4.2 Resistive Force Theory Based Locomotion

Home , Karmin

Development of a reduced-order modeling technique for granular locomotion by Shashank Agarwal B.Tech., Indian Institute of Technology Gandhinagar (2014)

Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of

Master of Science in Mechanical Engineering at the

Signature redacted

Author...... Department of Mechanical Engineering A!/ A / December 19, 2018 Signature redacted

Certified by... Kenneth Kamrin Associate Professor Thesis Supervisor Signature redacted

Accepted by...... MASSACHUSETTS INSTITUTE Prof. Nicolas Hadjiconstantinou OF TECHNOLOGY Chair, Graduate Program Committee FEB '2 5 2019

UiBiARIES ARCHIVES 2 Development of a reduced-order modeling technique for granular locomotion by Shashank Agarwal

Submitted to the Department of Mechanical Engineering on December 19, 2018, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering

Abstract The work is aimed towards the development and expansion of a reduced-order modeling technique called granular Resistive Force Theory(RFT) for modeling wheeled locomotion on granular beds. A combination of various modeling techniques, namely plasticity-based continuum modeling, discrete element method (DEM) modeling, along with RFT and collaborative experimentation are used for evaluation and expanding upon the capabilities of granular RFT. A dimensionless formulation corresponding to the onset of catastrophic rise in slipping of wheels during granular locomotion is proposed. This limit also corresponds to the limits of the existing form of RFT in modeling wheel-granular media interaction accurately. Along with granular locomotion, general problems of granular intrusion have also been studied extensively using continuum modeling to characterize the general force response of different granular media based on various system parameters.

Thesis Supervisor: Kenneth Kamrin Title: Associate Professor

3 4

:; .a---.-.-raM --.au ..-t-:3. m-3 ;--=-r: --a-.09.w -l--M---:- 5-? --... 6-3------:-, -. !2---.<--.--':.-- n--' -:':.-r ag.p~ ieqe:p g ru g -|-u:.:.gr g~..-P~P ..'-mk -:-:-y .:.:..- ..4-:,..:.-.-.. ..-.---.....+;..: ,-. . -,..- -.: 73 -;- :.-.-.- -.-- -.- -.-.: i;-g.( .- - Acknowledgments

The completion of this work would not have been possible without the guidance and support of my research and academic advisor Prof Kenneth Kamrin. Ken has continually encouraged me to be a better student and researcher providing me with his invaluable tutoring and guiding me how to look into the details of simple yet challenging problems from a perspective of a researcher. I thank my parents and elder sister for their encouragement and constant support from overseas. Special thanks to our collaborator Andy Karsai and Prof Daniel Goldman at Georgia Institute of Technology whose expertise in experimentation has been vital for this work. I am thankful for all of my peers MechE, and the members of the Kamrin Group for making this entire experience a positive one. I am also indebted to an innumerable number of friends who have been there for me throughout my time at MIT. In particular, though, I want to thank Maytee Chantharayukhonthorn, my lab-mate and roommate, whose friendship I could not do without.

5 6 Contents

1 Introduction: Granular media and its triple nature 13

2 Locomotion and wheels 15 2.1 History of development of wheels over time ...... 15 2.2 Differences of granular locomotion from rigid surface locomotion . .. 16 2.3 Granular intrusion: A mechanics viewpoint ...... 17 2.3.1 M icro-Inertia ...... 18 2.3.2 M acro-Inertia ...... 19 2.3.3 Elastic wave fluidization ...... 20

3 Traditional methods of modeling wheel locomotion 21 3.1 Bekker's pressure-sinkage relation ...... 21 3.2 Modified Bekker model: Wong and Rees model ...... 22

4 Resistive Force Theory (RFT): A new approach to model granular intrusion 25 4.1 Background ...... 25 4.2 Resistive Force Theory based locomotion ...... 26 4.3 Using RFT for force calculations ...... 27 4.4 Modeling wheel locomotion using implicit RFT code ...... 29

5 Computationally intensive methods of simulating granular media 33 5.1 Background ...... 33 5.2 Discrete Element Method (DEM I MD) ...... 33 5.3 Continuum modeling ...... 35 5.3.1 Constitutive laws ...... 35 5.3.2 Im plem entation ...... 37

6 Low speed Locomotion 39 6.1 Circular wheel locomotion ...... 39 6.1.1 Experimental details ...... 39 6.1.2 Continuum modeling using MPM ...... 43

7 6.1.3 RFT based modeling ...... 44 6.1.4 R esults ...... 45 6.1.5 Conclusion ...... 54 6.2 Non-Circular wheel locomotion ...... 56 6.2.1 Experimental details ...... 56 6.2.2 RFT based modeling ...... 58 6.2.3 R esults ...... 58 6.2.4 Conclusion ...... 58

7 Locomotion optimization with RFT 61 7.1 Introduction ...... 61 7.2 Experimental setup ...... 62 7.2.1 W heel ...... 62 7.2.2 Test rig ...... 62 7.3 D iscussion ...... 63

8 High speed locomotion 65 8.1 Experimental setup ...... 65 8.2 RFT based Modeling ...... 66 8.3 Continuum modeling (MPM) ...... 66 8.4 Observations and analysis ...... 68 8.5 Effects of initial boundary conditions on equilibrium state ...... 69 8.5.1 Effect of initial translation velocity ...... 70 8.5.2 Effect of angular velocity ramp rate ...... 71 8.6 Evaluation of critical velocity for grousered wheel locomotion . .... 73

9 Other general intrusions in sand 77 9.1 Background ...... 77 9.2 Plane-strain granular intrusion ...... 79 9.2.1 Projected area defines the resistive force magnitude ...... 79 9.2.2 Superposition of forces ...... 80 9.3 Force & flow transition in plowed granular media ...... 81 9.4 Depth based lift and drag variation on cylindrical objects ...... 83

8 List of Figures

1-1 A real life example of sand flow ...... 14

2-1 Evolution of wheel over time...... 15

3-1 Schematic of Bekker's Bevameter intrusment ...... 22 3-2 Stress field representation on rigid wheel as per TM model . ... . 23

4-1 Comparison of RFT and Plasticity theory results in plate intrusion 27 4-2 Fitted form of RFT force functions ...... 29 4-3 Sample RFT simulations for various wheel shapes ...... 31

5-1 Schematic of two particle interaction in DEM ...... 34 5-2 Schematic representation of Non-cohesive graular model ...... 35

6-1 Schematic and lab setup for force-slip experiments ...... 40 6-2 Wheels nomenclature for slow locomotion study ...... 42 6-3 A sample continuum simulation of wheel Type C ...... 43 6-4 A sample RFT modeling of wheel type B ...... 44 6-5 Wheel C on loose and compacted Poppy Seeds ...... 47 6-6 Wheel A and B on compacted Poppy Seeds ...... 50 6-7 Wheel C on MS and MMS ...... 51 6-8 Comparison of RFT and TM model for wheel C on MS ...... 54 6-9 Flapped wheel used in non-circular wheel locomotion study ...... 57 6-10 Observations made in non-circular wheel locomotion studies . .... 59

7-1 Representation of system parameters associated with the flapped wheel 62 7-2 Lab setup for flapped wheel experiments ...... 63 7-3 Normalized velocity variation with flapped wheel ...... 64

8-1 Grousered wheel test setup for high-speed locomotion studies . ... . 66 8-2 Sample continuum simulation of grousered wheel ...... 67 8-3 High speed locomotion performance of grousered wheel: ...... 69 8-4 Effect on initial velocity on equilibrium state for grousered wheels . . 70 8-5 Variation of system parameters for different initial Velocity ...... 71

9 8-6 Effect on the ramp rates on equilibrium state for grousered wheels . . 72 8-7 Dependence of critical angular velocity on various system parameters 74

9-1 Sokolovskii analytic form of Slip lines with characteristic log spiral .. 77 9-2 Continuum modeling: Flat plate intrusion in granular media . .... 78 9-3 Continuum modeling: Flat plate Intrusion in unconsolidated media 78 9-4 Continuum modeling: Intrusion of triangular intruder ...... 79 9-5 Continuum modeling: Unsymmetric Triangle intrusion flow pattern 80 9-6 Continuum modeling: Unsymmetric Triangle intrusion ...... 81 9-7 Force and flow transition in plowed granular media ...... 82 9-8 Lift and drag forces in granular media ...... 84 9-9 Lift and drag on cylinder: Variation of state variables ...... 85

~"- ~ ~- List of Tables

2.1 Evolution of wheel over time ...... 16

4.1 Generic values of fitting parameters in analytic form of RFT ..... 28

6.1 Granular material properties for slow locomotion studies ...... 41 6.2 Specifications of the wheels used in slow locomotion studies ...... 42 6.3 Comparison of RFT, MPM and TM models for wheel C on CPS . . . 48 6.4 Comparison of RFT, MPM and TM models for wheel A and B on CPS 49 6.5 Comparison of RFT and MPM models for wheel C on MS and MMS 52 6.6 Comparison of RFT and TM model for wheel C on MS ...... 55

8.1 Experimental and Simulation parameters for high-speed locomotion . 68

11 12 Chapter 1

Introduction: Granular media and its triple nature

A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (the most common example would be friction when grains slide). Although granular materials are very simple to describe as the evolution of the particles follows Newton's equations, with repulsive forces between particles acting only when there is a contact between particles, they exhibit a tremendous amount of complex behavior, much of which has not yet been satisfactorily explained. They behave differently than solids, liquids, and gases and yet quite similar to them in various scenarios which has led many to even characterize granular materials as a new form of matter. As an interesting fact, our world is full of granular materials, starting from naturally occurring substances like sand, snow, mud to food grains, which all fall under the category of granular media. In fact, the occurrence of granular media is not limited to naturally occurring substances and they actually constitute the second-most manipulated material (after water) in the industry. A huge variety of industrial and manmade materials fall under the category of granular materials including construction materials like cement and soil, pharmaceutical materials like medical pills and powders, food materials like grain, coffee beans etc.

The biggest hurdle and most the interesting challenge in modeling granular media is their triple nature of promptly evolving between three state of matter: solid, liquid and gas in closely similar scenarios. For instance, in the Figure 1-1 a golf club hits the ground at high speed, while the person himself is standing on the sand itself where it acts like a solid, the stream of sand in front of him is similar to the flow of a stream of liquid, at the same time the sand can also be seen to be diffusing in air like a gas around the stream of sand. Such is the nature of sand which makes it extremely complex and difficult to model efficiently by a single unified method.

13 Figure 1-1: Flow of sand during a golf shot. [1]

This work primarily focuses the general problem of granular intrusion with the aim of developing a shape generic, reduced-order semi-empirical model for evaluation of resistive forces acting on various shaped rigid bodies intruding into various kinds of granular medias. Various kinds of preexisting modeling methods like plasticity-based continnum modeling and discrete element method (DEM) based modeling along with experimental validation is used with the ultimate aim of developing a quick, reliable and accurate method of modeling of general problem of granular intrusion. The work is primarily focussed on two related phenomena: 1) Wheel locomotion and 2) Rigid body intrusion into the granular media. While the field of wheel locomotion on sand (also called terramechanics in general) is chosen due to its large scale application as well as availability of large amount of pre-existing literature ( [2, 3, 4, 5]), the general problem of intrusion into sand was taken because it is fundamental to understanding all type of solid body-granular media interactions. The more details of the fields and our work is given in later chapters.

14 Chapter 2

Locomotion and wheels

2.1 History of development of wheels over time

Beside numerous fields of application mentioned in the previous chapter, one of the most important fields where the understanding of granular media is still restricted and needs more work is vehicular locomotion. Locomotion is an important part of human existence. Humans made a huge leap in the field of locomotion as well as civilization itself upon invention of the wheel. Over the thousands of years, wheels have gone through a series of cycles of evolutions. The time line below [6] shows a high-level summary of important events which have taken place in the evolution of the wheel from its original form to current form.

Potter's Chariot Spoked Pneumatic Mordern wheel wheel wheel wheel wheel

Figure 2-1: Evolution of wheel over time. [6]

15 Table 2.1: Evolution of wheel over time 3500 BC First use of wheel in Mesopotamia as potter's wheel 3200 BC First use of wheel in chariot in Mesopotamia 1000 BC Invention of iron rim: starting of the use of wheels for transportation 1802 AD First wire tension spoke patented 1845 AD First pneumatic wheel invented (Never commercialized:high cost) 1888 AD Pneumatic tire reinvented (Commercialized: Economic design) 1910 AD Carbon added to rubber to prolong tire life 1911 AD Invention of tire-inner tube combination 1926 AD Inventions steel welded-spoke wheels 1954 AD Usage of tubeless tires started Modern wheels Steel and alloy wheels: High performance (efficient and economical) Special wheels: extraterrestrial and military applications)

2.2 Differences of granular locomotion from rigid surface locomotion

Even though the automobile industry has been a huge industry in itself for the past century, less work is done on the development of wheels meant specifically for granular locomotion. An excerpt from the book 'the eyes of the desert rats' by David Syrrett [7] which talks about adventures of Brigadier Ralph A. Bagnold in Libyan deserts in the late 1930s gives an apt feel of what it feels like to travel on the sand with conventional tires:

... I increased speed to forty miles an hour, feeling like a small boy on a horse about to take his first big fence.... Suddenly the light doubled in strength as if more suns had been switched on. A huge glaring wall of yellow shot up high into the sky a yard in front of us. The lorry tipped violently backwards-and we rose as in a lift, smoothly without vibration. We floated up and up on a yellow cloud. All the accus- tomed car movements had ceased; only the speedometer told us we were still moving fast... The paragraph is self-explanatory and says exactly what one observe when one tries to move over the sand (or any other granular media like dry snow for that matter). The place where locomotion of vehicles on any granular media vary from that on rigid roads is that while the performance of a vehicle on roads depends least on road itself (except in terms of energy losses in tires), the composite wheel performance while traveling on granular media is largely dictated by a relatively shallow, piece of ground that lies under the wheel and become directly involved in dynamic inter- play with vehicle's running gear (tires). The mechanical and deformation properties

16 of the driving media primarily define the complicit vehicle traveling behavior and hence the problem of vehicle locomotion expands from just vehicle dynamics to coupled vehicle and media dynamics. In the initial years of the 19th century during the development of automobiles, while the primary focus of tire development was on locomotion on rigid roads, people used their intuitions and experiences to develop ways of moving through sand. One such example is another excerpt from the same book by David Syrrett 171 where Brig. Bagnold gives some strategies to drive over the sand:

... It should be taken at full speed on the high gear as far as possible, only putting low gear when the speed slackens considerable and returning to high gear as soon as the sufficing acceleration has been obtained. Stopping anywhere on hard and slightly elevated ground should be avoided and getting into ruts of a preceding car is dangerous if the ground is sat all soft. A car will frequently go without difficulty in virgin ground but will stick if the same ground has been ploughed up by other cars...

