# RD-E: 0901 - Billiards (Pool)

A pool game is modeled to show the transmission of momentum between one impacting ball and 15 impacted balls.

The purpose of this example is to investigate the transmission of momentum between several balls. Contact with the various interfaces using the Penalty and Lagrange Multipliers' method is analyzed.

## Options and Keywords Used

• 16-node thick shell and sphere mesh (/PROP/TYPE20 (TSHELL))
• TYPE7 interface using the Lagrange Multipliers method and the Penalty method (/INTER/TYPE7)
• TYPE16 sliding and tied interface, and quadratic surface contact (/INTER/LAGMUL/TYPE16)
• Elastic shock
• Momentum transmission and shock wave
• Initial velocities (/INIVEL)
An initial velocity of 1.5 ms-1 in X direction is applied to all nodes of the white (cue) ball.

All nodes of the lower face of the table are completely fixed (translations and rotations).

Gravity is considered for all the balls nodes. A function defines the gravity acceleration in the Z direction compared with time. Gravity is activated using /GRAV.

## Input Files

Before you begin, copy the file(s) used in this example to your working directory.

## Model Description

Pool is a game consisting of 16 balls, each 50.8 mm in diameter. It is played on a small billiard table measuring 1800 mm x 900 mm. Fifteen (15) balls are placed in a triangle to enable their tight grouping. The initial velocity of the shooting (cue) ball is presumed equal to 1.5 ms-1. Elastic rebounds are observed.

Units: mm, g, N, MPa.

The material is subjected to a linear elastic law (/MAT/LAW1) with the following properties:
Material Properties: Balls (pheonolic resin)
Value
Initial density
0.00137 g.mm-3
Young's modulus
10500 $\left[\mathrm{MPa}\right]$
Poisson ratio
0.3
Material Properties: Frame (polymer)
Value
Initial density
0.001 g.mm-3
Young's modulus
1000 $\left[\mathrm{MPa}\right]$
Poisson ratio
0.49
Material Properties: Plate (slate)
Value
Initial density
0.0028 g.mm-3
Young's modulus
62000 $\left[\mathrm{MPa}\right]$
Poisson ratio
0

### Model Method

The balls are meshed with 16–node solid shells (quadratic elements) in order to improve the conditions of contact by taking into account the curvatures. The frame of the table is made of 16-node solid shells to comply with the interface used. The plate is modeled using only one solid element. The 16-node thick shells are considered as solid elements. They are defined by a thick TYPE20 shell property (number 16 solid formulation for quadratic 16-node thick shells, fully-integrated with 2x2x2 integration points).
The TYPE16 interface with the Lagrange Multipliers method is used to model the ball/ball and balls/table contacts. An interface must be defined for each ball (that is: 16 interfaces in total). An additional interface is used to define the contacts between the balls and the table (plate and frame).

Secondary nodes (red) are extracted from the external surfaces of the parts.

## Results

### Curves and Animations

Due to the faceting of the ball, contact between the impacting ball and the impacted balls is not perfectly symmetrical and momentum is not homogeneously transmitted among the balls. An apparent physical strike thus results.