Orientation effects in nanoindentation of single crystal copper

Numerical simulations and experimental results of nanoindentation on single crystal copper in three crystallographic orientations [(1 0 0), (0 1 1) and (1 1 1)] using a spherical indenter (3.4 μm radius) were reported. The simulations were conducted using a commercial finite element code (ABAQUS) with a user-defined subroutine (VUMAT) that incorporates large deformation crystal plasticity constitutive model. This model can take full account of the crystallographic slip as well as the orientation effects during nanoindentation. Distributions of the out-of-plane displacements and shear stresses as well as shear strains were obtained for indentation depths of up to 310 nm. The experimental studies were conducted using an MTS Nano Indenter (XP) system from which the load–displacement relationships were obtained while the surface topography as well as the surface profile along a line scan of indents were obtained using a Digital Instruments (Dimension 3100) atomic force microscope (AFM). The top views of the indent pile-up patterns under the spherical indenter show two-fold, three-fold, and four-fold symmetries for the (0 1 1), (1 1 1), and (1 0 0) orientations, respectively. Attempt was made to relate the anisotropic nature of the surface topographies around the indents in different crystallographic orientations of the single crystal copper specimens with the active slip systems and local texture variations. A reasonably good agreement had been obtained on several aspects of nanoindentation between the experimental and numerical results reported in this investigation as well as similar results reported in the literature. Thus, material properties of single crystal copper can be determined based on an appropriate numerical modeling of the nanoindentation on three crystallographic orientations.     

Nonlinear constitutive models from nanoindentation tests using artificial neural networks

This paper presents a new approach using Artificial Neural Networks (ANNs) models to simulate the response during nanohardness tests of a variety of materials with nonlinear behavior. The ANNs continuous input and output variables usually include material parameters, indentation deflection, and resisting force. Different ANN models, including dimensionless input/output variables, are generated and trained with discrete finite-element (FE) simulations with different geometries and nonlinear material parameters. Only the monotonic loading part of the load–displacement indentation response is used to generate the trained ANN models. This is a departure from classical indentation simulations or tests where typically the unloading portion is used to determine the stiffness and hardness. The experimental part of this study includes nanoindentation tests performed on a silicon (Si) substrate with and without a nanocrystalline copper (Cu) film. The new ANN models are used to back-calculate (inverse problem) the in situ nonlinear material parameters for different copper material systems. The results are compared with available data in the literature. The proposed FE–ANN modeling approach is very effective and can be used in calibrating and predicting the in situ inelastic material properties using the monotonic part of the indentation response and for depths above 50 nm where the overall resisting force represents a continuum response.     

Grain size,strain rate,and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: Synthesis,experiment, and constitutive modeling

For the first time, high quality bulk nanocrystalline (nc) fcc metals, with least amounts of imperfections, exhibiting high strength and ductility at room and different temperatures, under quasi-static and dynamic types of loading, were prepared and a comprehensive study on their post-yield mechanical properties was performed. This investigation included study of the effect of temperature on stress–strain responses of mechanically milled bulk nc Cu and Al. The samples after preparation through mechanical milling and consolidation processes were subjected to uniaxial compressive loading at quasi-static and dynamic strain rates of 10⁻² s⁻¹ and 1840–3105 s⁻¹, respectively, at temperatures ranging from 223 to 523 K. In both materials strong dependency of flow stress to temperature was observed; this dependency was rather more pronounced when the materials were tested at the quasi-static strain rate. Further, a new grain size and temperature dependent viscoplastic phenomenological constitutive equation, Khan–Liang–Farrokh (KLF) model was developed based on the Khan–Huang–Liang (KHL) constitutive equation. The model was featured to correlate different characteristic behaviors of polycrystalline materials in the plastic regime, as the result of grain refinement. In addition, the viscoplastic responses of bulk Cu and Al of different grain sizes (from sub-micron to nanometer range), and those from bulk nc Cu and Al at different strain rates (quasi-static to dynamic), recently published (21 and 22), were simulated using the newly developed equation. The results confirmed reasonable capability of the developed model to correlate a wide spectrum of the viscoplastic responses of these fcc metals.     

