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Dislocation structures in a single crystal microcantilever beam subjected to bending at the end of the analysis ( θ = 5%). ( a ) t = 0 . 01 ns. ( b ) t = 0 . 5 ns. ( c ) t = 0 . 5 ns and V cut − off = 20 m s − 1 . Solid and open triangles stand for dislocations of different signs.
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The effect of the integration time step and the introduction of a cut-off velocity for the dislocation motion was analysed in discrete dislocation dynamics (DD) simulations of a single crystal microbeam. Two loading modes, bending and uniaxial tension, were examined. It was found that a longer integration time step led to a progressive increment of...
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Context 1
... addition, dislocations were able to jump over the neutral axis of the beam (in which the applied stresses are low and the driving force for dislocation motion decreased) as a result of the long displacements in each time step, and they reached the upper-right region of the beam. This behaviour was unusual in the simulations carried out at t = 0.01 ns ( figure 3(a)). These problems, associated with the long jumps of the dislocations when long integration time steps were used, were cleverly solved by Cleveringa et al [7] by introducing a cut-off velocity of 20 m s −1 for the dislocation motion. ...
Context 2
... curves show that the cut-off velocity did not modify significantly the M-θ curve, which was very close to the one obtained with t = 0.01 ns and smoother than that computed with t = 0.5 ns without any limit for the dislocation velocity. Moreover, the dislocation structure at the end of the analysis, shown in figure 3(c), was very similar to that obtained with t = 0.01 ns. Obviously, the cut-off velocity reduced the maximum dislocation displacement in each time step by a factor of ≈ 20 and the position of the dislocations within the pile-ups were better resolved. ...
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A new formulation of the elastic boundary value problem of dislocations in bounded crystals is developed. This formulation is based on the ansatz that the stress field of dislocations in bounded domains can be constructed as the sum of a contribution corresponding to the classical infinite-domain solution plus a correction that is determined here f...
Citations
... This setting is consistent with that used in . The dislocation gliding away from free surfaces is similar to the cases discussed above, that is, a dislocation is deleted from the model and a permanent displacement is set along the slip plane (Segurado et al., 2007). ...
In Part II of this work we extend the method introduced in Part I to consider dislocation dynamic evolution through dislocations nucleation, glide, pile-up, and annihilation. The SP DDD-PD scheme is employed to investigate uniaxial tension in a single crystal and a polycrystal and verify its accuracy. The model is then used to simulate elastoplastic fracture by considering interactions between dislocations and crack growth. For Mode I elastoplastic fracture in a single crystal, we observe that the crack path is “attracted” towards regions of high density of gliding dislocations, leading to an undulating crack paths, as observed in experiments but never replicated by continuum-level computational models before. Tests on different sample sizes show how the proximity of constraints to the crack tip can lead to plastic hardening. Ductile-to-brittle transition happens naturally in this model when the crack, under Mode I displacement-controlled loading, approaches a free edge. A new way to calibrate the critical bond strain based on the material toughness or fracture energy is proposed. The present SP DDD-PD scheme can be used to investigate complicated elastoplastic fracture problems in which the interaction between dislocation motion and damage is critical.
... The seminal findings have included elucidating the crystal size effects [42][43][44][45][46][47][48], quantifying irreversibility induced surface roughness during cyclic loading as a precursor to crack initiation [49][50][51][52][53][54], and identifying emergent behavior by quantifying the interactions of dislocation junctions in the material hardening response [55]. Due to the inherent length-scale, DDD simulations have been compared to various forms of small-scale testing, including thin films [56][57][58][59], micropillars [60][61][62], micro-cantilever beam [63][64][65], and nanoindentation [66]. As shown in Fig. 3, the slip behavior and surface steps are captured from DDD for a micropillar [67]. ...
