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:

    “Men sometimes speak as if the study of the classics would at length make way for more modern and practical studies; but the adventurous student will always study classics, in whatever language they may be written and however ancient they may be. For what are the classics but the noblest recorded thoughts of man?... We might as well omit to study Nature because she is old.”
    —Henry David Thoreau (1817–1862)

    “A state without the means of some change is without the means of its conservation.”
    —Edmund Burke (1729–1797)