Calculate the elastic buckling collapse stress (the compressive strength), σ∗el, for an open-cell polystyrene foam with a relative density of 0.1. (Es for polystyrene = 3.5GPa). Recall that C2=0.05. Give the results in MPa.
σ∗el (in MPa):
The energy absorbed per unit volume in deforming a material is equal to the area under the stress strain curve. If the polystyrene foam has a flat stress plateau and densifies at a strain of 0.85, how much energy can it absorb per unit volume? Give your result in MJ/m3.
energy absorbed per unit volume (in MJ/m3):
stress=1.75
stress=1.75 MPa
E=1.4875 MJ/m3
To calculate the elastic buckling collapse stress (compressive strength), σ∗el, we can use the following formula:
σ∗el = (C2 * Es * (ρ / (1 - ρ))^2) / 1000
Where:
C2 is a constant (given as 0.05)
Es is the Young's modulus for polystyrene (given as 3.5 GPa)
ρ is the relative density of the foam (given as 0.1)
Substituting the given values into the formula, we have:
σ∗el = (0.05 * 3.5 GPa * (0.1 / (1 - 0.1))^2) / 1000
= (0.05 * 3.5 * 10^9 N/m^2 * (0.1 / 0.9)^2) / 1000
= (0.05 * 3.5 * 10^9 * (0.1 / 0.9)^2) / 1000
= (0.05 * 3.5 * 10^9 * 0.1^2) / (0.9^2 * 1000)
= (0.05 * 3.5 * 10^9 * 0.01) / 0.81
≈ 0.0765 GPa
To convert this into MPa, we multiply by 1000:
σ∗el = 0.0765 GPa * 1000
≈ 76.5 MPa
Therefore, the elastic buckling collapse stress (compressive strength) of the foam is approximately 76.5 MPa.
Now, let's calculate the energy absorbed per unit volume by the foam. Given that the foam densifies at a strain of 0.85 and has a flat stress plateau, the energy absorbed per unit volume can be determined by finding the area under the stress-strain curve.
The energy absorbed per unit volume can be calculated using the formula:
Energy absorbed per unit volume = σ∗el * εc
Where:
σ∗el is the elastic buckling collapse stress (compressive strength) we calculated earlier (76.5 MPa)
εc is the strain at which the foam densifies (given as 0.85)
Substituting the values into the formula, we have:
Energy absorbed per unit volume = 76.5 MPa * 0.85
= 65.025 MPa
To convert this into MJ/m³, we convert MPa to J/m³ and then divide by 10^6:
Energy absorbed per unit volume = 65.025 MPa * (10^6) / (10^6)
= 65.025 J/m³ / (10^6)
= 0.065025 MJ/m³
Therefore, the energy absorbed per unit volume in deforming the polystyrene foam is approximately 0.065025 MJ/m³.