A study on robust indentation techniques to evaluate elastic–plastic properties of metals

Spherical indentation is studied based on numerical analysis and experiment, to develop robust testing techniques to evaluate isotropic elastic–plastic material properties of metals. The representative stress and plastic strain concept is critically investigated via finite element analysis, and some conditions for the representative values are suggested. The representative values should also be a function of material properties, not only indenter angle for sharp indenter and indentation depth for spherical indenter. The pros and cons of shallow and deep spherical indentation techniques are also discussed. For an indentation depth of 20% of an indenter diameter, the relationships between normalized indentation parameters and load–depth data are characterized, and then numerical algorithm to estimate material elastic–plastic curve is presented. From the indentation load–depth curve, the new approach provides stress–strain curve and the values of elastic modulus, yield strength, and strain-hardening exponent with an average error of less than 5%. The method is confirmed to be valid for various elastic properties of indenter. Experimental validation of the approach then is performed by using developed micro-indentation system. For the material severely disobeying power law hardening, a modified method to reduce errors of predicted material properties is contrived. It is found that our method is robust enough to get ideal power law properties, and applicable to input of more complex physics.     

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.     

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.     

Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation

Knowledge of the relationship between the penetration depth and the contact radius is required in order to determine the mechanical properties of a material starting from an instrumented indentation test. The aim of this work is to propose a new penetration depth–contact radius relationship valid for most metals which are deformed plastically by parabolic and spherical indenters. Numerical simulation results of the indentation of an elastic–plastic half-space by a frictionless rigid paraboloïd of revolution show that the contact radius–indentation depth relationship can be represented by a power law, which depends on the reduced Young’s modulus of the contact, on the strain hardening exponent and on the yield stress of the indented material. In order to use the proposed formulation for experimental spherical indentations, adaptation of the model is performed in the case of a rigid spherical indenter. Compared to the previous formulations, the model proposed in the present study for spherical indentation has the advantage of being accurate in the plastic regime for a large range of contact radii and for materials of well-developed yield stress. Lastly, a simple criterion, depending on the material mechanical properties, is proposed in order to know when piling-up appears for the spherical indentation.     

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.     

基于微纳米压入法提取无铅焊料合金弹塑性力学参数的研究

本文通过微纳米压入法结合数值模拟研究了无铅焊料合金SnAg3.5 的弹塑性力学性能，分别采用圆柱形压头及两种不同锥角压头对无铅焊料合金进行压入测试：基于圆柱形压头测试过程中接触面积恒定的特点得到了无铅焊料的弹性模量，进一步采用塑性应变梯度理论对两种锥角压头的测试结果予以修正并通过数值模拟反分析得到相应的特征应力值，同时基于压入特征塑性应变与压头锥角的关系式得到两种不同锥角压头下的特征应变值，在此基础上经求解方程组得到焊料合金的初始屈服应力与应变强化因子，进而得到了焊料合金的幂强化弹塑性本构关系．该方法剔除了压入尺度效应的影响并保证了所得本构关系的唯一性，给出了一种通过原位压入测试表征金属材料弹塑性力学性能的有效方法．     

Multiple loading–unloading of an elastic–plastic spherical contact

A model for multiple repeated loading and unloading of an elastic–plastic sphere and a rigid flat is presented to cover a wide range of loading conditions far beyond the elastic limit. The sphere material is modeled as elastic linear isotropic hardening and follows the von Mises yield criterion. It is shown that although most of the plastic deformation occurs during the first loading, secondary plastic flow may evolve during the first unloading. The occurrence of this secondary plastic flow depends on the level of first loading and is strongly affected by the Poisson’s ratio and material hardening. The region of secondary plastic flow may propagate during the very first loading–unloading cycles, reaching a steady state after which the following loading–unloading cycles become fully elastic.     

Identification of residual stresses by instrumented elliptical indentation and inverse analysis

The residual stress tensor near the surface of a metal structural component or products of metallurgical processes (particularly welding) is considered in view of the estimation of its governing components. These are: the two principal stresses if the directions are “a priori” known as assumed here for the preliminary optimization of the indenter shape; otherwise two normal stresses and a shear stress according to a preselected reference system. In both situations the material is assumed to be endowed with known elastic–plastic properties. The novel parameter identification procedure investigated herein can be outlined as follows, when the unknown stresses are three: an instrumented indenter is adopted with elliptical section across the axis of an originally conical, or, as an alternative, of an originally spherical shape; three indentations are performed with the ellipse axis rotated by 45° in a sequence, in three locations near to each other at minimal distances apt to avoid interference; the three digitalized indentation curves (loading–unloading force versus penetration) are the source of the experimental data set used as input of inverse analysis; this is carried out by a fast method consisting of finite element simulations of the tests, “proper orthogonal decomposition”, “radial basis function” interpolation, and a first-order algorithm for the minimization of the discrepancy function. When the unknowns are the two principal stresses (directions known) two orthogonal indentations turn out to be sufficient.     

