Analysis of elastic and plastic deformation associated with indentation testing of thin films on substrates

A study has been made of the elastic and plastic deformation associated with submicrometer indentation of thin films on substrates using the finite element method. The effects of the elastic and plastic properties of both the film and substrate on the hardness of the film/substrate composite are studied by determining the average pressure under the indenter as a function of the indentation depth. Calculations have been made for film/substrate combinations for which the substrate is either harder or softer than the film and for combinations for which the substrate is either stiffer or more compliant than the film. It is found, as expected, that the hardness increases with indentation depth when either the yield strength or the elastic modulus of the substrate is higher than that of the film. Correspondingly, the hardness decreases with indentation depth when the yield strength or elastic modulus of the substrate is lower than that of the film. Functional equations have been developed to predict the hardness variation with depth under these different conditions. Finite element simulation of the unloading portion of the load displacement curve permits a determination of the elastic compliance of the film/substrate composite as a function of indentation depth. The elastic properties of the film can be separated from those of the substrate using this information. The results are in good agreement with King's analytical treatment of this problem.     

Effect of Varying Test Parameters on Elastic–plastic Properties Extracted by Nanoindentation Tests

A systematic experiment was performed in an effort to investigate how the levels of certain test parameters affect the values of elastic modulus, hardness, yield stress, and strain hardening constant obtained using nanoindentation test. Maximum applied load, loading (unloading) rate, and hold time at maximum load were varied at three levels. The effects of these testing parameters were investigated through a three-level, full factorial design of experiment. The experiments were conducted on ultrafine Al-Mg specimens that were mechanically extruded. Both longitudinal and transverse extrusion directions were examined to investigate effects of anisotropy on mechanical properties and evaluate the persistence of observed variations due to test parameters on different materials orientations. An indentation size effect (ISE) was observed demonstrating that maximum load—and thereby maximum indentation depth—can have a significant effect on values of hardness and elastic modulus. Hardness values decreased with higher loading rates, and higher rates of unloading resulted in higher values of elastic modulus (5–10 GPa increases). Strain-hardening exponent showed a decreasing trend with increasing loading rate while yield stress exhibited a consistent correlation to hardness across all studied parameters. The material exhibited very little creep during the hold period, and values of the calculated properties were not significantly altered by varying the length of the hold time. Anisotropy effect was observed, particularly in the values of yield strength. This is attributed to the preferred grain orientation due to extrusion.     

Anisotropic Mechanical Properties of SAC Solder Joints in Microelectronic Packaging and Prediction of Uniaxial Creep Using Nanoindentation Creep

In this paper, the mechanical properties and creep behavior of lead-free solder joints has been characterized by nano-mechanical testing of single grain SAC305 solder joints extracted from plastic ball grid array (PBGA) assemblies. The anisotropic mechanical properties characterized include the elastic modulus, hardness, and yield stress. An approach is suggested to predict tensile creep strain rates for low stress levels using nanoindentation creep data measured at very high compressive stress levels. The uniaxial creep rate measured on similarly prepared bulk (large) specimens was found to be of the same order-of-magnitude as the creep rate observed in single-grain BGA joints, with chararacteristically (slightly) higher creep strains measured during nanoindentation. This suggests that the same creep mechanism operates in both size domains. Electron backscattered diffraction (EBSD) and nanoindentation testing further showed that the modulus, hardness, and creep properties of solder joints are highly dependent on the crystal orientation.     

Region-dependent micro damage of enamel under indentation

The objective of this investigation is to explore the region-dependent damage behavior of enamel, as well as to develop a good understanding of the deformation mechanisms of enamel with numerical modeling. Nanoindentation experiments have been performed to investigate the load-penetration depth responses for outer and inner enamel. Results show that the unloading curve does not follow the loading curve, and degradation of stiffness in the unloading curve is observed. Based on the experimental data, a physical quantity, the chain density in protein, has been introduced to the Drucker-Prager plastic model. Numerical simulations show that the simulated load-penetration depth curves agree with the experiments, and the stiffness degradation behaviors of outer and inner enamel are captured by the numerical model. The region-dependent damage behavior of enamel could be revealed by the numerical model. The micro damage affected area at inner enamel is larger than that at outer enamel, indicating that the inner enamel experiences more micro damage than the outer one. Compared with its outer counterpart, the inner enamel which is rich in organic protein could break more internal protein chains to dissipate energy and to enhance its resistance to fracture accordingly.     

