Plane constrained uniaxial extension of a single crystal strip

Within continuum dislocation theory the plane constrained uniaxial extension of a single crystal strip deforming in single or double slip is analyzed. For the single and symmetric double slip, the closed-form analytical solutions are found which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Numerical solutions for the non-symmetric double slip are obtained by finite element procedures.     

A continuum model for initiation and evolution of deformation twinning

Within continuum dislocation theory the plastic deformation of a single crystal with one active slip system under plane-strain constrained shear is investigated. By introducing a twinning shear into the energy of the crystal, we show that in a certain range of straining the formation of deformation twins becomes energetically preferable. An energetic threshold for the onset of twinning is determined. A rough analysis qualitatively describes not only the evolving volume fractions of twins but also their number during straining. Finally, we analyze the evolution of deformation twins and of the dislocation network at non-zero dissipation. We present the corresponding stress-strain hysteresis, the evolution of the plastic distortion, the twin volume fractions and the dislocation densities.     

A dislocation-based constitutive law for pure Zr including temperature effects

In this work, a single crystal constitutive law for multiple slip and twinning modes in single phase hcp materials is developed. For each slip mode, a dislocation population is evolved explicitly as a function of temperature and strain rate through thermally-activated recovery and debris formation and the associated hardening includes stage IV. A stress-based hardening law for twin activation accounts for temperature effects through its interaction with slip dislocations. For model validation against macroscopic measurement, this single crystal law is implemented into a visco-plastic-self-consistent (VPSC) polycrystal model which accounts for texture evolution and contains a subgrain micromechanical model for twin reorientation and morphology. Slip and twinning dislocations interact with the twin boundaries through a directional Hall–Petch mechanism. The model is adjusted to predict the plastic anisotropy of clock-rolled pure Zr for three different deformation paths and at four temperatures ranging from 76 K to 450 K (at a quasi-static rate of 10⁻³ 1/s). The model captures the transition from slip-dominated to twinning-dominated deformation as temperature decreases, and identifies microstructural mechanisms, such as twin nucleation and twin–slip interactions, where future characterization is needed.     

Plane constrained shear of single crystal strip with two active slip systems

Within continuum dislocation theory the plane constrained shear of a single crystal strip with two active slip systems is considered. An analytical solution is found for symmetric double slip which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effects. Comparison with discrete dislocation simulations shows good agreement between the discrete and continuum approaches. Numerical procedures in the general case of non-symmetric double slip are proposed.     

Influence of single crystal orientation on homogeneous dislocation nucleation under uniaxial loading

Atomistic simulations are used to investigate how the stress required for homogeneous nucleation of partial dislocations in single crystal copper under uniaxial loading changes as a function of crystallographic orientation. Molecular dynamics is employed based on an embedded-atom method potential for Cu at 10 and 300 K. Results indicate that non-Schmid parameters are important for describing the calculated dislocation nucleation behavior for single crystal orientations under tension and compression. A continuum relationship is presented that incorporates Schmid and non-Schmid terms to correlate the nucleation stress over all tensile axis orientations within the stereographic triangle. Simulations investigating the temperature dependence of homogeneous dislocation nucleation yield activation volumes of ≈0.5- and activation energies of . For uniaxial compression, full dislocation loop nucleation is observed, in contrast to uniaxial tension. One of the main differences between uniaxial tension and compression is how the applied stress is resolved normal to the slip plane on which dislocations nucleate—in tension, this normal stress is tensile, and in compression, it is compressive. Last, the tension-compression asymmetry is examined as a function of loading axis orientation. Orientations with a high resolved stress normal to the slip plane on which dislocations nucleate have a larger tension-compression asymmetry with respect to dislocation nucleation than those orientations with a low resolved normal stress. The significance of this research is that the resolved stress normal to the slip plane on which dislocations nucleate plays an important role in partial (and full) dislocation loop nucleation in FCC Cu single crystals.     

