Nye Dislocation Density Tensor Research Articles

Resolving localized geometrically necessary dislocation densities in Al-Mg polycrystal via in situ EBSD

The distribution of geometrically necessary dislocation (GND) densities is critical to understanding the heterogeneous plastic deformation at intragranular scales in polycrystals. In this work, we performed an in situ electron backscatter diffraction (EBSD) measurement during the tensile test on a polycrystalline Al-Mg alloy. The EBSD patterns were processed through the integrated digital image correlation algorithm to enhance angular resolution. Based on the Nye dislocation density tensor, GND densities of 18 dislocation types in fcc crystals were resolved and mapped at macroscopic strains ranging from 5% to 16%. The evolution of GND distribution showed that GNDs were generated near grain boundaries at small strains and later localized at grain interiors along subgrain boundaries and slip bands at large strains. The inhomogeneous increase in GND densities of different dislocation types, representing dislocation substructures, along the subgrain boundaries was disclosed. Meanwhile, high GND densities were observed along the slip bands when a double noncoplanar slip system was activated inside the grains. The obstructed movement of dislocations by Lomer junctions explained the GND storage in different {111} slip planes. The existence of Lomer junctions was proven by Burgers circuit analysis with high-resolution transmission electron microscopy, which explained the increase in [101] screw dislocation density according to the dislocation reaction.

Atomistic to continuum mechanics description of crystal defects with dislocation density fields: Application to dislocations and grain boundaries

The atomic structure of crystal defects such as dislocations, grain or phase boundaries, control these defects’ properties: their mobility, ability to cross-slip, or solute segregation. These crystal defects can be conveniently studied by atomistic simulations and one then needs to transfer relevant information at the upper scale to model microstructures containing a large number of defects, e.g., a polycrystal. Here, we propose an atomistic to continuum mechanics crossover method that (i) represents the atomic structure of dislocations cores by an appropriate Nye dislocation density tensor field and (ii), captures quantitatively the short and long range mechanical fields of defects. For (i), we propose a modified and improved interpolation method based on the original work by Hartley and Mishin. For (ii), we use a field dislocation mechanics framework that rigorously calculates/evaluates the mechanical fields associated with any Nye dislocation density distribution. The transfer method relies on molecular static calculations using two energetic models — ab-initio for screw dislocation core simulations in tungsten, and EAM potential for low and large angle grain boundaries in copper. Our findings demonstrate the effectiveness of the proposed approach in reconstructing the Burgers vector, and continuous strain and rotation fields. The framework is further applied to analyze the elastic interactions between extrinsic edge dislocations and a low angle grain boundary in copper.

From electron tomography of dislocations to field dislocation mechanics: application to olivine

We propose a new procedure to extract information from electron tomography and use them as an input in a field dislocation mechanics. Dislocation electron tomography is an experimental technique that provides three-dimensional (3D) information on dislocation lines and Burgers vectors within a thin foil. The characterized 3D dislocation lines are used to construct the spatial distribution of the equivalent Nye dislocation density tensor. The model dislocation lattice incompatibility equation and stress balance equation are solved with a spectral code based on fast Fourier transform algorithms. As an output of the model, one obtains the 3D distribution of mechanical fields, such as strains, rotations, stresses, resolved shear stresses (RSSs) and energy, inside the material. To assess the potential of the method, we consider two regions from a previously compressed olivine sample. Our results reveal significant local variations in local stress fields and RSSs in various slip systems, which can impact the strong plastic anisotropy of olivine and the activation of different dislocation slip systems. It also evidences the built-up of kinematic hardening down to the nanometre scale.

Characterization of geometrically necessary dislocation evolution during creep of P91 steel using electron backscatter diffraction