岩石时频效应的实验研究

对饱和泵油南京砂岩和饱水大理岩在MTS上进行了单轴压缩实验和正弦波加载实验,求得两种饱和岩石的单轴抗压强度和屈服强度,并获得随正弦波频率的提高,饱和泵油南京砂岩的杨氏模量、纵横波速度增大,具有较明显的频散效应.饱和岩石中的微细观缺陷在正弦波加载中引起滞后,导致瞬时杨氏模量与应力成不对称"X"形,获得随正弦波频率增高,"X"交点在模量轴上的位置增高,杨氏模量增大,模量的频散增强.本文揭示了岩石的非线性时频响应的一些物理机理,对于岩石介质中的非线性波动研究以及地震、工程等理论研究和应用都具有重要意义.     

Nonlinearly Viscoelastic Nanoindentation of PMMA Under a Spherical Tip

The Oliver-Pharr method has been well established to measure Young’s modulus and hardness of materials without time-dependent behavior in nanoindentation. The method, however, is not appropriate for measuring the viscoelastic properties of materials with pronounced viscoelastic effects. One well-known phenomenon is the formation of unloading “nose” or negative stiffness during unloading that often occurs during slow loading-unloading in nanoindentation on a viscoelastic material. Most methods in literature have only considered the loading curve for analysis of viscoelastic nanoindentation data while the unloading portion is not analyzed adequately to determine the nonlinearly viscoelastic properties. In this paper, nonlinearly viscoelastic effects are considered and modeled using the nonlinear Burgers model. Nanoindentation was conducted on poly-methylmethacrylate (PMMA) using a spherical indenter tip. An inverse problem solving approach is used to allow the finite element simulation results to agree with the nanoindentation load–displacement curve during the entire loading and unloading stage. This approach has allowed the determination of the nonlinearly viscoelastic behavior of PMMA at submicron scale. In addition, the nanoindentation unloading “nose” has been captured by simulation, indicating that the negative stiffness in the viscoelastic material is the result of memory effect in time-dependent materials.     

基于塑性体积应变的岩石损伤变形特性实验研究

为分析岩石塑性变形与损伤的关系,在定义岩石的初始损伤和临界损伤,提出塑性体积应变分析方法,从而以塑性体积应变为损伤变量,采用归一化方法建立岩石的损伤本构模型。采用递增循环加载实验确定岩石损伤本构模型中的弹性卸载模量和弹性应变比例系数两个参数。通过实验和理论分析得出:当荷载较小时,普通单轴压缩状态下岩石损伤随荷载的增加具有减小趋势,荷载超过一定数值后,岩石损伤才开始增加;单轴递增循环压缩状态下当循环荷载大于约35%峰值强度后,卸载后岩石的损伤具有增加的趋势,小于该荷载之前具有减小的趋势。整个加载过程的理论应力-应变曲线能很好地与实验结果相吻合,在循环加载区间理论结果还能体现出岩石实验结果中的回滞环。     

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.     

COMP．B复合炸药动态力学性能和塑性流动本构关系的研究

利用自制的含能材料动态变温三轴压缩实验装置，采用准静态应变速率（１０－４／ｓ）和中等应变速率（３／ｓ），对国产复合炸药Ｃｏｍｐ．Ｂ进行了三轴压缩实验．测试了Ｃｏｍｐ．Ｂ在不同温度、不同应变速率条件下的杨氏模量Ｅ，泊松比ν和屈服强度Ｙ．实验结果表明，Ｃｏｍｐ．Ｂ具有明显的应变率相关和热软化效应．基于热激活模型，作了适当的改进，根据实验数据建立了含能材料塑性流动模型，分析表明该模型能合理地描述率相关材料的塑性流动，同时考虑了应变率和温度对塑性流动的影响．这些基础研究为含能材料动态力学性能的研究和炸药早爆机理的理论分析提供了依据     

Development of High Temperature Nanoindentation Methodology and its Application in the Nanoindentation of Polycrystalline Tungsten in Vacuum to 950 °C

