纳米压痕过程的三维有限元数值试验研究

采用有限元方法模拟了纳米压痕仪的加、卸载过程，三维有限元模型考虑了纳米压痕仪的标准Berkovich压头．介绍了有限元模型的几何参数、边界条件、材料特性与加载方式，讨论了摩擦、滑动机制、试件模型的大小对计算结果的影响，进行了计算结果与标准试样实验结果的比较，证实了模拟的可靠性．在此基础上，重点研究了压头尖端曲率半径对纳米压痕实验数据的影响．对比分析了尖端曲率半径r＝0与r＝100nm两种压头的材料压痕载荷—位移曲线．结果表明，当压头尖端曲率半径r≠0时，基于经典的均匀连续介质力学本构理论、传统的实验手段与数据处理方法，压痕硬度值会随着压痕深度的减小而升高．     

The correlation of the indentation size effect measured with indenters of various shapes

Experimental results are presented which show that the indentation size effect for pyramidal and spherical indenters can be correlated. For a pyramidal indenter, the hardness measured in crystalline materials usually increases with decreasing depth of penetration, which is known as the indentation size effect. Spherical indentation also shows an indentation size effect. However, for a spherical indenter, hardness is not affected by depth, but increases with decreasing sphere radius. The correlation for pyramidal and spherical indenter shapes is based on geometrically necessary dislocations and work-hardening. The Nix and Gao indentation size effect model (J. Mech. Phys. Solids 46 (1998) 411) for conical indenters is extended to indenters of various shapes and compared to the experimental results.     

An analysis of nanoindentation in linearly elastic solids

The conventional method to extract elastic properties in the nanoindentation of linearly elastic solids relies primarily on Sneddon’s solution (1948). The underlying assumptions behind Sneddon’s derivation, namely, (1) an infinitely large incompressible specimen; (2) an infinitely sharp indenter tip, are generally violated in nanoindentation. As such, correction factors are commonly introduced to achieve accurate measurements. However, little is known regarding the relationship between the correction factors and how they affect the overall accuracy. This paper first proposes a criterion for the specimen’s geometry to comply with the first assumption, and modifies Sneddon’s elastic relation to account for the finite tip radius effect. The relationship between the finite tip radius and compressibility of the specimen is then examined and a composite correction factor that involves both factors, derived. The correction factor is found to be a function of indentation depth and a critical depth is derived beyond which, the arbitrary finite tip radius effect is insignificant. Techniques to identify the radius of curvature of the indenter and to decouple the elastic constants (E and ν) for linear elastic materials are proposed. Finally, experimental results on nanoindentation of natural latex are reported and discussed in light of the proposed modified relation and techniques.     

考虑压头曲率半径的J2形变理论压痕有限元分析

采用经典J2形变理论,在考虑压头曲率半径的前提下,针对几组微压痕实验进行了有限元数值模拟,给出理论计算结果并与实验进行了系统比较．结果发现考虑了压头曲率半径后的经典J2形变理论得到的整段计算曲线不能和实验曲线吻合,可以推理出即使考虑压头曲率半径的的影响,经典J2形变理论也不能很好地解释实验现象．     

Size effect in the initiation of plasticity for ceramics in nanoindentation

The indentation size effect has been observed for many years and is usually associated with increasing hardness as the depth of indentation is reduced for pyramidal indenters. The indentation size effect for spherical indenters has recently been associated with an increase in the yield stress of metals proportional to the inverse cube root of indenter radius [Spary, I.J., Bushby, A.J., Jennett, N.M., 2006. On the indentation size effect in spherical indentation. Philos. Mag. 86 (33), 5581-5593]. Here we investigate ceramic materials where the yield point is high enough to be easily distinguished in nanoindentation tests. A robust method for determining the yield point from a nanoindentation test with spherical indenters is presented. The results for a range of ceramics confirm that the increase in yield pressure is directly proportional to the inverse cube root of indenter radius. Furthermore, the yield pressure is also shown to be proportional to the inverse square root of the contact radius. Revisiting data in the literature shows that this inverse square root relationship is also true for pyramidal indenters. This implies that the indentation size effect is driven by the contact area rather than by the depth of indentation or by the indenter radius.     

