Mechanics of indentation of plastically graded materials—I: Analysis

The introduction of controlled gradients in plastic properties is known to influence the resistance to damage and cracking at contact surfaces in many tribological applications. In order to assess potentially beneficial effects of plastic property gradients in tribological applications, it is essential first to develop a comprehensive and quantitative understanding of the effects of yield strength and strain hardening exponent on contact deformation under the most fundamental contact condition: normal indentation. To date, however, systematic and quantitative studies of plasticity gradient effects on indentation response have not been completed. A comprehensive parametric study of the mechanics of normal indentation of plastically graded materials was therefore undertaken in this work by recourse to finite element method (FEM) computations. On the basis of a large number of computational simulations, a general methodology for assessing instrumented indentation response of plastically graded materials is formulated so that quantitative interpretations of depth-sensing indentation experiments could be performed. The specific case of linear variation in yield strength with depth below the indented surface is explored in detail. Universal dimensionless functions are extracted from FEM simulations so as to predict the indentation load versus depth of penetration curves for a wide variety of plastically graded engineering metals and alloys for interpretation of, and comparisons with, experimental results. Furthermore, the effect of plasticity gradient on the residual indentation pile-up profile is systematically studied. The computations reveal that pile-up of the graded alloy around the indenter, for indentation with increasing yield strength beneath the surface, is noticeably higher than that for the two homogeneous reference materials that constitute the bounding conditions for the graded material. Pile-up is also found to be an increasing function of yield strength gradient and a decreasing function of frictional coefficient. The stress and plastic strain distributions under the indenter tip with and without plasticity gradient are also examined to rationalize the predicted trends. In Part II of this paper, we compare the predictions of depth-sensing indentation and pile-up response with experiments on a specially made, graded model Ni-W alloy with controlled gradients in nanocrystalline grain size.     

Conical indentation of strain-hardening solids

Conical indentation of strain hardening solids is examined within the spherical cavity expansion simulation pattern in finite strain plasticity. Analysis accounts for elastic compressibility and arbitrary strain hardening. Unlike existing studies of indentation processes that assume a definite yield point, the present formulation applies also to smooth elastoplastic transition. Approximate hardness formulae are derived, at different levels of accuracy, and compared with available finite element calculations. Effects of pile-up, or sink-in, and external friction have been ignored. It is suggested that test data over a range of cone angles can be used to reconstruct the axial stress–strain curve. The relation between cavitation and conical indentation is discussed. It is shown that the cylindrical Tresca cavitation yield stress serves as a natural scaling stress in estimating hardness of strain hardening solids.     

The Modified Rockwell Test: A New Probe for Mechanical Properties of Metals

In the present work a novel methodology is proposed, based on the combination of the Rockwell and the Vickers tests, to provide estimates of the mechanical properties of metal substrates. The analysis is based on some novel invariants obtained from the finite element solution of the Vickers indentation (the imprint diagonal relates to the maximum indentation depth and the residual indentation depth with the average pressure and the elastic modulus). Several other useful results are discussed and experiments are performed with a modified Rockwell apparatus on steel and aluminium alloys. The results are important for the interpretation of micro indentation tests. Inverting the indentation data, reasonably accurate results can be obtained for strain hardening properties for “power law” behaviour, whereas more complex strain hardening would require further investigation.     

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.     

Analysis of spherical indentation of materials with plastically graded surface layer

In the present work, a comprehensive parametric study for establishing contact mechanics of instrumented normal spherical indentation on homogeneous materials and materials with plastically graded surface layer (PGSL) was undertaken by dimensional analysis and finite element modeling. The spherical indentation response for homogeneous materials can be described only by two dimensionless parameters: strain hardening exponent and a unified parameter that can describe effects of both the normalized yield strength and the normalized indentation depth. The influences of these two parameters were investigated for a wide range of engineering materials, and the results may be used as an estimate of loading response and pile-up/sink-in behavior when the material properties are known. In the materials with PGSL, a linear gradient in yield strength, and no variation in elastic modulus and strain hardening exponent were explored. The indentation response of the materials with PGSL can be described only by three dimensionless parameters: the normalized indentation depth, the dimensionless strength gradient parameter, and the normalized PGSL thickness. The effects of these three parameters were studied systematically. The normalized pile-up/sink-in parameter is found to be an increasing function of the strength gradient parameter. The normalized pile-up/sink-in parameter increases (decreases) with increasing PGSL thickness for a fixed positive (negative) gradient case at large indentation depth. The results also indicate that the materials with positive PGSL can bear more loads and have significantly more resistance to contact crack formation.     

