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The arching phenomenon is intrinsic to granular materials and is a manifestation of the effects of their discrete microstructure. The fundamental mechanism of arching is the ability of dis...

Molecular dynamics simulations were performed to identify the underlying deformation mechanisms controlling the plastic behavior of nanoindented nanoscale multilayered Ti/TiN. MD simulations were conducted on pure Ti and pure TiN as well as on four different layer-thickness ratios of Ti/TiN multilayers, Ti:TiN = 1, 2.5, 4, and 7.5. The Ti layer thickness varied from 2 nm to 15 nm while the TiN layer thickness is kept constant of 2 nm. The plastic deformation of nanoindented pure Ti was dominated by the formation of dislocation loops resulting from basal partial dislocations, while very few perfect dislocations that tie dislocation loops together were observed. The plastic deformation of nanoindented pure TiN was controlled by the activation of perfect dislocation propagation along the ( 1 1 1 ) plane that dissociates into two partials. Depending on the thickness ratio, either dislocation pile-up or single dislocation crossing through the interface was the controlling plastic deformation mechanism of nanoindented Ti/TiN multilayers. For metal layer thicknesses above 5 nm, significant dislocation pile-ups were observed at the interface of the multi-layered samples. The Ti/TiN multilayer with a thickness ratio of 1:1 with individual layer thickness of 2 nm exhibited the highest strain-hardening rate. At this length scale, the activation of dislocation sources requires very high stresses, and the single dislocation crossing process is the most dominant deformation mechanism. The initiation of plasticity in the TiN layer occurs at a high level of stress since there is no dislocation pile-up at the interface.

Metal-ceramic multilayers have been reported to show high strength, measurable plasticity, and a high strain-hardening rate when the crystallographic layers are a few nanometers thick. In this work, large-scale molecular dynamics simulations are carried out in order to understand the deformation mechanisms of the Ti/TiN multilayer subjected to multiaxial loading. The yield behavior of the Ti/TiN multilayer is thoroughly explored by constructing the yield surface in the interface plane. The strong dependency of the yielding stresses on the loading direction highlights the anisotropic behavior of the structure. The Ti/TiN multilayer structure shows high strength and ductility under uniform compression loading. However, low strength and ductility are observed under tensile loading, which favors crack initiation and propagation. Unlike typical metal stress-strain curves, metal/ceramic multilayers show two main yield points. Furthermore, the Ti/TiN multilayer structure shows three distinctive peak points for compressive loading normal and parallel to the interface. Different slip planes are activated depending on loading directions. Two main mechanisms are found to control the plasticity of the Ti/TiN multilayer: (1) interface strengthening, in which, when the metal-ceramic multilayers are under compressive loading, the interface acts as a barrier and induces repulsive forces against the slip transmission from the Ti layer into the TiN layer; (2) interface softening, in which, when applying tensile loading on the metal-ceramic multilayer structure, the interfacial misfit dislocations act as sources for the emission of dislocations into the TiN layer or promote slip transmission from the Ti to the TiN layer.

Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distributions of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). The outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error, Comment: Submitted to the International Journal of Plasticity

Additive manufacturing (AM) methods such as Aerosol Jet (AJ) printing allow the fabrication of structures via sintering of micro and/or nanoparticles, leading to microstructures that consist of various combinations of pore and grain sizes. It has been reported that AJ printed and sintered silver micropillars show an unusual behavior of high stiffness and high strain-to-failure for structures with high porosity and vice versa (Saleh et al. 2018 [1]). This behavior, however, is accompanied by the stiffer structures having smaller grain sizes and softer structures having larger grain sizes. To explain the physics of this behavior where a trade-off between hardening caused by size effects (grain refinement and gradients) and softening caused by porosity is expected to play a critical role, a multi-scale modeling approach is proposed in this paper. The model formulation consists of a continuum dislocation dynamics (CDD) framework, coupled with continuum plasticity and finite element analysis. The dislocation dynamics formulation is introduced into a user material subroutine and coupled with a finite element commercial solver, in this case, LS-DYNA, to solve the model in three-dimensional scale with the same size as the AM micropillars. The results from the model capture the general trends observed in compression tests of AM micropillars. In particular, it is shown that the grain size and dislocation density have a disproportionately higher influence over the mechanical deformation of metallic structures when compared to the porosity. These results show that the behavior of AM structures in the plastic regime is dominated by grain size effects rather than porosity. Some limitations of the model and possible future refinements are discussed. The paper provides an important analytical framework to model the mechanical behavior of AM structures with internal porosity in the plastic regime.

The mechanical behavior of nanoscale ceramic–metallic (NbC/Nb) multilayer composites with different thickness ratios is investigated using molecular dynamics (MD) simulations. Based on the obtained stress–strain behavior and its dependence on temperature, strain rate, and loading path, the flow stress for the onset of plasticity is identified and modeled based on the nucleation theory, and the in-plane yield loci for different layer thicknesses are constructed. The results are used to establish the plastic flow potential for developing a continuum viscoplastic constitutive model for potential use in large-scale applications.

