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 indenter tip radius effect in micro- and nanoindentation hardness experiments.

Nix and Gao established an important relation between the microindentation hardness and indentation depth. Such a relation has been verified by many microindentation experiments (indentation depths in the micrometer range), but it does not always hold in nanoindentation experiments (indentation depths approaching the nanometer range). Indenter tip radius effect has been proposed by Qu et al. and others as possibly the main factor that causes the deviation from Nix and Gao's relationship. We have developed an indentation model for micro- and nanoindentation, which accounts for two indenter shapes, a sharp, conical indenter and a conical indenter with a spherical tip. The analysis is based on the conventional theory of mechanism-based strain gradient plasticity established from the Taylor dislocation model to account for the effect of geometrically necessary dislocations. The comparison between numerical result and Feng and Nix's experimental data shows that the indenter tip radius effect indeed causes the deviation from Nix-Gao relation, but it seems not be the main factor. The project supported by the National Natural Science Foundation of China (10121202) and the Ministry of Education of China (20020003023)     

血红细胞力学性能的纳米压痕实验和有限元模拟

采用纳米压痕技术和有限元方法研究了血红细胞的生物力学性能. 进行了血红细胞的纳米压痕实验, 得到了血红细胞的材料参数和变形形貌; 在实验基础上, 建立了血红细胞的三维有限元模型, 模拟了血红细胞的压痕载荷-位移曲线, 并考虑了参数效应. 数值模拟结果和实验数据符合很好. 通过改变压头与材料之间的摩擦系数和压头曲率半径等参数, 比较了载荷-位移曲线的变化情况. 研究表明摩擦系数对压痕载荷-位移曲线和应力分布影响很小, 而压头曲率对载荷-位移曲线的影响明显.     

Phenomenological constitutive model for a CNT turf

Carbon nanotubes (CNT), grown on a substrate, form a turf – a complex structure of intertwined, mostly nominally vertical tubes, cross-linked by adhesive contact and few bracing tubes. The turfs are compliant and good thermal and electrical conductors. In this paper, we consider the micromechanical analysis of the turf deformation reported earlier, and develop a phenomenological constitutive model of the turf. We benchmark the developed model using a finite element implementation and compare the model predictions to the results two different nanoindentation tests.The model includes: nonlinear elastic deformation, small Kelvin–Voigt type relaxation, caused by the thermally activated sliding of contacts, and adhesive contact between the turf and the indenter. The pre-existing (locked-in) strain energy of bent nanotubes produces a high initial tangent modulus, followed by an order of magnitude decrease in the tangent modulus with increasing deformation. The strong adhesion between the turf and indenter tip is due to the van der Waals interactions.The finite element simulations capture the results from the nanoindentation experiments, including the loading, unloading, viscoelastic relaxation during hold, and adhesive pull-off.     

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 analysis of nanoindentation in elasto-plastic solids

This paper examines the accuracy of the extracted elastic properties using the nanoindentation technique on elasto-plastic materials. The application of the correction factor evaluated in the linearly elastic case [Poon, B., Rittel, D., Ravichandran, G., 2008. An analysis of nanoindentation in linearly elastic solids. Int. J. Solids Structures 45 (24), 6018–6033.] on elastic–plastic materials is critically examined. It is then established that the accurate determination of the projected area of contact is found to be crucial for the accurate determination of elastic material properties. The conventional methods for the accurate determination of contact area are generally limited to ratios of Young’s modulus over yield stress, E/σ_y < 30 for elastic-perfectly plastic materials, which is too stringent for most materials. Thus, a new electrical resistance method is proposed to measure directly the projected contact area. Using numerical simulations, it was found that with the accurate determination of A, the error associated with the extracted elastic material properties is reduced by more than 50% in some cases. Using the newly proposed procedure, the error is also found to be independent of E/σ_y and the tip radius, ρ, and it is only a function of Poisson’s ratio, ν. This suggests that the errors might be due to the residual stresses at the plastic imprint that were found to depend on ν as well.     

