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In the preceding chapters we have discussed several computational approaches focused on the structure and motion of single dislocations. Here we turn our attention to collective motion of many dislocations, which is what the method of dislocation dynamics (DD) was designed for. Typical length and time scales of DD simulations are on the order of microns and seconds, similar to in situ transmission electron microscopy (TEM) experiments where dislocations are observed to move in real time. In a way, DD simulations can be regarded as a computational counterpart of in situ TEM experiments. One very valuable aspect of such a “computational experiment” is that one has full control of the simulation conditions and access to the positions of all dislocation lines at any instant of time. Provided the dislocation model is realistic, DD simulations can offer important insights that help answer the fundamental questions in crystal plasticity, such as the origin of the complex dislocation patterns that emerge during plastic deformation and the relationship between microstructure, loading conditions and the mechanical strength of the crystal. So far, two approaches to dislocation dynamics simulations have emerged. In the line DD method to be discussed in this chapter, dislocations are represented as mathematical lines in an otherwise featureless host medium. An alternative approach is to rely on a continuous field of eigenstrains, in which regions of high strain gradients reveal the locations of the dislocation lines. This representation leads to the phase field DD approach, which will be discussed in Chapter 11. Line DD has certain similarities with the models discussed in the previous chapters, but, at the same time, is rather different from all of them. For example, the representation of dislocations by line segments in line DD method is similar to the kinetic Monte Carlo (kMC) model of Chapter 9. However, having to deal with multiple dislocations on large length and times scales necessitates a more economical treatment of dislocations in the line DD method. Thus, line DD usually relies on less detailed discretization of dislocation lines and treats dislocation motion as deterministic.

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.