In this study, the change of geometrically necessary dislocation (GND) has been investigated during the elevated temperature creep of P91 steel. Firstly, the crept microstructures of P91 steel at 873 K and 165 MPa were characterized by electron backscatter diffraction (EBSD). Then, the resultant GND distributions were analyzed in detail. Meanwhile, the GND density was quantitatively estimated by two methods of kernel average misorientation (KAM) and Nye's dislocation density tensor, respectively. The obtained results indicated that during creep of P91 steel, the values of GND density calculated by the KAM method are highly consistent with those of Nye's dislocation density tensor method. Besides, the GND density increases significantly during the initial creep stage and attains the maximum approximately at the transition from initial creep to steady creep. During the subsequent creep, it gradually decreases. In addition, it suggested that the internal stress predicted by the obtained GND density is consistent with previous experimental data. Accordingly, it further demonstrated that the present characterization of GND evolution offers a convenient tool for the microstructure degradation analysis of P91 steel during creep. • The microstructure of P91 steel during creep has been investigated by EBSD method. • The change of GND density was quantitatively characterized by two methods of KAM and Ney's dislocation density tensor, respectively. • The GND densities obtained by the above methods are highly consistent. • The GND densities vary closely with the creep stage, which mainly results from the interaction between dislocation and precipitates.

The dependency of work hardening on dislocation statistics in cold rolled 1050 aluminum alloy

In this contribution, the phenomenon of work hardening is studied as a function of the nature of dislocation evolution with change in the level of deformation for commercially available 1050 Al alloy. To investigate the nature of dislocations in the cold-rolled aluminum alloy, total dislocation density was calculated using the indentation technique as well as by implementing two numerical approaches. For estimation of a statistically representative value of geometrically necessary dislocations (GND), electron backscattering diffraction measurements were performed over a large area (~ 0.5 mm2). The GND density of deformed samples was quantified by the implementation of a modified kernel average misorientation technique as well as by using Nye's dislocation density tensor for corresponding lattice curvature. The values of statistically stored dislocations (SSD) are obtained based on the assessment of the total dislocation density and GND values after different straining levels. The study provides both qualitative and quantitative illustrations of the mechanism of dislocation multiplication as thickness reduction increases, thereby, increasing the hardness of the samples. The results obtained reveal that the hardening of rolled materials is majorly governed by the SSD density at lower deformation. However, as the deformation level approaches the value of ~0.4, the density of GNDs rises and its contribution becomes significantly accountable for the strain hardening of the material. On the other hand, it has been observed that at lower straining levels the generated GNDs are trapped at grain boundaries and have a high contribution towards forest dislocations but with an increase of strain, the GNDs have a tendency to contribute nearly equally to mobile and forest dislocations. The estimated stored energies for samples subjected to rolling reduction ranging between 5.3 and 76% tend to change between the 5.6 kJ/m3 and 343.3 kJ/m3.

Grain size dependence of polycrystalline plasticity modeling in cylindrical indentation

Grain boundary strengthening effect for polycrystalline copper is studied numerically using crystal plasticity in conjunction with cylindrical indentation simulations under the plane strain condition. In order to compare with an isotropic, heterogeneous continuum model a new constitutive relation is developed. This new nonlocal continuum model that encompasses the heterogeneity in yield strength based on the exponentiated Weibull function can predict the plastic properties of materials in the micron length scale. The spatial description of the deformation gradient two-point tensor is utilized to capture the intrinsic size effect in line with the subsequent deformation measures. Moreover, the total geometrically necessary dislocation density is obtained from the non-zero components of Nye dislocation density tensor. From the simulation, the relationship between the effective Green–Lagrange strain and effective stress measures is explained using the persistent long-range order and intermittent short-range order. The observation derived from the analogy between the cylindrical indentation and the progress in cylindrical voids describes how different average grain sizes of polycrystalline materials are compared with the behavior of isotropic materials. The trajectories of directions of both principal stretch and maximum shear strain explain that the internal stresses induced by cylindrical indentation either hinder or reinforce the dislocation flow, forming strain localization sporadically. The grain size dependence of polycrystalline modeling incorporates the Hall–Petch strengthening as well as the homogenization of anisotropic polycrystalline metal into the isotropic effective medium. This is a physically-based model that is used to model failure characterization in metals.

Modelling the cyclic torsion of polycrystalline micron-sized copper wires by distortion gradient plasticity

ABSTRACT In this contribution, we show that the distortion gradient plasticity recently proposed by our group, characterised by a higher-order plastic potential leading to reliable predictions under non-proportional loading, can predict experimental data of literature on the cyclic torsion of copper wires of diameter ranging from 18 to 42 . To reach our goal, we plug our recent constitutive proposal in a framework that we have previously developed for the torsion problem, which is based on the pivotal theory established in 2004 by Gurtin, relying on Nye's dislocation density tensor to describe size effects in micron-scale metal plasticity. We implement the new model in a finite element code and identify its parameters by resorting to the Coliny evolutionary algorithm within the software Dakota.