The heterogeneous microstructure has a predominant effect on the mechanical behavior of polycrystalline material. In most instances, a homogenized parameter such as mean grain size is used to describe and to represent the microstructure. However, these models do not account for a measure of heterogeneity in the grain size and grain shape distributions. In this work, we introduce the grain size distribution into a multi-scale stress–strain-gradient model using a controlled Poisson Voronoi tessellation. The correlation between grain size distribution and strength is studied with various cases of grain size distribution with a fixed grid area and mean grain size. In addition, the effect of the spatial distribution of second phases and grain size on the material strength and ductility is also investigated. The results show that introducing heterogeneity into the microstructure can enhance the strength and ductility of the material compared with its equivalent homogeneous microstructure. In addition, different spatial distributions of phases can also lead to different mechanical responses.

The dynamic response of crystalline materials subjected to extreme shock compression is not well understood. The interaction between the propagating shock wave and the material’s defect occurs at the sub-nanosecond timescale which makes in situ experimental measurements very challenging. Therefore, computer simulation coupled with theoretical modelling and available experimental data is useful to determine the underlying physics behind shock-induced plasticity. In this work, multiscale dislocation dynamics plasticity (MDDP) calculations are carried out to simulate the mechanical response of copper reported at ultra-high strain rates shock loading. We compare the value of threshold stress for homogeneous nucleation obtained from elastodynamic solution and standard nucleation theory with MDDP predictions for copper single crystals oriented in the [0 0 1]. MDDP homogeneous nucleation simulations are then carried out to investigate several aspects of shock-induced deformation such as; stress profile c...

The mechanical behaviour of polycrystalline material is closely correlated to grain size. In this study, we investigate the size-dependent phenomenon in multi-phase steels using a continuum dislocation dynamic model coupled with viscoplastic self-consistent model. We developed a dislocation-based strain gradient plasticity model and a stress gradient plasticity model, as well as a combined model, resulting in a theory that can predict size effect over a wide range of length scales. Results show that strain gradient plasticity and stress gradient plasticity are complementary rather than competing theories. The stress gradient model is dominant at the initial strain stage, and is much more effective for predicting yield strength than the strain gradient model. For larger deformations, the strain gradient model is dominant and more effective for predicting size-dependent hardening. The numerical results are compared with experimental data and it is found that they have the same trend for the yield st...

This chapter, outlines a multiscale dislocation-based plasticity framework coupling discrete dislocation dynamics (DDD) with continuum dislocation-based plasticity. In this framework, and guided by DDD, a continuum dislocation dynamics (CDD) plasticity model involving a set of spatio-temporal evolution equations for dislocation densities representing mobile and immobile species is developed. The evolution laws consist of a set of components each corresponding to a physical mechanism that can be explicitly evaluated and quantified from DDD analyses. In this framework, stochastic events such as cross-slip of screw dislocations and uncertainties associated with initial microstructural conditions are explicitly incorporated in the continuum theory based on probability distribution functions defined by activation energy and activation volumes. The result is a multiscale dislocation-based plasticity model which can predict not only the macroscopic material mechanical behavior but also the corresponding microscale deformation and the evolution of dislocation patterns, size and gradient-dependent deformation phenomena, and related material instabilities at various length and time scales.

In strain-gradient plasticity, the length scale controlling size effect has been attributed to so-called geometrically necessary dislocations. This size dependency in plasticity can also be attributed to dislocation pileups in source-obstacle configurations. This has led to the development of stress-gradient plasticity models in the presence of stress gradients. In this work, we re-examine this pileup problem by investigating the double pileup of dislocations emitted from two sources in an inhomogeneous state of stress using both discrete dislocation dynamics and a continuum method. We developed a generalized solution for dislocation distribution with higher-order stress gradients, based on a continuum method using the Hilbert transform. We qualitatively verified the analytical solution for the spatial distribution of dislocations using the discrete dislocation dynamic. Based on these results, we developed a dislocation-based stress-gradient plasticity model, leading to an explicit expression for flow stress. Findings show that this expression depends on obstacle spacing, as in the Hall–Petch effect, as well as higher-order stress gradients. Finally, we compared the model with recently developed models and experimental results in the literature to assess the utility of this method.

In this paper we investigate the stochastic behavior in the mechanical response of polycrystalline materials consisting of few grains to hundreds of grains at micron size scales. We study the transition from stochastic (at small scale) to deterministic (at large scale) deformation behavior in polycrystalline samples using both simulation and nanoindentation experiments. Specifically, we develop a stochastic crystal plasticity model combining a Monte Carlo method with a polycrystal continuum dislocation dynamics model in a self-consistent viscoplasticity framework. Using this framework, we numerically calculate the mechanical properties of the polycrystal and gather randomized sampling data of the flow stress. The numerical results are compared to nanoindentation experimental data from three samples with ultra-fine grain structures manufactured via the severe plastic deformation method. The controlling mechanisms of the observed stochastic yield behavior of polycrystals are then discussed using simulations and experimental results. Our results suggest that it is the combination of stochastic plasticity at small scales (where the strength may vary from grain to grain) coupled with the effects of microstructural features such as grain size distribution and crystallite orientations that govern the uncertainty in the mechanical response of the polycrystalline materials. The extent of the uncertainty is correlated to the “effective cell size” in the sampling procedure of the simulations and experiments. The simulations and experimental results demonstrate similar quantitative behavior in terms of coefficient of variation within the same effective cell size.