The capability for high temperature nanoindentation measurements to 950 °C in high vacuum has been demonstrated on polycrystalline tungsten, a material of great importance for nuclear fusion and spallation applications and as a potential high temperature nanomechanics reference sample. It was possible to produce measurements with minimal thermal drift (typically ~0.05 nm/s at 750–950 °C) and no visible oxidative damage. The temperature dependence of the hardness, elastic modulus, plasticity index, creep, creep strain, and creep recovery were investigated over the temperature range, testing at 25, 750, 800, 850, 900 and 950 °C. The nanoindentation hardness measurements were found to be consistent with previous determinations by hot microhardness. Above 800 °C the hardness changes relatively little but more pronounced time-dependent deformation and plasticity were observed from 850 °C. Plasticity index, indentation creep and creep recovery all increase with temperature. The importance of increased time-dependent deformation and pile-up on the accuracy of the elastic modulus measurements are discussed. Elastic modulus measurements determined from elastic analysis of the unloading curves at 750–800 °C are close to literature bulk values (to within ~11 %). The high temperature modulus measurements deviate more from bulk values determined taking account of the high temperature properties of the indenter material at the point (850 °C) at which more significant time-dependent deformation is observed. This is thought to be due to the dual influence of increased time-dependency and pile-up that are not being accounted for in the elastic unloading analysis. Accounting for this time-dependency by applying a viscoelastic compliance correction developed by G. Feng and A.H.W. Ngan (J. Mater. Res. (2002) 17:660–668) greatly reduces the values of the elastic modulus, so they are agree to within 6 % of literature values at 950 °C.     

High Temperature Nanoindentation of PMR-15 Polyimide

This paper presents the high temperature nanoindentation experiments performed on an aerospace polymer resin–PMR-15 polyimide. The sharp-tipped Berkovich nanoindenter equipped with a hot-stage heating system was used. The indentation experiments were performed using the “hold-at-the-peak” method at various indenter holding times and unloading rates. The creep effect was seen to decrease with increasing holding time and/or unloading rate. Procedures used to minimize the creep effect are investigated at both ambient and elevated temperatures so that the correct contact depth (together with modulus and hardness) can be determined from nanoindentation load-depth curve. The temperature dependent mechanical properties of PMR-15 are measured through the current nanoindenter and results are consistent with those obtained from macroscopic tests.     

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.     

Determination of material properties using nanoindentation and multiple indenter tips

Nanoindentation is a useful method to probe the material properties of a solid. Its effective use lies in interpreting the data collected from a nanoindentation experiment with an associated analytical/numerical solution of the corresponding problem configuration. In this paper, a parametric finite element study has been performed to develop a new procedure for extracting elastic–plastic properties of a material through nanoindentation experiments with a substantially improved accuracy for the elastic properties of a elastic–plastic solid. The procedure involves data collected through the use of two, different, nanoindenter tips. Non-dimensional functions were constructed for two different indenter geometries to show that test results from multiple indenters, when appropriately manipulated, deliver superior results, compared to using one indenter. The material was assumed to be an isotropic elastic–plastic solid with power law hardening. Friction between the indenter and the material was included in the cases studied. The ratio of yield strength to elastic modulus was assumed to be in the range 0.0005–0.02 and the hardening coefficient was assumed to be between 0 and 0.4. Poisson’s ratio was fixed at 0.3.     

Determination of mechanical properties by instrumented indentation

The paper reviews the current state of the depth-sensing indentation (sometimes called nanoindentation), where the information on material behaviour and properties is obtained from the indenter load and depth, measured continuously during loading and unloading. It is shown how the contact parameters and principal characteristics are determined using pointed or spherical indenters. Indentation tests can be used for the measurement of hardness and elastic modulus, and also of the yield stress and for the construction of stress–strain diagrams, for the determination of the work of indentation and its components. Most devices use monotonic loading and unloading, but some also enable measurement under a small harmonic signal added to the basic monotonously increasing load. This makes possible continuous measurement of contact stiffness and the study of dynamic properties and the determination of properties of coatings. One section is devoted to the measurement on viscoelastic-plastic materials, where the delayed deforming must be considered during the measurement as well as in data evaluation. Instrumented indentation can also be used for the study of creep under high temperatures. The paper also discusses the errors arising in depth-sensing measurements and informs briefly about some other possibilities of the method.     

Effect of the relationship between the elastic modulus and plastic strain on residual stresses and strains in a tube

A semi-analytical solution of the problem of compression of a tube by an outer pressure with subsequent unloading is obtained. The effect of the relationship between the shear modulus and plastic strain on the residual stress and strain was evaluated using experimental data, according to which at an accumulated plastic strain of 0.25, the shear modulus decreases by 20%. It is found that despite the significant decrease in the shear modulus, its dependence on the accumulated strain has no significant effect on the residual strain. The effect of this dependence is manifested mainly in the distribution of the residual radial stress, but, in this case, too, it is extremely weak. The obtained general solution can be used to evaluate the effect of the relationship between the shear modulus and the accumulated plastic strain on the residual stress and strain for other materials.     