A potential for higher-order phenomenological strain gradient plasticity to predict reliable response under non-proportional loading.

We propose a plastic potential for higher-order (HO) phenomenological strain gradient plasticity (SGP), predicting reliable size-dependent response for general loading histories. By constructing the free energy density as a sum of quadratic plastic strain gradient contributions that each transitions into linear terms at different threshold values, we show that we can predict the expected micron-scale behaviour, including increase of strain hardening and strengthening-like behaviour with diminishing size. Furthermore, the anomalous behaviour predicted by most HO theories under non-proportional loading is avoided. Though we demonstrate our findings on the basis of Gurtin (Gurtin 2004 J. Mech. Phys. Solids 52, 2545-2568, doi:10.1016/j.jmps.2003.11.002) distortion gradient plasticity, adopting Nye's dislocation density tensor as primal HO variable, we expect our results to hold qualitatively for any HO SGP theory, including crystal plasticity.

Geometrically necessary dislocation density measurements at a grain boundary due to wedge indentation into an aluminum bicrystal

An aluminum bicrystal with a symmetric tilt Σ43 (3 3 5)[1 1 0] coincident site lattice grain boundary was deformed plastically via wedge indentation under conditions that led to a plane strain deformation state. Plastic deformation is induced into both crystals and the initially straight grain boundary developed a significant curvature. The resulting lattice rotation field was measured via Electron Backscatter Diffraction (EBSD). The Nye dislocation density tensor and the associated Geometrically Necessary Dislocation (GND) densities introduced by the plastic deformation were calculated. The grain boundary served as an impediment to plastic deformation as quantified through a smaller lattice rotation magnitude and smaller GND density magnitudes in one of the crystals. There is evidence that the lattice rotations in one grain brought a slip system in that grain into alignment with a slip system in the other grain, upon which the impediment to dislocation transmission across the grain boundary was reduced. This allowed the two slip systems to rotate together in tandem at later stages of the deformation. Finite element crystal plasticity simulations using classical constitutive hardening relationship capture the general features observed in the experiments.

Spatial distribution of the net Burgers vector density in a deformed single crystal

A two-dimensional deformation field on an indented single crystal, where the only non-zero lattice rotation occurs in the plane of deformation and only three effective in-plane slip systems are activated, is investigated both experimentally and numerically. Electron Backscatter Diffraction (EBSD) is utilized to probe the lattice rotation field on the sample. The lattice rotation field is utilized to calculate the two non-zero components of Nye's dislocation density tensor, which serves as a link between plastic and elastic deformation states. The enhanced accuracy of EBSD enabled measurements of the net Burgers vector density, and a new quantity β, which monitors the activity of slip systems in the deformed zone. The β-field is compared to the slip system activity obtained by analytical solution and also by crystal plasticity simulations. A qualitative comparison of the three methods confirms that the β-field obtained experimentally agrees with the slip system activity obtained analytically and by numerical methods.

Modelling the torsion of thin metal wires by distortion gradient plasticity

Under small strains and rotations, we apply a phenomenological higher-order theory of distortion gradient plasticity to the torsion problem, here assumed as a paradigmatic benchmark of small-scale plasticity. Peculiar of the studied theory, proposed about ten years ago by Morton E. Gurtin, is the constitutive inclusion of the plastic spin, affecting both the free energy and the dissipation. In particular, the part of the free energy, called the defect energy, which accounts for Geometrically Necessary Dislocations, is a function of Nye's dislocation density tensor, dependent on the plastic distortion, including the plastic spin. For the specific torsion problem, we implement this distortion gradient plasticity theory into a Finite Element (FE) code characterised by implicit (Backward Euler) time integration, numerically robust and accurate for both viscoplastic and rate-independent material responses. We show that, contrariwise to other higher-order theories of strain gradient plasticity (neglecting the plastic spin), the distortion gradient plasticity can predict some strengthening even if a quadratic defect energy is chosen. On the basis of the results of many FE analyses, concerned with (i) cyclic loading, (ii) switch in the higher-order boundary conditions during monotonic plastic loading, (iii) the use of non-quadratic defect energies, and (iv) the prediction of experimental data, we mainly show that (a) including the plastic spin contribution in a gradient plasticity theory is highly recommendable to model small-scale plasticity, (b) less-than-quadratic defect energies may help in describing the experimental results, but they may lead to anomalous cyclic behaviour, and (c) dissipative (unrecoverable) higher-order finite stresses are responsible for an unexpected mechanical response under non-proportional loading.

