.04 1 0.0024 10 0.0086
20 0.021
30 0.0321
0.5 0
Run 2 4.6x107 0.24 28.7 0.359 0.33 0.04 1 0.0001 10 0.0086
20 0.021
30 0.0321



Drucker Prager Model Verification

The Drucker-Prager model was used to perform FEM simulations on the results of

a triaxial test. The results are used to compare the way perfect plasticity approximates

powder behavior. The specimen was simulated as a deformable body (height=15cm,

diameter=7cm) and discritized into 1063 tetrahedral elements as seen in Figure 6.2 a). A

confining pressure of 10 OKPa was applied around the specimen and the specimen was

deformed axially by 3cm (up to 20% strain) as seen in Figure 6.2 b).














a) Undeformed snecimen b) Deformed snecimen

Figure 6.2: FEM simulation of a triaxial test using the Drucker-Prager Model









The parameters for the model where determined from the shear tests and are shown

in Table 6.1. The axial stress was calculated from the average of the stress at the top of

the specimen. The axial stress-strain curves from the simulation were compared with the

results that were obtained experimentally as seen in figure 6.3.


40 Silica @1lmnvmin
35 -
30
25 -
20 ** ***** *

15 Experimental

5
0
0 0.02 0.04 0.06 0.08 0.1
Strain


Figure 6.3: Axial stress-strain curves for experiment and FEM using Drucker-Prager the
model

Drucker Prager Model Hopper FEM Results

The results from the shear tests above were used to run an ABAQUS FEM simulation

on the diamondback hopper in the static case (during storage). As described in section

3.2, the powder is simulated as a deformable body with quadratic tetrahedron (C3D10OM)

elements. The powder was divided into 11 different sets and the gravitational force was

applied to each set to simulate filing. The steel wall is simulated as a discrete rigid body

that required discritization. The boundary conditions where the tangential Coulomb

friction of [t=0.4 on the walls and a vertical displacement of zero at the outlet. The

loading is a gravitational constant of 9.81m/s2. Figure 6.4 shows a side view of the









calculated contact stresses at the end of each filling step (Saada, 2005). They show the

increase in stress at the bottom of the hopper and moving up as the force is applied.






















Figure 6.4: FEM calculated contact stresses at various filling steps

It is important to note that the stress concentrations in the vertical section of the hopper

(above the transition to a converging section), are not a powder phenomena but are rather

due to FEM error in the contact analysis procedure. This is due to the fact that some of

the elements in the powder penetrate the wall and cause stress concentrations. Also the

non-symmetry in the simulation could be due the fact that the meshing is not 100%

symmetrical (automatic meshing techniques were used). Another explanation is probably

due to the fact that the simulation of filling is not completely static and there is movement

of powder nodes on wall nodes causing non-symmetry.

To simulate discharge the surface at the bottom of the powder was moved by a distance

of 5cm. Figure 6.5 shows the contact stresses at 4 different intervals (Saada, 2005). Note























Figure 6.5: FEM calculated contact stresses at various discharge steps
the stress change close to the outlet and in the converging section. The comments above

explain the non-symmetry of the simulations. The area where the stress measurements are

being compared and analyzed is shown in Figure 6.6. The FEM section shown shows the

nodes where contact stresses are recorded. They are then compared with the Tekscan

measurements along the corresponding paths shown as white lines in the plot.


TekScan


FEM

Fi




...'--^ t.. ... .* *. '.^ .
Ir



I.


Figure 6.6: FEM vs. TekScan area of interest









It is important to note that the reasons the white lines are curved is due to the fact that the

pad is curved inside the hopper and is presented a flat section in the figure. Also due to

the complexity of the contact analysis simulation, a balance between mesh density,

element types and computer simulation time an appropriate mesh had to be chosen. The

mesh chosen had 14 nodes across each of the paths that were analysed. These are seen as

the black dots in the FEM contour plot.

Figure 6.7 shows the comparisons of the static wall stresses at seven locations

(from 2.8cm to 19.6cm below the hopper transition). The plots here show a similar trend.

They capture the general magnitude of the wall stresses that are higher at the middle of

the flat section but are decreasing as we go round in the curved section. The simulations

show some oscillations, but fail to calculate the same oscillations that are recorded by the

tekscan pads. The magnitudes of the stresses are better approximated at the flat section

(between 0 and 40 deg) but are slightly higher at the range of 40 deg to 90 deg. It is

possible that the oscillations in the pad readings are due to that fact that the powder is not

completely static and there is some movement while the powder is settling and the

measurement are being made.

As explained earlier the simulation of flow was done by applying a displacement of

5 cm to the bottom surface of the powder. Another method attempted was the removal of

the boundary conditions at the surface this however produced large deformations of the

elements and caused the simulation to abort. The data analysis was conducted at various

locations in the flow simulation. Figure 6.8 shows the comparisons at two seven paths











SMeasured Static Wall Stress
0 FEM Static Wall Stress


a-

0
o I
B w


45 Angle (deg) 90


M Measured Static Wall Stress
0 FEM Static Wall Stress


45 Angle (deg) 90


45 Angle (deg)


0 o o0 @14cm

45 Angle(deg) 90
0





45 Angle (deg) 90


45 Angle (deg) 90


000
0o @19.6cm







A0 90
45 Angle (deg) 9


Figure 6.7: FEM calculated static wall stresses vs. measured wall stresses at seven
locations


Angle (deg)


(@8.4 cm)
OO
000 0 0 0
NIIII0 0












SMeasured Flow Wall Stress
- X FEM Flow Wall Stress at 0.25 cm
........... ::G::::::: FEM Flow Wall Stress at 5 cm


"R" 6
=5(
S4)
3


45 Angle (deg) 90


Measured Flow Wall Stress
- X FEM Flow Wall Stress at 0.25 cm
O ii::::*: FEM Flow Wall Stress at 5 cm












45 90
Angle (deg)


45 Angle (deg) 90


45
Angle (deg)


45 Angle (deg) 90

12


45 Angle (deg) 90


10 @19.6 cm


6
4O
2


0 45 Angle (deg) 90
Figure 6.8: FEM calculated contact stresses vs. measured wall stresses at seven locations
during flow










and at two simulation displacements (0.25cm and 5cm). All the results show that the

Drucker-Prager model approximates the wall stresses during flow better than during

storage. The stresses follow the same pattern of decreasing as the angle is increased. The

simulation captures a good estimate of magnitude and some of the oscillatory pattern. We

see that the magnitude of the oscillations increases as the displacement is increased. This

is probably due to the increase in contact wall friction criteria between the nodes at the

wall and the nodes at the surface of the powder.

Several other parameters that affect the model were also investigated. The effect of

flow rate is shown in Figure 6.9. The 5 cm displacement was applied at times of 1 second

and 0.1 seconds. The figure shows the stresses at 5.6 cm and 8.4 cm transitions.

MEASURED Flow Wall Stress I MEASURED Flow Wall Stress
9 X- FEM at Discharge time=0.1 sec X FEM at Discharge time=0.1 sec
"iiiiiiiiiiiii EM at Discharge time=1 sec 7 0 FEM at Discharge time=1 sec
8
7 6












0 20 40 60 80 0 I
Anle (deg) 0 20 An4ge (deg) 60 80

a) At 5.6cm from hopper transition b) At 8.4cm from hopper transition


Figure 6.9: FEM calculated contact stresses vs. measured wall stresses at two locations
during flow at two different times