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Fabric Materials

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

C=[cxxxxcxxyy0cyyyy0symmcxyxy]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

ϵ=C1σ\epsilon = C^{-1} \sigma

where C1C^{-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

E~x\tilde E_x the elastic modulus in the warp direction per unit width
E~y\tilde E_y the elastic modulus in the fill (weft) direction per unit width
νxy\nu_{xy} the Poissons ratio in the warp-fill direction
G~xy\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

νyxE~x=νxyE~y\nu_{yx} \tilde E_x = \nu_{xy} \tilde E_y

These material properties are defined relative to a material principal directions mm giving a compliance matrix Cm1C_m^{-1}.

Cm1=[1E~xνyxE~y0νxyE~x1E~y0001G~xy]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

E~x,E~y,G~xy>0\tilde E_x, \tilde E_y, \tilde G_{xy} > 0
(1νxyνyx)>0(1-\nu_{xy}\nu_{yx}) > 0

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

νyx2<E~yE~xνxy2<E~xE~y\begin{matrix} {\nu_{yx}}^2<\dfrac{\tilde E_y}{\tilde E_x} & {\nu_{xy}}^2<\dfrac{\tilde E_x}{\tilde E_y}\\ \end{matrix}