  In the perspective of homogenization theory, strain-gradient elasticity is a
strategy to describe the overall behaviour of materials with coarse
mesostructure. In this approach, the effect of the mesostructure is described
by the use of three elasticity tensors whose orders vary from 4 to 6.
Higher-order constitutive tensors make it possible to describe rich physical
phenomena. However, these objects have intricate algebraic structures that
prevent us from having a clear picture of their modeling capabilities. The
harmonic decomposition is a fundamental tool to investigate the anisotropic
properties of constitutive tensor spaces. For higher-order tensors (i.e.
tensors of order $n\geq$3), its establishment is generally a difficult task. In
this paper a novel procedure to obtain this decomposition is introduced. This
method, that we have called the \textit{Clebsch-Gordan Harmonic Algorithm},
allows to obtain \emph{explicit} harmonic decompositions satisfying good
properties such as orthogonality and unicity. The elements of the decomposition
also have a precise geometrical meaning simplifying their physical
interpretation. This new algorithm is here developed in the specific case of 2D
space and applied to Mindlin's Strain-Gradient Elasticity. We provide, for the
first time, the harmonic decompositions of the fifth- and sixth-order
elasticity tensors involved in this constitutive law.
