Editing Composite material (section)

=== Stiffness and Compliance Elasticity ===
Composite materials are generally [[anisotropic]], and in many cases are [[Orthotropic material|orthotropic]]. [[Voigt notation]] can be used to reduce the rank of the stress and strain tensors such that the [[stiffness]] <math>C</math> (often also referred to by <math>Q</math>) and compliance <math>S</math> can be written as a [[Matrix (mathematics)|matrix]]:<ref>{{cite book |last1=Lekhnit͡skiĭ |first1=Sergeĭ Georgievich |title=Theory of Elasticity of an Anisotropic Elastic Body |date=1963 |publisher=Holden-Day |oclc=652279972 }}{{pn|date=January 2025}}</ref>

<math>\begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} =
  \begin{bmatrix}
  C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix}
  \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix}</math> and <math>\begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} =
  \begin{bmatrix}
  S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\
S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\
S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36} \\
S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46} \\
S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56} \\
S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix}
  \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix}</math>

When considering each ply individually, it is assumed that they can be treated as thi lamina and so out–of–plane stresses and strains are negligible. That is <math>\sigma_3 = \sigma_4 = \sigma_5 = 0</math> and <math>\varepsilon_4 = \varepsilon_5 = 0</math>.<ref name=":0">{{cite book |doi=10.1007/978-94-011-4489-6 |title=Mechanics of Composite Materials and Structures |date=1999 |isbn=978-0-7923-5871-8 |editor-last1=Soares |editor-last2=Soares |editor-last3=Freitas |editor-first1=Carlos A. Mota |editor-first2=Cristóvão M. Mota |editor-first3=Manuel J. M. }}</ref> This allows the stiffness and compliance matrices to be reduced to 3x3 matrices as follows:

<math>C = 
  \begin{bmatrix}
    \tfrac{E_{\rm 1}}{1-{\nu_{\rm 12}}{\nu_{\rm 21}}} & \tfrac{E_{\rm 2}{\nu_{\rm 12}}}{1-{\nu_{\rm 12}}{\nu_{\rm 21}}} & 0 \\
    \tfrac{E_{\rm 2}{\nu_{\rm 12}}}{1-{\nu_{\rm 12}}{\nu_{\rm 21}}} & \tfrac{E_{\rm 2}}{1-{\nu_{\rm 12}}{\nu_{\rm 21}}} & 0 \\
    0 & 0 & G_{\rm 12} \\
    \end{bmatrix} \quad </math> and <math> \quad S = 
  \begin{bmatrix}
    \tfrac{1}{E_{\rm 1}} & - \tfrac{\nu_{\rm 21}}{E_{\rm 2}} & 0 \\
    -\tfrac{\nu_{\rm 12}}{E_{\rm 1}} & \tfrac{1}{E_{\rm 2}} & 0 \\
    0 & 0 & \tfrac{1}{G_{\rm 12}} \\
    \end{bmatrix} </math>
[[File:Transform coordinate system.png|thumb|331x331px|Two different coordinate systems of material. The structure has a (1-2) coordinate system. The material has a (x-y) principal coordinate system.]]
For fiber-reinforced composite, the fiber orientation in material affect anisotropic properties of the structure. From characterizing technique i.e. tensile testing, the material properties were measured based on sample (1-2) coordinate system. The tensors above express stress-strain relationship in (1-2) coordinate system. While the known material properties is in the principal coordinate system (x-y) of material. Transforming the tensor between two coordinate system help identify the material properties of the tested sample. The [[transformation matrix]] with <math>\theta      </math> degree rotation is <ref name=":0" />

<math>T(\theta)_\epsilon = 
\begin{bmatrix} \cos^2 \theta & \sin^2 \theta & \cos \theta\sin \theta
\\ sin^2 \theta & \cos^2 \theta & -\cos \theta\sin \theta
\\ -2\cos \theta\sin \theta & 2\cos \theta\sin \theta & \cos^2 \theta - \sin^2 \theta \end{bmatrix}      </math> for <math>\begin{bmatrix} \acute{\epsilon} \end{bmatrix} = T(\theta)_\epsilon \begin{bmatrix} \epsilon \end{bmatrix}

      </math><math>T(\theta)_\sigma = 
\begin{bmatrix} \cos^2 \theta & \sin^2 \theta & 2\cos \theta\sin \theta
\\ sin^2 \theta & \cos^2 \theta & -2\cos \theta\sin \theta
\\ -\cos \theta\sin \theta & \cos \theta\sin \theta & \cos^2 \theta - \sin^2 \theta \end{bmatrix}      </math> for <math>\begin{bmatrix} \acute{\sigma} \end{bmatrix} = T(\theta)_\sigma \begin{bmatrix} \sigma \end{bmatrix}      </math>