Molecular dynamics (MD) simulations were performed to explore the effect of interface structure on the inception of plastic deformation in Nb/NbC nanolayered composites. Using the atomistically informed Frank–Bilby method and disregistry analysis, we characterized the structure of the Nb/NbC interface, including misfit dislocations, dislocation nodes and three coherent interface structures. According to the crystallographic analysis of the interface, four possible coherent interface structures were identified. However, study of the interface energy showed that only three of these are energetically stable. After the relaxation of the interface, the unstable coherent region, which features Nb atoms in the Nb layer on the top of the Nb atoms in the NbC layer, evolves into a condensed interface dislocation node. Three stable coherent interface regions are retained in association with the formation, glide and reaction of interface misfit dislocation loops. Disregistry analysis of the Nb/NbC interface revealed that (i) all misfit dislocations are edge type, and (ii) misfit dislocations enclosing the coherent extended nodes have Burgers vectors along 〈1 1 0〉 Nb . The role of the interface structure in the plastic deformation of Nb/NbC nanolayered composites was studied under two loading conditions, i.e. uniform compression and nanoindentation. Under uniform compression, lattice dislocations primarily nucleate from the condensed nodal regions where the local strains are the highest. Dislocations propagate in two {1 1 0} slip planes with the same Schmid factors. Under nanoindentation, by proper positioning of the indenter, lattice dislocations nucleate from the segments of the misfit dislocations and propagate in {1 1 2} slip planes. MD simulations also show cross-slip of lattice dislocations from {1 1 2} planes into {1 1 0} planes and the formation of vacancies as a result of climb of dislocation jogs.

Experimental studies show that metal–ceramic multilayers can have high strength, high strain hardening and measurable plasticity when the ceramic layer is a few nanometers thick. Using molecular dynamics simulations we studied deformation mechanisms in metal–ceramic multilayers and the role of interface structure and layer thickness on mechanical behavior. NbC/Nb multilayers were investigated numerically using the molecular dynamics (MD) method with empirical interatomic potentials. The interface dislocation structure was characterized by combining MD simulations and atomically informed Frank–Bilby theory. Two sets of pure edge misfit dislocations have been identified. Plastic deformation in NbC/Nb multilayers commences first in the metal layers by nucleation and glide of lattice dislocations initiating from interface misfit dislocations. These dislocations glide in the Nb layer and are deposited at the interface. The deposited dislocations facilitate slip transmission from the Nb layer to the NbC layer. The critical strain corresponding to dislocation nucleation is insensitive to layer thickness but depends on interface dislocation structure. The strain hardening and the peak flow strength of NbC/Nb multilayers are associated with the slip transmission from Nb to NbC, and are correlated to the interfacial dislocations, Nb layer thickness, and NbC layer thickness. The flow strength decreases with increasing Nb layer thickness and decreasing the NbC layer thickness.

Nanoindentation was used to evaluate the effect of three surface preparation techniques—mechanical polishing, electropolishing, and ion polishing—on experimental measurements of incipient plasticity in commercially pure Ni 200. Surface preparation techniques are linked to defect densities, estimated with image quality (IQ) and kernel average misorientation (KAM) data obtained from electron backscatter diffraction patterns and the Taylor relation. Minimum yield pressures are insensitive to surface preparation, while mean yield pressure depends on dislocation density, and the maximum yield pressure is likely influenced by defects other than dislocations. KAM coupled with IQ may be a useful non-destructive parameter to relate surface defect density to the resulting changes in the spatial variability of incipient plasticity during a nanoindentation experiment. This analysis makes the assumption that geometrically necessary dislocation density is proportional to total dislocation density; in cases where this condition is not satisfied, the KAM analysis may not be valid.

Deformation twinning, a common and important plastic deformation mechanism, is the key contributor to the excellent combination of strength and ductility in twinning-induced plasticity (TWIP) steel. In the open literature, a significant amount of research has been reported on the microstructural characteristics of deformation twinning and its influence on the overall deformation behavior of TWIP steel. In this study, we examine the feasibility of a mechanism-based crystal plasticity model in simulating the microstructural level deformation characteristics of TWIP steel. To this end, a model considering both double-slip and double-twin is developed to investigate the stress–strain behavior and local microstructural features related to the formation and growth of micro-twins in low stacking fault energy (SFE) TWIP steel. The twin systems are described as pseudo-slips that can be activated when their resolved shear stress reaches the corresponding critical value. A hardening law that accounts for the interaction among the slip and twin systems is also developed. Numerical simulations for different mesh sizes and single crystal patch tests under different loading modes are carried out to verify the modeling procedure. Our simulation results reveal that, despite its simple nature, the double-slip/double-twin model can capture the key deformation features of TWIP steel, including twin volume fraction evolution, continuous strain hardening, and the final fracture in the form of strain localization.

