Science and Technology of Nuclear Installations

Volume 2016, Article ID 4649870, 9 pages

http://dx.doi.org/10.1155/2016/4649870

## The Definition Method and Optimization of Atomic Strain Tensors for Nuclear Power Engineering Materials

College of Architecture and Environment, Sichuan University, Chengdu 610065, China

Received 5 June 2016; Accepted 11 July 2016

Academic Editor: Yan Yang

Copyright © 2016 Xiangguo Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A common measure of deformation between atomic scale simulations and the continuum framework is provided and the strain tensors for multiscale simulations are defined in this paper. In order to compute the deformation gradient of any atom , the weight function is proposed to eliminate the different contributions within the neighbor atoms which have different distances to atom , and the weighted least squares error optimization model is established to seek the optimal coefficients of the weight function and the optimal local deformation gradient of each atom. The optimization model involves more than 9 parameters. To guarantee the reliability of subsequent parameters identification result and lighten the calculation workload of parameters identification, an overall analysis method of parameter sensitivity and an advanced genetic algorithm are also developed.

#### 1. Introduction

Titanium alloys have been largely used as nuclear power engineering materials [1], and it is important and significant to analyze the atomic-level strain distribution of these materials. The strain tensors are commonly defined by the local deformation of the continuum. Unlike displacement, strain is not a physical quantity that can be measured directly, and it is calculated from a definition that relies on the gradient of the continuous displacement field. At the microscale, it is difficult to define the local deformation according to the position of each atom which is obtained from the adjacent discrete time interval, so there is no universally accepted definition of strain tensors of atomic scale so far.

Many engineering problems involving physical phenomena need to calculate strain tensors at atomic scale. Wang et al. [2] pointed that it was important to analyze the atomic-level strain distribution and get the atomic stress-strain curve while studying the mechanical behavior of Zr-based metallic glass under indentation. Hirth et al. [3] considered that the computation of the deformation gradient and strain tensors made the approach useful for evaluation of continuum models, development of microstructure and mechanical property relationships, and identification of dislocations and disclinations, as well as for quantification of plastic spin and strain gradients.

In recent years, many researchers are challenging to provide a common measure of deformation between atomic scale simulations and the continuum framework and define the strain tensors for multiscale simulations. Zimmerman et al. [4] defined the slip vector according to the positions of atoms and successfully identified the lattice distortion and the formation of dislocation structures, but these measures could not be utilized in the continuum framework. Mott et al. [5] presented a definition of the local atomic strain increments in three dimensions and an algorithm for computing them. First, an arbitrary arrangement of atoms was tessellated into Delaunay tetrahedra, and then the deformation gradient increment tensor for interstitial space was obtained from the displacement increments of the corner atoms of Delaunay tetrahedra. However, it was complicated to establish the tetrahedral elements of atoms. Gullett et al. [6] proposed an atomic strain tensor that is based on the definition of a discrete equivalent to the continuum deformation gradient that accounts for the relative motion of an atom and its neighbors in a nonlocal fashion. This method was computationally efficient, because the deformation gradient arose from an optimization procedure that did not rely on a geometric decomposition, and the strain tensors were computed directly from the deformation gradient and were appropriate for general finite, multiaxial deformation states. When the deformation gradient at an atom was formed, a weight function should be built to eliminate the different contributions within the neighbor atoms which had different distances to the atom. However, Gullett et al. [6] did not study the establishment and optimization method of the weight function which played an important role in the formulation of the discrete deformation but just used the invariant weight function of the artificial assumption to calculate the discrete deformation gradient at the atom.

By summarizing the shortcomings of the existing methods mentioned above, the work done by this paper can be categorized into three parts: first, the strain tensors for multiscale simulations are defined, and the weighted least squares error optimization model involving more than 9 parameters is established to seek the optimal coefficients of the weight function and the optimal local deformation gradient of each atom; next, to guarantee the reliability of subsequent parameters identification result and also to lighten the calculation workload of parameters identification, an overall analysis method of parameter sensitivity, based on Latin Hypercube Sampling method and Spearman rank correlation method, is proposed; furthermore, on the fundamentals of the result of parameter sensitivity and basic genetic algorithm (GA), an advanced genetic algorithm based on the advanced niche genetic algorithm, global peak value determination strategy, and local accurate searching techniques is developed. Finally, taking alpha titanium as an example, the strain tensors of atoms are computed by means of the method proposed in this paper, and the method is proved to be correct and feasible by comparing with the results got by other methods from the existing reference.

#### 2. Modeling Approach

##### 2.1. Deformation Gradient and Strain Tensors

In order to describe the positions of atoms at the initial time and at the current time , we assume a fixed Cartesian coordinate system as shown in Figure 1.