Non-local Crystal Plasticity Model Research Articles

Unraveling size-affected plastic heterogeneity and asymmetry during micro-scaled deformation of CP-Ti by non-local crystal plasticity modeling

Titanium alloy is prominent in manufacturing of high-performance miniature devices, while size-affected plastic heterogeneity and asymmetry in micro-scaled deformation are not well understood due to its complex deformation mechanism. How the interplay among dislocation slip, deformation twinning, and strain gradient affects the size-dependent plastic heterogeneity and asymmetry is an intractable and non-eluded issue that needs to be unraveled. In this research, a non-local crystal plasticity model (NLCPM) considering discrete twins was established and implemented into a finite element framework. The proposed NLCPM successfully represented the size-dependent asymmetric mechanical response, primary and secondary twinning, and texture evolution. The size-dependent fracture behavior, plastic heterogeneity, and T-C asymmetry during micro-scaled deformation of commercial pure titanium (CP-Ti) were explored by coupling micro-tension/compression test and full-field simulation. Results showed that T-C asymmetry in flow stress was enhanced in the coarse-grained and small-diameter samples. For the sample with D≥0.6 mm, the ductility of CP-Ti was improved with the increase of grain size as there are more prevailing twins and the concentrating stress could be relieved by the abundant slip rather than cracking. Grain-level strain is more homogeneous in compression when d was increased from 60 to 110 μm since more non-prismatic slips were activated that accommodated strain along various directions. Meanwhile, plastic strain was allocated within discrete twin bands and thus alleviated strain concentration. Furthermore, the hardening effect of twin boundaries cancelled out the softening effect near the free surface, resulting in a narrower softening layer in the coarse-grained samples compared to its grain size. With the decrease of the sample size, the proportion of the softening layer increased, while the volume fraction of twinning decreased, leading to a reduction in macroscopic stress and fracture strain. This work enhances the in-depth understanding of size-affected plastic heterogeneity and asymmetry during micro-scaled deformation of CP-Ti and supports the forming of high-quality micro components under complex deformation path.

Geometrically necessary dislocations and related kinematic hardening in gradient grained materials: A nonlocal crystal plasticity study

Gradient grained metals whose microstructure is characterized by a spatially graded grain size distribution show a better strength-ductility combination than their homogeneous counterparts. Kinematic hardening associated with geometrically necessary dislocations (GNDs) is considered to be a dominant strengthening mechanism in gradient grained metals. However, the precise kinematics of GND accumulation and the nature of the back stress fields remain unclear, restricting the understanding of their deformation mechanisms. In this work, a nonlocal crystal plasticity model which explicitly accounts for the interaction between dislocations and grain boundaries is developed. The nonlocal feature is achieved by introducing a flux term to account for the spatial redistribution of dislocations due to their motion. In addition, back stress produced by the spatial variation of GND density introduces an explicit internal length scale into the model. The nonlocal nature of the model on the slip system level enables the direct investigation of strain gradient effects caused by internal deformation heterogeneities. Furthermore, the interaction between dislocations and grain boundaries leads to the formation of pileups near grain boundaries, which is key to studying the grain size effects in polycrystals. Finite element implementation of the model for polycrystals with different grain sizes quantitatively captures the grain size effect. Simulation results of gradient grained materials and their homogeneous counterparts demonstrate that smaller grains lead to higher GND density and enhanced back stress. Small grains significantly contribute to the GND-induced isotropic hardening and GND-induced kinematic hardening in gradient grained metals. This investigation helps to understand the underlying strengthening mechanisms of gradient grained metals, and the model can be readily applied to other kinds of heterogeneous materials.

