A novel yield function for single crystals based on combined constraints optimization

A novel yield function representing the overall plastic deformation in a single crystal is developed using the concept of optimization. Based on the principle of maximum dissipation during a plastic deformation, the problem of single crystal plasticity is first considered as a constrained optimization problem in which constraints are yield functions for slip systems. To overcome the singularity that usually arises in solving the above problem, a mathematical technique is used to replace the above constrained optimization problem with an equivalent problem which has only one constraint. This single constraint optimization problem, the so-called combined constraints crystal plasticity (CCCP) model, is implemented into a finite element code and the results of modeling the uniaxial tensions of the single crystal copper along different crystallographic directions and also hydroforming of aluminum tubes proved the capability of the proposed CCCP model in accurately predicting the deformation in polycrystalline materials.     

Quantifying dislocation microstructure evolution and cyclic hardening in fatigued face-centered cubic single crystals

Discrete dislocation dynamics simulations were performed to investigate the dislocation microstructure evolution and cyclic hardening during the early stages of fatigue loading in nickel single crystals. The effects of the crystal size and initial dislocation densities on both the mechanical response and the evolution of dislocation microstructure were quantified. Crystals having an initial dislocation density of 10¹²  m⁻² and diameter less than $2.0\mathrm{\mu }\mathrm{m}$ do not show any dislocation density multiplication or cyclic hardening. In contrast, crystals having the same initial dislocation density and diameters larger than $2.0\mathrm{\mu }\mathrm{m}$ show a significant dislocation density accumulation in the form of dislocation cell-like structures, even after only a few number of loading cycles. This dislocation density accumulation was also accompanied by considerable cyclic hardening. The dislocation cell size and its wall thickness increase with increasing crystal size. With increasing dislocation density the critical crystal size, at which dislocation cell-structures form, decreases. The information theoretic entropy is utilized as a metric to quantify the extent of dislocation patterning and the formation and evolution of dislocation cell structures over time. Cross-slip was found to play a dominant role in the dislocation cell-structure formation. Further insights on the mechanisms contributing to the observed behavior are presented and discussed.     

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.     

On the uniqueness of a rate-independent plasticity model for single crystals

This paper presents the uniqueness and existence conditions for a rate-independent plasticity model for single crystals under a general stress state. The model is based on multiple slips on three-dimensional slip systems. The uniqueness condition for the plastic slips in a single crystal with nonlinear hardening is derived using the implicit function theorem. The uniqueness condition is the non-singularity of a matrix defined by the Schmid tensors, the elasticity, and the hardening rates of the slip systems. When this matrix becomes singular, the limitations on the loading paths that can be accommodated by the active slip systems (i.e., the existence conditions) are also given explicitly. For the compatible loading paths, a particular solution is selected by requiring the solution vector to be orthogonal to the null space of the singular coefficient matrix. The paper also presents a fully implicit algorithm for the plasticity model. Numerical examples of an fcc copper single crystal under cyclic loadings (pure shear and uniaxial strain) are presented to demonstrate the main features of the algorithm.     

HILL屈服准则与晶体塑性模型对FCC单晶材料塑性各向异性描述能力的比较

采用宏观HILL模型和晶体塑性模型对面心立方单晶(FCC)材料的非均匀交形进行了数值模拟，意在比较两种不同尺度的模型对塑性各向异性的描述能力的差异。为了使两种模型具有可比性，对于FCC单晶材科，本文提出一种用晶体塑性模型来确定HILL模型中各向异性参数的标定方法。数值分析表明，两类模型对单晶体塑性各向异性的描述能力存在着差异。对FCC单晶材料，HILL模型对各向异性的预测能力没有晶体塑性模型细致，晶体塑性模型更能追踪塑性各向异性的变化。但两种模型对应力应变响应预测的趋势是一致的。对两种模型描述的差异，做了详细的分析。     

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.     

An effective computational algorithm for rate-independent crystal plasticity based on a single crystal yield surface with an application to tube hydroforming

Up to now, several computational methods have been proposed for crystal plasticity models. The main objective of these computational methods has been to overcome the problem with the non-uniqueness of active slip systems during the plastic deformation of a single crystal. Crystal plasticity models based on a single crystal yield function have been proposed as alternative algorithms to overcome this problem. But the problem with these models is that they use a highly non-linear yield function for the crystal, which makes them computationally expensive. In this paper, a computational method is proposed that would modify a single crystal yield function in order to make it computationally efficient. Also to better capture experimental data, a new parameter is introduced into the single crystal yield function to make it more flexible. For verification, this crystal plasticity model was directly applied for the simulation of hydroforming of an extruded aluminum tube under complex strain paths. It was found that the current model is considerably faster than the previous crystal plasticity model based on a power-law type single crystal yield surface. Due to its computational efficiency, the current crystal plasticity model can also be used to calculate the anisotropy coefficients of phenomenological yield functions.     

