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We investigate the ability of frame-invariant amplitude equations [G. H.Gunaratne, Q. Ouyang, and H. Swinney, Phys. Rev. E {\bf 50}, 2802 (1994)] todescribe quantitatively the evolution of polycrystalline microstructures and weextend this approach to include the interaction between composition and stress.Validations for elemental materials include studies of theAsaro-Tiller-Grinfeld morphological instability of a stressed crystal surface,polycrystalline growth from the melt, grain boundary energies over a wide rangeof misorientation, and grain boundary motion coupled to shear deformation.Amplitude equations with accelerated strain relaxation in the solid are shownto model accurately the Asaro-Tiller-Grinfeld instability. Polycrystallinegrowth is also well described. However, the survey of grain boundary energiesshows that the approach is only valid for a restricted range of misorientationsas a direct consequence of an amplitude expansion. This range coversapproximately half the complete range allowed by crystal symmetry for somefixed reference set of density waves used in the expansion. Over this range,coupled motion to shear is well described by known geometrical rules and atransition from coupling to sliding motion is also reproduced. Amplitudeequations for alloys are derived phenomenologically in a Ginzburg-Landauspirit. Vegard's law is shown to be naturally described by seeking a gaugeinvariant form of those equations under a transformation that corresponds to alattice expansion and deviations from Vegard's law can be easily incorporated.Those equations realistically describe the dilute alloy limit and have the sameflexibility as conventional phase-field models for incorporating arbitraryfree-energy/composition curves...

Amplitude equations for polycrystalline materials with interaction between composition and stress