Hardness Of Single Crystals Research Articles

Mesoscale slip behavior in single crystal and bicrystal tantalum

Single-crystal and bicrystal tensile specimens of various orientations were prepared from coarse-grained, pure Ta to investigate fundamental BCC slip behavior at the mesoscale, that is, without considering discrete dislocation mechanisms. Single crystal yield behavior was well-represented by concurrent {110} and {112} slip modes with the initial CRSS for {110} 6% lower than for {112}. A new metric quantitatively representing slip multiplicity in a crystal visco-plastic context was introduced. Initial strain hardening correlates positively with this “activity index” tending to confirm the dominant role of latent hardening in single crystals. Anelastic hysteresis at least as significant as in polycrystal/polyphase alloys was observed, with loop width increasing with pre-strain rather than solely with flow stress as previously supposed. Tensile tests of “triplets” consisting of a bicrystal specimen and its two constituent single crystal specimens revealed the differential effects of grain boundaries (GB) on strength and anelastic hysteresis. Two triplets followed the rule-of-mixtures (ROM); two triplets showed synergistic strength and anelasticity. Crystal plasticity (CP) simulations with two slip modes having different hardening behaviors predicted the single crystal stress-strain curves reasonably for various orientations. Plastic deformation incompatibility was identified as the likely source of deviation from rule-of-mixture behavior. CP simulations predicted the appearance of additional slip systems near the GBs in the synergistic triplets, but did not predict the observed strength and anelastic hysteresis.

Bauschinger effect and latent hardening under cyclic micro-bending of Ni-base Alloy 718 single crystals: Part II. Single crystal plasticity modeling with latent kinematic hardening

In this paper, the Bauschinger effect and latent hardening of single crystals are assessed in finite element calculations using a single crystal plasticity model with kinematic hardening. To this end, results of cyclic micro-bending experiments on single crystal Alloy 718 in different crystal orientations (single slip and multi slip) with respect to the loading direction are used to determine the slip system related material properties of the single crystal plasticity model. Two kinematic hardening laws are considered: a kinematic hardening law describing latent hardening and a kinematic hardening law without latent hardening. For the determination of material properties for both hardening laws, a gradient-based optimization method is used. The results show that the different strength levels observed for micro-bending tests on different crystal orientations can only be described with latent kinematic hardening well, whereas the pronounced Bauschinger effect is described well by both kinematic hardening laws. It is concluded that cyclic micro-bending experiments on single crystals using different crystal orientations give an appropriate data base for the determination of the slip system related material properties of the single crystal plasticity model with latent kinematic hardening.

Manipulating deformation mechanisms with Y alloying of Mg

The effect of Y concentration on the slip and twinning mechanisms in binary Mg–Y alloys are investigated using transmission electron microscopy, electron backscattered diffraction, and visco-plastic self-consistent polycrystal constitutive modeling. Four concentrations of Y are studied in hot-rolled and recrystallized sheet material. The materials were deformed in tension and compression in the rolling direction and compression in the normal direction in order to invoke distinct proportions of slip and twin mechanisms with each test. Within the single crystal hardening model used in polycrystal modeling, a slip-twin interaction law is introduced to account for dislocation density reductions due to dislocation absorption during twin boundary migration. We show that increasing Y concentration reduces the intensities of both the initial and deformation textures. During deformation, the plastic anisotropy in yield stress, the tension-compression asymmetry, and amount of 101‾2⟨1‾011⟩ twinning is shown to decrease with increasing Y. For each alloy, the model identifies a single set of material parameters that successfully reproduced all measured stress-strain curves and achieved agreement with measured deformation textures and twin area fractions. Transcending texture effects, the model interpretation of the flow responses suggests that increased concentrations of Y increase the critical resolved shear stress for basal slip but have negligible effects on the other slip modes. The reduced plastic anisotropy with increases in Y is explained by a concomitant decrease in the prismatic-to-pyramidal slip critical resolved shear stress ratio. The model suggests that their nearly equivalent critical resolved shear stress values lead to the enhanced non-basal activity of Mg–Y alloys, which was confirmed by transmission electron microscopy. The calculations suggest that beyond any texture differences, this reduction in twinning can be attributed to a slightly increased resistance for 101‾2⟨1‾011⟩ twin propagation, particularly in the binary with the highest Y content.