Size effects under homogeneous deformation of single crystals: A discrete dislocation analysis

Mechanism-based discrete dislocation plasticity is used to investigate the effect of size on micron scale crystal plasticity under conditions of macroscopically homogeneous deformation. Long-range interactions among dislocations are naturally incorporated through elasticity. Constitutive rules are used which account for key short-range dislocation interactions. These include junction formation and dynamic source and obstacle creation. Two-dimensional calculations are carried out which can handle high dislocation densities and large strains up to 0.1. The focus is laid on the effect of dimensional constraints on plastic flow and hardening processes. Specimen dimensions ranging from hundreds of nanometers to tens of microns are considered. Our findings show a strong size-dependence of flow strength and work-hardening rate at the micron scale. Taylor-like hardening is shown to be insufficient as a rationale for the flow stress scaling with specimen dimensions. The predicted size effect is associated with the emergence, at sufficient resolution, of a signed dislocation density. Heuristic correlations between macroscopic flow stress and macroscopic measures of dislocation density are sought. Most accurate among those is a correlation based on two state variables: the total dislocation density and an effective, scale-dependent measure of signed density.     

Wedge indentation into elastic-plastic single crystals. 2: Simulations for face-centered cubic crystals

The asymptotic stress and deformation fields associated with the contact point singularity of a nearly-flat wedge indenter impinging on a specially-oriented single face-centered cubic crystal are derived analytically in a companion paper. In order to investigate the extent of the asymptotic fields, the indentation process is simulated numerically using single crystal plasticity. The quasistatically translating asymptotic fields consist of four angular elastic sectors separated by plastically deforming sector boundaries, as predicted in the companion paper. The asymptotic stress distributions are in accord with the analytical predictions. In addition, simulations are performed for a wedge indenter with a 90° included angle in order to investigate the consequences of finite deformation and finite lattice rotation. Several salient features of the deformation field for the nearly-flat indenter persist in the deformation field for the 90° wedge indenter. The existence of the salient features is validated by comparison to experimental measurements of the lower bound on geometrically necessary dislocation (GND) densities.     

Active slip-band separation and the energetics of slip in single crystals

This research supports recent efforts to provide an energetic approach to the prediction of stress–strain relations for single crystals undergoing single slip and to give precise formulations of experimentally observed connections between hardening of single crystals and separation of active slip-bands. Non–classical, structured deformations in the form of two-level shears permit the formulation of new measures of the active slip-band separation and of the number of lattice cells traversed during slip. A formula is obtained for the Helmholtz free energy per unit volume as a function of the shear without slip, the shear due to slip, and the relative separation of active slip-bands in a single crystal. This formula is the basis for a model, under preparation by the authors, of hardening of single crystals in single slip that is consistent with the Portevin-Le Chatelier effect and the existence of a critical resolved shear stress.     

Mechanical annealing under low-amplitude cyclic loading in micropillars

Mechanical annealing has been demonstrated to be an effective method for decreasing the overall dislocation density in submicron single crystal. However, simultaneously significant shape change always unexpectedly happens under extremely high monotonic loading to drive the pre-existing dislocations out of the free surfaces. In the present work, through in situ TEM experiments it is found that cyclic loading with low stress amplitude can drive most dislocations out of the submicron sample with virtually little change of the shape. The underlying dislocation mechanism is revealed by carrying out discrete dislocation dynamic (DDD) simulations. The simulation results indicate that the dislocation density decreases within cycles, while the accumulated plastic strain is small. By comparing the evolution of dislocation junction under monotonic, cyclic and relaxation deformation, the cumulative irreversible slip is found to be the key factor of promoting junction destruction and dislocation annihilation at free surface under low-amplitude cyclic loading condition. By introducing this mechanics into dislocation density evolution equations, the critical conditions for mechanical annealing under cyclic and monotonic loadings are discussed. Low-amplitude cyclic loading which strengthens the single crystal without seriously disturbing the structure has the potential applications in the manufacture of defect-free nano-devices.     

