models of plastic deformation are extensively used in engineering practice, their range of application is limited by the underlying database. The reliability of continuum plasticity descriptions is dependent on the accuracy and range of available experimental data. Under complex loading situations, however, the database is often hard to establish. Moreover, the lack of a characteristic length scale in continuum plasticity makes it difficult to predict the occurrence of critical localized deformation zones. In small volumes, or in situations where submicrometer reso­lution of material deformation is required (e. g., in thin films, micropillars, near dislocations, grain boundaries, etc.), the use of continuum plasticity models would be questionable.

Although homogenization methods have played a significant role in determining the elastic properties of new materials from their constituents (e. g., composite materials), the same methods have failed to describe plasticity. It is widely appreciated that plastic strain is fundamentally heterogenous, displaying high strains concentrated in small material volumes, with virtually undeformed regions in between. Experimental obser­vations consistently show that plastic deformation is heterogeneous at all length scales. Depending on the deformation mode, heterogeneous dislocation struc­tures appear with definitive wavelengths. A satisfactory

description of realistic dislocation patterning and strain localization has been rather elusive. Attempts aimed at this question have been based on statistical mechanics, reaction-diffusion dynamics, or the theory of phase transitions. Much of the effort has aimed at clarifying the fundamental origins of inhomogeneous plastic deformation. On the other hand, engineering descrip­tions of plasticity have relied on experimentally ver­ified constitutive equations.

Planar dislocation arrays are formed under mono­tonic stress deformation conditions, and are com­posed of parallel sets of dislocation dipoles. While persistent slip bands (PSBs) are found to be aligned in planes with their normal parallel to the direction of the critical resolved shear stress (CRSS), planar arrays are aligned in the perpendicular direction. Dislocation cell structures, on the other hand, are honeycomb configurations in which the walls have high dislocation density, while the cell interiors have low dislocation density. Cells can be formed under both monotonic and cyclic deformation conditions. However, dislocation cells under cyclic deformation tend to appear after many cycles. Direct experimental observations of these structures have been reported for many materials.

Two main approaches have been advanced to model the mechanical behavior in this meso length scale. The first is based on statistical mechanics meth­ods. In this approach, evolution equations for statisti­cal averages (and possibly for higher moments) are to be solved for a complete description of the deforma­tion problem. The main challenge in this regard is that, unlike the situation encountered in the develop­ment of the kinetic theory of gases, the topology of interacting dislocations within the system must be included. The second approach, commonly known as dislocation dynamics (DD), was initially motivated by the need to understand the origins of hetero­geneous plasticity and pattern formation. An early variant of this approach (the cellular automata) was first developed by Lepinoux and Kubin,1 and this was followed by the proposal of DD by Ghoniem and Amodeo.2-4

In these early efforts, dislocation ensembles were modeled as infinitely long and straight in an isotro­pic, infinite elastic medium. The method was further expanded by a number of researchers,5 with appli­cations demonstrating simplified features of defor­mation microstructure. DD has now become an important computer simulation tool for the descrip­tion of plastic deformation at the micro — and meso — scales (i. e., the size range of a fraction of a micrometer to tens of micrometers). The method is based on a hierarchy of approximations that enable the solution of relevant problems with today’s computational resources. In its early versions, the collective behav­ior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations.5 Recently, sev­eral research groups have extended the DD method­ology to the more physical, yet considerably more complex case of three-dimensional (3D) simulation. The method can be traced back to the concepts of internal stress fields and configurational forces. The more recent development of 3D lattice DD by Kubin and coworkers6-8 has resulted in greater confidence in the ability of DD to simulate more complex defor­mation microstructures. More rigorous formulations of 3D DD have contributed to its rapid development and applications in many systems.9-15

Many experimental observations have shown that neutron irradiation of metals and alloys at temper­atures below recovery stage V causes a substantial increase in the upper yield stress (radiation hardening) and, beyond a certain dose level, induces a yield drop and plastic instability (see Chapter 1.03, Radiation — Induced Effects on Microstructure and Chapter 1.04, Effect of Radiation on Strength and Ductility of Metals and Alloys). Furthermore, the postdeformation microstructure of a specimen showing an upper yield point has demonstrated two significant features. First, the onset of plastic deformation is generally found to coincide with the formation of‘cleared’ channels, where practically all plastic deformation takes place. The sec­ond feature refers to the fact that the material volume in between cleared channels remains almost undeformed (i. e., no new dislocations are generated during deforma­tion). In other words, the initiation of plastic defor­mation in these irradiated materials occurs in a very localized fashion. This specific type ofplastic flow local­ization is considered to be one of many possible plastic instabilities in both irradiated and unirradiated materials.16,17

