Basic mechanical equations

Stress tensor

$$\sigma =\left[\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{yx}& {\sigma }_y& {\tau }_{yz}\\ {\tau }_{zx}& {\tau }_{zy}& {\sigma }_z\end{array}\right]$$

Herein, ${\tau }_{xy}={\tau }_{yx},{\tau }_{xz}={\tau }_{zx},{\tau }_{yz}={\tau }_{zy}$, so there are 6 stress componets in total, ${\sigma }_x,{\sigma }_y,{\sigma }_z,{\tau }_{xy},{\tau }_{yz},{\tau }_{zx}$.

Strain tensor

$$\epsilon =\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& {\epsilon }_{zx}\\ {\epsilon }_{xy}& {\epsilon }_y& {\epsilon }_{yz}\\ {\epsilon }_{zx}& {\epsilon }_{yz}& {\epsilon }_z\end{array}\right]$$

Herein, ${\epsilon }_{xy}=\frac{1}{2}{\gamma }_{xy},{\epsilon }_{yz}=\frac{1}{2}{\gamma }_{yz},{\epsilon }_{zx}=\frac{1}{2}{\gamma }_{zx}$ are shear strains。 ${\gamma }_{xy},{\gamma }_{yz},{\gamma }_{zx}$ are engineering shear strains. ${\epsilon }_x,{\epsilon }_y,{\epsilon }_z$ are linear strains. So there are 6 strain componets in total.

The relationship between stress and strain

$$\begin{array}{l}{\sigma }_x=C_{11}{\epsilon }_x+C_{12}{\epsilon }_y+C_{13}{\epsilon }_z+C_{14}{\gamma }_{yz}+C_{15}{\gamma }_{zx}+C_{16}{\gamma }_{xy}\\ {\sigma }_y=C_{21}{\epsilon }_x+C_{22}{\epsilon }_y+C_{23}{\epsilon }_z+C_{24}{\gamma }_{yz}+C_{25}{\gamma }_{zx}+C_{26}{\gamma }_{xy}\\ {\sigma }_z=C_{31}{\epsilon }_x+C_{32}{\epsilon }_y+C_{33}{\epsilon }_z+C_{34}{\gamma }_{yz}+C_{35}{\gamma }_{zx}+C_{36}{\gamma }_{xy}\\ {\tau }_{yz}=C_{41}{\epsilon }_x+C_{42}{\epsilon }_y+C_{43}{\epsilon }_z+C_{44}{\gamma }_{yz}+C_{45}{\gamma }_{zx}+C_{46}{\gamma }_{xy}\\ {\tau }_{zx}=C_{51}{\epsilon }_x+C_{52}{\epsilon }_y+C_{53}{\epsilon }_z+C_{54}{\gamma }_{yz}+C_{55}{\gamma }_{zx}+C_{56}{\gamma }_{xy}\\ {\tau }_{xy}=C_{61}{\epsilon }_x+C_{62}{\epsilon }_y+C_{63}{\epsilon }_z+C_{64}{\gamma }_{yz}+C_{65}{\gamma }_{zx}+C_{66}{\gamma }_{xy}\end{array}$$

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:

$$\begin{array}{ll}Stress& Strain\\ {\sigma }_x\to {\sigma }_1& {\epsilon }_x\to {\epsilon }_1\\ {\sigma }_y\to {\sigma }_2& {\epsilon }_y\to {\epsilon }_2\\ {\sigma }_z\to {\sigma }_3& {\epsilon }_z\to {\epsilon }_3\\ {\tau }_{yz}\to {\sigma }_4& {\gamma }_{yz}=2{\epsilon }_{yz}\to {\epsilon }_4\\ {\tau }_{zx}\to {\sigma }_5& {\gamma }_{zx}=2{\epsilon }_{zx}\to {\epsilon }_5\\ {\tau }_{xy}\to {\sigma }_6& {\gamma }_{xy}=2{\epsilon }_{xy}\to {\epsilon }_6\end{array}$$