Precipitation strengthening of stress-aged Al-Cu-Mg-Ag alloy single crystals

Al-Cu-Mg-Ag single crystals with five different orientations were used to study the precipitation behavior during stress aging under a stress range from 0 to 150 MPa. TEM characterization of the aged samples indicated that three kinds of precipitates Ω, S and θ′ formed, and the number density of S precipitates increased with the applied stress. The distribution of two Ω variants Ω1/Ω2 behaved preferentially oriented under higher applied stress and the anisotropic degree was various between 0.3 and 1.4, which depends on the plane orientation of single crystal. As the stress loading directions of single crystals with (6 5‾ 10), (3‾ 1 3) and (2 1 2) plane orientations are more perpendicular to the dislocation slide planes {111}, the distribution of Ω1/Ω2 in these single crystals is more homogeneous with the number ratio between 1.0 and 1.2. Considering the coupling effects of crystallographic anisotropy, stress-oriented precipitates and stress induced dislocations on precipitates, the size and distribution of Ω, S and θ’ were quantitatively counted and analyzed. The increments of precipitation strengthening were calculated and compared with the measured hardness results, indicating that the hardness of single crystal with (6 5‾ 10) orientation increase by most with 15 HV after stress aging. The trend of hardness changes with applied stress and plane orientations was well modeled in stress-aged Al-Cu-Mg-Ag single crystals.

Strain hardening and features of slowing down the movement of dislocations in single crystals

The study of deformation hardening of single crystals for modern engineering is an important task due to the emergence of new promising methods of processing metal materials. Although in practice we usually deal with polycrystalline metals and alloys, it is advisable to start the analysis with simpler objects – single crystals of pure metals, where you can most clearly and fully identify the main patterns of deformation hardening. The article considers the temperature range limited by the interval of intensive thermal return.

Springback prediction for a mechanical micro connector using CPFEM based numerical simulations

The bending process of an industrial connector is considered and investigated via numerical simulation using a crystal plasticity finite element model (CPFEM). The process consists of sequentially bending a 0.1 mm thick copper-based alloy (CuBe2) with progressive tools into a miniature cylindrical connector of around 1 mm in diameter. The paper focuses on the prediction of springback through the influence of several key-parameters of the numerical simulations. The finite element characteristics, single crystal plasticity model features and the number of grains in the sheet thickness are investigated in order to highlight relevant and influential parameters in CPFEM based microforming process simulations. The influence of the elastic properties is analyzed and a modification of the Peirce-Asaro-Needleman single crystal hardening law is taken into account in order to improve the description of reverse strain path changes. Finally, the numerical results are discussed and compared to the springback measured during the industrial process of the cylindrical connector. It is demonstrated, through the very good agreement with the experimental results, that such approach can be useful to simulate industrial processes.

Experimental Characterisation and Numerical Modelling of the Effect of Cold Rolling on the Nanoindentation Response of Pure Zinc Grains

In this study, the orientation-dependent response of as-received annealed cold-rolled pure zinc and material with thickness reduction rate of 50% grains using instrumented indentation tests is investigated. The experiments were characterized by orientation microscopy and atomic force microscopy scans to quantify the orientation-dependent mechanical response during nanoindentation. The single crystal hardening parameters are then identified for each family of slip system by using crystal plasticity finite element (CPFE) simulations. Comparison between experimental and numerical results in terms of “load-penetration depth” curves show a good agreement. The increased percentage of cold reduction increases the identified critical resolved shear stress (CRSS). Finally, the accuracy of the model is evaluated by comparing experimental and numerical data issued from nanoindentation response grains of distinct crystalline orientations involving different slip systems activity rates.

DAMASK – The Düsseldorf Advanced Material Simulation Kit for modeling multi-physics crystal plasticity, thermal, and damage phenomena from the single crystal up to the component scale