We developed a plane-strain crystal plasticity finite element (CPFE) numerical model to predict the tensile behavior of twinning-induced plasticity (TWIP) steel with both slip and mechanical twinning as the main deformation modes. Our CPFE model may not only predict well the tensile stress versus strain (S-S) curve but also capture the variation in the volume fraction of twins with a reasonable accuracy. The nucleation of mechanical twin is obviously controlled by the stress concentration. At the same time, the growth of twin may either lead to a stress relaxation in the matrix or cause a local stress concentration around twin, which depends on the deformation condition.

A geometrically linear continuum mechanics framework is proposed for gradient plasticity combining ‘strain gradients’ and, with a novel approach, ‘stress gradients’. Thereby the duality of kinematic and kinetic quantities is exploited in view of the ‘div-grad-curl orthogonality’ in continuum field theories. On the one hand the non-integrability of the plastic distortion results in the well-established dislocation density - often denoted as the geometrically-necessary-dislocation (GND) density - that enters the energy storage function. On the other hand - as entirely novel concept introduced in this contribution - the non-equilibrium of the plastic stress results in the disequilibrium density that parameterizes the dual dissipation potential within the convex analysis setting of plasticity. Consequently both, the dislocation density as well as the disequilibrium density contribute in modelling the size-dependent hardening state of a material in a continuum mechanics setting. The novel approach is eventually elucidated in much detail for the specific case of single crystal plasticity.

Continuum dislocation dynamics (CDD) with a novel constitutive law based on dislocation density evolution mechanisms was developed to investigate the deformation behaviors of single crystals. The dislocation density evolution law in this model is mechanism-based, with parameters predicted by lower-length scale models or measured from experiments, not an empirical law with parameters back-fitted from the flow curves. Applied on iron single crystal, this model was validated by experimental data and compared with traditional single crystal constitutive models using a Hutchinson-type hardening law or a dislocation-based hardening law. The CDD model demonstrated higher fidelity than other constitutive models when anisotropic single crystal deformation behaviors were investigated. The traditional Hutchinson type hardening laws and other constitutive laws based on a Kocks formulated dislocation density evolution law will only succeed in a limited number of loading directions. The main advantage of CDD is the novel physics-based dislocation density evolution laws in describing the meso-scale microstructure evolution. Another advantage of CDD is on cross-slip, which is very important when loading conditions activate only one primary slip system. In addition to the dislocation hardening, CDD also takes into consideration dislocation defect interactions. Irradiation hardening of iron single crystal was simulated with validation from experimental results.

Recent studies of micro- and nano-scale metallic structures have exposed considerable statistical distribution, in addition to significant size dependencies, in the yield strength. This intrinsic statistical variation is particularly evident in the micro-compression and microtension thin film tests. This work investigates the relationship between the initial dislocation density, the heterogeneous initial spatial dislocation distribution, and the resulting localized deformation with multiscale discrete dislocation dynamics simulations. This relationship is examined separately from commonly reported external factors affecting observed strength, such as variations in specimen geometry and base support. Towards this end, we performed multiscale dislocation dynamics simulations of geometries commonly employed in micro-scale testing techniques, including micro-pillar compression, microtensile thin film, and microbulge tests. The statistical variation of yield strengths from all three simulation geometries is in agreement with experimental data from the corresponding loading techniques. We show that the onset of plasticity is stochastic in small volumes containing a small density of dislocations: a contrast to classical deterministic plasticity theory. The yield stress in these small volumes is stochastic, not deterministic, because of statistical variation of the initial dislocation content. The numerical results exhibit a localized deformation process and demonstrate a strong dependence of the yield stress on the initial dislocation density, the initial dislocation spatial distribution, and the specimen geometry size. Leveraging nucleation theory, a stochastic model for the onset of plasticity in micro- and nano-scale structures is developed based on these results.

The strengthening effect of precipitates in metals is investigated within a multiscale approach that utilizes models of various length scales; namely, molecular mechanics (MM), discrete dislocation dynamics (DD), and an equivalent inclusion method (EIM). In particular, precipitates are modeled as particles whose stress fields interact with dislocations. The stress field resulting from the elastic mismatch between the particles and the matrix is accounted for by using the EIM, whereas the MM method is employed to develop rules for the DD method for short range interactions between a single dislocation and an inclusion. The DD method is used to predict the strength of the composite structure resulting from the interaction between ensembles of dislocations and particles. As an application to this method, the mechanical behavior of advanced high strength steel is investigated and the results are compared to the experimental data published in previous studies. The results show that the finely dispersiv...

Nanoporous gold (NP-Au) exhibits microscale plasticity, but macroscopically fails in a relatively brittle manner. This current study suggests that a core-shell structure can increase both ductility and strength of NP-Au. A core Au foam structure was created using conventional dealloying methods with average ligament size of 60 nm. Nickel was then electroplated on to the NP-Au with layer thicknesses ranging from 2.5 nm to 25 nm. Nanoindentation demonstrated a significant increase in the hardness of the coated Np-Au, to about five times of that of the pure Np-Au, and a decrease in creep by increasing the thickness of the coated Ni layer. Molecular dynamics simulations of Au–Ni ligaments show the same trend of strengthening behavior with increasing Ni thickness suggesting that the strengthening mechanisms of the Np-Au are comparable to those for fcc nano ligaments. The simulations demonstrate two different strengthening mechanisms with the increased activity of the twins in plated Au–Ni ligaments, which lead...

