  We derive the Green tensor of Mindlin's anisotropic first strain gradient
elasticity. The Green tensor is valid for arbitrary anisotropic materials, with
up to 21 elastic constants and 171 gradient elastic constants in the general
case of triclinic media. In contrast to its classical counterpart, the Green
tensor is non-singular at the origin, and it converges to the classical tensor
a few characteristic lengths away from the origin. Therefore, the Green tensor
of Mindlin's first strain gradient elasticity can be regarded as a physical
regularization of the classical anisotropic Green tensor. The isotropic Green
tensor and other special cases are recovered as particular instances of the
general anisotropic result. The Green tensor is implemented numerically and
applied to the Kelvin problem with elastic constants determined from
interatomic potentials. Results are compared to molecular statics calculations
carried out with the same potentials.
