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.     

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 the strain hardening and texture evolution in high manganese steels: Experiments and numerical investigation

We present a systematic investigation on the strain hardening and texture evolution in high manganese steels where twinning induced plasticity (TWIP) plays a significant role for the materials' plastic deformation. Motivated by the stress–strain behavior of typical TWIP steels with compositions of Fe, Mn, and C, we develop a mechanistic model to explain the strain-hardening in crystals where deformation twinning dominates the plastic deformation. The classical single crystal plasticity model accounting for both dislocation slip and deformation twinning are then employed to simulate the plastic deformation in polycrystalline TWIP steels. While only deformation twinning is activated for plasticity, the simulations with samples composed of voronoi grains cannot fully capture the texture evolution of the TWIP steel. By including both twinning deformation and dislocation slip, the model is able to capture both the stress–strain behaviors and the texture evolution in Fe–Mn–C TWIP steel in different boundary-value problems. Further analysis on the strain contributions by both mechanisms suggests that deformation twinning plays the dominant role at the initial stage of plasticity in TWIP steels, and dislocation slip becomes increasingly important at large strains.     

A model for the deformations and thermodynamics of liquids involving a dislocation kinetics

A model for the deformation and thermodynamics of liquids is developed that depends on dislocation kinetics. The approach uses concepts from statistical mechanics to model a stochastic evolution equation for a scalar dislocation density function. The dislocation density is used in an idealized model for the discrete discontinuous deformation due to dislocation motion and dislocation creation kinetics. The total deformation functional for a liquid is modelled as a continuum deformation of an idealized lattice structure plus the discontinuous deformation due to dislocation kinetics. This results in a thermodynamic model that has an elastic response from the continuum lattice structure and a fluid response from the dislocation kinetics.In the thermodynamics, a generalized internal energy functional is assumed to exist and to have a dependence on the functions of entropy, continuum lattice strain, scalar dislocation density, velocity, and mass density. The continuum lattice strain is termed the recoverable strain and its conjugate variable is the thermodynamic stress. The conjugate variable to the scalar dislocation density is the thermodynamic chemical potential for a dislocation configuration, somewhat analogous to Gibbs' treatment of chemical potential for various mass species.This model implies that a liquid and a crystalline solid have analogous deformation and thermodynamic responses. Their differences appear in the dislocation densities and in the dislocation chemical potentials. To illustrate the deformation response analogy, some solutions are developed for simple laminar shear flows. Also, using some concepts primarily from Kuhlmann-Wilsdorf's melting model, a definition for a specific dislocation creation heat equivalent is given. This thermodynamic formalism suggests that the melting process can be modelled as the consequence of a continuous change in the dislocation density function.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-ENG-48.     

基于物理变量的热粘塑性本构模型

在位错的运动和产生与塑性变形的一般关系及考虑到热激活与粘性阻尼效应的位错集体运动的统一理论基础上，通过对结构参量的演化规律的具体建议，提出了一种基于物理变量的热粘塑性本构模型。在此模型的基础上，讨论了金属材料动态力学行为的微观机理。     

A dislocation density based material model to simulate the anisotropic creep behavior of single-phase and two-phase single crystals

The primary and secondary creep behavior of single crystals is observed by a material model using evolution equations for dislocation densities on individual slip systems. An interaction matrix defines the mutual influence of dislocation densities on different glide systems. Face-centered cubic (fcc), body-centered cubic (bcc) and hexagonal closed packed (hcp) lattice structures have been investigated. The material model is implemented in a finite element method to analyze the orientation dependent creep behavior of two-phase single crystals. Three finite element models are introduced to simulate creep of a γ′ strengthened nickel base superalloy in 〈1 0 0〉, 〈1 1 0〉 and 〈1 1 1〉 directions. This approach allows to examine the influence of crystal slip and cuboidal microstructure on the deformation process.     

Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics

A new numerical dislocation climb model based on incorporating the pipe diffusion theory (PDT) of vacancies with 3D discrete dislocation dynamics (DDD) is developed. In this model we hold that the climb rate of dislocations is determined by the gradient of the vacancy concentration on the segment, but not by the mechanical climb force as traditionally believed. The nodal forces on discrete dislocation segments in DDD simulation are transferred to PDT to calculate the vacancy concentration gradient. This transfer establishes a bridge connecting the DDD and PDT. The model is highly efficient and accurate. As verifications, two typical climb-involved examples are predicted, e.g. the activation of a Bardeen-Herring source as well as the shrinkage and annihilation of prismatic loops. Finally, the model is applied to study the breakup process of an infinite edge dislocation dipole into prismatic loops. This coupling methodology provides us a useful tool to intensively study the evolution of dislocation microstructures at high temperatures.     

