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Mechanics

We start with an outline of classical mechanics, to provide a framework for the discrete ele-

ment method (DEM). While most of the material in this chapter can be found scattered in

various books on mechanics, no text seems to be available which covers concisely the con-

cepts needed for DEM simulation. This chapter is intended as a crash course in theoretical

mechanics, with an emphasis on issues relevant to computer implementation and testing. We

give a list of secondary literature that the reader may refer to for further details.

1.1 Degrees of freedom

Before discussing the dynamics of a mechanical system, we need to understand the nature of

the variables in the system. There are independent variables on the one hand, usually called

‘degrees of freedom’, and then there are dependent variables which depend on the degrees of

freedom, via algebraic relations or derivatives.

1.1.1 Particle mechanics and constraints

The concept of a ‘mass point’ means that we neglect the size of the mass and are interested

only in its trajectory. The position of a single mass point moving along the Cartesian x-axis

is described by the value of x, which corresponds to a single degree of freedom. A point

moving in the xy-plane has two degrees of freedom, r

=(x, y), and a point moving in three-

dimensional real space will have three degrees of freedom, r

= (x,y,z).Although we can

describe the motion of a point in three-dimensional space by four ‘space–time coordinates’

using the tuple (x,y,z,t),in classical mechanics t is not considered a degree of freedom but

rather a parameter, i.e. an independent variable which cannot be inﬂuenced.

Two mass points moving independently along the x-axis represent two degrees of freedom,

and r

(here and in the following, we assume equal masses). If we ‘glue’ these two particles

together at distance d =r

−r

as in Figure 1.1, one degree of freedom gets lost, and we are

Understanding the Discrete Element Method: Simulation of Non-Spherical Particles for Granular

and Multi-body Systems, First Edition. Hans-Georg Matuttis and Jian Chen.

Companion website: www.wiley.com/go/matuttis

2 Understanding the Discrete Element Method

dof

=2∙2–1=3

dof

=3·2–3=3

dof

=4·2–5=3

Figure 1.1 In two dimensions, the number of degrees of freedom n

dof

for1,2,3or4constrained

particles with an increasing number of constraints introduced. Newly added constraints are in black;

previous constraints are in gray.

left with only a single degree of freedom; in this case we can use either of r

, r

or the average

)/2 to determine the position of both particles uniquely. This means that one constraint

between two position variables eliminates one degree of freedom.

In two dimensions, for two point particles at r

= (x

) and r

= (x

) we have four

degrees of freedom, x

and y

. If we again ﬁx the distance between the particles at a

constant distance d, so that



− x

)

+ (y

− y

)

= d, (1.1)

we can choose any three variables from {x

} and the fourth will then be determined

from (1.1) by elementary geometry. Alternatively, we can introduce new variables, such as

the position of the center of mass, (x, y) = (r

+ r

)/2 for particles of the same mass, the

displacement (x, y) = (x

− x

− y

) between the particles, and the angle θ that the

line segment between the two particles makes with the x-axis. In any case, we end up with

three independent variables to describe the positions of the two particles fully. This means

that a single constraint (1.1) reduces the number of degrees of freedom, i.e. the number of

independent variables in the system, by 1.

In three-dimensional space, for two particles at positions (x

) and (x

) as

shown in Figure 1.2, a constraint



− x

)

+ (y

− y

)

+ (z

− z

)

= d (1.2)

will again reduce the number of degrees of freedom by 1, so if we want to work with the

center of mass

(x,y,z) =

{

) + (x

)

}

we need two angles, φ and θ say, to describe the orientation of the ‘rod’ in space. Rotation

around the orientation of the rod is not a degree of freedom, as it does not change the positions

of the two points. In principle, it does not matter how one deﬁnes the degrees of freedom,

whether it is with six variables and one constraint (1.2), with three Cartesian coordinates for

Mechanics 3

dof

=3∙2–1=5

dof

=3·4–6=6

dof

=3·3–3=6 n

dof

=3·5–9=6

Figure 1.2 In three dimensions, the number of degrees of freedom n

dof

for 1, 2, 3, 4 or 5 particles

constrained so that the resulting cluster has no internal degrees of freedom. Newly added constraints are

in black; previous constraints are in gray.

the center of mass and two angles, or with three Cartesian coordinates for one endpoint and

two angles. In each case the number of degrees of freedom is the same, namely 5.

