Texture averaging of irradiation growth ($\overline{\mathbf{\epsilon }}={\sum }_g{v}_g{R}_g{\mathbf{\epsilon }}^{(g)}{R}_g^T$)

An ideal orientation average $\overline{\mathbf{\epsilon }}={\sum }_g{v}_g{R}_g{\mathbf{\epsilon }}^{(g)}{R}_g^T$. Random orientation averages a traceless single-grain growth tensor to zero, while crystallographic texture can retain a directional bulk strain. crystal grain constraints can modify this simple independent-grain estimate.

In an ideal axisymmetric volume-conserving crystal grain model, directional growth is proportional to $1-3⟨{\cos }^2\theta ⟩$, where $\theta$ is the angle between that direction and the crystal c-axis. Random crystallographic texture has $⟨{\cos }^2\theta ⟩=1/3$.