Poisson's Ratio - Length Change

Length Change

For a cube stretched in the x-direction (see figure 1) with a length increase of in the x direction, and a length decrease of in the y and z directions, the infinitesimal diagonal strains are given by:

$d\varepsilon_x=\frac{dx}{x}\qquad d\varepsilon_y=\frac{dy}{y}\qquad d\varepsilon_z=\frac{dz}{z}$

Integrating the definition of Poisson's ratio:

$-\nu \int_L^{L+\Delta L}\frac{dx}{x}=\int_L^{L-\Delta L'}\frac{dy}{y}=\int_L^{L-\Delta L'}\frac{dz}{z}$

Solving and exponentiating, the relationship between and is found to be:

$\left(1+\frac{\Delta L}{L}\right)^{-\nu} = 1-\frac{\Delta L'}{L}$

For very small values of and, the first-order approximation yields:

$\nu \approx \frac{\Delta L'}{\Delta L}$

Read more about this topic: Poisson's Ratio

Famous quotes containing the words length and/or change:

“Twenty-four-hour room service generally refers to the length of time that it takes for the club sandwich to arrive. This is indeed disheartening, particularly when you’ve ordered scrambled eggs.”
—Fran Lebowitz (b. 1950)

“Suddenly I’m dogmatic. Why does my status change every time you get a new woman, Jack?”
—Warren Beatty (b. 1937)