Mechanics of Dislocation Fields

By Claude Fressengeas

Accompanying the present trend of engineering systems aimed at size reduction and design at microscopic/nanoscopic length scales, Mechanics of Dislocation Fields describes the self-organization of dislocation ensembles at small length scales and its consequences on the overall mechanical behavior of crystalline bodies.

The account of the fundamental interactions between the dislocations and other microscopic crystal defects is based on the use of smooth field quantities and powerful tools from the mathematical theory of partial differential equations. The resulting theory is able to describe the emergence of dislocation microstructures and their evolution along complex loading paths. Scale transitions are performed between the properties of the dislocation ensembles and the mechanical behavior of the body.

Several variants of this overall scheme are examined which focus on dislocation cores, electromechanical interactions of dislocations with electric charges in dielectric materials, the intermittency and scale-invariance of dislocation activity, grain-to-grain interactions in polycrystals, size effects on mechanical behavior and path dependence of strain hardening.

• Language: English
• Publisher: Wiley
• Release date: Sep 25, 2017
• ISBN: 9781118578186

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Introduction

I.1. Background and motivation

The plasticity of crystalline materials is a dynamic phenomenon resulting from the motion under stress of crystal defects known as dislocations. Such a statement is grounded on numerous convincing observations, and it is widely accepted by the scientific community for materials having a sufficient number of independent slip systems and, in polycrystals, if grain size is sufficiently large to allow dislocation glide. Nevertheless, the conventional theories of plasticity have been using macroscopic variables whose definition does not involve the notion of dislocation. This paradoxical situation arises from the enormous range covered by the length scales used in the description of plasticity, from the elementary lattice spacing in atomistic descriptions to the meter scale in engineering studies. It may indeed have seemed impossible to account for the astounding complexity of the dynamics of dislocation ensembles at microscale in describing the mechanical properties of engineering structures. Justifications offered for such a simplification have usually found their origin in perfect disorder assumptions. Namely, plastic strain has been regarded as resulting from a very large number of randomly distributed elementary dislocation glide events, showing no order whatsoever at any intermediate length scale and time scale. Hence, deriving the mechanical properties arising from the mutual interactions of dislocations simply requires averaging procedures on any space and time domain.

Dislocation climb and grain boundary mechanisms, such as grain boundary migration, grain boundary rotation and dislocation emission, also contribute to plasticity to some extent depending on the local temperature and strain rate, on the slip system availability, the grain size and orientation, the defects density and mobility, the loading history, etc. For example, when the grain size lies in the tens of nanometers range as in nanocrystalline metallic materials, the role of dislocation glide is limited and grain boundary-mediated plasticity prevails, because the fraction of matter directly affected by grain boundaries becomes very large. Similarly, the lack of independent slip systems hampers the role of dislocation glide. According to the von Mises criterion [VON 28], at least five independent slip systems are needed for arbitrary plastic flow to occur homogeneously by dislocation glide, and this requirement is relaxed to four slip systems if the flow is inhomogeneous [HUT 83]. For instance, dislocation glide is restricted to only three independent slip systems in orthorhombic olivine, by far the most abundant (about 60–70%) and the weakest mineral in Earth’s upper mantle under a wide range of thermo-mechanical conditions. Therefore, olivine aggregates do not fulfill Hutchinson’s relaxed criterion, and additional plasticity mechanisms are needed to accommodate arbitrary deformation of the upper mantle [COR 14].

A straightforward jump by simple averaging from microscopic to macroscopic scale has long been the prevailing idea in mechanical sciences as well as in the materials science community when dislocation glide is predominant. This point of view may be justified, for example in bcc metals at low temperature, where the motion of dislocations is subject to large lattice friction. It reaches its limits when the elastic interactions between dislocations become the order of the interactions with other obstacles to their motion (lattice friction, solute atmospheres, precipitates, etc.). Since dislocation densities commonly increase during material loading, such a situation is met sooner or later as plastic strain increases. The field of elastic interactions between dislocations then becomes able to generate collective behavior and self-organized phenomena in the form of dislocation patterns emerging through ordered spatio-temporal dynamic regimes, with characteristic length scales and time scales [KUB 02]. Numerous examples of dislocation patterns, involving dislocation-rich and dislocation-poor regions, are observed in optical or electronic microscopy. Such is the case of the dislocation walls formed in cyclic loading (see Figure I.1), of dislocation cells (Figure I.2) and localized slip bands on the surface of single crystals (Figure I.3), with characteristic length scales in the μ m range.

Figure I.1. Dislocation walls in Si single crystal cyclically loaded in tension - compression at high temperature [LEG 04]

Figure I.2. Optical micrography of giant dislocation cells after GaAs crystal growth. Note that the average cell size varies in inverse proportion to stress. Inset: dislocation cells through X-ray imaging, dark areas are the images of lattice distortion around dislocations [NEU 01]

Figure I.3. Slip lines on the surface of Cu30at%Zn single crystal strained in tension at 19.4% and 77K [ZAI 06]

Similar spatial structures can also be inferred from the complex temporal behavior inherent to deformation curves in certain metallic alloys (Portevin-Le Chatelier effect, Lüders bands, etc.) [KUB 02]. In such conditions, the simple averaging procedures alluded to above are no longer justified, and the conventional theories of elastoplasticity are unable to account for the emerging patterns because they lack the relevant internal length scales.

Although time intermittency of plasticity was described as early as 1932 in Zn single crystals [BEC 32], the prevailing interpretation in the material science literature has also been that intermittent fluctuations add at random to a smooth net response in time, when averaging over sufficiently large time scales. This is again consistent with an assumption of perfect disorder of the plastic activity. A fundamentally different understanding emerged during the last few decades when statistical analysis of these fluctuations became available, that of a scale-invariant phenomenon characterized by power law distributions of fluctuation size, and correlations in space and time [BRI 08, DIM 06, WEI 97, MIG 01, WEI 07]. Thus, simple averaging procedures in time have again been dismissed and elasto-plastic theories able to account for such correlations have been promoted.

