S. Bulent Biner

Modeling the Ductile Brittle Fracture Transition in Reactor Pressure Vessel Steels Using a Cohesive Zone Model Based Approach

Fracture properties of Reactor Pressure Vessel (RPV) steels show large variations with changes in temperature and irradiation levels. Brittle behavior is observed at lower temperatures and/or higher irradiation levels whereas ductile mode of failure is predominant at higher temperatures and/or lower irradiation levels. In addition to such temperature and radiation dependent fracture behavior, significant scatter in fracture toughness has also been observed. As a consequence of such variability in fracture behavior, accurate estimates of fracture properties of RPV steels are of utmost importance for safe and reliable operation of reactor pressure vessels.A cohesive zone based approach is being pursued in the present study where an attempt is made to obtain a unified law capturing both stable crack growth (ductile fracture) and unstable failure (cleavage fracture). The parameters of the constitutive model are dependent on both temperature and failure probability. The effect of irradiation has not been considered in the present study. The use of such a cohesive zone based approach would allow the modeling of explicit crack growth at both stable and unstable regimes of fracture. Also it would provide the possibility to incorporate more physical lower length scale models to predict DBT. Such a multi-scale approach would significantly improve the predictive capabilities of the model, which is still largely empirical.Copyright

Phase-Field Crystal Modeling of Material Behavior

The phase-field crystal, PFC, method introduced by Elder and coworkers [1–3], can be viewed as multiscale simulation algorithm that bridges the classical molecular dynamics, MD, simulations and the phase-field methods covered in the previous chapters. PFC method introduces an order parameter defined as the local-time-averaged atomic number density which is able to produce periodicity of crystal lattices. In the model, any perturbation or lattice defects result in an increase in the free energy, thus enabling to obtain the information which has been only possible by the atomistic simulations previously. In addition, PFC method produces various atomistic events in much larger spatial and temporal dimensions that are not easily accessible with current MD simulation techniques. Therefore, PFC method has emerged as an attractive simulation approach.

Solving Phase-Field Equations with Finite Elements

The finite element method, FEM, and sometimes also called finite element analysis, FEA, was originally developed in the aircraft industry in 1960s [1, 2]. Therefore, it is a very mature algorithm and widely used in engineering and science as a general numerical approach for the solution of PDEs subject to known boundary and initial conditions. The use of piecewise continuous functions over subregions of domain to approximate the unknown function was first introduced by Courant [3]. This approach was later formalized [4, 5] and term finite elements for these subregions was introduced by Clough [6]. Therefore, similar to finite difference technique the FEM is also local in nature. However, FEM has superior and unique characteristics to describe very complex geometries and boundaries of domains. There are plenty of textbook and online materials covering both theoretical and practical aspects of FEM or FEA and some of them are listed in the reference section.

Preliminaries About the Codes

The Matlab/Octave programming language was chosen for the codes presented in the book. In the development of the codes, it is assumed that the reader has some degree of experience in computer programming. There are differences in syntax between Matlab/Octave programming and the other traditional programming languages such as Fortran, C, and C++. The readers who are not familiar with Matlab/Octave programming may find useful to consult the extensive documentations that are provided at their websites. In addition, there are plenty of books and online tutorials available, of which some of them are listed in reference section.

Introduction to Numerical Solution of Partial Differential Equations

Many of the fundamental theories of physics and engineering, including the phase-field models, are expressed by means of systems of partial differential equations, PDEs. A PDE is an equation which contains partial derivatives, such as

Solving Phase-Field Models with Fourier Spectral Methods

\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial {x}^2}

Multi-scale modeling of inter-granular fracture in UO2

in which u is regarded as function of length x and time t. There is no real unified theory for PDEs. They exhibit their own characteristics to express the underlying physical phenomena as accurately as possible. Since PDEs can be hardly solved analytically, their solutions relay on the numerical approaches. A brief of summary of the numerical techniques involving their spatial and temporal discretization is given below. These techniques will be applied to solving the equations of the various phase-field models throughout the book and their detailed descriptions and implementations are given in relevant chapters. There are numerous textbooks also available on the subjects, of which some of them are listed in the references.

Modeling of Late Blooming Phases and Precipitation Kinetics in Aging Reactor Pressure Vessel (RPV) Steels

Finite difference algorithms offer a more direct approach to the numerical solution of partial differential equations than any other method. Finite difference algorithms are based on the replacement of each derivative by a difference quotient. Finite difference algorithms are simple to code, economic to compute, and easy to parallelize for the distributed computing environments. However, they also have disadvantages in terms of accuracy and imposing complex boundary conditions. For better understanding of the method, the solution of one-dimensional transient heat conduction is given as an example together with the source code in this section.