Notations
The following notations are employed throughout the present documentation set.
Vectors are denoted by boldface type, such as v. The length or magnitude is denoted ‖v‖ or simply v. The addition of a hat denotes a unit vector, such that ˆv=v/‖v‖. The addition of a dot denotes a time derivative, such as ˙v=∂v/∂t.
There is a global coordinate system (xyz). The vector v can be expressed in the global coordinate system by the relations:
(1)v=v(x,y,z)=vxˆi+vyˆj+vzˆkwith vx=v⋅ˆi,vy=v⋅ˆj,vz=v⋅ˆk
where ˆi, ˆj, and ˆk are unit vectors directed along the positive x, y, and z axes, respectively. The global coordinate system is used by both the 2D and 3D models. The 2D model is oriented to lie in the xy plane, resulting in the following constraints in PFC (see Figure 1):
Particles and contacts remain centered in the xy plane.
Particles consist of unit-thickness disks.
Forces act in the xy plane such that
(2)F=F(x,y,z)=Fxˆi+Fyˆj+Fzˆk(Fz≡0).Moments act perpendicular to the xy plane such that
(3)M=M(x,y,z)=Mxˆi+Myˆj+Mzˆk(Mx≡My≡0).
There are no such constraints on the 3D model.
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