Size-dependent energy in crystal plasticity and continuum dislocation models.

In the light of recent progress in coarsening the discrete dislocation mechanics, we consider two questions relevant for the development of a mesoscale, size-dependent plasticity: (i) can the phenomenological expression for size-dependent energy, as quadratic form of Nye's dislocation density tensor, be justified from the point of view of dislocation mechanics and under what conditions? (ii)how can physical or phenomenological expressions for size-dependent energy be computed from dislocation mechanics in the general case of elastically anisotropic crystal? The analysis based on material and slip system symmetries implies the negative answer to the first question. However, the coarsening method developed in response to the second question, and based on the physical interpretation of the size-dependent energy as the coarsening error in dislocation interaction energy, introduces additional symmetries. The result is that the equivalence between the phenomenological and the physical expressions is possible, but only if the multiplicity of characteristic lengths associated with different slip systems, is sacrificed. Finally, we discuss the consequences of the assumption that a single length scale governs the plasticity of a crystal, and note that the plastic dissipation at interfaces has a strong dependence on the length scale embedded in the energy expression.

Estimations of bulk geometrically necessary dislocation density using high resolution EBSD

Characterizing the content of geometrically necessary dislocations (GNDs) in crystalline materials is crucial to understanding plasticity. Electron backscatter diffraction (EBSD) effectively recovers local crystal orientation, which is used to estimate the lattice distortion, components of the Nye dislocation density tensor (α), and subsequently the local bulk GND density of a material. This paper presents a complementary estimate of bulk GND density using measurements of local lattice curvature and strain gradients from more recent high resolution EBSD (HR-EBSD) methods. A continuum adaptation of classical equations for the distortion around a dislocation are developed and used to simulate random GND fields to validate the various available approximations of GND content.

Nonlocal continuum crystal plasticity with internal residual stresses

We derive a three-dimensional constitutive theory accounting for length-scale dependent internal residual stresses in crystalline materials that develop due to a non-homogeneous spatial distribution of the excess dislocation (edge and screw) density. The second-order internal stress tensor is derived using the Beltrami stress function tensor φ that is related to the Nye dislocation density tensor. The formulation is derived explicitly in a three-dimensional continuum setting for elastically isotropic materials. The internal stresses appear as additional resolved shear stresses in the crystallographic visco-plastic constitutive law for individual slip systems. Using this formulation, we investigate two boundary value problems involving single crystals under symmetric double slip. In the first problem, the response of a geometrically imperfect specimen subjected to monotonic and cyclic loading is investigated. The internal stresses affect the overall strengthening and hardening under monotonic loading, which is mediated by the severity of initial imperfections. Such imperfections are common in miniaturized specimens in the form of tapered surfaces, fillets, fabrication induced damage, etc., which may produce strong gradients in an otherwise nominally homogeneous loading condition. Under cyclic loading the asymmetry in the tensile and compressive strengths due to this internal stress is also strongly influenced by the degree of imperfection. In the second example, we consider simple shear of a single crystalline lamella from a layered specimen. The lamella exhibits strengthening with decreasing thickness and increasing lattice incompatibility with shearing direction. However, as the thickness to internal length-scale ratio becomes small the strengthening saturates due to the saturation of the internal stress. Finally, we present the extension of this approach for crystalline materials exhibiting elastic anisotropy, which essentially depends on the appropriate Green function within φ.

Experimental lower bounds on geometrically necessary dislocation density

A single nickel crystal is indented with a wedge indenter such that a two-dimensional deformation state with three effective plane strain slip systems is induced. The in-plane lattice rotation of the crystal lattice is measured with a three micrometer spatial resolution using Orientation Imaging Microscopy (OIM). All non-zero components of the Nye dislocation density tensor are calculated from the lattice rotation field. A rigorous analytical expression is derived for the lower bound of the total Geometrically Necessary Dislocation (GND) density. Existence and uniqueness of the lower bound are demonstrated, and the apportionment of the total GND density onto the effective individual slip systems is determined. The lower bound solution reduces to the exact solution under circumstances in which only one or two of the effective slip systems are known to have been activated. The results give insight into the active slip systems as well as the dislocation structures formed in the nickel crystal as a result of the wedge indentation.

