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.     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

An elasto-plastic theory of dislocation and disclination fields

A linear theory of the elasto-plasticity of crystalline solids based on a continuous representation of crystal defects – dislocations and disclinations – is presented. The model accounts for the translational and rotational aspects of lattice incompatibility, respectively associated with the presence of dislocations and disclinations. The defects content relates to the incompatible plastic strain and curvature tensors. The stress state is described by using the conjugate variables to strain and curvature, i.e., the stress and couple-stress tensors. Defect motion is described by two transport equations. A dynamic interplay between dislocations and disclinations results from a disclination-induced source term in the transport of dislocations. Thermodynamic guidance provides the driving forces conjugate to dislocation and disclination velocity in a continuous context, as well as admissible constitutive relations for the latter. When dislocation and disclination velocity vanish, the model reduces to deWit’s elasto-static theory of crystal defects. It also reduces to Acharya’s linear elasto-plastic theory for dislocation fields when the disclination density is ignored. The theory is intended for use in instances where rotational defects matter, such as grain boundaries. To illustrate its applicability, a finite high-angle tilt boundary is modeled using a disclination dipole and its behavior under tensile loading normal to the boundary is shown.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.     

Multiscale modelling of the plastic anisotropy and deformation texture of polycrystalline materials

A hierarchical multilevel method is presented for the plastic deformation of polycrystalline materials with texture-induced anisotropy. It is intended as a constitutive material model for finite element codes for the simulation of metal forming processes or for the prediction of forming limits. It consists of macroscopic models of which the parameters are to be identified using the results of two-level (meso/macro) or three-level (micro/meso/macro) models. A few such two-level models are presented, ranging from the full-constraints Taylor model to the crystal-plasticity finite element models, including the grain interaction models GIA, LAMEL and ALAMEL. Validation efforts based on experimental cold rolling textures obtained for steel and aluminium alloys are shortly discussed. An assessment is also given of the assumptions of the LAMEL and ALAMEL models concerning stress and strain rate heterogeneity at grain boundaries, based on the results of a crystal plasticity finite element study. Finally a recent three-level model which also looks at the microscopic level (dislocation substructure) is discussed.     

Dislocation nucleation in Σ3 asymmetric tilt grain boundaries

Atomistic simulations were used to investigate dislocation nucleation from Σ3 asymmetric (inclined) tilt grain boundaries under uniaxial tension applied perpendicular to the boundary. Molecular dynamics was employed based on embedded atom method potentials for Cu and Al at 10 K and 300 K. Results include the grain boundary structure and energy, along with mechanical properties and mechanisms associated with dislocation nucleation from these Σ3 boundaries. The stress and work required for dislocation nucleation were calculated along with elastic stiffness of the bicrystal configurations, exploring the change in response as a function of inclination angle. Analyses of dislocation nucleation mechanisms for asymmetric Σ3 boundaries in Cu show that dislocation nucleation is preceded by dislocation dissociation from the boundary. Then, dislocations preferentially nucleate in only one crystal on the maximum Schmid factor slip plane(s) for that crystal. However, this crystal is not simply predicted based on either the Schmid or non-Schmid factors. The synthesis of these results provides a better understanding of the dislocation nucleation process in these faceted, dissociated grain boundaries.     

Multiscale dislocation dynamics simulations of shock compression in copper single crystal

Ultra short pulse shock wave propagation, plastic deformation and evolution of dislocations in copper single crystals with (0 0 1), (0 1 1) and (1 1 1) orientations are investigated using multiscale dislocation dynamics plasticity analyses. The effects of peak pressure, pulse duration, crystal anisotropy and the nonlinear elastic properties on the interaction between shock wave and dislocations are investigated. The results of our calculations show that the dislocation density has a power law dependence on pressure with a power of 1.70 and that the dislocation density is proportional to pulse duration and sensitive to crystal orientation. These results are in very good agreement with the analytical predications of Meyers et al. [Meyers, M.A., Gregori, F., Kad, B.K., Schneider, M.S., Kalantar, D.H., Remington, B.A., Ravichandran G., Boehly, T., Wark, J., 2003. Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Materialia 51, 1211–1228] and the experimental results of Murr [Murr, L.E., 1981. Residual microstructure-mechanical property relationships in shock loaded metals and alloys. In: Meyers, M.A., Murr, L.E. (Eds.), Shock Waves and High Strain Rate Phenomena in Metals. Plenum, New York, pp. 607–673]. It is shown also that incorporating the effect of crystal anisotropy in the elastic properties results in orientation dependent wave speed and peak pressure. The relaxed configurations of dislocation microstructures show the formation of microbands coincident with the slip planes.     

