Equivalent Indentation Research Articles

A novel model for obtaining both equi-biaxial and uniaxial residual stress of metallic materials by instrumented sharp indentation

As a new method, the indentation technique has become popular for measuring residual stress and mechanical properties at surfaces due to its simplicity of use, in-situ applicability, and nano-to-macro range of measurements. In this study, a novel model for obtaining the surface residual stress of metallic materials is proposed by instrumented sharp indentation. The metallic material containing surface residual stress is equivalent to another non-stressed material with a different yield stress in Hollomon law from the perspective of indentation response. Thus, a unified equivalent material indentation (UEMI) model is established for measuring both equi-biaxial and uniaxial residual stresses within metallic structural materials. Additionally, only five parameters need to be calibrated via finite element analysis (FEA) in this model. Moreover, the novel model is successfully verified using lots of simulated materials from FEA and seven metallic materials experimental data found in the references. The results show that the predicted residual stress agrees well with the pre-applied residual stress which meeting the needs of engineering.

Strain rate sensitivity of a Cu60Zr40 metallic and nanoglass

The strain rate sensitivity, m, of a binary Cu60 Zr40 nanoglass (NG) and metallic glass (MG) are investigated using nanoindentation. Indentations were performed at different loading rates in the range of 0.26–8 mN/s, which gives equivalent indentation strain rates over three decades. The load vs. displacement curves of MG exhibited noticeable displacement bursts at low loading rates, which gradually decreased with increasing loading rate suggesting a transition from more to less severe heterogeneous plastic flow. While in the case of NG, no noticeable displacement bursts are present at any of the loading rates suggesting a near homogeneous plastic flow. In both NG and MG, the hardness decreases with increasing loading rate, resulting in negative strain rate sensitivity, m. The m for NG is higher than the MG, indicating a more homogeneous flow underneath the indentation. Interface indentation experiments and subsequent analysis of the deformation zone showed a larger number of fine secondary shear bands (SSBs) in NG as compared to the primary shear bands (PSBs), while the plastic flow in MG is accommodated mostly by the PSBs. The findings of the current study will help improve the understanding of the plastic deformation behavior of NGs and provide insights for designing the novel microstructural architecture of amorphous alloys with improved ductility.

A continuous contact force model for impact analysis

Hunt and Crossley proposed a general expression of the contact force in 1975. The dissipative force term of the general expression has two exponents: the exponent of indentation depth and that of velocity. Because it is almost impossible to obtain an analytical solution based on the general expression, more than twenty continuous contact models have been developed based on the simplification of the general expression. In these studies, the exponent of the indentation depth was set as 0.25, 0.5, 0.65, 1.0 or 1.5, and the exponent of the velocity was set to 1.0. This paper proposes a new continuous contact force model with arbitrary values of the exponents of the indentation depth and velocity. The model is based on the general expression of the contact force. Considering the rule of energy equivalence, an approximatedynamic equation is developed by introducing the equivalent indentation and equivalent velocity. Subsequently, a new continuous contact force model is constructed based on the system dynamic equation and approximate dynamic equation. The influences of the two exponents on the performance of the continuous contact force models are investigated by analyzing the simulation results. Moreover, the validity of the new model is demonstrated by comparing the simulation results with two published experimental datasets. The comparison also indicates that the performance of the continuous contact model can be enhanced by selecting appropriate values of the two exponents.

A generalized Maugis-Dugdale solution for adhesion of power-law graded elastic materials

The present paper provides a generalized Maugis-Dugdale analytical model of adhesive contact between an axisymmetric rigid punch of arbitrary shape and a graded elastic half-space with Young's modulus varying with depth according to a power-law. Based on the principle of superposition and an equivalent indentation method to solve an axisymmetric external crack problem, a series of closed-form solutions are derived for power-law punch profiles which reduce to the existing solutions for homogeneous isotropic materials and for paraboloidal geometries as special cases. A complete transition between the JKR and DMT type models is captured through a generalized Tabor parameter. Finite element analysis is performed to validate the obtained analytical solutions and examine the effect of interaction potential form on the pull-off force. Further important findings from this study include: (1) The pull-off force for non-paraboloidal punch profiles is found to be largely dependent on the form of the interaction potential, which is distinct from the familiar paraboloid case; (2) For the so-called Gibson solid, the independence of the pull-off force on the Tabor parameter is valid only for the paraboloidal punch profiles; (3) A three-dimensional adhesion map has been constructed to aid the appropriate choice of different contact models involving power-law graded elastic materials.