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.     

Material Characterization at High Strain by Adapted Tensile Tests

The strength of materials at high strain levels has been determined using the so-called Continuous-Bending-under-Tension (CBT) test. This is a modified tensile test where the specimen is subjected to repetitive bending at the same time. This test enables to create high levels of uniform strain. A wide variety of materials has been tested this way. The strength of the material after CBT testing has been measured in different ways: by secondary tensile tests, by interrupted CBT tests, and directly from the fracture in the CBT test. All methods yield similar results: the strength is largely unaffected by the cyclic pre-deformation and mainly depends on the overall increase in length. Only for multi-phase materials the strength shows a minor influence of CBT test conditions. The hardening follows the extrapolated hardening observed in a conventional tensile test, except for brass. This test method can potentially be used for measuring hardening curves at high strain levels.     

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.     

Oliver–Pharr indentation method in determining elastic moduli of shape memory alloys—A phase transformable material

Instrumented indentation test has been extensively applied to study the mechanical properties such as elastic modulus of different materials. The Oliver–Pharr method to measure the elastic modulus from an indentation test was originally developed for single phase materials. During a spherical indentation test on shape memory alloys (SMAs), both austenite and martensite phases exist and evolve in the specimen due to stress-induced phase transformation. The question, “What is the measured indentation modulus by using the Oliver–Pharr method from a spherical indentation test on SMAs?” is answered in this paper. The finite element method, combined with dimensional analysis, was applied to simulate a series of spherical indentation tests on SMAs. Our numerical results indicate that the measured indentation modulus strongly depends on the elastic moduli of the two phases, the indentation depth, the forward transformation stress, the transformation hardening coefficient and the maximum transformation strain. Furthermore, a method based on theoretical analysis and numerical simulation was established to determine the elastic moduli of austenite and martensite by using the spherical indentation test and the Oliver–Pharr method. Our numerical experiments confirmed that the proposed method can be applied in practice with satisfactory accuracy. The research approach and findings can also be applied to the indentation of other types of phase transformable materials.     

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.     

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．     

Identification of Post-Necking Hardening Phenomena in Ductile Sheet Metal

A combined theoretical/experimental approach accurately quantifying post-necking hardening phenomena in ductile sheet materials that initially exhibit diffuse necking in tension is presented. The method is based on the minimization of the discrepancy between the internal and the external work in the necking zone during a quasi-static tensile test. The main focus of this paper is on the experimental validation of the method using an independent material test. For this purpose, the uniaxial tube expansion test is used to obtain uniaxial strain hardening behavior beyond the point of maximum uniform strain in a tensile test. The proposed method is used to identify the post-necking hardening behavior of a cold rolled interstitial-free steel sheet. It is demonstrated that commonly adopted phenomenological hardening laws cannot accurately describe all hardening stages. An alternative phenomenological hardening model is presented which enables to disentangle pre- and post-necking hardening behavior. Additionally, the influence of the yield surface on the identified post-necking hardening behavior is scrutinized. The results of the proposed method are compared with the hydraulic bulge test. Unlike the hydraulic bulge test, the proposed method predicts a decreased hardening rate in the post-necking regime which might be associated with probing stage IV hardening. While inconclusive, the discrepancy with the hydraulic bulge test suggests differential work hardening at large plastic strains.     

A computational study on the instrumented sharp indentations with dual indenters

Finite element analysis was performed to investigate the indentation response of elasto-plastic solids for conical indenters of half included angles of 60° and 70.3°. The interdependence indentation parameters resulting from a single indentation load–depth curve is considered. Regarding dimensional analysis, several dimensionless relationships are constructed as functions of the reduced elastic modulus-loading curvature ratio E¹/C and the strain hardening exponent n. Further, the duality between corresponding parameters with dual indenters is explored. Finally, a new method based on dual indenters is proposed to extract the strain hardening exponent and the reduced elastic modulus of an indented material. The accuracy of this method is verified and discussed with experimental data from the literature and representative materials.     