Wedge indentation into elastic-plastic single crystals, 1: Asymptotic fields for nearly-flat wedge

Asymptotic stress and deformation fields under the contact point singularities of a nearly-flat wedge indenter and of a flat punch are derived for elastic ideally-plastic single crystals with three effective in-plane slip systems that admit a plane strain deformation state. Face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal-close packed (HCP) crystals are considered. The asymptotic fields for the flat punch are analogous to those at the tip of a stationary crack, so a potential solution is that the deformation field consists entirely of angular constant stress plastic sectors separated by rays of plastic deformation across which stresses change discontinuously. The asymptotic fields for a nearly-flat wedge indenter are analogous to those of a quasistatically growing crack tip fields in that stress discontinuities can not exist across sector boundaries. Hence, the asymptotic fields under the contact point singularities of a nearly-flat wedge indenter are significantly different than those under a flat punch. A family of solutions is derived that consists entirely of elastically deforming angular sectors separated by rays of plastic deformation across which the stress state is continuous. Such a solution can be found for FCC and BCC crystals, but it is shown that the asymptotic fields for HCP crystals must include at least one angular constant stress plastic sector. The structure of such fields is important because they play a significant role in the establishment of the overall fields under a wedge indenter in a single crystal. Numerical simulations—discussed in detail in a companion paper—of the stress and deformation fields under the contact point singularity of a wedge indenter for a FCC crystal possess the salient features of the analytical solution.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

A nonclassical constitutive model for crystal plasticity and its application

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Size effects on void growth in single crystals with distributed voids

The effect of void size on void growth in single crystals with uniformly distributed cylindrical voids is studied numerically using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter. A plane strain cell model is analyzed for a single crystal with three in-plane slip systems. It is observed that small voids allow much larger overall stress levels than larger voids for all the stress triaxialities considered. The amount of void growth is found to be suppressed for smaller voids at low stress triaxialities. Significant differences are observed in the distribution of slips and on the shape of the deformed voids for different void sizes. Furthermore, the orientation of the crystalline lattice is found to have a pronounced effect on the results, especially for the smaller void sizes.     

Bending of single crystal microcantilever beams of cube orientation: Finite element model and experiments

The aim of this work is to investigate the microstructure evolution, stress-strain response and strain hardening behavior of microscale beams. For that purpose, two single crystal cantilever beams in the size dependent regime were manufactured by ion beam milling and beams were bent with an indenter device. A crystal plasticity material model for large deformations was implemented in a finite element framework to further investigate the effect of boundary constraints. Simulations were performed using bulk material properties of single crystal copper without any special treatment for the strain gradients. The difference between the slopes of the experimental and the simulated force displacement curves suggested negligible amount of strain gradient hardening compared to the statistical hardening mechanisms.     

A stochastic approach to capture crystal plasticity

A stochastic crystal plasticity model is proposed and applied within the rate-independent regime. As opposed to conventional deterministic algorithms wherein multiple slip systems are activated and redundant constraints may exist, the new Monte Carlo plasticity (MCP) paradigm is based on a stochastic chain of singly activated slip systems and thus avoids the possible ill-condition associated with multi-slip algorithms. The choice of the activated slip system is made at each Monte Carlo (MC) step based on the Metropolis algorithm. The MCP model is implemented within a Material Point Method (MPM) as a constitutive model to capture the elasto-plastic behavior of polycrystalline materials. A comparison with a commonly used singular value decomposition (SVD) algorithm indicates that MCP offers superior computational efficiency while maintaining comparable accuracy.     

A dislocation density-based single crystal constitutive equation

Single crystal constitutive equations based on dislocation density (SCCE-D) were developed from Orowan’s strengthening equation and simple geometric relationships of the operating slip systems. The flow resistance on a slip plane was computed using the Burger’s vector, line direction, and density of the dislocations on all other slip planes, with no adjustable parameters. That is, the latent/self-hardening matrix was determined by the crystallography of the slip systems alone. The multiplication of dislocations on each slip system incorporated standard 3-parameter dislocation density evolution equations applied to each slip system independently; this is the only phenomenological aspect of the SCCE-D model. In contrast, the most widely used single crystal constitutive equations for texture analysis (SCCE-T) feature 4 or more adjustable parameters that are usually back-fit from a polycrystal flow curve. In order to compare the accuracy of the two approaches to reproduce single crystal behavior, tensile tests of single crystals oriented for single slip were simulated using crystal plasticity finite element modeling. Best-fit parameters (3 for SCCE-D, 4 for SCCE-T) were determined using either multiple or single slip stress–strain curves for copper and iron from the literature. Both approaches reproduced the data used for fitting accurately. Tensile tests of copper and iron single crystals oriented to favor the remaining combinations of slip systems were then simulated using each model (i.e. multiple slip cases for equations fit to single slip, and vice versa). In spite of fewer fit parameters, the SCCE-D predicted the flow stresses with a standard deviation of 14 MPa, less than one half that for the SCCE-T conventional equations: 31 MPa. Polycrystalline texture simulations were conducted to compare predictions of the two models. The predicted polycrystal flow curves differed considerably, but the differences in texture evolution were insensitive to the type of constitutive equations. The SCCE-D method provides an improved representation of single-crystal plastic response with fewer adjustable parameters, better accuracy, and better predictivity than the constitutive equations most widely used for texture analysis (SCCE-T).     

Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the cud of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The rele- vance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.