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.     

Spherical indentation of incompressible rubber-like materials

In recent years, indentation tests have been proven very useful in probing mechanical properties of small volumes of materials. However, a class of materials that very little has been done in this direction is rubber-like materials (elastomers). The present work investigates the spherical indentation of incompressible rubber-like materials. The analysis is performed in the context of second-order hyperelasticity and is accompanied by finite element computations and an extensive experimental program with spherical indentors of different radii. Uniaxial tensile tests were also performed and it was found that the initial elastic modulus correlates well with the indentation response. The experiments suggest stiffer indentation response than that predicted by linear elasticity, which is somehow counter-intuitive, if the uniaxial material response is to be considered. Regarding the uniqueness of the inverse problem, that is to establish material properties from spherical indentation tests, the answer is disappointing. We prove that the inverse problem does not give unique answer regarding the constitutive relation, except for the initial stiffness.     

Analysis of spherical indentation of superelastic shape memory alloys

Dimensional analysis and the finite element method are applied in this paper to study spherical indentation of superelastic shape memory alloys. The scaling relationships derived from dimensional analysis bridge the indentation response and the mechanical properties of a superelastic shape memory alloy. Several key variables of a superelastic indentation curve are revealed and examined. We prove that the bifurcation force in a superelastic indentation curve only relies on the forward transformation stress and the elastic properties of the initial austenite; and the return force in a superelastic indentation curve only relies on the reverse transformation stress and the elastic properties of the initial austenite. Furthermore, the dimensionless functions to determine the bifurcation force and the return force are proved to be identical. These results not only enhance our understanding of spherical indentation of superelastic shape memory alloys, but also provide the theoretical basis for developing a practicable method to calibrate the mechanical properties of a superelastic material from the spherical indentation test.     

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.     

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.     

Indentation of plastically graded substrates by sharp indentors

The present paper investigates the indentation of plastically graded substrates by sharp indentors. Contact analysis of plastically graded surfaces can be particularly useful in the design of load-bearing devices such as gears, rollers and electric contacts found in many macro- and micro-electro-mechanical systems. Substrates made of plastically graded materials are often encountered in nature or are artificially produced as a result of chemical and/or physical surface treatments. The variation of the plastic properties depends on micro-structural or compositional changes of the material with depth. The analysis of indentation of plastically graded substrates by sharp indentors provide the load-penetration response, as well as the strains and stresses inside the substrate, at maximum loading and at complete unloading. The parametric analysis of the solutions enables the direct correlation of the plastic properties and the load-penetration curves obtained from instrumented indentation tests. The variation of the plastic properties can subsequently be related to important micro-structural parameters such as particle composition, dislocation density and grain size. The results of this work show how surface modifications can induce plastic graded properties that strengthen substrates against contact-induced damage.     

纳米晶铜单向拉伸变形的分子动力学模拟

纳米材料是由尺度在1－100nm的微小颗粒组成的体系，由于它具有独特的性能而备受关注。本文简要地回顾了分子动力学在纳米材料研究中的应用，并运用它模拟了平均晶粒尺寸从1．79－5．38nm的纳米晶体的力学性质。模拟结果显示：随着晶粒尺寸的减小，系统与晶粒内部的原子平均能量升高，而晶界上则有所下降；纳米晶体的弹性模量要小于普通多晶体，并随着晶粒尺寸的减小而减小；纳米晶铜的强度随着晶粒的减小而减小，显示了反常的Hall-Petch效应；纳米晶体的塑性变形主要是通过晶界滑移与运动，以及晶粒的转动来实现的；位错运动起着次要的、有限的作用；在较大的应变下（约大于5％），位错运动开始起作用；这种作用随着晶粒尺寸的增加而愈加明显。     

Nanoindentation of Pb/Sn solder alloys; experimental and finite element simulation results

A thermodynamics based damage mechanics coupled constitutive model is used to simulate the nanoindentation experiment on eutectic commercial Pb/Sn solder alloys. The study is aimed at explaining the apparent size dependent behavior observed at small indentation depths for this class of highly rate dependent materials. Interest is at small depths of indentation and a small-strains small-displacements theory is used. The study is motivated by the fact, that in the case of rate independent materials such a size dependence of the response is explained in terms of storage of the so-called geometrically necessary dislocations. Presently it is not clear if this is the source of size effects in a rate dependent material. Moreover, the storage of dislocations is difficult in eutectic solder alloys due to thermally activated diffusion processes that promote dislocation mobilization as opposed to storage. Therefore, a classical theory constitutive model should be able to capture such a size dependent feature of the response. The paper describes the constitutive model and its integration scheme. Simulations of the nanoindentation problem are performed at different temperatures and indentation strain rates and numerical results are compared with experiments.     

