Rise op correlations of transformation strains in RANDOM polycrystals

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Abstract

We investigate the statistics of the transformation strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo marten- sitic phase transitions. In our laminated polycrystal model the orientation of the n grains (crystallites) is given by an uncorrelated random array of the orientation angles O i i = l.,n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains σ i i = 1., n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables θi = 1., ,n are uncorrelated, the random variables ej, i = 1.n may be correlated. This issue is central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely long grains (laminated polycrystal of height L =∞); (ii) Grains of finite but large height (L :>1); and (iii) Chain of short grains (L = l 0/(2n), l 0 < 1). With references to de Finetti's theorem, Riesz' rearrangement inequality, and near neighbor approximations, our analyses establish that under the scaling limits (i), (ii), and (iii) the arrays of transformation strains arising from given boundary conditions exhibit no correlations, long-range correlations, and exponentially decaying short-range correlations, respectively.

Original languageEnglish (US)
Pages (from-to)1550-1584
Number of pages35
JournalSIAM Journal on Mathematical Analysis
Volume40
Issue number4
DOIs
StatePublished - 2008

All Science Journal Classification (ASJC) codes

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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