Question 16.9: Determination of Initial Stress in a Polymer A band of polyi......
Determination of Initial Stress in a Polymer
A band of polyisoprene is to hold together a bundle of steel rods for up to one year. If the stress on the band is less than 1500 psi, the band will not hold the rods tightly. Determine the initial stress that must be applied to a polyisoprene band when it is slipped over the steel. A series of tests showed that an initial stress of 1000 psi decreased to 980 psi after six weeks.
Although the strain on the elastomer band may be constant, the stress will decrease over time due to stress relaxation. We can use Equation 16-5
σ = σ_{0} \exp (-t/λ) (16-5)
and our initial tests to determine the relaxation time for the polymer:
σ = σ_{0} \exp (- \frac{t}{λ})
980 psi = 1000 psi \exp(-\frac{6 \ weeks}{λ})
– \frac{6}{λ} = \ln(\frac{980}{1000}) = \ln (0.98) = -0.0202
λ = \frac{6}{0.0202} =297 \ weeks
Now that we know the relaxation time, we can determine the stress that must be initially placed onto the band so that it will still be stressed to 1500 psi after 1 year (52 weeks).
1500 = σ_{0} \exp (-52/297) = σ_{0} \exp(-0.1751) = 0.8394 \ σ_{0}
σ_{0} = \frac{1500}{0.8394} = 1787 \ psi
The polyisoprene band must be made significantly undersized so it can slip over the materials it is holding together with a tension of 1787 psi. After one year, the stress will still be 1500 psi.