A physics-based crystal plasticity model for the prediction of the dislocation densities in micropillar compression

• Generalized constitutive relations that incorporate the thermally activated and drag mechanisms. • Continuum description of crystal plasticity framework combined with the geometrically necessary dislocation densities and the mobile/immobile statistically stored dislocation densities. • Computation of the geometrically necessary dislocation densities based on the L 1 -norm minimization scheme to provide physical interpretation of the lower bound on overall GND population. • Dislocation mechanisms depending on generation, annihilation, interactions, trapping, and recovery. • Evolution laws of dislocation densities with mechanism-based material parameters. A new nonlocal crystal plasticity model is developed to study the mechanical behavior of face-centered cubic single crystals under heterogeneous inelastic deformation through a crystal plasticity finite element method. The main feature of this work is generalized constitutive relations that incorporate the thermally activated and drag mechanisms to cover different kinetics of viscoplastic flow in metals at a variety of ranges of stresses and strain rates. The constitutive laws are founded upon integrating continuum description of crystal plasticity framework with dislocation densities which is relevant to the geometrically necessary dislocation densities and the statistically stored dislocation densities. The model describes the plastic flow and the yielding of face-centered cubic single-crystal employing evolution laws of dislocation densities with mechanism-based material parameters passed from experiments or small-scale computational models. The geometrically necessary dislocation densities evolve on account of the curl of the plastic deformation gradient where its associated closure failure of the Burgers circuit exists. A minimization scheme termed L 1 -norm is utilized to secure the physical values of the geometrically necessary dislocation densities on slip systems. The evolution equations of statistically stored dislocation densities describe the complex interactions between two distinct dislocation populations, mobile, and immobile statistically stored dislocation dislocations, relying on generation, annihilation, interactions, trapping, and recovery. The experiments of a micropillar compression for the copper single crystal are compared to the computational results obtained using the formulation. The physics-based model clarifies the complex microstructural evolution of dislocation densities in metals and alloys, allowing for more accurate prediction. .

Influence of Crystal Plasticity Parameters on the Strain Hardening Behavior of Polycrystals

The effective mechanical properties of a polycrystal depend directly on the single-crystal properties of each grain and its crystallographic orientation with respect to the load axis. While the micromechanical approach has been used quite extensively to study the influence of grain shape and crystallographic texture on the resulting mechanical behavior of a polycrystal, the influence of the crystal plasticity parameters, which describe the constitutive behavior of the single crystal, requires to be investigated systemically because, typically, these parameters are fitted to describe a given material behavior. In the current research, this gap is filled by systemically studying the effect of changes in crystal plasticity parameters on the effective mechanical properties of polycrystals. The numerical model employed here consists of a representative volume element of 100 grains, and the material properties are described by using a non-local crystal plasticity model. A proper homogenization technique was used to homogenize the micromechanical results to an effective macroscopic material response. The equivalent stress versus equivalent plastic strain curve was obtained numerically by introducing the Voce-type hardening law, mimicking the material behavior in uniaxial tensile tests. The four parameters of the Voce-type hardening law were fitted to the macroscopic stress-strain curves, and the correlation between the crystal plasticity parameters and the Voce parameters has been studied, which is an efficient way to study the influence of microscopic material descriptions on the macroscopic behavior of polycrystals.

Influence of Pore Characteristics on Anisotropic Mechanical Behavior of Laser Powder Bed Fusion–Manufactured Metal by Micromechanical Modeling

In recent times, additive manufacturing (AM) has proven to be an indispensable technique for processing complex 3D parts because of the versatility and ease of fabrication it offers. However, the generated microstructures show a high degree of complexity due to the complex solidification process of the melt pool. In this study, micromechanical modeling is applied to gain deeper insight into the influence of defects on plasticity and damage of 316L stainless steel specimens produced by a laser powder bed fusion (L‐PBF) process. With the statistical data obtained from microstructure characterization, the complex AM microstructures are modeled by a synthetic microstructure generation tool. A damage model in combination with an element deletion technique is implemented into a nonlocal crystal plasticity model to describe anisotropic mechanical behavior, including damage evolution. The element deletion technique is applied to effectively model the growth and coalescence of microstructural pores as described by a damage parameter. Numerical simulations show that the shape of the pores not only affects the yielding and hardening behavior but also influences the porosity evolution itself.

Robust Optimization Scheme for Inverse Method for Crystal Plasticity Model Parametrization.

