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．     

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)     

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.     

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.     

Comparison between Berkovich,Vickers and conical indentation tests: A three-dimensional numerical simulation study

Three-dimensional numerical simulations of Berkovich, Vickers and conical indenter hardness tests were carried out to investigate the influence of indenter geometry on indentation test results of bulk and composite film/substrate materials. The strain distributions obtained from the three indenters tested were studied, in order to clarify the differences in the load–indentation depth curves and hardness values of both types of materials. For bulk materials, the differentiation between the results obtained with the three indenters is material sensitive. The indenter geometry shape factor, β, for evaluating Young’s modulus for each indenter, was also estimated. Depending on the indenter geometry, distinct mechanical behaviours are observed for composite materials, which are related to the size of the indentation region in the film. The indentation depth at which the substrate starts to deform plastically is sensitive to indenter geometry.     

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.     

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.     

考虑压头曲率半径和应变梯度的微压痕分析

在压头尖端曲率半径取100nm的前提下，采用Chen和Wang的应变梯度理论，对微压痕实验进行了系统的数值分析. 首先通过拟合载荷-位移实验曲线的后半段来确定材料的屈服应力和幂硬化指数值,然后用有限元方法数值模拟压痕实验，并将计算得到的整段载荷-位移曲线及硬度-位移曲线和实验结果进行了比较. 结果表明应变梯度理论所预测的计算结果和实验结果很好地符合,包括压痕深度在亚微米和微米范围内的整段曲线.     

Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments

The enhanced gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. They are successful in explaining the size effects encountered in many micro- and nano-advanced technologies due to the incorporation of an intrinsic material length parameter into the constitutive modeling. However, the full utility of the gradient-type theories hinges on one's ability to determine the intrinsic material length that scales with strain gradients, and this study aims at addressing and remedying this situation. Based on the Taylor's hardening law, a micromechanical model that assesses a nonlinear coupling between the statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs) is used here in order to derive an analytical form for the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro- and nano-indentation experiments using both spherical and pyramidal indenters. The deviation of the Nix and Gao [Mech. Phys. Solids 46 (1998) 411] and Swadener et al. [J. Mech. Phys. Solids 50 (2002) 681; Scr. Mater. 47 (2002) 343] indentation size effect (ISE) models’ predictions from hardness results at small depths for the case of conical indenters and at small diameters for the case of spherical indenters, respectively, is largely corrected by incorporating an interaction coefficient that compensates for the proper coupling between the SSDs and GNDs during indentation. Experimental results are also presented which show that the ISE for pyramidal and spherical indenters can be correlated successfully by using the proposed model.     

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.     

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.     

SIZE EFFECT AND GEOMETRICAL EFFECT OF SOLIDS IN MICRO—INDENTATION TEST

Micro-indentation tests at scales of the order of sub-micron show that the measured hardness increases strongly with decreasing indent depth or indent size,which is frequently referred to as the size effect.At the same time,at micron or sub-micron scale,another effect,which is referred to as the geometrical size effects such as crystal grain size effect,thin flim thickness effect,etc.,also influences the measured material hardness.However,the trends are at odds with the size-independence implied by the conventional elastic-plastic theory.In the present research,the strain gradient plasticity theory(Fleck and Hutchinson)is used to model the composition effects(size effect and geometrical effect) for polycrystal material and metal thin film/ceramic substrate systems when materials undergo micro-indenting.The phenomena of the “pile-up“ and “sink-in“ apeared in the indentation test for the polycrystal materials are also discussed.Meanwhile,the micro-indentation experiments for the polycrystal Al and for the Ti/Si3N4 thin film/substrate system are carried out.By comparing the theoretical predictions with experimental measuremtns.the values and the variation trends of the micro-scale parameter included in the strain gradient plasticity theory are predicted.     

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．     

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.     

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C⁰ axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.     

