The Constitutive Equations of Anisotropic Elasticity

Basic mechanical equations

Stress tensor

$$ \boldsymbol{\sigma}=\left[\begin{array}{ccc} \sigma_{x} & \tau_{x y} & \tau_{x z} \\ \tau_{y x} & \sigma_{y} & \tau_{y z} \\ \tau_{z x} & \tau_{z y} & \sigma_{z} \end{array}\right] $$

Herein, $\tau_{x y}=\tau_{y x}, \tau_{x z}=\tau_{z x}, \tau_{y z}=\tau_{z y}$, so there are 6 stress componets in total, $\sigma_{x}, \sigma_{y}, \sigma_{z}, \tau_{x y}, \tau_{y z}, \tau_{z x}$.

Strain tensor

$$ \boldsymbol{\varepsilon}=\left[\begin{array}{ccc} \varepsilon_{x} & \varepsilon_{x y} & \varepsilon_{z x} \\ \varepsilon_{x y} & \varepsilon_{y} & \varepsilon_{y z} \\ \varepsilon_{z x} & \varepsilon_{y z} & \varepsilon_{z} \end{array}\right] $$

Herein, $\varepsilon_{x y}=\frac{1}{2} \gamma_{x y}, \varepsilon_{y z}=\frac{1}{2} \gamma_{y z}, \varepsilon_{z x}=\frac{1}{2} \gamma_{z x}$ are shear strains。 $\gamma_{x y}, \gamma_{y z}, \gamma_{z x}$ are engineering shear strains. $\varepsilon_{x}, \varepsilon_{y}, \varepsilon_{z}$ are linear strains. So there are 6 strain componets in total.

The relationship between stress and strain

$$ \begin{rcases} \begin{array}{l} \sigma_{x}=C_{11} \varepsilon_{x}+C_{12} \varepsilon_{y}+C_{13} \varepsilon_{z}+C_{14} \gamma_{y z}+C_{15} \gamma_{z x}+C_{16} \gamma_{x y} \\ \sigma_{y}=C_{21} \varepsilon_{x}+C_{22} \varepsilon_{y}+C_{23} \varepsilon_{z}+C_{24} \gamma_{y z}+C_{25} \gamma_{z x}+C_{26} \gamma_{x y} \\ \sigma_{z}=C_{31} \varepsilon_{x}+C_{32} \varepsilon_{y}+C_{33} \varepsilon_{z}+C_{34} \gamma_{y z}+C_{35} \gamma_{z x}+C_{36} \gamma_{x y} \\ \tau_{y z}=C_{41} \varepsilon_{x}+C_{42} \varepsilon_{y}+C_{43} \varepsilon_{z}+C_{44} \gamma_{y z}+C_{45} \gamma_{z x}+C_{46} \gamma_{x y} \\ \tau_{z x}=C_{51} \varepsilon_{x}+C_{52} \varepsilon_{y}+C_{53} \varepsilon_{z}+C_{54} \gamma_{y z}+C_{55} \gamma_{z x}+C_{56} \gamma_{x y} \\ \tau_{x y}=C_{61} \varepsilon_{x}+C_{62} \varepsilon_{y}+C_{63} \varepsilon_{z}+C_{64} \gamma_{y z}+C_{65} \gamma_{z x}+C_{66} \gamma_{x y} \end{array} \end{rcases} $$

Herein,

The relationship between stress and strain

Here we use 1, 2, 3 axis instead of x, y, z axis to simplify the symbols of stress and strain component. The corresponding substitution relationship is as follows:

$$ \begin{array}{ll} Stress & Strain\\ \sigma_{x} \rightarrow \sigma_{1} & \varepsilon_{x} \rightarrow \varepsilon_{1} \\ \sigma_{y} \rightarrow \sigma_{2} & \varepsilon_{y} \rightarrow \varepsilon_{2} \\ \sigma_{z} \rightarrow \sigma_{3} & \varepsilon_{z} \rightarrow \varepsilon_{3} \\ \tau_{y z} \rightarrow \sigma_{4} & \gamma_{y z}=2 \varepsilon_{y z} \rightarrow \varepsilon_{4} \\ \tau_{z x} \rightarrow \sigma_{5} & \gamma_{z x}=2 \varepsilon_{z x} \rightarrow \varepsilon_{5} \\ \tau_{x y} \rightarrow \sigma_{6} & \gamma_{x y}=2 \varepsilon_{x y} \rightarrow \varepsilon_{6} \end{array} $$