The equivalent axisymmetric model for Berkovich indenters in power-law hardening materials

Nix and Gio [Nix, W.D., Gao, H.J., 1998. Indentation size effects in crystalline materials: a law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids 46, 411–425] established an important relation between the micro-indentation hardness and indentation depth for axisymmetric indenters. For the Berkovich indenter, however, this relation requires an equivalent cone angle. Qin et al. [Qin, J., Huang, Y., Xiao, J., Hwang, K.C., 2009. The equivalence of axisymmetric indentation model for three-dimensional indentation hardness. Journal of Materials Research 24, 776–783] showed that the widely used equivalent cone angle from the criterion of equal base area leads to significant errors in micro-indentation, and proposed a new equivalence of equal cone angle for iridium. It is shown in this paper that this new equivalence holds for a wide range of plastic work hardening materials. In addition, the prior equal-base-area criterion does not hold because the Berkovich indenter gives much higher density of geometrically necessary dislocations than axisymmetric indenter. The equivalence of equal cone angle, however, does not hold for Vickers indenter.     

NANOINDENTATION OF THIN-FILM-SUBSTRATE SYSTEM： DETERMINATION OF FILM HARDNESS AND YOUNG’S MODULUS

In the present paper, the hardness and Young‘s modulus of film-substrate systems are determined by means of nanoindentation experiments and modified models. Aluminum film and two kinds of substrates, i.e. glass and silicon, are studied. Nanoindentation XP Ⅱ and continuous stiffness mode are used during the experiments. In order to avoid the influence of the Oliver and Pharr method used in the experiments, the experiment data are analyzed with the constant Young‘s modulus assumption and the equal hardness assumption. The volume fraction model (CZ model) proposed by Fabes et al. (1992) is used and modified to analyze the measured hardness. The method proposed by Doerner and Nix (DN formula) (1986) is modified to analyze the measured Young‘s modulus. Two kinds of modified empirical formula are used to predict the present experiment results and those in the literature, which include the results of two kinds of systems, i.e., a soft film on a hard substrate and a hard film on a soft substrate. In the modified CZ model, the indentation influence angle, φ, is considered as a relevant physical parameter, which embodies the effects of the indenter tip radius, pile-up or sink-in phenomena and deformation of film and substrate.     

Indentation by nominally flat or conical indenters with rounded corners

Axisymmetric indentation of a flat surface is considered: specifically, the case of flat-ended indenter with rounded edges, and the case of a shallow cone with a rounded tip. Analytical solutions are obtained for the normal and sequential tangential loading, in both full or partial slip conditions (with the Cattaneofn9Mindlin approximation) , and for the complete interior stress field in all these conditions.Implications for strength of the contact are discussed with reference to metallic or brittle materials, with the intention to shed more light in particular to the understanding of common fretting fatigue or indentation testings with nominally flat or conical indenters. It is found that the strength of the contact, which is nominally zero for perfectly sharp flat or conical indenters, is well defined even for a small radius of curvature. This is particularly true for the flat indenter, for which the strength is even significantly higher than for the classical Hertzian indenter for a wide range of geometrical and loading conditions, rendering it very attractive for design purposes.     

The Influence of Indenter Tip Imperfection and Deformability on Analysing Instrumented Indentation Tests at Shallow Depths of Penetration on Stiff and Hard Materials

We report on the difficulties of extracting plastic parameters from constitutive equations derived by instrumented indentation tests on hard and stiff materials at shallow depths of penetration. As a general rule, we refer here to materials with an elastic stiffness more than 10 % of that of the indenter and a yield strain higher than 1 %, as well as to penetration depths less than ～ 5 times the characteristic tip defect length of the indenter. We experimentally tested such a material (an amorphous alloy) by nanoindentation. To describe the mechanical response of the test, namely the force-displacement curve, it is necessary to consider the combined effects of indenter tip imperfections and indenter deformability. For this purpose, an identification procedure has been carried out by performing numerical simulations (using Finite Element Analysis) with constitutive equations that are known to satisfactorily describe the behaviour of the tested material. We propose a straightforward procedure to address indenter tip imperfection and deformability, which consists of firstly taking account of a deformable indenter in the numerical simulations. This procedure also involves modifying the experimental curve by considering a truncated length to create artificially the material’s response to a perfectly sharp indentation. The truncated length is determined directly from the loading part of the force-displacement curve. We also show that ignoring one or both of these issues results in large errors in the plastic parameters extracted from the data.     