The conventional elastic constitutive relationships are referred to as local because they relate the stresses and elastic strains at the same point in the body. Such relationships have been found insufficient to account for the emergence of self-organization phenomena at intermediate length scales, because the solutions they induce to boundary value problems are scale independent. The fundamental reason for scale independence is, as suggested above, the lack of an internal physical length scale to be compared with the body’s dimensions. As opposed to local relationships, nonlocal elastic constitutive laws link the stresses at a given point in the body to the elastic strains in a neighborhood of this point. The extent of the neighborhood provides (or limits) the characteristic length scale of the elastic response of the body. Convolution integrals may be used to further formalize nonlocality [ERI 02], but a first approach has consisted of simply using strain gradients obtained from Taylor expansions of these integrals, and introducing the necessary length scales in a phenomenological way into the constitutive equations [AIF 84, FLE 94, FOR 97, NIX 98]. Such approaches are usually referred to as strain gradient theories of elastoplasticity. They may be useful in the characterization of the emerging patterns, but the identification of the involved length scales may sometimes raise difficulties. The notion that appropriate, physically based ingredients for a dynamic elasto-plastic description of the emerging patterns could be the dislocation density fields is quite recent [ACH 01, TEO 70], although these measures of crystal distortion incompatibility had been defined and used much earlier in elasto-static calculations [KOS 79, KRÖ 58, KRÖ 80, MUR 63, NYE 53]. Being areal renditions of a vectorial closure defect along a closed path integral, namely the Burgers vector obtained in integrating the plastic distortion along the Burgers circuit, dislocation densities are scale-dependent quantities. Hence, when used in the solution of boundary value problems, they induce a characteristic ratio between the resolution length scale adopted for their introduction and the size of the envisioned body. Clearly, this resolution length scale has to be much smaller than the size of the crystal defect pattern to be described in order to obtain accurate results on the spatio-temporal dynamics of the latter. However, there is no mandatory rule, and the choice of the resolution length scale depends on the accuracy demanded from the description. Hence, a phenomenon deemed non-local in a fine-scale solution scheme may well be seen as local when the scale of resolution is vastly enlarged. Thus, the dislocation density-based framework is intrinsically nonlocal, but the resolution length must be properly chosen, depending on the problem at hand. In addition, nonlocality of the framework cannot be substituted for nonlocal material behavior.

Further interest in using continuous dislocation densities for elasto-plastic modeling arises from the dynamic framework that derives from the conservation of the Burgers vector across arbitrary patches during dislocation motion through the body. Indeed, the plastic distortion rate can be construed as transport of dislocation densities [KRÖ 58, MUR 63]. Being unquestionable from a kinematic point of view, the transport framework provides a natural basis for the description of plasticity through dislocation motion. With coarse-graining issues to be clarified in the following, this feature allows for the description of plasticity to be shifted from overall shear strain rates not accounting for the existence of its elementary carriers (the dislocations) to plastic distortion rates documented with dislocation densities and velocities. Complemented by admissible dislocation mobility laws and elastic constitutive laws, as well as balance equations and boundary/initial conditions, the transport scheme enables a well-posed set of partial differential equations to be formulated for the solution of boundary value problems, where the unknown fields are the displacement and dislocation density fields. Being of hyperbolic character, the transport equations confer propagative properties to the predicted plastic activity, in agreement with recent experimental observations [FRE 09]. Furthermore, the stability of their numerical solutions through finite element approximations or fast Fourier transform methods requires specific algorithms, as detailed in [DJA 15, ROY 05, VAR 06].

I.2. Objectives

Due to the nature of the set of partial differential equations alluded to above, with unknown fields comprising both the dislocation density and displacement fields, the primary objective assigned to the dynamic theory of dislocation fields is to describe the emergence of inhomogeneous dislocation distributions at some mesoscopic (intermediate) length scale and time scale, and to explore the consequences of their presence and evolutions on the mechanical behavior of the body along a given loading path. In this process, scale transitions are clearly performed between the micro-scale properties of the dislocation ensembles and the macro-scale mechanical properties of the body on the one hand, and between the fine time scales used for investigating dislocation motion and the averaged engineering time scales on the other. Several variants of this overall objective will be examined in this book, with specific characteristics depending on the scale of the envisioned dislocation ensemble, and whether it pertains to single crystals or polycrystals.

– At the smallest scale, two fundamental issues regarding dislocation cores in a crystal lattice will be raised from the standpoint of the mechanical theory of dislocation fields, the stability of their equilibrium configuration under no load, and the Peierls stress, i.e. the applied stress needed to set the dislocation cores into motion. The Peierls-Nabarro model [PEI 40, NAB 47] was the first attempt at describing the core of an edge dislocation as a continuous density distribution, whose integration over the core area yields the Burgers vector. An equilibrium spatial distribution of the dislocation density field was found when the shear stress field arising from the elastic interactions in the dislocation core, which tends to spread the core, is counterbalanced by a restoring stress field opposing dislocation core expansion. In the original model, the restoring stresses originate from a complementary misfit energy reflecting the resistance of the crystal to shear and originally taken as a sinusoidal function of the misfit, thus conferring non-convex character to the total free energy. The Peierls stress vanishes in this model because the total free energy is invariant in a translation of the dislocation. The present framework will allow for a dynamic elasto-plastic analysis of the stability of the equilibrium state and of the existence of a Peierls