Viscoelastic solutions for conical indentation

The aim of indentation analysis is to link indentation data, typically an indentation force vs. indentation depth curve, P–h, to meaningful mechanical properties of the indented material. While well established for time independent behavior, the presence of a time dependent behavior can strongly affect both the loading and the unloading responses. The paper presents a framework of viscoelastic indentation analysis based on the method of functional equations, developed by Lee and Radok [1960, The contact problem for viscoelastic bodies, J. Appl. Mech. 27, 438–444]. While the method is restricted to monotonically increasing contact areas, we show that it remains valid at the very beginning of the unloading phase as well. Based on this result, it is possible to derive closed form solutions following the classical procedure of functional formulations of viscoelasticity: (1) the identification of the indentation creep function, which is the indentation response to a Heaviside load; and (2) a convolution integral of the load history over the indentation creep function. This is shown here for a trapezoidal loading by a conical indenter on three linear isotropic viscoelastic materials with deviator creep: the 3-parameter Maxwell model, the 4-parameter Kelvin–Voigt model and the 5-parameter combined Kelvin–Voigt–Maxwell model. For these models, we derive closed form solutions that can be employed for the back-analysis of indentation results from the loading and holding period and for the definition of unloading time criteria that ensure that viscous effects are negligible in the unloading response.     

Improved reverse analysis for material characterization with dual sharp indenters

It was illustrated by the author in the previous work that combinations between material properties and indentation parameters can be used as mixed parameters in dimensionless functions to capture the indentation response of materials to single and dual sharp indenters. These issues are further extended in the present study. A parametric finite element analysis was performed to investigate the conical indentation response of elasto-plastic solids. Frictional effects are studied. Conical indenters of half-included angles from 50° to 88° are considered to examine several fundamental features of instrumented sharp indentation within the frame work of limit analysis. Regarding dimensional analysis, it is found that a Taylor series expansion according to the elastic indentation work-total indentation work ratio W_e/W_t can be used to improve dimensionless functions. Within this context, a new set of dimensionless functions is explicitly constructed for hardness and indentation parameters of single and dual indenters. Based on formulated functions, a reverse analysis with dual sharp indenters, which was previously proposed by the author, is improved to extract mechanical properties of materials.     

周期性蜂窝材料微极等效模型及其微观应力的一种快速算法

将周期性蜂窝材料等效为具有非局部本构的微极连续介质,以解释实验中出现的尺度效应和边界层效应.在评论相关的多种不同方法(能量法、体积平均的均匀化法等)之后,提出了一种基于位移连续和单胞力平衡的推导微极等效本构参数的新方法.以正方形单胞制成的结构为例,在不同的结构与单胞尺寸比下,考虑承受集中点载荷、均布轴力和均布剪力三种载荷工况,比较了离散完全计算、经典连续介质等效和不同微极连续体等效本构的计算结果,建议了较好的微极本构参数值.数值模拟表明,集中点载荷和剪切载荷作用时,在加载点附近和边界部分,微极等效可以显著提高计算精度.最后,给出了一种映射算法,可以根据微极等效连续体分析的结果,快速计算出对应微观单胞构件的应力,以开有圆孔的方板应力集中为例,验证并考察了所提快速算法的有效性和计算精度.     

Nano-Indentation Hardness and Strain Hardening of Silicon,Sodium Chloride and Tungsten Crystals

Background

A previous review of micro- and nano-indentation hardness tests and their analyses gave emphasis to obtaining measurements of continuous nano-indentation load, P, versus, depth, h, recordings that monitor the full elastic–plastic deformation behavior of a localized crystal volume [1].

Objective

Attention is given to determining the complete, indentation-based, elastic–plastic deformation properties of the local volume, including the initial crystal elastic deformation behavior and, especially to evaluation of post pop-in plastic strain hardening.

Method

Stress–strain calculations are presented for an initial Hertzian elastic loading and follow-on crystal micro- and nano-scale plastic deformation responses [2].

Results

Applied load, P, dependencies on contact diameters, d_i, of silicon crystals are compiled on the basis of elastic, plastic and cracking predictions, giving indication at the lowest P values of an indentation size effect (ISE) for the crystal hardness. Elastic–plastic stress–strain curves are presented for sodium chloride and tungsten crystals. The hardness and strain hardening calculations also demonstrate an influence of the ISE.

Conclusions

The exceptional plastic strain hardening behaviors scale dimensionally with corresponding dislocation interactions and sessile reactions within the very localized plastic indentation zones. There is usefulness in determining elastic modulus values from the initial loading record. Micro- and nano-scale dislocation interactions/reactions account for the high stress and strain hardening levels as well as the occurrence of an ISE.