Effect of oxygen content and microstructure on the thermo-mechanical response of three Ti–6Al–4V alloys: Experiments and modeling over a wide range of strain-rates and temperatures

Results from a series of experiments on three different titanium alloys, under quasi-static and dynamic loading conditions are presented. The Ti–6Al–4V titanium alloys include the ELI version and two with higher oxygen contents. The strain-rates are varied from 10⁻⁶ to 3378 s⁻¹ while observations are made at temperatures from 233 to 755 K. The alloys initial and deformed photomicrographs and various deformation mechanisms responsible for the induced plastic deformation, are presented and discussed. Differences in the responses of these alloys are observed in terms of thermal softening, work hardening, and strain-rate and temperature sensitivities. The Khan–Huang–Liang (KHL) model is used to effectively simulate the observed responses obtained from these experiments. The model, with the constants determined from these experiments, is then used to predict strain-rate jump experimental results, and also high temperature dynamic experiments for one of the alloys; the predictions are found to be very close to the observations.     

Ductile–brittle transitions in the fracture of plastically-deforming,adhesively-bonded structures. Part I: Experimental studies

Rate effects for adhesively-bonded joints in steel sheets failing by mode-I fracture and plastic deformation were examined. Three types of test geometries were used to provide a range of crack velocities between 0.1 and 5000 mm/s: a DCB geometry under displacement control, a wedge geometry under displacement control, and a wedge geometry loaded under impact conditions. Two fracture modes were observed: quasi-static crack growth and dynamic crack growth. The quasi-static crack growth was associated with a toughened mode of failure; the dynamic crack growth was associated with a more brittle mode of failure. The experiments indicated that the fracture parameters for the quasi-static crack growth were rate independent, and that quasi-static crack growth could occur even at the highest crack velocities. Effects of rate appeared to be limited to the ease with which a transition to dynamic fracture could be triggered. This transition appeared to be stochastic in nature, it did not appear to be associated with the attainment of any critical value for crack velocity or loading rate. While the mode-I quasi-static fracture behavior appeared to be rate independent, an increase in the tendency for dynamic fracture to be triggered as the crack velocity increased did have the effect of decreasing the average energy dissipated during fracture at higher loading rates.     

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.     

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〉.     

A study of different modes of collapse in metallic hemispherical shells resting on flat platen and compressed with hemispherical nosed indenter

The paper presents experimental and analytical studies on axial compression of aluminium spherical shells having Radius/wall thickness (R/t) ratios between 23 and 135. Quasi-static compressive load was applied centrally and with offset through a indenter having diameter of 22 mm. Testing was carried out on an INSTRON machine having 250 T capacity. Shells having different radius and wall thickness were tested, to classify their modes of collapse and their corresponding energy absorption mechanism. In experiments shells of lower R/t values were found to collapse due to formation of an inward dimple associated with a rolling plastic hinge in central as well as in offset loading. On the other hand, shells of higher R/t values were collapsed initially with formation of an axisymmetric inward dimple, but in later stage of compression showed buckling of non-symmetric shape consisting of integral number of lobes and stationary plastic hinges. The stationary hinges were formed between consecutive lobes. Experimental observations are used to propose an analytical model for prediction of load–compression and energy–compression curves. The results obtained from analytical model compared with the experimental results and found match fairly well.     

Multiaxial and non-proportional loading responses,anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures

Results from a series of multiaxial loading experiments on the Ti–6Al–4V titanium alloy are presented. Different loading conditions are applied in order to get the comprehensive response of the alloy. The strain rates are varied from the quasi-static to dynamic regimes and the corresponding material responses are obtained. The specimen is deformed to large strains in order to study the material behavior under finite deformation at various strain rates. Torsional Kolsky bar is used to achieve shear strain rates up to 1000 s⁻¹. Experiments are performed under non-proportional loading conditions as well as dynamic torsion followed by dynamic compression at various temperatures. The non-proportional loading experiments comprise of an initial uniaxial loading to a certain level of strain followed by biaxial loading, using a channel-type die at various rates of loadings. All the non-proportional experiments are carried out at room temperature. Experiments are also performed to investigate the anisotropic behavior of the alloy. An orthotropic yield criterion [proposed by Cazacu, O., Plunkett, B., Barlat, F., 2005. Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 22, 1171–1194.] for anisotropic hexagonal closed packed materials with strength differential is used to generate the yield surface. Based on the definition of the effective stress of this yield criterion, the observed material response for the different loading conditions under large deformation is modeled using the Khan–Huang–Liang (KHL) equation assuming isotropic hardening. The model constants used in the present study, were pre-determined from the extensive uniaxial experiments presented in the earlier paper [Khan, A.S., Suh, Y.S., Kazmi R., 2004. Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys. International Journal of Plasticity 20, 2233–2248]. The model predictions are found to be extremely close to the observed material response.     

