In mechanics stress, strain, stiffness and compliance are written in tensors notation essentially, because of their nature. The stress in 3D can be written in 2x2 tensor as follows

$$ \sigma_{ij} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} $$

and the strain can be written as $$ \sigma_{ij} =
\begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \epsilon_{13}\\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23}\\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33} \end{bmatrix} $$

The stress and strain are related by stiffness, in tensor notation as

$$ \sigma = C:\epsilon \label{stiffness_strain_tensor} $$

The stiffness tensor is a 4th rank tensor which may not be very easily visualized on a paper. But here I will try to elaborate a little bit, using its resemblance with 4-D array, or a 2-D matrix of 2-D matrix. It gets even simpler to understand when these 2-D matrices are square with dimension 3x3. We would write

$$C_{ijkl} = \begin{bmatrix} \begin{bmatrix} C_{1111} & C_{1112} & C_{1113}\\ C_{1121} & C_{1122} & C_{1123}\\ C_{1131} & C_{1132} & C_{1133} \end{bmatrix} & \begin{bmatrix} C_{1211} & C_{1212} & C_{1213}\\ C_{1221} & C_{1222} & C_{1223}\\ C_{1231} & C_{1232} & C_{1233} \end{bmatrix} & \begin{bmatrix} C_{1311} & C_{1312} & C_{1313}\\ C_{1321} & C_{1322} & C_{1323}\\ C_{1331} & C_{1332} & C_{1333} \end{bmatrix} \\ \begin{bmatrix} C_{2111} & C_{2112} & C_{2113}\\ C_{2121} & C_{2122} & C_{2123}\\ C_{2131} & C_{2132} & C_{2133} \end{bmatrix} & \begin{bmatrix} C_{2211} & C_{2212} & C_{2213}\\ C_{2221} & C_{2222} & C_{2223}\\ C_{2231} & C_{2232} & C_{2233} \end{bmatrix} & \begin{bmatrix} C_{2311} & C_{2312} & C_{2313}\\ C_{2321} & C_{2322} & C_{2323}\\ C_{2331} & C_{2332} & C_{2333} \end{bmatrix} \\ \begin{bmatrix} C_{3111} & C_{3112} & C_{3113}\\ C_{3121} & C_{3122} & C_{3123}\\ C_{3131} & C_{3132} & C_{3133} \end{bmatrix} & \begin{bmatrix} C_{3211} & C_{3212} & C_{3213}\\ C_{3221} & C_{3222} & C_{3223}\\ C_{3231} & C_{3232} & C_{3233} \end{bmatrix} & \begin{bmatrix} C_{3311} & C_{3312} & C_{3313}\\ C_{3321} & C_{3322} & C_{3323}\\ C_{3331} & C_{3332} & C_{3333} \end{bmatrix} \end{bmatrix} \label{4tensor}$$

The same can be written in components form, as $$\sigma_{ij} = C_{ijkl}\epsilon_{kl}$$ This components form (called as indices notation) means\

$$ \begin{equation} \sigma_{11} = C_{1111}\epsilon_{11} + C_{1112} \epsilon_{12} + C_{1113}\epsilon_{13} + C_{1121} \epsilon_{21} + C_{1122} \epsilon_{22} + C_{1123}\epsilon_{11} + C_{1131} \epsilon_{31} + C_{1132} \epsilon_{32} + C_{1133}\epsilon_{33} \label{comp_long_eq} \end{equation} $$

or it can be thought of a component wise multiplication of 3x3 stiffness matrix (2nd rank tensor) and 3x3 strain matrix (2nd rank tensor), as follows
$$\sigma_{11} = \begin{bmatrix} C_{1111} & C_{1112} & C_{1113}\\ C_{1121} & C_{1122} & C_{1123}\\ C_{1131} & C_{1132} & C_{1133} \end{bmatrix}: \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \epsilon_{13}\\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23}\\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33} \end{bmatrix}$$
and the other components of stress matrix (or 2nd order tensor) can be written as
$$\sigma_{ij} = \begin{bmatrix} C_{ij11} & C_{ij12} & C_{ij13}\\ C_{ij21} & C_{ij22} & C_{ij23}\\ C_{ij31} & C_{ij32} & C_{ij33} \end{bmatrix}: \begin{bmatrix} \epsilon_{11} & \epsilon_{12} & \epsilon_{13}\\ \epsilon_{21} & \epsilon_{22} & \epsilon_{23}\\ \epsilon_{31} & \epsilon_{32} & \epsilon_{33} \end{bmatrix}$$ Remember, each of these colon (:) signed operation is summation over component wise multiplication, and not the traditional matrix rotation. Of course, this type of multiplication is not very new for all of us, so an analogy can be created, by writing this 3x3 2nd order tensor in the form of a 1D array (vector) or 1x9 matrix. Writing stiffness tensor for one component $\sigma_{11}$ in a row vector and the strain tensor as column vector, the above relations can be written in the form of simple vector notation as follows $$\sigma_{ij} = \begin{bmatrix} C_{ij11} & C_{ij12} & C_{ij13}& C_{ij21} & C_{ij22} & C_{ij23}& C_{ij31} & C_{ij32} & C_{ij33} \end{bmatrix} \begin{bmatrix} \epsilon_{11} \\ \epsilon_{12} \\ \epsilon_{13}\\ \epsilon_{21} \\ \epsilon_{22} \\ \epsilon_{23}\\ \epsilon_{31} \\ \epsilon_{32} \\ \epsilon_{33} \end{bmatrix}$$

