Page 323 - Mechanics of Asphalt Microstructure and Micromechanics

Applications of Discrete Element Method 315

spheres to represent irregular particles even using the burn algorithms. The methods in
Chapter 3 include one on the computation of the volume and mass center coordinates
of irregular particles. For translational motion, the motion equation is the same as that
of spheres. However, the mass momentum for irregular particles is quite different from
that of spheres. The question is, for irregular particles, how different are the momen-
tums from those of equivalent ellipsoids? The following presents the work by Wang et
al. (2007) to answer this question.

9.6.1 Moment of Inertia Tensor
The momentum of inertia is a term used to describe the capacity of a cross-section to
resist bending. It is a ma thematical property of a section concerning the shape and
mass distribution about a set of reference axes. The reference axes usually run through
the mass center of the objects. The momentum of inertia in a solid body with density
r(r) with respect to a given axis is expressed as I and defined by the volume integral:
∫
I = ρ() 2 (9-32)
r r dV

Where r is the distance from a volume element dV to the axis of rotation. The mo-
mentum of inertia of an object depends on its shape and distribution of mass within the
object. It is directly related to rotational kinetic energy and angular momentum by the
following relation:
1
K = Iω 2 L = Iω (9-33)
2
Where w is the angular velocity; K is the rotational kinetic energy; and L is angular
momentum.
For 3D motion, a general case, the “moments of inertia,” about three rotation axes,
are not enough to describe the shape and mass distribution. It generally requires six
“products of inertia.” The total of nine “moments of inertia” form a 3 3 matrix and is
called moment of inertia tensor. It can be expressed as:
jk ∫
I = ρ() 2 − x x dV (9-34)
)
r r δ(
V jk j k
Where r is the distance of a point to the axis; d jk is the Kronecker delta. Expanding
this equation in terms of Cartesian axes gives the following equation:
⎡ ⎡ y + z 2 − xy − xz ⎤ I ⎡ 11 I 12 I ⎤
2
13
I = ∫ ρ(, ⎢ ⎢ − xy z + x 2 − yz ⎥ ⎥ dxdydz ⎢ ⎢ I 21 I 22 I 23 ⎥ ⎥ (9-35)
2
x y z , )
V
⎢ − xz − yz x + y 2 ⎥ ⎣ I ⎢ I I ⎥
2
33 ⎦
⎣ ⎦ 31 322
The diagonal elements of the tensor, I 11 , I 22 , and I 33 , are just simple moments of iner-
tia about x, y, and z axes. The other elements of the inertia tensor, such as I 12 , are the
“products of inertia,” which make the moment of inertia tensor always symmetric. Us-
ing the boundary coordinates of the particle, the moment of inertia tensor for a solid
body can be obtained numerically by summing the products of all the voxels contained
in a particle. The moment of inertia tensor can be expressed as:
⎡ y + z 2 − xy − xz ⎤ ⎤
2
p ∑
I = m s ⎢ ⎢ − xy z + x 2 − yz ⎥ ⎥ (9-36)
2
⎢ − xz − yz x + y 2 ⎥
2
⎣ ⎦