The relationship between stress and strain can be written as follows: $$ \begin{rcases} \begin{array}{l} \sigma_{1}=C_{11} \varepsilon_{1}+C_{12} \varepsilon_{2}+C_{13} \varepsilon_{3}+C_{14} \varepsilon_{4}+C_{15} \varepsilon_{5}+C_{16} \varepsilon_{6} \\ \sigma_{2}=C_{21} \varepsilon_{1}+C_{22} \varepsilon_{2}+C_{23} \varepsilon_{3}+C_{24} \varepsilon_{4}+C_{25} \varepsilon_{5}+C_{26} \varepsilon_{6} \\ \sigma_{3}=C_{31} \varepsilon_{1}+C_{32} \varepsilon_{2}+C_{33} \varepsilon_{3}+C_{34} \varepsilon_{4}+C_{35} \varepsilon_{5}+C_{36} \varepsilon_{6} \\ \sigma_{4}=C_{41} \varepsilon_{1}+C_{42} \varepsilon_{2}+C_{43} \varepsilon_{3}+C_{44} \varepsilon_{4}+C_{45} \varepsilon_{5}+C_{46} \varepsilon_{6} \\ \sigma_{5}=C_{51} \varepsilon_{1}+C_{52} \varepsilon_{2}+C_{53} \varepsilon_{3}+C_{54} \varepsilon_{4}+C_{55} \varepsilon_{5}+C_{56} \varepsilon_{6} \\ \sigma_{6}=C_{61} \varepsilon_{1}+C_{62} \varepsilon_{2}+C_{63} \varepsilon_{3}+C_{64} \varepsilon_{4}+C_{65} \varepsilon_{5}+C_{66} \varepsilon_{6} \end{array} \end{rcases} $$

Completely anisotropic (21 elastic constants)

In a homogeneous elastomer, if each point has different elastic properties in different directions, this kind of elastomer is called general anisotropic body.

$$ \begin{Bmatrix}\begin{array}{l} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{array}\end{Bmatrix}=\left[\begin{array}{llllll} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{21} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\\ C_{31} & C_{32} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{41} & C_{42} & C_{43} & C_{44} & C_{45} & C_{46} \\ C_{51} & C_{52} & C_{53} & C_{54} & C_{55} & C_{56} \\ C_{61} & C_{62} & C_{63} & C_{64} & C_{65} & C_{66} \end{array}\right] \begin{Bmatrix}\begin{array}{l} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{array}\end{Bmatrix} $$

With one elastic symmetry plane (13 elastic constants)

$$ \begin{Bmatrix}\begin{array}{l} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{array}\end{Bmatrix}=\left[\begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & C_{16} \\ C_{21} & C_{22} & C_{23} & 0 & 0 & C_{26} \\ C_{31} & C_{32} & C_{33} & 0 & 0 & C_{36} \\ 0 & 0 & 0 & C_{44} & C_{45} & 0 \\ 0 & 0 & 0 & C_{54} & C_{55} & 0 \\ C_{61} & C_{62} & C_{63} & 0 & 0 & C_{66} \end{array}\right]\begin{Bmatrix}\begin{array}{l} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{array}\end{Bmatrix} $$

With two elastic symmetry plane (Orthotropic, 9)

$$ \begin{Bmatrix}\begin{array}{l} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{array}\end{Bmatrix}=\left[\begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{21} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{31} & C_{32} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{array}\right]\begin{Bmatrix}\begin{array}{c} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{array}\end{Bmatrix} $$

With one elastic symmetry axis (Transversely isotropic, 5 elastic constants)

$$ \begin{Bmatrix}\begin{array}{l} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{array}\end{Bmatrix}=\left[\begin{array}{cccccc} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{21} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{31} & C_{31} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & \left(C_{11}-C_{12}\right) / 2 \end{array}\right]\begin{Bmatrix}\begin{array}{l} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{array}\end{Bmatrix} $$

Isotropic (2 elastic constants)

$$ \begin{Bmatrix}\begin{array}{c} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3} \\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{array}\end{Bmatrix}=\left[\begin{array}{cccccc} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{21} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{21} & C_{21} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & \left(C_{11}-C_{12}\right) / 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & \left(C_{11}-C_{12}\right) / 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & \left(C_{11}-C_{12}\right) / 2 \end{array}\right]\begin{Bmatrix}\begin{array}{l} \varepsilon_{1} \\ \varepsilon_{2} \\ \varepsilon_{3} \\ \varepsilon_{4} \\ \varepsilon_{5} \\ \varepsilon_{6} \end{array}\end{Bmatrix} $$