While these strategies were extremely valuable in those days, they were more based on intuition and experience than mechanics based technical know-how. In the later years of the 1960s, interest towards locomotion over rough terrains (mainly sand deserts) increased especially with the work of Bekker 121 who gave a semi-empirical formulation for modeling rigid cylindrical wheel motion over the on non-cohesive sands. In the later years various authors [8, 4, 9] expanded on the work of Bekker to develop various variations of Bekker theory for modeling wheel motion on different materials like drysnow [10], clay [11] etc. The most notable contribution to wheel- terrain modeling is the work by Wong and Reece which has become the de facto model of rigid cylindrical wheels on soft terrain [12, 13]. The model introduced by Wong and Reece derives wheel torque, thrust, and sinkage by estimating the stress distributions along the wheel-terrain contact region. The model is based upon the Bekker pressure-sinkage relation and the Janosi-Hanamoto shear-displacement equation [9]. A general formulation for modeling cylindrical wheel locomotion given by Wong et al [12, 13] is explained in the next chapter.

2.3 Granular intrusion: A mechanics viewpoint

In this section, we carry out a general discussion about the various flow regimes a granular material encounters during a general intrusion process. A better understanding of these regimes is vital for understanding the dynamics of wheel locomotion because locomotion is essentially a large-scale non-linear intrusion process at the mi- croscopic level. Dense, confined granular assemblies in quasistatic, slow shear flow are usually described as solids abiding by elastoplastic, rate-independent, constitutive laws [14].

17 Far from a confined granular state at large enough shear strains, where the material reaches the critical state, the material in a granular assembly is characterized by an internal friction coefficient y. In such a state the shear stress (T) in the medium is proportional to the normal stress (P), with a coefficient of proportionality [.

;T < pP (2.1)

= o'/v21 and P = (1/3)tr() (2.2) where o- is 3-dimensional stress tensor, t is the equivalent shear stress in the material, P is hydrostatic pressure, and y is called the internal coefficient of friction. While in the quasi-static critical state regime the inertia of the grains is too small to be relevant, with increasing shear rates, various inertial effects start affecting the system. Major among them are discussed below:

2.3.1 Micro-Inertia At low-pressure and/or large-shear rates, the inertia of grains starts affecting the effective strength of the medium [15]. This behavior is captured by a dimensionless number called the inertial number, I [161, which represents the ratio of inertial forces to confining forces. It has been found [17] that p is solely dependent upon I for stiff particles at steady state. The various parameters in the collisional and non-quasistatic regime obey: T= P(I)P (2.3)

p+(I) P 2 - Ps (2.4) I0/I+ 1

I = (2.5) 2P/p 8 where, 7 is equivalent shear stress in the material (given by equation 2.2), P is hydrostatic pressure (given by equation 2.2), /u(I) is the internal coefficient of friction,

I , P 2 , andyit, are material constants, y is shear strain rate, d is the mean particle diameter, and Ps is the particle density. It is important to note here that the above mentioned inertial effect is due to the inertia of individual grains relative to each other, which affects the coefficient of

18 friction of medium, and thus represents micro-inertial effects in the system. When the pressure, P , in the system is large and deformation rates are low, the medium is expected to be far from a collisional regime, and hence micro-inertial effects do not contribute significant variations in the effective coefficient of friction (p) of the medium.

2.3.2 Macro-Inertia The macro-inertial effect associated with the bulk of grains affects the media's response when bulk velocity changes are high. A dimensionless macro-inertial number -M, which is directly proportional to the inertia of the grains as a bulk and inversely to hydrostatic pressure in the system: IM = (pv 2 )/P represents those effects in the system. The basic governing equations for nearly incompressible granular media in the rigid-plastic limit can be given as:

pi = -V.o- + pg (2.6) i (2.7) a-=-PI + P(p, + Il ) I0/I + I ID I where p is density of the medium, b is the material acceleration, g is the acceleration due to gravity vector, o- is 3-dimensional stress tensor, P is hydrostatic pressure, p, is static coefficient of friction, Yd is upper coefficient of friction, I is a material constant, I is the micro-inertial number, and D is the strain-rate tensor, which is presumed to be roughly traceless during dense flow where a constant-volume flow is expected. Based on the formulation for IM provided above, it can be observed that that macro- inertial effects in the system are expected to be high only in regimes of high velocity or low pressure or both. The cases of high-speed intrusions correspond to such cases and hence dynamics of high-speed locomotion is more complex.

2.3.3 Elastic wave fluidization The two effects described in previous sections do not account for any influence of grain stiffness. The particle stiffness, even if large, could also affect the strength of the medium during rapid impact if elastic waves through the medium cause contact

19 fluctuations. These fluctuations can increase fluidization as in [18, 191, and are of a different nature from the inertial effect of grain collisions described above. The effect could be quantified in terms of speed intrusion to speed of sound, V = /E/p, for elastic modulus, E. This effect is more likely to be observed at local flow velocities near V and not away from it.

20 Chapter 3

Traditional methods of modeling wheel locomotion

3.1 Bekker's pressure-sinkage relation

As mentioned earlier, Bekker was among the very first authors who proposed a semi- empirical method of solving wheel locomotion whose underlying modeling approach relied on the analysis of two fundamental relations: the pressure-sinkage relation, and the shear stress-shear deformation relation. In the context of wheeled mobility, the pressure-sinkage relation governs the depth that a wheel will sink into the terrain when subjected to load, and consequently how much resistance it will encounter during driving. The shear stress-shear displacement relationship governs the amount of traction that a wheel will generate when driven, and therefore how easily it will progress through terrain and surmount obstacles. The pressure-sinkage relationship described by Bekker [2] was in the form of a semi-empirical equation that relates sinkage with the normal pressure of a plate pushed into the soil. The proposed relation is commonly referred to as the Bekker equation, and provides a link between the displacement (sinkage) and stress (pressure) of a plate (which can be viewed as a proxy for a wheel or track if one discretizes the leading surface of a wheel into sufficiently small sub-surfaces):

P= + k) zn (3.1)

Parameters kc, ko, n are empirical constants that are dependent on soil properties, while b corresponds to the plate width. These parameters can be obtained from field tests conducted with a device called a bevameter (Figure 3-1 [20, 5]).

21 LoeGng cybnders

TOrSem Toem n t p pe e 3.2 M e dWmkX aBe Lker

9' press"r M gouge

Pz e~ ~ PeersnPae

Figure 3-1: Schematic of Bevameter intrusment proposed by Bekker [2]

3.2 Modified Bekker model: Wong and Rees model

Over the past four decades, the original framework introduced by Bekker [2] has been expanded and modified by several researchers, and has found applications in many studies of wheeled and tracked vehicle's mobility [21, 22]. The most notable contribution to wheel-terrain modeling is the work by Wong and Reece which has, become the de facto model of rigid cylindrical wheels on soft terrain [12, 13]. The model introduced by Wong and Reece derives wheel torque, thrust, and sinkage by estimating the stress distributions along the wheel-terrain contact region. The model is based upon the Bekker pressure-sinkage relation and the Janosi-Hanamoto shear-displacement equation [9]. The details of the model are mentioned next.

The stress field under a wheel can be divided into two components (assuming a two-dimensional model, temporarily ignoring out of plane motion): normal stress and tangential stress. A schematic representation of the stress distribution at a wheel-terrain interface is presented in Figure 3-2.

Normal stress (-) can be calculated by beginning with Bekker's pressure-sinkage relation, then introducing a scaling function to satisfy the zero-stress boundary conditions present at the fore and aft points of contact of the wheel with the terrain (known as "soil entry" and "soil exit"). The equation is expressed as a piecewise

22 ME A

-V

fi'~ r J _r hj*

Figure 3-2: (a)Normal and Tangential stress profile along a wheel surface and (b) Various associated parameters wrt. Wong and Rees Model

function, as:

< Of or,1= ( L,+ k ) Z n 0 M< 0 92 = (k + ko ) Zn 0, < 0 < 0,

zi = r(cos 0 - cos Of)

Z2 = r cos (Of - Orn o (0, - O) - cos Of) (3.2) Om - or

where Of is the soil entry angle, 0, is the exit angle, and 0m is the angle at which the maximum normal stress occurs. This angle can be calculated as:

Om = (cI + c2 * s)Of (3.3)

where ci and c 2 are experimentally obtained constant parameters defined in [8]. s represents the slip and is defined as:

V V - V V sw =t 1t -- - (3.4) where, V is the actual forward speed of the wheel, Vt is the theoretical speed which can be determined from the angular speed w and the radius r of the wheel and V is the speed of slip of the wheel with reference to the ground. The shear stress in the longitudinal direction is the primary source of driving traction. The shear stress T is a function of o-, soil parameters and the measured shear displacement, J: -F= (c + -tan0<) 1eK (3.5)

23 where c and # are the cohesion and the angle of internal shearing resistance of the terrain, respectively, and K is the shear displacement modulus which may be considered as a measure of the magnitude of the shear displacement required to develop the maximum shear stress (see [10]). J represents the shear displacement of the wheel edge with respect to the adjacent soil and is given as / Of dO -Vtdt Vt (3.6) 0 0 where V is the tangential slip velocity given earlier in equation 3.4. The thrust, T, is computed as the sum of all shear force components in the direction of forward wheel motion.

T br] Tcos OdO (3.7)

Compaction resistance, R,, is then computed as the result of all normal force components acting to resist forward motion.

RC = br -sin OdO (3.8)

Drawbar pull,F , is calculated as the net longitudinal force (i.e. the difference between the thrust force and resistance force). F, is the resultant force that can either accelerate the wheel or provide a pulling force at the vehicle axle.

F =T - Rc (3.9)

Driving torque can be obtained by integrating the shear stress along the wheel contact patch: M = br2 TdO (3.10)

This set of equations constitutes the backbone of the model proposed by Wong and Reece, and it will be referred from here on as the TM model.

24 Chapter 4

Resistive Force Theory (RFT): A new approach to model granular intrusion

4.1 Background

The traditional terramechanics approaches are formulated based on the understanding of the physical nature of vehicle-terrain interaction, and detailed analyses of the mechanics of wheel-terrain interaction. These models rely on a set of parameters that include intrinsic soil properties such as cohesion and internal angle of friction, along with semi-empirical parameters including the shear modulus and the sinkage coefficients and hence are often over-parametrized and require adhoc terrain testing. Hence the usage of these modes for modeling wheel locomotion typically results in limited performance when wheel geometry is modified, when operational conditions diverge from nominal conditions (e.g., the high slip condition), and when parameter estimation from wheel performance data is attempted. On the other hand, approaches based on RFT have the advantage of relying on a compact set of parameters and can be applied to a wide range of wheel geometries. While the traditional terramechanics approaches by various authors like Bekker [2], Janosi and Hanamoto [9J, Wong and Reece [121 etc are based on detailed understanding of the physical nature and mechanics of wheel-terrain interaction, major caveats of all of them includes their specialization for locomotor shapes (In most of the cases cylindrical rigid wheel geometries assuming plane-strain deformation in sand) and dependence on various intrinsic material properties (like cohesion and internal angle of friction in wong model) and 'semi-empirical' parametric properties which are specialized for specific wheel geometries (like angle of maximum normal pressure, fore and aft angles etc in wong model). While having specificity to shapes limits applicability of these models to 'modified' wheel shapes in various situations, over parameterization like 10 parameters in case of Wong and Reece model (namely K, , K0 , n , ao, ai,#, c, kx, 6 e, 0m) limits their performance in case of variation in operation conditions because

25 re-calibrations require large numbers of experiments to be performed using some specialized experimental setups like the Bevameter in Bekker model. For countering all these caveats, the main focus of our research, granular Resistive Force Theory, comes at rescue due to its compact closed loop formulation. More details of the method are mentioned in the next section.

4.2 Resistive Force Theory based locomotion

RFT was originally a theory introduced in the literature of fluid mechanics used to approximate Stokesian behavior in order to model undulatory and flagellar propulsion in viscous fluids at low Reynolds numbers [23]. For an object which locomotes by swimming through fluids (such that the velocity on each part of the swimmer takes different values), while finding an analytical expression of the total drag forces is difficult (from the Navier-Stokes equations) using Numerical computation is expensive. Hence, Gray and Hancock [23] approximated a solution to such problems by postulating that the force field on an infinitesimal element of a slender body (whose radius of curvature is significantly larger than the width) is hydro-dynamically de- coupled from the rest of its body. Thus the drag force on an element (of very simple geometry) can be computed from its tangent direction i (or normal ft) and local velocity. Once the force on each individual element is known, the net drag for the swimmer is given by a linear superposition. Following on the same lines, granular RFT was recently adapted by Maladen et al [24] to solve the problems of subsurface swimming in granular media. Unlike viscous fluids, for an intruder moving slowly in granular materials, the drag force is dominated by friction: it is independent of the moving speed and increases with penetration depth. The RFT formula then takes the form:

Fd = J kpzI[fl,(V- -t)i + f(v -n)i]ds, (4.1) where kpgz| is the local overburden pressure on a small element ds of the intruder at the depth IzI. When granular RFT was first developed, the functional forms of f' and fii were determined from experimental trials [24, 25]. To make the formulation more structured, components fi and fil are refined in form of a, and oz which represents force per unit area per unit depth in x and z directions in the lab reference frame (gravity in the positive z-direction). More details about the integral form of RFT are discussed in the next section. Though the initial force function of RFT (in form of a,,) were proposed using experiments, Askari and Kamrin [26] later successfully verified that the functional form obtained using experiments earlier can also be derived using conventional Mohr-Coulomb plasticity (shown in figure 4-1, giving an indication of possible explanation of origins of RFT.

26 A

.1W2 3 x/2 a(ICM ) 0.21

- SC 9g :Z C

W2 .7 C) 2

-.12 -W12 0 1

Figure 4-1: (A)Illustration of RFT associated parameters on a small plate intrusion geometry. (B,C) Results of plasticity theory simulation vs. experimental measurement of resistive forces in granular media using a small plate element, showing the horizontal (a, ) and vertical (a, ) stresses per unit depth in a bed of glass beads.

It is also interesting to note here that though granular RFT has its inspiration from its fluid counterpart, granular RFT has demonstrated to be more effective in modeling the forces in granular interactions with intruders than its viscous counterpart in locomotion in the quasistatic locomotor regime, and jumping on granular media (27, 28].

