Anomalous Grain Boundary Diffusion: Fractional Calculus Approach

Grain boundary (GB) diffusion in engineering materials at elevated temperatures often determines the evolution of microstructure, phase transformations, and certain regimes of plastic deformation and fracture. Interpreting experimental data with the use of the classical Fisher model sometimes encounters contradictions that can be related to violation of Fick’s law. Here, we generalize the Fisher model to the case of non-Fickian (anomalous) diffusion ubiquitous in disordered materials. The process is formulated in terms of the subdiffusion equations with time-fractional derivatives of order α∈(0,1] and β∈(0,1] for grain volume and GB, respectively. It is shown that propagation along GB for the case of a localized instantaneous source and weak localization in GB (β>α/2) is approximately described by distributed-order subdiffusion with exponents α/2 and β. The mean square displacement is calculated with the use of the alternating renewal process model. The tail of the impurity concentration profiles along the z axis is approximately described by the dependence ∝exp(-Az6/5) for all 0<α≤1, as in the case of normal GB diffusion, so the 6/5-law itself can serve as an identifier of a more general phenomenon, namely, anomalous GB diffusion.

EXACT DISTRIBUTION OF INTERMITTENTLY CHANGING POSITIVE AND NEGATIVE COMPOUND POISSON PROCESS DRIVEN BY AN ALTERNATING RENEWAL PROCESS AND RELATED FUNCTIONS

Alternating renewal processes have been widely used to model social and scientific phenomenal where independent “on” and “off” states alternate. In this paper, we study a model where the value of a process cumulates and declines according to two modes of compound Poisson processes with respect to an underlying alternating renewal process. The model discussed in the present paper can be used as a revenue management model applied to inventory or to finance. The exact distribution of the process is derived as well as the double Laplace transform with respect to the level and time of the process.

Homogeneous Discrete Time Alternating Compound Renewal Process: A Disability Insurance Application

Discrete time alternating renewal process is a very simple tool that permits solving many real life problems. This paper, after the presentation of this tool, introduces the compound environment in the alternating process giving a systematization to this important tool. The claim costs for a temporary disability insurance contract are presented. The algorithm and an example of application are also provided.