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

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