The relationship between stress and strain can be written as follows:

$$\begin{array}{l}{\sigma }_1=C_{11}{\epsilon }_1+C_{12}{\epsilon }_2+C_{13}{\epsilon }_3+C_{14}{\epsilon }_4+C_{15}{\epsilon }_5+C_{16}{\epsilon }_6\\ {\sigma }_2=C_{21}{\epsilon }_1+C_{22}{\epsilon }_2+C_{23}{\epsilon }_3+C_{24}{\epsilon }_4+C_{25}{\epsilon }_5+C_{26}{\epsilon }_6\\ {\sigma }_3=C_{31}{\epsilon }_1+C_{32}{\epsilon }_2+C_{33}{\epsilon }_3+C_{34}{\epsilon }_4+C_{35}{\epsilon }_5+C_{36}{\epsilon }_6\\ {\sigma }_4=C_{41}{\epsilon }_1+C_{42}{\epsilon }_2+C_{43}{\epsilon }_3+C_{44}{\epsilon }_4+C_{45}{\epsilon }_5+C_{46}{\epsilon }_6\\ {\sigma }_5=C_{51}{\epsilon }_1+C_{52}{\epsilon }_2+C_{53}{\epsilon }_3+C_{54}{\epsilon }_4+C_{55}{\epsilon }_5+C_{56}{\epsilon }_6\\ {\sigma }_6=C_{61}{\epsilon }_1+C_{62}{\epsilon }_2+C_{63}{\epsilon }_3+C_{64}{\epsilon }_4+C_{65}{\epsilon }_5+C_{66}{\epsilon }_6\end{array}$$

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.

$$\left\{\begin{array}{c}\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\end{array}\right\}=\left[\begin{array}{llllll}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ C_{21}& C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ C_{31}& C_{32}& C_{33}& C_{34}& C_{35}& C_{36}\\ C_{41}& C_{42}& C_{43}& C_{44}& C_{45}& C_{46}\\ C_{51}& C_{52}& C_{53}& C_{54}& C_{55}& C_{56}\\ C_{61}& C_{62}& C_{63}& C_{64}& C_{65}& C_{66}\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\end{array}\right\}$$

With one elastic symmetry plane: 13 elastic constants

$$\left\{\begin{array}{c}\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& C_{16}\\ C_{21}& C_{22}& C_{23}& 0& 0& C_{26}\\ C_{31}& C_{32}& C_{33}& 0& 0& C_{36}\\ 0& 0& 0& C_{44}& C_{45}& 0\\ 0& 0& 0& C_{54}& C_{55}& 0\\ C_{61}& C_{62}& C_{63}& 0& 0& C_{66}\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\end{array}\right\}$$

With two elastic symmetry plane: Orthotropic, 9

$$\left\{\begin{array}{c}\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{21}& C_{22}& C_{23}& 0& 0& 0\\ C_{31}& C_{32}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\end{array}\right\}$$

With one elastic symmetry axis: Transversely isotropic, 5 elastic constants

$$\left\{\begin{array}{c}\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{21}& C_{11}& C_{13}& 0& 0& 0\\ C_{31}& C_{31}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& (C_{11}-C_{12})/2\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\end{array}\right\}$$

Isotropic: 2 elastic constants

$$\left\{\begin{array}{c}\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\sigma }_4\\ {\sigma }_5\\ {\sigma }_6\end{array}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{12}& 0& 0& 0\\ C_{21}& C_{11}& C_{12}& 0& 0& 0\\ C_{21}& C_{21}& C_{11}& 0& 0& 0\\ 0& 0& 0& (C_{11}-C_{12})/2& 0& 0\\ 0& 0& 0& 0& (C_{11}-C_{12})/2& 0\\ 0& 0& 0& 0& 0& (C_{11}-C_{12})/2\end{array}\right]\left\{\begin{array}{c}\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\epsilon }_4\\ {\epsilon }_5\\ {\epsilon }_6\end{array}\end{array}\right\}$$