  Higher-order tensors appear in various areas of mechanics as well as physics,
medicine or earth sciences. As these tensors are highly complex, most are not
well understood. Thus, the analysis and the visualization process form a highly
challenging task. In order to solve this problem a suitable tensor
decomposition for arbitrary tensors is desirable. Since many real-world
examples are described in the three-dimensional space, this paper will also
focus on this space. Tensors of order zero and one are well understood and for
second order tensors the well-known spectral decomposition exists. However,
there is no standard decomposition for tensors of order higher than two. This
work will focus on summarizing facts about the deviatoric decomposition and on
its calculation in a recursive way. It decomposes a tensor of arbitrary order
into a sum of orthogonal, irreducible parts. Thus, into parts that are
orthogonal and invariant under rotation. These irreducible parts are based on
so-called deviators. The calculations and results of the deviatoric
decomposition are not broadly understood in the engineering community. To
change this situation, information about this decomposition from a range of
literature sources was gathered and will be summarized and explained in this
work to make further analysis possible. Different applications and examples for
often used physical tensors will also be given. The goal of this work is to
give other researchers the chance to also use this knowledge.