基于应变率阶跃测试的晶体铜压入应变率敏感性研究

纳米压入测试可以原位获取材料的诸多力学性能,包括弹性模量,硬度,屈服应力,应变率敏感指数等。本文利用应变率阶跃测试技术对多晶铜试样的应变率敏感性进行测试分析,硬度-位移曲线表明压头下方所存在的变形梯度对各阶跃应变率下的硬度值存在明显影响;采用基于晶体细观机制的塑性应变梯度理论对压入变形梯度效应予以修正,比较了修正与未修正数据所得的应变率敏感指数,在有效剔除压入变形梯度影响的基础上,应变率阶跃测试可实现单次压入下材料应变率敏感性的测试表征。     

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.     

Use of the hole-drilling method for measuring residual stresses in highly stressed shot-peened surfaces

The same shot-peening treatment was applied to five steels with different mechanical properties. The induced residual stress profiles were analyzed using X-ray diffraction and incremental hole drilling (IHD). The results of both techniques showed that IHD can still be successfully used for measuring shot-peening residual stresses, even if these exceed the yield strength of the bulk material. Expected errors due to the plasticity effect are reduced by the strain hardening of the surface. For an assessment of the reliability of IHD data, strain-hardening variation was quantified by microhardness measurements to estimate the yield strength of the plastified layer. All the main calculation methods for IHD were applied. The results were compared and discussed with respect to the characteristics of each method.     

Interfacial gradient plasticity governs scale-dependent yield strength and strain hardening rates in micro/nano structured metals

The effect of the material microstructural interfaces increases as the surface-to-volume ratio increases. It is shown in this work that interfacial effects have a profound impact on the scale-dependent yield strength and strain hardening of micro/nano-systems even under uniform stressing. This is achieved by adopting a higher-order gradient-dependent plasticity theory [Abu Al-Rub, R.K., Voyiadjis, G.Z., Bammann, D.J., 2007. A thermodynamic based higher-order gradient theory for size dependent plasticity. Int. J. Solids Struct. 44, 2888–2923] that enforces microscopic boundary conditions at interfaces and free surfaces. Those nonstandard boundary conditions relate a microtraction stress to the interfacial energy at the interface. In addition to the nonlocal yield condition for the material’s bulk, a microscopic yield condition for the interface is presented, which determines the stress at which the interface begins to deform plastically and harden. Hence, two material length scales are incorporated: one for the bulk and the other for the interface. Different expressions for the interfacial energy are investigated. The effect of the interfacial yield strength and interfacial hardening are studied by analytically solving a one-dimensional Hall–Petch-type size effect problem. It is found that when assuming compliant interfaces the interface properties control both the material’s global yield strength and rates of strain hardening such that the interfacial strength controls the global yield strength whereas the interfacial hardening controls both the global yield strength and strain hardening rates. On the other hand, when assuming a stiff interface, the bulk length scale controls both the global yield strength and strain hardening rates. Moreover, it is found that in order to correctly predict the increase in the yield strength with decreasing size, the interfacial length scale should scale the magnitude of both the interfacial yield strength and interfacial hardening.     

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.     

金属材料的强度与应力-应变关系的球压入测试方法

压入法获取材料单轴应力-应变关系和抗拉强度对服役结构完整性评价有重要的基础意义.假定材料均匀连续、各向同性、应力应变关系符合Hollomon律,基于能量等效假定,即代表性体积单元(representativevolume element, RVE)的vonMises等效和有效变形域内能量中值等效假定,本文提出了关联材料载荷、深度、球压头直径和Hollomon律的四参数半解析球压入(semi-analyticalspherical indentation,SSI)模型.通过球压入载荷-深度试验关系获得材料的应力-应变关系和抗拉强度.考虑压入过程中的损伤效应,针对金属材料提出了用于球压入测试的材料弹性模量修正模型.对11种延性金属材料完成了球压入试验,采用本文提出的球压入试验方法测到的弹性模量、应力-应变关系和抗拉强度与单轴拉伸试验结果吻合良好.