Continuous contact force model with an arbitrary damping term exponent: Model and discussion

More than twenty continuous contact force models have been presented for modeling contact-impact problems; however, there was no uniformity regarding the value of the damping term exponent in these models. A new type of continuous contact force model is proposed in this research, in which the value of the damping term exponent can be arbitrary. Then, the effect of the value of the damping term exponent on the model accuracy is investigated by comparing the simulation and experimental results. As it is almost impossible to obtain an analytical solution based on the system dynamic equation, according to the rule of energy equivalence, an approximate dynamic equation is developed by introducing an equivalent indentation and equivalent velocity. Based on the system dynamic equation and the approximate dynamic equation, a primary formula for the hysteresis damping factor of the model can be obtained. Through nondimensional analysis, new models for different values of the damping term exponent are established by modifying the primary formula. The comparison between the simulation results and published experimental data demonstrates the validities of the new models and reveals that the influence of the value of the damping term exponent on the model accuracy can be considered to be negligible.

Analysis of Viscoelastic Behavior of Polypropylene/Carbon Nanotube Nanocomposites by Instrumented Indentation.

In this work, the viscoelastic behavior of polypropylene (PP)/multi-walled carbon nanotube (MWCNT) nanocomposites was investigated by indentation testing and phenomenological modeling. Firstly, indentation tests including two-cycle indentation were carried out on PP/MWCNT nanocomposite with three MWCNT loadings (1, 3 and 5 wt %). Next, the Maxwell–Voigt–Kelvin model coupled with two-cycle indentation tests was used to predict the shear creep compliance function and the equivalent indentation modulus. The indentation hardness and elastic modulus of the PP/MWCNT nanocomposites extracted based on the Oliver and Pharr method were compared with the equivalent indentation modulus predicted based on the Maxwell–Voigt–Kelvin mode. The experimental results indicated that the addition of nanotubes into the polypropylene has a positive effect on the micro-mechanical properties of PP/MWCNT nanocomposites. Indentation hardness and elastic modulus increased significantly with increasing MWCNT loading. The creep resistance at the micro-scale of the PP/MWCNT nanocomposites improved with the addition of MWCNTs, with creep displacement reduced by up to 20% by increasing the carbon nanotube loading from 1 to 5 wt %. The Maxwell–Voigt–Kelvin model with three and five Voigt–Kelvin units accurately predicted the shear creep function and its change with increasing MWCNT loading. However, the equivalent indentation modulus was found to be sensitive to the number of Voigt–Kelvin units: the more Voigt–Kelvin units, the better the model predicts the equivalent indentation modulus.

An Equivalent Indentation Method for the External Crack with a Dugdale Cohesive Zone

An equivalent indentation method is developed for the external crack problem with a Dugdale cohesive zone in the both axisymmetric and two-dimensional (2D) cases. This is achieved based on the principle of superposition by decomposing the original problem into two simple boundary value problems, with one considering action of a constant traction within the cohesive zone, and the other corresponding to indentation by a rigid concave punch. Closed-form expressions are derived for the distributions of displacement and traction on the crack interface, which are consistent with the classical results in fracture mechanics. Results show that the interfacial traction distributions in the axisymmetric and 2D cases share the similar mathematical forms except for different coordinate parameters. Finite element analysis is performed to validate the obtained analytical solutions. The proposed method relies solely on a few contact solutions on the surface irrespective of a general elasticity solution in the whole body, and it may find applications in the external crack analysis and adhesive contact model involving functionally graded elastic solids or piezoelectric materials.

Relationship of Stiffness-Based Indentation Properties Using Continuous-Stiffness-Measurement Method.

The determination of elastic modulus (E) and hardness (H) relies on the accuracy of the contact area under the indenter tip, but this parameter cannot be explicitly measured during the nanoindentation process. This work presents a new approach that can derive the elastic modulus (E) and contact depth (hc) based on measured experiment stiffness using the continuous-stiffness-measurement (CSM) method. To achieve this, an inverse algorithm is proposed by incorporating a set of stiffness-based relationship functions that are derived from combining the dimensional analysis approach and computational simulation. This proposed solution considers both the sink-in and pile-up contact profiles; therefore, it provides a more accurate solution when compared to a conventional method that only considers the sink-in contact profile. While the proposed solution is sensitive to Poisson’s ratio (ν) and the equivalent indentation conical angle (θ), it is not affected by material plasticity, including yield strength (σy) and work hardening (n) for the investigated range of 0.001 < σy/E < 0.5. The proposed stiffness-based approach can be used to consistently derive elastic modulus and hardness by using stiffness and the load-and-unload curve measured by the continuous-stiffness-measurement (CSM) method.