含冷却孔镍基合金次级取向效应的应变梯度晶体塑性有限元研究

单晶镍基合金具有优异的耐高温、高强、高韧等性能, 这些力学性能受制造过程引入的次级取向和冷却孔的影响. 已有研究大多关注单孔薄板的变形机理和力学性能, 而工程中应用的往往是多孔薄板, 当前亟需阐明多孔的塑性滑移带变形机理、次级取向效应以及冷却孔引起的应变梯度效应. 文章采用基于位错机制的非局部晶体塑性本构模型对含冷却孔镍基单晶薄板的单拉变形进行了数值模拟. 此模型基于塑性滑移梯度与几何必需位错的关系引入了位错流动项, 因此可有效刻画非均匀变形过程中的应变梯度效应. 为了全面揭示含孔镍基薄板的次级取向效应, 系统研究了[100]和[110]取向(两种次级取向)下镍基薄板的单拉变形行为, 并重点探究了在两种次级取向下冷却孔数量对薄板塑性行为的影响. 此外, 还分析了镍基合金板变形过程中各个滑移系上分切应力变化、主导滑移系开动以及几何必需位错密度的演化过程, 并讨论了塑性滑移量及其分布特征对不同次级取向镍基合金板强度的影响. 研究表明, 单孔和多孔的[110]薄板抗拉强度均低于[100]薄板, 多孔薄板的塑性变形过程比单孔薄板更为复杂且受次级取向影响更大, 并且发生滑移梯度位置主要位于冷却孔附近以及塑性滑移带区域. 研究结果可为工程中镍基合金的设计和服役提供理论指导.      

Non-local crystal plasticity model with intrinsic SSD and GND effects

A strain gradient-dependent crystal plasticity approach is presented to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. In order to be capable of predicting scale dependence, the heterogeneous deformation-induced evolution and distribution of geometrically necessary dislocations (GNDs) are incorporated into the phenomenological continuum theory of crystal plasticity. Consequently, the resulting boundary value problem accommodates, in addition to the ordinary stress equilibrium condition, a condition which sets the additional nodal degrees of freedom, the edge and screw GND densities, proportional (in a weak sense) to the gradients of crystalline slip. Next to this direct coupling between microstructural dislocation evolutions and macroscopic gradients of plastic slip, another characteristic of the presented crystal plasticity model is the incorporation of the GND-effect, which leads to an essentially different constitutive behaviour than the statistically stored dislocation (SSD) densities. The GNDs, by their geometrical nature of locally similar signs, are expected to influence the plastic flow through a non-local back-stress measure, counteracting the resolved shear stress on the slip systems in the undeformed situation and providing a kinematic hardening contribution. Furthermore, the interactions between both SSD and GND densities are subject to the formation of slip system obstacle densities and accompanying hardening, accountable for slip resistance. As an example problem and without loss of generality, the model is applied to predict the formation of boundary layers and the accompanying size effect of a constrained strip under simple shear deformation, for symmetric double-slip conditions.     

Thermodynamically consistent phase field approach to dislocation evolution at small and large strains

A thermodynamically consistent, large strain phase field approach to dislocation nucleation and evolution at the nanoscale is developed. Each dislocation is defined by an order parameter, which determines the magnitude of the Burgers vector for the given slip planes and directions. The kinematics is based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions. The relationship between the rates of the plastic deformation gradient and the order parameters is consistent with phenomenological crystal plasticity. Thermodynamic and stability conditions for homogeneous states are formulated and satisfied by the proper choice of the Helmholtz free energy and the order parameter dependence on the Burgers vector. They allow us to reproduce desired lattice instability conditions and a stress-order parameter curve, as well as to obtain a stress-independent equilibrium Burgers vector and to avoid artificial dissipation during elastic deformation. The Ginzburg–Landau equations are obtained as the linear kinetic relations between the rate of change of the order parameters and the conjugate thermodynamic driving forces. A crystalline energy coefficient for dislocations is defined as a periodic step-wise function of the coordinate along the normal to the slip plane, which provides an energy barrier normal to the slip plane and determines the desired, mesh-independent height of the dislocation bands for any slip system orientation. Gradient energy contains an additional term, which excludes the localization of a dislocation within a height smaller than the prescribed height, but it does not produce artificial interface energy. An additional energy term is introduced that penalizes the interaction of different dislocations at the same point. Non-periodic boundary conditions for dislocations are introduced which include the change of the surface energy due to the exit of dislocations from the crystal. Obtained kinematics, thermodynamics, and kinetics of dislocations at large strains are simplified for small strains and rotations, as well.     

Mechanism-based strain gradient crystal plasticity—I. Theory

We have been developing the theory of mechanism-based strain gradient plasticity (MSG) to model size-dependent plastic deformation at micron and submicron length scales. The core idea has been to incorporate the concept of geometrically necessary dislocations into the continuum plastic constitutive laws via the Taylor hardening relation. Here we extend this effort to develop a mechanism-based strain gradient theory of crystal plasticity. In this theory, an effective density of geometrically necessary dislocations for a specific slip plane is introduced via a continuum analog of the Peach-Koehler force in dislocation theory and is incorporated into the plastic constitutive laws via the Taylor relation.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

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.     