A theory of radiation hardening was proposed by Seeger18 in terms of dislocation interaction with radiation-induced obstacles (referred to as depleted zones). Subsequently, Foreman19 performed computer simulations of loop hardening, in which the elastic interaction between the dislocation and loops was neglected. The model is based on Orowan’s mecha­nism, which assumes that the obstacles are indestructi­ble. In this view of matrix hardening, the stress necessary to overcome localized interaction barriers leads to the increase in the yield strength, while long-range elastic interactions are completely ignored. These short-range dislocation-barrier interaction models lead to an estimate of the increase of the CRSS of the form: At = a mb/l, where a is a numerical constant represent­ing obstacle strength, m the shear modulus, b the mag­nitude of the Burgers vector, and l is the average interobstacle distance. Kroupa,20 on the other hand, viewed hardening as a result of the long-range elastic interaction between slip dislocations and prismatic loops. In their model of friction hardening, the force necessary to move a rigid, straight dislocation on its glide plane past a prismatic loop was estimated. In these two classes of hardening models, all dislocation sources are assumed to be simultaneously activated at the yield point, and plastic deformation is assumed to be homogeneous throughout the material volume. Thus, they do not address the physics of plastic flow localization. Singh et al. introduced the concept of ‘stand-off distance’ for the decoration of dislocations with small self-interstitial atom (SIA) loops, and pro­posed the ‘cascade-induced source hardening’ (CISH) model, in analogy with Cottrell atmospheres.

The central question of the formation of dislocation decoration was treated analytically by Trinkaus etal23 Subsequent detailed elasticity calculations showed that glissile defect clusters that approach dislocation cores within a stand-off distance are absorbed, while clusters can accumulate just outside this distance. Trinkaus etal. concluded that dislocation decoration is a consequence of defect cluster mobility and trapping in the stress field of grown-in dislocations. The CISH model was used to calculate the stress necessary to free decorated dislocations from the atmosphere ofloops around them (upper yield, followed by yield drop), so that these freed dislocations can act as dislocation sources. The CRSS increase in irradiated Cu was shown by Singh et al21 to be given by: At ~ 0.1m(b/l)(d/y)2, where m is the shear modulus, and b, l, d, andy are the Burgers vector, interdefect distance, defect diameter, and stand­off distance, respectively. Assuming that y ~ d, and l~ 35b, they estimated At/m ~ 5 x 10~3. The phe­nomenon of yield drop was proposed to result from the unpinning of grown-in dislocations, decorated with small clusters or loops of SIAs. Neutron irradiation of pure copper at 320 K leads to an increase in the upper yield stress and causes a prominent yield drop and the initiation of plastic flow localization.

In this chapter, we first present a description of the mathematical and computational foundations of DD. We then assess the physical mechanisms, that are responsible for the initiation of plastic insta­bility in irradiated face-centered cubic (fcc) metals by detailed numerical simulations of the interaction between dislocations and radiation-induced defect clusters. Two distinct problems that are believed to cause the onset of plastic instability are addressed in the present study. First, we aim to determine the mechanisms of dislocation unlocking from defect cluster atmospheres as a result of the long-range elastic interaction between dislocations and sessile prismatic interstitial clusters situated just outside the stand-off distance. The main new feature of this analysis is that dislocation deformation is explicitly considered during its interaction with SIA clusters. Second, we investigate the mechanisms of structural softening in flow channels as a consequence of dis­location interaction with stacking fault tetrahedra (SFTs). Based on these numerical simulations, a new mechanism of channel formation is proposed, and the magnitude of radiation hardening is also computed. As an important application of DD simu­lations, we then present a model that describes the shift in the ductile-to-brittle transition temperature (DBTT) as a result of neutron irradiation, and com­pare this physically based model to experimental data. Finally, we discuss the future outlook for DD simulations and the role they may play in under­standing radiation effects on the mechanical proper­ties of structural materials.