  A definition of the Gr\"uneisen parameters for anisotropic materials is
derived based on the response of phonon frequencies to uniaxial stress
perturbations. This Gr\"uneisen model relates the thermal expansion in a given
direction ($\alpha_{ii}$) to one element of the elastic compliance tensor,
which corresponds to the Young's modulus in that direction ($Y_{ii}$). The
model is tested through ab initio prediction of thermal expansion in zinc,
graphite, and calcite using density functional perturbation theory, indicating
that it could lead to increased accuracy for structurally complex systems. The
direct dependence of $\alpha_{ii}$ on $Y_{ii}$ suggests that materials which
are flexible along their principal axes but rigid in other directions will
generally display both positive and negative thermal expansion.
