A spherical rubber balloon has an initial thickness of 0.45mm, and diameter 250mm. Final diameter of 1000mm.
Find final thickness, true stress in the plane of the balloon wall, Air pressure required.
(Assume rubber is idea and obeys Guassian statistics, and take the shear modulus to be 1MPa.)
To find the final thickness of the balloon, we can use the principle of volume conservation. The volume of the balloon remains the same before and after inflation.
The initial volume of the balloon can be calculated using the formula for the volume of a sphere:
V_initial = (4/3) π r_initial^3
where r_initial is the initial radius of the balloon (half of the initial diameter). Plugging in the values, we have:
V_initial = (4/3) π (250/2)^3 = (4/3) π 125000
The final volume of the balloon can also be calculated using the same formula, but with the final diameter:
V_final = (4/3) π (1000/2)^3 = (4/3) π 250000
Since the volumes are the same, we can set them equal to each other:
V_initial = V_final
Simplifying the equation:
(4/3) π 125000 = (4/3) π 250000
Dividing both sides by (4/3) π:
125000 = 250000
This shows that the equation is not valid. Therefore, we cannot achieve a final diameter of 1000mm using this initial thickness of 0.45mm.
As for the true stress in the plane of the balloon wall, we can find it using the formula:
True stress = Shear stress * (Initial thickness / Final thickness)
The shear stress can be calculated using Hooke's Law:
Shear stress = Shear modulus * Shear strain
Given that the shear modulus is 1MPa (megapascal), we need to calculate the shear strain. The shear strain can be defined as the change in thickness divided by the initial thickness:
Shear strain = (Final thickness - Initial thickness) / Initial thickness
Plugging in the values, we have:
Shear strain = (t_final - t_initial) / t_initial
The air pressure required to achieve the final diameter can be calculated using the formula for the hoop stress in a thin-walled pressure vessel:
Hoop stress = (Pressure * Final diameter) / (2 * Final thickness)
Given that the hoop stress is equal to the true stress, we can rearrange the formula to solve for the air pressure:
Pressure = (True stress * 2 * Final thickness) / Final diameter
Substituting the known values, we have:
Pressure = (True stress * 2 * t_final) / D_final
Unfortunately, we cannot proceed with the calculations since the final diameter cannot be achieved with the given initial thickness.