Kinematic theory for scale-invariant patterns in acicular martensitas

We present a kinematic theory which explains the emergence of scale-invariant patterns in acicular martensites which occur, for example, in Fe---C and Fe---Ni alloys. Scale-invariant structures emerge naturally as a consequence of competition between the average tip-velocity, ν, and the rate of nucleation, I, of the martensite grains. Martensite growth is analyzed in terms of fixed points of a well-defined renormalization group. It is shown that the stationary probability distribution of the martensite grain size P(l), is governed by two fixed points - an unstable (noncritical) fixed point at ν/I = 0, characterized by a Gamma distribution and a stable (critical) fixed point at ν/I = ∞, characterized by a Lévy distribution, P(l) l−α. A universal crossover function describes the SPD at intermediate values of ν/I. The analysis may also be used to compute the tip-velocity from optical micrograph pictures of martensite grains.