In this study, we developed a physically based mesoscale model for dislocation dynamics systems to predict the deformation and spontaneous formation of spatio-temporal dislocation patterns over microscopic space and time. Dislocations and dislocation patterns are emblematic of plastic deformation, a nonlinear, dissipative process involving the dynamics of underlying dislocations as carriers of plastic deformation. The mesoscale model includes a set of nonlinear partial differential equations of reaction–diffusion type. Here, we consider the equations within a one-dimensional framework and analyze the stability of steady-state solutions for these equations to elucidate the associated patterns with their intrinsic length scale. The numerical solution to the model yields the spatial distribution of dislocation patterns over time and provides respective stress–strain curves. Finally, we compare the stress–strain curves associated with the dislocation patterns with the experimental results noted in the literature.

The effect of crystallographic orientation and sample volume on incipient plasticity of commercially pure polycrystalline nickel 200 was investigated using electron back scatter diffraction (EBSD) and nanoindentation. A nickel sample was annealed, and grain orientations were determined using EBSD. Specifically oriented grains were indented using tips of nominal radii 100, 1000, and 1300 nm. The onset of plasticity in relatively defect-free small volumes is characterized by a sharp “pop-in” event during load-controlled nanoindentation. Grains in the (001) orientation yield at higher pressures; larger radii tips are more sensitive to orientation effects but yield at lower stresses. Subsurface defects may result in the dominance of tip radius over orientation, indicated by statistical analysis of yield points. An activation volume analysis, in conjunction with the yield pressure as a function of tip size, suggests that dislocation loops may play a critical role in causing these effects.

The size dependent deformation of Cu single crystal micropillars with thickness ranging from 0.2 to 2.5 μm subjected to uniaxial compression is investigated using a Multi-scale Dislocation Dynamics Plasticity (MDDP) approach. MDDP is a hybrid elasto-viscoplastic simulation model which couples discrete dislocation dynamics at the micro-scale (software micro3d ) with the macroscopic plastic deformation. Our results show that the deformation field in these micropillars is heterogeneous from the onset of plastic flow and is confined to few deformation bands, leading to the formation of ledges and stress concentrations at the surface of the specimen. Furthermore, the simulation yields a serrated stress–strain behavior consisting of discrete strain bursts that correlates well with experimental observations. The intermittent operation and stagnation of discrete dislocation arms is identified as the prominent mechanism that causes heterogeneous deformation and results in the observed macroscopic strain bursts. We show that the critical stress to bow an average maximum dislocation arm, whose length changes during deformation due to pinning events, is responsible for the observed size dependent response of the single crystals. We also reveal that hardening rates, similar to that shown experimentally, occur under relatively constant dislocation densities and are linked to dislocation stagnation due to the formation of entangled dislocation configuration and pinning sites.

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed., NOTICE: This is the author's version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structual formatting and other quality control mechanisms , may not be reflected in this document. Changes may have been to this work since it was submitted for publication., application/pdf

This paper presents a double-slip/double-twin polycrystal plasticity model using finite element solution to investigate the kinetics of deformation twinning of medium manganese (Mn) twinning-induced plasticity (TWIP) steels. Empirical equations are employed to estimate the stacking fault energy (SFE) of TWIP steels and the critical resolved shear stress (CRSS) for dislocation slip and deformation twinning, respectively. The results suggest that the evolution of twinning in Fe–xMn–1.4Al–0.6 C (x = 11.5, 13.5, 15.5, 17.5, and 19.5 mass%) TWIP steels, and its relation to the Mn content, can explain the effect of Mn on mechanical properties. By comparing the double-slip/double-twin model to a double-slip model, the predicted results essentially reveal that the interaction behavior between dislocation slip and deformation twinning can lead to an additional work hardening. Also, numerical simulations are carried out to study the influence of boundary conditions on deformation behavior and twin formation. The nucleation and growth of twinning are found to depend on internal properties (e.g., mismatch orientation of grains and stress redistribution) as well as on external constraints (e.g., the applied boundary conditions) of the material.

The deformation process in copper and aluminium single crystals under shock loading is investigated using a multiscale model of plasticity that couples discrete dislocation dynamics and finite element analyses. Computer simulations are carried out to mimic loading condition of high strain rates ranging from 105 to 107 s−1, and short pulse durations of few nanoseconds involved in recent laser based experiments. The effects of strain rate, shock pulse duration and the nonlinear elastic properties are investigated. Relaxed configurations using dislocation dynamics show formation of dislocation micro bands and weak dislocation cells. Statistical analyses of the dislocation microstructures are preformed to study the characteristics of the local dislocation densities and the distribution of the instantaneous dislocations velocities.

Results pertaining to the formation and dynamics of planar dislocation boundaries in deformed fcc single crystals using a multi-scale analysis are presented. A pure tilt boundary and experimentally observed extended geometrically necessary boundaries (GNBs) are constructed within the representative volume element (RVE) for multi-scale simulations. The model couples discrete dislocation dynamics analysis with continuum finite element to correct for the boundary conditions and image stress. It is shown that the right boundary condition of the RVE is critical in modeling GNBs and their long-range stresses. Effects of various numerical factors such as domain length and mesh sensitivity are also discussed. The effect of changing the spacing between two dislocation boundaries on the self-stress field and the stability, particularly in the space between the two dislocation boundaries, is presented. Relaxed configurations using dislocation dynamics show formation of a uniform network stabilized by formation of junctions and dipoles.