Investigation of Indentation Zone on K9 Glass under Ultrasonic Vibration Condition Using an Equivalent Mean Contact Pressure

Ultrasonic vibration-assisted grinding (UVAG) is a promising processing method for parts made of hard-brittle material, as a green machining. The material removal mechanism of hard-brittle material in UVAG is not yet fully understood. The mean contact pressure in the mechanical model of material deformation/fracture that is induced by abrasive grit reveals the deformation/fracture level during the material removal of the indentation of hard-brittle material. To investigate the material removal mechanism of K9 glass in UVAG, a new variable, namely, the equivalent mean contact pressure (EMCP) of glass indentation under ultrasonic vibration was proposed as a corresponding variable for mean contact pressure. A series of indentation experiments with/without ultrasonic vibration was carried out to analyze the material fracture of glass as induced by single random-shaped diamond grits. To obtain the EMCP, an equivalent characteristic size of ultrasonic vibration indentation of K9 glass was defined and measured in the experiments. The vertical load, EMCP and the profile features of the indentation zone, such as the equivalent characteristic size and indentation depth, were investigated in the indentation experiments. Subsurface damage of the K9 glass was also investigated by using UVAG and traditional grinding experiments to analyze the significance of the EMCP on material fracture during machining. The experimental results show the EMCP correlated better with the subsurface damage from UVAG than vertical load. Therefore, compared with the load, the EMCP is a better variable to reveal the difference of material removal mechanism between the UVAG and traditional grinding of glasses.

Analysis of the equivalent indenter concept used to extract Young’s modulus from a nano-indentation test: some new insights into the Oliver–Pharr method

In this paper a thorough analysis of the equivalent indenter concept applied to nano-indentation is carried out, motivated by the fact that previous works in the field have not considered the requirement of a consistent relation between contact depth and projected contact area. Dimensional analysis is initially used to prove that the shape of the axisymmetric equivalent indenter can be regarded as a material property, provided that size-effects are negligible. Subsequently, it is shown that such shape can effectively be employed to describe the nano-indentation unloading stage by means of Sneddon’s elastic solution which is formally valid only for indentation into a flat surface. This allows for formulating the problem of extracting Young’s modulus from the unloading curve as an optimization problem. However, it is proved that the latter does not have a unique solution, due to the particular mathematical structure of the underlying equations; hence, additional constraints are needed to set restrictions on the admissible equivalent indenter shapes. An example of such constraint is hidden in some apparent inconsistencies of the well-known Oliver–Pharr method, which is demonstrated to be based on an equivalent conical indenter whose semi-apical angle depends on the ratio between residual and total penetration. Specifically, this angle tends to 90° when the material exhibits extensive inelastic deformation, whereas it reduces to the one characteristic of the real indenter for a perfectly elastic material. This provides a new physical explanation for the relatively good accuracy of the method even in presence of a non-negligible residual contact impression on the sample.

J0230104 肩こりのハンディタイプ押込試験システムを用いたYoung率による計測評価

Spherical indentation testing is studied to evaluate the muscle stiffness of human shoulder. Here, the Hertzian contact theory is functionally extended to evaluate indentations for the thin tissues on the human body. In the expansions, the technique used for evaluating the thickness of finite specimens is first explained by analyzing the experimental results of indentations. Then, the elasticity of soft materials with finite thickness is theoretically derived by defining an equivalent indentation strain for the analysis of the indentation process. The expansions are examined to evaluate its reliability by applying them to measure the stiffness of some soft materials. Furthermore, this technology is applied to the elasticity investigation of the human body. Especially, the measurement results of elastic characteristics of the body of human body are shown and the availability of developed device is discussed to reveal the deformation mechanics of human body in this report.