4.3 Using RFT for force calculations

We now explain the basic strategy used in the calculation of forces acting on an arbitrarily shaped body using RFT formulation presented by Li et al [25]. As mentioned above the basic philosophy of RFT is decoupling of forces on various sub-surfaces of the body and using linear superposition to calculate the net forces. Hence, for a convex-shaped solid slowly intruding into granular material, one can decompose its intruding surface into arbitrarily small leading surfaces. Then, the force for each sub-surface is calculated as a function of the intruder's orientation angle / and the intrusion velocity vector angle -y (refer to Figure 4-1(A)). By simply summing over the forces experienced by the small leading surfaces, the net resistive force on the whole intruder is calculated. In two dimensions, the resistive force(f , f,) on an intruder can be calculated as:

27 (fx, fZ) = ((Ox ( , -y), a ,(0, -y)) H(z) IzIds (4.2)

where H(z) is the Heaviside function whose value goes to 1 only when a surface is submerged in ground else its zero and z is the intrusion depth from the surface of the ground and is always positive. The stresses upon these intruding surfaces are defined by the functions ax,, which represent force per unit area per unit depth. RFT takes an empirical approach by determining these functions through repeated slow experimental intrusions of a small flat plate submerged a unit depth in the tested media at various values of / and -y (figure 4-1(A)), creating a force diagram from force measurements for these angled surface elements (figure 4-1(B,C)). These small elements obey linear superposition and can be applied to any desired intrusion surface. Thus this simple closed form equation makes granular RFT a useful tool for estimating the forces on both submerged and intruding bodies in granular media. While the original dependence of ax,, was obtained in form of experimental data, Li et al [25] derived a trigonometrical form of generic values of the same as a function of 3 and -y by performing a Fourier analysis on the graph shown in figure 4-1(B,C). Before performing the analysis, axz were converted into two components owing to the fact that general form of force response for most of the non-cohesive materials is same, differing only by a constant factor among themselves. The components consist of a grain-structure interaction parameter, , and the generic values of age. Force function splitting for ax,z, functional forms for agez and corresponding fitting graphs are shown below: aX,= * gen (4.3)

~en 1 m# ny m# n+y age" = [Am,ncos2( + ) + Bm,nsin2r( + )] (4.4) m=-1 n=O 1 1 genm n-y i (m# n-y (45 CXen = [C,ncos27( + -) + D,nsir2r( + )] (4.5) m=-1 n=O In the above formulation, considering terms whose magnitudes are greater than 0.05 A0,0 , only following 9 terms were found to be significant (With the value of each component mentioned in the table)

Table 4.1: Generic values of fitting parameters in analytic form of RFT

Parameter A 0 ,0 A 1,0 B1,1 Bo,1 B 1 ,1 C1,1 C0,1 C_1,1 D1,o Values 0.206 0.169 0.212 0.358 0.055 -0.124 0.253 0.007 0.088

It is important to note here that though RFT has a more compact form, terramechanics models can be utilized for broader terrain types (granted proper characteri-

28 3 A ageneric (Ncm ) fz B (N/cm n/2 1n/2 0I.4

0*0

I ~1-1 120.4 0 n/2 Z 0 ,t/2 Ck

Figure 4-2: Fitted form of RFT force functions obtained using Forier analysis by Li et al [251 formulation zation) and velocity regimes where the applicability of granular RFT (especially to cohesive soils) is yet to be verified.

4.4 Modeling wheel locomotion using implicit RFT code

Following on the lines of the RFT formulation mentioned in the previous section, the following section explains the core functioning of an iterative implicit RFT scheme implemented in MATLAB®. In terms of material/system properties, a single scaling parameter() is required which is sufficient to characterize complete wheel-terrain interaction properties. The formulation is applicable for any wheel shape and some examples are shown at the end of the section. The overall system consists of two parts, an intruding body, and the media being intruded. To begin with, utilizing the rigid wheel assumption, the wheel surface is discretized into smaller sub-surfaces that together approximated the total geometry of the wheel. The orientation, velocity direction, depth, and area of each sub-surface along with normalized force per unit depth (oz") from Li et al [25] and associated scaling coefficients ( ) is used for finding the resistive forces from the media on each subsurface. The net resistive force and moment on the wheel are then calculated by using a linear superposition principle mentioned previously along all the leading surfaces of the body as shown representatively in equations below. In performing all these simulations, a 'leading edge hypothesis' was used which made sure that

29 the resistive forces experienced by the wheel consisted contributions from only those surfaces which were moving 'into' the sand (and not other way round as the other case would have represented the cohesive nature of sand). That is,

Fresistive = f ((ax(, -y), a.(, y))H(z)IzIds (4.6)

ax,(P, ) = a ((, Y) (4.7) where, FResistive is the total resistive force acting on the wheel (intruding body), az,x are material specific force functions [Stress/depth], Z," are universal generic force functions [Stress/depth], ( is the grain-structure interaction parameter (Scaling parameter), # is angles of inclination [Fig 4-11, y is angles of intrusion velocity [Fig 4-11, H(z) is the heaviside function ( = 1 if surface is submerged, 0 otherwise), z is the intrusion depth and is always positive, Once the net resistive force is known, the wheel's motion is then captured using a momentum balance in the x-z lab frame coordinates

M'bcm =F gravity + FResistive + Fexternal (4.8) where, M is the mass of the wheel,vcm is the velocity of the wheel's center of mass, Fgravity is the force vector on the wheel due to gravity and FExternal is the external force vector on the wheel (e.g. pulling forces etc). It is to be noted that a special kind of forced slip experiments were also simulated using the RFT implementation (Chapter 6). In such cases, the wheel's translation, as well as angular velocities were predefined. Hence for such cases, a momentum balance in the x lab frame coordinate and angular momentum balance along the axis of the wheel, gave the values of total drawbar pull and torque (respectively) which wheel required to sustain the given velocity conditions. The vertical motion (Sinkage) of the wheel was captured by doing a momentum balance in z lab frame coordinate.

30 Red: Velocities Red: Velocities Blue: Forces Blue: Forces

Z x

Granular Media Granular Media

Red: Velocities Red: Velocities e: Forces Blue: Forces

Granular Media Granular Media

Figure 4-3: Simulations of various wheel shapes A) Elliptic wheel, B) Grousered wheel, C) Tank treads and D)Cylindrical wheel) using implicit RFT code

31 32 Chapter 5

Computationally intensive methods of simulating granular media

5.1 Background

The two semi-empirical approaches mentioned so far, namely terramechanical as well as RFT, have the advantage of being extremely efficient in the terms of computational effort required. But the low computational cost in both the methods comes at the expense of lesser information about the system. Both the methods provide with the stress at the surface of the wheel and do not provide any information about the media in which intrusion itself is taking place. The scenario is not undesirable as the objective of the methods was to evaluate the forces only but for expanding upon their fields and scenarios of applications of both the methods, a full system modeling is required. We use two different methods for full system modeling:

" Discrete element Method based Molecular Dynamics (MD) modeling

" Material Point Method(MPM) based Continuum modeling

5.2 Discrete Element Method (DEM I MD)

Discrete Element method is a well-established method of particle modeling which takes into account each particle of the system for modeling the system. We use Molecular Dynamics (MD) based LAMMPS software from modeling grains as and when required. The basic details of particle modeling are given further. The model being used in our work uses an inbuilt granular model based on work on Brinliantov and others [29, 30, 31]. The methods involve solving various system parameters at the individual particle level, taking into account the normal and tangential forces acting on adjacent, interacting particles, numerically integrated to find positions and

33 velocities. The model uses Hookean contacts for force calculation between a pair of particles and velocity verlet algorithm for position and velocity update (second-order accurate in time).

A B

Figure 5-1: (A)Schematic of two particle interaction [29], (B) A sample 2D plane- strain plate intrusion simulation using LAMMPS

For the calculation of forces following formulation is utilized:

F = (knij - meff nvn) - (ktAst + meff ytvt) (5.1)

Ft < pF, (5.2) The first equation above is Hookean contact definition. Force in normal direction is directly proportional to overlapping distance 6n1 and force in tangential direction is proportional to history-dependent tangential overlap Asj. The force also contains the effect of damping in the normal and tangential direction in form of N2 and 'yt coefficients. The second equation is the coulomb friction condition. While this method can be highly precise, large scale differences between the size of the simulated system and the individual particle size make the number of individual particles involved to be very large. For example, a 10cm x 10cm x 10cm system of particles of size 0.5mm would have 8x106 particles and 48x10 6 variables to solve (3 positions and 3 velocities) for each time step. Despite its computational cost, DEM remains the most reliable and proven technique at hand for modeling complex granular interactions. We primarily use this method for performing 2 and 3-dimensional intrusion scenarios at different velocities. Figure 5-2 shows a sample 2D flat plate intrusion experiment done using LAMMPS. Quantitative results are shown in relevant sections.

34 AJ

5.3 Continuum modeling

5.3.1 Constitutive laws While DEM methods are highly accurate but are computationally expensive, mean- while, a continuum model based computation can solve the same system by considering carefully selected chunks of material as blocks of uniform state to save large amounts of computation time. As mentioned earlier, the work of Askari and Karmin [26], in being successful to be able to regenerate current form of RFT was indicative of the fact that continuum modeling is a reliable tool of modeling low-velocity granular locomotion. A quick overview of the method and implementation is given later in this chapter. For more details, their paper can be referred [26]. Motivated by their work we implement following three constitutive models using 2D Material Point Method code developed by Dunatunga et al [32].

" Non-cohesive Granular media model

* Critical State Model for granular media

Non Cohesive Granular media model This model was included in the original work of Dunatunga et al [32]. The main properties of granular media being captured by the model are two: 1) Shear strength of the material is proportional to Hydrostatic Pressure (only for positive pressure). 2) The material loses all its strength if the density of the medium goes below the critical density. The free expansion prevents the system from ever having tension. Given below is the mathematical form of the model. Note that this sand model was obtained from p(I) by assuming I ~ 0 and that grains separate into a stress-free medium when below a critical density:

Figure 5-2: Schematic representation of Non-cohesive graular model. (Left) shear strength of material depends on Pressure. (Right) Material can not support tension.

ri < ,P and P = Pc if P>0 (5.3)

35 P = 0 if P < Pc (5.4) = 0 if P=O (5.5) where, = o'/v 21 Equivalent shear stress, P -1/3(tr(o)) Hydrostatic Pressure, ao= 0 + P1 Deviatoric part of stress tensor AND pc is the critical density of the material.

Linear Critical State Model for granular media

This model is an improved version of the previous non-cohesive sand model because this model allows for density variation in media [331. As an important observation, it can be observed from the constitutive equations that the gradient of packing fraction (#) over time always tries to evolve system towards the critical state density (in case of deforming media): Ac = Ps + (0 - #ss)X (5.6) do= (# - #")#x3 (5.7) dt where, 0sC" is Steady state critical packing fraction p," is Coefficient of Internal friction at steady state critical packing fraction x Dimensionless scaling constant i Equivalent shear strain rate For the purpose of implementation, the packing fraction was converted into the ratio of local medium density and grain density

= P/Pgrain (5.8)

Using the above substitution, the evolution equation for 4 gives: dp(C S)#oi (5.9) 1 - Pgrain dt

dp dt (5.10) (p - Osjpgrain)p Pgrain Using Pn = m/vn for a material point, where Vn is volume and m is mass of material point at time step n. And integrating both sides from tn to tn1

ln(1 - Sc Pgrainv V" i - (5.11) m V 36 Implicit volume update at each step is found as: /3( m Vne = 3vn + (1 - ) (5.12) Pgrain&bs where, 13 = exp(-Xy#O"At) This volume update (along with the density value below which material losses all its strength) is the only additional update in the previous non-cohesive model whose details can be seen in Dunatunga et al [32].

5.3.2 Implementation Attempts are being made by various authors including Dunatunga and Kamrin [32] to use the Material Point Method (MPM), a derivative of the fluid-implicit-particle (FLIP) method [341, based on the particle-in-cell (PIC) method [35] to perform granular plasticity simulations. The key idea behind MPM is that the state of the simulation is contained in Lagrangian material points, while the equations of motion are solved on a background computational mesh in a manner similar to FEM. Since the state is saved at each material point, the mesh is reset at the beginning of each computational step allowing for large deformations without mesh distortion. The basic computational layout for granular media is extensively discussed in Sulsky et al. [36]. Successful implementation, along with experimental verification of MPM, was done by Dunatunga et al for scenarios in high-speed intrusions. The model being developed by them for dry non-cohesive sand could be used in simulating locomotion and intrusion scenarios in various granular media and has been utilized here. For the implementation of various different intrusion scenarios (geometry and motion), evaluating and updating specific system properties and implementation of new material models, the materials files are updated as and when required.

37 38 Chapter 6

Low speed Locomotion

6.1 Circular wheel locomotion

With all the methodologies discussed in the previous chapters, we start with the problem of modeling a plane strain cylindrical wheel motion on a test-bed of non- cohesive granular material moving under low-velocity forced-slip conditions. While low-velocity locomotion represents the velocity regime in which the sand under the wheel can be considered to be in quasi-static state (more discussion about how to decide and define 'low velocity' limits, is done in next chapter on high-velocity speed locomotion), the forced-slip condition means that the wheel moves at a fixed predefined sets of angular velocity (w)and transnational velocity(v) conditions throughout the run. The output parameters being studied were drawbar force (required to be applied on the wheel to maintain input translation velocity), torque (required to be applied to maintain input angular velocity) and sinkage (resulted as a function of the constrained motion of the wheel). Figure 6-1 shows an equivalent system schematic as well as actual experimental setup being used for the study. A brief summary of the experimental setup and input parameters is given in the next section.

6.1.1 Experimental details The data collection for experiments was done by a previous student in collaboration with Crab Lab at GeorgiaTech, Atlanta. Hence limited details about the data collection are given in this document but more details about the same can be obtained in the relevant paper.

Testing Rig The 1m long, 0.6m wide and 16cm deep Lexan Testbed shown in Figure 6-1 B. A carriage slides on two low-friction rails attached to the main aluminum frame allows

39 A B

Drawbar pull

Sinkage Torque

Granular bed

Figure 6-1: A) Schematic and B) Experimental setup of forced-slip Terramechanics rig used in the study, where translation (v) and angular velocities (w) are controlled while Torque, Drawbar pull and Sinkage (z-direction motion is free) are measured for controlling longitudinal translation of the wheel. The wheel mount was allowed to freely translate in the vertical direction. Thus the translational and the angular velocity of the wheel as well as applied vertical load were controlled. As mentioned earlier, the experiments were conducted under a forced slip condition, the wheel angular velocity w and wheel longitudinal velocity v were controlled according to:

slip = 1 - V (6.1) rw where slip is the desired slip ratio and r is the nominal wheel radius. In the experiments wheel-angular-velocity was held constant while longitudinal velocity was varied to achieve the desired slip ratio. In terms of output parameters, draw-wire encoders were used for horizontal and vertical displacements, a 6-axis force torque transducer mounted between the wheel mount and the carriage was used to measure the vertical load and traction generated by the wheel and an ange-to-ange reaction torque sensor was used to measure the driving torque applied to the wheel. The rig was capable of providing approx 1m run, 120 mm/s maximum translation velocity, and 400/s of wheel angular velocity. For the experiments with Poppy Seeds, a similar setup with controlled packing fraction was used. The vertical loads were varied between 18 N and 190 N mentioned in related experiments.