Nanoindentation of polymers

This present work focuses on the surface characterization of reactor-thermoplastic-polyolefin copolymers (R-TPO) for automotive applications to deduce an appropriate parameter as a measure of the surface stickiness behavior. Therefore, nanoindentation experiments are performed and the well-known mechanical contact properties have been determined. Additional, it has been tried to measure an adhesion force between the indenter tip and the material’s surface. The adhesion force has been measured as negative force during unloading and is the tensile force needed to bring the indenter tip to the initial position. Moreover, the verification and correction of the viscoelastic effects have been considered.     

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.     

Indentation responses of piezoelectric films

Frictionless indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter (i.e., punch), conical indenter (i.e., cone), and spherical indenter (i.e., sphere) are investigated. Both insulating and conducting indenters are considered. The technique of Hankel transformation is employed to derive the corresponding dual integral equations for the mixed boundary value indentation problems. For the two limiting cases of infinitely thick and infinitely thin piezoelectric films, closed-form solutions are obtained. For piezoelectric films of finite thickness, a numerical method is constructed to solve the dual integral equations and semi-empirical models having only two unknown parameters are proposed for the responses of indentation force, electric charge and electric potential, and contact radius. With the two parameters inferred from the numerical results, the semi-empirical formulae are found to provide good estimates of the indentation responses for the two limiting cases of infinitely thick and thin piezoelectric films, as well as those in between. The inferred parameters in the proposed semi-empirical formulae for normalized indentation force and electric charge are checked against four different piezoelectric materials and are found to be insensitive to the selection of piezoelectric materials. It is believed that the proposed semi-empirical indentation formulae are useful in developing experimental indentation techniques to extract the material properties of piezoelectric films.     

纳米压痕试验中压头尺寸效应的准连续介质法分析

采用准连续介质法模拟了单晶铝纳米压痕试验过程,分析了不同宽度的刚性矩形压头所引起的初始塑性变形特点,获得了载荷-压深、应变能-位移和硬度-压深曲线.从位错理论的角度分析了压头尺寸对纳米压痕测试结果的影响.研究发现:随着压头宽度的不断增大,压头下方位错形核所需要的载荷和压深程度增大,需要的应变能增加,应变能的变化速率递增,纳米硬度值减小,呈现出明显的尺寸效应.同时表明在一定的压入深度下,硬度与压头尺寸之间存在着一定的比例关系,不同尺寸压头获得的硬度值可以相互换算,但当矩形刚性压头宽度大于或等于120时这种尺寸效应消失.研究结果为纳米压痕实验过程中压头尺寸的选择提供了参考依据.     

Two new expanding cavity models for indentation deformations of elastic strain-hardening materials

Two expanding cavity models (ECMs) are developed for describing indentation deformations of elastic power-law hardening and elastic linear-hardening materials. The derivations are based on two elastic–plastic solutions for internally pressurized thick-walled spherical shells of strain-hardening materials. Closed-form formulas are provided for both conical and spherical indentations, which explicitly show that for a given indenter geometry indentation hardness depends on Young’s modulus, yield stress and strain-hardening index of the indented material. The two new models reduce to Johnson’s ECM for elastic-perfectly plastic materials when the strain-hardening effect is not considered. The sample numerical results obtained using the two newly developed models reveal that the indentation hardness increases with the Young’s modulus and strain-hardening level of the indented material. For conical indentations the values of the indentation hardness are found to depend on the sharpness of the indenter: the sharper the indenter, the larger the hardness. For spherical indentations it is shown that the hardness is significantly affected by the strain-hardening level when the indented material is stiff (i.e., with a large ratio of Young’s modulus to yield stress) and/or the indentation depth is large. When the indentation depth is small such that little or no plastic deformation is induced by the spherical indenter, the hardness appears to be independent of the strain-hardening level. These predicted trends for spherical indentations are in fairly good agreement with the recent finite element results of Park and Pharr.     

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.     

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．     

NANOINDENTATION SIMULATION OF PE/POSS UNDER DIFFERENT SHAPES OF INDENTERS

In this paper,the nanoindentation simulation on the two models of neat polyethylene(PE) and the polyethylene incorporated with 25wt% POSS(POSS-PE) is performed to reveal the reinforcing mechanism of the mechanical properties.The influence of the indenter shapes on nanoindentation is researched by using three different shapes of diamond indenters(cube-corner indenter,cylindrical indenter with spherical tip and cylindrical indenter with flat tip).The molecular mechanics method is adopted to eliminate the temp...     