A bottom-up material modeling based on a nonlocal crystal plasticity model requires information of a large set of physical and phenomenological parameters. Because of the many material parameters, it is inherently difficult to determine the nonlocal crystal plasticity parameters. Therefore, a robust method is proposed to parameterize the nonlocal crystal plasticity model of a body-centered cubic (BCC) material by combining a nanoindentation test and inverse analysis. Nanoindentation tests returned the load–displacement curve and surface imprint of the considered sample. The inverse analysis is developed based on trust-region-reflective algorithm, which is the most robust optimization algorithm for the considered non-convex problem. The discrepancy function is defined to minimize both the load–displacement curves and the surface topologies of the considered material under applying varied indentation forces obtained from numerical models and experimental output. The numerical model results based on the identified material properties show good agreement with the experimental output. Finally, a sensitivity analysis performed changing the nonlocal crystal plasticity parameters in a predefined range emphasized that the geometrical factor has the most significant influence on the load–displacement curve and surface imprint parameters.

Micromechanical Modeling of DP600 steel: From Microstructure to The Sheet Metal Forming Process

This study proposes a micromechanical modeling scheme to predict relevant mechanical behavior of DP600 steel for the sheet metal forming process. This study can be divided into two parts which are the prediction of the advanced anisotropic initial yield function by means of microstructure-based simulations and the investigation of microstructure changes during the sheet metal forming process. Firstly, based on the quantitative microstructure characterization of DP600 steel by EBSD analysis, the obtained statistical information of important microstructural features is used to generate a microstructure model with the help of an advanced dynamic microstructure generator (ADMG), which combines a particle simulation method with radical Voronoi tessellation. In the next step, finite element simulations with a non-local crystal plasticity model for the individual grains are conducted. With the help of these simulations, the crystal plasticity parameters are adapted to match the experiments. The resulting parameterized microstructure model of DP600 steel is then applied to various loading conditions to investigate the corresponding mechanical responses. For the second part, macroscopic simulations of the bending process are performed and local deformation fields of the location of interest are captured and imposed as boundary conditions on the microstructure model to study the changes in the microstructural features.

Hydrogen distribution in metallic polycrystals with deformation

To understand hydrogen embrittlement and predict probable damage sites based on high hydrogen segregation, it is important to identify hydrogen distribution mechanisms in metallic microstructure with deformation. Hydrogen segregation in metals is affected by various factors such as grain-size, the character of grain boundaries (GBs), loading direction and strain-rate. To this end, a computational framework consisting of non-local dislocation density-based crystal plasticity model coupled with slip-rate based hydrogen transport model is presented to study the role of (i) grain-size, (ii) loading direction, (iii) strain-rate and (iv) GB character on hydrogen distribution and segregation in the pre-charged metallic microstructure. The computational framework is capable of accounting for the change in local hydrogen concentration due to prevailing hydrostatic pressure, trapping by dislocations, GB energetics and local slip-rates in the metallic microstructure. The difficulty in dislocation motion at the inter-granular regions of polycrystal due to high cross-hardening offered by geometrically necessary dislocations (GNDs) is incorporated in the model as additional isotropic hardening, whereas the back stress due to the GND pileups is included as a kinematic term in the flow rule. In addition, GNDs act as hydrogen trap sites leading to increased hydrogen concentration in the inter-granular regions. The strain-rate factor initially provided by Krom & Bakker (1999) is modified in the present work to make it compatible with the dislocation density-based coupled framework, which is able to calculate trapped hydrogen in dislocations along the slip systems. Plastically deformed polycrystals of various grain-sizes, along different directions, with varying strain rates and containing Σ3[11¯0](111) and Σ5[001](210) GBs show that the gradients of hydrostatic pressure and GB character are major factors controlling hydrogen segregation in the microstructure.

