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# Statistical Mechanics: Undamaged ⇐-radiation-⇒ Damaged Atomic Lattice Density Evolution and Stochastic Deformation Functional

## Abstract

The deformation kinematics for radiation damage response of bulk materials is presently semi-empirical and phenomenological based on a continuum mechanics supposition: there exists a function space of continuous functions to describe material displacement, strain, strain-rate metrics by using the mathematics of differential calculus. Existing data being assembled from tests on nano-length-scale(NLS) samples provide objective evidence that the continuum mechanics supposition is not an adequate generic mathematical description for radiation damage response in surface-dominated material structures at NLS. An alternative approach will be described that uses concepts and methods from classical statistical mechanics and describe deformation kinematics as a stochastic accumulation of discrete damage events at atomic lattice nano-length-scales. Although radiation damage deformation at a lattice length-scale in solids is mechanistically different from velocity scattering developed by Boltzmann for a kinetic theory of gases, the two problem areas are technically similar and in some simply cases there are useful mathematical analogs. The technical similarities and mathematical analogs will be used to define probability density functions (number per unit volume functions) for undamaged and damaged “size and size change” lattice species (similarity to a probability density function for atomic velocities in gas theory). In general, equations for undamaged and damaged lattice density function evolution are Boltzmann-type equations, which can be approximated and solved for simple cases of radiation induced material damage. Using a path integral approach and the two probability density functions, a stochastic functional will be derived for the relative deformation between any two arbitrary spatial points in a radiation damaged material. Given the relative deformation as an explicit functional of the undamaged and damaged lattice density functions, the kinematics metrics of relative velocity, strain, strain rate, etc., will also be functionals. In the case of radiation damage and annealing, the two lattice density functions are analog expressions to those commonly used to model “birth and death” population evolution.

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# Statistical Mechanics: Undamaged ⇐-radiation-⇒ Damaged Atomic Lattice Density Evolution and Stochastic Deformation Functional

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