纳米铜丝尺寸效应的分子动力学模拟

采用EAM原子势函数对不同截面尺寸纳米铜丝拉伸性能进行了零温分子动力学模拟。研究表明截面变化对纳米丝拉伸性能有明显影响。由于表明原子弛豫，纳米丝存在表面张应力以及与之相平衡的内部应力。这种本征应力的存在是纳米丝尺寸效应的根源。纳米丝截面减小，则拉伸强度提高、屈服推迟、初始拉伸模量的软化程度增加。     

Particle fracture in high-volume-fraction ceramic-reinforced metals: Governing parameters and implications for composite failure

Weibull parameters of angular alumina particles are determined from experimental tensile test data on high-ceramic-content metal matrix composites using a micromechanical model that accounts for internal damage in the form of particle cracking, the dominant damage mode in these composites. The fraction of broken particles is assessed from the drop of Young's modulus and particle fracture is assumed to be stress controlled. Two extreme load-sharing modes, namely a purely local and a global load-sharing mode, are considered to account for the load redistribution due to particle fracture. Consistent powder strength parameters can be thus “back-calculated” for particles that are embedded in different Al-Cu matrices. On the other hand, this calculation fails for pure Al matrix composites, which exhibit a much larger strain to failure than Al-Cu matrix composites. It is shown that for Al matrix composites, the role of plastic (composite) strain on particle fracture constitutes a second parameter governing particle damage. This finding is rationalized by particle-particle interactions in these tightly packed ceramic particle-reinforced composites, and by the increase of matrix stress heterogeneity that is brought with increasing plastic strain. Failure of the alloyed matrix composites is well described by the (lower bound) local load-sharing micromechanical model, which predicts a catastrophic failure due to an avalanche of damage. The same model predicts failure of pure aluminium matrix composites to occur at the onset of tensile instability, also in agreement with experimental results once the role of plastic strain on damage accumulation is accounted for.     

Delamination of an elastic film from an elastic–plastic substrate during adhesive contact loading and unloading

Adhesive contact between a rigid sphere and an elastic film on an elastic–perfectly plastic substrate was examined in the context of finite element simulation results. Surface adhesion was modeled by nonlinear springs obeying a force-displacement relationship governed by the Lennard–Jones potential. A bilinear cohesive zone law with prescribed cohesive strength and work of adhesion was used to simulate crack initiation and growth at the film/substrate interface. It is shown that the unloading response consists of five sequential stages: elastic recovery, interface damage (crack) initiation, damage evolution (delamination), film elastic bending, and abrupt surface separation (jump-out), with plastic deformation in the substrate occurring only during damage initiation. Substrate plasticity produces partial closure of the cohesive zone upon full unloading (jump-out), residual tensile stresses at the front of the crack tip, and irreversible downward bending of the elastic film. Finite element simulations illustrate the effects of minimum surface separation (i.e., maximum compressive surface force), work of adhesion and cohesive strength of the film/substrate interface, substrate yield strength, and initial crack size on the evolution of the surface force, residual deflection of the elastic film, film-substrate separation (debonding), crack-tip opening displacement, and contact instabilities (jump-in and jump-out) during a full load–unload cycle. The results of this study provide insight into the interdependence of contact instabilities and interfacial damage (cracking) encountered in layered media during adhesive contact loading and unloading.     

泡沫金属压痕试验的数值模拟及其反演

在理论研究的基础上，将泡 沫金属压痕试验的有限元数值模拟结果与用无量纲分析法构造出的一系列无量纲函数相结 合，建立了泡沫金属压痕试验中载荷-压痕深度关系曲线与泡沫金属的弹塑性材料参数之 间的联系. 利用这种联系，就可以实现用压痕试验通过反演分析来确定泡沫金属的材料参数. 研究结果表明，泡沫金属材料的杨氏模量，屈服强度及塑性可压缩因子等参数均可由其压痕 试验唯一的确定，但其塑性平台区终点应变的确定还需进一步的研究.     

Mechanics of indentation of plastically graded materials—II: Experiments on nanocrystalline alloys with grain size gradients

A systematic study of depth-sensing indentation was performed on nanocrystalline (nc) Ni-W alloys specially synthesized with controlled unidirectional gradients in plastic properties. A yield strength gradient and a roughly constant Young's modulus were achieved in the nc alloys, using electrodeposition techniques. The force vs. displacement response from instrumented indentation experiments matched very well with that predicted from the analysis of Part I of this paper. The experiments also revealed that the pile-up of the graded alloy around the indenter is noticeably higher than that for the two homogeneous reference alloys that constitute the bounding conditions for the graded material. These trends are also consistent with the predictions of the indentation analysis.