Influence of the mode of deformation on recrystallisation kinetics in Nickel through experiments,theory and phase field model

Abstract

In this paper, we report the influence of the mode of deformation on recrystallisation kinetics through experiments, theory and a phase field model. Ni samples of 99.6% purity are subjected to torsion and rolling at two equivalent plastic strains and the recrystallisation kinetics and microstructure are compared experimentally. Due to significant differences in the distributions of the nuclei and stored energy for the same equivalent strain, large differences are observed in the recrystallisation kinetics of rolled and torsion-tested samples. Next, a multi-phase field model is developed in order to understand and predict the kinetics and microstructural evolution. The coarse-grained free energy parameters of the phase field model are taken to be a function of the stored energy. In order to account for the observed differences in recrystallisation kinetics, the phase field mobility parameter is a required constitutive input. The mobility is calculated by developing a mean field model of the recrystallisation process assuming that the strain free nuclei grow in a uniform stored energy field. The activation energy calculated from the mobilities obtained from the mean field calculation compares very well with the activation energy obtained from the kinetics of recrystallisation. The recrystallisation kinetics and microstructure as characterised by grain size distribution obtained from the phase field simulations match the experimental results to good accord. The novel combination of experiments, phase field simulations and mean field model facilitates a quantitative prediction of the microstructural evolution and kinetics.     

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λ_n = [-n, n] ^d ∩ ℤ ^d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ ^d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ ^d . In the case d > 4, we also make use of Wilson’s method. An erratum to this article is available at .     

Random Logistic Maps. I

Let {C _i} ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n} ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1–X _n). It is shown here that: (i) E ln C ₁<0X _n0 w.p.1. (ii) E ln C ₁=0X _n0 in probability (iii) E ln C ₁>0, E |ln(4–C ₁)| such that (0, 1)=1 and is invariant for {X _n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C ₁>0 and – ln(1–x) (dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4–C ₁)|< and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀0.     

Quantitative Recurrence and Large Deviations for Teichmuller Geodesic Flow

We prove quantitative recurrence and large deviations results for the Teichmuller geodesic flow on connected components of strata of the moduli space Q _g of holomorphic unit-area quadratic differentials on a compact genus g ≥ 2 surface.     

Iterations of random and deterministic functions

Letf be a probability generating function on [0, 1]. The convergence of its iteratesf _n to fixed points is studied in this paper. Results include rates forf andf ^-1. Also iterates of independent identically distributed stable processes are studied and a trichotomy based on the order of the stability is established. Dedicated to the memory of Professor K G Ramanathan     

Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

We introduce a new method for proving the estimate

where solves the equation . The method can be applied to the Laplacian on . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

     

Entropy maximization

It is shown that (i) every probability density is the unique maximizer of relative entropy in an appropriate class and (ii) in the class of all pdf f that satisfy ∝ fh _i dμ = λ _i for i = 1, 2, ..., ... kthe maximizer of entropy is an f ₀ that is proportional to exp(Σc _i h _i ) for some choice of c _i . An extension of this to a continuum of constraints and many examples are presented.