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):
I worked with this equation:
sigma=C_2*E_s*0.1^2 = 750000[Pa]
Then for the energy:
U=sigma*s_densification= 750000*0.8=600000[J/m^3]
But the answer is incorrect.
Also, I know that E=C_1*E_s*0.1^2 =15[MPa] But I don't know how can I use it.
Some help please
600000
answer is 600000
Homework Posting Tips
Please show your work. Tutors will not do your homework for you. Please show your work for any question that you are posting.
not sure, I found it was 500000 J/m³
To find the energy per unit volume that the foam can absorb, we need to use the strain energy density equation:
U = (1/2) * C1 * (strain)² + (1/2) * C2 * (strain)⁴
In this equation, U represents the strain energy density, C1 and C2 are material constants, and strain is the deformation the material undergoes.
Given:
C1 = 1.0
C2 = 0.05
strain = 80% = 0.8 (since strain is usually expressed as a decimal)
Now, we can plug in the values into the equation to calculate the strain energy density:
U = (1/2) * C1 * (strain)² + (1/2) * C2 * (strain)⁴
U = (1/2) * 1.0 * (0.8)² + (1/2) * 0.05 * (0.8)⁴
U = 0.4 * 0.64 + 0.0256 * 0.4096
U = 0.256 + 0.01048576
U = 0.26648576 J/m³
Therefore, the foam can absorb approximately 0.266 J/m³ of energy per unit volume when loaded in uniaxial compression to a strain of 80%.