This paper reviews recent studies, that not only includes both experiments and modeling components, but celebrates a close coupling between these techniques, in order to provide insights into the plasticity and failure of polycrystalline metals. Examples are provided of studies across multiple-scales, including, but not limited to, density functional theory combined with atom probe tomography, molecular dynamics combined with in situ transmission electron miscopy, discrete dislocation dynamics combined with nanopillars experiments, crystal plasticity combined with digital image correlation, and crystal plasticity combined with in situ high energy X-ray diffraction. The close synergy between in situ experiments and modeling provides new opportunities for model calibration, verification, and validation, by providing direct means of comparison, thus removing aspects of epistemic uncertainty in the approach. Further, data fusion between in situ experimental and model-based data, along with data driven approaches, provides a paradigm shift for determining the emergent behavior of deformation and failure, which is the foundation that underpins the mechanical behavior of polycrystalline materials.
... The effect of introducing a cut off velocity was discussed in e.g. Ref. [44]. Finite deformations effects, as studied in Ref. [45], are neglected in this study; a small strain assumption has been made. ...
Planar discrete dislocation plasticity (DDP) calculations that simulate thin single crystal films bonded to a rigid substrate indented by a rigid wedge are performed for different values of film thickness and dislocation source density. As in prior studies, an indentation size effect (ISE) is observed when indentation depth is sufficiently small relative to the film thickness. The dependence of the ISE on dislocation source density is quantified in this study, and a modified form of the scaling law for the dependence of hardness on indentation depth, first derived by Nix and Gao, is proposed, which is valid over the entire range of indentation depths and correlates the length scale parameter with the average dislocation source spacing. Nano-indentation experimental data from the literature are fitted using this formula, which further verifies the proposed scaling of indentation pressure on dislocation source density.
... [56,57]. The introduction of a cut-off velocity aims to improve the simulation efficiency without inducing side effects, as described in [58]. Annihilation of a pair of dislocations of opposite sign on the same slip plane occurs when the distance between them is less than the critical annihilation distance L e = 6b. ...
... The inclusion of schemes for capturing and correctly modelling the dislocation flux at the DDP/CPFE interface (following e.g. [58]) will be the subject of future investigations. Further efforts will also focus on the use of the proposed technique to develop a physically-based relationship to explain the transition in hardness of thin films on substrates across the scales, i.e. dislocation dominated size effects at smaller indentation depths transitioning to the layer-dominated and substrate-dominated hardness observed at larger indentation depths, as described in previous studies [20]; nano-sliding and fretting problems will also be modelled using this method in a forthcoming study. ...
A method of concurrent coupling of planar discrete dislocation plasticity (DDP) and a crystal plasticity finite element (CPFE) method was devised for simulating plastic deformation in large polycrystals with discrete dislocation resolution in a single grain or cluster of grains for computational efficiency; computation time using the coupling method can be reduced by an order of magnitude compared to DDP. The method is based on an iterative scheme initiated by a sub-model calculation, which ensures displacement and traction compatibility at all nodes at the interface between the DDP and CPFE domains. The proposed coupling approach is demonstrated using two plane strain problems: (i) uniaxial tension of a bi-crystal film and (ii) indentation of a thin film on a substrate. The latter was also used to demonstrate that the rigid substrate assumption used in earlier DDP studies is inadequate for indentation depths that are large compared to the film thickness, i.e. the effect of the plastic substrate modelled using CPFE becomes important. The coupling method can be used to study a wider range of indentation depths than previously possible using DDP alone, without sacrificing the indentation size effect regime captured by DDP. The method is general and can be applied to any problem where finer resolution of dislocation mediated plasticity is required to study the mechanical response of polycrystalline materials, e.g. to capture size effects locally within a larger elastic/plastic boundary value problem.
... It is useful to obtain estimates of these two time-scales in a system where the characteristic microstructural size, say d (e.g. grain size or the average TP source spacing), is in the submicron to micron regime. As far as t nuc is concerned, there seems to be no quantitative data available from experiments, but MD and discrete dislocation dynamics simulations indicate that t nuc scales inversely with applied strain rate (Van der Giessen and Needleman, 1999;Segurado et al., 2007;Zhu et al., 2008;Benzerga, 2008) so that t nuc $ 10 1 À10 À 3 s in the quasi-static loading regime. Using Fig. 2 as a reference, if a TP were to nucleate in the immediate proximity of A, then the time required for it to glide at an average velocity v to the point P located at a distance x from A would be t glide ¼ x=v . ...
