Determination of the hardness and elastic modulus from continuous vickers indentation testing

Continuous Vickers (H _v) indentation tests were performed on different materials (ion crystals, metals, ceramics, silica glass and plastic). Load-indentation depth curves were taken during the loading as well as during the unloading period by a computer controlled hydraulic mechanical testing machine (MTS 810). The indentation work measured both the loading and the unloading periods, and these were used for the evaluation of parameters characterizing the materials. It was found empirically that there were linear connections between the maximum load to the power 3/2 and the indentation work. These connections were used to relate the conventional hardness number, H _v, and Young's modulus, E, with the work performed during loading and unloading. This work can be determined with great accuracy from the measurements. The values of the Young's modulus and the Vickers hardness determined this way agree well with those obtained by conventional methods. On the basis of continuous indentation tests, materials can be easily classified into the isomechanical groups introduced by Ashby. For this classification the H _v/E ratio is generally used. As a substitute for H _v/E another parameter is recommended which can be determined easily from a single measurement.     

Representative indentation elastic modulus evaluated by unloading of nanoindentation made with a point sharp indenter

The conventional method to extract elastic modulus from the nanoindentation on isotropic linearly elastic solids is based on Sneddon’s solution (1965). However, it is known that the solution is valid only for incompressive elastic solids with the Poisson’s ratio ν of 0.5. This paper first proposes the modification of the solution in a wide range of ν from 0 to 0.5 through the numerical analysis on the unloading behavior of a simulated conical nanoindentation with a finite element method. As a result of the modification, the coefficient of linearity between the indentation elastic parameter k_e and Young’s modulus E is empirically given as a function of ν and the inclined face angle of the indenter, β, where k_e is defined as k_e ≡ P/h² with the indentation load P and penetration depth of the indenter h. According to the linear relationship between k_e and E, it is found that elastic rebound during unloading of a nanoindentation is uniquely characterized by a representative indentation elastic modulus E¹ defined in terms of E, ν and β, and that the value of E¹ can be evaluated from the P–h relationship with k_e and β. For an isotropic elastoplastic solid, the indentation unloading parameter k₂ defined as k₂ ≡ P/(h–h_r)² for a residual depth h_r is different from k_e even though a linearly elastic solid with k_e and elastoplastic solid with k₂ have a common E¹. In order to evaluate E¹ of an elastoplastic solid, the corresponding k_e is estimated from k₂ with an empirical equation as a function of the relative residual depth ξ defined as ξ ≡ h_r/h_max for the maximum penetration depth h_max. A nanoindentation experiment confirmed the validity of the numerical analysis for evaluating the elastic modulus.     

AFM indentation method used for elastic modulus characterization of interfaces and thin layers

Atomic force microscopy (AFM) is increasingly being used as a nanoindentation tool to measure local elastic properties of surfaces. In this article, a method based on AFM in force volume (force curve mapping) mode is employed to measure the elastic modulus distribution at the interface of a glass flake-reinforced polypropylene sample and at a lead-free Cu–solder joint. Indentation arrays are performed using a diamond AFM tip. The processing of experimental AFM indentation data is automated by customized software that can analyse and calibrate multiple force curves. The analysis algorithm corrects the obtained force curves by selecting the contact point, discarding the non-contact region and subtracting the cantilever deflection from the measured force curve in order to obtain true indentation curves. A Hertzian model is then applied to the resulting AFM indentation data. Reference materials are used to estimate the tip radius needed to extract the elastic modulus values. With the proposed AFM measurement method, we are able to obtain high-resolution maps showing elastic modulus variations around a composite interface and a Cu–solder joint. No distinct interphase region is detected in the composite case, whereas a separate intermetallic layer (1–2 μm thick) of much higher Young’s modulus (~131 GPa) than Cu and solder material is identified in the Cu–solder joint. Elastic modulus results obtained for the Cu (~72 GPa), solder (~50 GPa) and glass (~65 GPa) materials are comparable to the results obtained by instrumented indentation [~73, ~46 and ~61 GPa], which accentuates the potential of this method for applications requiring high lateral resolution.     