Modelling of Fatigue Crack Initiation in Hydrogen Charged Polycrystalline Nickel

Hydrogen Embrittlement (HE) leads to deterioration of the fracto-mechanical properties of metals. In spite of vast literature, it is still not clearly understood and demands significant research on this topic. For better understanding of the hydrogen effect on fatigue behaviour of metals, present work focuses on developing a computational framework for fatigue crack initiation studies in metals in the presence of hydrogen. The developed framework consists of a nonlocal crystal plasticity model coupled with hydrogen transport model to study the fatigue behaviour of hydrogen charged metals. The nonlocal crystal plasticity model accounts for the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) in polycrytalline metal. Hydrogen transport model, on the other hand, accounts for diffusion and trapping behavior of hydrogen due to concentration gradient, pressure gradient, plastic strain-rate with dislocations as the only trapping sites along the slip systems. A polycrystalline representative volume element (RVE) with periodic boundary conditions is used in this study. Fatigue crack initiation criterion is proposed for the simulated RVE with controlled microstructure by considering a critical value of the fatigue indicator parameter (FIP). FIP is formulated based on the experimental observations of several crack initiation sites along the grain boundaries, their normal direction with respect to loading direction and the accumulated plastic strain in nickel polycrystalline samples. Developed simulation framework correctly accounts cyclic stress-strain behavior and multiple fatigue crack initiation sites observed experimentally in the presence of hydrogen.

Formation of GNDs in copper single crystal under pure bending considering planar double slip

In the present work a finite strain non-local crystal plasticity model is implemented via UMAT in ABAQUSTM. The main objective of this work is to study the distribution of geometrically necessary dislocations in pure bending. Ashby (1969) with his hypothesis showed that the gradient of strains caused by inhomogeneous deformation plays a major role in the mechanical behavior of crystals. In this work we study the evolution of geometrically necessary dislocations (GNDs) in case of pure bending of Cu single crystals. In the present work we develop a crystal plasticity model considering the evolution of GNDs. For verification of current implementation, we have used a planar double slip model of FCC crystals derived by Asaro (1979). We show that in the absence of screw dislocations the total GNDs are in form of only edge type dislocations. The distribution of statistically stored dislocations (SSDs), GNDs and their components are in good agreement with Dai (1997).

Indentation size effects in spherical nanoindentation analyzed by experiment and non-local crystal plasticity

Nanoindentation has become a rather mature tool to characterize and measure mechanical properties of materials on micro- and nanometer length scales. However, the different types of indentation size effects (ISE) hamper the assessment of true, i.e., scale independent, material properties by nanoindentation. In the present study, the mechanisms of the ISE in spherical indentation have been investigated and quantified. Furthermore, it has been shown that the determination of non-local crystal plasticity parameters from nanoindentation results is possible. The thus-parameterized non-local crystal plasticity describes the higher-order size effect occurring in spherical indentation correctly and offers the possibility to understand the mutual contributions of geometrically necessary dislocations (GND) and statistically stored dislocations (SSD) to the material response. Nanoindentation experiments have been performed into a single grain of an ARMCO iron specimen, where the load displacement curves have been recorded, and the post-indentation surface topologies have been characterized. Furthermore, transmission electron microscopy has been performed to analyse the plastically deformed volume beneath the indentation. By comparing experimental and numerical nanoindentation results with respect to the plastic zone and dislocation structures, we validate the non-local crystal plasticity model of indentation and increase our understanding of the underlying mechanisms of the ISE in spherical indentation.

Experimental and modelling study of fatigue crack initiation in an aluminium beam with a hole under 4-point bending

Slip band formation and crack initiation during cyclic fatigue were investigated by in-situ experiments and non-local CPFEM simulations systematically. Experimental techniques including EBSD, digital image correlation (DIC) and SEM have been used to obtain consistent grain orientations, local strains, as well as the locations where slip bands and micro-cracks form on the sample surface. The realistic microstructure based on the EBSD map has been generated and used for finite element modelling. An advanced non-local crystal plasticity model, which considers the isotropic and kinematic hardening of the plastic strain gradient, has been adopted. The simulation results match well the corresponding experimental results. It was found that total strain and averaged slip on all slip systems, combined with accumulated slip on specific slip planes help predict the location and orientation of slip bands and micro-crack initiation correctly. Furthermore, a fatigue indicating parameter based on competition between maximum slip and the total slip has been proposed to reproduce the experimental observations.

