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Journal ArticleDOI

A contact dynamics code implementation for the simulation of asteroid evolution and regolith in the asteroid environment

15 Jul 2021-Icarus (Academic Press)-Vol. 363, pp 114441

TL;DR: The Contact Dynamics method is described, a class of DEM based on non-smooth mechanics, and its implementation in the open-source software LMGC90, and a parallelized kd-tree is implemented to monitor the performance of the code as it simulates a number of granular systems.

AbstractOver the last decades, simulations by discrete elements methods (DEM) have proven to be a reliable analysis tool in various domains of science and engineering. By providing access to the local physical mechanisms, DEM allows the exploration of microscopic based phenomena related to particles properties and interactions in various conditions and to revisit constitutive equations consequently. The growing computer power and memory now allow us to handle large collections of grains of various shapes and sizes by DEM simulations and in particular, it offers new perspectives in the exploration of the behavior of asteroids seen as self-gravitating and cohesive granular aggregates. In this paper we describe the Contact Dynamics (CD) method, a class of DEM based on non-smooth mechanics, and its implementation in the open-source software LMGC90. In contrast to more classical approach, Hard- and Soft-Sphere DEM, the CD method is based on an implicit time integration of the equations of motion and on a non-regularized formulation of mutual exclusion between particles. This numerical strategy is particularly relevant to the study of dense granular assemblies (even of large size) because it does not introduce numerical artifacts due to contact stiffness. So that it can be used for Small Body research, we implement a parallelized kd-tree and monitor the performance of the code as it simulates a number of granular systems. We provide examples of the simulation of the accretion of self-gravitating aggregates as well as their rotational disruption. We use the routines at our disposal in the code to monitor their behavior through the entire evolution and find agreement with previous research.

Summary (4 min read)

1. Introduction

  • During the last two decades, research about asteroids in general and small asteroids in particular has increased dramatically.
  • The pictures obtained of asteroid Itokawa revealed a small body covered in dust, pebbles, rocks and boulders going from micrometers to tens of meters in size [8, 9].
  • DEM make it possible to multiply numerical experiments in order to study the influence of particles characteristics (particle size, morphology), interactions (friction, cohesion) and loading, on the local and collective behaviour of the system.
  • In Smooth approaches [11, 12], interaction forces between particles can be written as a function linking contact forces to contact kinematics.
  • Then, some specificities for the modelling of granular asteroid are in-60 troduced, including cohesion, particle shape and self-gravity.

2.1. The Non-Smooth Philosophy

  • This situation can be described as non-smooth for at least three reasons:.
  • The geometrical conditions of non-interpenetration of the different objects (non-smoothness in space), the contact forces governed by non-regularized laws (non-smoothness in law) and velocity jumps due to collisions between bodies (non-smoothness in time).
  • This stiffness imposes very small discretization steps and often artificial inertias or viscosities are introduced to ensure numerical stability [14, 44].75.
  • A contact resolution algorithm ensuring the respect of constraints related to the choice of non-smooth interaction laws.
  • To use CD within a DEM philosophy, both previous ingredients are combined in a contact detection algorithm to deal with potential interactions between particles.

2.2.1. In the body frame

  • On the other hand, for complex shapes, the Euler equation remains non-linear and depends implicitly on the orientation of the objects.
  • The system is updated according to the second Equation of system (4).
  • Note that, the update110 of the orientation of each particle when dealing with non-spherical particle will be discussed in details in Sec.2.5.1. 2.2.2.
  • To find a solution to the contact problem, a relation between these two unknowns should be added through the definition of a contact law, detailed in the next section.
  • The time index is omitted to135 make pleasant reading.

2.3. Contact laws

  • When considering a system composed of rigid bodies, the physical behaviour of the system depends entirely on the laws of interaction between particles.
  • The contact unilaterality is described by the well-known Signorini Condition [54] relating the normal component of the contact relative velocity vn and the normal component of the contact impulse rn.
  • According to the set of inequalities (9), when a collision occurs between two bodies, the relative velocity vanishes.
  • In order to allow bounces in the simulation, one can introduce, for example, a restitution coefficient, linking the velocity before and after shock (vi+1n = −envin).
  • The Amontons-Coulomb friction law is one of the simplest way to consider a resistance in the tangential direction.