Crystal Plasticity (CP) modeling is a powerful and well established computational materials science tool to investigate mechanical structure–property relations in crystalline materials. It has been successfully applied to study diverse micromechanical phenomena ranging from strain hardening in single crystals to texture evolution in polycrystalline aggregates. However, when considering the increasingly complex microstructural composition of modern alloys and their exposure to—often harsh—environmental conditions, the focus in materials modeling has shifted towards incorporating more constitutive and internal variable details of the process history and environmental factors into these structure–property relations. Technologically important fields of application of enhanced CP models include phase transformations, hydrogen embrittlement, irradiation damage, fracture, and recrystallization. A number of niche tools, containing multi-physics extensions of the CP method, have been developed to address such topics. Such implementations, while being very useful from a scientific standpoint, are, however, designed for specific applications and substantial efforts are required to extend them into flexible multi-purpose tools for a general end-user community. With the Düsseldorf Advanced Material Simulation Kit (DAMASK) we, therefore, undertake the effort to provide an open, flexible, and easy to use implementation to the scientific community that is highly modular and allows the use and straightforward implementation of different types of constitutive laws and numerical solvers. The internal modular structure of DAMASK follows directly from the hierarchy inherent to the employed continuum description. The highest level handles the partitioning of the prescribed field values on a material point between its underlying microstructural constituents and the subsequent homogenization of the constitutive response of each constituent. The response of each microstructural constituent is determined, at the intermediate level, from the time integration of the underlying constitutive laws for elasticity, plasticity, damage, phase transformation, and heat generation among other coupled multi-physical processes of interest. Various constitutive laws based on evolving internal state variables can be implemented to provide this response at the lowest level. DAMASK already contains various CP-based models to describe metal plasticity as well as constitutive models to incorporate additional effects such as heat production and transfer, damage evolution, and athermal transformations. Furthermore, the implementation of additional constitutive laws and homogenization schemes, as well as the integration of a wide class of suitable boundary and initial value problem solvers, is inherently considered in its modular design.

A differential-exponential hardening law for non-Schmid crystal plasticity finite element modeling of ferrite single crystals

Crystal plasticity finite element (CPFE) modeling of multi-phase, third generation advanced high strength steel (3GAHSS) requires finding the hardening parameters of slip systems operating in different phases (e.g. FCC, BCC, BCT). It is common to see the Schmid law used to model the deformation of BCC crystals. However, researches by Bassani and others have shown that BCC crystals could obey the non-Schmid law. In this paper we examined the differences between using a CPFE model based on the Schmid versus a non-Schmid law to model the uniaxial compression of single crystal ferrite micropillars with distinct orientations carved out of dual phase DP980 and three-phase QP980 steel sheets. To accurately model the changing hardening rate of the single crystal, a new exponential hardening model was developed that would differentially harden slip systems. Criteria for transitioning from stage I to stage II hardening of single crystals were also developed and verified.Finally, it was shown that it is not sufficient to use only one single crystal micropillar compression force-displacement curve for the calibration of the non-Schmid CPFE model. The predictions of the resulting model would be unreliable, and its accuracy highly dependent on the orientation of the micropillar chosen for calibration. However, when two mircopillar compression force-displacement curves were used for calibration, the model's predictions were consistently accurate. Using more than two curves for calibration did not result in significant improvements.

A strain-rate and temperature dependent constitutive model for BCC metals incorporating non-Schmid effects: Application to tantalum–tungsten alloys

In this work, we present a multiscale physically based constitutive law for predicting the mechanical response and texture evolution of body-centered cubic (BCC) metals as a function of strain-rate and temperature. In the model, deformation of individual single crystals results not only from the resolved shear stress along the direction of slip (Schmid law) but also from shear stresses resolved along directions orthogonal to the slip direction as well as the three normal stress components (non-Schmid effects). We account for coupled Schmid and non-Schmid effects through the modification of the resolved shear stress for both 1/2〈11¯1〉{110} and 1/2〈111¯〉{112} slip systems and the modification of the slip resistance for 1/2〈111¯〉{112} slip systems. The single crystal model is implemented into a self-consistent homogenization scheme containing a hardening law for crystallographic slip. The hardening law is based on the evolution of dislocation densities that incorporates strain-rate and temperature effects through the Peierls stress, thermally activated recovery, dislocation substructure formation and dislocation interactions. The polycrystal model is calibrated and validated using a set of mechanical and texture data collected on a tantalum–tungsten alloy, Ta–10W, at temperatures ranging from 298K to 673K and strain-rates from 10−3s−1 to 2400s−1. We show the model effectively captures the anisotropic hardening rate and texture evolution for all data using a single set of single-crystal hardening parameters. Comparisons between predictions and measured data allow us to discuss the role of slip on {110} and {112} in determining plasticity and texture evolution in Ta–10W.