A generalized stress intensity factor to be applied to rounded V-shaped notches

In the presence of sharp (zero radius) V-shaped notches the notch stress intensity factors (N-SIFs) quantify the intensities of the asymptotic linear elastic stress distributions. They are proportional to the limit of the mode I or II stress components multiplied by the distance powered 1 − λ_i from the notch tip, λ_i being Williams’ eigenvalues. When the notch tip radius is different from zero, the definition is no longer valid from a theoretical point of view and the characteristic, singular, sharp-notch field diverges from the rounded-notch solution very next to the notch. Nevertheless, N-SIFs continue to be used as parameters governing fracture if the notch root radius is sufficiently small with respect to the notch depth.Taking advantage of a recent analytical formulation able to describe stress distributions ahead of rounded V-notches, the paper gives a generalized form for the notch stress intensity factors, in which not only the opening angle but also the tip radius dimension is explicitly involved. Such parameters quantify the stress redistribution due to the root radius with respect to the sharp notch case.     

Small scale,grain size and substrate effects in nano-indentation experiment of film–substrate systems

The mechanical properties of film–substrate systems have been investigated through nano-indentation experiments in our former paper (Chen, S.H., Liu, L., Wang, T.C., 2005. Investigation of the mechanical properties of thin films by nano-indentation, considering the effects of thickness and different coating–substrate combinations. Surf. Coat. Technol., 191, 25–32), in which Al–Glass with three different film thicknesses are adopted and it is found that the relation between the hardness H and normalized indentation depth h/t, where t denotes the film thickness, exhibits three different regimes: (i) the hardness decreases obviously with increasing indentation depth; (ii) then, the hardness keeps an almost constant value in the range of 0.1–0.7 of the normalized indentation depth h/t; (iii) after that, the hardness increases with increasing indentation depth. In this paper, the indentation image is further investigated and finite element method is used to analyze the nano-indentation phenomena with both classical plasticity and strain gradient plasticity theories. Not only the case with an ideal sharp indenter tip but also that with a round one is considered in both theories. Finally, we find that the classical plasticity theory can not predict the experimental results, even considering the indenter tip curvature. However, the strain gradient plasticity theory can describe the experimental data very well not only at a shallow indentation depth but also at a deep depth. Strain gradient and substrate effects are proved to coexist in film–substrate nano-indentation experiments.     

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.     

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.     

含缺陷双层石墨烯的纳米压痕模拟研究

石墨烯具有独特的力学、电学性能,被誉为是具有战略意义的新材料,具有广泛的应用前景. 目前生产的石墨烯含有各种缺陷,相较于完美石墨烯,其仍有较大应用价值. 因此有必要研究和掌握缺陷对石墨烯性能的影响,以便在目前的生产技术下,推动其工业化应用. 采用Tersoffff 势来模拟C—C 共价键的相互作用,Lernnard-Jones 势来模拟非成键碳原子之间相互作用力,基于分子动力学模拟了金刚石压头压入含缺陷双层石墨烯的纳米压痕过程,讨论了Lernnard-Jones 势函数的截断半径最佳值以及得到了典型的载荷-位移曲线. 重点探讨了Stone-Thrower-Wales、空位(包括单空位和双空位缺陷) 以及圆孔缺陷当位置不同和数目不同时对石墨烯力学性能的影响. 得出结论:薄膜中心存在缺陷时,破坏强度下降幅度特别明显. 空位缺陷在压头半径范围内存在时,临界载荷与缺陷与薄膜中心的距离成线性关系;缺陷数目越多,其杨氏模量、破坏强度等就越低. 圆孔缺陷数目在压头范围外达到一定浓度后会使石墨烯的力学性质显著降低. 本文结论也说明石墨烯结构稳定,对小缺陷不敏感,缺陷石墨烯仍具有较好的性能和使用价值.