  The theory of first strain gradient elasticity (SGE) is widely used to model
size and non-local effects observed in materials and structures. For a material
whose microstructure is centrosymmetric, SGE is characterized by a sixth-order
elastic tensor in addition to the classical fourth-order elastic tensor. Even
though the matrix form of the sixth-order elastic tensor is well-known in the
isotropic case, its complete matrix representations seem to remain unavailable
in the anisotropic cases. In the present paper, the explicit matrix
representations of the sixth-order elastic tensor are derived and given for all
the 3D anisotropic cases in a compact and well-structured way. These matrix
representations are necessary to the development and application of SGE for
anisotropic materials