2.4. Resolution of the local frictional contact problem

  • If the authors make the difference of two successive iterations appear to the left hand side, then the equation can be written as follows: Wαα(rk+1α − rkα) = vk+1α − vα, f ree − nc∑ β<α Wαβrk+1β − nc∑ β≥α Wαβrkβ (12) The convergence of the Gauss-Seidel algorithm makes the right-hand side member tend towards zero.
  • This resolution method is independent of the particle geometry and, in the case of collections of rigid bodies, the approximation performed on the W matrix does not disturb the convergence of the Gauss-Seidel algorithm [20].
  • 5. Extensions of interest to model granular asteroids180.
  • The next subsections will detailed these specific features in the framework of190 Contact Dynamics.

2.5.1. Modelling regolith as polyhedral particles

  • Grain angularity modeling is of major importance to model realistic granular asteroids in order to better predict their behavior.
  • Recent reports show that particle shape strongly195 affects the strength and dilatancy properties of the granular media.
  • The second, called accurate, can use a method of intersection of triangular faces [47] or a method of plane separator [44].
  • Without any modification of the contact law, a face-edge contact (i.e. contact215 line) can be represented by two points whereas a face-face (i.e. contact surface) can be replaced by three points since they involve an equivalent number of geometrical unilateral constraints.
  • (17) Figure 4(a) shows the evolution of ∆ET for the first benchmark (reduced in this case to rotational energy) after 150 minutes of simulated time, as a function of the time step and the initial rotation speed.

2.5.3. Gravitational forces

  • The implementation of gravitational forces is of paramount importance when simulating granular asteroids.
  • This method has been called a “static domain decomposition technique.” pykdgrav, which has been validated in [80, 81], is a package that implements the Barnes-Hut method [78] for computing the combined gravitational field and/or potential of N particles.
  • The kd-tree is implemented as a numba jitclass to achieve much higher performance than the equivalent pure Python implementation.
  • The package implements OpenMP multithreading, but no support for higher parallelism290 is implemented at the moment.

2.6. LMGC90 Implementation

  • The CD framework has been implemented in the LMGC904 open-source platform under open-source license CECILL, initially developed by Jean and Dubois ([43]5) to address multi-contact problems.
  • The software benefits today from a number of contributions such as contact detection between polyhedra by Saussine et al.[47], solver295 parallelization with OpenMP by Renouf et al.[50], thermal and electrical coupling effects by Renouf [49], coupling with fluids [82] or various type of contact interactions such cohesive zone models [83] for simulating, for example, breakable particles.
  • In fact, contacts between two particles and their behaviour are also considered as objects.
  • Practically, the code proposes a global pattern to model and solve the problem, with various functionalities, which the user may enrich with its own routines through plug points.
  • First, the accretion process is presented, detailing the different parameters and initial conditions used, both for aggregates made of spheres or polyhedra.

3.2. Particle accretion process.

  • Intergranular friction and grain shape on different physical quantities.the authors.
  • Numerical efficiency is also discussed in terms of CPU time spent in each phase of the CD-algorithm.
  • Note that, during the accretion simulation cohesion between the particles is set to 0.345.

3.2.1. Kinetic energy

  • Figure 7 shows different snapshot taken during the accretion for the 8000-particle polyhedron assembly.
  • At the end of the accretion process, in absence of inter-particle cohesion and anisotropic shape of the particles or anisotropic shape of the containing box, the shape of the aggregate is spherical as illustrated in Fig.7(e) for aggregate composed of polyhedral particles.
  • The rapid increases of 〈Ec〉 is mainly due to the very loose nature of the initial-state packing.
  • Then, the kinetic energy relaxes as the time is increased and reach a constant plateau with very small values from t > 125min for all ns.
  • The evolution of the kinetic energy is closely related to various dissipative mechanisms at the particles scale, which can be understood through the evolution of the particle connectivity within the assembly.