Spherical indentation of composite laminates with controlled gradients in elastic anisotropy

Three-dimensional elastic analyses and experiments of indentation of thick laminated plates of carbon fiber reinforced epoxy are presented. Pointwise, the material is characterized as a linear elastic orthotropic material. The in-plane orientation of the carbon fibers is systematically varied as a function of depth. The influence of fiber orientation as a function of depth on the indentation response is considered along with the relationship between the indenter force vs depth. The fiber orientation profiles considered are those of a continuous linear variation between 90° at the outer surfaces and 0° at the center plane of the laminate, and a cross ply laminate involving alternating 90° and 0° layers through thickness. Experimentally, it is found that for the case of a cross-ply laminate, the indentation produces delaminations localized at the interfaces that separate planes of dissimilar orientation. For this case, stress concentrations at interfaces between plies of dissimilar orientation coincide with the observed sites of delamination. For the graded case, evidence of enhanced nonlinear deformation is found, without the nucleation of cracks. Computations show that for the graded material, tensile stresses perpendicular to fibers are suppressed significantly, possibly explaining the absence of matrix cracks in this material. Measured and computed indenter force-depth variations were found to be in good agreement. Experiments and computations also reveal that the orientation-graded material is more compliant when subjected to indentation than the conventional cross-ply laminate.     

功能梯度材料涂层半空间的轴对称光滑接触问题

求解了功能梯度材料涂层半空间的轴对称光滑接触问题,其中梯度层剪切模量按照线性变化,利用Hankel积分变换方法求解微分方程,将问题化为具有Cauchy型奇异核的积分方程.数值方法求解表明:功能梯度材料涂层半空间在刚性柱形压头和球形压头作用下,接触表面分布应力,接触半径以及最大压痕受材料梯度效应的影响较大.     

A new reverse analysis to determine the constitutive response of plastically graded case hardened bearing steels

A new reverse analysis is presented that determines the plastic response of both nongraded and plastically graded materials (PGMs) without the need for traditional tension or compression tests. The method utilizes the concepts of expanding cavity model for strain hardening materials, Tabor’s rule of converting Vickers hardness to flow stress, representative plastic strain induced by indentation, and finite element modeling of the macro indentation process. A unique flow curve is determined when the experimentally measured increase in micro Vickers hardness matches that predicted by the proposed method within the plastic zone of a macro Vickers indent. The method is validated for a nongraded stainless steel first and then extended to determine the gradient in flow curves of plastically graded, case-hardened M50 NiL material, a widely used bearing steel. Such knowledge of the plastic response of the case hardened region will help optimize the design of case hardened bearings with longer rolling contact fatigue life and higher thrust load capabilities.     

A Load-Based Multiple-Partial Unloading Micro-Indentation Technique for Mechanical Property Evaluation

A load-based multiple-partial unloading micro-indentation technique has been developed for evaluating mechanical properties of materials. Comparing to the current prevailing nano/micro-indentation methods, which require precise measurements of the indentation depth and load, the proposed technique only measures indentation load and the overall indentation displacement (i.e. including displacement of the loading apparatus). Coupled with a multiple-partial unloading procedure during the indentation process, this technique results in a load-depth sensing indentation system capable of determining Young’s modulus of metallic alloys with flat, tubular, or curved architectures. Test results show consistent and correct elastic modulus values when performing indentation tests on standard alloys such as steel, aluminum, bronze, and single crystal superalloys. The proposed micro-indentation technique has led to the development of a portable load-depth sensing indentation system capable of on-site, in-situ material property measurement.     

Orientation effect on the Boussinesq indentation of a transversely isotropic piezoelectric material

Three-dimensional finite element analysis was used to study the effect of the angle between the loading direction and the axisymmetric direction on the indentation behavior of a transversely isotropic piezoelectric half-space by a cylindrical indenter of flat end. Two cases were considered in the analysis, which included (a) the indentation by an insulating indenter, and (b) the indentation by a conducting indenter. Both the indentation load and the indentation-induced potential were found to be proportional to the indentation depth. Using the simulation results and the analytical relationship for the indentation by a rigid, insulating indenter, semi-analytical relationships were developed between the indentation load and the indentation depth and between the indentation-induced potential on the indenter and the indentation depth for the conducting indenter, respectively. The proportionality between the indentation-induced potential and the indentation depth is only a function of the angle between the loading direction and the poling direction, independent of the type of indenters, which may be used to measure the relative direction of the loading axis to the axisymmetric axis of transversely piezoelectric materials from the indentation test.     

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.     