Determination of hardness to elastic modulus ratios using Knoop indentation measurements and a model based on loading and reloading half-cycles

A model is developed which results in an expression relating the residual in-surface dimension of the minor diagonal of a Knoop indenter through a residual-width parameter (b _R/b^*)² to the hardness-to-modulus ratio (H/E) of elastic-plastic materials. The relationship is shown to predict within reasonable accuracy both the intercept and slope of a plot of (b _R/b^*)² against (H/E), using data for a variety of materials. The work supplements concepts presented in two previous investigations.     

获取材料本构关系和硬度的压入法研究

依据锥形压入试验和弹塑性接触有限元分析,提出基于能量原理预测金属材料本构关系的CR-EMI方法。通过该方法揭示锥形压入能量比与表征应力之间存在线性律,提出通过压入曲线(P-h曲线)获取材料本构关系的关系式,根据Hollomon本构关系模型预测硬度的H-EMI方法。通过对多种金属材料进行压入试验和有限元分析,验证CR-EMI方法和H-EMI方法的有效性与精确性。     

On the determination of elastic modulus in very stiff materials by depth sensing indentation

Depth sensing indentation tests were carried out in stiff ceramics like Al₂O₃, AlN, SiC and B₄C, using a diamond Berkovich tip. The experiments show that the accuracy of the data depends on the stiffness ratio between material and indenter. An iterative calibration procedure is proposed to get a reliable estimation of the elastic modulus.     

Fracture toughness, hardness and elastic modulus of hydrogenated amorphous carbon films deposited by chemical vapor deposition

Fracture toughness, hardness and elastic modulus of hydrogenated amorphous carbon thin films produced from butene gas deposited by chemical vapor deposition on silicon were measured by depth sensing indentation. Voltage bias varying from − 60 to − 400 V and pressures of 2 and 8 Pa were used for deposition. Cube corner indentation produces film chipping at loads lower than 400 mN. A new approach to measure toughness was proposed to determine the energy released during film chipping. Fracture toughness results from this new approach are in between of the ones obtained from methods proposed in the literature.     

Effects of process and source on elastic modulus of single cellulose fibrils evaluated by atomic force microscopy

A test method to measure cellulose fibril elastic modulus using atomic force microscopy was used to investigate the effects of process and source on the moduli of single cellulose fibrils. The cellulose fibrils were generated from cellulose by mechanical treatments. Individual fibrils were suspended over a micro scale groove etched on a silicon wafer. A nano-scale three-point bending test was performed to obtain the elastic moduli. The results indicated that the elastic moduli of cellulose fibrils were not significantly different between 30 min and 60 min of high intensity ultrasonic treatment for Lyocell fiber, between isolation methods of ultrasonic and homogenizer treatment for pure cellulose fiber, and between different cellulose sources of pulp fibers treated by homogenizer regardless the effects of sample size coupled with inherent variation in the raw material. The elastic modulus of Lyocell fibrils with diameters from 150 to 180 nm was evaluated to be 98 ± 6 GPa. Modulus values decreased dramatically when the diameter was more than 180 nm.     

Young's modulus of lignin from a continuous indentation test

The deformation of lignin in a continuous ball indentation test was almost entirely elastic up to a stress of 2.2 × 10⁸ Pa (22 kg mm^–2). The load versus depth of indentation curve of the lignin followed closely the classical Hertz equation thus enabling the Young's modulus of lignin to be calculated. From these results a stress-strain curve for lignin was drawn.Visitor from Mechanical Engineering Department, University of Maryland, College Park, Md 20742, USA.     