An expanding cavity model incorporating strain-hardening and indentation size effects

An expanding cavity model (ECM) for determining indentation hardness of elastic strain-hardening plastic materials is developed. The derivation is based on a strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic power-law hardening material. Closed-form formulas are provided for both conical and spherical indentations. The indentation radius enters these formulas with its own dimensional identity, unlike that in classical plasticity based ECMs where indentation geometrical parameters appear only in non-dimensional forms. As a result, the newly developed ECM can capture the indentation size effect. The formulas explicitly show that indentation hardness depends on Young’s modulus, yield stress, strain-hardening exponent and strain gradient coefficient of the indented material as well as on the geometry of the indenter. The new model reduces to existing classical plasticity based ECMs (including Johnson’s ECM for elastic–perfectly plastic materials) when the strain gradient effect is not considered. The numerical results obtained using the newly developed model reveal that the hardness is indeed indentation size dependent when the indentation radius is very small: the smaller the indentation, the larger the hardness. Also, the indentation hardness is seen to increase with the Young’s modulus and strain-hardening level of the indented material for both conical and spherical indentations. The strain-hardening effect on the hardness is observed to be significant for materials having strong strain-hardening characteristics. In addition, it is found that the indentation hardness increases with decreasing cone angle of the conical indenter or decreasing radius of the spherical indenter. These trends agree with existing experimental observations and model predictions.     

基于能量密度等效的超弹性压入模型与双压试验方法

针对超弹性材料压入问题, 本文基于能量密度中值等效原理, 提出了描述球、平面、锥3类压头独立压入下载荷、深度、压头几何尺寸和Mooney-Rivlin本构关系参数之间关系的半解析超弹性压入模型(semi-theoretical hyperelastic-material indentation model, SHIM), 进而提出了球、平面、锥压入组合的双压试验方法(indentation method due to dual indenters, IMDI). 正向验证表明, 基于系列超弹性材料的本构关系参数, 由SHIM分别预测的球、平面、锥3类压入下的载荷-位移曲线与有限元分析(finite element analysis, FEA)结果之间密切吻合; 反向验证表明, 基于系列超弹性材料的FEA条件本构关系下3类压入的载荷-位移曲线, 由双压试验方法预测的Mooney-Rivlin本构关系与FEA条件本构关系密切吻合. 针对3种超弹性橡胶, 完成了球、平面、锥压入试验, 应用双压试验方法获得的3组Mooney-Rivlin本构关系均与单轴拉伸试验结果吻合良好.      

An Instrumented Indentation Method for Young’s Modulus Measurement with Accuracy Estimation

The relationships between indentation responses and Young’s modulus of an indented material were investigated by employing dimensional analysis and finite element method. Three representative tip bluntness geometries were introduced to describe the shape of a real Berkovich indenter. It was demonstrated that for each of these bluntness geometries, a set of approximate indentation relationships correlating the ratio of nominal hardness/reduced Young’s modulus H _n /E _r and the ratio of elastic work/total work W _e/W can be derived. Consequently, a method for Young’s modulus measurement combined with its accuracy estimation was established on basis of these relationships. The effectiveness of this approach was verified by performing nanoindentation tests on S45C carbon steel and 6061 aluminum alloy and microindentation tests on aluminum single crystal, GCr15 bearing steel and fused silica.     

Effect of indenter tip radius on the load deflection behavior of thin plates

The effect of indenter radius on the load-deflection behavior of a clamped, thin isotropic and homogeneous plate is discussed. A depth-sensing nanoindenter was used to apply a transverse load to the plate center and measure the corresponding plate center deflections. The applied stress is assumed to be uniformly distributed over the contact area. This study shows that the central deflection of the plate is insensitive to small changes of indenter tip radius when the normalized indenter tip radius is smaller than about 0.10 (here, normalized indenter tip radius is defined as the ratio of the indenter tip radius to the plate radius). For these small normalized indenter tip radii, the indenter radius effect is the same in both the small (linear) deflection range and in the large (nonlinear) deflection range. Blunt indenter tips help to minimize material nonlinearity in the region of contact due to localized plastic deformation and/or microcracks, allowing geometrically nonlinear deflection data to be obtained in a less ambiguous manner.     