... However, it should be noted that our results mostly deal with the very first stages of plasticity, where interactions among dislocations are not as relevant. Nevertheless, most of these limitations can also be applied to 2D DD, which has been a very useful tool to study size effects in plasticity [34][35][36][37][38][39]. ...
The mechanisms of growth of a circular void by plastic deformation were studied by means of molecular dynamics in two dimensions (2D). While previous molecular dynamics (MD) simulations in three dimensions (3D) have been limited to small voids (up to ≈10 nm in radius), this strategy allows us to study the behavior of voids of up to 100 nm in radius. MD simulations showed that plastic deformation was triggered by the nucleation of dislocations at the atomic steps of the void surface in the whole range of void sizes studied. The yield stress, defined as stress necessary to nucleate stable dislocations, decreased with temperature, but the void growth rate was not very sensitive to this parameter. Simulations under uniaxial tension, uniaxial deformation and biaxial deformation showed that the void growth rate increased very rapidly with multiaxiality but it did not depend on the initial void radius. These results were compared with previous 3D MD and 2D dislocation dynamics simulations to establish a map of mechanisms and size effects for plastic void growth in crystalline solids.
... We are particularly interested in analyzing the effect of the strip size on its mechanical response. The details of the DD code here employed are reported in [1,2] 4 . The SGCP model consists of an extension of the model developed in [5][6][7] (see also Gurtin et al [8,9]): it is of the higherorder and work-conjugate type (see, e.g., [10] and references therein) and it involves both energetic and dissipative higher-order terms; in the limit of vanishing material length scales, the SGCP model particularizes to the crystal viscoplasticity model of Peirce et al [11], within the small strain range 5 . ...
... The DD results have been obtained by means of the code developed as described in Segurado et al [1] (see also [2]). The method has been formerly established by Kubin and co-workers (see, e.g., [3]) and van der Giessen and Needleman [4]. ...
We aim at understanding the multislip behaviour of metals subject to irreversible deformations at small-scales. By focusing on the simple shear of a constrained single-crystal strip, we show that discrete Dislocation Dynamics (DD) simulations predict a strong latent hardening size effect, with smaller being stronger in the range [1.5 µm, 6 µm] for the strip height. We attempt to represent the DD pseudo-experimental results by developing a flow theory of Strain Gradient Crystal Plasticity (SGCP), involving both energetic and dissipative higher-order terms and, as a main novelty, a strain gradient extension of the conventional latent hardening. In order to discuss the capability of the SGCP theory proposed, we implement it into a Finite Element (FE) code and set its material parameters on the basis of the DD results. The SGCP FE code is specifically developed for the boundary value problem under study so that we can implement a fully implicit (Backward Euler) consistent algorithm. Special emphasis is placed on the discussion of the role of the material length scales involved in the SGCP model, from both the mechanical and numerical points of view.
... In most DD simulations a maximum time step t = 5 × 10 −10 s has been used along with a maximum dislocation velocity 20 m s −1 [24,30,31] along with an additional underrelaxation scheme. Segurado et al [32] have investigated the effects of the velocity cut-off and the time step showing that the macroscopic uniaxial tensile response and the resulting dislocation structures can be significantly influenced. Furthermore, we have found that when the plastic flow stress is controlled mainly by the obstacles with ensuing pileups, the time step, velocity cut-off and underrelaxation can unduly influence the dislocation unpinning from obstacles. ...
The application of discrete dislocation (DD) dynamics methods to study materials with realistic yield stresses and realistic cohesive strengths requires new algorithms. Here, limitations of the standard algorithms are discussed, and then new algorithms to overcome these limitations are presented and their successes demonstrated by example. With these new methods, the stability, accuracy and robustness of the DD methodology for the study of deformation and fracture is significantly improved.
