Poisson's Ratio - Orthotropic Materials

Orthotropic Materials

For orthotropic materials such as wood, Hooke's law can be expressed in matrix form as

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

where

is the Young's modulus along axis

is the shear modulus in direction on the plane whose normal is in direction

is the Poisson's ratio that corresponds to a contraction in direction when an extension is applied in direction .

The Poisson's ratio of an orthotropic material is different in each direction (x, y and z). However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent. There are only nine independent material properties; three elastic moduli, three shear moduli, and three Poisson's ratios. The remaining three Poisson's ratios can be obtained from the relations

$\frac{\nu_{\rm yx}}{E_{\rm y}} = \frac{\nu_{\rm xy}}{E_{\rm x}}~, \qquad \frac{\nu_{\rm zx}}{E_{\rm z}} = \frac{\nu_{\rm xz}}{E_{\rm x}}~, \qquad \frac{\nu_{\rm yz}}{E_{\rm y}} = \frac{\nu_{\rm zy}}{E_{\rm z}}$

From the above relations we can see that if then . The larger Poisson's ratio (in this case ) is called the major Poisson's ratio while the smaller one (in this case ) is called the minor Poisson's ratio. We can find similar relations between the other Poisson's ratios.

Read more about this topic: Poisson's Ratio

Famous quotes containing the word materials:

“Our job is now clear. All Americans must be prepared to make, on a 24 hour schedule, every war weapon possible and the war factory line will use men and materials which will bring, the war effort to every man, woman, and child in America. All one hundred thirty million of us will be needed to answer the sunrise stealth of the Sabbath Day Assassins.”
—Lyndon Baines Johnson (1908–1973)