I have a question about Voigt notation. I've searched the internet and found a lot of sites describing how to preform Voigt notation on 3x3 matrix. The problem is that all of those examples are shown on the symmetric 3x3 tenosr - like stress or strain tensor.
Can anyone tell me how to use Voigt notation on nonsymmetric 3x3 tensor in order to get vector of 9 components?
Yes, you can use Voigt notation for nonsymmetric tensors. If you have two 2nd order 3x3 tensors, $\sigma_{ij}$, and $\epsilon_{ij}$ and a 4th order 3x3x3x3 linear map between them, $C_{ijkl}$, you can represent the relationship in the following form:
$$ \left\{ \begin{array}{c} \sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\\ \sigma_{21}\\ \sigma_{31}\\ \sigma_{32} \end{array}\right\} = \left[ \begin{array}{ccccccccc} C_{1111}&C_{1122}&C_{1133}&C_{1112}&C_{1113}&C_{1123}&C_{1121}&C_{1131}&C_{1132}\\ C_{2211}&C_{2222}&C_{2233}&C_{2212}&C_{2213}&C_{2223}&C_{2221}&C_{2231}&C_{2232}\\ C_{3311}&C_{3322}&C_{3333}&C_{3312}&C_{3313}&C_{3323}&C_{3321}&C_{3331}&C_{3332}\\ C_{1211}&C_{1222}&C_{1233}&C_{1212}&C_{1213}&C_{1223}&C_{1221}&C_{1231}&C_{1232}\\ C_{1311}&C_{1322}&C_{1333}&C_{1312}&C_{1313}&C_{1323}&C_{1321}&C_{1331}&C_{1332}\\ C_{2311}&C_{2322}&C_{2333}&C_{2312}&C_{2313}&C_{2323}&C_{2321}&C_{2331}&C_{2332}\\ C_{2111}&C_{2122}&C_{2133}&C_{2112}&C_{2113}&C_{2123}&C_{2121}&C_{2131}&C_{2132}\\ C_{3111}&C_{3122}&C_{3133}&C_{3112}&C_{3113}&C_{3123}&C_{3121}&C_{3131}&C_{3132}\\ C_{3211}&C_{3222}&C_{3233}&C_{3212}&C_{3213}&C_{3223}&C_{3221}&C_{3231}&C_{3232} \end{array}\right] \left\{ \begin{array}{c} \epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{12}\\ \epsilon_{13}\\ \epsilon_{23}\\ \epsilon_{21}\\ \epsilon_{31}\\ \epsilon_{32} \end{array}\right\} $$
With symmetric tensors this can be simplified down to a 6x6 matrix. Without symmetry you just won't be able to simplify the matrix.