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Regions in which the effect of the displacement discontinuities is localized. Three discontinuities are depicted.
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The standard strategy developed by Van der Giessen and Needleman (1995 Modelling Simul. Mater. Sci. Eng. 3 689) to simulate dislocation dynamics in two-dimensional finite domains was modified to account for the effect of dislocations leaving the crystal through a free surface in the case of arbitrary non-convex domains. The new approach incorporate...
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... In our model, the only stage considered is when the loop has filled all the pillar cross section. The displacement jump due to a full plane slip has been considered in the past using a strong discontinuity approach in the context of finite elements and discrete dislocation dynamics (Romero et al., 2008) . However, introducing this discontinuity in other numerical frameworks as FFT imply smoothing out the jump to make the model numerically tractable. ...
A stochastic discrete slip approach is proposed to model plastic deformation in submicron domains. The model is applied to the study of submicron pillar (D <= 1 um) compression experiments on tungsten (W), a prototypical metal for applications under extreme conditions. Slip events are geometrically resolved in the specimen and considered as eigenstrain fields producing a displacement jump across a slip plane. This novel method includes several aspects of utmost importance to small-scale plasticity, i.e. source truncation effects, surface nucleation effects, starvation effects, slip localization and an inherently stochastic response. Implementation on an FFT-spectral solver results in an efficient computational 3-D framework. Simulations of submicron W pillars (D <= 1 um) under compression show that the method is capable of capturing salient features of sub-micron scale plasticity. These include the natural competition between pre-existing dislocations and surface nucleation of new dislocations. Our results predict distinctive flow stress power-law dependence exponents as well as a size-dependence of the strain-rate sensitivity exponent. The results are thoroughly compared with experimental literature.
... The influence of the shape of the void on the nature of its growth can be determined using other advanced research methods. In [126], a research methodology was used to complement the discrete dislocation plasticity (DDP) method with calculations using XFEM [127,128]. Higher stress levels, strain hardening and void growth rates occurred under biaxial loading (compared to uniaxial loading). With a constant initial proportion of void volume, it was observed that elliptical-shaped voids showed larger surface areas Figure 15. ...
... The influence of the shape of the void on the nature of its growth can be determined using other advanced research methods. In [126], a research methodology was used to complement the discrete dislocation plasticity (DDP) method with calculations using XFEM [127,128]. Higher stress levels, strain hardening and void growth rates occurred under biaxial loading (compared to uniaxial loading). With a constant initial proportion of void volume, it was observed that elliptical-shaped voids showed larger surface areas relative to cylindrical voids. ...
The paper presents a literature review on the development of microvoids in metals, leading to ductile fracture associated with plastic deformation, without taking into account the cleavage mechanism. Particular emphasis was placed on the results of observations and experimental studies of the characteristics of the phenomenon itself, without in-depth analysis in the field of widely used FEM modelling. The mechanism of void development as a fracture mechanism is presented. Observations of the nucleation of voids in metals from the turn of the 1950s and 1960s to the present day were described. The nucleation mechanisms related to the defects of the crystal lattice as well as those resulting from the presence of second-phase particles were characterised. Observations of the growth and coalescence of voids were presented, along with the basic models of both phenomena. The modern research methods used to analyse changes in the microstructure of the material during plastic deformation are discussed. In summary, it was indicated that understanding the microstructural phenomena occurring in deformed material enables the engineering of the modelling of plastic fracture in metals.
... However, as the voided crystals are a non-convex domain and the slip planes are discontinuous at the void boundary, the slip step formed after a dislocation leaves the void boundary cannot be continuous i.e. throughout the material. Therefore, for non-convex domains, i.e. crystals containing voids, cracks, notches, the discontinuous slip step left behind by the leaving dislocation can be incorporated by coupling the methodology either with the method of embedded discontinuities (Romero et al., 2008;Simo et al., 1993) or the Extended Finite Element Method (XFEM) (Belytschko and Black, 1999;Liang et al., 2019). As an example, the difference in deformation between conventional DDP and XFEM-DDP formulation for some non-convex domains is illustrated in Fig. 1. ...
... Therefore, for non-convex domains, i.e. crystals containing voids, cracks, notches, the discontinuous slip step left behind by the leaving dislocation can be incorporated by coupling the methodology either with the method of embedded discontinuities (Romero et al., 2008;Simo et al., 1993) or the Extended Finite Element Method (XFEM) (Belytschko and Black, 1999;Liang et al., 2019). As an example, the difference in deformation between conventional DDP and XFEM-DDP formulation for some non-convex domains is illustrated in Fig. 1. Romero et al. (2008) proposed a new DDP formulation coupled with the method of embedded discontinuities to simulate the deformation of a voided crystal. Later, Segurado and Llorca (Segurado and Llorca, 2009;Segurado and LLorca, 2010) used the proposed methodology of Romero et al. (2008) to study the effect of void size and crystal lattice orientation on growth of voids with circular cross-section. ...