3.2.2. Connectivity and packing fraction

  • Figure 9(a) shows the evolution of the coordination number Z (defined as the mean number of contact per particle)370 as a function of time in assembly of spheres.
  • Note that for the sake of clarity the evolution of Z in aggregates of polyhedra is not shown as the curves are basically the same.
  • But390 the authors see here, that for gravitational aggregates, Zmax is largely dependent on ns; therefore, they must carefully calibrate the number of particles in order to carry out the most realistic simulations.
  • Along the same line, the variations of the final state value of the packing fraction ν (defined from the total volume of the particles divided by the volume of the encompassing sphere and averaged over t > 125min), is shown in Fig. 10(b) as a function of the number of particles ns, both for frictionless spheres and polyhedra aggregates.
  • In contrast to the evolution of the final-state coordination number, the packing fraction remains slightly dependent on the number of particles.

3.2.3. Stress transmission

  • It is also interesting to evaluate the evolution of the mean pressure within the assembly during the accretion process.
  • Thus, and just for the sake of convenience, the volume the authors consider is that of the encompassing sphere at each instant t.
  • Interestingly, the data collapses along a single common straight line regardless430 of grain shape.
  • Let’s consider the ideal case of a “non-granular” asteroid, i.e. an asteroid made of a single monolithic rock with a homogeneous distribution of its density.
  • As the authors can see in Figure 11, the prediction given by Eq.23 is in good agreement with their numerical data both for aggregates of spheres and polyhedra.

3.2.4. Numerical efficiency445

  • Finally, in this section the authors discuss more numerical details about the time spent in each step of the Contact Dynamics algorithm in order to evaluate the numerical method (and its implementation).
  • In spheres assemblies the CPU-time spent for gravitational forces calculation is longer than the CPU-time spent in contact detection.
  • This point has been identified for a long time and has led to the implementation of parallel computing techniques to reduce the time consumed by this part of the method [50, 90].
  • The evolution of the time spent in solving the contact problem is460 less linear (in log-log scale) than the other two curves.
  • Indeed, the number of iterations of the Gauss-Seidel algorithm is not constant over time, strong variations of this number can disturb the mean value, which explains the non-linear behaviour observed on the Figs 12a and Fig. 12b.

3.3. Spin-up process

  • The test conditions are close to those used in the literature [31].
  • As cohesion is increased, a slower decrease in the value of the semi-axis in the direction of the axis of rotation and485 a reduction in the gap between the two main semi-axes occurs.
  • This results in greater510 variations in the evolution of the solid fraction, leading to a less compact system than low cohesive systems.
  • For the two lowest values, there is an almost rapid decrease up to half of the simulation time, then this decrease is combined with a periodic fluctuation, where the period is related to the velocity increments given to the aggregate.
  • The authors have provided a simplified description of the mathematical formalism that was introduced by the creators of the method, its implementation in the LMGC90 simulations code, the parallelisation method that has been implemented and a few examples of gravitational accretion and rotational disruption of asteroid size cohesive aggregates.

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A contact dynamics code implementation for the
simulation of asteroid evolution and regolith in the
asteroid environment
Paul Sánchez, Mathieu Renouf, Emilien Azéma, Rémy Mozul, Frédéric Dubois
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Paul Sánchez, Mathieu Renouf, Emilien Azéma, Rémy Mozul, Frédéric Dubois. A contact dynamics
code implementation for the simulation of asteroid evolution and regolith in the asteroid environment.
Icarus, Elsevier, 2021, 363, pp.114441. �10.1016/j.icarus.2021.114441�. �hal-03185129�

A Contact Dynamics Code Implementation for the Simulation of Asteroid
Evolution and Regolith in the Asteroid Environment
Paul S
´
anchez
1
Colorado Center for Astrodynamics Research, University of Colorado Boulder, 431 UCB, Boulder, Colorado, USA
Mathieu Renouf
LMGC, Universit´e de Montpellier, CNRS, Montpellier, France
Emilien Az
´
ema
LMGC, Universit´e de Montpellier, CNRS, Montpellier, France
Institut Universitaire de France (IUF), Paris, France
R
´
emy Mozul
LMGC, Universit´e de Montpellier, CNRS, Montpellier, France
Fr
´
ed
´
eric Dubois
LMGC, Universit´e de Montpellier, CNRS, Montpellier, France
Abstract
Over the last decades, simulations by discrete elements methods (DEM) have proven to be a reliable analysis tool in
various domains of science and engineering. By providing access to the local physical mechanisms, DEM allows the
exploration of microscopic based phenomena related to particles properties and interactions in various conditions and
to revisit constitutive equations consequently. The growing computer power and memory now allow us to handle large
collections of grains of various shapes and sizes by DEM simulations and in particular, it oers new perspectives in
the exploration of the behavior of asteroids seen as self-gravitating and cohesive granular aggregates. In this paper we
describe the Contact Dynamics (CD) method, a class of DEM based on non-smooth mechanics, and its implementation
in the open-source software LMGC90. In contrast to more classical approach, Hard- and Soft-Sphere DEM, the CD
method is based on an implicit time integration of the equations of motion and on a non-regularized formulation of
mutual exclusion between particles. This numerical strategy is particularly relevant to the study of dense granular
assemblies (even of large size) because it does not introduce numerical artefacts due to contact stiness. So that it can
be used for Small Body research, we implement a parallelised kd-tree and monitor the performance of the code as it
simulates a number of granular systems. We provide examples of the simulation of the accretion of self-gravitating
aggregates as well as their rotational disruption. We use the routines at our disposal in the code to monitor their
behaviour through the entire evolution and find agreement with previous research.
Keywords: celestial mechanics - granular - methods: numerical - minor planets, asteroids: general

1. Introduction
During the last two decades, research about asteroids in general and small asteroids in particular has increased
dramatically. The first near-Earth asteroid (NEA), (433) Eros was discovered 1898, but it would be a long time before
its first pictures were taken by the NEAR Shoemaker mission almost 20 years ago. In the mean time, asteroids passed
from being cursed as “the vermin of the sky” [1] by astronomers trying to observe distant stars and galaxies, to be the5
target of a number of space missions. The most recent finished one is the JAXA Hayabusa mission to asteroid (25143)
Itokawa [2] and the two that are taking place at the time of this writing are the JAXA Hayabusa2 mission to asteroid
(162173) Ryugu [3] and the NASA OSIRIS-REx mission to asteroid (101955) Bennu [4]. One common objective in
all these missions is the return of a sample of the particles on their surfaces for analysis. For the last two, the use of
numerical methods to simulate the interaction of the spacecrafts with the asteroid surfaces has been a very important10
part of the research eorts [5, 6, 7].
The pictures obtained of asteroid Itokawa revealed a small body covered in dust, pebbles, rocks and boulders
going from micrometers to tens of meters in size [8, 9]. The sample that was returned to Earth in 2010 revealed its
chemical composition as well as the varied shapes of the few dust grains that were trapped in the sample canister
after the failure of the sampling mechanism [10, 8]. These and previous findings confirmed the idea that asteroids15
had a granular structure and that their bodies were held together by gravitational attraction. This being so, the use
of the numerical tools and theoretical approaches to study granular matter became a necessity, especially codes that
implemented dierent Discrete Element Methods (DEM).
By definition, DEM aim to model the behaviour of a collection of distinct interacting particles. As any numerical
method, it constitutes a significant support to experimentation in the sense that they give us access to information that20
is dicult to obtain experimentally (i.e., packing disorder, contacts orientation, force distributions). DEM make it
possible to multiply numerical experiments in order to study the influence of particles characteristics (particle size,
morphology), interactions (friction, cohesion) and loading, on the local and collective behaviour of the system. The
common denominator of DEM is to consider the degrees of freedom associated with the elements (grains), considered
as rigid objects, and to integrate the equations of motion for these degrees of freedom. DEM can be classified into25
two main families of approaches: Smooth and Non Smooth.
In Smooth approaches [11, 12], interaction forces between particles can be written as a function linking contact
forces to contact kinematics. Typically, the normal reaction is taken as proportional to particle penetration (Hertzian
contact model [13]). The particle motion is smooth (at least as twice dierentiable) and, therefore, the equations of
dynamics are written as ordinary dierential equations that can be integrated by conventional methods such as the30
Gear method for Molecular Dynamics (MD) [12], the Newmark [11], or Verlet scheme [14].
In Non Smooth approaches [15, 16, 17, 18, 19], there is no explicit relation between contact forces and contact
kinematics. Contact conditions are described by complementarity relations between the contact forces and displace-
1
Corresponding Author: diego.sanchez-lana@colorado.edu
2