SUMMARY The flow stress in the yield surface of plastic constitutive equation is modified with a higher order gradient term of the effective plastic strain to model the effect of inhomogeneous deformation in granular materials. The gradient constitutive model has been incorporated into the finite element code ABAQUS and used to simulate biaxial shear tests on dry sand. It is shown that the shape of the post-peak segment of the load displacement curve predicted by the numerical analysis is dependent on the mesh size when gradient term is not used. Use of an appropriate gradient coefficient is shown to correct this and predict a unique shape of the load displacement curve regardless of the mesh size. The gradient coefficient required turns out to be approximately inversely proportional to the mesh elemental area. Use of the strain gradient term is found to diffuse the concentration of plastic strains within shear band resulting in its consistent width. The coefficient of the higher gradient term appears as a function of the grain size, the mean confining stress, and the plastic softening modulus. Copyright # 2004 John Wiley & Sons, Ltd.

The problem of dislocation patterning and interaction of threading dislocations with immobile dislocation loops and defects is investigated analytically and computationally based on a statistical analysis and a recently developed model of discrete stochastic dislocation dynamics (SDD), respectively. The statistical analysis is based on the Friedel–Kocks model and shows the validity of the Friedel relation for the critical resolved stress while a power law with different stress dependence is obtained for the average pinning distance on a stable dislocation array. The difference of the stress dependence is attributed to each model assumptions, such as stable dislocation configurations in athermal system or meta-stable configurations in thermally activated system. The SDD computational study includes thermal and strain fluctuation, predicting non-trivial fractal instability of the plastic strain. The height difference correlations of the plastic strain show that the external load causes a multifractality, and enhances the instability at higher order moments.

A unified phenomenological model is developed to study the dislocation glide through weak obstacles during the first stage of plastic deformation in metals. This model takes into account both the dynamical responses of dislocations during the flight process and thermal activations while dislocations are bound by obstacle arrays. The average thermal activation rate is estimated using an analytical model based on the generalized Friedel relations. Then, the average flight velocity after an activation event is obtained numerically by discrete dislocation dynamics (DD). To simulate the dynamical dislocation behavior, the inertia term is implemented into the equation of dislocation motion within the DD code. The results from the DD simulations, coupled with the analytical model, determine the total dislocation velocity as a function of the stress and temperatures. By choosing parameters typical of the face centered cubic metals, the model reproduces both obstacle control and drag control motion in low and high velocity regimes, respectively. As expected by other string models, dislocation overshoots of obstacles caused by the dislocation inertia at the collisions are enhanced as temperature goes down.

Deformation in metals is size dependent over length scales ranging from a few nanometers to 100 μm. Two frameworks, dislocation dynamics and gradient plasticity, each having a length scale with different interpretation are discussed. It is suggested that various dislocation arrangements produce different physical behavior with different gradients and length scales.

A framework for investigating size-dependent small-scale plasticity phenomena and related material instabilities at various length scales ranging from the nano-microscale to the mesoscale is presented. The model is based on fundamental physical laws that govern dislocation motion and their interaction with various defects and interfaces. Particularly, the multi-scale framework merges two scales, the nano-microscale where plasticity is determined by explicit three-dimensional dislocation dynamics analysis providing the material length-scale, and the continuum scale where energy transport is based on basic continuum mechanics laws. The result is a hybrid elasto-viscoplastic simulation model coupling discrete dislocation dynamics with finite element analyses. With this hybrid approach, one can address complex size-dependent problems including, dislocation boundaries, dislocations in heterogeneous structures, dislocation interaction with interfaces and associated shape changes and lattice rotations, as well as deformation in nano-structured materials, localized deformation and shear bands.

Superplastic forming is a valuable metal working technique because of the extreme ductility that can be achieved. However, it is limited in application due to the presence of small voids that grow and coalesce during the forming process, often causing premature failure. In order to understand and control this phenomenon accurate constitutive models must be developed which account for void parameters that affect the macroscopic behavior of the material. This paper looks specifically at the effect of void size and spacing on the ductility and flow stress of viscoplastic materials. Based on the gradient-dependent theory of plasticity, a model is proposed that accounts for size effects by incorporating strain gradient terms into a continuum based constitutive equation. Both experimental testing and finite element (FE) modeling were performed on Pb–Sn, tensile specimens with small holes drilled in them in random patterns. The experimental tests indicate that a decrease in void size results in an increase in ductility. The FE results demonstrate that the gradient terms strengthen the material by diffusing the strain in areas of high strain concentration and delay failure by slowing void growth. In addition, the model predicted an increase in ductility and flow stress with decreasing void size.

