The phase field method (PFM) can be used as an approach to dislocation dynamics simulations alternative to the line DD method discussed in Chapter 10. The degrees of freedom in PFM are continuous smooth fields occupying the entire simulation volume, and dislocations are identified with locations where the field values change rapidly. As we will see later, as an approach to dislocation dynamics simulations PFM holds several advantages. First, it is easier to implement into a computer code than a line DD model. In particular, the complex procedures for making topological changes (Section 10.4) are no longer necessary. Second, the implementation of PFM can take advantage of well-developed and efficient numerical methods for solving partial differential equations (PDEs). Another important merit of PFM is its applicability in a wide range of seemingly different situations. For example, it is possible to simulate the interaction and co-evolution of several types of material microstructures, such as dislocations and alloying impurities, within a unified model. PFM has become popular among physicists and materials scientists over the last 20 years, but as a numerical method it is not new. After all, it is all about solving PDEs on a grid. Numerical integration of PDEs is a vast and mature area of computational mathematics. A number of efficient methods have already been developed, such as the finite difference method [121], the finite element method [122], and spectral methods [123], all of which have been used in PFM simulations. The relatively new aspects of PFM are associated with the method’s formulation and applications, which are partly driven by the growing interest in understanding material microstructures. In Section 11.1, we begin with the general aspects of PFM demonstrated by two simple applications of the method not related to dislocations. Section 11.2 describes the elements required to adapt PFM to dislocation simulations. There we will briefly venture into the field of micromechanics and consider the concept of eigenstrain. The elastic energy of an arbitrary eigenstrain field is derived in Section 11.3. Section 11.4 discusses an example in which the PFM equations for dislocations are solved using the fast Fourier transform method.
Periodic Boundary Conditions for Dislocation Dynamics Simulations in Three Dimensions