... Romero et al. (2008) proposed a new DDP formulation coupled with the method of embedded discontinuities to simulate the deformation of a voided crystal. Later, Segurado and Llorca (Segurado and Llorca, 2009;Segurado and LLorca, 2010) used the proposed methodology of Romero et al. (2008) to study the effect of void size and crystal lattice orientation on growth of voids with circular cross-section. However, in the authors' opinion, the Extended Finite Element method (X-FEM) is increasingly preferred over the method of embedded discontinuities because of the former's versatility, ease of implementation and availability in commercial software. ...
Voids are one of the many material defects present at the microscopic length scale. They are primarily responsible for the formation of cracks and hence contribute to ductile fracture. Circular voids tend to deform into elliptical voids just before their coalescence to form cracks. The principle aim of this study is to investigate the effect of void shape on the micro-mechanism of void growth by using Discrete Dislocation Plasticity simulations. For voided crystals, conventional DDP produces a continuous slip step throughout the material even if a dislocation escapes from a non-convex domain. To overcome this issue, the Extended Finite Element Method (XFEM) is used here to incorporate the displacement discontinuity. Different aspect ratios of elliptical voids are considered under uniaxial and biaxial deformation boundary conditions. The results suggest that voids having the largest surface area tend to have maximum growth rate as compared to void with lower surface area, i.e. “larger is faster”. Under biaxial loading, a higher magnitude of strain hardening, and void growth rate are observed as compared to uniaxial loading. The results also suggest that the orientation of slip planes as well as voids, affect the overall plastic behavior of the voided-ductile material. Furthermore, circular void tends to induce minimum growth rate but have the maximum strain hardening effect as compared to other void shapes under both loading conditions. The results of this study provide a deeper understanding of ductile fracture with applications in manufacturing industry, aerospace industry and in the design of nano/micro-electromechanical devices i.e. NEMS/MEMS.
... DDP has been used to study many dislocation-mediated problems, most typically finite-size problems where plane strain conditions apply and crossslip is not expected to be a major mechanism: size effects in plastic flow (Balint, Deshpande, Needleman, & Van der Giessen, 2006;Nicola, Van der Giessen, & Needleman, 2003), geometrical effects (Romero, Segurado, & Lorca, 2008), fracture mechanics (Deshpande, Needleman, & Van der Giessen, 2003;O'Day & Curtin, 2005; Van der Giessen, Deshpande, Cleveringa, & Needleman, 2001), crack growth (Cleveringa, Van der Giessen, & Needleman, 2000), fatigue (Deshpande, Needleman, & Van der Giessen, 2002), creep (Ayas, van Dommelen, & Deshpande, 2014), etc. Thus, despite the obvious shortcoming, DDP still offers valuable insight into many problems. ...
This chapter concerns with dynamic discrete dislocation plasticity (D3P), a two-dimensional method of discrete dislocation dynamics aimed at the study of plastic relaxation processes in crystalline materials subjected to weak shock loading. Traditionally, the study of plasticity under weak shock loading and high strain rate has been based on direct experimental measurement of the macroscopic response of the material. Using these data, well-known macroscopic constitutive laws and equations of state have been formulated. However, direct simulation of dislocations as the dynamic agents of plastic relaxation in those circumstances remains a challenge. In discrete dislocation dynamics (DDD) methods, in particular the two-dimensional discrete dislocation plasticity (DDP), the dislocations are modeled as discrete discontinuities in an elastic continuum. However, current DDP and DDD methods are unable to adequately simulate plastic relaxation because they treat dislocation motion quasi-statically, thus neglecting the time-dependent nature of the elastic fields and assuming that they instantaneously acquire the shape and magnitude predicted by elastostatics. This chapter reproduces the findings by Gurrutxaga-Lerma et al. (2013), who proved that under shock loading, this assumption leads to models that invariably break causality, introducing numerous artifacts that invalidate quasi-static simulation techniques. This chapter posits that these limitations can only be overcome with a fully time-dependent formulation of the elastic fields of dislocations. In this chapter, following the works of Markenscoff & Clifton (1981) and Gurrutxaga-Lerma et al. (2013), a truly dynamic formulation for the creation, annihilation, and nonuniform motion of straight edge dislocations is derived. These solutions extend the DDP framework to a fully elastodynamic formulation that has been called dynamic discrete dislocation plasticity (D3P). This chapter describes the several changes in paradigm with respect to DDP and DDD methods that D3P introduces, including the retardation effects in dislocation interactions and the effect of the dislocation’s past history. The chapter then builds an account of all the methodological aspects of D3P that have to be modified from DDP, including mobility laws, generation rules, etc. Finally, the chapter explores the applications D3P has to the study of plasticity under shock loading.