材料维氏硬度仪器化压入方法准确度研究

针对仪器化压入测试识别材料维氏硬度的精度问题,该文采用有限元数值分析方法获得不同材料的维氏硬度理论计算值,以此为基础,分别对仪器化压入识别材料维氏硬度的3种代表性方法——"ISO方法"、"Kang方法"和"Ma方法 "进行理论精度分析,并进行实验验证。结果表明:1)3种方法识别的维氏硬度理论误差均随材料比功We/Wt增加呈先减小后增大的趋势;其中,"ISO方法"识别的维氏硬度值相比理论值普遍偏大,"Kang方法"识别的维氏硬度值相比理论值均偏小。2)"Ma方法"基于仪器化压入识别材料维氏硬度的理论误差最小,理论准确度相对较高。3)当被测材料的材料比功在0.01We/Wt0.3时,对应η和n不同取值下的3种方法各自识别的维氏硬度误差值较为离散;当0.3We/Wt0.85时则较为集中。该文工作,为下一步研究维氏硬度仪器化压入新方法提供一定的理论基础。     

Influence of visco-elasto-plastic properties of magnetite on the elastic modulus: Multicyclic indentation and theoretical studies

In the present work, we study the indentation behaviour of the magnetite coexisting with hematite in a natural dual-phase crystal. In particular we show the influence of cycling indentation conditions on the elastic modulus measurement in relation to the visco-elasto-plastic properties of the material. Elastic properties of Fe₃O₄ are investigated using Oliver and Pharr's technique, which is based on depth-sensing indentation (DSI) analysis. Depending on the visco-elasto-plastic properties of the material, the indentation test conditions (monotonic, cyclic, loading and unloading rates, dwell time at peak load, …) can modify the shape of the load–depth curve and, subsequently, the results. Molecular dynamics simulation based on shell model potential, is used to determine elastic quantities including elastic modulus, bulk modulus and Young modulus.     

Surface morphology,elastic modulus and hardness of thin film cathodes for Li-ion rechargeable batteries

In this study, surface morphology, elastic modulus and hardness of two thin film cathode materials, namely layered structured LiNi_1/3Co_1/3Mn_1/3O₂ and spinel structured LiMn₂O₄, during the charge/discharge cycles, are measured by using Scanning Electron Microscopy, Atomic Force Microscopy and nanoindentation experiments. Furthermore, the effects of depth of discharge (DOD) and charging rate (current density) on the changes of elastic modulus and hardness of the spinel structured LiMn₂O₄ are also investigated. The results have shown that both elastic modulus and hardness of the thin film cathodes have been significantly affected by the charge/discharge cycles as well as the condition of the charge/discharge processes. These results suggest the importance of the mechanical properties of the cathode materials to the reliability and integrity of the cathode materials to be used for the Li-ion batteries. The possible mechanisms of the changes in mechanical properties are also discussed.     

Cell-wall hardness and Young's modulus of melamine-modified spruce wood by nano-indentation

Samples of spruce wood were infiltrated with a melamine–formaldehyde resin. After curing of the resin, a melamine concentration of 24% (v/v) was measured in the secondary cell walls of melamine treated wood. Nano-indentation tests revealed an average Young's modulus of 16.1 GPa and a hardness of 0.24 GPa for untreated secondary cell walls. In the melamine treated cell walls, an increase in the Young's modulus of 33% to 21.4 GPa was observed. With 115%, i.e. 0.52 GPa, the increase in longitudinal hardness due to melamine–formaldehyde treatment was even more pronounced. This proves clearly that melamine treatment of wood improves mechanical properties of cell walls. Thus, treatment of wood with melamine–formaldehyde resin shows a considerable potential to improve mechanical properties, as desired for applications where large stresses normal to grain arise.     

分形维数和弹性模量衰减表征2D-C/SiC的拉伸蠕变损伤

真空条件对2D—C／SiC复合材料在1300℃和1500℃进行了高温拉伸蠕变试验，蠕变进行到0、0．5h、2h、10h、25h、50h中断试验，用SEM观察表面形貌，用盒维数法计算试样表面裂纹的分形维数；同时测量试样的弹性模量。结果表明，由于2D—C／SiC特有的蠕变损伤形式，所形成的损伤尺度都较短，其分形维数介于0～1之间。用分形维数和弹性模量衰减都可表征2D—C／SiC的蠕变损伤，两种损伤参量所描述的蠕变损伤总的发展趋势基本一致，即蠕变开始阶段损伤发展较快，随后进入缓慢发展的第二阶段。在第二阶段中，分形维数表征的损伤持续单调增加；而用弹性模量衰减表征的损伤在该阶段出现先下降随后升高的现象。以基体裂纹为主要损伤形式的条件下。分形维数主要反映蠕变试样局部的损伤，而弹性模量衰减反映的是蠕变试样整体的损伤。     