Unloading of an elastic–plastic loaded spherical contact

The process of unloading of an elastic–plastic loaded sphere in contact with a rigid flat is studied by finite element method. The sphere material is assumed isotropic with elastic-linear hardening. The numerical simulations cover a wide range of material properties and sphere radius. The contact load, stresses, and deformation in the sphere during both loading and unloading, are calculated for a wide range of interferences. Analytical dimensionless expressions are presented for the unloading load–deformation relation, the residual interference and the residual curvature of the sphere after complete unloading. A new measure termed elastic–plastic loading index is introduced to indicate the plasticity level of the loaded sphere. Some ideas regarding reversibility of the unloading process and elasticity of multiple loading unloading are also presented.     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

A numerical approach to indentation technique to evaluate material properties of film-on-substrate systems

Spherical indentation approach (Lee et al., 2005, Lee et al., 2010) for the evaluation of bulk material properties is extended to that for elastic–plastic properties of film-on-substrate systems. Our interest focuses on single isotropic, metallic, and elastic–plastic film on a substrate, and we do not consider the size effects in plasticity behavior. We first determine the optimal data acquisition location, where the strain gradient is the least and the effect of friction is negligible. Dimensional analysis affords the mapping parameters as functions of normalized indentation variables. An efficient way is further introduced to reduce both the number of analyses and the regression order of mapping functions. The new numerical approach to the film indentation technique is then proposed by examining the finite element solutions at the optimal point. With the new approach, the values of elastic modulus, yield strength, and strain-hardening exponent of film materials are successfully obtained from the spherical indentation tests. We have shown that the effective property ranges such as indenter properties, substrate modulus, and E/E_s ratio can be extended without additional simulations and even loss of accuracy. For other ranges of variables or other properties, which are not dealt with in this study, this methodology is applicable through resetting FEA variables and finding proper normalized parameters.     

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.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

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.     

Dynamic coupling of fluid and structural mechanics for simulating particle motion and interaction in high speed compressible gas particle laden flow

In this work, structural finite element analyses of particles moving and interacting within high speed compressible flow are directly coupled to computational fluid dynamics and heat transfer analyses to provide more detailed and improved simulations of particle laden flow under these operating conditions. For a given solid material model, stresses and displacements throughout the solid body are determined with the particle–particle contact following an element to element local spring force model and local fluid induced forces directly calculated from the finite volume flow solution. Plasticity and particle deformation common in such a flow regime can be incorporated in a more rigorous manner than typical discrete element models where structural conditions are not directly modeled. Using the developed techniques, simulations of normal collisions between two 1 mm radius particles with initial particle velocities of 50–150 m/s are conducted with different levels of pressure driven gas flow moving normal to the initial particle motion for elastic and elastic–plastic with strain hardening based solid material models. In this manner, the relationships between the collision velocity, the material behavior models, and the fluid flow and the particle motion and deformation can be investigated. The elastic–plastic material behavior results in post collision velocities 16–50% of their pre-collision values while the elastic-based particle collisions nearly regained their initial velocity upon rebound. The elastic–plastic material models produce contact forces less than half of those for elastic collisions, longer contact times, and greater particle deformation. Fluid flow forces affect the particle motion even at high collision speeds regardless of the solid material behavior model. With the elastic models, the collision force varied little with the strength of the gas flow driver. For the elastic–plastic models, the larger particle deformation and the resulting increasingly asymmetric loading lead to growing differences in the collision force magnitudes and directions as the gas flow strength increased. The coupled finite volume flow and finite element structural analyses provide a capability to capture the interdependencies between the interaction of the particles, the particle deformation, the fluid flow and the particle motion.     

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.     

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．     

A thermodynamic formulation of finite-deformation elastoplasticity with hardening based on the concept of material isomorphism

This paper presents a thermodynamic formulation of a model for finite deformation of materials exhibiting elastoplastic material behaviour with non-linear isotropic and kinematic hardening. Central to this formulation is the notion that the form of the elastic constitutive relation be unaffected by the plastic deformation or transformation in the material, as commonly assumed in particular in the context of crystal plasticity. When generalized to the phenomenological context, this implies that the internal variable representing plastic deformation is an elastic material isomorphism. Among other things, this requirement on the plastic deformation leads directly to the standard elastoplastic multiplicative decomposition of the deformation gradient. In addition, a dependence of the plastic part of the free energy on the plastic deformation itself yields a thermodynamic form for the centre of the elastic range of the material, i.e. the back stress. Finally, we show how this approach can be applied to formulate thermodynamic forms for linear, and non-linear Armstrong-Frederick, kinematic hardening models.     

高温仪器化压入测试中热接触对位移测量漂移的影响

基于对室温压头和热试样接触后传热过程的分析, 重点研究热接触引起的压头基托热膨胀对高温仪器化压入测试中位移测量漂移的影响. 首先, 通过热传导理论分析, 获得热接触后基托内温度场分布的解析解, 进而研究基托热膨胀引起的位移测量漂移量; 然后, 建立有限元分析模型, 数值模拟高温仪器化压入过程, 验证理论模型的准确性. 研究发现, 压头与热试样接触面间的热传导性质显著影响基托内的温度场分布; 对于不同材料的试样, 接触面间传热性能不同, 基托的热膨胀量差异可以达到几个数量级. 研究结果有助于优化高温压入测试程序, 提高测试的可靠性.      

Contact laws between solid particles

This paper presents a comprehensive study for the contact laws between solid particles taking into account the effects of plasticity, strain hardening and very large deformation. The study takes advantage of the development of a so-called material point method (MPM) which requires neither remeshing for large deformation problems, nor iterative schemes to satisfy the contact boundary conditions. The numerical results show that the contact law is sensitive to impact velocity and material properties. The contact laws currently used in the discrete element simulations often ignore these factors and are therefore over-simplistic. For spherical particles made of elastic perfectly plastic material, the study shows that the contact law can be fully determined by knowing the relative impact velocity and the ratio between the effective elastic modulus and yield stress. For particles with strain hardening, the study shows that it is difficult to develop an analytical contact law. The same difficulty exists when dealing with particles of irregular shapes or made of heterogeneous materials. The problem can be overcome by using numerical contact laws which can be easily obtained using the material point method.