On Brinell and Boussinesq indentation of creeping solids

As an alternative to traditional tensile testing of materials subjected to creep, indentation testing is examined. Axisymmetric punches of shapes defined by smooth homogeneous functions are analysed in general at power law behaviour both from a theoretical and a computational point of view. It is first shown that by correspondence to nonlinear elasticity and self-similarity the problem to determine time-dependent properties admits reduction to a stationary one. Specifically it is proved that the creep rate problem posed depends only on the resulting contact area but not on specific punch profiles. As a consequence the relation between indentation depth and contact area is history independent. So interpreted, the solution for a flat circular cylinder (Boussinesq) is not only of intrinsic interest but serves as a reference solution to generate results for various punch profiles. This is conveniently carried out by cumulative superposition and in particular ball indentation (Brinell) is analysed in depth. A carefully designed finite element procedure based on a mixed variational principle is used to provide a variety of explicit results of high accuracy pertaining to stress and deformation fields. Universal relations for hardness at creep are proposed for Boussinesq and Brinell indentation in analogy with the celebrated formula by Tabor for indentation of strain-hardening plastic materials. Quantitative comparison is made with a diversity of experimental data attained by earlier writers and the relative merits of indentation strategies are discussed.     

Hertz contact at finite friction and arbitrary profiles

Axisymmetric contact at finite Coulomb friction and arbitrary profiles is examined analytically and numerically for dissimilar linear elastic solids. Invariance and generality are aimed at and an incremental procedure is developed resulting in a reduced benchmark problem corresponding to a rigid flat indentation of an elastic half-space. The reduced problem, being independent of loading and contact region, was solved by a finite element method based on a stationary contact contour and characterized by high accuracy. Subsequently, a tailored cumulative superposition procedure was developed to resolve the original problem to determine global and local field values. Save for the influence of the coefficients of friction and contraction ratio, it is shown that at partial slip the evolving relative stick-slip contour is independent of any convex and smooth contact profile at monotonic loading. For flat and conical profiles with rounded edges and apices, results are illustrated for relations between force, depth and contact contours together with surface stress distributions. The solution for dissimilar solids in a full space is transformed to a half-space problem and solved for a combination of material parameters in order to first determine interface traction distributions. Subsequently, full field values for the two solids were computed individually. In order to predict initiation of fracture and plastic flow, results are reported for the location and magnitude of maximum tensile stress and effective stress, respectively, for a range of geometrical and material parameters. In two illustrations, predicted results are compared with experimental findings related to initiation of brittle fracture and load-depth relations at nanoindentation.     

On oblique contact of creeping solids

Oblique indentation of power-law creeping solids by a rigid die is analysed in three dimensions with perfectly plastic behaviour emerging as an asymptotic case. Indenter profiles are prescribed to be axisymmetric for simplicity but not by necessity. Invariance and generality is aimed at, as the problem is governed by only four essential parameters, i.e. the die profile, p, the indentation angle, γ, the power-law exponent, n, and the coefficient of friction, μ. The solution strategy is based on a self-similar transformation resulting in a reduced problem corresponding to flat die indentation of complete contact. The reduced auxiliary problem, being independent of loading, history and time, was solved by a three-dimensional finite element analysis characterized by high accuracy. Subsequently, cumulative superposition was used to resolve the original problem and global and invariant relations between force, depth and contact area were determined. Detailed results are given for the location and shape of the contact region and stick/slip contours as well as for local states of surface stresses and deformation at flat and spherical indenters. Due to the asymmetry prevailing, it was found that in the spherical case, contact contours proved to be oval and shifted, although with normal and tangential forces only weakly coupled. Finite friction as compared to full adhesion proved to have only a minor effect on global relations. The framework laid down may be applied to the contact of structural assemblies subjected especially to elevated temperatures and also to various issues such as compaction of powder aggregates, flattening of rough surfaces and plastic impact.     

