A multi-scale model for stresses, strains and swelling of reactor components under irradiation

Abstract

Predicting strains, stresses and swelling in nuclear power plant components exposed to irradiation directly from the observed or computed defect and dislocation microstructure is a fundamental problem of fusion power plant design that has so far eluded a practical solution. We develop a model, free from parameters not accessible to direct evaluation or observation, that is able to provide estimates for irradiation-induced stresses and strains on a macroscopic scale, using information about the distribution of radiation defects produced by high-energy neutrons in the microstructure of materials. The model exploits the fact that elasticity equations involve no characteristic spatial scale, and hence admit a mathematical treatment that is an extension to that developed for the evaluation of elastic fields of defects on the nanoscale. In the analysis given below we use, as input, the radiation defect structure data derived from ab initio density functional calculations and large-scale molecular dynamics simulations of high-energy collision cascades. We show that strains, stresses and swelling can be evaluated using either integral equations, where the source function is given by the density of relaxation volumes of defects, or they can be computed from heterogeneous partial differential equations for the components of the stress tensor, where the density of body forces is proportional to the gradient of the density of relaxation volumes of defects. We perform a case study where strains and stresses are evaluated analytically and exactly, and develop a general finite element method implementation of the method, applicable to a broad range of predictive simulations of strains and stresses induced by irradiation in materials and components of any geometry in fission or fusion nuclear power plants.

Influence of the stress state on the cross-slip free energy barrier in Al: An atomistic investigation

“…V 0 ij = −∂H int el /∂σ ij and Ω ijkl = −∂ 2 H int el /(∂σ ij ∂σ kl ) are two tensors, of second and fourth rank, respectively, that determine the change of the energy landscape due to the applied stress and to the interaction between the applied stress and the internal stress associated with the defect, respectively. The polarization tensor Ω ijkl can be defined as the difference of curvature in the free energy landscape basin because of the interaction of the stress fields, which stands for the difference of the compliance tensor in the vicinity of the defect [39,40].…”

Relaxation volumes of microscopic and mesoscopic irradiation-induced defects in tungsten

“…Recently we have shown that it is possible to compute stresses and strains in reactor components on the macroscopic scale of centimetres and metres from the distribution of irradiation-induced defects [13]. As a source term, this model requires the spatially varying density of relaxation volumes of defects, and so allows the direct simulation of volumetric radiation-induced swelling and the associated stresses from an atomistic or object-based model.…”

“…One outstanding issue, which this paper is intended to address, is to have good data for the relaxation volumes of lattice defects, as while accessible from simulation for years, reporting this information has been somewhat neglected in favour of establishing accurate values of formation energies and ground-state structures. For example, a comprehensive compilation of data on relaxation volumes of individual self-interstitial and vacancy point defects derived from DFT calculations has been reported only recently [18,13,19].…”

“…We present simple empirical formulae for the relaxation volumes of defects that might be used for predicting stresses and strains in engineering components containing these defects [13]. As tungsten is (nearly) elastically isotropic, it suffices to present results in terms of a single relaxation volume, and the relaxation volume anisotropy parameter, defined as the ratio of the smallest to largest partial relaxation volumes.…”

Atomistic-object kinetic Monte Carlo simulations of irradiation damage in tungsten

“…This requires the bulk elastic Lamé parameters λ and µ, which we compute at the beginning of the simulation using the empirical potential supplied. The centreof-position of an object is taken to be [57] R = n a=1 P a R a n a=1 P a…”