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I was given the following formula to relate the change of area against the change of length of an electric wire with a Poisson ratio:

$ {\Delta A \over A} = -2 \nu {\Delta L \over L} $

where $ \Delta A \over A $ represents the change in cross-sectional area of the wire due to the transverse strain as the wire gets pulled longitudinally stretching length $ L $ to $ L+\Delta L$.

I don't get how this equation is derived. The Poisson ratio is defined by $ \nu = -{\epsilon_{lateral} \over {\epsilon_{longitudinal}} } = - {{\Delta d / d} \over {\Delta L / L}} $ where $ d $ is the diameter of the cross section. Then the ratio of the area:

$ {\Delta A \over A} = {{0.25\pi(d+\Delta d)^2 - 0.25\pi d^2} \over {0.25\pi d^2}} = {{2d \Delta d} \over d^2} + {{\Delta d^2} \over {d^2}} = -2 \nu { \Delta L \over L} + \big( \nu {\Delta L \over L} \big)^2 \neq -2 \nu {\Delta L \over L} $

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  • $\begingroup$ One man's fish is another man's poisson. $\endgroup$ Jan 18 at 0:06
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Since you are essentially using infinitesimal changes, then higher order differences can be neglected.

I.e. following from your equation $${\Delta A \over A} = {{0.25\pi(d+\Delta d)^2 - 0.25\pi d^2} \over {0.25\pi d^2}} = {{2d \Delta d} \over d^2} + {{\Delta d^2} \over {d^2}} = -2 \nu { \Delta L \over L} + \big( \nu {\Delta L \over L} \big)^2 $$

because $\left(\frac{\Delta L}{L}\right)^2$ is a second order difference, you can assume that $\left( \nu {\Delta L \over L} \right)^2\approx 0$.

Therefore: $$ -2 \nu { \Delta L \over L} + \underbrace{\left( \nu {\Delta L \over L} \right)^2}_{\approx 0}\approx -2 \nu { \Delta L \over L} $$

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  • $\begingroup$ Is there some kind of rule of thumb to decide when I should or shouldn't ignore higher order terms? $\endgroup$
    – KMC
    Jan 17 at 18:38
  • $\begingroup$ @KMC -- perhaps when the first order terms are systematically approaching zero but the higher order terms are non-zero, which happens at certain kinds of equilibrium points $\endgroup$
    – Pete W
    Jan 17 at 20:14
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    $\begingroup$ I would argue that if $\frac{\Delta L}{L} $ is less than to 0.001 (which is common for materials in the elastic region) then its safe to approximate that to zero. Because in that case $\left(\frac{\Delta L}{L}\right)^2 $ would be less than 0.000001. However, I've never seen that written down as a rule of thumb. $\endgroup$
    – NMech
    Jan 17 at 21:41

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