A multi-scale computational model of crystal plasticity at submicron-to-nanometer scales

Plastic flow in crystal at submicron-to-nanometer scales involves many new interesting problems. In this paper, a unified computational model which directly combines 3D discrete dislocation dynamics (DDD) and continuum mechanics is developed to investigate the plastic behaviors at these scales. In this model, the discrete dislocation plasticity in a finite crystal is solved under a completed continuum mechanics framework: (1) an initial internal stress field is introduced to represent the preexisting stationary dislocations in the crystal; (2) the external boundary condition is handled by finite element method spontaneously; and (3) the constitutive relationship is based on the finite deformation theory of crystal plasticity, but the discrete plastic strains induced by the slip of the newly nucleated or propagating dislocations are calculated by dislocation dynamics methodology instead of phenomenological evolution equations used in conventional crystal plasticity. These discrete plastic strains are then localized to the continuum material points by a Burgers vector density function proposed by us. Various processes, such as loop dislocation evolution, dislocation junction formation etc., are simulated to verify the reliability of this computational model. Specifically, a uniaxial compression test for micro-pillars of Cu is simulated by this model to investigate the ‘dislocation starvation hardening’ observed in the recent experiment.     

On dislocation models of grain boundaries

Dislocation models of grain boundaries was suggested by Bragg (Proc Phys Soc 52:54–55, 1940) and Burgers (Proc Phys Soc 52:23–33, 1940). The first quantitative study of these models was given by Read and Shockley (Phys Rev 78(3):275–289, 1950). They obtained a formula for the dependence of the grain boundary energy on the misorientation of the neighboring grains, which became a cornerstone of the grain boundary theory. The Read–Shockley formula was based on a proposition that the grain boundary energy is the sum of energies of the two sets of dislocations that come from the two neighboring grains. This proposition was proved under an assumption on a quite special geometry of the slip planes. This paper aims to show that the assumption is not necessary and the proposition holds for arbitrary geometry of slip planes. Another goal of this paper is to provide all basic formulas of the theory: though the dislocation model of grain boundaries is considered in all treatises on dislocation theory, a complete analysis, including the relations for lattice rotations and displacements, has not been given. This analysis shows, in particular, that continuum theory does not yield the proper relations for the lattice misorientations, and these relations must be introduced by an independent ansatz.     

A discrete dislocation analysis of bending

Bending of a strip in plane strain is analyzed using discrete dislocation plasticity where the dislocations are modeled as line defects in a linear elastic medium. At each stage of loading, superposition is used to represent the solution in terms of the infinite medium solution for the discrete dislocations and a complementary solution that enforces the boundary conditions, which is non-singular and obtained from a linear elastic, finite element solution. The lattice resistance to dislocation motion, dislocation nucleation and dislocation annihilation are incorporated into the formulation through a set of constitutive rules. Solutions for cases with multiple slip systems and with a single slip system are presented. The bending moment versus rotation relation and the evolution of the dislocation structure are outcomes of the boundary value problem solution. The effects of slip geometry, obstacles to dislocation motion and specimen size on the moment versus rotation response are considered. Also, the evolution of the dislocation structure is studied with emphasis on the role of geometrically necessary dislocations. The dislocation structure that develops leads to well-defined slip bands, with the slip band spacing scaling with the specimen height.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

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 checker-board 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.     

基于微结构动态演化机制的单晶镍基高温合金晶体塑性本构及其有限元模拟

因其优异的高温力学性能,镍基单晶高温合金在航空航天和能源等领域得到了广泛的应用.镍基单晶高温合金优异的高温性能来源于其特有的两相微结构.基于代表体胞模型及分块均匀化方法,以位错密度为主要内变量,发展了一个包含两相微结构和位错演化信息的单晶镍基高温合金塑性行为的本构模型.该本构模型充分考虑了镍基单晶合金中位错在基体相和沉淀增强相中的多种演化机制,例如,基体位错八面体滑移、立方滑移、位错攀移、交滑移、位错弓出、位错切过沉淀增强相以及位错Kear-Wilsdolf(K-W)锁形成与解锁等.在商用有限元软件ABAQUS的框架下,编制了UMAT用户材料子程序.利用该用户子程序,对单晶和多晶镍基高温合金在不同温度、不同加载方向下的单调塑性、循环塑性、蠕变等典型行为进行了计算模拟.结果表明:该晶体塑性本构模型能"统一地"刻画镍基高温合金在不同温度、不同方向下的多种变形行为,并与实验结果具有良好的一致性.     

Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach

Thermodynamically consistent, three-dimensional (3D) phase field approach (PFA) for coupled multivariant martensitic transformations (PTs), including cyclic PTs, variant–variant transformations (i.e., twinning), and dislocation evolution is developed at large strains. One of our key points is in the justification of the multiplicative decomposition of the deformation gradient into elastic, transformational, and plastic parts. The plastic part includes four mechanisms: dislocation motion in martensite along slip systems of martensite and slip systems of austenite inherited during PT and dislocation motion in austenite along slip systems of austenite and slip systems of martensite inherited during reverse PT. The plastic part of the velocity gradient for all these mechanisms is defined in the crystal lattice of the austenite utilizing just slip systems of austenite and inherited slip systems of martensite, and just two corresponding types of order parameters. The explicit expressions for the Helmholtz free energy and the transformation and plastic deformation gradients are presented to satisfy the formulated conditions related to homogeneous thermodynamic equilibrium states of crystal lattice and their instabilities. In particular, they result in a constant (i.e., stress- and temperature-independent) transformation deformation gradient and Burgers vectors. Thermodynamic treatment resulted in the determination of the driving forces for change of the order parameters for PTs and dislocations. It also determined the boundary conditions for the order parameters that include a variation of the surface energy during PT and exit of dislocations. Ginzburg–Landau equations for dislocations include variation of properties during PTs, which in turn produces additional contributions from dislocations to the Ginzburg–Landau equations for PTs. A complete system of coupled PFA and mechanics equations is presented. A similar theory can be developed for PFA to dislocations and other PTs, like reconstructive PTs and diffusive PTs described by the Cahn–Hilliard equation, as well as twinning and grain boundaries evolution.     

Constitutive modeling of strain rate effects in nanocrystalline and ultrafine grained polycrystals

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivity in nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain interior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-type homogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is successfully captured by the proposed model.     

Continuum framework for dislocation structure,energy and dynamics of dislocation arrays and low angle grain boundaries

We present a continuum framework for dislocation structure, energy and dynamics of dislocation arrays and low angle grain boundaries that are allowed to be nonplanar or nonequilibrium. In our continuum framework, we define a dislocation density potential function on the dislocation array surface or grain boundary to describe the orientation dependent continuous distribution of dislocations in a very simple and accurate way. The continuum formulations incorporate both the long-range dislocation interaction and the local dislocation line energy, and are derived from the discrete dislocation model. The continuum framework recovers the classical Read–Shockley energy formula when the long-range elastic fields of the low angle grain boundaries are canceled out. Applications of our continuum framework in this paper are focused on dislocation structures on static planar and nonplanar low angle grain boundaries and misfitting interfaces. We present two methods under our continuum framework for this purpose, including the method based on the Frank׳s formula and the energy minimization method. We show that for any (planar or nonplanar) low angle grain boundary, the Frank׳s formula holds if and only if the long-range stress field in the continuum model is canceled out, and it does not necessarily hold for a total energy minimum dislocation structure.     

Simulation of polycrystal deformation with grain and grain boundary effects

Modeling the strengthening effect of grain boundaries (Hall-Petch effect) in metallic polycrystals in a physically consistent way, and without invoking arbitrary length scales, is a long-standing, unsolved problem. A two-scale method to treat predictively the interactions of large numbers of dislocations with grain boundaries has been developed, implemented, and tested. At the first scale, a standard grain-scale simulation (GSS) based on a finite element (FE) formulation makes use of recently proposed dislocation-density-based single-crystal constitutive equations (“SCCE-D”) to determine local stresses, strains, and slip magnitudes. At the second scale, a novel meso-scale simulation (MSS) redistributes the mobile part of the dislocation density within grains consistent with the plastic strain, computes the associated inter-dislocation back stress, and enforces local slip transmission criteria at grain boundaries.Compared with a standard crystal plasticity finite element (FE) model (CP-FEM), the two-scale model required only 5% more CPU time, making it suitable for practical material design. The model confers new capabilities as follows:

(1)

The two-scale method reproduced the dislocation densities predicted by analytical solutions of single pile-ups.

(2)

Two-scale simulations of 2D and 3D arrays of regular grains predicted Hall-Petch slopes for iron of 1.2 ± 0.3 MN/m^3/2 and 1.5 ± 0.3 MN/m^3/2, in agreement with a measured slope of 0.9 ± 0.1 MN/m^3/2.

(3)

The tensile stress-strain response of coarse-grained Fe multi-crystals (9-39 grains) was predicted 2-4 times more accurately by the two-scale model as compared with CP-FEM or Taylor-type texture models.

(4)

The lattice curvature of a deformed Fe-3% Si columnar multi-crystal was predicted and measured. The measured maximum lattice curvature near grain boundaries agreed with model predictions within the experimental scatter.

     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.