Poisson's Ratio - Transversely Isotropic Materials

Transversely Isotropic Materials

Transversely isotropic materials have a plane of symmetry in which the elastic properties are isotropic. If we assume that this plane of symmetry is, then Hooke's law takes the form

$\begin{bmatrix} \epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy} \end{bmatrix} = \begin{bmatrix} \tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & 0 & 0 & 0 \\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm y}} & 0 & 0 & 0 \\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm y}} & 0 & 0 & 0 \\ 0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} & 0 \\ 0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\ \end{bmatrix} \begin{bmatrix} \sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy} \end{bmatrix}$

where we have used the plane of symmetry to reduce the number of constants, i.e., .

The symmetry of the stress and strain tensors implies that

$\cfrac{\nu_{\rm xy}}{E_{\rm x}} = \cfrac{\nu_{\rm yx}}{E_{\rm y}} ~,~~ \nu_{\rm yz} = \nu_{\rm zy} ~.$

This leaves us with six independent constants . However, transverse isotropy gives rise to a further constraint between and which is

$G_{\rm yz} = \cfrac{E_{\rm y}}{2(1+\nu_{\rm yz})} ~.$

Therefore, there are five independent elastic material properties two of which are Poisson's ratios. For the assumed plane of symmetry, the larger of and is the major Poisson's ratio. The other major and minor Poisson's ratios are equal.

Read more about this topic: Poisson's Ratio

Famous quotes containing the word materials:

“In how few words, for instance, the Greeks would have told the story of Abelard and Heloise, making but a sentence of our classical dictionary.... We moderns, on the other hand, collect only the raw materials of biography and history, “memoirs to serve for a history,” which is but materials to serve for a mythology.”
—Henry David Thoreau (1817–1862)