Defect-induced incompatibility of elastic strains: Dislocations within the Landau theory of martensitic phase transformations

In dislocation-free martensites the components of the elastic strain tensor are constrained by the Saint-Venant compatibility condition which guarantees continuity of the body during external loading. However, in dislocated materials the plastic part of the distortion tensor introduces a displacement mismatch that is removed by elastic relaxation. The elastic strains are then no longer compatible in the sense of the Saint-Venant law and the ensuing incompatibility tensor is shown to be proportional to the gradients of the Nye dislocation density tensor. We demonstrate that the presence of this incompatibility gives rise to an additional long-range contribution in the inhomogeneous part of the Landau energy functional and to the corresponding stress fields. Competition among the local and long-range interactions results in frustration in the evolving order parameter (elastic) texture. We show how the Peach-Koehler forces and stress fields for any distribution of dislocations in arbitrarily anisotropic media can be calculated and employed in a Fokker-Planck dynamics for the dislocation density. This approach represents a self-consistent scheme that yields the evolutions of both the order parameter field and the continuous dislocation density. We illustrate our method by studying the effects of dislocations on microstructure, particularly twinned domain walls, in an Fe-Pd alloy undergoing a martensitic transformation.

A comparison between crystal and isotropic strain gradient plasticity theories with accent on the role of the plastic spin

In the small deformation range, we consider crystal and isotropic “higher-order” theories of strain gradient plasticity, in which two different types of size effects are accounted for: (i) that dissipative, entering the model through the definition of an effective measure of plastic deformation peculiar of the isotropic hardening function and (ii) that energetic, included by defining the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., [Gurtin, M.E., 2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]). In order to compare the two modellings, we recast both of them into a unified deformation theory framework and apply them to a simple boundary value problem for which we can exploit the Γ-convergence results of [Bardella, L., Giacomini, A., 2008. Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity. J. Mech. Phys. Solids 56 (9), 2906–2934], in which the crystal model is made isotropic by imposing that any direction be a possible slip system. We show that the isotropic modelling can satisfactorily approximate the behaviour described by the isotropic limit obtained from the crystal modelling if the former constitutively involves the plastic spin, as in the theory put forward in Section 12 of [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568]. The analysis suggests a criterium for choosing the material parameter governing the plastic spin dependence into the relevant Gurtin model.

Influence of material parameters and crystallography on the size effects describable by means of strain gradient plasticity

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ -convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.

Stress-free states of continuum dislocation fields: Rotations, grain boundaries, and the Nye dislocation density tensor

We derive general relations between grain boundaries, rotational deformations, and stress-free states for the mesoscale continuum Nye dislocation density tensor. Dislocations generally are associated with long-range stress fields. We provide the general form for dislocation density fields whose stress fields vanish. We explain that a grain boundary (a dislocation wall satisfying Frank's formula) has vanishing stress in the continuum limit. We show that the general stress-free state can be written explicitly as a (perhaps continuous) superposition of flat Frank walls. We show that the stress-free states are also naturally interpreted as configurations generated by a general spatially-dependent rotational deformation. Finally, we propose a least-squares definition for the spatially-dependent rotation field of a general (stressful) dislocation density field.

Mesoscale Theory of Grains and Cells: Crystal Plasticity and Coarsening

Crystals with spatial variations in their axes naturally evolve into cells or grains separated by sharp walls. At high temperatures, polycrystalline grains form from the melt and coarsen with time: the dislocations can both climb and glide. At low temperatures under shear the dislocations (which allow only glide) form into cell structures. We present here a mesoscale theory of dislocation motion. It provides a quantitative description of deformation and rotation, grounded in a microscopic order parameter field exhibiting the topologically conserved quantities. The topological current of the Nye dislocation density tensor is derived from a microscopic theory of glide driven by Peach-Koehler forces between dislocations using a simple closure approximation. The resulting theory is shown to form sharp dislocation walls in finite time, both with and without dislocation climb.