by Stefan Müller
Microstructure in crystals
Young measures are also helpful for a better understanding of microstructure in crystals. However, in this case there is an additional difficulty, in that two different crystalline structures can be joined together only if the two lattices agree on the interface.
This geometric condition can be abstractly included in the theory in a simple way, but many relevant questions are still open. This apparently harmless geometric condition leads naive averaging methods to failure, because in the averaging geometrically prohibited structures can arise. In the last years it has become clear that this problem is strictly connected with fundamental problems from other areas of mathematics. One question which can be easily understood is the following. One often observes a layered microstructure (Figure 5), possibly with various hierarchical levels. Occasionally different structures appear (e.g. hexagonal). The question is whether any optimal microstructure can be realized via hierarchical layering.
Figure 8 Two crystal lattices can be stress-free matched only if they differ only by a shear along the interface (left picture), in the right picture this condition is violated, and there are too many atoms of the right structure on the interface.
It was proven that this question is identical to a fundamental problem in the calculus of variations, which was open since the pioneering work of C. B. Morrey in the 50ies. Vladimir Sverák, with whom our group has a close collaboration, obtained 40 years later a negative answer. He showed that there are certain three-dimensional periodic structures whose effective properties are not described by any layered structure (Sverak). In two dimensions the question is still open. Under relatively strong assumptions on the geometry our group obtained recently a positive answer: certain planar structures can be always approximated by layered structures (Müller).
Through construction of complex layered structures we were able to solve a long-standing open problem in the theory of elliptic partial differential equations (which describe e. g. elastic deformation or spatial diffusion of chemical concentrations). This shows that rough structures appear much more frequently than one would expect.
In its simpler, idealized form the theory ignores all effects which limit the fineness of the structure (as in the sailing example). In reality the interfaces between different phases also contribute to the energy, limiting the fineness. A precise quantitative understanding of this phenomenon is a big open problem for the future, up to now we were able to show in some models that interesting new effects, as e. g. a self-similar refining close to interfaces, can appear (Figure 9, see also Figure 5) (KohnMüller,Conti).
Figure 9 Simplified models are enough to predict a refinement of microstructures close to interfaces, cfr. Figure 5.
New materials from mathematical analysis?
A very interesting class of materials are magneto-elastic materials, which expand or contract under application of a magnetic field. They find application as actuators (e. g. loudspeakers) or sensors. Experimental research motivated by a mathematical theory developed by DeSimone and James (DeSimone-James), lead recently to discovery of a new material (a Nickel-Manganese-Gallium alloy) whose magneto-elastic effect is significantly larger as in the best previously known material, Terfenol-D (a special alloy, whose magneto-elastic effect was called "gigantic", because it is much bigger than in all natural compounds). This property is however apparent only if the experimental sample is cut along specific crystallographic directions. These directions, which are difficult to find through simple trial and error, have been predicted by the mathematical theory. This is an example of how mathematical models can expand the classical methodologies in the development of new materials.
A more complete mathematical discussion of these topics can be found in
S. Müller, Variational models for microstructure and phase transitions, CIME Lecture Notes
Cited mathematical literature:
(Sverak) V. Sverák, Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. Edinburgh 120 (1992), 185-189
(DeSimone-James) A. DeSimone and R. D. James, A theory of magnetostriction oriented towards applications, J. Appl. Phys. 81 (1997) 5706-5708
(Müller) S. Müller, Rank-one convexity implies quasiconvexity on diagonal matrices, Int. Math. Res. Not. (1999), 1087-1095
(Kohn-Müller) R. V. Kohn and S. Müller, Surface energy and microstructure in coherent phase transitions, Comm. Pure Appl. Math. 47 (1994), 405-435.