Axisymmetric frictionless contact of functionally graded materials

The main interest of this study is a new method to solve the axisymmetric frictionless contact problem of functionally graded materials (FGMs). Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into a series of sub-layers with shear modulus varying linearly in each sub-layer and continuous at the sub-interfaces. With this model, the axisymmetric frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically. An erratum to this article can be found at     

Hardness-packing density scaling relations for cohesive-frictional porous materials

Recent progress in instrumented nanoindentation makes it possible today to test in situ phase properties and structures of porous materials that cannot be recapitulated ex situ in bulk form. But it requires a rigorous indentation analysis to translate indentation data into meaningful mechanical properties. This paper reports the development and implementation of a multi-scale indentation analysis based on limit analysis, for the assessment of strength properties of cohesive-frictional porous materials from hardness measurements. Based on the separation-of-scale condition, we implement an elliptical strength criterion which results from the nonlinear homogenization of the strength properties of the constituents (cohesion and friction), the porosity and the microstructure, into a computational yield design approach to indentation analysis. We identify the resulting upper bound problem as a second-order conical optimization problem, for which advanced optimization algorithms became recently available. The upper bound yield design solutions are benchmarked against solutions from comprehensive elastoplastic contact mechanics finite element solutions and lower bound solutions. Furthermore, from a detailed parameter study based on intensive computational simulations, we identify characteristic hardness-packing density scaling relations for cohesive-frictional porous materials. These scaling relations which are developed for two pore-morphologies, a matrix-pore morphology and a polycrystal (perfect disordered) morphology, are most suitable for the reverse analysis of the strength parameters of cohesive-frictional solids from indentation hardness measurements.     

Evaluation of the stress–strain curve of metallic materials by spherical indentation

A method for deducing the stress–strain uniaxial properties of metallic materials from instrumented spherical indentation is presented along with an experimental verification.An extensive finite element parametric analysis of the spherical indentation was performed in order to generate a database of load vs. depth of penetration curves for classes of materials selected in order to represent the metals commonly employed in structural applications. The stress–strain curves of the materials were represented with three parameters: the Young modulus for the elastic regime, the stress of proportionality limit and the strain-hardening coefficient for the elastic–plastic regime.The indentation curves simulated by the finite element analyses were fitted in order to obtain a continuous function which can produce accurate load vs. depth curves for any combination of the constitutive elastic–plastic parameters. On the basis of this continuous function, an optimization algorithm was then employed to deduce the material elastic–plastic parameters and the related stress–strain curve when the measured load vs. depth curve is available by an instrumented spherical indentation test.The proposed method was verified by comparing the predicted stress–strain curves with those directly measured for several metallic alloys having different mechanical properties.This result confirms the possibility to deduce the complete stress–strain curve of a metal alloy with good accuracy by a properly conducted instrumented spherical indentation test and a suitable interpretation technique of the measured quantities.     

Indentation of a hard film on a soft substrate: Strain gradient hardening effects

The strain gradient work hardening is important in micro-indentation of bulk metals and thin metallic films, though the indentation of thin films may display very different behavior from that of bulk metals. We use the conventional theory of mechanism-based strain gradient plasticity (CMSG) to study the indentation of a hard tungsten film on soft aluminum substrate, and find good agreement with experiments. The effect of friction stress (intrinsic lattice resistance), which is important in body-center-cubic tungsten, is accounted for. We also extend CMSG to a finite deformation theory since the indentation depth in experiments can be as large as the film thickness. Contrary to indentation of bulk metals or soft metallic films on hard substrate, the micro-indentation hardness of a hard tungsten film on soft aluminum substrate decreases monotonically with the increasing depth of indentation, and it never approaches a constant (macroscopic hardness). It is also shown that the strain gradient effect in the soft aluminum substrate is insignificant, but that in the hard tungsten thin film is important in shallow indentation. The strain gradient effect in tungsten, however, disappears rapidly as the indentation depth increases because the intrinsic material length in tungsten is rather small.     

Effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials

Up to now, the indentation of hyperelastic soft materials has not been completely understood. In this paper, the spherical indentation on hyperelastic soft solids was systematically investigated through theoretical analysis and finite element method (FEM). The validation and application of the Hertzian load-displacement relation for indentation of hyperelastic soft materials were clarified, the effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials were analyzed and discussed. It was found that the complicated indentation behaviors of hyperelastic soft solids mainly depended on the coupling interactions of large deformation and material nonlinearity. Besides, we proposed two new nonlinear elastic contact models to separate the effects of large deformation and material nonlinearity on spherical indentation responses of hyperelastic soft solids. Our efforts might help to enhance the understanding of hyperelastic indentation problems and provided necessary instructions for the mechanical characterization of hyperelastic soft materials.