... At a given stage of loading, the stress and strain fields in the crystal are obtained by the superposition of the two fields, the first one given by the sum of those induced by the individual dislocations in the current configuration and the second one, which corrects for the actual boundary conditions. This term is computed by solving a linear elastic boundary value problem using the finite element method with the appropriate boundary conditions, as detailed in Van der Giessen and Needleman (1995) and Segurado et al. (2007). ...
... Then, the velocities of the dislocations are obtained from the corresponding resolved shear stress, as given by (2) and the new positions are computed from these velocities. More details about the numerical implementation of the model can be found in Segurado et al. (2007) and Romero et al. (2008). It should be noticed that the current version of the program has been parallelized to improve performance in multiprocessor shared-memory computers. ...
... The time step in the Euler forward time-integration algorithm was constant and equal to 0.05 ns and the maximum velocity of the dislocations was limited to 100 m/s. The values of these two numerical parameters have been validated in previous studies (Segurado et al., 2007). ...
The micromechanisms of plastic deformation and void growth were analyzed using discrete dislocation dynamics in an isolated FCC single crystal deformed in-plane strain in the plane. Three different stress states (uniaxial tension, uniaxial deformation and biaxial deformation) were considered for crystals oriented in different directions and with a different number of active slip systems. It was found that strain hardening and void growth rates depended on lattice orientation in uniaxial tension because of anisotropic stress state. Crystal orientation did not influence, however, hardening and void growth when the crystals were loaded under uniaxial or biaxial deformation because the stress state was more homogeneous, although both (hardening and void growth rates) were much higher than under uniaxial tension. In addition, the number of active slip systems did not substantially modify the mechanical behavior and the void growth rate if plastic deformation along the available slip systems was compatible with overall crystal deformation prescribed by the boundary conditions. Otherwise, the incompatibility between plastic deformation and boundary conditions led to the development of large hydrostatic elastic stresses, which increased the strain hardening rate and reduced the void growth rate.
... The above conclusions suggest that the actual forms of defect energy and plastic potential are of fundamental importance, and a large effort should be put in order to obtain simple and effective forms of them, for instance, by extensively comparing the results of Discrete Dislocation simulations (e.g., [42,43]) with those obtained from a robust finite element implementation of strain gradient models (see, e.g., Ostien and Garikipati [44] for a discontinuous Galerkin implementation of Gurtin's model [1] without dissipative strain gradients). Such an implementation would also allow one to solve other interesting boundary value problems. ...
In the small deformation range, we consider and discuss the phenomenological (or isotropic) “higher-order” theory of strain gradient plasticity put forward in Section 12 of Gurtin [1], which includes the dissipation due to the plastic spin through a material parameter called χ. In fact, χ weighs the square of the plastic spin rate into the definition of an effective measure of plastic flow peculiar of the isotropic hardening function. Such a model has been identified by Bardella [2] as a good isotropic approximation of a crystal model to describe the multislip behaviour of a single grain, provided that χ be set as a specific function of other material parameters involved in the modelling, including the length scales. The main feature of the underlying gradient approach is the accounting for both dissipative and energetic strain gradient dependences, with related size effects. The dissipative strain gradients enter the model through the definition of the above mentioned effective measure of plastic flow, whereas the energetic strain gradients are involved in the modelling by defining the defect energy, a function of Nye’s dislocation density tensor added to the free energy to account for geometrically necessary dislocations (see, e.g., Gurtin [1]). By exploiting the deformation theory approximation, we apply the model to a simple boundary value problem so that we can discuss the effects of (a) the criterium derived by Bardella [2] for choosing χ and (b) non-quadratic forms of the defect energy. We show that both χ and the nonlinearity chosen for the defect energy strongly affect quality and magnitude of the energetic size effect which is possible to predict.
