For an orthotropic material the constitutive matrix simplifies to
C = [ c x x x x c x x y y c x x z z 0 0 0 c y y y y c y y z z 0 0 0 c z z z z 0 0 0 c x y x y 0 0 s y m m c y z y z 0 c z x z x ]
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} C = c xxxx c xx yy c yyyy sy mm c xx zz c yyzz c zzzz 0 0 0 c x y x y 0 0 0 0 c yzyz 0 0 0 0 0 c z x z x however it is easier to define the inverse of the constitutive relationship
ϵ = C − 1 σ \epsilon = C^{-1} \sigma ϵ = C − 1 σ where C − 1 C^{-1} C − 1
ν j i E i = ν i j E j \nu_{ji} E_i = \nu_{ij} E_j ν ji E i = ν ij E j These material properties are defined relative to a material principal directions m m m C m − 1 C_m^{-1} C m − 1
C m − 1 = [ 1 E x − ν y x E y − ν z x E z 0 0 0 − ν x y E x 1 E y − ν z y E z 0 0 0 − ν x z E x − ν y z E y 1 E z 0 0 0 0 0 0 1 G x y 0 0 0 0 0 0 1 G y z 0 0 0 0 0 0 1 G z x ] 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} C m − 1 = E x 1 − E x ν x y − E x ν x z 0 0 0 − E y ν y x E y 1 − E y ν yz 0 0 0 − E z ν z x − E z ν zy E z 1 0 0 0 0 0 0 G x y 1 0 0 0 0 0 0 G yz 1 0 0 0 0 0 0 G z x 1 A stable material must satisfy the following conditions
E x , E y , E z , G x y , G y z , G z x > 0 E_x, E_y, E_z, G_{xy}, G_{yz}, G_{zx} > 0 E x , E y , E z , G x y , G yz , G z x > 0 ( 1 − ν x y ν y x ) , ( 1 − ν y z ν z y ) , ( 1 − ν z x ν x z ) , > 0 (1-\nu_{xy}\nu_{yx}), (1-\nu_{yz}\nu_{zy}), (1-\nu_{zx}\nu_{xz}), > 0 ( 1 − ν x y ν y x ) , ( 1 − ν yz ν zy ) , ( 1 − ν z x ν x z ) , > 0 1 − ν x y ν y x − ν y z ν z y − ν z x ν x z − 2 ν x z ν z y ν y x > 0 1-\nu_{xy}\nu_{yx} - \nu_{yz}\nu_{zy} - \nu_{zx}\nu_{xz} - 2 \nu_{xz}\nu_{zy}\nu_{yx} >0 1 − ν x y ν y x − ν yz ν zy − ν z x ν x z − 2 ν x z ν zy ν y x > 0 Using these relationships leads to the following conditions which apply to the Poisson's ratios
ν y x 2 < E y E x ν x y 2 < E x E y ν z y 2 < E z E y ν y z 2 < E y E z ν x z 2 < E x E z ν z x 2 < E z E x \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} ν y x 2 < E x E y ν zy 2 < E y E z ν x z 2 < E z E x ν x y 2 < E y E x ν yz 2 < E z E y ν z x 2 < E x E z Note: if E x = E y = E z E_x = E_y = E_z E x = E y = E z ν x y = ν y z = ν z x \nu_{xy} = \nu_{yz} = \nu_{zx} ν x y = ν yz = ν z x G x y = G y z = G z x G_{xy} = G_{yz} = G_{zx} G x y = G yz = G z x