Orthotropic Materials

For an orthotropic material the constitutive matrix simplifies to

$$C = \begin{bmatrix} c_{xxxx} & c_{xxyy} & c_{xxzz} & 0 & 0 & 0\\ & c_{yyyy} & c_{yyzz} & 0 & 0 & 0\\ & & c_{zzzz} & 0 & 0 & 0\\ & & & c_{xyxy} & 0 & 0\\ & symm & & & c_{yzyz} & 0\\ & & & & & c_{zxzx}\\ \end{bmatrix}$$

however it is easier to define the inverse of the constitutive relationship

$$\epsilon = C^{-1} \sigma$$

where $C^{-1}$ is the compliance matrix. Elastic orthotropic behaviour is governed by nine independent elastic constants: three elastic moduli, three Poisson's ratios and three shear moduli. Three further (dependent) Poisson's ratios are defined through the relationship

$$\nu_{ji} E_i = \nu_{ij} E_j$$

These material properties are defined relative to a material principal directions $m$ giving a compliance matrix $C_m^{-1}$.

$$C_m^{-1} = \begin{bmatrix} \dfrac{1}{E_x} & -\dfrac{\nu_{yx}}{E_y} & -\dfrac{\nu_{zx}}{E_z} & 0 & 0 & 0\\[2pt] -\dfrac{\nu_{xy}}{E_x} & \dfrac{1}{E_y} & -\dfrac{\nu_{zy}}{E_z} & 0 & 0 & 0\\[2pt] -\dfrac{\nu_{xz}}{E_x} & -\dfrac{\nu_{yz}}{E_y} & \dfrac{1}{E_z} & 0 & 0 & 0\\[2pt] 0 & 0 & 0 & \dfrac{1}{G_{xy}} & 0 & 0 \\[2pt] 0 & 0 & 0 & 0 & \dfrac{1}{G_{yz}} & 0 \\[2pt] 0 & 0 & 0 & 0 & 0 &\dfrac{1}{G_{zx}} \\[2pt] \end{bmatrix}$$

A stable material must satisfy the following conditions

$$E_x, E_y, E_z, G_{xy}, G_{yz}, G_{zx} > 0$$

$$(1-\nu_{xy}\nu_{yx}), (1-\nu_{yz}\nu_{zy}), (1-\nu_{zx}\nu_{xz}), > 0$$

$$1-\nu_{xy}\nu_{yx} - \nu_{yz}\nu_{zy} - \nu_{zx}\nu_{xz} - 2 \nu_{xz}\nu_{zy}\nu_{yx} >0$$

Using these relationships leads to the following conditions which apply to the Poisson's ratios

$$\begin{matrix} {\nu_{yx}}^2<\dfrac{E_y}{E_x} & {\nu_{xy}}^2<\dfrac{E_x}{E_y}\\ {\nu_{zy}}^2<\dfrac{E_z}{E_y} & {\nu_{yz}}^2<\dfrac{E_y}{E_z}\\ {\nu_{xz}}^2<\dfrac{E_x}{E_z} & {\nu_{zx}}^2<\dfrac{E_z}{E_x}\\ \end{matrix}$$

Note: if $E_x = E_y = E_z$, $\nu_{xy} = \nu_{yz} = \nu_{zx}$ and $G_{xy} = G_{yz} = G_{zx}$ then the orthotropic material reduces to an isotropic material.