Sample boundary effect in nanoindentation of nano and microscale surface structures

Nanoindentation is a widely used technique to characterize mechanical properties of materials in small volumes. When the sample size is comparable to the indent size, the indentation-induced plastic zone can be affected by the sample boundary which may cause inaccurate interpretation of the mechanical properties. In this study, the sample boundary effect is investigated by performing experiments and atomistic simulations of nanoindentation into nano- and micro-scale Au pillars and bulk Au (0 0 1) surfaces. In experiments, a more compliant deformation is observed in pillar indentations compared to bulk Au. The elastic modulus decreases with increasing indent size over sample size ratio. Atomistic simulations are performed to gain insights on the mechanisms of pillar deformation and pillar boundary effect. The reduced modulus has a similar trend of decrease with increasing indent size over sample size ratio. Significantly different dislocation activities and dislocation interactions with the pillar boundary contribute to the lower value of the reduced modulus in the pillar indentation. The presence of the free surface would allow the dislocations to annihilate, causing a higher plastic recovery during the pillar unloading process.     

Activation volume and deviation from Cottrell–Stokes law at small grain size

Dependence of activation volume with flow stress is examined for metals with grain size lower than 0.3 μm and larger than few tens of nanometers, where plastic deformation is most likely to be governed by a combination of grain boundary sliding and dislocations activity. The experimentally observed deviation from the classic linear behavior given by Cottrell–Stokes law [Basinski, Z.S., 1974. Forest hardening in face centered cubic metals. Scripta Metallurgica 8, 1301–1308] is analyzed, thanks to a modified Orowan equation taking into account of the grain boundaries sliding coupled to dislocations activity. These results are compared to experimental measurements of the activation volume, between room temperature and 120 °C, for a copper nanostructure with a grain size of 100 nm. A constant activation volume is observed at low stress (or high temperature) followed by an increase of activation volume with stress (inverse Cottrell–Stokes behavior). This analysis follows our initial experiments on this fine grained metal prepared by powder metallurgy, which exhibits ductility at near constant stress and strain rate [Champion, Y., Langlois, C., Guérin-Mailly, S., Langlois, P., Bonnentien, J.-L., Hÿtch, M.J., 2003. Near-perfect elastoplasticity in pure nanocrystalline copper. Science 300, 310–311].     

Numerical validation of the shear compression specimen. Part II: Dynamic large strain testing

Part I of this work addressed quasi-static loading of the shear compression specimen (SCS), which has been especially developed to investigate the shear dominant response of materials at various strain rates. The stress and strain states were characterized numerically. Approximations were presented to reduce the measured load,P, and displacement,d, into equivalent stress and strain . This paper addresses dynamic loading of the SCS. Several simulations were made for representative materials, whose stress-strain behavior is assumed to be rate-independent. The results show that stress wave loading induces strong oscillations in theP-d curve. However, the curve remains smooth in the gage section. The oscillations are about the quasistatic load values, so that with suitable filtering of the dynamicP-d curves, the quasi-static ones are readily recovered. Consequently, the approach that was developed for quasi-static loading of the SCS is now extended to dynamic loading situations. The average strain rate is rather constant and scales linearly with the prescribed velocity. As the plastic modulus becomes smaller, the strain rate reaches higher values. Friction at the end pieces of the specimen is also investigated, and shown to have a small overall influence on the determined mechanical characteristics. This paper thereby confirms the potential of the SCS for large strain testing of materials, using a unified approach, over a large range of strain rates in a seamless fashion.     

Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account

Motivated by recent progress in viscoelastic indentation analysis, the identification of viscoelastic properties from nanoindentation test data taking the real tip geometry into account is presented in this paper. Based on the elastic solution of the indentation problem, the corresponding viscoelastic solution is obtained by the application of the method of functional equations. This general solution, which accounts for the real geometric properties of the indenter tip, is specialized for the case of a trapezoidal load history, commonly employed in nanoindentation testing. Three deviatoric creep models, the single dash-pot, the Maxwell, and the three-parameter model are considered. The so-obtained expressions allow us to determine viscoelastic model parameters via back calculation from the measured load–penetration history. The presented approach is illustrated by the identification of short-term viscoelastic properties of bitumen. Hereby, the influence of loading rate, maximum load, and temperature on the model parameters is investigated.     

