Formation of Microstructure

Solid phase transformations in alloys and other materials, which involve a change of shape in the underlying crystal lattice at a critical temperature, typically lead to complicated patterns of microstructure comprising twinned laminates and other features.
Martensitic microstructure in CuZnAl
(M. Morin, INSA de Lyon) The morphology of such microstructures is crucial for determining the macroscopic behaviour of the material. For martensitic transformations a key contribution was the crystallographic theory of martensite (Wechsler, Lieberman & Read, 1953), which highlighted the important role played by compatibility of gradients in the formation and morphology of microstructure. This theory was given an energetic derivation in the multi-well nonlinear elastostatic model of Ball & James (1987, 1992), which has led both to an increased understanding of specific microstructures, and to a series of developments by outstanding mathematicians concerning the interplay between microstructure and the calculus of variations. However, elastostatics has only a limited bearing on the dynamic process by which patterns of microstructure are formed under cooling or stress. We aim to make a concerted theoretical and computational attack on this technologically important dynamic pattern-formation problem, using ideas from nonlinear partial differential equations and the associated dynamical systems theory.

Important for the pattern formation problem, but also of intrinsic interest, is how best to model interfacial energy, which selects and sets length-scales for the microstructure. Twin boundaries can be diffuse or atomistically sharp, and models of interfacial energy may predict either or both behaviours. We will investigate various models (second gradient, ‘free discontinuity’, nonlocal), their relation to atomistic theories, and how their predictions for interfacial structure and metastability compare with experiment.

The formation of microstructure, now at a slightly larger scale, also underpins the theory of fatigue in metals. Under cyclic loading of a material such as copper it is observed that “ladders” of dislocation bands form with a characteristic spacing between regions of high dislocation density and regions of near pristine material (Ahmed, Wilkinson & Roberts, 1997, 2001). While the properties of the material can be predicted given the local configuration of the bands, there is at present no mathematical theory which can predict the formation of these ladders.

The specific projects associated with this theme are

Models for interfacial energy
Well-posedness of dynamics
Microstructure evolution and morphology
Modelling dislocation ladders in fatigue.

References

J. Ahmed, A.J. Wilkinson and S.G. Roberts. Characterising dislocation structures in bulk fatigued copper single crystals using electron channelling contrast imaging (ECCI). Philosophical Magazine Letters, 76:237–245, 1997.
J. Ahmed, A.J. Wilkinson and S.G. Roberts. Electron channelling contrast imaging characterisation of dislocation structures associated with extrusion and intrusion systems and fatigue cracks in copper single crystals. Philosophical Magazine A, 81:1473–1488, 2001.
J.M. Ball and R.D. James. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100:13–52, 1987.
J.M. Ball and R.D. James. Proposed experimental tests of a theory of fine microstructure, and the two-well problem. Phil. Trans. Roy. Soc. London A, 338:389–450, 1992.
M.S. Wechsler, D.S. Lieberman and T.A. Read. On the theory of the formation of martensite. Trans. Amer. Inst. Min. (Metall.) Engrs., 197:1503–1515, 1953.