Materials

Three simulants were used in this work: Quikrete 1962 medium sand (MS, predom- inantly silica particles of size 0.3-0.8mm), Mars Mojave Simulant (MMS, mixture of

40 finely crushed and sorted granular basalt with 80% of particles within 10,pm intended to mimic, Mars soil characteristics) and Poppy Seeds (PS). Various soil properties measured using standard plate penetration tests and terra-mechanical methods [10 are presented in Table 6.1. Specifically, MS and MMS were characterized through plate penetration tests and direct shear tests. The RFT constant for these simulants was extrapolated from the plate penetration tests. The Poppy Seeds (PS) on the other hand, were only characterized by plate intrusion experiments.

Table 6.1: Granular material properties for slow locomotion studies MS MMS PS (# = 0.58) PS (# = 0.60) k, [kN/mn+1J -2.05e+4 846 -2.06e+5 -3.24e+5 k. [kN/mn+2] 3.13e+6 6708 7.07e+6 1.11e+7 n 1.0 1.4 1 1 c [Pa] 1500 600 0 0 4D [deg] 34 35 36 45 kx [m] 0.0006 0.0006 0.045 0.045 RFT Constant [N/cm 3l 2.02 3.05 0.35 0.55 K [m] 0.0006 0.0006 0.045 0.045 3 Pgrain [kg/M ] 2600 2875 1100 1100 Packing Fraction # 0.6 0.6 0.580 0.605 MPM : Pinternal 0.53 0.50 0.53 0.54 MPM : Psurface C: 0.55 C: 0.55 C: 0.55 C: 0.55 B: 0.35 B: 0.35 A: 0.60 A: 0.60

For the MPM based continuum modeling, it was assumed that the motion of all the wheels considered in this study can be modeled as plane strain problems (which is a justifiable assumption to take if the out-of-plane depth of contact area between wheel and sand is larger than its width). The plastic flow parameters for the simulations were calibrated by matching zero-slip experimental data to zero-slip plane-strain MPM simulations. Since the actual deformation in experiments was not always plane-strain, we accept potential inaccuracy brought about by the plane- strain simplifying assumption. The MPM simulations were found to be most sensitive to the internal coefficient of friction of sand (MS/MMS/PS). The effective internal friction values (plinternal) for each media was evaluated by finding the value of Pinternal which when used in MPM simulation results in the same sinkage as found experimentally. This matching was done once (for the zero-slip case) for each media and once that value was obtained, the same value was used for all the simulations throughout the study. Values for all four materials are shown in Table 6.1. Calibration trials for deciding the surface friction

41 coefficient (Psurf ace) between the wheels and the sand were found to be accurate enough with the use of original 3D friction coefficients and hence various original wheel-sand pair values (reported in Table: 6.1) were used.

Wheels Experiments were conducted with three different wheels with aspect ratios (i.e., width/radius) of 0.5, 1.05 and 1.23. The wheels are shown in Figure 6-2, while wheel dimensions are given in Table 6.2. Wheels A, B, and C have been tested on PS, while wheel C has been tested also on MS and MMS.

Figure 6-2: Wheels nomenclature for slow locomotion study (Image not to scale).

Table 6.2: Specifications of the wheels utilized in this study and summary of the experiment conducted. Surface coating, 60 grit, PLA, MMS. A B C Type Smooth Wheel Lugged Wheel Smooth Wheel Radius [mm] 101.6 72.5 (to lug tips) 130 Aspect Ratio 0.5 1.05 1.23 PS V V MS - / MMS - / Vertical Loads [N] 20 18 80-190

Wheel A is a Nylon wheel with a narrow aspect ratio. The wheel surface was coated with 60 grit sandpaper in order to guarantee sufficient friction at the wheel- terrain interface. Wheel B was manufactured using a MakerBot Replicator II 3D printer using PLA filament. The wheel has 15 lugs, equally spaced, 10 mm tall and 11 mm thick, which span the whole width of the wheel. This wheel has no sandpaper coating, as the presence of the lugs guarantees sufficient wheel-terrain engagement. Finally, wheel C is an aluminum cylinder coated with MMS. For continuum modeling of wheel-media surface interaction, the coefficient of surface friction for wheel C with

42 all the simulants was taken as 0.55 and for wheel A and B (which were experimented only with PS), the values were 0.60 and 0.35 respectively.

6.1.2 Continuum modeling using MPM

Displacement Magnitude Equivalent plastic strain rate 1.0e-4 1.0e-3 1.0e-2 1.00-1 1.0e+0 1.09-2 1.00-1 1.00+0 1.0e+1

5t=to

t = t1

Figure 6-3: A sample MPM implementation of wheel Type C in LPS at negative slip ( i = -0.3) velocity condition. Time to corresponds to state when the wheel is freely resting on the medium, t1 corresponds to the transition state and t 2 corresponds to time instance when the equilibrium sinkage condition is met.

The MPM algorithm described in Dunatunga and Kamrin 132] was used to implement the set of constitutive equations given in section 5.3.1. The values of relevant material properties for various simulants used in this study are provided in Table 6.1. The wheel was modeled as an elastic solid with fixed horizontal translation speed and a fixed angular velocity, which are instantaneously applied on the wheel explicitly. In terms of simulation resolution, a 200 x 200 grid was used to represent a domain size of im x im with 2 x 2 linear material points seeded per grid cell at the beginning of the simulation. Figure 6-3 shows a sample simulation done using the

43 MPM implementation. As a common feature of plasticity based solutions to granular intrusions, an intermittent shear-band structure is seen to emerge surrounding the wheel, though the displacement itself was observed to be smooth.

6.1.3 RFT based modeling Similar to that explained earlier in Chapter 4, RFT simulations were implemented using an implicit iterative scheme in MATLAB. Utilizing the rigid wheel assumption, wheel surfaces were discretized into smaller subsurfaces that together approximated the total geometry. Being a forced-slip study, the structuring of the code was little different than earlier. The orientation, velocity direction, depth, and area of each sub-surface along with normalised force per unit depth (from Li et al [25]) and associated scaling coefficients from Table 6.1 were used for finding the resistive forces from the media on each subsurface. The net resistive force and moment on the wheel were calculated using the RFT superposition principle mentioned previously. As the wheel's x-translational motion was predefined (forced slip tests), a momentum balance in the x lab frame coordinate and angular momentum balance along the axis of the wheel, gave the values of total drawbar pull and torque (respectively) required to sustain the given velocity conditions. The vertical motion (sinkage) of the wheel was captured by balancing momentum in the lab frame z coordinate. In performing all these simulations, a 'leading edge hypothesis' was used which made sure that the resistive forces experienced by the wheel consisted of contributions from only those surface elements which were moving 'into' the sand, i.e. surfaces whose outward normal and velocity make a positive dot product. A sample RFT simulation setup for wheel-type B is shown in Figure 6-4.

Time: 3.5 sec

Red: Velocities Blue: Forces

Granular Media

Figure 6-4: A sample implicit RFT implementation (wheel Type B) in MATLAB where red arrows represent the normalized velocity vectors of the wheel surface elements, and blue arrows show the normalized resistive force vectors on each subsection.

44 6.1.4 Results

The results section is structured as follows: The performance of wheel C on PS prepared under various packing states is described first. These experiments have two primary aims, first is to study the sensitivity of the RFT model to granular material density, and the second is to analyze the capability of MPM-based continuum modeling in capturing system dynamics. Subsequently, the performance of wheels A and B on PS are presented. These experiments are aimed at investigating the ability of both the aforementioned methods in predicting the performance of wheels with diverse thickness-to-diameter aspect ratios. Finally, the performances of wheel C on MS and MMS sands are presented. These experiments are aimed at examining the capabilities of RFT to accurately model wheel performance when the force response surfaces for the granular material are not directly available, while also examining the capability of MPM continuum modeling for the cases. Each experiment was performed at least five times, with the boxplots (Figures: 6- 5, 6-6, 6-7 and 6-8) showing the average and standard deviation. In order to quantify the performance of the various methods involved, several error metrics defined below were evaluated. Each of these metrics can help understand a particular aspect of the correlation between the model predictions and measured data. The metrics under consideration are the mean absolute error, the coefficient of correlation, and the coefficient of variation. The mean absolute error A is defined as follows:

Ik A = |Xe - XIi (6.2) where X, is the experimental average (either traction F, torque M, or sinkage z), Xm is the model prediction, and k is the number of data points used in the evaluation. The mean absolute error provides an estimate of the absolute deviations and has the dimensions of the quantity under investigation. The coefficient of correlation R is used to evaluate the correlation between the trends of the modeled predictions and the measured values. The coefficient of correlation R is defined as

k Ek XeXm - Ek R= X- X. En X. (6.3)

k X2 -( X ) k X2 -- ( Xe)]

A value of 1.0 for the coefficient of correlation R, indicates a perfect correlation between the trends of the predicted and measured data. The correlation will generally be regarded as strong if the value of R is greater than 0.8. With a value of R less than 0.5, the correlation is usually regarded as weak. Finally, the coefficient of variation

45 CV is defined as follows:

2 c _ CV =k64 (Xe - X) (6.4) k ((Xe2

Where, Xe and X, are experimental and model predicted values respectively and k represents the total number of slip values at which experiments are done for a given load value. The CV provides a normalized measure of deviations. If the value of CV is zero, the predicted and measured data will have a perfect match, representing a zero deviation of the model from experiments.

Sensitivity to Poppy Seeds packing state Figure 6-5 presents experimental results of the wheel C on PS. To highlight potential quantitative differences in wheel performance during travel on a soil in loose and compact states, the granular material was prepared at two packing fractions that were chosen to span the onset of dilatancy. For the packing states selected, the poppy seeds show different behavior under plate penetration tests, which results in different RFT properties. However, experiments show that wheel performance is moderately affected by terrain preparation. Drawbar measurements for the loose and dense states are within 25% of each other. In absolute terms, the difference between loose and dense packing does not exceed 4 N for any tested slip level. Torque measurements stay within a 7% difference, while the sinkages' average variation is 11%. The fact that wheel performance is unaffected by terrain preparation is surprising since on firmer terrain one would expect less sinkage and thus increased traction. The high difference in the angle of repose of LPS and CPS confirms large differences in the initial medium state; the small difference in wheel performance is surprising. It is possible that this is a result peculiar to the poppy seeds' mechanical properties or it could be the low penetration of wheel into the medium which causes these effects. Table 6.3 presents the values of mean absolute error, the coefficient of correlation, and the coefficient of variation for RFT, MPM, and TM models. Excepting drawbar outcomes, RFT and MPM consistently show lower mean absolute error values, high coefficient of correlation values, and lower coefficient of variation values than TM. In particular, the RFT performance improves when high density terrain parameters are used, especially when sinkage is considered. The high coefficients of correlation show that all the models follow the trends of experimental data. The lower coefficient of variation highlights how MPM performs better than other two models, especially for torque measurement. It should be reiterated that the MPM simulations were conducted assuming plane strain conditions in the wheel locomotion. The approximation is less valid at high wheel sinkages due to the reduced aspect ratio of wheel-media interface area, hence

46 20r 13 Experiment RFT MPM 0 TM z I- U- -20-

op .0 -40-

(a) (b)

U U U U U U 6 U.. U, z U U. -- U. I- 'U 2 (D U * - S

0 -2

60F -' 40 U, f 20-

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Sil) SilD (e) (f)

Figure 6-5: Wheel C on Poppy Seeds. (a), (c) and (e) correspond to dense poppy seed state (# - 0.60) while (b), (d) and (f) correspond to loose poppy seed state (# ~ 0.58). Experiments were performed five times and boxplots present the average reading and one standard deviation. Nominal vertical load is 120 N. Resistive force theory (RFT), Continuum modeling(MPM) and terramechanics (TM) approaches produce similar predictions for drawbar, while the RFT and MPM outperform the TM model when sinkage (at dense state) is evaluated. Torque predictions show visible deviation for all the models with the RFT and MPM producing estimates closer to measured values. 47 Table 6.3: Comparison of resistive force theory (RFT), continuum modeling (MPM) and terramechanics (TM) models' predictions for wheel C on Poppy Seeds under 120 N nominal load. The mean absolute error A has the dimension of [NJ for drawbar, [NmJ for torque, and [mm] for sinkage. The coefficient of correlation R and the coefficient of variation CV are unitless. Compacted Loose RFT MPM TM RFT MPM TM Drawbar A 5.05 6.84 3.41 5.64 7.80 3.58 R 0.92 0.94 0.99 0.96 0.93 0.99 CV 0.08 0.12 0.08 0.07 0.11 0.05 Torque A 1.68 0.81 4.30 1.68 0.64 4.46 R 1.00 0.97 0.98 1.00 0.99 0.99 CV 0.32 0.16 0.78 0.33 0.15 0.88 Sinkage * 2.73 3.16 26.07 9.94 4.41 26.48 f 0.97 0.96 0.93 0.93 0.96 0.86 CV 0.11 0.10 0.80 0.30 0.14 0.76 an exact matching of results for high sinkage cases is not expected. The terrain parameters for the TM model(only for PS) were also not calculated according to standard terramechanics practices. According to terramechanics guidelines, the dimensions of the intruder used for finding TM fitting parameters should approximately be the same as the average contact patch area of the wheels. But in the above analysis, wheels had a much different contact patch area than that of the intruder (2.5 x 3.8 cm2 ). Hence, this could partially explain the poor performance shown by the TM model. In order to obtain meaningful drawbar predictions, the shear displacement modulus was set to 0.04 m which is larger (by a factor of two) than any value found in the literature. A large shear modulus means that larger deformations are needed to generate shear stress which can be consistent with the nature of poppy seeds.

Sensitivity to wheel geometry on Poppy Seeds Figure 6-6 presents the results obtained with wheels A and B on dense poppy seeds. These wheels have different aspect ratios and geometries, with wheel B being a lugged wheel and wheel A being a smooth wheel. Table 6.4 presents the values of mean absolute error, the coefficient of correlation, and coefficient of variation for the RFT, continuum model (MPM), and the TM model. For all the outputs considered here, RFT consistently shows lower mean absolute error, higher coefficient of correlation,

48 and lower coefficient of variation values than TM model for torque and sinkage.

