= Solution
<crystal grain> structure controls both the source of incompatibility and its conversion into a bulk dimensional change. Separate \b[orientation distribution, <grain size>, <crystal grain> shape and defect structure] rather than treating “fine grained” as a universal cure.
In thermal cycling, different <crystal grain> orientations rotate the <thermal expansion tensor> into different laboratory directions. Bonding prevents each <crystal grain> from following its independent free <strain>. Coarse <crystal grains> can produce larger surface relief and longer local mismatch distances; finer <crystal grains> can distribute the accommodation more uniformly. However, changing <grain size> also changes strength, boundary-mediated deformation and <creep (deformation)>. A fine but strongly textured material can still show directional growth, and enhanced boundary accommodation can itself be irreversible. <crystallographic texture> and the way mismatch is accommodated must therefore be measured separately.
For <irradiation growth>, <crystallographic texture> determines whether directional single-grain changes reinforce or cancel. An illustrative independent-grain average is
$$
\overline{\boldsymbol\varepsilon}^{g}=\sum_jv_jR_j\boldsymbol\varepsilon_j^{g}R_j^T,
$$
where $v_j$ is a <crystal grain> volume fraction and $R_j$ its orientation. A random orientation average of a traceless growth tensor is zero, whereas <crystallographic texture> can retain a preferred bulk <strain>. For an ideal hexagonal <crystal grain> with principal growth <strains> $(g,g,-2g)$, the <strain> along a direction making angle $\theta$ to the c-axis is
$$
\varepsilon_d=g(1-3\cos^2\theta),\qquad
\langle\varepsilon_d\rangle=g\big(1-3\langle\cos^2\theta\rangle\big).
$$
Random <crystallographic texture> has $\langle\cos^2\theta\rangle=1/3$ and cancels this ideal directional growth. This is an orientation model, not proof that real <crystal grain> constraints or volume-changing processes vanish.
<grain boundaries> also absorb mobile defects. Boundary spacing and aspect ratio change directional sink distances, while <dislocation> <density>, <cold working> and <recrystallisation> change competing sinks and residual <stresses>. These affect the single-grain response before it is averaged. In one model compared with zirconium-alloy experiments, increasing <grain size>, reducing aspect ratio or reducing <dislocation> <density> decreased predicted growth. That result is a useful counterexample to assuming that smaller <crystal grains> always reduce <irradiation growth>.<Effects of Microstructural Factors on Irradiation Growth in Zirconium-Base Alloys by Noriyuki Sadaoka and Motomasa Fuse>[Transactions of the Japan Institute of Metals 29, 970–976, abstract and stated microstructural trends]{ref}
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