In this verification case, two frictionless spheres with the same radius, , are stacked between two fixed walls, causing them to be compressed. The walls are positioned at m and , with the particles centers initially located at m and m. This setup, illustrated in Figure 2.23: Compressed spheres stacked between two fixed walls, ensures the particles remain in contact and under compression.

Figure 2.23: Compressed spheres stacked between two fixed walls

A general expression for the acceleration of particle 1 is as follows:

(2–21)

where is the gravity force, is the particle 1-wall spring force, is the particle 1-wall damping force, is the particle 1-particle 2 spring force and is the particle 1-particle 2 damping force.

The expressions for each of these forces are:

(2–22)

(2–23)

(2–24)

(2–25)

(2–26)

where refers to particle mass, is the gravity acceleration, refers to spring coefficients, and to the damping coefficients.

The is calculated by:

(2–27)

And the individual is computed as:

(2–28)

where refers to Young's Modulus and is the particle diameter.

The damping coefficient is determined directly by the relation shown in the figure below. Refer to Rocky DEM Technical Manual to see this relation formulation.

Figure 2.24: Relation between the damping coefficient and the restitution coefficient .

The acceleration for particle 1 can be written as:

(2–29)

Similarly, for particle 2:

(2–30)

where the expressions for each of the forces are:

(2–31)

(2–32)

(2–33)

(2–34)

(2–35)

The acceleration for particle 2 can be written as:

(2–36)

This system of equations (Equation 2–29 and Equation 2–36) does not have any known analytical solution so far. However, to be compared with Rocky, it can be numerically solved using the fourth-order Runge-Kutta method.

The equations shown in the last section can be resolved and equivalent results can be calculated by Rocky considering the same input data and boundary conditions. The input parameters for this verification case setup are presented in Table 2.10: Input parameters for the compressed spheres case validation..

Table 2.10: Input parameters for the compressed spheres case validation.

Parameter	Value	Unit
Physical Model:
Thermal	Disabled	-
Normal Force	Linear Spring Dashpot	-
Adhesive Force	None	-
Restitution Coefficient	1	-
Gravity Y	-9.81
Particle 1:
Sphere Radius	0.0005
Sphere Density	20000
Friction Coefficient	0	-
Young Modulus	2E06	Pa
Particle 2:
Sphere Radius	0.0005
Sphere Density	10000
Friction Coefficient	0	-
Young Modulus	2E06	Pa
Bottom Wall:
Friction Coefficient	0	-
Young Modulus	2E06	Pa
Top Wall:
Friction Coefficient	0	-
Young Modulus	2E06	Pa
Solver Parameters:
Simulation Duration	0.001
Time Interval	1E-05

After running the simulations using the parameters described in the previous section, it is possible to compare the results between Rocky and the Runge-Kutta solution. The evolution of the particles positions (measured from their centers) is shown in Figure 2.25: Evolution of the position of particle 1 over simulation time. and Figure 2.26: Evolution of the position of particle 2 over simulation time..

Figure 2.25: Evolution of the position of particle 1 over simulation time.

Figure 2.26: Evolution of the position of particle 2 over simulation time.

The absolute and relative errors for the particles positions are compared in Figure 2.27: Absolute error between Rocky and Runge-Kutta solution for the particles position. and Figure 2.28: Relative error between Rocky and Analytical solution for the particles position., respectively. The absolute error is maximum for particle 1 around 8.0E-07 m, representing a 0.18% relative error, while for particle 2 it is roughly 1.8E-06 m, representing a 0.13% relative error.

Figure 2.27: Absolute error between Rocky and Runge-Kutta solution for the particles position.

Figure 2.28: Relative error between Rocky and Analytical solution for the particles position.