Table 6.4: Comparison of resistive force theory (RFT), continuum modeling(MPM) and terramechanics (TM) models predictions for wheel A and B on Poppy Seeds under 20 N and 18 N nominal load respectively. The mean absolute error A has the dimension of [NJ for drawbar, [Nm] for torque, and [mmJ for sinkage. Coefficients of correlation R and coefficients of variation CV are unitless. A B RFT MPM TM RFT MPM TM Drawbar A 0.94 1.39 1.02 1.65 2.53 1.26 R 1.00 0.98 0.99 0.99 0.96 0.98 CV 0.07 0.11 0.07 0.10 0.16 0.08 Torque A 0.10 0.21 0.41 0.05 0.08 0.26 R 0.99 0.87 0.98 0.99 0.91 0.98 CV 0.21 0.43 0.74 0.20 0.29 0.82 Sinkage ,A 1.60 5.15 17.73 3.10 4.49 10.98 R 1.00 0.87 0.83 0.87 0.64 0.75 CV 0.08 0.24 0.76 0.21 0.25 0.66

On the other hand, considering drawbar force, TM performs better than RFT (though the difference is not high with both methods having CV below 0.10 and R above 90%). Thus based on the requirement of high R and low CV, it can be concluded that for the cases considered here, in general RFT shows a better performance than the TM model. The performance of MPM appears to be on par with the TM model in all of the above cases. Considering torque, while the CV values are high, the absolute value of the mean error is within 0.3 Nm. Sinkage comparisons show a better performance (lower mean absolute error) of MPM than TM, but the correlation coefficient values were low. As mentioned before, performing continuum simulations for wheels of narrow aspect ratios under assumption plane-strain conditions is another possible source of error.

Sensitivity to vertical load on MS and MMS sands Set of experiments with wheel C on MS and MMS, for a wide range of vertical loads ranging between 80 N and 190 N were performed. For this data set testbed did not allow for material preparation to a specified packing state. However, the terrain was

49 41 c Experiment - RFT ------MPM 0 --- TM z U- X L1L -4

-8 0 F.

(a) (b)

1 U U

U 0.6- ......

z U -

0.2- U -

0 I- -0.2-

-0.6( (C) (d)

4 0r

30-

~,20- U ------C 10

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Slio Slio (e) (f)

Figure 6-6: Wheel A (a,c,e) and wheel B (b,d,f) on dense poppy seeds. Experiments were performed five times and boxplots present the average reading and one standard deviation. Nominal vertical load is 20 N for wheel A, and 18 N for wheel B. All experiments are conducted for a packing state of 0.60. Resistive force theory (RFT), continuum modeling (MPM) and terramechanics (TM) approaches produce similar predictions for drawbar, while the RFT outperforms MPM and the TM model when sinkage is evaluated. Torque predictions show visible deviation for all the models with the RFT and MPM producing estimates more closer to measured values. 50 Experiment

50r o Experiment SExperiment - RFT -RFT - - -MPM - - -MPM

0 X rT U- U

-50 0

(a) (b)

15r o Experiment c Experiment - RFT - RFT - - - MPM - - MPM 10 E

0 0

o Experiment Experiment - 50 - - RFT RFT --- MPM -MPM

~30

C 10

-0.5 0 :j -0.5 0 0.5 1 -1 -0.5 0 0.5 1 slip Slip (e) (f)

Figure 6-7: Wheel C on MS (a,c,e) with vertical loads of 80N, 130N,150N,190N (light to dark) and MMS (b,d,f) with vertical loads of 80N, 110N,150N,190N (light to dark). MS experiments were performed ten times and boxplots present the average reading and one standard deviation. The relevance of these results lies in the fact that terrain characterization for the MS and MMS was not performed according to standard procedures utilized by RFT. Hence, this analysis shows the full potential of RFT application to generic granular materials, wheel geometry, and loading conditions.

51 Table 6.5: Performance metrics for the RFT predictions of wheel C on MS and MMS. The mean absolute error A has the dimension of [NJ for drawbar, [Nm] for torque, and [mm] for sinkage. The coefficient of correlation R and coefficient of variation CV are unitless. MS 80 N 130 N 150 N 190 N RFT MPM RFT MPM RFT MPM RFT MPM Drawbar A 5.55 6.42 6.16 9.74 6.33 11.78 7.11 14.21 R 1.00 0.96 0.99 0.97 0.99 0.97 0.98 0.96 CV 0.09 0.10 0.08 0.11 0.07 0.12 0.07 0.12 Torque A 1.24 0.93 1.79 1.17 2.11 1.39 2.82 1.38 R 0.99 0.97 1.00 0.97 1.00 0.97 1.00 0.98 CV 0.39 0.29 0.36 0.26 0.36 0.26 0.38 0.20 Sinkage A 2.74 2.70 3.62 3.45 3.67 2.41 2.81 1.69 R 0.86 0.93 0.72 0.86 0.85 0.97 0.92 0.95 CV 0.28 0.30 0.28 0.23 0.26 0.14 0.18 0.09

MMS 80 N 110 N 150 N 190 N RFT MPM RFT MPM RFT MPM RFT MPM Drawbar A 11.42 12.11 13.03 16.26 14.01 21.91 15.11 23.11 R 0.97 0.93 0.97 0.93 0.97 0.95 0.95 0.93 CV 0.15 0.17 0.14 0.18 0.13 0.21 0.14 0.21 Torque A 0.93 1.43 1.42 1.98 1.76 2.20 2.22 2.62 R 0.98 0.94 0.99 0.94 1.00 0.94 0.99 0.95 CV 0.31 0.56 0.36 0.46 0.33 0.41 0.32 0.40 Sinkage A 5.75 5.33 5.85 3.52 7.47 4.86 18.41 10.80 R 0.50 0.44 0.70 0.88 0.85 0.97 0.93 0.90 CV 1.37 1.45 0.59 0.45 0.55 0.31 0.56 0.33

carefully prepared between tests in order to achieve repeatable consistent loosely packed conditions. The relevance of using RFT for these experiments lies in the fact that terrain characterization for the MS and MMS was not performed according

52 to the standard procedures utilized by RFT for poppy seeds. The force response surfaces for these materials were obtained using scaling of similar response surfaces for PS using corresponding scaling parameters presented in table 6.1. Hence, this analysis shows the full potential of RFT application to generic granular materials. For this analysis, TM modeling was not done but continuum analysis (MPM) for these experiments was done in a similar fashion as before. Figure 6-7 presents the results for four vertical loads (80 N, 110/130 N, 150 N, 190N) for wheel C traveling on MS (a,c,e) and MMS (b,d,f). For the MS sand, both the RFT and MPM underestimate (in absolute value) drawbar pull, while they both underestimate torque for positive slip only. Sinkage predictions are accurate for the whole slip range with the mean absolute error in the range of 4-5 mm. For the MMS sand, similar trends are observed with less accuracy and larger absolute errors at high positive slips. Table 6.5 presents the values of mean absolute error, the coefficient of correlation, and the coefficient of variation for the RFT and MPM model. Regardless of the quantity under consideration, RFT performs better at lower vertical loads in terms of mean absolute error for both the models. This is also partially true for sinkage. The coefficient of correlation is above 0.85 in all cases except one. In general, the coefficient of variation decreases with increasing load for both the models. Increased variability in the sinkage measurements comes from the uncertainty in controlling the terrain's free surface level and flatness. With increasing load, sinkage increases, which then leads to decreased relative quantity of aforementioned errors. The performance of MPM is comparable to RFT in most cases and is observed to be better in a few cases (based on CV and R data).

Comparison between RFT and TM Although the MS and MMS sands were characterized following best practices for TM models, results obtained with the TM model remain inaccurate when using the shear modulus obtained from direct shear tests. As discussed in 1221, the shear modulus calculated from direct shear tests is in the order of tenths of millimeters, creating unrealistically high drawbar and torque predictions. However, even if the shear modulus is treated as a tuning parameter, TM predictions generally remain less accurate than RFT. For this analysis, predictions for the TM models are provided with the measured shear displacement modulus and a modulus of 0.015 m (the corresponding results are labeled TM*). The discussion below is based on the results obtained with the larger shear modulus. Results presented in Figure 6-8 show performance for wheel C under 130 N of the vertical load while traveling on MS. Table 6.6 presents the values of mean absolute error, the coefficient of correlation, and coefficient of variation for the RFT and the TM models. When analyzing drawbar, the RFT has a similar coefficient of correla-

53 150 20 40 150l Experiment 20 - Experiment 40 Experiment RIFT '- RFT .....-...- RFT 75 T . .-..... TM . 30 ...... TM Z TM Z ...... U- -TMTM E .. TM* 0 10 20---TM

-75 -10 - 10

-150 ...... -.201 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Slip Slip Slip (a) (b) (c)

Figure 6-8: Comparison of RFT and TM model for wheel C on MS. Experiments were repeated ten times and boxplots present the average reading and one standard deviation. Nominal vertical load is 130 N. Resistive force theory (RFT) and terramechanics (TM) approaches produce similar predictions for drawbar and torque at positive slip, while the RFT outperform the TM model when sinkage is considered. tion, but lower mean absolute error and coefficient of variation than the TM* model. The TM* model deviates significantly from measured data at negative slip. The situation is similar for torque. However, in this case, the RFT underestimates torque readings for the whole range, even if it maintains a high coefficient of correlation at 0.99. The analysis is more intricate when sinkage is considered. Qualitatively, the TM* model accurately describes the data at low negative slip, while RFT predictions are closer at positive slip level. As a result, the TM* and the RFT model have similar metrics with a mean absolute error close to 4 mm, a coefficient of variation below 0.4, and a coefficient of correlation above 0.7 for both. TM* model performance for drawbar and torque predictions is similar to the RFT when only positive slip is considered. This is relevant because, for design and evaluation purposes, the performances between 10% and 30% slip are typically used as indicators. However, as shown by the wheel-terrain configurations previously discussed, sinkage predictions were inaccurate when TM* model was used.

6.1.5 Conclusion In this work we analyzed the performance of resistive force theory (RFT) and MPM based continuum modeling to the problem of predicting rigid wheel-dry granular media interaction. Upon comparison of experimental data for three differently shaped rigid wheels under forced-slip and variable load conditions, we concluded that though RFT was originally developed for studying legged locomotion on granular media, it can also be used as a qualitatively and quantitatively accurate model for the locomotion of rigid wheels on granular materials. The current work also es-

54 Table 6.6: Comparison of resistive force theory (RFT) and terramechanics (TM) models predictions for wheel C on MS under 130 N nominal load. Columns labeled TM* refer to the TM model with shear modulus k = 0.015 m. The mean absolute error A has the dimension of [NI for drawbar, [Nm] for torque, and [mm] for sinkage. Coefficients of correlation R and coefficients of variation CV are unitless. Drawbar Torque Sinkage RFT TM TM* RFT TM TM* RFT TM TM* A 6.16 71.84 18.30 1.79 9.66 3.59 3.62 4.84 4.52 R 0.99 0.93 1.00 1.00 0.91 0.97 0.72 0.57 0.85 CV 0.08 0.76 0.23 0.36 2.14 0.98 0.28 0.39 0.36

tablished plasticity-based continuum modeling using an MPM implementation as a suitable candidate for predicting wheel performance. MPM studies give complete flow and stress fields which gives deeper insight about the system and is of vital importance for improving our understanding of locomotion processes. Quantitative comparison was done via comparing drawbar, torque and sinkage with model predictions. When compared to the TM model, the torque and sinkage predicted by RFT as well as the continuum model show lower mean absolute errors and coefficients of variation for wheels of different aspect ratios moving on loosely/closely packed poppy seeds with MPM having higher accuracy than RFT in predicting torque as well as sinkage values in general (except for wheel A on PS). Drawbar values calculated with RFT, MPM as well as TM models are close to the experimental measurement (within 25%), with TM having better accuracy than other two methods and RFT having better accuracy than MPM. Considering the empirical nature of the TM model, it should be possible to obtain better predictions by performing an ad-hoc characterization of the PS simulant. In fact, by performing pressure-sinkage experiments using a plate with an area comparable with the contact patch of the wheels under consideration, it should be possible to obtain pressure-sinkage parameters that result in more accurate TM model predictions. By extrapolating the response surfaces from poppy seeds, we used RFT to model forced-slip experiments of rigid wheels on two natural sands (MS and MMS). RFT predictions showed to be qualitatively in agreement with experiment, exhibiting overall better performance when compared to the TM model. However, quantitative dis- agreement between models and experiments across the whole slip range remains. For example, drawbar in RFT is generally overestimated (in absolute value) at large slip ratios (> 0.4 or - < 0.4), and underestimated at low positive slip ratios (between 0 and 0.2). While RFT calculated sinkage values are accurate for MS for the whole slip range under all wheel loadings, for MMS the predictions show significant deviations, particularly under large loadings. For MS and MMS, it remains to be seen how the accuracy of RFT predictions is affected if the response surfaces are directly obtained

55 (in the current study response surfaces were extrapolated). It is to be noted that though the MPM simulation results did in this study have some variations from actual experiments, a major reason could be the fact that the MPM implementation here was done assuming the granular motion to be plane- strain in all the test cases. The assumption can be justified in low sinkage cases where the out-of-plane width of the contact patch of the wheel interface is much larger than it is in-plane width. A fully 3-dimensional model could help eliminate the issue. Thus, while this study focused on quasi-static, low velocity, forced-slip wheel motion, future studies are planned to experimentally and computationally explore the angular velocity driven, free locomotion of rigid wheels (in 3D) at wider ranges of angular velocities on Poppy seeds as well as other simulants to explore high speed locomotion dynamics as well as the capability of these methods in modeling the different scenarios.

6.2 Non-Circular wheel locomotion

In the previous study, the performance of RFT and continnum modeling was proven for largely circular wheel geometries at the equilibrium states. In order to further understand the RFT capabilities, studies were done to verify the accuracy of RFT in modeling more complex wheel shapes. It is to be noted that for these studies, continuum modeling was not done. A five flapped wheel shape was chosen as the wheel shape and is shown in Figure 6-9. Each flap angle was fixed at 700 wrt. inner circumference for the entire duration of the study. While the previous studies were done under forced slip conditions, true locomotion was used for this study i.e. only the angular velocities of the wheel was controlled while translation velocity, as well as vertical displacement (sinkage) was decided by system dynamics. The performance of RFT in modeling the wheel was measured in terms of accuracy in modeling variation of three output variables over time, namely: 1) Observed sinkage 2) Torque requirement measured at the motor and 3) Power requirement of the motor to drive wheel at a required angular velocity. Although the experiments were repeated for a range of velocities, the results below are shown only for a velocity of 20"/sec. Specific details of various aspects of the study are mentioned below:

6.2.1 Experimental details Test Rig The experimental setup used in the study of the circular wheel was also used here with one additional update of using flapped wheel instead of a fixed shape wheel. For controlling the flaps of the wheel, a 90 W Maxon DC Motor in combination with ADS 50/10 4-Q-DC servoamplifier was used. Control and measurement signals

56 are handled by a NI PCIe-6363 card through Labview software. The apparatus was capable of approximately 1 m of total horizontal displacement with a maximal wheel angular velocity of approximately 400 s. The testbed length was 1 m, width was 0.6 m and the soil depth was 0.16 i.