In this work, we discuss a thermodynamic theory of plasticity for self-organization of collective dislocations in FCC metals. The theory is described by geometrical tensor quantities of crystal defect fields such as dislocation density tensor, representing net mobile dislocation density and geometrically necessary boundaries, and the incompatibility tensor representing immobile dislocation density. Conservation laws for the two kinds of dislocation density are formulated with dislocation products and interactions terms. Based on the second law of thermodynamics, we drive basic constitutive equations for the dislocation flux, production and interaction terms of dislocations. We also derive a set of reaction-diffusion equations for the dislocation density tensor and incompatibility tensor which describes the vein and persistent slip band (PSB) ladder structures. These equations are analyzed using linear stability and bifurcation analysis. An intrinsic mesoscopic length scale is determined which provides an estimate for the wavelength of the PSBs. The basic aspects of the model are motivated and substantiated by analyzing the stress fields of various possible dislocation configurations using discrete dislocation dynamics.

A dislocation dynamics (DD) model for plastic deformation, connecting the macroscopic mechanical properties to basic physical laws governing dislocation mobility and related interaction mechanisms, has been under development. In this model there is a set of critical reactions that determine the overall results of the simulations, such as the stress-strain curve. These reactions are, annihilation, formation of jogs, junctions, and dipoles, and cross-slip. In this paper we discuss these reactions and the manner in which they influence the simulated stress- strain behavior in fcc and bcc metals. In particular, we examine the formation (zipping) and strength of dipoles and junctions, and effect of jogs, using the dislocation dynamics model. We show that the strengths (unzipping) of these reactions for various configurations can be determined by direct evaluation of the elastic interactions. Next, we investigate the phenomenon of hardening in metals subjected to cascade damage dislocations. The microstructure investigated consists of small dislocation loops decorating the mobile dislocations. Preliminary results reveal that these loops act as hardening agents, trapping the dislocations and resulting in increased hardening.

Constitutive modeling for the particle size effect on the strength of particulate-reinforced metal matrix composites is investigated. The approach is based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique that is used to calculate the overall flow properties of composites. It is shown that the strain gradient term introduces a spatial length scale in the constitutive equations for composites, and the dependence of the flow stress on the particle size/spacing can be obtained. Moreover, a nondimensional analysis along with the numerical result yields an explicit relation for the strain gradient coefficient in terms of particle size, strain, and yield stress. Typical results for aluminum matrix composites with ellipsoidal particles are calculated and compare well with data measured experimentally.

A simplified theory of instability of plastic flow is applied in this paper to analyze the formation of shear localized chips in orthogonal machining. A flow localization parameter is expressed in terms of associated cutting conditions and properties of the workpiece material. The analysis is used to investigate the effect of cutting conditions on the onset of shear localization and the formation of adiabatic shear banding in metal cutting. Comparisons are made between the analysis and experiments in which the flow localization parameter is obtained for several workpiece materials. The results of this investigation are thought to lend a strong justification for the analysis and its potential benefits in analyzing and/or remedying problems associated with chip formation and temperature generated in metal cutting.

A simplified theory of instability of plastic flow is applied to analyze the formation of shear localized chips in orthogonal machining. A flow localization parameter is expressed in terms of associated cutting conditions and properties of the workpiece material. The analysis, which indicates the important parameters in the cutting process, is used to investigate the effect of cutting conditions on the onset of shear localization and the formation of adiabatic shear banding in metal cutting. Comparisons are made between the analysis and experiments in which the flow localization parameter is obtained for several workpiece materials. The results of this investigation seem to support the analysis and its potential benefits in analyzing and/or remedying problems associated with chip formation and temperature generated in metal cutting.

A micromechanically based continuum model is developed to analyze the enhancement of plastic properties of particulate-reinforced metal-matrix composites over matrix materials. The composite is idealized as uniformly distributed periodic arrays of unit cells. Each unit cell consists of a rigid inclusion surrounded by a plastically deforming material. An energy method is adopted to obtain the overall constitutive relation for the composite on the basis of the local nonuniform deformation fields. Effects of particle volume fractions and shapes (e.g. whiskers, discs, etc.) as well as the matrix properties on the flow properties of the composite are obtained. The results are in good agreement with experimental observations and finite element analyses found in the literature. An explicit expression is also proposed, providing a means for evaluating various factors affecting the strength of composites.

In latent interactions of dislocations, junction formation is one of the most important phenomena that contribute to the evolution of strength. In this work, the latent hardening coefficients for pure aluminum are estimated using 3D multiscale dislocation dynamics program (MDDP). Three well-known junction configurations, namely, the Hirth lock, the glissile junction, and the Lomer lock, are studied using 3D discrete dislocation dynamics simulations. The evolution of strength is discussed as a function of the resolved shear stress (RSS) and the number of junctions for the three junctions investigated. Hirth lock and Lomer lock are found to be the weakest and strongest junctions, respectively. Collinear reaction of dislocations does not form a junction but causes a higher strength than a Lomer lock. Quantitative and qualitative results are compared with those found in the literature.

A model of bowed-out tilt-wall array has been recently proposed by the authors to provide a correlation between the flow stress and particle size/spacing of metal matrix composites. Here, the authors elaborate on the tilt-wall model and show that the number of dislocations needed to predict the experimental result, for aluminum alloys reinforced with alumina or TiB[sub 2] particulates, is not affected by the dislocation spacing for realistic pile-ups with 2 < d/[bar d] < 7, where [bar d] is the spacing between two adjacent slip planes. The model is also used to explain new experimental results for an Al-Si alloy with Si particles of different sizes.

