Question

determine the poisson's ratio of the material of a wire whose volume remains constant under an external normal stress

Answer 1

Volume = Area x Length
a l = (a - Δa ) (l + Δl)
a = πr²
Δa = 2πrΔr
πr²l = (πr² - 2πrΔr) (l + Δl)
1 = [(πr² - 2πrΔr)/πr²] [(l + Δl)/l]
1 = (1 - Δr/r) (1 + Δl/l)
1 = 1 - (Δr/r) (Δl/l) - Δr/r + Δl/l
(Δr/r) (Δl/l) is the smallest term and hence can be neglected.
So, Δr/r = Δl/l
(Δr/r) / (Δl/l) = 1
which shows that the Poisson's ratio is the ratio of lateral strain to longitudinal strain.


thankuuu