Kinematic theory for scale-invariant patterns in acicular martensitas
✍ Scribed by Madan Rao; Surajit Sengupta
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 483 KB
- Volume
- 224
- Category
- Article
- ISSN
- 0378-4371
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✦ Synopsis
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, v, 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(/), is governed by two fixed points -an unstable (noncritical) fixed point at v/I = 0, characterized by a Gamma distribution and a stable (critical) fixed point at v/1 = oo, characterized by a I_,6vy distribution, P(I) ~ l -a. A universal crossover function describes the SPD at intermediate values of v/l. The analysis may also be used to compute the tip-velocity from optical micrograph pictures of martensite grains.
Scale-invariant patterns [ 1 ] are common in nature. A variety of systems, like the human lung [2], cracks in brittle materials [3], distribution of galaxies in the universe [4] and martensitic microstructures [5], show scale invariance over reasonably large length scales. However, in most cases, the origin of this scale invariance is not fully understood, In this paper, we furnish an explanation for the origin of scale-invariant microstructures in acicular martensites [6,8] as a straight-forward example of a crossover process involving a flow towards a "self-organized critical" (SOC) fixed point of a renormalisation group (RG) transformation. The reader will find a detailed account of this work in Ref. [9].
The name "martensite" has been used to denote metastable configurations arising from structural transitions in a wide variety of systems [6,8] ranging from metals to polymeric crystals and crystalline membranes [ 10]. To be specific, we therefore specialize to the well-studied example of acicular martensites occurring in Fe-C and Fe-Ni systems.