Micromechanical modeling approach to derive the yield surface for BCC and FCC steels using statistically informed microstructure models and nonlocal crystal plasticity

In order to describe irreversible deformation during metal forming processes, the yield surface is one of the most important criteria. Because of their simplicity and efficiency, analytical yield functions along with experimental guidelines for parameterization become increasingly important for engineering applications. However, the relationship between most of these models and microstructural features are still limited. Hence, we propose to use micromechanical modeling, which considers important microstructural features, as a part of the solution to this missing link. This study aims at the development of a micromechanical modeling strategy to calibrate material parameters for the advanced analytical initial yield function Barlat YLD 2004-18p. To accomplish this, the representative volume element is firstly created based on a method making use of the statistical description of microstructure morphology as input parameter. Such method couples particle simulations to radical Voronoi tessellations to generate realistic virtual microstructures as representative volume elements. Afterwards, a nonlocal crystal plasticity model is applied to describe the plastic deformation of the representative volume element by crystal plasticity finite element simulation. Subsequently, an algorithm to construct the yield surface based on the crystal plasticity finite element simulation is developed. The primary objectives of this proposed algorithm are to automatically capture and extract the yield loci under various loading conditions. Finally, a nonlinear least square optimization is applied to determine the material parameters of Barlat YLD 2004-18p initial yield function of representative volume element, mimicking generic properties of bcc and fcc steels from the numerical simulations.

Investigating the influence of crystal orientation on bending size effect of single crystal beams

Influence of crystal orientation on bending size effect has been numerically investigated for single crystal beams. This work is inspired by the experimental observations of Hayashi et al. (2011), where they observed a significant difference in bending size effect for two different crystal orientations. We have used a higher order non-local crystal plasticity model which can account for different hardening contributions by SSDs (statistically stored dislocations) and GNDs (geometrically necessary dislocations) simultaneously. It was found that strain hardening together with an additional kinematic hardening caused by accumulation of GNDs and the number of activated slip systems can be seen as the origin of the orientation dependence of bending size effect. We have also observed a pronounced orientation dependence of spring back size effect, which can be explained on the basis of number of the activated slip systems and equivalent plastic strain. Simulation results showing enhanced or diminished bending size effect for different crystal orientations reveal the importance of crystal orientation for precise micro-bending operations.

Plastic deformation modelling of tempered martensite steel block structure by a nonlocal crystal plasticity model

The plastic deformations of tempered martensite steel representative volume elements with different martensite block structures have been investigated by using a nonlocal crystal plasticity model which considers isotropic and kinematic hardening produced by plastic strain gradients. It was found that pronounced strain gradients occur in the grain boundary region even under homogeneous loading. The isotropic hardening of strain gradients strongly influences the global stress–strain diagram while the kinematic hardening of strain gradients influences the local deformation behaviour. It is found that the additional strain gradient hardening is not only dependent on the block width but also on the misorientations or the deformation incompatibilities in adjacent blocks.

Continuum simulation of the evolution of dislocation densities during nanoindentation

When nanoindenting dislocation-free regions of single crystals a so-called pop-in phenomenon is commonly observed. Molecular dynamics (MD) studies have revealed homogeneous nucleation of dislocations in a perfect crystal as a mechanism causing such pop-in behavior. In this work we transfer this knowledge gained on the atomic scale into a dislocation nucleation model that is applied within a dislocation density based crystal plasticity description. Furthermore, we develop a non-local formulation of a crystal plasticity model that is devised to yield a valid description of plasticity also in situations where the dislocation density is small or even vanishing and where conventional plasticity models fail. This is accomplished by studying the evolution of statistically stored and geometrically necessary dislocation densities separately. We apply this non-local crystal plasticity model to investigate the evolution of dislocation densities in the early stages of nanoindentation. The results of our continuum model show good agreement with MD simulations for cases where nanoindentation into an initially dislocation-free crystal is studied, i.e. where a pop-in occurs when the critical stress underneath the indenter reaches the critical value for homogeneous dislocation nucleation. After thus validating our model we study the influence of pre-existing homogeneous and local dislocation densities. Both cases show a good qualitative agreement with recent experimental findings and it is concluded that pre-existing local dislocations densities reduce the load at which a pop-in occurs and – more importantly – change the mechanism from homogeneous dislocation nucleation to rapid dislocation multiplication. In general, our results show that continuum plasticity formulations can be extended such that applications to nanoscale volumes become possible.

Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories

This paper focuses on the unification of two frequently used and apparently different strain gradient crystal plasticity frameworks: (i) the physically motivated strain gradient crystal plasticity models proposed by Evers et al. [2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. Journal of the Mechanics and Physics of Solids 52, 2379–2401; 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209–5230] and Bayley et al. [2006. A comparison of dislocation induced back stress formulations in strain gradient crystal plasticity. International Journal of Solids and Structure 43, 7268–7286; 2007. A three dimensional dislocation field crystal plasticity approach applied to miniaturized structures. Philosophical Magazine 87, 1361–1378] (here referred to as Evers–Bayley type models), where a physical back stress plays the most important role and which are further extended here to deal with truly large deformations, and (ii) the thermodynamically consistent strain gradient crystal plasticity model of Gurtin (2002–2008) (here referred to as the Gurtin type model), where the energetic part of a higher order micro-stress is derived from a non-standard free energy function. The energetic micro-stress vectors for the Gurtin type models are extracted from the definition of the back stresses of the improved Evers–Bayley type models. The possible defect energy forms that yield the derived physically based micro-stresses are discussed. The duality of both type of formulations is shown further by a comparison of the micro-boundary conditions. As a result, this paper provides a direct physical interpretation of the different terms present in Gurtin's model.

Distribution based model for the grain boundaries in polycrystalline plasticity

This contribution focuses on the development of constitutive models for the grain boundary region between two crystals, relying on the dislocation based polycrystalline model documented in (Evers, L.P., Parks, D.M., Brekelmans, W.A.M., Geers, M.G.D., 2002. Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. J. Mech. Phys. Solids 50, 2403–2424; Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004a. Non-local crystal plasticity model with intrinsic SSD and GND effects. J. Mech. Phys. Solids 52, 2379–2401; Evers, L.P., Brekelmans, W.A.M., Geers, M.G.D., 2004b. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. Int. J. Solids Struct. 41, 5209–5230). The grain boundary is first viewed as a geometrical surface endowed with its own fields, which are treated here as distributions from a mathematical point of view. Regular and singular dislocation tensors are introduced, defining the grain equilibrium, both in the grain core and at the boundary of both grains. Balance equations for the grain core and grain boundary are derived, that involve the dislocation density distribution tensor, in both its regular and singular contributions. The driving force for the motion of the geometrically necessary dislocations is identified from the pull-back to the lattice configuration of the quasi-static balance of momentum, that reveals the duality between the stress and the curl of the elastic gradient. Criteria that govern the flow of mobile geometrically necessary dislocations (GNDs) through the grain boundary are next elaborated on these bases. Specifically, the sign of the projection of a lattice microtraction on the glide velocity defines a necessary condition for the transmission of incoming GNDs, thereby rendering the set of active slip systems for the glide of outgoing dislocations. Viewing the grain boundary as adjacent bands in each grain with a constant GND density in each, the driving force for the grain boundary slip is further expressed in terms of the GND densities and the differently oriented slip systems in each grain. A semi-analytical solution is developed in the case of symmetrical slip in a bicrystal under plane strain conditions. It is shown that the transmission of plastic slip occurs when the angle made by the slip direction relative to the grain boundary normal is less than a critical value, depending on the ratio of the GND densities and the orientation of the transmitted dislocations.

Predicting the Hall–Petch effect in fcc metals using non-local crystal plasticity

Many conventional continuum approaches to solid mechanics do not address the size sensitivity of deformation to microstructural features like grain boundaries, and are therefore unable to capture much of the experimentally observed behavior of polycrystal deformation. We propose a non-local crystal plasticity model, in which the geometrically necessary dislocation (GND) density is calculated using a non-local integral approach. The model is based on augmented F e F p kinematics, which account for the initial microstructure (primarily grain boundaries) present in the polycrystal. With the augmented kinematics, the initial GND and the evolving GND state are determined in a consistent manner. The expanded kinematics and the non-local crystal plasticity model are used to simulate the tensile behavior in copper polycrystals with different grain sizes ranging from 14 μm to 244 μm. The simulation results show a grain size dependence on the polycrystal’s yield strength, which are in good agreement with the experimental data.

A semi-discrete and non-local crystal plasticity model for nanocrystalline metals

The application of traditional continuum crystal plasticity models to nanocrystalline metals is discussed and compared to a novel semi-discrete crystal plasticity model. The new model exhibits larger stress heterogeneities and greater amounts of intragranular plasticity. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.