Simulant Quikrete medium sand with grain density of 2600kg/nm 3 and grain size in the range of 0.3-0.8 mm range was used. For RFT simulations, the scaling coefficient ( ) was taken to be 2.02 (same as the previous study, more details can be obtained from Table: 6.1).

Wheel

Figure 6-9: Top and side view of flapped wheel

The wheel used for this study (shown in figure 6-9) consisted of a 10cm wide circular base of 8cm outer diameter (close flap diameter) with five identical flaps attached to the circumference with each of them having a radius of curvature same as radius of the wheel such that in the completely-closed state, they form the circumference of the wheel. The mass of the wheel was 8 kg with a vertical force of 66.7N acting vertically up on it. The mass of the carriage assembly carrying the wheel on the horizontal friction-less carriage was 9kg. For the entire duration of the study the flaps were kept at 700 wrt tangent to the circle at the base of the flaps. The Wheel had different effective inertia(s) in the vertical and horizontal direction(s) due to the fact that in the horizontal direction, it needs to accelerate itself as well as the platform, while in the vertical direction, the weight of only the wheel is required to be supported (on frictionless rails). The exact formulation for incorporating this difference is given in the next section.

57 6.2.2 RFT based modeling The RFT simulations were done in an exact way explained in chapter 4. The force balance equations considering the difference in the inertia of the wheels in two directions are shown below. A sample simulation frame for this wheel is shown in figure 6-10.

Un+1[ U(n sF.(n+1, 7n+1)) /mX UflX + = t -- mZg + F, + f9s F2(On+1, (6.5)

where, x,z are displacement and ux,u, are axis velocities in corresponding directions. mx is effective mass in horizontal direction with mx = mwheel -+ mcarriage mZ is effective mass in vertical direction with mz = mwheel OS represents integral over leading edge surface of wheel 3 and -y are orientation and velocity direction angles (Ref Chapter 4). n is current time step, n + 1 is next time step and At is time step size. Fup is net external upward force on the wheel. g is acceleration due to gravity.

6.2.3 Results Figure: 6-10 shows the variation of sinkage and torque requirement of the wheel over time when it is rotated at 20"/sec. It was observed that while the accuracy of RFT in capturing the dynamic variation of sinkage as well as Torque requirement of locomotion is very high, the average value of horizontal translation velocity is also quite accurate. The absence of an exact match between RFT and experiments can be attributed to mechanical constraints on the performance of the motor in terms of the application of torque to maintain the given angular velocity.

6.2.4 Conclusion This study further verifies the robustness of RFT in modeling slow-speed, near- surface intrusion in non-cohesive granular media for objects of various shapes and sizes (until the particle size effects does not come into the play for objects with sizes within 5 to 7 times grain diameter). In the further studies, the layout of developing smarter control systems making use of robust RFT and performance of granular RFT in high-speed intrusion /locomotion process is discussed.

58 -1

Red: Velocities Blue: Forces

Grr~ Mock

0 1-26 xemea ea Evpenwenal Resumts 12Expwn tal Rss RFT SdIuADc Pes4*s 20 RFT on Rssdh 00- 007

0 0

001

S5 10 Is a 6 10 ts 0 6 10 1 Time(sec) TOWl(S-C) Ton

Figure 6-10: Experimentation with Flapped Wheel: Verification of capability of RFT in modeling complex variables like torque and sinkage for non-circular wheel ( flaps being opened at 700 through out the experiment ) moving at angular velocity of 20 deg/sec. A) Experiment and B) RFT simulations of free locomotion. C) quantitative results: Sinkage of center of wheel, velocity being achieved and torque required to maintain the given angular velocity.

59 60 Chapter 7

Locomotion optimization with RFT

7.1 Introduction

Besides numerous applications of developing a fast, reliable and accurate method of modeling resistive forces on bodies intruding through a granular media, one of the most important and direct applications of the same is in locomotion optimization. The first step in optimization comes in the form of evaluation of expected velocity of locomotor based on various system inputs. Taking wheel as the locomotor under consideration, the motion of a wheel can be represented in functional form (41)) with resultant velocity v as the output and various system parameters as inputs as:

V =

61 7.2 Experimental setup

7.2.1 Wheel

For the purpose of experimental testing of this work, the flapped wheel introduced in section 6.2 was used. In reference to this wheel, while all other variables introduced in equation 7.1 have their usual meaning and are represented in figure 7-1, 0 is an unusual parameter in the equation 7.1 and it corresponds to the angle at which the flaps are open wrt direction of tangent to the circle at the base of the corresponding flap, thereby representing the shape of the wheel. Note that the above formulation ignores other wheel-shape related parameters like diameter, number of. flaps etc as they are expected to remain constant in all the condition while parameters like mass and gravity can vary based on locomotion conditions. A schematic representation of all the associated parameters in the formulation is shown in figure 7-1.

V FTowing

Figure 7-1: Representation of various parameters associated with flapped wheel(left) and closed & open (right) state of the wheel (based on desired angle requirement).

7.2.2 Test rig

For running these experiments an in-house experimental setup was developed and is represented in figure 7-2. This setup allows for running length of 10ft and is 3ft wide. The basic structure of the bed is similar to that explained in the discussion of non-circular wheel locomotion (section 6.2) with the major difference being that of a longer test bed than that previously used. Though some of the basic test cases were run on this setup, the results are not presented here as proper analysis and calibration are yet to be done.

62 B*~i

Figure 7-2: (A) Schematic and (B)Lab setup used for flapped wheel locomotion studies. (C) shows Carriage assembly for the wheel. Inset of (B) shows the side view of the flapped wheel in open state.

7.3 Discussion

Reconsidering the equation 7.1, the problem reduces to evaluation of the function 4<. Conventionally, the solution can be obtained from a terra-mechanical Bekker type theory or a full plasticity based continuum simulation or a discrete element solution. While the later two methods are numerically slow to be evaluated on a real-time basis, a better alternative to terra-mechanical formulation is available in form of RFT now. Being a non-linear system, the function

63 sidering the following scenario. We assumed that we are given a fixed value of mass, gravity, dimensions, and drawbar pull associated with a wheel. And the aim of the wheel controller is to achieve a fixed translation velocity. In such scenarios, there are two variables that the controller can vary: flap angle(6) and angular velocity(w). Now, as it can be observed that the solution to this problem is not unique. There are multiple possible combinations of 0 and w which can lead to the same translation velocity. While a conventional controller (getting real-time feedback from the wheel) will choose a state which is first encountered, a RFT based controller will know all the possible states beforehand and choose a state based on additional requirements like the state which can be achieved in minimum time (based on reaction time of motors in flap and rotor), a state which results in minimum power requirement, a state which results in minimum opening of flaps etc. Just to illustrate, Figure 7-3 shows the variation of translation velocity (normalized by closed flap circumferential velocity, RbaseW) for different combinations of angular velocity, flap angles and drawbar pull. Each of the data points can be obtained using RFT on a real-time basis (using implicit RFT implementation as used by us in earlier works) and hence can be used for smarter optimization.

Drawbar [NJ 2 0 (A=40'Is (o=60"/s 22 44 66

110

132

0.5 154

2 r 0 =80*ls = 100*/s 28 56 1 I 84 112

140

168 0.5 L%6 40 60 80 100 120 40 60 80 100 120 I Flap Angle. 0 [De(

Figure 7-3: Normalized velocity variation with flap angle(x-axis), pulling force(see color bars) and angular velocities (legends)

64 Chapter 8

High speed locomotion

In the previous chapter, studies on verifying the capability of RFT in modeling quasi- static or slow speeds were done and it was verified that RFT is highly robust in modeling wheel locomotion in low-velocity or Quasi-static regimes. But the question about how to define slow-velocity or quasi-static regime and what happened if one goes out of this regime remains. To answer these question a collaborative study in collaboration with Goldman Group at Georgia Tech is done. For this study, instead of completely circular wheels, grousered wheels (shown in figure 8-1) were used to get more traction from the test-bed media.

8.1 Experimental setup

The setup (shown in figure 8-1) consists of the wheel mounted on a gearmotor assembly. The gearmotor-wheel assembly is mounted on a friction-less vertical rail such that former is free to move in the vertical plane. The vertical rail itself is attached to a horizontal sliding platform which can freely slide in the horizontal direction. Thus the vertical motion experiences the inertia of wheel only due to the wheel gearmotor assembly while the horizontal motion has inertia due to gearmotor-wheel assembly as well as the horizontal sliding platform. The wheel motion is completely restricted from moving in the direction parallel to the axis of the wheel. The primary input variable is angular velocity ( besides other constant system parameters like the mass of wheel, additional vertical force on wheel due to counter pulley, the mass of sliding platform, state of material etc). The primary output variables are horizontal velocity fop wheel and sinkage over time. A representative image of the wheel, as well as the experimental setup, are shown in Figure:8-1. The exact details of system parameters including wheel geometry are given in Table: 8.1.

65 Inner Diameter: 23.5 cm Total grousers: 30 nos B Unear Bet Grouser Height: 1.5 cm Carriate and Grouser Width: 0.6 cm Ple yt in-plane depth 16 cm N

3-0 Printed Wheel -...

Figure 8-1: A) Grousered wheel with size specifications, B) 3D model and C) Ex- perimental setup of test rig used for high-speed locomotion studies done at Georgia Tech. Only the angular velocity of the wheel is controlled. The mass of the wheel can be controlled by adding additional mass on gear motor assembly. The gear motor assembly which includes the wheel is free to slide on the frictional-less rails. The Horizontal inertia of the system includes the mass of sliding plate-form which con- nect the gearmotor-wheel assembly to horizontal(frictionless) sliding rails via vertical frictionless sliding rails.

8.2 RFT based Modeling

For RFT based modeling, implicit code mentioned earlier was utilized. Similar to non-circular locomotion case, this study also traces true locomotion of wheel and hence both the horizontal and vertical motion are independently and implicitly solved for evaluation output variables (sinkage and translation velocity). Various input parameters used for numerical modeling in this study are given in Table: 8.1.

8.3 Continuum modeling (MPM)

For modeling the wheels using MPM based continuum modeling, a methodology similar to that explained in slow speed locomotion studies was used. Similar to earlier studies, the problem is modeled in 2D assuming plane-strain conditions. Thus the variables like the density of media and wheel were converted to 2 Dimensional values. Fixed angular velocity boundary condition was applied on the wheel explicitly in the relevant material file. In terms of simulation resolution, a 200 x 200 grid was used to represent a domain size of 1m x1m with 2x2 linear material points per grid cell. The details of the system parameters are given in Table: 8.1. A sample simulation of the wheel moving at 62RPM for different time instances in shown in figure 8-2

66 MagnWUc* D. :+ ,.e19m Ma V*Iocffy (A) 0.400 0.019 0.037 ~I (B) 1.0*-0 0.02 M i.06 1.

Eqiv doWn Pasf Strdn (D) Eq-#a.M sftfitUn Iaf- (C) 1.00-01 0.2 0 1 24 5 1.00401 1.-0-1 0h 1 2 5 10 20 1.0402

0.4 T 0.6- a U 0.3 - 0.4- E 0.2 I 0) -2 0.1 0. t3 a. t2: 0 0.2 04 06 0.8 1 1.2 04 1.6 1!8 2 0 0.2 0.4 0.6 08 1 12 1.4 1.6 18 2' Time [secl Time [sec

Figure 8-2: Sample MPM simulation of Grousered wheel: A)Displacement B) Veloc- ity C) Equivalent plastic strain and D) Equivalent plastic strain rate magnitude(s) for a grousered wheel moving at 62RPM on Closely packed poppy seeds.

67 Table 8.1: System parameters and Mechanical properties of the wheel and granular materials considered in this study 3D properties Grousered wheel Wheel inner diameter 23.5 cm Number of grousers 30 nos Wheel width 16.0 cm Grouser width 16.0 cm Grouser height 1.5 cm Grouser thickness 0.6 cm Wheel mass 18.5 kg Vertical force 72.8 N Granular media: Compacted Poppy Seeds Density, PGrainJ3D 1100 kg/M 3 Packing fraction (4) 0.605

Plane-strain properties (MPM) Grousered Wheel Wheel inner diameter 23.5 cm Number of grousers 30 nos Grouser height 1.5 cm Grouser thickness 0.6 cm Vertical force 72.8 N In-plane width 16 cm Density, Pweel ~ 427 kg/m 2 Granular media:Compacted Poppy Seeds Density, Pmedia 106.48 kg/M 2 (Continuum property includes 4')

RFT scaling coefficient, : 0.55

8.4 Observations and analysis

For this analysis, as stated earlier, to test the limits of RFT, variation of wheel horizontal velocity with angular velocity was studied. A plot of horizontal velocity variation with angular velocity variation from 0 RPM to 100 RPM is plotted in 8-3.

Graphs (A) in Figure 8-3, shows that while for low velocity or quasi-static limits (below approx ~ 50 RPM), RFT, as well as MPM simulations, converge and accurately predicts the variation of translation velocity with angular velocity. But above a certain velocity, which we call critical angular velocity or the turnover angular velocity (~ 50 RPM in this case), the methods appears to diverge. While RFT continues to predict the linear increase in velocity with angular velocity, continuum modeling (MPM), predicts a turn over similar to experimental observations. Similarly, the accuracy of continuum modeling in capturing the variation of sinkage is much higher than RFT and the later one completely breaks down after ~ 50 RPM. The linear translation velocity variation of RFT is not unexpected due to its rate-insensitive

68 700 A B 600 50

500 E0' * E40

400 - " 30 C n300 E 020- . 200 4 e Experiment 10 * * * MPM Sims 100 - MPM Sims * RFT Sims *RFTSims * Experiment 0 1 0 0 20 40 60 80 100 0 20 40 60 80 100 RPM RPM

Figure 8-3: High speed performance of Grousered wheel: Variation of (A) Translation velocity and (B) Sinkage with angular velocity for grousered wheels. Specifications of system are given in Table 8.1. nature and the fact that its current form does not capture the inertial effects in the deforming media. The capability of continuum modeling which does not uses micro-inertial model for modeling grains (p(I) rheology was set to be inactive in these simulations) indicates the fact that the turnover can be largely explained in terms of macro-inertia of the system and micro-inertial effects (discussed in section 2.3) are negligible here. In order to pinpoint the phenomenon responsible for the turnover, various parameter swap studies were done and the preliminary results are presented further.

