A helmet has an open cell polystyrene foam liner that is designed to absorb the kinetic energy from an impact. The foam has a relative density of 10% and the Young's modulus of the solid polystyrene that the foam is made from is 1.5GPa. How much energy per unit volume, in J/m3, can the foam absorb if it is loaded in uniaxial compression to a strain of 80%? Please assume that C1=1.0 and C2=0.05.
U (in J/m3):
To calculate the energy per unit volume (U) that can be absorbed by the foam, we can use the following equation:
U = C1 * σc * ε + C2 * ε^2
Where:
- U is the energy per unit volume in J/m3
- C1 is a material constant
- σc is the compressive stress in Pa
- ε is the strain
Given:
- C1 = 1.0
- C2 = 0.05
- σc is the compressive stress (which we need to determine)
- ε = 80%
To find σc, we can use Hooke's Law for uniaxial compression:
σc = E * ε
Where:
- E is the Young's modulus in Pa (1.5 GPa = 1.5 * 10^9 Pa)
Plugging in the values:
σc = (1.5 * 10^9 Pa) * 0.8
Now we can substitute σc and ε into the equation for U:
U = (1.0) * [(1.5 * 10^9 Pa) * 0.8] * 0.8 + (0.05) * (0.8)^2
Calculating:
U = (1.2 * 10^9) * 0.8 + 0.05 * 0.64
U = 0.96 * 10^9 + 0.032
Simplifying:
U = 9.6 * 10^8 + 0.032
U ≈ 9.6 * 10^8 J/m3
Therefore, the foam can absorb approximately 9.6 * 10^8 Joules of energy per cubic meter.