Simulation of micro-indentation hardness of FCC single crystals by mechanism-based strain gradient crystal plasticity

The size effect observed in the micro-indentation of FCC single crystal copper is modelled by the employment of mechanism-based strain gradient crystal plasticity (MSG-CP). The total slip resistance in each active system is assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The MSG-CP constitutive model is implemented into the Abaqus/Standard FE platform by developing the User MATerial subroutine UMAT. The simulation of micro-indentation hardness on (0 0 1) and (1 1 1) single crystal copper, with a conical indenter having a sharp tip, a conical indenter with a spherical tip and a three-sided Berkovich indenter, is undertaken. The phenomena of pile-up and sink-in have been observed in the simulation and dealt with by appropriate use of the contact analysis function in Abaqus. These phenomena have been taken into account in the determination of the contact areas and hence the average indentation depth for anisotropic single crystals. The depth dependence of the micro-indentation hardness, the size effect, is calculated. The micro-hardness results from the simulation are compared with those of the published experimental ones in the literature and a good agreement is found.     

Orientation dependence of the plastic slip near notches in ductile FCC single crystals

Results from experiments conducted on copper FCC single crystals are reported. Two symmetric crystallographic orientations and four nonsymmetric crystallographic orientations were tested. The slip line fields that form near a pre-existing notch in these specimens were observed. The changes in these patterns as the orientation of the notch in the crystal is rotated in an {101} plane are discussed. Sectors of similar slip line patterns are identified and the type of boundaries between these sectors are discussed. A type of sector boundary called mixed kink is identified. Specimen orientations that differ by 90° are found to have different slip line patterns, contrary to the predictions of perfectly plastic slip line theory. The locations of the first slip lines to form are compared to the predictions obtained using anisotropic linear elastic stress field solutions and the initial plane-strain yield surfaces. It is found that comparison of these surface slip line fields to plane strain crack tip solutions in the annular region between 350 and is justified. The differences in anisotropic elastic solutions for orientations that are 90° apart explain the lack of agreement with perfectly plastic slip line theory.     

Strain hardening in bent copper foils

A series of systematic tensile and microbend tests were conducted on copper foil specimens with different thicknesses. The specimens were made of a copper foil having almost unidirectional crystal orientations that was considered to be nearly single-crystal. In order to investigate the effects of slip system interactions, two different crystal orientations relative to the tensile direction were considered in the tests: one is close to coplanar double-slip orientation, and the other is close to the ideal cube orientation (the tensile direction nearly coincides to [0 0 1]) that yields multi-planar multi-slip deformation. We extended the microbend test method to include the reversal of bending, and we attempted to divide the total amount of strain-hardening into isotropic and kinematic hardening components. In the tensile tests, no systematic tendency of size dependence was observed. In the microbend tests, size-dependent kinematic hardening behavior was observed for both the crystal orientations, while size dependence of isotropic hardening was observed only for the multi-planar multi-slip case. We introduce an extended crystal plasticity model that accounts for the effects of the geometrically necessary dislocations (GNDs), which correspond to the spatial gradients of crystallographic slips. Through numerical simulations performed using the model, the origin of the size-dependent behavior observed in the microbend tests is discussed.     

Indentation Schmid factor and orientation dependence of nanoindentation pop-in behavior of NiAl single crystals

Instrumented nanoindentation techniques have been widely used to characterize the small-scale mechanical behavior of materials. The elastic-plastic transition during nanoindentation is often indicated by a sudden displacement burst (pop-in) in the measured load-displacement curve. In defect-free single crystals, the pop-in is believed to be the result of homogeneous dislocation nucleation because the maximum shear stress corresponding to the pop-in load approaches the theoretical strength of the materials and because the statistical distribution of pop-in stresses is consistent with what is expected for a thermally activated process of homogeneous dislocation nucleation. This paper investigates whether this process is affected by crystallography and stress components other than the resolved shear stress. A Stroh formalism coupled with the two-dimensional Fourier transformation is used to derive the analytical stress fields in elastically anisotropic solids under Hertzian contact, which allows the determination of an indentation Schmid factor, namely, the ratio of maximum resolved shear stress to the maximum contact pressure. Nanoindentation tests were conducted on B2-structured NiAl single crystals with different surface normal directions. This material was chosen because it deforms at room temperature by {1 1 0}〈0 0 1〉 slip and thus avoids the complexity of partial dislocation nucleation. Good agreement is obtained between the experimental data and the theoretically predicted orientation dependence of pop-in loads based on the indentation Schmid factor. Pop-in load is lowest for indentation directions close to 〈1 1 1〉 and highest for those close to 〈0 0 1〉. In nanoindentation, since the stress component normal to the slip plane is typically comparable in magnitude to the resolved shear stress, we find that the pressure sensitivity of homogeneous dislocation nucleation cannot be determined from pop-in tests. Our statistical measurements generally confirm the thermal activation model of homogeneous dislocation nucleation. That is, the extracted dependence of activation energy on resolved shear stress is almost the same for all the indentation directions considered in this study, except for those close to 〈0 0 1〉. Because very high pop-in loads are measured for orientations close to 〈0 0 1〉, which implies a large contact area at pop-in, there is a higher probability of activating pre-existing dislocations in these orientations, which may explain the discrepancy near 〈0 0 1〉.     