ments or velocities (as the well know Signorini condition [16]). No regularization is required. The equations of
motion are rewritten as non-dierentiable relations involving velocity jumps and impulse [16]. Equations of motion35
can be driven using time stepping or Event Driven (ED) [20] and discretized by the θ-method [17] or Leap Frog (LF)
method [21]. The ED method, associated with irregular time discretization, is well suited for collections of rigid
bodies in which the time to cover the mean free path is much greater than the contact time of a collision between two
bodies. The method then assumes that the collision time is eectively zero and so, by construction, only binary, and
not multi-body, collisions can be simulated.40
At the end fo the 90s, a first ED (Hard-Sphere DEM) model is proposed to the study of self-gravitating particles
of meters in size [21, 22]. The authors developed a code named PKDgrav [21], originally used to study star clus-
ters, which has been used also to study planetary rings, planetesimal formation, binary asteroid formation, rotational
evolution of asteroids and asteroid collisions to name a few topics [23, 24, 25, 26, 27, 28].
More recently (2008), an MD (Soft-Sphere DEM) has been introduced to model self-gravitating granular systems45
[29, 30]. Based on a Smooth DEM formalism, the code developed by the authors has been used to study the rotational
evolution of asteroids, internal structure and strength of small asteroids, binary asteroid and asteroid pair formation,
and penetrometry in the asteroid environment [31, 32, 33, 34, 35, 36, 37]. This last point was largely applied to the
simulation of the Touch-and-Go Sample Acquisition Mechanism (TAGSAM) for the OSIRIS-REx mission [6]. Note
that a SSDEM version has been available in the PKDgrav code mentioned previously since 2012 [38].50
Later on, in 2014, a first use of a Non Smooth approach was proposed to study regolith processes [39]. More re-
cently, in 2018, the Contact Dynamics (CD) method, was applied to analyze the strength properties of self-gravitating
aggregates of spheres [40]. A year later, Ferrrari, et al [41, 42] used the same method to study asteroid aggregation
problems with angular particles.
The implicit formulation of the method and the introduction of nonsmooth laws in iterative or direct algorithms55
makes the CD method less accessible for computer implementation than other DEM methods based on explicit for-
mulation. Thus, the aim of this paper is to present the spirit of the CD method, some details of its implementation in
the LMGC90 open-source platform ([43]
2
) together with a direct application of self-gravity. In this paper, to underline
the numerical eciency of the CD method for modelling granular asteroids, the CD method is applied to model the
accretion of spherical and polyhedral particles. Then, some specificities for the modelling of granular asteroid are in-60
troduced, including cohesion, particle shape and self-gravity. The main results of accretion simulations are discussed
and some perspectives are given regarding forthcoming research avenues.
2
https://git-xen.lmgc.univ-montp2.fr/lmgc90/lmgc90 user
3

2. The Contact Dynamics Method
2.1. The Non-Smooth Philosophy
The objective here is to represent an asteroid as a collection of rigid solids subject to conditions of mutual non-65
interpenetrability and friction in case of contact. This situation can be described as non-smooth for at least three
reasons: The geometrical conditions of non-interpenetration of the dierent objects (non-smoothness in space), the
contact forces governed by non-regularized laws (non-smoothness in law) and velocity jumps due to collisions be-
tween bodies (non-smoothness in time).
One way to get rid of this non-smoothness is the use of regularization technics: This is what the smooth-DEM70
methods do when the non-interpenetrability of bodies is replaced by suciently sti repulsion laws that apply when
two of them approach each other. We are thus reduced to dierential equations that can use classical numerical
techniques. But, in each case, a compromise must be accepted between the requirement of precision and the stiness
of the approximate equations. This stiness imposes very small discretization steps and often artificial inertias or
viscosities are introduced to ensure numerical stability [14, 44].75
To overcome this kind of process, Moreau relies on the framework of convex analysis [45] to develop what he will
call the Contact Dynamics (CD) approach [16]. It is based on two main ingredients: non-smoothed contact and non
smooth dynamics. It implies:
1. An implicit time integration scheme to describe the dynamic evolution of the particles and to manage in the
same time contacts and shocks;80
2. A contact resolution algorithm ensuring the respect of constraints related to the choice of non-smooth interaction
laws.
To use CD within a DEM philosophy, both previous ingredients are combined in a contact detection algorithm to deal
with potential interactions between particles.
2.2. Dynamical Equations85
2.2.1. In the body frame
The motion of a set of N
p
rigid particles is governed by the Newton-Euler’s equations:
M
¨
q = F
ext
+ R,
J
˙
ω + ω × Jω = M
ext
+ M,
(1)
involving the second time derivative of the configuration parameter q, the spin ω expressed in a frame attached to the
body, the external force and moment resultants (for example gravitational forces), denoted F
ext
and M
ext
respectively,
and the contact reaction forces (exerted by neighboring particles) called R. M and J represent respectively the mass90
and the inertia matrices.
4

