Elastic softening and its relation to martensitic transitions has been a long-standing controversy in the literature. I don't want to take sides here, but let me sketch my understanding of the current understanding.
Martensites are materials whose crystal structure undergoes a shape transformation as they are cooled. This can be a simple stretch in one direction (with coupled squishings in the other directions), or can be a modulation with layers of atoms being pushed back and forth in a regular pattern. The interesting morphology (patterns of domains) found in martensites are formed to minimize the strain needed to make the locally deformed regions fit together (either at the transformation temperature when they need to fit into the undeformed material, or below when the domains need to fit into a prescribed shape for the object as a whole).
Sometimes (but not always!) this transformation is heralded above the transition temperature by the softening of the elastic constant along the eventual direction of deformation. This makes some sense: if it's about to stretch spontaneously in some direction, it might be easier to stretch. It does not have to happen, and often doesn't: just like ice doesn't have to become squishy or water stiff as it approaches 32 F.
(Above the transformation temperature, if you pull hard enough, you can transform the material early into the low-temperature, deformed state. This is called ``superelasticity'' or ``superplasticity'' or something like that, and isn't what I'm discussing. Ice near the freezing point will melt if you squeeze it too, which makes it slippery under ice-skates.)
The controversy came when a bunch of physicists started claiming that the elastic softening caused the martensitic transformation. Maybe they were partly right for some systems; surely they weren't right for all systems. In any case, the elastic softening is definitely a precursor: it happens above the transition, hinting of things to come.
Sivan Kartha, Jim Krumhansl, and I suspect that (at least in some cases) the tweed precursors cause the elastic softening! After all, the tweed can deform a lot under external strain, and will deform most easily in the direction of the martensitic transformation. Our prediction would be that systems without tweed (like pure materials) won't have elastic softening: when you add impurities that make tweed form, you'll get softening coming in with the tweed. We haven't got a quantitative theory, so nothing is published; on the other hand, that's why I put ``elastic softening'' along with the central peak as a precursor.
James P. Sethna, firstname.lastname@example.org
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).