Dislocation emission from nanovoid with the effect of neighboring nanovoids and surface stresses

In the light of the vital mechanism for nanovoid evolution depending strongly on the effect of neighboring nanovoids, a generalized self-consistent model is suggested to describe nanovoid growth by the dislocation emission from nanovoid surface accounting for the effect of neighboring nanovoids and surface stresses in ductile porous materials. The explicit solution of the critical stress for dislocation emission is derived by means of the complex variable method. Analysis shows the advanced model can be implemented as the effective means to address the strong dependence of the nanovoid growth by the dislocation emission upon the size and volume fraction of nanovoid, growth/shrinkage of the neighboring nanovoid, remote applied stress as well as the surface effect.     

On dislocation patterning: Multiple slip effects in the rate equation approach

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development of various types of dislocation microstructures (dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc.), depending on the externally applied loading and the intrinsic lattice constraints. The term “dislocation patterning” was introduced over 20 years ago by the third author and a corresponding “gradient dislocation dynamics” framework was suggested to describe such phenomena. In the W–A model proposed at that time by the last two authors, it was shown how coupled nonlinear evolution equations of the reaction-diffusion type for the forest (immobile) and gliding (mobile) dislocation densities can generate dislocation microstructures which correspond to walls perpendicular to the slip direction for Cu-crystals oriented for single slip under cyclic loading conditions. This model is adapted to the multiple slip case here. Weakly nonlinear analysis predicts that dislocation patterns should correspond to domains of walls perpendicular to each slip direction and separated by domain walls in the same orientations. This result is confirmed by numerical analysis and experimental observations. The present model generalizes the original W–A model to the case of multiple slip and considers also explicitly gradient effects by allowing for non-uniform dislocation velocities and internal stress effects.     

Dislocation model of localized plastic deformation initiated with a flat punch

We consider a classical contact mechanics problem, namely, the indentation of a ductile half-plane by a rigid flat punch (in plane strain), and revisit it using the dislocation mechanics approach. The dislocation nucleation and dislocation interaction beneath the indenter are examined. The threshold load for dislocation nucleation and the dislocation emission angle are obtained in analytical form. Moreover, based on the consideration of dislocation interaction, we explore the mechanism of contact load evolution (hardening). A triangular “dead zone” beneath the indenter, which could not be thus far accurately explained by traditional continuum models, is predicted in good agreement with the results of careful experiments that are reported in the literature. The proposed model is likely to be useful for the analysis of contacts at both the micro- and macro-scales.     

基于微结构演化的V5Cr5Ti合金塑性本构模型

为合理描述V5Cr5Ti合金的塑性变形行为,本文建立了基于微结构演化的塑性本构模型。首先,采用小尺寸试样开展了V5Cr5Ti合金单轴拉伸试验,并对其在不同应变程度下的微结构演化特征进行了分析。研究发现,影响V5Cr5Ti合金塑性变形行为的主要因素是位错密度演化以及团簇状和弥散析出相。据此建立了位错密度演化方程、组分相含量体积分量演化方程,并考虑团簇状和弥散状第二相对V5Cr5Ti合金流动应力的影响,进一步建立了包括非热应力、热激活应力和弥散相强化应力的流动应力关系式。最后,根据隐式应力更新算法对新模型进行了有限元实现,并与实验结果进行比较,验证了新模型的合理性和预测精度。     

Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys

Discrete dislocation dynamics (DDD) has been used to model the deformation of nickel-based single crystal superalloys with a high volume fraction of precipitates at high temperature. A representative volume cell (RVC), comprising of both the precipitate and the matrix phase, was employed in the simulation where a periodic boundary condition was applied. The dislocation Frank-Read sources were randomly assigned with an initial density on the 12 octahedral slip systems in the matrix channel. Precipitate shearing by superdislocations was modelled using a back force model, and the coherency stress was considered by applying an initial internal stress field. Strain-controlled loading was applied to the RVC in the [0 0 1] direction. In addition to dislocation structure and density evolution, global stress-strain responses were also modelled considering the influence of precipitate shearing, precipitate morphology, internal microstructure scale (channel width and precipitate size) and coherency stress. A three-stage stress-strain response observed in the experiments was modelled when precipitate shearing by superdislocations was considered. The polarised dislocation structure deposited on the precipitate/matrix interface was found to be the dominant strain hardening mechanism. Internal microstructure size, precipitate shape and arrangement can significantly affect the deformation of the single crystal superalloy by changing the constraint effect and dislocation mobility. The coherency stress field has a negligible influence on the stress-strain response, at least for cuboidal precipitates considered in the simulation. Preliminary work was also carried out to simulate the cyclic deformation in a single crystal Ni-based superalloy using the DDD model, although no cyclic hardening or softening was captured due to the lack of precipitate shearing and dislocation cross slip for the applied strain considered.     

基于微结构演化的V5Cr5Ti合金塑性本构模型

为合理描述V5Cr5Ti合金的塑性变形行为,本文建立了基于微结构演化的塑性本构模型。首先,采用小尺寸试样开展了V5Cr5Ti合金单轴拉伸试验,并对其在不同应变程度下的微结构演化特征进行了分析。研究发现,影响V5Cr5Ti合金塑性变形行为的主要因素是位错密度演化以及团簇状和弥散析出相。据此建立了位错密度演化方程、组分相含量体积分量演化方程,并考虑团簇状和弥散状第二相对V5Cr5Ti合金流动应力的影响,进一步建立了包括非热应力、热激活应力和弥散相强化应力的流动应力关系式。最后,根据隐式应力更新算法对新模型进行了有限元实现,并与实验结果进行比较,验证了新模型的合理性和预测精度。     

Continuum slip viscoplasticity with the Haasen constitutive model: Application to CdTe single crystal inelasticity

Small strain constitutive equations are developed for the thermomechanical behavior of semiconductor single crystals, including dislocation density as an evolving parameter. The model of Haasen, Alexander and coworkers is modified (extended) to include evolution of coefficients in the definition of internal stress. These account for an evolving dislocation substructure. The resulting model is applied in a continuum slip framework to allow multiple slip orientations. Slip system interaction rules are adapted to include slip system interaction for multiple slip conditions and to suppress secondary slip and dislocation density generation for single slip orientations. The approach is discussed relative to other models for viscoplasticity of single crystals and is examined in the context of thermodynamics with internal state variables. The framework is used to correlate experimental data from compression tests of single crystals of the compound semiconductor CdTe from room temperature to near the melting point. Sensitivity of the model to uncertainties such as initial dislocation density is explored.     

Micromechanical modeling of the elastic–viscoplastic behavior of polycrystalline steels

A self-consistent model developed to describe the elastic–viscoplastic behavior of heterogeneous materials is applied to low carbon steels to simulate tensile tests at various strain rates in the low temperature range. The choice of crystalline laws implemented in the model is discussed through the viscoplastic flow rule and several strain-hardening laws. Comparisons between three work-hardening models show that the account of dislocation annihilation improves the results on simulations at large strains. The evolution of the Lankford coefficients and texture development are also successfully simulated. Some microstructural aspects of deformation such as the stored energy and the evolution of the flow rates are discussed. By including the dislocation density on each slip system as internal variable, intragranular heterogeneities are underscored.     

A thermodynamically consistent model of static and dynamic recrystallization

We propose a thermodynamically consistent model of static and dynamic recrystallization for metals during and after severe plastic deformations that is capable of predicting the evolution of dislocation density as well as mean grain size.     

Alloying effects on dislocation substructure evolution of aluminum alloys

The constitutive response of aluminum alloys is controlled by the evolution of dislocation substructure including mobile and forest dislocation density, cell size distribution and morphology, and misorientation angle between neighboring cells. The present study focuses upon the small strain regime and compares the measured microstructural evolution of 3003, 5005, and 6022 aluminum alloys during deformation. Room temperature tensile deformation experiments were performed on industrially manufactured specimens of each alloy and the evolving microstructure was compared with the mechanical response. The dislocation structure evolution was characterized using transmission electron microscopy and orientation imaging of deformed specimens. It was observed that structural evolution is a function of lattice orientation and the character of neighboring grains. In general, the dislocation cell size and misorientation angle between dislocation cells evolves systematically with deformation at relatively small strain levels.