... Finite bodies experiencing smallscale plasticity must therefore account for the inhomogeneous distribution of image forces stemming from the placement and shape of free surfaces. Domain geometries can vary greatly, and previous efforts have examined a range of shapes from simple thin films [19, 20] and cylinders [21, 22] to complex, concave shapes such as microvoids [23, 24], undulating surfaces [25] and the intricate geometries of electronic devices [26]. Accounting for free surfaces is, in principle, similar to solving the discrete DD problem in bulk materials—a quasistatically evolving elastic boundary value problem (BVP) must be solved while the DD equations are integrated numerically in time. ...
Discrete dislocation dynamics (DD) approaches have proven useful in modeling the dynamics of large ensembles of dislocations. Continuing interest in finite body effects via image stresses has extended DD numerical approaches to improve the handling of surfaces. However, a physically accurate, yet computationally scalable, implementation has been elusive. This paper presents a new framework and implementation of a finite element-based discrete DD code that (1) treats arbitrarily shaped non-convex surfaces through image tractions, (2) allows for systematic refinement of the finite element mesh both in the bulk and on the surface and (3) provides a platform to scale to relatively larger and lengthier simulations. The approach is based on the capabilities of the Parallel Dislocation Simulator coupled through a distributed shared memory implementation for the calculation of large numbers of dislocation segments interacting with an independently large number of surface finite elements. Surface tracking approaches enable topological features at surfaces to be modeled. We verify the computed results via comparisons with analytical solutions for an infinite screw dislocation and prismatic loop near a surface and examine surface effects on a Frank–Read source. Convergence of the image force error with h- and p-refinement is shown to indicate the computational robustness. Additionally, through larger numerical experiments, we demonstrate the new capabilities in a three-dimensional elastic body of finite extent.
... 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.
... This necessarily limits the scope of the method to infinite straight edge dislocations so that plane strain conditions apply. The DDP method proposed by Van der Giessen & Needleman [9] enabled interesting questions associated with dislocation-mediated plasticity and failure to be investigated, such as size effects in plastic flow [10,11], geometrical effects [12], fracture mechanics [13][14][15], crack growth [16], fatigue [17] and more. Thus, despite the obvious shortcomings (e.g. ...
In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (10⁶ s⁻¹ or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of time-dependent plasticity during shock loading is outlined here and will be presented in detail elsewhere.
... 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]. ...
... Our identification based on the DD results for a strip with zero-inclined slip systems (see the black plots in figures 5 and 6) provides τ s /τ 0 ≈ 1.7, while h 0 /τ 0 ≈ 42. About the value q = 0.6 of the conventional latent hardening parameter, we note that a value larger than ≈1 makes the plastic slip γ (2) (x) numerically vanish. However, q does not have a big influence on the predictions of the SGCP model, in which γ (2) (x) mostly affects the results through its gradient. ...
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.
... The authors are aware of the limitations of equation (1.2), something that will be addressed in a future paper. effects [12], fracture mechanics [13][14][15], crack growth [16], fatigue [17] and more. Thus, despite the obvious shortcomings (e.g. ...
In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (10 6 s -1 or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of time-dependent plasticity during shock loading is outlined here and will be presented in detail elsewhere.
... The simulation domain consists of a periodic arrangement of through thickness cylindrical holes. Even though our model is 3D we choose this geometry to compare our results with previous 2D dislocation dynamics simulations [37]. We apply an increasing external stress under which dislocations nucleate and evolve following equation (3.4). ...
... The stress-strain responses of the three different configurations are depicted in figure 7(a). The stress-strain curve of the domain with the larger void radius has the smallest elastic slope and the lowest yield stress in very good agreement with the simulations in the work by Romero et al [37]. We also show the simulated elastic response as dashed lines for comparison. ...
We present a phase-field approach that couples the representation of material heterogeneities with discontinuities in the displacement field to describe defects and damage evolution. This method in contrast to discrete models does not require topological changes in the mesh representation reducing the complexity in the implementation of the numerical simulation. Both defects and material inhomogeneities are described in terms of phase fields and their evolution and interaction follows from a set of analogous equations delivering a unified theory that couples the response of heterogeneous materials with displacement discontinuities seamlessly. We show the effectiveness of the model by predicting dislocation structures in a 3D periodic array of voids in nickel single crystal and the nucleation and evolution of crazes in polymethyl methacrylate.
