A Practical Meso-Scale Polycrystal Model to Predict Dislocation Densities and the Hall–Petch Effect (original) (raw)
Related papers
A dislocation density based constitutive model for the face centered cubic crystal structure has been implemented into a crystal-plasticity finite element framework and extended to consider the mechanical interaction between mobile dislocations and grain boundaries by the authors [Ma, A., Roters, F., Raabe, D., 2006a. A dislocation density based constitutive model for crystal-plasticity FEM including geometrically necessary dislocations. Acta Materialia 54, 2169–2179; Ma, A., Roters, F., Raabe, D., 2006b. On the consideration of interactions between dislocations and grain boundaries in crystalplasticity finite element modeling – theory, experiments, and simulations. Acta Materialia 54, 2181–2194]. The approach to model the grain boundary resistance against slip is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of internal interfaces. This energy barrier is derived from the assumption of thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems. In this study we focus on the influence of the one remaining model parameter which can be used to scale the obstacle strength of the grain boundary.
Dislocation impediment by the grain boundaries in polycrystals
Acta Mechanica, 2021
Thermodynamic dislocation theory incorporating dislocation impediment by the grain boundaries is developed to analyze the shear test of polycrystals. With a small set of physics-based material parameters, we are able to simulate the stress-strain curves for the load and its reversal, which are consistent with the experimental curves of Thuillier and Manach (Int J Plast 25:733-751, 2009). Representative distributions of plastic slip under load and its reversal are presented, and their evolution explains the extended length of the transition stage during load reversal.
International Journal of Plasticity, 2008
The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checkerboard pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall-Petch type relation is predicted with Hall-Petch exponents ranging from %0.3 to %1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall-Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall-Petch effect in the simulations.
Acta Materialia 54 (2006) 2181
"We suggest a dislocation based constitutive model to incorporate the mechanical interaction between mobile dislocations and grain boundaries into a crystal plasticity finite element framework. The approach is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of grain boundaries. The energy barrier is derived by using a geometrical model for thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems and is formulated as a vectorial conservation law. The new model is applied to the case of 50% (frictionless) simple shear deformation of Al bicrystals with either a small, medium, or large angle grain boundary parallel to the shear plane. The simulations are in excellent agreement with the experiments in terms of the von Mises equivalent strain distributions and textures. The study reveals that the incorporation of the misorientation alone is not sufficient to describe the influence of grain boundaries on polycrystal micro-mechanics. We observe three mechanisms which jointly entail pronounced local hardening in front of grain boundaries (and other interfaces) beyond the classical kinematic hardening effect which is automatically included in all crystal plasticity finite element models owing to the change in the Schmid factor across grain boundaries. These are the accumulation of geometrically necessary dislocations (dynamic effect; see [Ma A, Roters F, Raabe D. A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations. Acta Mater 2006;58:2169–79]), the resistance against slip penetration (dynamic effect; this paper), and the change in the orientation spread (kinematic effect; this paper) in the vicinity of grain boundaries."
2009
Plastic deformation of crystalline solids depends to a high degree on the mechanisms related to the dislocation network. In order to accommodate plastic deformation and to reduce the crystal's energy, new dislocations are nucleated and pile up near the grain or phase boundaries, thereby giving rise to material strengthening. The nucleation and motion of dislocations is hence an essential mechanism to explain plastic yielding, work hardening as well as size and hysteresis effects in crystal plasticity and needs embedding into the constitutive framework of modeling materials with microstructure. An important aspect of modeling dislocation microstructures by a continuum approach lies in a sensible representation of those effects stemming from the characteristics of the discrete crystal lattice which, in particular, prohibits high local dislocation concentrations. Such a saturation behavior gives rise to numerous experimentally observed effects. In particular, experimental investigations hint at an essential size-effect of many properties of elasto-plastic crystals (e.g., the size-dependence of the indentation force during nano-indentation experiments, the grain-size dependence of the yield stress of Hall-Petch or other type, etc.)