Contact with stick zone between an indenter and a thin incompressible layer

The dominant asymptotic term for the indentation of a thin elastic incompressible layer by an axisymmetric rigid indenter is considered. Complete adhesion is supposed everywhere in the contact area or else in a given inner region surrounded by an annular frictionless zone. Both the problems are formulated in the form of systems of coupled dual integral equations. Using operators transforming kernels of the Hankel transform into kernels of the Weber–Orr transform, the dual integral equations are reduced to systems of Fredholm integral equations of the second kind whose structures permit deriving asymptotic solutions. Simple expressions for the contact stresses, the penetration depth, and the contact radius in the case of an unknown contact area are obtained. Explicit formulae, derived for the flat and power law indenter profiles, allow us to analyze how stick and frictionless zones affect mechanical characteristics. Results manifest that the punch penetration exhibits strong sensitivity to contact conditions inspite of the fact that the radial traction is small. A conical indenter is less sensitive than flat-ended and spherical indenters.     

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.     

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.     

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.     

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.     

An Inverse Identification Method for the Characterization of Elastic Conforming Contact Behavior During Flat Punch Indentation

Experimental Mechanics - The flat punch indentation problem is a typical prototype of conforming contact which can be frequently encountered in many applications. Its characterization is of great...     

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.     

Measurement of Sub-Surface Core Damage in Sandwich Structures Using In-situ Hertzian Indentation During X-ray Computed Tomography

Composite sandwich structures with honeycomb cores show varying properties in geometry and mechanical behavior depending on the studied scale. Herein a new test and evaluation method for sub-surface core damage in the indentation area of honeycomb sandwich structures using computed tomography is presented. The combination of X-ray micro-computed tomography (X-μCT) and an image analysis procedure adjusted to the detection of core deformation mechanisms allows the extraction and quantification of externally invisible, sub-surface damage in the sandwich composite. For this specific contact or indentation loading case on the sandwich face sheet an in-situ device is introduced, enabling a 3D analysis of the structural change during progressing indentation depth.     

Effect of permanent indentation on the delamination threshold for small mass impact on plates

Small mass impacts on composite structures are common cases caused by hailstones and runway debris. Small mass impactors usually result in a wave controlled local response, which is independent of boundary conditions. This response occurs before the reflection of waves from the boundaries and cannot be modeled by large mass drop weight tests. An elasto-plastic contact law, which accounts for permanent indentation and damage effects, was used here to study small mass impact on laminated composite plates. By comparing with results from the Hertzian contact law, it was found that damage can change the dynamic response of the structure significantly with increasing impact velocity. Due to smaller contact force generated for the case of using elasto-plastic contact, the central displacement of the plate is also less than the one using Hertzian contact law. The linearized version of the contact law was then used to derive the closed-form approximations of the contact force, indentation and plate central displacement for the impact loading of composite laminates. The threshold velocity for delamination onset under small mass impact was predicted analytically based on the obtained peak contact forces by combining with an existing quasi-static delamination threshold load criterion. A good agreement was found between the predicted threshold values and published experimental results.     

Delamination mechanism maps for a strong elastic coating on an elastic–plastic substrate subjected to contact loading

Hard wear resistant coatings that are subjected to contact loading sometimes fail because the coating delaminates from the substrate. In this report, systematic finite element computations are used to model coating delamination under contact loading. The coating and substrate are idealized as elastic and elastic–plastic solids, respectively. The interface between coating and substrate is represented using a cohesive zone law, which can be characterized by its strength and fracture toughness. The system is loaded by an axisymmetric, frictionless spherical indenter. We observe two failure modes: shear cracks may nucleate just outside the contact area if the indentation depth or load exceeds a critical value; in addition, tensile cracks may nucleate at the center of the contact when the indenter is subsequently removed from the surface. Delamination mechanism maps are constructed which show the critical indentation depth and force required to initiate both shear and tensile cracks, as functions of relevant material properties. The fictitious viscosity technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces allows us to explore a wider parametric space that a conventional cohesive model cannot handle. Numerical results have also been compared to analytical analyses of asymptotic limits using plate bending and membrane stretching theories, thus providing guidelines for interpreting the simulation results.     

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.     

