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I needed to know the Young’s modulus of the ballistics gel we were using in order to calculate things like deformation on impact. To measure this we found a formula on Wikipedia that gives the force required to cause an indentation by a sphere into a surface of given Young’s modulus. The equation gives the force as a result of the effective elastic modulus (E^*), the depth of deformation (d), and the radius of the sphere (R)

F = 4/3 E^* R^1/2 d^3/2

(Eqn. 1)

 

The E^* is a function of the Young’s modulus of both the surface being indented (E₁) and the sphere (E₂), and the Poisson’s ratios of the surface (v₁) and the sphere (v₂).

 

1 / E^*  =  ( ( 1 - v₁² ) / E₁ )  +  ( ( 1 - v₂² ) / E₂ )

 

(Eqn. 2)

 

I used a chromium steel sphere to make the indentation and looked up its properties.

 

v₂ = 0.3           E₂ = 200 x 10⁹ Pa

 

This makes the second fraction of equation 2 essentially zero, so I was able to ignore that part of Eqn 2. and solve for E₁, the Young’s modulus of the ballistics gel like this.

 

E^* = E₁ / (1-v₁²)

(Eqn. 3)

 

Now recognizing that the force in Eqn 1 is simply F = mg, and substituting Eqn 3. Into Eqn 1. Solving for E₁ gives the formula

 

E₁ = 3mg (1-v₁²) / 4 R^1/2 d^3/2

 

The Poisson’s ratio (v₁) for gelatin = 0.5. And for the rest of this paper, the Young’s modulus of the ballistics gelatin will be noted as E_y and its Poisson’s ratio as v_p.

   

Setup

For this test I got out the egg crusher again so that I could easily add a known mass. On the bottom of the crusher I put a one-inch slab of ballistics gel, the two inch steel ball went on top of that, and the top board of the crusher went on top of that. This allowed me to easily add together the masses of the ball, board, and any other weight for the test.

Once the rig was set up I held the ball and board so that they were just at the surface of the ballistics gel and marked on the vertical polls the original position of the board. Then I added five pounds of weight to the top board causing the ball to sink into the gel, I then marked the final position of the board and measured the difference to give the depth of penetration.

Results

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