Phenomenological constitutive equation for plastic deformation in metallic alloys are usually developed with the basic assumption of uniformity of the deformation field, or by the homogenization of the field over a macroscopic representative element. This, in turn, coupled with simple nondimensional analysis within the classical continuum theory of plasticity, leads to the development of homogeneous constitutive equations that relate the flow stress to some internal variables, such as strain hardening, strain rate, and volume fraction of second phase particles in metal matrix composites. However, there are circumstances where the size of the microstructure significantly influences the overall mechanical properties of the material. This becomes even more crucial when the size of the plastic deformation zone and the magnitude of the strain gradient within it become comparable to the size of the underlying dislocation structure. For example, predicting the width of the shear band in metallic alloys and its scaling with the size of the microstructure is an important issue from a fundamental point of view. A constitutive model that can capture this phenomenon can also provide an explanation of other observed phenomena such as the size of the plastic zone near a crack tip, the dependence of flow stress onmore » particle size in metal matrix composite, the effect of specimen size on the stress-strain curve in pure copper, and the significant increase in flow stress observed in indentation tests when the size of the indenter is in the nanometer range. The objective of this paper is to provide a possible explanation of how a local microscopic strain inhomogeneity resulting from a heterogeneous microstructure can generate a strain gradient effect when the deformation field is averaged over a macroscopic representative element, leading to a specific predictive relation for the strain gradient coefficient c in terms of the materials elastic-plastic moduli and grain size.« less

After a brief review of a recently developed gradient-dependent theory of plasticity various questions related to the yield function and the loading-unloading condition in the presence of higher order strain gradients and the determination of the corresponding phenomenological coefficients are addressed. For rate-independent materials, we construct as before an analytical solution for the strain profile in the postlocalization regime providing the shear band thickness and strain within it but we now compare these results to recently obtained experimental data by assigning appropriate values to the gradient coefficients. We also address some questions recently raised in the literature regarding our nonlinear shear band analysis. For rate-dependent materials, the resulting spatio-temporal differential equation for the strain is solved numerically using the finite difference method. It is shown that the band width does not depend on the grid size, as long as the the grid size is smaller than a certain characteristic length. Various initial imperfections of different amplitudes and sizes are examined, and the possibility of simultaneous development of two shear bands and their interaction is investigated.

Discrete dislocation dynamics is a numerical tool developed to model the plasticity of crystalline materials at an intermediate length scale, between the atomistic modeling and the crystal plasticity theory. In this review we show, using examples from the literature, how a discrete dislocation model can be used either in a hierarchical or a concurrent multiscale framework. In the last section of this review, we show through the uniaxial compression of microcrystal application, how a concurrent multiscale model involving a discrete dislocation framework can be used for predictive purposes.

An experimental method using a digital image processing technique is developed for the purpose of characterizing material behavior at large elastoplastic deformations and the associated phenomenon of localization of plastic flow into shear bands. This allows for a detailed description of the evolution of the nonuniform deformation pattern in the post-localization regime. The experimental results are utilized to calibrate a recently developed gradient-dependent constitutive equation which takes into account the effect of heterogeneous plastic flow, anisotropy and large deformations. The measured values of the gradient coefficients are of small magnitude suggesting that higher order gradients are important only in the highly inhomogeneous region as expected. Moreover, it is found that anisotropic effects become significant in the post-localization regime where the anisotropy ratio changes considerably.

The macroscopic behavior of materials subjected to large deformations is investigated by considering the structure and mechanics of the single slip. Physically based constitutive relations for the plastic flow and material spin are then rigorously derived by resorting to the concept of non-coaxiality; yielding a class of vertex-type plasticity models incorporating features like noncoaxiality and large material rotation. It is shown that these phenomena are inherently coupled through the concept of plastic spin as it relates to the persistence of noncoaxiality. The implication of such phenomena to large deformations is investigated by examining their influence on the development of axial effects accompanying finite shear deformation, as well as on the onset of localized shear bands.

It is well known that the mechanical behavior of nanoscale multilayered composites is strongly governed by single dislocation mechanisms and dislocation-interface interactions. Such interactions are complex and multiscale in nature. In this work, two such significant effects are modeled within the dislocation dynamics-continuum plasticity framework: elastic properties mismatch (Koehler image forces) and interface shearing in the case of weak interfaces. The superposition principle is used to introduce the stress fields due to both effects solved for by finite elements. The validation of both methodologies is presented. Furthermore, it was found that the layer-confined threading stress of a dislocation in hair-pin configuration increases if the layer is surrounded by layers made of a stiffer material and that this strengthening effect grows more significant as the layer thickness decreases. The observation made through molecular dynamics, that weak interfaces act as dislocation sinks, was also captured with our approach. A dislocation is attracted to the interface independent of its sign or character. Also the force increases sharply as the dislocation approaches the interface. These findings agree with published molecular dynamics simulations and dislocation-based equilibrium models of this type of interaction.