A new insight into the work-of-indentation approach used in the evaluation of material’s hardness from nanoindentation measurement with Berkovich indenter

The work-of-indentation approach employed to extract the hardness of a material from the load–displacement data is often misinterpreted and improperly compared with the Oliver and Pharr method. A theoretical basis is presented to show that the hardness values evaluated using these two methods are fundamentally different, but are interrelated. As such, an expression that relates one type of hardness to another is derived by considering an energy-based relationship between the contact and the maximum penetration depths for an ideally sharp Berkovich indenter. This modified work-of-indentation approach is first validated using the load–displacement data obtained from the finite element simulation of the indentation contact performed with a Berkovich equivalent conical indenter on materials having a wide range of elastic recovery. The real load–displacement data – with and without dwelling – from ceramics oxides and metals corresponding to a peak indentation load lying in the range 20–120mN are also considered to validate the proposed method. It has been found that a correction due to the amount of work done during dwelling is required if the load–displacement data also feature a dwelling phase. The hardness value so obtained is found to be in close agreement with its Oliver and Pharr analogue for each material considered thereby indicating that the modified approach, in the present form, is applicable even for blunt indenter when the tip radius is very small as compared to the maximum penetration depth. Further refinement of this methodology is, however, required to take the effect of material’s pile-up, sink-in, and tip bluntness on the measured hardness into account.

Plastic deformation behaviors of Ni- and Zr-based bulk metallic glasses subjected to nanoindentation

Plastic deformation behaviors of Ni42Ti20Zr21.5Al8Cu5Si3.5 and Zr51Ti5Ni10Cu25Al9 bulk metallic glasses at room temperature were studied by nanoindentation testing and atomic force microscopy under equivalent indentation experimental conditions. The different chemical composition of these two bulk metallic glasses produced variant tendencies for displacement serrated flow to occur during the loading process. The nanoindentation strain rate was calculated as a function of indentation displacement in order to verify the occurrence of displacement serrated flow at different loading rates. Atomic force microscopy revealed decreasing numbers of discrete shear bands around the indentation sites as loading rates increased from 0.025 to 2.5mNs−1. Variations in plastic deformation behaviors between Ni and Zr-based glasses materials can be explained by the different metastable microstructures and thermal stabilities of the two materials. The mechanism governing plastic deformation of these metallic glasses was analyzed in terms of an established model of the shear transformation zone.

An approximate model of cylindrical and spherical contact on composite laminates of finite thickness

An empirical force–contact area relationship for the two-dimensional (2D) plane strain line contact between a stiff cylindrical indenter and a composite laminate of finite thickness supported on a rigid substrate is presented. The validity of a 2D Hertzian line contact force–contact length relation originally meant for isotropic contact problems on laminates of finite thickness is first investigated. The same relationship can be applied to orthotropic composite plates provided that an equivalent indentation modulus is known. The indentation modulus can then be corrected for the finite thickness of the plates. The results correlate well with experimental observation. Similar procedures can be performed to find the corresponding correction for a 3D axisymmetric contact problem.

Extracting Elastic Properties of Heterogeneous Layered Materials using Atomic Force Microscopy (AFM)

AFM nano-indentation is used to evaluate micro-mechanical properties of biological samples. However, such samples often have multi-components with heterogeneous properties, whereas traditional data analysis yields an aggregate modulus that may not represent the individual constituents. This study aimed to develop an analysis technique that can extract individual elastic moduli from AFM indentation tests on heterogeneous multi-layered samples. Building on a modified homogenization theory (Li, Acta Mechanica, 2005), finite element modeling (FEM) was used to simulate the indentation response of layered samples and determine the relationship between equivalent volume ratio (fΩ) versus indentation depth (D), and to examine whether the relative substrate (E1) and inclusion (E2) properties could be extracted from depth-dependent apparent modulus values (Epw vs. D). To validate the FEM, two-layered polydimethylsiloxane (PDMS) samples were fabricated with a top 5-um thick inclusion on a 50-um substrate layer, and were indented to 3-um depth with a 15-um spherical-tipped silicon probe. Our computational studies indicated that fΩ vs. D was highly influenced by the inclusion and indenter geometries, but was insensitive ( E1). Thus, our proposed method extracted heterogeneous properties of double-layered PDMS, and provides an approach to study layered biological samples using AFM indentation.