1.1.2 From point particles to rigid bodies

When we introduce one more point mass at (x

) to our set-up, we have 9 variables in

total. If we connect this new point to both ends of our rod with the additional constraints



− x

)

+ (y

− y

)

+ (z

− z

)

= d

, (1.3)



− x

)

+ (y

− y

)

+ (z

− z

)

= d

, (1.4)

we get a triangle, as in the middle diagram of Figure 1.2. Again, we can give an alternative

description of its position in space using the center of mass, and use three angles, φ,θ and ψ,

to describe the orientation. So the formula

(degrees of freedom)



 

= (variables)



 

−(constraints)



 

again holds. If we connect a fourth particle rigidly to the cluster of three particles so that it

does not lie in the plane described by the other three, as shown in the fourth diagram from

the left in Figure 1.2, then the three extra constraints exactly compensate for the additional

three coordinates (x

) of the new particle. In fact, for four or more spatially connected

particles, the total number of degrees of freedom is always 6. Note that the rigid body formed

by the connected particles need not be three-dimensional; for example, although a triangle

is a two-dimensional shape, if it can rotate in three dimensions, then it also has six degrees

of freedom. Through the reasoning above, we have derived that an extended rigid body has

six degrees of freedom, irrespective of its size. The angular degrees of freedom φ,θ,ψ are

obtained from the rectilinear degrees of freedom (x

), (x

), . . . of the particles

upon introducing constraints of ﬁnite length between the particles.

4 Understanding the Discrete Element Method

‘Mathematically’ one can deﬁne a point particle as an object having ‘zero extension’ and a

rigid body as one having ‘zero deformation’. A more pragmatic deﬁnition of a point particle is

an object whose extent is much smaller than the distances that it covers in the processes under

investigation; after all, the Earth is pretty extended, but t he point-mass approach to describing

its trajectory around the sun works rather well. Likewise, a rigid body is an object for which

the deformations are much smaller than the scales that are of interest in the processes being

investigated.

1.1.3 More context and terminology

In principle, a ‘continuum’ has inﬁnitely many degrees of freedom; but in order to solve

continuum problems with a computer, we have to ﬁrst discretize the continuum to obtain a

ﬁnite number of degrees of freedom. We could, for instance, decompose the continuum into

representative mass points and model the elasticity by springs between the mass points. The

deformation of a spring can be computed from the positions of the bodies, so the springs

will not be degrees of freedom, while the coordinates of the mass points will be degrees

of freedom. With a ﬁnite element discretization, we decompose the elastic continuum into

a space-ﬁlling partition of elements for which elastic stress relations hold, and the degrees

of freedom are the nodes of the elements. Depending on the choice of boundary conditions,

there may be as many nodes as there are elements, or more; therefore, from the nodes one can

calculate the center of mass of the elements, but not vice versa. Describing the physics via the

motion of particles, for example of centers of mass, is called the ‘Lagrangian representation’.

This approach is natural for particulate systems, so we will adopt it in this book. Formulating

the physics for a reference system in which, e.g., density amplitudes change is called the

‘Eulerian representation’; this representation is preferable for many continuum problems. In

a Lagrangian representation, velocities of mechanical bodies are not degrees of freedom: they

can be obtained as the time derivatives of the positions on which they depend. On the other

hand, when we simulate a ﬂuid volume where velocities are assigned to the nodes of a ﬁnite

element or ﬁnite difference approximation in ‘Eulerian representation’, it is the velocities that

are the degrees of freedom.

In the previous two subsections, we introduced constraints as algebraic relations between

positions, but we remark here that constraints (whose associated functions are usually denoted

by g in formulae) can also be imposed on velocities. For a pendulum of length l swing-

ing around the origin as in Figure 1.3(a), the constraint g(x, y) stating that the bob (whose

diameter we will neglect) stays at constant distance from the origin is

+ y

= l

. (1.5)

In § 2.8 we will discuss the numerical solution of a problem where, in addition to constraints

on x and y, constraint relations for ˙x and ˙y are also in effect. In undergraduate mechanics,

it is common to circumvent solving the equations of motion of a constrained system with

variables (x, y) that simultaneously satisfy (1.5) by transforming into plane polar coordinates

(φ, r) so that r is eliminated. For more complicated mechanical systems, such a simplifying

transformation may not be possible any more, for instance if the pendulum is connected with

a unidirectionally moving body as in Figure 1.3(b).

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