8.5 Effects of initial boundary conditions on equilibrium state

Implementation of all the above simulations, as well as experiments, was done on a test-bed whose total available run length was ~ 1m. The limitation on bed-length limits the total time a wheel gets to reach an equilibrium velocity and sinkage. Hence the detailed analysis of the effect of two initial parameters namely initial velocity and ramp rate (the rate at which angular velocity is creased from zero to final angular velocity) on equilibrium state (sinkage and translation velocity) of the wheel state was done.

69 8.5.1 Effect of initial translation velocity

For this study, initial velocities of [620, 520, 420, 320 and 220] mm/sec were imparted to the wheel in directions same as well as opposite to the motion of wheel (in separate trials) at the starting of the simulation along with a constant angular velocity of 50 RPM. The wheel used for this study had a base diameter of 25 cm with 30 grousers of 11 mm height and 6mm width across its circumference. The effective weight of the wheel was 18.5 kg and a vertical load 72 N in the upward direction was applied to the wheel. The internal coefficient of friction of the granular media was taken to be 0.61 and the surface friction between the wheel surface and granular media was taken to be 0.35.

70 r 800 r

60 600 - 400 50 E 200 -vint= -620 mm/s E40- E -vini = '520 mm/~s e 0 -- vit = -420 mm/s 30- -vini= -320 mm/s -200 vi= -220 mm/s -V - -620 mm/s v t = 220 mm/s v- -ii220 mm/s 20 r 1 0 - V = -520 mm/s - vi = 320 mm/s -400 -vin m 320 mm/s - nh_= -420 mm/s - vi = 420 mm/s ----vinit = 420 mm/~s 10 _-v = -320 mm/s - vi m 520 mm/s -600 -vini = 520 mm/~s -vi = -220 mm/s - v,-- = 620 mm/s I-vinit n 620 mm/s 0 -800 0 1 2 3 4 5 0 1 2 3 4 5 Time [sec] Time [sec]

Figure 8-4: Variation of Sinkage (left) and Translation velocity (right) over time for wheels being imparted with various initial translation velocities (and a constant angular velocity of 50RPM).

The variation of sinkage and translation velocity are plotted in figure 8-4. As can be observed from the figure that the study verifies that the initial velocity conditions on the wheel do not affect the final equilibrium state of the wheel in terms of sinkage and translation velocity values. Hence it can be concluded that application of some initial translation velocity to the wheel in order to achieve equilibrium state faster, is a valid method of speeding up the simulations. For visualization purposes, the snapshots of variation of displacement magnitude in the system for velocities of 420mm/s in same as well as opposed to the expected direction of locomotion is plotted figure 8-5.

70 VO = -420 m/s v, = 420 m/s

roman

Figure 8-5: Variation of displacement in the system at various time instances in the system for initial translation velocities 420m/s in the opposite (left) and in the same direction (right) of expected locomotion based on angular velocity. The angular velocity was kept constant throughout the experiment in both cases (at 50RPM). Note that time instances in left and right cases are independent of each other.

8.5.2 Effect of angular velocity ramp rate

For this study, ramp rates of 120, 30, 40, 60, 100 and 150] RPM/sec were imparted to the wheel to achieve a final angular velocity of 50 RPM. The wheel used for this study also had an outer base diameter of 25 cm with 30 grousers of 11 mm height and 6mm width across its circumference. The effective weight of the wheel was 18.5 kg and a vertical load 72 N in the upward direction was applied to the wheel. The internal coefficient of friction of the granular media was taken to be 0.61 and the surface friction between the wheel surface and granular media was taken to be 0.35. The variation of sinkage and translation velocity are plotted in figure 8-6. As can be seen, that study verified that the ramp rate does not affect the final equilibrium state of the wheel. It should be noted that for higher ramp rates, wheel does sinks

71 to larger depths and later on recovers to equilibrium value whereas for lower ramp rates, it sinkage never exceeds the final sinkage which also indicates that total physical distance traveled to achieve equilibrium state is lesser for lower ramp rates (from the instant of reaching final angular velocity).

500 500

400 400 C., E 300 E 300 E E 20 RPM/s Z 200 -30 RPMs o200 0 -40 RPM/s 0 200-- RPM/s -- 60 RPMs 250 RPM/s 100 100 RPM/s 100 300 RPM/s 0 0 X 150 RPM/s I0 350 RPM/s 0 0 1 2 3 4 0 1 2 3 4 Time [sec] Time [sec]

50 -

40 - -20 RPM/s -30 RPM/s E 40 RPM/s 30 - 60 RPM/s -100 RPM/s .Ecc 20 - 150 RPM/s 200 RPM/s 250 RPM/s 10 300 RPM/s - 350 RPM/s 0 0 1 2 3 4 5 Time [sec]

Figure 8-6: Variation of Translation velocity (top) and Sinkage (below) over time for wheels being imparted with a final angular velocity of 50RPM at ramp rates varying from 20RPM/sec to 350 RPM/sec. The translation velocity graphs (top) are divided into left and right for clarity.

72 8.6 Evaluation of critical velocity for grousered wheel locomotion

The various potential parameters capable of affecting the translation velocity response (Vtranslation) of the system wrt. angular velocity (w) are R: Inner Radius of the wheel L: Width of wheel h: Height of grousers Pwheel Density of wheel Pmedium Density of granular medium Isurface Coefficient of internal friction of wheel-medium interface Pmedium Coefficient of internal friction of granular medium Internal g Gravity

Using Buckingham-pi theorem (dimensional analysis), the value of critical angular velocity is expected to be:

9 wheel hR = -- , h medium, Psurface, RAc t (8.1) Rpmedium R L Based on the assumptions of plane-strain motion and the fact that motion was found to be very weakly dependent on surface friction (Psurface). The above form reduces to c = ,hmedium) (8.2) -W (Pmredium,1R Amei Independent of the simulation data, we discuss expected trends of variation of critical angular velocity with various system parameters based on physical understanding of the process of locomotion.

Internal coefficient of friction: As the internal coefficient of friction is indicative of relative resistance of a media wrt. shear, increasing internal coefficient of friction is expected to lead to an increase in the strength of the material with the material acting like a rigid body/surface at p -+ 00. Hence with increasing internal friction value, critical angular velocity is expected to increase.

Density of the wheel and the medium: Increased density of the medium is expected to make medium act stronger with p -> 00 representing a rigid body. Increase in the density of the medium is also expected to scale up the pressure in the system leading to a increased shear strength. Hence, with an increasing density of the medium, critical angular velocity is expected to increase. An increase in wheel density is expected to increase the pressure below the wheel as well as an increase

73 Effects of various system paramters on wC and Vra 1 2 100 1 200

80 0.8- 150 0 8 60 Z 0.6- 0.6 100 40 L 0.4- 0.4 0-2 0,2 20 -

A __ 045 0.5 0.55 0.6 0.65 0.7 0.75 0.6 0.8 D 1.2 1 4 Internal Coefficient of friction (is) Wheel Density Fraction (ps

0.9 r ,90 0.6 0.55 08~ 80 0.5 - 07 70 0.45 s0 0.6 60 $ 0.4 60 0.5 e'O 0.35 0 03 04 1 1.2 1.4 1.6 1.8 2 0.14 0.6 0.8 1 1.2 14 Medium Density Fraction (plp,) Radius Fraction

0.42 65 0 55

0-4 60 70 55 ^ 0.5 - 0.38 50 60S 0.36 45 0.45

0.34 40

0.32 35 0.5 1 1.5 2 0.5 2 Grouser Height Fraction Gravity Fraction

Figure 8-7: Dependence of critical angular velocity for grousered wheel locomotion on various relevant system parameters. All the parameters (like radius, wheel and medium density etc) are varied in reference to the base case presented in figure 8-3 in terms of dimensions as well as properties. in the sinkage. Hence a clear trend can not be stated based on intuition. But the scaling relations indicate critical velocity trends with wheel density to be opposite to that with medium density. Hence, increasing wheel density is expected to decrease the critical angular velocity.

Gravity: Increasing gravity is expected to increase the pressure in the system and thus make the material act stronger (zero gravity means no strength in the system due to absence of any hydrostatic pressure.) Also from the scaling argument

74 stated above, increasing gravity is expected to increase the critical angular velocity.

Radius: Based on scaling arguments, it is expected that the increasing radius will result in a decrease in the critical angular velocity.

Height of grousers: Increasing grouser height is expected to give a better grip to the wheel over the sand and thus expected to increase critical velocity. But at the same time, increased grouser height increases the effective radius of the wheel, hence a clear trend can't be predicted directly.

It can be observed from figure 8-7 that in general, the trends are in line with the explanations given above. The exact order of the trends is not reported here as the results appear to be affected by numerical variations. Future studies are planned to carry out a more detailed analysis of trends with the more calibrated numerical simulations.

75 76 Chapter 9

Other general intrusions in sand

In this chapter, we explain our work on understanding the dynamics of the process of granular intrusion at a more fundamental level. We study the intrusion of simple shapes (like a rectangle, triangle, circle etc) into a granular media making use of various computational tools we have discussed earlier to observe and interpret various resistive forces in different cases.

9.1 Background

The basic formulation for characterisation of forces and flow around a flat plate body moving into a granular media under a static load boundary conditions in the absence of any bodyforces (like gravity) was first given by Sokolovskii [37]. Figure 9-1 shows symmetric half of slip lines given by Sokolovskii formulation, more details about which can be seen found on Page 66 of [37]. Over the years, the shape of the slip lines has been experimentally [38, 39] as well as numerically verified and studied, but most of these studies have been restricted to flat plate intrusions.

Intruding surface

Uniformly distributed Pressure boundary condition

Solid Zone Log Spiral Families of Slip Lines

Figure 9-1: Sokolovskii analytic form of Slip lines with characteristic log spiral

77 In this work, we start with modeling of intrusion of a flat plate into a bed of granular media as a verification study for evaluating the robustness of the MPM based continuum modeling. Once the qualitative verification is done, quantitative experiments with various intruder shapes are done to observe the development of flow profile under various intruder shapes and explore the simplistic ways of estimating intruding forces on various kinds of intruders into the granular media.

Equivalent Plastic Strain Rate 2.000.-03 0.01 0.02 0.06 2.000e-01 uIii I I -

Figure 9-2: Continuum modeling of plate intrusion.

Plastic Strain Rate Platic Strain 5.0.-02 C.2 0.5 1 2 5 10 6.00+01 6.0.-02 0.2 0.6 1 2 5 10 5.0e+01

Density (kg/m3) DIsplacernrt 6.4.+02 WvJ1111ius5 S75 693 6.19+02 0.09+00 01 0.2 0.3 4.0.-01

Figure 9-3: Critical state model based flat plat intrusion in unconsolidated media.

78 Figure 9-2 shows the variation of equivalent plastic strain rate in the material upon intrusion by a flat plate intruder at two different instances. While the formation of the characteristic spiral log profile at a instant just when intruder intrudes into the media was a capability-verification of method against the existing theoretical knowledge, the capability of method in modeling the collapsing of the material back onto the intruder as it plunges through was an indicative of robustness of the method in modeling the granular intrusion. Based on all these verification, further studies were done using different intruder shapes and material models. The exact details of the material properties used in these studies are not mentioned here as the properties were generic in nature and the overall aim of the study was to obtain general trends in force response and flow profiles in the media. For the same reason no direct comparison/calibration to any experimental study was done. Another important finding of this work was that for unconsolidated granular media using the critical state model explained in section 5.3.1, the characteristic spiral log shape was found to be absent (shown in figure 9-3). Further research is planned to be carried out to explain the phenomenon.

9.2 Plane-strain granular intrusion

9.2.1 Projected area defines the resistive force magnitude

strain rate IipzO T iangular shapes used in study Equivalent Plastic

60D0

-50--

L40D0

300-

- H = 28mm W=48mm H = 38mm W=48mm 1000 - H = 4mm W=48mm

- H = 58mm W=48mm 0 0.05 0.1 0.15 0.2 Distance [m]

Figure 9-4: A MPM implementation of triangular intruders:(A) Variation of vertical resistive force with depth, (B) Variation of strain rate for different triangular shapes.

Realizing the formation of the triangular solid zone of no shearing just below the flat impacting surface and analytical fit by many 138, 39, 40] to approximate the forces

79 on different impact surface shapes, we explore if the main source of resistive forces on the bodies intruding vertically into the sand can be attributed as the slip zones, the shape of which is mainly dependent on projected area. Based on this assumption, the force on a simple intruding body should depend only on the projected area. To verify the claim we take 4 triangular shapes of the same width but different heights resulting in systems of triangles with the same projected area but different impact angles. If our assumption is correct, the vertical forces experienced by each shape should be the same. Figure 9-4 shows the variation of vertical forces over depth (A) as well as equivalent plastic strain rate during intrusion (B) of 4 triangular shapes into the media. It was observed from figure 9-4 that net vertical forces, as well as effective shear jammed zone (the triangular shear jammed region below the intruder), are in line with our assumptions.

9.2.2 Superposition of forces

Figure 9-5: Plastic strain variation around a unsymmetric triangular shape.

Proving this result is an important breakthrough in establishing the validity of basic assumptions used in RFT. Though Askari and Ken did show similar results for superposition of drag forces on symmetric objects moving horizontally through the granular media, we show these results for unsymmetric objects( a sample simulation shown in figure 9-5) with an explanation of the reason behind the phenomenon in terms of flow of media and variation of stress-strain in the media. Figure 9-6(a) shows the variation of forces obtained during intrusion of symmetric triangles and (b) shows the forces obtained on triangles formed by the combination of half triangles from the initial triangles. Visualization of the flow field around the intruding triangle gives an indication

80 4 x10 Cone of variable cone angle Cones of Various Half Angle 2 16000 -Un-symmetric 1. a=,500 2. a = 60 \ 1.8 0 3. a =70 14000 1. a, = 60' a = 500 0 2 0 1.6 2. a, = 60 a 2 = 70 _ 3. a, = 70' a = 50' 12000 2 1.4 TVX

10000 1.2 I - z 1 8000

0 0 LL U- 1 0.8 6000

0.6 --- Cone 4 a= 60 , a2 50 4000 - -Cones: al =60 , a2= 70 0 0.4 -- Cone : a,= 70, a2 50 0 - / ... ~ -- Cone ,:a =60 0 .SuperposMon resuft Cone + 0 2000 I Cone 2 - -- Cone a= 50 0.2 2: - SuperposMon resut: Cone 1 + Cone . -- Cone : a =70 0 -Superposfflon result : Cone 2+ Cone , n 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 - 0.2 0.25 0.3 Distance [m] Distance [m]

Figure 9-6: Variation of forces on triangular intruders. (A) Forces on 3 different triangles with same height but different half angles (B) Comparison of forces on unsymmetric triangles measured directly with those obtained using superposition. that the flow on two halves of the triangle is independent of each other. Observing figure 9-6 it can be concluded that the total force on the unsymmetric intruders is the summation of forces on individual parts. It is also to be noted that though the force profiles have different shapes, after the development of the full velocity profile one can also obtain the total vertical force on these shapes using projected area method as done in the previous section. A more detailed analysis is planned to be done in the future.