Online Use of Physically Based Plasticity Models for Steady State Cold Rolling Processes

A procedure has been developed to incorporate computationally costly physically based crystal plasticity models to calculate texture and anisotropy for steady state forming processes online. When using these models, at every point in the deformed zone, an average and a nonlinear solution procedure for stresses and/or strains in all these grains is required. The online calculation cost is avoided by offline creating a database with texture and anisotropy data for all possible deformation modes of the process. The case studied is a cold rolling process, but can easily be extended to any type of forming process, when the deformation field is known in advance. Textures and anisotropy data are predicted using a viscoplastic self-consistent model, but the method is suitable for any kind of crystal plasticity model. Single crystal plastic parameters, such as the critical resolved shear stress, the single crystal hardening parameters, and the strain-rate sensitivity, have been calibrated based on mechanical tests by means of a direct search simplex algorithm. The online calculated deformation history is compared to the histories stored in the database and the best match is selected. The deformation history is divided in two zones, the one before the neutral point where forward shearing occurs and the one after the neutral point where backward shearing occurs. One online deformation generation and selection procedure requires 0.005 s of CPU time for a database with a division in deformation gradients fine enough to accurately cover all deformations. The method allows calculating yield surfaces at any point in space based on microstructural effects modeled by crystal plasticity, without incremental material updating and necessity to define a kinematic and isotropic hardening, which makes the method suitable for fast models to calculate rolling forces and torques online.

Deformation behavior of the cobalt-based superalloy Haynes 25: Experimental characterization and crystal plasticity modeling

The deformation behavior of a wrought, cobalt-based superalloy, Haynes 25, was studied using a combination of experimental techniques and crystal plasticity modeling. The microstructure was examined by transmission electron microscopy and electron backscattered diffraction to determine the deformation and hardening mechanisms contributing to the mechanical behavior of the alloy. A high density of stacking faults within the grains, consistent with planar glide, was found to be the predominant defect mechanism with no evidence of deformation twinning observed. A strain-rate and temperature-sensitive hardening law that includes effects of planar glide was developed for the material and implemented in a self-consistent homogenization scheme. The hardening of individual crystals is based on the evolution of dislocation densities per plane and includes the effects of strain rate and temperature through thermally activated recovery and dislocation interactions. The model is validated on a comprehensive set of compression tests performed at temperatures ranging from 298 to 673K and strain rates ranging from 10−3 to 2600s−1 and found capable of reproducing the stress–strain response and texture for all tests with a unique set of single-crystal hardening parameters.

Modeling mechanical response and texture evolution of α-uranium as a function of strain rate and temperature using polycrystal plasticity

We present a polycrystal plasticity model based on a self-consistent homogenization capable of predicting the macroscopic mechanical response and texture evolution of α-uranium over a wide range of temperatures and strain rates. The hardening of individual crystals is based on the evolution of dislocation densities and includes effects of strain rate and temperature through thermally-activated recovery, dislocation substructure formation, and slip-twin interactions. The model is validated on a comprehensive set of compression tests performed on a clock-rolled α-uranium plate at temperatures ranging from 198 to 573K and strain rates ranging from 10−3 to 3600s−1. The model is able to reproduce the stress–strain response and texture for all tests with a unique set of single-crystal hardening parameters. We elucidate the role played by the slip and twinning mechanisms and their interactions in large plastic deformation of α-uranium as a function of strain rate and temperature.

Structure and hardness of octahedral natural diamond single crystals depending on the HPHT treatment conditions

The structure evolution of octahedral natural diamond single crystals has been studied depending on the HPHT treatment using Raman scattering and infrared spectroscopy. It has been found that the formation of a polycrystalline diamond capsule around a single crystal at p = 8 GPa and T = 1500°C gives rise to a combined structural-stressed state in the single crystal due to its plastic strain. This state has been manifested by a significantly (more than double) broadening of the characteristic line of diamond (1332 cm−1) in the Raman spectrum and the increase of a single crystal hardness from 105 to 120 GPa.

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 ( Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.

Plastic yielding and work hardening of single crystals in a soft device

An analytical solution to the problem of an anti-plane constrained shear of single crystals placed in a soft device within the continuum dislocation theory is found. The dependence of the nucleation stress on the grain size exhibits a modest deviation from the Hall–Petch relation. It is shown that, as soon as the dissipation is taken into account, the hardening behavior becomes nearly identical to that of single crystals in a hard device. To cite this article: K.C. Le, Q.S. Nguyen, C. R. Mecanique 337 (2009).

