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:

“Punishment followed on a grand scale. For ten days, an unconscionable length of time, my father blessed the palms of his child’s outstretched, four-year-old hands with a sharp switch. Seven strokes a day on each hand; that makes one hundred forty strokes and then some. This put an end to the child’s innocence.”
—Christoph Meckel (20th century)

“When we have to change our mind about someone, we hold the inconvenience he has caused us very much against him.”
—Friedrich Nietzsche (1844–1900)