It is quite basic of mechanics, that the shear stains are symmetric as $\epsilon_{21} = \epsilon_{12}$, $\epsilon_{31} = \epsilon_{13}$ and $\epsilon_{23} = \epsilon_{32}$, and knowing the same symmetry for stiffness 2nd order tensors, Eq. (\ref{comp_long_eq}) can be rewritten as $$ \begin{equation} \sigma_{11} = C_{1111}\epsilon_{11} + C_{1122} \epsilon_{22} + C_{1133}\epsilon_{33} + 2C_{1123}\epsilon_{11} + 2C_{1113}\epsilon_{13} + 2C_{1112} \epsilon_{12} \label{comp_long2_eq} \end{equation} $$

With this equation the vector form can be contracted as $$\sigma_{ij} = \begin{bmatrix} C_{ij11} & C_{ij22} & C_{ij33}& 2C_{ij23} & 2C_{ij13} & 2C_{ij12} \end{bmatrix} \begin{bmatrix} \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33}\\ \epsilon_{23} \\ \epsilon_{13} \\ \epsilon_{12} \end{bmatrix}$$

As, overall result for any component $\sigma_{ij}$ is just a summation, so the order of these components in Eq. (\ref{comp_long2_eq}) and Eq. (\ref{comp_long_eq}) is arbitrary. But it is the most followed notation of order which is used here. And eventually this these indices are also contracted as per the vector notation as follows
$$ \begin{array}{cc} 11 & \longrightarrow 1 \\ 22 & \longrightarrow 2 \\ 33 & \longrightarrow 3 \\ 23 & \longrightarrow 4 \\ 13 & \longrightarrow 5 \\ 12 & \longrightarrow 6 \end{array} $$

Now, one last step is left to make this tensor notation of stress, strain and stiffness, pretty simple $$\sigma_{i} = \begin{bmatrix} C_{i1} & C_{i2} & C_{i3}& 2C_{i4} & 2C_{i5} & 2C_{i6} \end{bmatrix} \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \epsilon_{3}\\ \epsilon_{4} \\ \epsilon_{5} \\ \epsilon_{6} \end{bmatrix}$$

Remember, we had the choice to write this equation as follows as well

$$\sigma_{i} = \begin{bmatrix} C_{i1} & C_{i2} & C_{i3}& C_{i4} & C_{i5} & C_{i6} \end{bmatrix} \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \epsilon_{3}\\ 2\epsilon_{4} \\ 2\epsilon_{5} \\ 2\epsilon_{6} \end{bmatrix}$$

Each one of these shear strain elements is sometimes written as $2\epsilon_i = \gamma_i$.

Note: In both of the above equations i = 1 to 6 and not 1 to 3 as in all previous equations and relations
Now we have everything sorted out so, we can define our stress strain relationship (for all $\sigma_i$) in vector and matrix form

$$\begin{bmatrix} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3}\\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13}& C_{14} & C_{15} & C_{16}\\ C_{21} & C_{22} & C_{23}& C_{24} & C_{25} & C_{26}\\ C_{31} & C_{32} & C_{33}& C_{34} & C_{35} & C_{36}\\ C_{41} & C_{42} & C_{43}& C_{44} & C_{45} & C_{46}\\ C_{51} & C_{52} & C_{53}& C_{54} & C_{55} & C_{56}\\ C_{61} & C_{62} & C_{63}& C_{64} & C_{65} & C_{66} \end{bmatrix} \begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \epsilon_{3}\\ 2\epsilon_{4} \\ 2\epsilon_{5} \\ 2\epsilon_{6} \end{bmatrix}$$ Similarly, the compliance is defined as $$\epsilon = S:\sigma \label{stiffness_stress_tensor}$$