Combined numerical simulation and nanoindentation for determining mechanical properties of single crystal copper at mesoscale

Constitutive laws are critical in the investigation of mechanical behavior of single crystal or polycrystalline materials in applications spanning from microscale to macroscale. In this investigation, a combined FEM simulation and experimental nanoindentation approach was taken to determine the mechanical behavior of single crystal copper incorporating the mesoplastic constitutive model. This model was implemented in a user-defined subroutine in 3D ABAQUS/Explicit code. Nanoindentation was modeled using the multiscale modeling technique involving mesoplasticity and elasticity, i.e., mesoplastic constitutive model was used near the local nanoindentation region (where the dislocations are generated) while elastic constitutive model was used in rest of the region in the workmaterial. The meso-mechanical behavior of the crystalline structure and the effect of the mesoplastic parameters on the nanoindentation load-displacement relationships were investigated in the FEM analysis. Nanoindentation tests were conducted on single crystal copper to determine load-displacement relationships. Appropriate mesoplastic parameters were determined by fitting the simulated load-displacement curves to the experimental data. The mesoplastic model, with appropriate parameters, was then used to determine the stress-strain relationship of a single crystal copper at meso-scale. The effect of indenter radius (3.4-) on material hardness under nanoindentation was simulated and found to match the experimental data for several indenter radii (3.4, 10 and ). A comparison of the topographies of nanoindentation impressions in the experiments with FEM results showed a reasonably good agreement.     

Deformation and failure mechanisms of single crystal superalloy under temperature gradient

The crystallographic constitutive model under temperature gradient is developed and introduced to study the deformation and failure mechanisms of single crystal superalloy. Tensile tests of thin-walled pipe specimen at different temperatures without cooled air flow were carried out. Based on the experimental results, the temperature dependence of constitutive model was studied and the basic parameters of constitutive model were obtained. To investigate the deformation and failure mechanisms, the thin-walled pipe specimen with cooled air flow under temperature gradient were tested. Considered the fluid-solid interface (FSI), a finite element method (FEM) was proposed to simulate the process of tension. In FEM, the activation rate of slip system was defined as the failure law of specimen under temperature gradient. The simulation result was in good agreement with the experiment result. Furthermore, the fracture surface of the specimen was observed by the scanning electron microscopy (SEM). The microstructure revealed that the slip deformation belonged to {1 1 1} crystalplane is a principal failure mechanism of single crystal superalloy under temperature gradient. The results of the SEM also implied that the proposed FEM method can be used to systemically study the deformation and failure behavior of single crystal superalloy cooled blade.     

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.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

Multiscale modelling of the induced plastic anisotropy in bcc metals

This paper presents a new framework to predict the qualitative and quantitative variation in local plastic anisotropy due to crystallographic texture in body-centered cubic polycrystals. A multiscale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal plasticity formulation into the subroutine of an explicit dynamic FEM code (LS-DYNA). Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Optimal results were obtained when {1 1 0}〈1 1 1〉, {1 1 2}〈1 1 1〉, and {1 2 3}〈1 1 1〉 slip systems were considered to be potentially active. The predicted material heterogeneity can be utilised for research involving any texture-dependent work hardening behaviour, such as surface roughening.     

Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper

Nanoscale contact of material surfaces provides an opportunity to explore and better understand the elastic limit and incipient plasticity in crystals. Homogeneous nucleation of a dislocation beneath a nanoindenter is a strain localization event triggered by elastic instability of the perfect crystal at finite strain. The finite element calculation, with a hyperelastic constitutive relation based on an interatomic potential, is employed as an efficient method to characterize such instability. This implementation facilitates the study of dislocation nucleation at length scales that are large compared to atomic dimensions, while remaining faithful to the nonlinear interatomic interactions. An instability criterion based on bifurcation analysis is incorporated into the finite element calculation to predict homogeneous dislocation nucleation. This criterion is superior to that based on the critical resolved shear stress in terms of its accuracy of prediction for both the nucleation site and the slip character of the defect. Finite element calculations of nanoindentation of single crystal copper by a cylindrical indenter and predictions of dislocation nucleation are validated by comparing with direct molecular dynamics simulations governed by the same interatomic potential. Analytic 2D and 3D linear elasticity solutions based on the Stroh formalism are used to benchmark the finite element results. The critical configuration of homogeneous dislocation nucleation under a spherical indenter is quantified with full 3D finite element calculations. The prediction of the nucleation site and slip character is verified by direct molecular dynamics simulations. The critical stress state at the nucleation site obtained from the interatomic potential is in quantitative agreement with ab initio density functional theory calculation.     

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.     

纳米压痕试验中压头尺寸效应的准连续介质法分析

采用准连续介质法模拟了单晶铝纳米压痕试验过程,分析了不同宽度的刚性矩形压头所引起的初始塑性变形特点,获得了载荷-压深、应变能-位移和硬度-压深曲线.从位错理论的角度分析了压头尺寸对纳米压痕测试结果的影响.研究发现:随着压头宽度的不断增大,压头下方位错形核所需要的载荷和压深程度增大,需要的应变能增加,应变能的变化速率递增,纳米硬度值减小,呈现出明显的尺寸效应.同时表明在一定的压人深度下,硬度与压头尺寸之间存在着一定的比例关系,不同尺寸压头获得的硬度值可以相互换算,但当矩形刚性压头宽度大于或等于120A时这种尺寸效应消失.研究结果为纳米压痕实验过程中压头尺寸的选择提供了参考依据.     

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.     

The Nano-Jackhammer effect in probing near-surface mechanical properties

Because of its ease of implementation and insensitivity to indenter drift, dynamic indentation techniques have been frequently used to measure mechanical properties of bulk and thin film materials as a function of indenter displacement. However, the actual effect of the oscillating tip on the material response has not been examined. Recently, it has been shown that the oscillation used with dynamic indentation techniques alters the measured hardness value of ductile metallic materials, especially at depths less than 200 nm. The alteration in the hardness is due to the added energy associated with the oscillation which assists dislocation nucleation. Atomistic simulations on nickel thin films agree with experiments that more dislocations are nucleated during dynamic indents than with quasi-static indents. Through the analysis of quasi-static and dynamic indents made into nickel single crystals and thin films, a theory to describe this phenomenon is presented. This is coined the Nano-Jackhammer effect, a combination of dislocation nucleation and strain rate sensitivity caused by indentation with a superimposed dynamic oscillation.     

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).     

Crystallographic orientation and size dependence of tension-compression asymmetry in molybdenum nano-pillars

Uniaxial tension and compression experiments on [0 0 1] and [0 1 1] oriented molybdenum nano-pillars exhibit tension-compression asymmetry, a difference in attained stresses in compression vs. tension, which is found to depend on crystallographic orientation and sample size. We find that (1) flow stresses become higher at smaller diameters in both orientations and both loading directions, (2) compressive flow stresses are higher than tensile ones in [0 0 1] orientation, and visa versa in [0 1 1] orientation, and (3) this tension-compression asymmetry is in itself size dependent. We attribute these phenomena to the dependence of twinning vs. antitwinning deformation on loading direction, to the non-planarity of screw dislocation cores in Mo crystals, and to the possibly lesser role of screw dislocations in governing nano-scale plasticity compared with bulk Mo.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

压头不同晶向下纳米压痕的多尺度研究

为了研究压头晶体各向异性对纳米压痕的影响,采用多尺度准连续介质(QC)法模拟了不同晶向Ni压头与Ag薄膜的纳米压痕过程。通过对比不同晶向下压头在薄膜上触发的原子滑移,发现压头的晶向引起的界面失配位错在很大程度上决定薄膜开启初始原子滑移系的难易。然后对比了压头在不同晶向下测得的薄膜纳米硬度,发现其计算值是一样的。最后研究压头表面和压痕表面的正应力和切应力的分布,分析了应力分布与原子滑移系的关系。     

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.     

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.