Figures (19)
Citations
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01 Jan 2008
TL;DR: It is found that mass shed from the equator of a critically spinning body accretes into a satellite if the material is collisionally dissipative and the primary maintains a low equatorial elongation.
Abstract: Many asteroids and trans-neptunian objects have satellites: the tally stands at over 150 on http://tinyurl.com/dweqf . The smallest of these binary systems are main-belt and near-Earth asteroids, but the environments of these two types of object are very different, making it difficult to work out a common mechanism to explain their formation. Now Walsh et al. present a model that fits the bill. Properties of the observed main-belt and near-Earth asteroids with satellites are matched by simulations involving the slow spinup of a 'rubble pile' asteroid via the thermal YORP effect (where radiation from an irregular body exerts a net force on that body). The mass shed from the equator of a spinning body accretes into a satellite if the material consists of particles undergoing energy-dissipating collisions. Binary asteroids are created by the slow spin up of a 'rubble pile' asteroid via the thermal YORP effect (where radiation from an irregularly shaped body exerts a net force on the body). The mass shed from the equator of a critically spinning body accretes into a satellite if the material is collisionally dissipative. Asteroids with satellites are observed throughout the Solar System, from subkilometre near-Earth asteroid pairs to systems of large and distant bodies in the Kuiper belt. The smallest and closest systems are found among the near-Earth and small inner main-belt asteroids, which typically have rapidly rotating primaries and close secondaries on circular orbits. About 15 per cent of near-Earth and main-belt asteroids with diameters under 10 km have satellites1,2. The mechanism that forms such similar binaries in these two dynamically different populations was hitherto unclear. Here we show that these binaries are created by the slow spinup of a ‘rubble pile’ asteroid by means of the thermal YORP (Yarkovsky–O’Keefe–Radzievskii–Paddack) effect. We find that mass shed from the equator of a critically spinning body accretes into a satellite if the material is collisionally dissipative and the primary maintains a low equatorial elongation. The satellite forms mostly from material originating near the primary’s surface and enters into a close, low-eccentricity orbit. The properties of binaries produced by our model match those currently observed in the small near-Earth and main-belt asteroid populations, including 1999 KW4 (refs 3, 4).