采用梯度功能方法的IPMC弹性模量改进模型

采用梯度功能材料力学方法,通过改进层合板理论模型来预测材料的力学性能。将饱和掺杂金属原子的外层作为表面镀层,将无掺杂的基体作为中间层,两层之间掺杂的金属原子含量逐渐变化,作为梯度层;使用EDS方法得到了Pt型金属/离子聚合物复合材料(IPMC)沿厚度方向的Pt原子含量分布,证明了模型分层的合理性;采用ASTM标准测定了IPMC材料的拉伸和弯曲弹性性能。本模型采用Mori-Tanaka方法预测表面镀层和梯度层的弹性性能,采用梯度力学方法,最终得到IPMC材料整体的弹性性能。用本文中模型预测常态下和含水饱和态下的IPMC拉伸性能与实验值相比,误差分别为 0.69%和-2.05%;预测常态下IPMC的表观弯曲弹性模量与实验值相比,误差为-0.99%。     

Bulk modulus and hardness of chalcopyrite structured solids

The bulk modulus and microhardness can be represented by an empirical linear relation that is a simple function of melting temperature T_m, atomic volume Ω and product of ionic charges (Z₁Z₂Z₃). Values of bulk modulus B and microhardness H of A^IB^IIIC₂^VI and A^IIB^IVC₂^V chalcopyrite semiconductors exhibit a linear relationship when plotted against the k_BT_m/Ω (k_B = Boltzmann's constant), but fall on two straight lines according to the product of ionic charges of the compounds. This correlation is similar in form to other correlations in the literature for diffusion data of materials that indicate the significance of the melting temperature as a scaling or lattice dynamic properties of materials. The calculated results are compared with available experimental data and previous calculations based on phenomenological models.     

Weibull modulus of nano-hardness and elastic modulus of hydroxyapatite coating

Here we report the microstructural dependence of nano-hardness (H) and elastic modulus (E) of microplasma sprayed (MIPS) 230 μm thick highly porous, heterogeneous hydroxyapatite (HAP) coating on SS316L. The nano-hardness and Young’s modulus data were measured on polished plan section (PS) of the coating by the nanoindentation technique with a Berkovich indenter. The characteristic values of nano-hardness and Young’s modulus were calculated through the application of Weibull statistics. Both nano-hardness and the Young’s modulus data showed an apparent indentation size effect. In addition, there was an increasing trend of Weibull moduli values for both the nano-hardness and the Young’s modulus data of the MIPS-HAP coating as the indentation load was enhanced from 10 to 1,000 mN. An attempt was made in the present work, to provide a qualitative model that can explain such behavior.     

Analysis of data from various indentation techniques for thin films intrinsic hardness modelling

Due to the influence of the substrate, direct measurement of the hardness of thin films by standard micro-indentation tests is not always possible. In such situation, determination of the intrinsic film hardness requires the analysis of a set of experimental apparent hardness values obtained for different indentation loads. A number of mathematical equations based on various assumptions were proposed in literature for that purpose.Most of the models were established on the basis of standard Vickers indentation. Using these models to process the data obtained by Knoop indentation does not provide the same intrinsic hardness value, even after Knoop/Vickers standard conversion, than the one obtained from Vickers indentation. The same problem arises when processing the data coming from depth-sensing indentation. A method to obtain comparable hardness values is proposed in the present work by considering an “equivalent” Vickers hardness in the case of Knoop indentations and the corresponding Martens hardness for depth-sensing indentation. This method has been used to determine the intrinsic hardness of titanium nitride film.