Each cell within the model is subject to several physical constraints, which re- strict the allowable distributions of forces based on the stress of the cell. These constraints arise as a result of applying Newton’s Laws of Motion (laws that all physical objects obey) to the cells within the model (along with the overdamping assumption mentioned earlier). They say that the forces on the cell must exactly cancel one another out. They also require the turning forces on the cell to cancel out so that there can be no net torque on a cell.
More constraints arise from the fact that the shape of the cell must reflect the forces on it. For example, if it is elongated, it must be experiencing forces to hold it in this elongated state. This constraint can be derived and combined with the force and torque balance constraints to calculate the viscous forces from the contact and active forces within the model.
In this section, we will derive the algebraic forms of these constraints based on the surface force density function T(θ), which defines the force vector at a given angle θ on the surface of the cell.
We now introduce the notation that will be used throughout this section. We assume when applying forces that cells are circular for the purposes of this analysis, rather than elliptical, greatly simplifying the algebra. If any cell within a simulation reaches an aspect ratio of more than 3 : 1, at which point the circular approximation starts to break down, we discard the simulation from the analysis. The boundary of a circular cell can be defined by a radius R and an angular parameter θ taken from the x-axis, thus every point on the perimeter is of the form Rˆer(θ) = (R cos θ, R sin θ), where ˆer(θ) is the unit radial vector at angle θ.
The unit tangent vector at a given angle θ given by ˆeθ(θ) = (− sin θ, cos θ).
functions. Here we derive the constraints in terms of the surface density function before substituting its explicit form into the derived constraints.
Cells are modelled as objects with no inertia, so that forces and torques on the cell must be balanced. This assumption coupled with the fact that the forces on a cell must be consistent with its stress leads to three constraints:
• Force Balance RR2π
0 T(θ)dθ = 0
• Torque Balance RR2π
0 T(θ).ˆeθ(θ)dθ = 0
• Stress
The third constraint means that the distribution of forces acting on a cell must match its current stress at any time. We derive this constraint now.
3.4.1
Stress Constraint
Cauchy’s Stress Theorem (Appendix Equation 1) states that the traction vector (or force-density vector) W(θ) is given by the stress tensor σ acting on the normal at that point ˆer(θ):
W(θ) = σˆer(θ) (3.8)
The expression allows us to calculate the function W(θ) from the stress tensor σ.
Cell i within the model can experience any distribution of forces due to its interactions with neighbouring cells. We define this distribution of forces using a surface force density vector as Ti(θ). In this section, we would like to determine
which components of this general surface force density vector affect the overall stress of the cell.
This relationship is expressed as an integral (as with force balance) which guarantees that the total distribution of forces on the cell is consistent with its stress.
To calculate this relationship, we note that the unit radial vector in 2D, ˆer(θ),
has the property (using suffix notation and ei to represent a component of the
unit radial vector):
1 π
Z 2π
0
where δij is the Kronecker delta. Starting from Equation3.8 in suffix notation: Ti(θ) = σijej(θ) 1 π Z 2π 0 Ti(θ)ek(θ)dθ = σij 1 π Z 2π 0 ej(θ)ek(θ)dθ σik = 1 π Z 2π 0 Ti(θ)ek(θ)dθ
Or equivalently in vector notation, the stress constraint equation is:
σ = 1 π Z 2π 0 T(θ) ⊗ ˆer(θ)dθ = 1 π Z 2π 0 Txcos θ Txsin θ Tycos θ Tysin θ ! dθ (3.9)
where T(θ) = (Tx(θ), Ty(θ)). Examples of different functions T(θ) and the re-
sulting stresses and shape changes on the circle are given in Figure 3.6.
3.4.2
Calculating the Viscous Forces from Contact Forces
The constraints on the form of the surface force density function Ti(θ), which
ensure force balance, torque balance and a shape that is consistent are:
0 = Z 2π 0 T(θ)dθ (3.10) σ = 1 π Z 2π 0 Txcos θ Txsin θ Tycos θ Tysin θ ! dθ (3.11)
We now describe how these equations can be used to derive the viscous forces from the contact and active forces. We limit our discussion to only contact forces but the active forces are treated in the same way as the contact forces.
To derive the equations relating the viscous forces, we use a modified form of Equation 3.1: Ti(θ) = 1 Ri s π√3 6 X j (Fcij + Fvij)δ(θ − θij)
where Fc and Fv are the contact and viscous forces respectively. This equation is substituted into Equations 3.10 and 3.11. For cell i, with neighbours indexed
Figure 3.6: Different forms of T(θ) and the associated stress tensors as calculated from Equation 3.11.
by j, this gives the form of the constraints as: −X j Fcij =X j Fvij s 6π √ 3Riσ − X j Fcij ⊗ ˆer(θij) = X j Fvij ⊗ ˆer(θij)
These formulae represent a system of simultaneous linear equations that must be solved to determine the viscous forces. In order for this system to be solvable
by some procedure (either by a best fit or exact solution), the rank of the matrix representing this linear system must be at least the number of unknowns that we are solving for.
Counting the degrees of freedom (unknowns) and comparing it to the number of constraints, we see that the 2 formulae above represent 6 constraints per cell.
For a system of N cells with periodic boundary conditions, there are 3N contacts each with one viscous force vector to be determined, i.e. 2 degrees of freedom, so 6N unknown quantities.
The number of constraints exactly matches the number of unknowns so we can solve the linear system of equations. The procedure used to solve this system of equations is described in the next section.