# The Green tensor of Mindlin's anisotropic first strain gradient elasticity

## TL;DR

This paper derives a non-singular Green tensor for Mindlin's anisotropic strain gradient elasticity, applicable to arbitrary anisotropic materials, and demonstrates its numerical implementation and validation against molecular statics results.

## Contribution

It provides the first general derivation of the Green tensor for anisotropic first strain gradient elasticity, including numerical implementation and validation.

## Key findings

- Green tensor is non-singular at the origin.
- The tensor converges to classical form away from the origin.
- Numerical results agree with molecular statics calculations.

## Abstract

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.