And the 4th rank tensor is written as

$$S_{ijkl} = \begin{bmatrix} \begin{bmatrix} S_{1111} & S_{1112} & S_{1113}\\ S_{1121} & S_{1122} & S_{1123}\\ S_{1131} & S_{1132} & S_{1133} \end{bmatrix} & \begin{bmatrix} S_{1211} & S_{1212} & S_{1213}\\ S_{1221} & S_{1222} & S_{1223}\\ S_{1231} & S_{1232} & S_{1233} \end{bmatrix} & \begin{bmatrix} S_{1311} & S_{1312} & S_{1313}\\ S_{1321} & S_{1322} & S_{1323}\\ S_{1331} & S_{1332} & S_{1333} \end{bmatrix} \\ \begin{bmatrix} S_{2111} & S_{2112} & S_{2113}\\ S_{2121} & S_{2122} & S_{2123}\\ S_{2131} & S_{2132} & S_{2133} \end{bmatrix} & \begin{bmatrix} S_{2211} & S_{2212} & S_{2213}\\ S_{2221} & S_{2222} & S_{2223}\\ S_{2231} & S_{2232} & S_{2233} \end{bmatrix} & \begin{bmatrix} S_{2311} & S_{2312} & S_{2313}\\ S_{2321} & S_{2322} & S_{2323}\\ S_{2331} & S_{2332} & S_{2333} \end{bmatrix} \\ \begin{bmatrix} S_{3111} & S_{3112} & S_{3113}\\ S_{3121} & S_{3122} & S_{3123}\\ S_{3131} & S_{3132} & S_{3133} \end{bmatrix} & \begin{bmatrix} S_{3211} & S_{3212} & S_{3213}\\ S_{3221} & S_{3222} & S_{3223}\\ S_{3231} & S_{3232} & S_{3233} \end{bmatrix} & \begin{bmatrix} S_{3311} & S_{3312} & S_{3313}\\ S_{3321} & S_{3322} & S_{3323}\\ S_{3331} & S_{3332} & S_{3333} \end{bmatrix} \end{bmatrix} \label{4tensor_C}$$

$$\epsilon_{ij} = \begin{bmatrix} S_{ij11} & S_{ij12} & S_{ij13}\\ S_{ij21} & S_{ij22} & S_{ij23}\\ S_{ij31} & S_{ij32} & S_{ij33} \end{bmatrix}: \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13}\\ \sigma_{21} & \sigma_{22} & \sigma_{23}\\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}$$

$$\epsilon_{ij} = \begin{bmatrix} S_{ij11} & S_{ij12} & S_{ij13}& S_{ij21} & S_{ij22} & S_{ij23}& S_{ij31} & S_{ij32} & S_{ij33} \end{bmatrix} \begin{bmatrix} \sigma_{11} \\ \sigma_{12} \\ \sigma_{13}\\ \sigma_{21} \\ \sigma_{22} \\ \sigma_{23}\\ \sigma_{31} \\ \sigma_{32} \\ \sigma_{33} \end{bmatrix}$$

$$\epsilon_{ij} = \begin{bmatrix} S_{ij11} & S_{ij22} & S_{ij33}& 2S_{ij23} & 2S_{ij13} & 2S_{ij12} \end{bmatrix} \begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix}$$

$$\begin{bmatrix} \epsilon_{1} \\ \epsilon_{2} \\ \epsilon_{3} \\ 2\epsilon_{4} \\ 2\epsilon_{5} \\ 2\epsilon_{6} \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} & S_{13}& 2S_{14} & 2S_{15} & 2S_{16}\\ S_{21} & S_{22} & S_{23}& 2S_{24} & 2S_{25} & 2S_{26}\\ S_{31} & S_{32} & S_{33}& 2S_{34} & 2S_{35} & 2S_{36}\\ 2S_{41} & 2S_{42} & 2S_{43}& 4S_{44} & 4S_{45} & 4S_{46}\\ 2S_{51} & 2S_{52} & 2S_{53}& 4S_{54} & 4S_{55} & 4S_{56}\\ 2S_{61} & 2S_{62} & 2S_{63}& 4S_{64} & 4S_{65} & 4S_{66} \end{bmatrix} \begin{bmatrix} \sigma_{1} \\ \sigma_{2} \\ \sigma_{3}\\ \sigma_{4} \\ \sigma_{5} \\ \sigma_{6} \end{bmatrix}$$