Effect of secondary orientation on notch-tip plasticity in superalloy single crystals

Numerical and experimental evolutions of slip fields in notched Ni-Base Single Crystal superalloy tensile specimens are presented as a function of secondary crystallographic orientation. The numerical predictions based on three-dimensional anisotropic elasticity and crystal plasticity are compared with experimental observations. The results illustrate the strong dependence of the slip patterns and the plastic zone size and shape on the secondary orientation of notches, which can have important consequences on crack initiation. Specific orientations or non-symmetric notch geometries lead to non-symmetric patterns on both sides of the sample. The computations show that strongly different plastic zones are expected in the core of the sample and at free surfaces. The ability of the anisotropic elastic model to anticipate the plastic domains, based on identifying dominant slip systems, is confirmed by the crystal plasticity computations, at low load levels. An important observation is that kink shear banding is a real deformation mode operating at crack tips and notches in high strength nickel-based single crystal superalloys for specific orientations.     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源, 从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程. 本文基于晶体塑性理论, 从滑移系的受力和变形出发研究材料的细观损伤, 建立了考虑滑移面分解正应力的细观损伤模型, 为晶体材料解理断裂的分析提供了新方法. 首先, 在晶体弹塑性变形构型的基础上引入损伤变形梯度张量的概念, 从变形运动学着手建立了考虑损伤能量耗散的本构方程, 并推导了塑性流动方程与损伤演化方程; 然后, 建立了相应的数值计算方法, 给出了应力与状态变量的更新算法, 推导了Jacobian矩阵的表达式; 接着, 以$[100]$取向的单晶铜材料为例, 通过有限元计算与试验结果的对比, 并采用粒子群优化算法标定了11个材料细观参数; 最后, 将所提细观损伤模型应用于RVE单轴拉伸过程的模拟, 得到了考虑损伤影响的应力应变曲线, 并分析了材料的塑性流动与损伤演化过程. 结果表明, 本文所提模型能够计算材料在受载过程中的损伤累积效应, 合理反映晶体材料的细观损伤机理.      

Multiscale modeling of the plasticity in an aluminum single crystal

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.     

An atomistically-informed dislocation dynamics model for the plastic anisotropy and tension-compression asymmetry of BCC metals

Atomistic simulations have shown that a screw dislocation in body-centered cubic (BCC) metals has a complex non-planar atomic core structure. The configuration of this core controls their motion and is affected not only by the usual resolved shear stress on the dislocation, but also by non-driving stress components. Consequences of the latter are referred to as non-Schmid effects. These atomic and micro-scale effects are the reason slip characteristics in deforming single and polycrystalline BCC metals are extremely sensitive to the direction and sense of the applied load. In this paper, we develop a three-dimensional discrete dislocation dynamics (DD) simulation model to understand the relationship between individual dislocation glide behavior and macro-scale plastic slip behavior in single crystal BCC Ta. For the first time, it is shown that non-Schmid effects on screw dislocations of both {110} and {112} slip systems must be implemented into the DD models in order to predict the strong plastic anisotropy and tension-compression asymmetry experimentally observed in the stress-strain curves of single crystal Ta. Incorporation of fundamental atomistic information is critical for developing a physics-based, predictive meso-scale DD simulation tool that can connect length/time scales and investigate the underlying mechanisms governing the deformation of BCC metals.     

Nucleation of partial dislocations at a crack and its implication on deformation mechanisms of nanostructured metals

Nucleation of partial dislocations at a crack is analyzed based a multiscale model that incorporates atomic information into continuum-mechanics approach. The crack and the slip plane as the extension of the crack are modeled as a surface of displacement discontinuities embedded in an elastic medium. The atomic potential between the adjacent atomic layers along the slip plane is assumed to be the generalized stacking fault energy, which is obtained based on atomic calculations. The relative displacements along the slip plane, corresponding to the configurations of partial dislocations and stacking faults, are solved through the variational boundary integral method. The energetics of partial dislocation nucleation at the crack in FCC metals Al and Cu are comparatively studied for their distinctive difference in the intrinsic stacking fault energy. Compared with nucleation of perfect dislocations in previous studies, several new features have emerged. Among them, the critical stress and activation energy for nucleation of partial dislocations are markedly lowered. Depending on the value of stacking fault energy and crack configuration, the saddle-point configurations of partial dislocations can be vastly different in terms of the nucleation sequence and the size of the stacking fault. These findings have significant implication for identifying the fundamental dislocation and grain-boundary-mediated deformation mechanisms, which may account for the limiting strength and the high strain rate sensitivity of nanostructured metals.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.