# Fabric Materials

A fabric material is a variant on an orthotropic material but only exists for the 2D case.

$C = \begin{bmatrix} c_{xxxx} & c_{xxyy} & 0 \\ & c_{yyyy} & 0 \\ symm & & c_{xyxy} \\ \end{bmatrix}$

As wth an orthotropic material it is easier to define the inverse of the constitutive relationship

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

where $C^{-1}$ is the compliance matrix. Fabric orthotropic behaviour is governed by four independent elastic constants: two elastic moduli, a Poisson's ratios and a shear modulus. These are

$\tilde E_x$ the elastic modulus in the warp direction per unit width
$\tilde E_y$ the elastic modulus in the fill (weft) direction per unit width
$\nu_{xy}$ the Poissons ratio in the warp-fill direction
$\tilde G_{xy}$ the shear modulus in the warp-fill direction per unit width

The further (dependent) Poisson's ratio is defined through the relationship

$\nu_{yx} \tilde E_x = \nu_{xy} \tilde E_y$

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

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

A stable material must satisfy the following conditions

$\tilde E_x, \tilde E_y, \tilde G_{xy} > 0$
$(1-\nu_{xy}\nu_{yx}) > 0$

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

$\begin{matrix} {\nu_{yx}}^2<\dfrac{\tilde E_y}{\tilde E_x} & {\nu_{xy}}^2<\dfrac{\tilde E_x}{\tilde E_y}\\ \end{matrix}$