17 citations


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CHAPTER 3: SOIL MINERALOGY. 3.1 Importance of Soil Mineralogy in Geotechnical Engineering. 3.2 Atomic Structure. 3.3 Interatomic Bonding. 3.4 Secondary Bonds. 3.5 Crystals and Their Properties. 3.6 Crystal Notation. 3.7 Factors Controlling Crystal Structures. 3.8 Silicate Crystals. 3.9 Surfaces. 3.10 Gravel, Sand, and Silt Particles. 3.11 Soil Minerals and Materials Formed by Biogenic and Geochemical Processes. 3.12 Summary of Nonclay Mineral Characteristics. 3.13 Structural Units of the Layer Silicates. 3.14 Synthesis Pattern and Classification of the Clay Minerals. 3.15 Intersheet and Interlayer Bonding in the Clay Minerals. 3.16 The 1:1 Minerals. 3.17 Smectite Minerals. 3.18 Micalike Clay Minerals. 3.19 Other Clay Minerals. 3.20 Summary of Clay Mineral Characteristics. 3.21 Determination of Soil Composition. 3.22 X-ray Diffraction Analysis. 3.23 Other Methods for Compositional Analysis. 3.24 Quantitative Estimation of Soil Components. 3.25 Concluding Comments. Questions and Problems. CHAPTER 4: SOIL COMPOSITION AND ENGINEERING PROPERTIES. 4.1 Introduction. 4.2 Approaches to the Study of Composition and Property Interrelationships. 4.3 Engineering Properties of Granular Soils. 4.4 Dominating Influence of the Clay Phase. 4.5 Atterberg Limits. 4.6 Activity. 4.7 Influences of Exchangeable Cations and pH. 4.8 Engineering Properties of Clay Minerals. 4.9 Effects of Organic Matter. 4.10 Concluding Comments. Questions and Problems. CHAPTER 5: SOIL FABRIC AND ITS MEASUREMENT. 5.1 Introduction. 5.2 Definitions of Fabrics and Fabric Elements. 5.3 Single-Grain Fabrics. 5.4 Contact Force Characterization Using Photoelasticity. 5.5 Multigrain Fabrics. 5.6 Voids and Their Distribution. 5.7 Sample Acquisition and Preparation for Fabric Analysis. 5.8 Methods for Fabric Study. 5.9 Pore Size Distribution Analysis. 5.10 Indirect Methods for Fabric Characterization. 5.11 Concluding Comments. Questions and Problems. CHAPTER 6: SOIL-WATER-CHEMICAL INTERACTIONS. 6.1 Introduction. 6.2 Nature of Ice and Water. 6.3 Influence of Dissolved Ions on Water. 6.4 Mechanisms of Soil-Water Interaction. 6.5 Structure and Properties of Adsorbed Water. 6.6 Clay-Water-Electrolyte System. 6.7 Ion Distributions in Clay-Water Systems. 6.8 Elements of Double-Layer Theory. 6.9 Influences of System Variables on the Double Layer. 6.10 Limitations of the Gouy-Chapman Diffuse Double Layer Model. 6.11 Energy and Force of Repulsion. 6.12 Long-Range Attraction. 6.13 Net Energy of Interaction. 6.14 Cation Exchange-General Considerations. 6.15 Theories for Ion Exchange. 6.16 Soil-Inorganic Chemical Interactions. 6.17 Clay-Organic Chemical Interactions. 6.18 Concluding Comments. Questions and Problems. CHAPTER 7: EFFECTIVE, INTERGRANULAR, AND TOTAL STRESS. 7.1 Introduction. 7.2 Principle of Effective Stress. 7.3 Force Distributions in a Particulate System. 7.4 Interparticle Forces. 7.5 Intergranular Pressure. 7.6 Water Pressures and Potentials. 7.7 Water Pressure Equilibrium in Soil. 7.8 Measurement of Pore Pressures in Soils. 7.9 Effective and Intergranular Pressure. 7.10 Assessment of Terzaghi's Equation. 7.11 Water-Air Interactions in Soils. 7.12 Effective Stress in Unsaturated Soils. 7.13 Concluding Comments. Questions and Problems. CHAPTER 8: SOIL DEPOSITS-THEIR FORMATION, STRUCTURE, GEOTECHNICAL PROPERTIES, AND STABILITY. 8.1 Introduction. 8.2 Structure Development. 8.3 Residual Soils. 8.4 Surficial Residual Soils and Taxonomy. 8.5 Terrestrial Deposits. 8.6 Mixed Continental and Marine Deposits. 8.7 Marine Deposits. 8.8 Chemical and Biological Deposits. 8.9 Fabric, Structure, and Property Relationships: General Considerations. 8.10 Soil Fabric and Property Anisotropy. 8.11 Sand Fabric and Liquefaction. 8.12 Sensitivity and Its Causes. 8.13 Property Interrelationships in Sensitive Clays. 8.14 Dispersive Clays. 8.15 Slaking. 8.16 Collapsing Soils and Swelling Soils. 8.17 Hard Soils and Soft Rocks. 8.18 Concluding Comments. Questions and Problems. CHAPTER 9: CONDUCTION PHENOMENA. 9.1 Introduction. 9.2 Flow Laws and Interrelationships. 9.3 Hydraulic Conductivity. 9.4 Flows Through Unsaturated Soils. 9.5 Thermal Conductivity. 