# Notations

The following notations are employed throughout the present documentation set.

Vectors are denoted by boldface type, such as $$\mathbf{v}$$. The length or magnitude is denoted $$\left\| \mathbf{v} \right\|$$ or simply $$v$$. The addition of a hat denotes a unit vector, such that $$\hat{\mathbf{v}} = \mathbf{v} / \left\| \mathbf{v} \right\|$$. The addition of a dot denotes a time derivative, such as $$\dot{\mathbf{v}} = \partial \mathbf{v} / \partial t$$.

There is a global coordinate system ($$xyz$$). The vector $$\mathbf{v}$$ can be expressed in the global coordinate system by the relations:

(1)$\begin{split}\begin{array}{l} \mathbf{v} = \mathbf{v}(x,y,z) = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}} \\ \mbox{with } v_x = \mathbf{v} \cdot \hat{\mathbf{i}},\; v_y = \mathbf{v} \cdot \hat{\mathbf{j}},\; v_z = \mathbf{v} \cdot \hat{\mathbf{k}} \end{array}\end{split}$

where $$\hat{\mathbf{i}}$$, $$\hat{\mathbf{j}}$$, and $$\hat{\mathbf{k}}$$ 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):

1. Particles and contacts remain centered in the $$xy$$ plane.

2. Particles consist of unit-thickness disks.

3. Forces act in the $$xy$$ plane such that

(2)$\mathbf{F} = \mathbf{F}(x,y,z) = F_x \hat{\mathbf{i}} + F_y \hat{\mathbf{j}} + F_z \hat{\mathbf{k}} \quad (F_z \equiv 0).$
4. Moments act perpendicular to the $$xy$$ plane such that

(3)$\mathbf{M} = \mathbf{M}(x,y,z) = M_x \hat{\mathbf{i}} + M_y \hat{\mathbf{j}} + M_z \hat{\mathbf{k}} \quad (M_x \equiv M_y \equiv 0).$

There are no such constraints on the 3D model.

Figure 1: Global coordinate system and orientation of the 2D model, which consists of unit-thickness disks centered in the $$xy$$ plane.