Micromechanics of pyramidal indentation in fcc metals: Single crystal plasticity finite element analysis

This work concerns analysis of Vickers and Berkovich indentation experiments through extensive crystal plasticity finite element simulations. The simulations are performed by recourse to the Bassani and Wu hardening model for pure fcc crystals undergoing both easy-glide stage I and stage II deformations, as well as with the model proposed by Pierce, Asaro and Needleman for precipitation hardened fcc crystals, deforming initially under stage II which also undergo strong hardening saturation in stage III. Simulations are also conducted with a model based on the hardening description by Bassani and Wu, whose physical basis and predictive capability have been enhanced for pure copper crystals. Based upon the activity of the slip systems and the strength of dislocation interactions, this work provides a fundamental insight into the influence of prior work hardening in single crystal indentation. Discussions are given on the role of the latent hardening description upon the development of material pileup and sinking-in at the contact boundary as well as on the correlation between the single crystal and polycrystalline contact responses. The present investigation further illustrates on the influence of the orientation of the slip systems with respect to the pyramidal tip upon the formation of irregular imprint morphologies. Extraction of the single crystal hardening parameters from instrumented indentation P– h s curves is also briefly addressed. Finally, the contact deformation regimes ruling the response of isotropic strain hardening media are examined in light of the simulations for single crystal indentation.

Ellipticity loss analysis for tangent moduli deduced from a large strain elastic–plastic self-consistent model

In order to investigate the impact of microstructures and deformation mechanisms on the ductility of materials, the criterion first proposed by Rice is applied to elastic–plastic tangent moduli derived from a large strain micromechanical model combined with a self-consistent scale-transition technique. This approach takes into account several microstructural aspects for polycrystalline aggregates: initial and induced textures, dislocation densities as well as softening mechanisms such that the behavior during complex loading paths can be accurately described. In order to significantly reduce the computing time, a new method drawn from viscoplastic formulations is introduced so that the slip system activity can be efficiently determined. The different aspects of the single crystal hardening (self and latent hardening, dislocation storage and annihilation, mean free path, etc.) are taken into account both by the introduction of dislocation densities per slip system as internal variables and the corresponding evolution equations. Comparisons are made with experimental results for single and dual-phase steels involving linear and complex loading paths. Rice’s criterion is then coupled and applied to this constitutive model in order to determine the ellipticity loss of the polycrystalline tangent modulus. This criterion, which does not need any additional “fitting” parameter, is used to build Ellipticity Limit Diagrams (ELDs).

A self-consistent boundary element, parametric dislocation dynamics formulation of plastic flow in finite volumes

We present a self-consistent formulation of 3-D parametric dislocation dynamics (PDD) with the boundary element method (BEM) to describe dislocation motion, and hence microscopic plastic flow in finite volumes. We develop quantitative measures of the accuracy and convergence of the method by considering a comparison with known analytical solutions. It is shown that the method displays absolute convergence with increasing the number of quadrature points on the dislocation loop and the surface mesh density. The error in the image force on a screw dislocation approaching a free surface is shown to increase as the dislocation approaches the surface, but is nevertheless controllable. For example, at a distance of one lattice parameter from the surface, the relative error is less than 5% for a surface mesh with an element size of 1000 × 2000 (in units of lattice parameter), and 64 quadrature points. The Eshelby twist angle in a finite-length cylinder containing a coaxial screw dislocation is also used to benchmark the method. Finally, large scale 3-D simulation results of single slip behavior in cylindrical microcrystals are presented. Plastic flow characteristics and the stress–strain behavior of cylindrical microcrystals under compression are shown to be in agreement with experimental observations. It is shown that the mean length of dislocations trapped at the surface is the dominant factor in determining the size effects on hardening of single crystals. The influence of surface image fields on the flow stress is finally explored. It is shown that the flow stress is reduced by as much as 20% for small single crystals of size less than 0.15 μ m .

Examining crystallographic orientation dependence of hardness of silica stishovite

We conducted nanoindentation to explore the hardness and elastic properties of silica stishovite, synthesized at high pressure and quenched to ambient conditions. A total of 10 crystallographic orientations were examined on selected grains with a maximum load of 4 or 20 mN. We observed discontinuity in the load–displacement curve (pop-in) for the [ 2 5 1 ¯ ] and [ 6 2 1 ¯ ] grains subjected to a maximum load of 20 mN. The single-crystal hardness at high plastic deformation is quasi-isotropic with an average of 32 ± 1 GPa , similar to the polycrystalline hardness reported earlier; the theoretical hardness determined from the experiments is about 54 ± 3 GPa . These two hardnesses suggest that stishovite is one of the hardest oxides. The measured indentation moduli are close to the predictions at low load (minor plasticity) but are considerably lower at high load (high plasticity). Both indentation hardness and modulus decrease with increasing plasticity. Our results underscore the necessity of considering the degree of plastic deformation when interpreting hardness and elastic moduli from indentation experiments.