9.6 Electrical Conductivity. 9.7 Diffusion. 9.8 Typical Ranges of Flow Parameters. 9.9 Simultaneous Flows of Water, Current, and Salts Through Soil-Coupled Flows. 9.10 Quantification of Coupled Flows. 9.11 Simultaneous Flows of Water, Current, and Chemicals. 9.12 Electrokinetic Phenomena. 9.13 Transport Coefficients and the Importance of Coupled Flows. 9.14 Compatibility-Effects of Chemical Flows on Properties. 9.15 Electroosmosis. 9.16 Electroosmosis Efficiency. 9.17 Consolidation by Electroosmosis. 9.18 Electrochemical Effects. 9.19 Electrokinetic Remediation. 9.20 Self-Potentials. 9.21 Thermally Driven Moisture Flows. 9.22 Ground Freezing. 9.23 Concluding Comments. Questions and Problems. CHAPTER 10: VOLUME CHANGE BEHAVIOR. 10.1 Introduction. 10.2 General Volume Change Behavior of Soils. 10.3 Preconsolidation Pressure. 10.4 Factors Controlling Resistance to Volume Change. 10.5 Physical Interactions in Volume Change. 10.6 Fabric, Structure, and Volume Change. 10.7 Osmotic Pressure and Water Adsorption Influences on Compression and Swelling. 10.8 Influences of Mineralogical Detail in Soil Expansion. 10.9 Consolidation. 10.10 Secondary Compression. 10.11 In Situ Horizontal Stress (K 0 ). 10.12 Temperature-Volume Relationships. 10.13 Concluding Comments. Questions and Problems. CHAPTER 11 STRENGTH AND DEFORMATION BEHAVIOR. 11.1 Introduction. 11.2 General Characteristics of Strength and Deformation. 11.3 Fabric, Structure, and Strength. 11.4 Friction Between Solid Surfaces. 11.5 Frictional Behavior of Minerals. 11.6 Physical Interactions Among Particles. 11.7 Critical State: A Useful Reference Condition. 11.8 Strength Parameters for Sands. 11.9 Strength Parameters for Clays. 11.10 Behavior After Peak and Strain Localization. 11.11 Residual State and Residual Strength. 11.12 Intermediate Stress Effects and Anisotropy. 11.13 Resistance to Cyclic Loading and Liquefaction. 11.14 Strength of Mixed Soils. 11.15 Cohesion. 11.16 Fracturing of Soils. 11.17 Deformation Characteristics. 11.18 Linear Elastic Stiffness. 11.19 Transition from Elastic to Plastic States. 11.20 Plastic Deformation. 11.21 Temperature Effects. 11.22 Concluding Comments. Questions and Problems. CHAPTER 12: TIME EFFECTS ON STRENGTH AND DEFORMATION. 12.1 Introduction. 12.2 General Characteristics. 12.3 Time-Dependent Deformation-Structure Interaction. 12.4 Soil Deformation as a Rate Process. 12.5 Bonding, Effective Stresses, and Strength. 12.6 Shearing Resistance as a Rate Process. 12.7 Creep and Stress Relaxation. 12.8 Rate Effects on Stress-Strain Relationships. 12.9 Modeling of Stress-Strain-Time Behavior. 12.10 Creep Rupture. 12.11 Sand Aging Effects and Their Significance. 12.12 Mechanical Processes of Aging. 12.13 Chemical Processes of Aging. 12.14 Concluding Comments. Questions and Problems. List of Symbols. References. Index.

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Q1. What have the authors contributed in "A contact dynamics code implementation for the simulation of asteroid evolution and regolith in the asteroid environment" ?

By providing access to the local physical mechanisms, DEM allows the exploration of microscopic based phenomena related to particles properties and interactions in various conditions and to revisit constitutive equations consequently. In this paper the authors describe the Contact Dynamics ( CD ) method, a class of DEM based on non-smooth mechanics, and its implementation in the open-source software LMGC90. In contrast to more classical approach, Hardand Soft-Sphere DEM, the CD method is based on an implicit time integration of the equations of motion and on a non-regularized formulation of mutual exclusion between particles. This numerical strategy is particularly relevant to the study of dense granular assemblies ( even of large size ) because it does not introduce numerical artefacts due to contact stiffness. So that it can be used for Small Body research, the authors implement a parallelised kd-tree and monitor the performance of the code as it simulates a number of granular systems. The authors provide examples of the simulation of the accretion of self-gravitating aggregates as well as their rotational disruption.