Failure behavior of composite sandwich structures under local loading

Usually when analyzing the mechanical response of foam-cored fiber-reinforced composite sandwich structures to localized static loading, the face sheets are treated as a linear-elastic material and no damage initiation and growth is considered. However, practice shows that at higher indentation magnitudes damage develops in the face sheet in the area of contact with the indentor, which could lead to local failure of the face laminate due to the loss of bending stiffness and strength. Therefore, the main objective of the present study is to develop a damage model for predicting the local failure in the composite face sheet and its influence on the load–displacement behavior of sandwich structures under local loading. For this purpose, the Hoffman failure criterion is incorporated into a finite element modeling procedure using the ABAQUS program system. Results deducted from the modeling procedure are compared with experimental data obtained in the case of static indentation tests performed on sandwich beam specimens using steel cylindrical indentors. It is shown that taking into account the damage in the face sheet leads to a substantial improvement in the performance of the model when simulating the mechanical behavior of the sandwich structures at higher indentation values.     

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.     

Effects of the moving speed of a rigid stamp on contact behaviors of anisotropic materials based on real fundamental solutions

The paper studies contact problem of a rigid stamp moving at a constant speed over the surface of anisotropic materials. The solution method is based on Galilean transformation, Fourier transform and singular integral equation. The stated mixed boundary value problem is reduced to a Cauchy type singular integral equation based on real fundamental solutions, which is solved exactly in the case of a rigid flat or cylindrical stamp. Explicit expressions for various stresses are obtained in terms of elementary functions. In particular, explicit formula is derived to determine the unknown contact region for the cylindrical stamp. For a flat stamp, detailed calculations are provided to show the influences of dimensionless moving speed on the normal and in-plane stress. For a cylindrical stamp, the effects of dimensionless moving speed, the mechanical loading and the radius on the contact region, the normal and in-plane stress are analyzed in detail.     

Hertzian fracture at unloading

Hertzian fracture through indentation of flat float glass specimens by steel balls has been examined experimentally. Initiation of cone cracks has been observed and failure loads together with contact and fracture radii determined at monotonically increasing load but also during unloading phases. Contact of dissimilar elastic solids under decreasing load may cause crack inception triggered by finite interface friction and accordingly the coefficient of friction was determined by two different methods. In order to make relevant predictions of experimental findings, a robust computational procedure has been developed to determine global and local field values in particular at unloading at finite friction. It was found that at continued loading it is possible to specify in advance how the contact domain divides into invariant regions of stick and slip. The maximum tensile stress was found to occur at the free surface just outside the contact contour, the relative distance depending on the different elastic compliance properties and the coefficient of friction. In contrast, at unloading invariance properties are lost and stick/slip regions proved to be severely history dependant and in particular with an opposed frictional shear stress at the contact boundary region. This causes an increase of the maximum tensile stress at the contour under progressive unloading. Predictions of loads to cause crack initiation during full cycles were made based on a critical stress fracture criterion and proved to be favourable as compared to the experimental results.     

Stress measurement of SS400 steel beam using the continuous indentation technique

An analytical model is presented for determining surface residual stress using continuous indentation. The elastic residual stress is assumed to have no influence on contact area or hardness and to be uniform over a volume that is several times larger than the indentation mark. A step-by-step analysis for the residual-stress-induced load difference at a given depth is outlined here and such concepts as stress interaction, stress-sensitive contact morphology, and reversible contact recoveries during a stress relaxation are described. Finally, the proposed method is applied to the interpretation of the continuous indentation results obtained from an SS400 steel beam in which controlled bending stresses are generated. The stress estimated, however, showed a high scatter due to plastic pile-up deformation. When the optically measured contact area is used as an alternative of the contact area calculated from the unloading curve, the re-evaluated stress agrees well with the already known applied stress.     

纳米压痕仪接触投影面积标定方法的研究

基于Oliver与Pharr方法的纳米压痕实验以其简单方便获得广泛的应用，但众多因素对压 痕实验结果的影响范围并无明确的结论. 其中压痕接触面积的确定是一个重要环节，该因素 对实验结果，特别是小深度下的实验结果具有重要影响. 仔细分析了Oliver与Pharr 方法并进行了几种材料的纳米压痕实验，针对该方法在接触深度确定、不同深度范围下方法 的适用性进行了说明. 分析结果表明，对所有的材料使用统一的面积公式，只有在大压痕深 度时才是适用的，而在小压痕深度时可能带来较大的误差. 因此，应慎重使用由Oliver与 Pharr方法得到的小压痕深度的硬度数据.