Modelling and Simulation in Materials Science and Engineering
In this paper we present a simple and effective numerical method which allows a fast FFTbased evaluation of stress generated by dislocations with arbitrary directions and Burgers vectors if the (site-dependent) dislocation density is known. Our method allows the evaluation of the dislocation stress using rectangular grid with shape-anisotropic discretization cells without employing higher multipole moments of the dislocation interaction coefficients. Using the proposed method, we first simulate the stress created by relatively simple nonhomogeneous distributions of vertical edge and so called 'mixed' dislocations in a disk-shaped sample, what is necessary to understand the dislocation behaviour in more complicated systems. The main part of our research is devoted to the stress distribution in polycrystalline layers with the dislocation density rapidly varying with the distance to the layer bottom. Considering GaN as a typical example of such systems, we investigate dislocation-induced stress for edge and mixed dislocations, having random orientations of Burgers vectors among crystal grains. We show that the rapid decay of the dislocation density leads to many highly non-trivial features of the stress distributions in such layers and study in detail the dependence of these features on the average grain size. Finally we develop an analytical approach which allows to predict the evolution of the stress variance with the grain size and compare analytical predictions with numerical results.
Plastic deformation of crystalline solids depends to a high degree on the mechanisms related to the dislocation network. In order to accommodate plastic deformation and to reduce the crystal's energy, new dislocations are nucleated and pile up near the grain or phase boundaries, thereby giving rise to material strengthening. The nucleation and motion of dislocations is hence an essential mechanism to explain plastic yielding, work hardening as well as size and hysteresis effects in crystal plasticity and needs embedding into the constitutive framework of modeling materials with microstructure. An important aspect of modeling dislocation microstructures by a continuum approach lies in a sensible representation of those effects stemming from the characteristics of the discrete crystal lattice which, in particular, prohibits high local dislocation concentrations. Such a saturation behavior gives rise to numerous experimentally observed effects. In particular, experimental investigations hint at an essential size-effect of many properties of elasto-plastic crystals (e.g., the size-dependence of the indentation force during nano-indentation experiments, the grain-size dependence of the yield stress of Hall-Petch or other type, etc.)
Acta Materialia 54 (2006) 2169
A dislocation density based constitutive model for face-centred cubic crystals is introduced and implemented into a crystal plasticity finite element framework. The approach assumes a homogeneous dislocation structure and tracks the dislocation evolution on each slip system. In addition to the statistically stored dislocations, the geometrically necessary dislocation density is introduced in order to consider strain gradients and thus render the model size sensitive. Furthermore, we develop a consistent algorithm for the updating of the geometrically necessary dislocation density. A simple shear experiment of an aluminium single crystal is used to calibrate the material parameters of the model and demonstrate its size sensitivity.
International Journal of Solids and Structures, 2015
A coarse-grained extension of a recent nanoscale elasto-plastic model of polar dislocation and disclination density fields is developed to model grain boundary-mediated plasticity in polycrystals. At a small resolution length scale, the polar dislocation/disclination densities render continuously the discontinuities of the elastic displacements/rotations across grain boundaries. When the resolution length scale increases, the net polarities of a crystal defect ensemble decrease, perhaps to the point where no strain/curvature incompatibility is left in the body. The defect densities are then labeled ''statistical''. However both polar and statistical dislocation/disclination densities contribute to plastic flow, and a coarse-grained mesoscopic plastic curvature rate needs to be defined. In addition, whereas it is overlooked at nanoscale where grain boundaries are seen as continua, tangential continuity of the elastic/plastic curvature/strain rates across grain boundaries needs to be considered at mesoscale, because the latter are seen as singular discontinuity interfaces. It induces long-range, grain-to-grain, elastic/plastic interactions across interfaces. The mesoscale model allow preserving the essential features of the lower scale approach. In particular, it is shown that it allows accounting for such plastic deformation mechanisms as grain boundary migration and grain boundary misorientation variation by disclination motion and concurrent dislocation nucleation, when plasticity by dislocation glide is unavailable. Accumulation of polar defect densities in the vicinity of the grain boundaries and triple lines, leading to long-range inter-granular activation of slip and grain size effects, are also predicted by the model.