9.3 Force & flow transition in plowed granular media

This work takes inspiration from the work of Gravish et al [40] who carried out drag plate experiments in a granular media for various initial packing fractions and observed the phenomenon of occurrence of bifurcation in the force and flow response of a granular media at the onset of dilatancy,qc. They observed that below this critical density (0,), rapid fluctuations in the drag force, FD, were observed while above 0c, fluctuations in FD are periodic and increase with,. Using velocity field measurements, they explained the origin'of fluctuations to be in periodic formation and destruction of stable shear bands above, #c. They also proposed a friction-based

81 Plain strain drag variation using continuum modeling a) __s)

7 S0.618 c))) _0

C) d) 4N 0

0-. 603 3- ~0.603

100 40 0w-N, 0 1 2 ie2 3 4 0 0.1 0.2 0.3 OA 0.5 0.6 0.7 0.1 Distance

soo Eq. Plastic strain .00 400 K .5 300 ______10.02S

2oo 4g 00 S800 000 120 1100W 3 F.gsrr Density (kg/Fw ) Loe-+ I LG .403

IZ :0.2

0- 0 1487.5 I . eO-031. *

vertical~ ~I shqt) Plght Uppr: ainuu plnrtansmuainrsls o aito Figure 9-7: Force and flow transition in plowed granular media: Left upper: Experimental overview [40] a)drag force, surface deformation and (b) velocity fields measured as a function of prepared volume fraction, 0. (c) Experimental setup (d) Temporal fluctuations in drag force become periodic as q5 is increased (arbitrary vertical shift). Right Upper:Continuum plane strain simulation results for variation of drag forces over distance traveled for different initial packing fractions. Down: Detailed material state for over compacted case (0 = 0.603),obtained using continuum modeling. (A) Drag force (N/m) vs time steps. (B)Equivalent plastic strain, (C) equivalent plastic strain rate and (D) density variation over the domain. model to captures the dynamics of the phenomenon. Their work is summarized in figure 9-7. Though the experiments done by Gravish et al 140] had a strong 3-dimensional component to it, we carry out the similar simulation in plane-strain conditions to explore if a constitutive model based on a basic critical state theory model explained in 5.3.1 can capture similar observations. The aim of this work was to observe the phenomenon and not match the exact results and hence the generic material properties were used.

82 Observation: As shown in the continuum modeling based graph in figure 9-7, the phenomenon of increasing force fluctuations with packing fractions was also observed in continuum simulations. The formation of shear bands was also observed as stated by Gravish et al[401. The fluctuations in under-compacted material (# =0.581) towards the end can be attributed to the presence of the wall (causing compaction of material near it). Thus, this work verifies the utility of use continuum modeling in phenomenon captured by Gravish et al [40] experiments and that it is not a size- dependent effect. A more detailed calibration and 3-dimensional study along with plane strain experiments are planned to be performed in future for a comprehensive study of the phenomenon.

9.4 Depth based lift and drag variation on cylindrical objects

This work takes inspiration from the work of Guillard et al [411, whose work was an experimental and numerical study of the forces experienced by a cylinder moving horizontally at various depths in a granular bed being loaded with and without gravity. The force opposite to the direction of gravity is said to be 'lift' while the force opposite to the direction of velocity is named 'drag'. The first observation they made was that despite the symmetry of the object, they observed strong lift forces on the cylinder. The second observation they made was whereas the drag force was found to increase linearly with depth, the lift force was shown to saturate at depths much greater than the cylinder diameter. Their work is summarized in figure 9- 8(a). As stated in their conclusive paragraph, to address the question of drag and lift forces in granular media in the framework of continuum modeling, we use MPM based simulations. Guillard et al did their experiments with different lengths of the cylinders with the same diameter. The linear scaling of drag and lift forces observed (refer inset of Figure 9-8 (a))verifies the applicability of plane strain simulations for the problem. It is to be noted that similar to the previous study of the drag on vertical plates, the aim of the study was to understand the generic force response of the media. Hence, the detailed calibration of properties between the experiments and simulations was not performed. Figure 9-8(b) shows the continuum simulations results for in-plane width of 100cm for cylinder. Observations Observing the flow profile of the material around the plate gives an indication that the primary source of generation of lift and drag on the plates are the flow along the flow lines. As the depths of plates are increased into the bed of sand(used interchangeably with granular media), the equilibrium flow profiles began to evolve differently as shown in figure 9-8. Besides the drag and lift forces, velocity profile in the medium using MPM was also observed to be similar that reported in the paper. The general explanation of the phenomenon appears to be

83 - I- (a) G-OL=6cm . 0.8 G-0 L=9cm 4 - .0- M L=12 cm

15 - z 0.6- - (C)

10 - 0 1000 20. - pgh .. 0.4-

5- 5 ~ 0.2 -- 0-0 L--6 cm () G-0 L=9 cm. . 1 40 L=012 cm h, 0. h (m) 0.2

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 h (in) h (mi) (a)

40000 60

35000 5 . 50 30000 -

40- 25000 40--

-J - -- p20000 - - 30

LL 15000 L-J20

10000 -

* - 7 - 10 t~- 2 5000

0 -,0 0 500 1000 1500 2000 0 0.05 0.1 0.15 pgh Depth,h [m] (b)

Figure 9-8: Variation of Lift and Drag on cylindrical object of 4mm diameter with depth from (a)Guillar et al experiments [41] (in-plane cylinder length [6,9,12]cm and (b) plane strain continuum simulations (Comparison ot be done with inset images of upper graphs) similar to that explained by Guillar et al [42] which is: while the origin of drag forces on the cylinder are shear force along the slip lines which are continuously encountering hydro-statically loaded virgin material, the lift forces have origin in slip lines as well as the material state above the cylinder. An increasing depth of cylinder linearly increases the hydrostatic pressure on the material which in turn causes a linear increase in shear-strength of the material and hence drag forces. The variation of lift is a more complex phenomenon. Observing the flow profile of the media around the cylinder it can be observed that the flow of media around the

84 -1

Eqaiaet Plasic strain

Equivabont Plastic strain fate 1:7MnVo

to-

Velac*K Magnitude

0 10

203

Figure 9-9: Variation of various state variable (Velocity, Eq plastic strain and Eq plastic strain rate) on the cylindrical object of 4mm diameter at depths of [1,2,3,4,5,6]cm using plane strain continuum simulations

85 plate is a local phenomenon where the material is continuous getting dumped on the moving cylinder. Initially with increasing depth, the size of this flow-zone begins to increase and hence lift force increases but as the depth goes above a certain depth, the flow profile gets completely localized. In this region, the material above the plate is continuously flowing from the front of the cylinder to the top of it and a local saturation pressure occurs. It is evident from the lift variation graphs that this region achieves uniform pressure state at a certain ratio (of cylinder diameter). This region is indirectly responsible for generation of lift forces on the plate and hence causes drag force to reach a plateau region. This work was just a preliminary work for verification of capability of continuum modeling in capturing the phenomenon i.e. a verification of the fact it is not size effect. A further deeper analysis of the same based on force and energy balance arguments is planned to be done in future work.

86 Bibliography

[1] Wikipedia. Lee westwood making a bunker shot at the 2008 open, 2016.

121 Mieczyslaw Gregory Bekker. Introduction to terrain-vehicle systems. University of Michigan Press, 1969.

[3] M. G. Bekker. Theory of Land Locomotion; the mechanics of vehicle mobility. The University of Michigan Press, Ann Arbor, 1956.

[4] J Y Wong, M Garber, and J Preston-Thomas. Theoretical prediction and experimental substantiation of the ground pressure distribution and tractive performance of tracked vehicles. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 198(4):265-285, 1984.

[5] J. Y. Wong. Theory Of Ground Vehicles. John Wiley & Sons, New York, 3rd edition, 2001.

[6] David waters. The untold history of the wheel and its evolution, 2017.

[7] David Syrett. The Eyes of the Desert Rats: British Long-Range Reconnaissance Operations in the North African Desert 1940-43. Helion and Company, 2014.

[8] Jo-Yung Wong and AR Reece. Prediction of rigid wheel performance based on the analysis of soil-wheel stresses part i. performance of driven rigid wheels. Journal of Terramechanics, 4(1):81-98, 1967.

191 Z. Janosi and B. Hanamoto. Analytical determination of drawbar pull as a function of slip for tracked vehicles in deformable soils. In Proceedings of the 1st International Conference on Terrain-Vehicle Systems, Turin, Italy, 1961.

[10] J. Y. Wong. Terramechanics and Off-Road Vehicle Engineering. Elsevier, UK, 2nd edition, 2010.

[11] R. G. Pope. The effect of sinkage rate on pressure sinkage relationships and rolling resistance in real and artificial clays. Journal of Terramechanics, 6(4):31- 38, 1969.

87 [121 J. Y. Wong and A. R. Reece. Prediction of rigid wheel performance based on analysis of soil-wheel stresses, Part I. Performance of driven rigid wheels. Journal of Terramechanics, 4(1):81-98, 1967.

1131 J. Y. Wong and A. R. Reece. Prediction of rigid wheel performance based on analysis of soil-wheel stresses, Part II. Performance of towed rigid wheels. Journal of Terramechanics, 4(2):7-25, 1967.

114] M. Prochnow F. Da Cruz, S. Emam, J.N. Roux, and F. Chevoir. Rheophysics of dense granular materials: Discrete simulation of plane shear flows. PHYSICAL REVIEW E 72, 021309, 2005.

115] GDrMiDi. On dense granular flows. The European Physical Journal E, vol. 14, pp. 341-365, 8, 2004.

[16] Y. Forterre B. Andreotti and 0. Pouliquen. Granular media. Cambridge University Press, 2009.

[171 Y. Forterre P. Jop and 0. Pouliquen. A new constitutive law for dense granular flows. IUSTI, CNRS UMR 6595, Universite de Provence, 5 rue Enrico Fermi, 13465 Marseille, 2005.

[18J L. T. H. Van Dellen 0. Dauchot J. A. Dijksman, G. H. Wortel and M. Van Hecke. Jamming, yielding, and rheology of weakly vibrated granular media,. 10.1103/PhysRevLett.107.108303, 2011. [191 Grest Silbert, Ertas, Halsey, Levine, and Plimpton. Self-fluidization of subaerial rapid granular flows. Powder Technology 182 (2008) 323 - 333,, 2008.

[20] M. G. Bekker. Introduction to Terrain-Vehicle Systems. The University of Michigan Press, Ann Arbor, 1969.

[21] G. Ishigami, A. Miwa, K. Nagatani, and K. Yoshida. Terramechanics-based for steering maneuver of planetary exploration rovers on loose soil. Journal of Field Robotics, 24(3):233-250, 2007.

122] F. Zhou, R. E. Arvidson, K. Bennett, B. Trease, R. Lindemann, P. Bellutta, K. lagnemma, and C. Senatore. Simulations of mars rover traverses. Journal of Field Robotics, 31(1):141-160, 2014.

[23] J Gray and GJ Hancock. The propulsion of sea-urchin spermatozoa. Journal of Experimental Biology, 32(4):802-814, 1955.

[241 Ryan D Maladen, Yang Ding, Chen Li, and Daniel I Goldman. Undula- tory swimming in sand: subsurface locomotion of the sandfish lizard. science, 325(5938):314-318, 2009.

88 [251 Chen Li, Tingnan Zhang, and Daniel I Goldman. A terradynamics of legged locomotion on granular media. Science, 339(6126):1408-1412, 2013.

[261 Hesam Askari and Ken Kamrin. Intrusion in heterogeneous materials: Simple global rules from complex micro-mechanics. arXiv preprint arXiv:1510.02966, 2015.

[27] J. Aguilar and D. I. Goldman. Robophysical study of jumping dynamics on granular media. Nature Physics, vol. 12, pp. 278-283, 2015.

[281 T. Zhang and D. I. Goldman. The effectiveness of resistive force theory in granular locomotion. Physics of Fluids, vol. 26, p. 101308, 2014.

[291 H. P. Zhang and H. A. Makse. Jamming transition in emulsions and granular materials. Phys. Rev. E, vol. 72, no. 1, p. 011301, 7, 2005.

[301 Hertzsch Brilliantov, Spahn and Poschel. Model for collisions in granular gases. Phys Rev E, 53, p 5382-5392, 1996.

1311 Grest Silbert, Ertas, Halsey, Levine, and Plimpton. Granular flow down an inclined plane: Bagnold scaling and rheology. Phys Rev E, 64, p 051302, 2001.

132] Ken Dunatunga, Sachith andKamrin. Continuum modeling and simulation of granular flows through their many phases. arXiv preprint arXiv:1411.5447, 2015.

1331 Soil behaviour and critical state soil mechanics. Cambridge university press, 1990.

134] D. Kothe J. Brackbill and H. RuppelL. Flip: a low-dissipation, particle-in-cell method for fluid flow. Comput. Phys. Commun.48, 25-38, 1988.

[351 F. H. Harlow. The particle-in-cell computing method for fluid dynamics. Meth. Comput. Phys, 1964.

[36j Z. Chen D. Sulsky and H. L. Schreyer. A particle method for history-dependent materials. Comput. Meth. Appl. Mech. Engng118 (1-2), 179-196, 1994.

[37] Statics of Granular Media. Pergamon, 1965.

[38J Ivo R. Petersi Endao Hani and Heinrich M. Jaeger. High-speed ultrasound imaging in dense suspensions reveals impact-activated solidification due to dynamic shear jamming. 10.1038/ncommsl2243, 2015.

[391 S. R. Waitukaitis and H M Jaeger. Impact-activated solidification of dense suspensions via dynamic jamming fronts. Nature 487, 205-209, 2012.

89 [401 Nick Gravish, Paul B Umbanhowar, and Daniel I Goldman. Force and flow transition in plowed granular media. Physical review letters, 105(12):128301, 2010.

1411 Frangois Guillard, Yo81 Forterre, and Olivier Pouliquen. Lift forces in granular media. Physics of Fluids, 26(4):043301, 2014.

[42] Frangois Guillard, Yo81 Forterre, and Olivier Pouliquen. Origin of a depth- independent drag force induced by stirring in granular media. Physical Review E, 91(2):022201, 2015.