Finite Element Simulation of Hot Nanoindentation in Vacuum

A finite element model is developed to investigate technical issues associated with hot nanoindentation measurements in vacuum, e.g. thermal expansion-induced drift and temperature variations at the contact region between the cold indenter tip and hot specimen. With heat conduction properly accounted for, the model is able to reasonably reproduce experimental indentation measurements on fused silica and copper—two materials with significantly different thermal and mechanical properties—at several temperatures. Temperature and loading rate effects on thermal drift are established using this model and an analytical expression for predicting thermal drift is numerically calibrated. The model also captures details of the indentation process that are not directly accessible experimentally, and reaffirms the need for operational refinements in order to acquire high temperature indentation data of high quality, especially in a vacuum environment. Such information can guide experiments aimed at understanding thermally-activated phenomena in materials.     

Cylindrical nano-indentation on metal film/elastic substrate system with discrete dislocation plasticity analysis: A simple model for nano-indentation size effect

The cylindrical nano-indentation on metal film/elastic substrate is computationally studied using two-dimensional discrete dislocation plasticity combined with the commercial software ANSYS®, with a focus on the storage volume for geometrically necessary dislocations (GNDs) inside the films and the nano-indentation size effect (NISE). Our calculations show that almost all GNDs are stored in a rectangular area determined by the film thickness and the actual contact width. The variations of indentation contact width with indentation depth for various film thicknesses and indenter radii are fitted by an exponential relation, and then the GND density underneath the indenter is estimated. Based on the Taylor dislocation model and Tabor formula, a simple model for the dependence of the nano-indentation hardness of the film/substrate system on the indentation depth, the indenter radius and the film thickness is established, showing a good agreement with the present numerical results.     

Size effects in the conical indentation of an elasto-plastic solid

The size effect in conical indentation of an elasto-plastic solid is predicted via the Fleck and Willis formulation of strain gradient plasticity (Fleck, N.A. and Willis, J.R., 2009, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057). The rate-dependent formulation is implemented numerically and the full-field indentation problem is analyzed via finite element calculations, for both ideally plastic behavior and dissipative hardening. The isotropic strain-gradient theory involves three material length scales, and the relative significance of these length scales upon the degree of size effect is assessed. Indentation maps are generated to summarize the sensitivity of indentation hardness to indent size, indenter geometry and material properties (such as yield strain and strain hardening index). The finite element model is also used to evaluate the pertinence of the Johnson cavity expansion model and of the Nix–Gao model, which have been extensively used to predict size effects in indentation hardness.     

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.     

Finite element study for conical indentation of elastoplastic micropolar material

Numerous experiments have repetitively shown that the material behavior presents effective size dependent mechanical properties at scales of microns or submicrons. In this paper, the size dependent behavior of micropolar theory under conical indentation is studied for different indentation depths and micropolar material parameters. To illustrate the effectiveness of the micropolar theory in predicting the indentation size effect (ISE), an axisymmetric finite element model has been developed for elastoplastic contact analysis of the micropolar materials based on the parametric virtual principle. It is shown that the micropolar parameters contribute to describe the characteristic of ISE at different scales, where the material length scale regulates the rate of hardness change at large indentation depth and the value of micropolar shear module restrains the upper limit of hardness at low indentation depth. The simulation results showed that the indentation loads increase as the result of increased material length scale at any indentation depth, however, the rate of increase is higher for lower indentation depth, relative to conventional continuum. The numerical results are presented for perfectly sharp and rounded tip conical indentations of magnesium oxide and compared with the experimental data for hardness coming from the open literature. It is shown that the satisfactory agreement between the experimental data and the numerical results is obtained, and the better correlation is achieved for the rounded tip indentation compared to the sharp indentation.     

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.     

The capillary force in micro- and nano-indentation with different indenter shapes

The influence of the indenter shapes and various parameters on the magnitude of the capillary force is studied on the basis of models describing the wet adhesion of indenters and substrates joined by liquid bridges. In the former, we consider several shapes, such as conical, spherical and truncated conical one with a spherical end. In the latter, the effects of the contact angle, the radius of the wetting circle, the volume of the liquid bridge, the environmental humidity, the gap between the indenter and the substrate, the conical angle, the radius of the spherical indenter, the opening angle of the spherical end in the truncated conical indenter are included. The meniscus of the bridge is described using a circular approximation, which is reasonable under some conditions. Different dependences of the capillary force on the indenter shapes and the geometric parameters are observed. The results can be applicable to the micro- and nano-indentation experiments. It shows that the measured hardness is underestimated due to the effect of the capillary force.