Analysis methods and size effects in the indentation fracture toughness assessment of very thin oxide coatings on glass

Very thin oxide coatings (<100 nm) which are used as anti-reflection and barrier layers in low emissivity architectural glass have been studied by nanoindentation methods to determine the effect of coating thickness on fracture toughness. Traditional microindentation-derived methods to determine the fracture toughness are unsuitable for assessing very thin coatings (<500 nm) and alternative energy-based models are required depending on what features are visible in indentation load–displacement curves. In cases where radial cracks are formed and grow in a discontinuous manner there are excursions in the load–displacement curve which can be the basis for analysis. In cases where picture frame cracks are observed there are no such features and an alternative approach based on assessment of irreversible work of indentation is required. This paper reviews the methods for obtaining fracture toughness data for very thin coatings and assesses the existence of size effects in the mechanical response of oxide coatings with different thickness on a glass substrate. For oxide coatings in the thickness range 100 to 400 nm no size effects in fracture toughness were observed.     

A robust integration algorithm for implementing rate dependent crystal plasticity into explicit finite element method

Severe numerical instability in the integration of rate dependent crystal plasticity (RDCP) model is one of the main problems for implementing RDCP into finite element method (FEM), especially for simulating dynamic/transient forming process containing complicated contact conditions under large step length, large strain and high strain rate. In order to overcome the problem, an implicit model is deduced with the primary unknowns of shear strain increments of slip systems under the corotational coordinate system in the paper. The homotopy auto-changing continuation method combined with the Newton–Raphson (N–R) iteration is adopted. The subroutine VUMAT is developed for implementing RDCP model in ABAQUS/Explicit. Simulation results show that the algorithm is stable and accurate in 3D FE simulations on both dynamic simple loading and complicated loading process containing nonlinear contacts under the conditions of the maximal step length of 3.5 × 10⁻⁶ s, the maximal strain of 1.05, the maximal loading speed of 120 mm s⁻¹, and the minimal material rate sensitivity coefficient of 0.01. The predictions of the model on crystal behaviors of anisotropy, rate sensitivity and elasticity, as well as ear profiles in deep cup drawing are in agreement with experiments.     

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.     

准脆性单晶材料动态压痕试验研究

本研究设计了由钛合金压杆与金刚石压头组成的动态压痕试验装置,研究准脆性单晶材料(冰糖和RDX含能晶体)等的高应变率响应。试验装置引入了动量缓冲块,可以避免压杆造成的多次压痕加载,对样品施加产生80μs脉冲宽度的动态压痕载荷。对比了多次与单次加载压痕加载的冰糖单晶压痕坑,其动态硬度值约为5.18MPa。RDX单晶在压痕载荷作用下非常容易发生断裂破碎,测得动态硬度值为1.304MPa。相比RDX单晶,冰糖单晶的塑性变形能力更好。用光学显微镜观察到了压痕坑的径向裂纹和侧边裂纹,试验研究给出了准脆性晶体在动态压痕载荷下的裂纹扩展模式。     

Phase and velocity distributions in vertically upward high-viscosity two-phase flow

The two-phase pressure drop in vertical industrial pipes is mainly determined by gravitation and acceleration of the fluid, which means that the void fraction is key an important parameter in any model to predict pressure drops. Typically, these models are applied in industry to size pumps and, e.g., emergency relief systems. There is a shortage of void fraction data in the literature for liquids with a dynamic viscosity above 1000 mPa s. Adiabatic experiments have been performed of mixtures of nitrogen and solutions of polyvinylpyrrolidone (Luviskol^®) in water with dynamic viscosities in the range 900–7000 mPa s. Inner tube diameter was 54.5 mm. Mass flux and quality were varied in a wide range: 8–3500 kg/m²/s and 0–82%, respectively. The corresponding superficial velocities were 0.005–3.4 m/s for the liquid and 0–30 m/s for the nitrogen. For comparison, reference measurements were taken of mixtures of nitrogen with water (1 mPa s). Care has been taken to measure only well-developed flows.