Fast determination of parameters describing manufacturing imperfections and operation wear of nanoindenter tips

Nanoindentation is an effective technique for determining mechanical properties of bulk materials and thin films. Prevailing measurement uncertainties in nanoindentations by Vickers or Berkovich diamond pyramids are commonly caused by manufacturing imperfections of the indenter's side angles and tip sharpness. Moreover, the tip geometry changes over the indenter operation time, due to diamond wear.The paper presents a fast method for estimating diamond indenters' tip nano and micro geometry. This method is based on a combination of nanoindentations on Si(100), used as reference material, with FEM supported calculations of Martens hardness. The hardness calculations are conducted using an equivalent indenter tip geometry with geometrical characteristics, which may vary for a specific set of parameters, describing the real indenter with manufacturing imperfections. These parameters are varied in successive iterations until the calculated hardness converges with that of the reference material.For a quick determination of these parameters, the software package “TIDE” (TΙp Deviations Estimation) has been developed. “TIDE” is based on numerous FEM supported simulation results of nanoindentations, conducted on the reference material, varying the indenter tip geometry. By this software package, a quick prediction of nanoindenters' tip equivalent geometry is facilitated, also for anticipating changes due to wear of the diamond over time.

Toward the development of continuum single-crystal contact mechanics analyses to microindentation experiments

This investigation concerns three-dimensional continuum crystal plasticity finite element simulations of spherical and pyramidal (Berkovich) indentation experiments performed in fcc single and polycrystals. Our fundamental purpose is to examine and extend the correlations between hardness and the uniaxial stress–strain characteristics found from the seminal work by Tabor and from contact mechanics analyses to the case of single-crystal micro/nanoindentations. In doing so, isotropic equivalents to the single-crystal responses are examined in conjunction with the concept of Tabor’s equivalent (representative) indentation stress. The paper also focusses in the self-similar responses of two types of crystals, illustrating influence of the critical resolved shear stress for dislocation gliding on the development of a transitional elasto-plastic regime and a fully-plastic response. A discussion is then given on the potential applicability of the present framework to the analysis of indentation size effects. Clear evidence is finally provided supporting conception that single-crystal sharp and spherical indentations render similar responses at a sufficient penetration of the tip.

Mechanical Characterisation at Nanometric Scale of a New Design of SOFCs

AbstractThe mechanical stability is an important parameter to gain knowledge in the potential applications of a novel design of electrolyte‐supported SOFC, based on yttria‐stabilised zirconia (YSZ) and NiO‐YSZ composites, with cross‐linked channels (∼90 μm of diameter). In this experimental work, the mechanical properties (hardness, H and Young's modulus, E) at different applied loads have been studied using the nanoindentation technique and the equivalent indenter method. On the other hand, the different fracture mechanisms have been determined using atomic force microscopy (AFM), observing the plastic behaviour that takes place during the indentation process. H value for YSZ is higher than that for NiO‐YSZ, while E values for YSZ and NiO‐YSZ are 260 ± 15 and 205 ± 20 GPa, respectively. Only YSZ samples present several radial cracks at the corners nucleated by sharp indentation, thus indicating that H values have been underestimated.

球圧子押込試験による相当押込ひずみを用いたYoung率計測法の軟材料への適用性評価

Spherical indentation testing is used to measure the mechanical properties of materials because of its convenience and low-invasiveness. A method based on the Hertz theory is one of the wellknown methods used to verify the reliability of this testing technique, but it cannot be applied in the case of soft materials with a finite thickness and large deformation. In this paper, Young's modulus measurement using equivalent indentation strain in spherical indentation testing is carried out to evaluate the applicability of this measurement method for various specimens such as soft materials, biological tissues, and so on. In particular, the applicability of this method is validated by using some type of indenters and materials under different experimental conditions. The effects of the types on the relationship between the specimen thickness, indenter diameter, and specimen hardness are studied to formulate a function of Young's modulus and the diameter. The formulated results are evaluated by comparing between experimental and simulation results, and it is shown that the Young's modulus of soft materials with a finite thickness can be measured by the spherical indentation testing technique with the measurement method using equivalent indentation strain.

The role of the contact geometry in wheel–rail impact due to wheel flats: Part II

A classification of wheel flats according to the different stages of their growth is given, along with the characteristic features of the dynamic wheel–rail interaction for each category. Mathematical expressions and frequency spectra of the corresponding wheel mass trajectories are derived. Difference is made between the subcritical and the transcritical speed regime. A criterion is derived for contact loss for worn flats. Simulations show that the dynamic wheel–rail interaction is governed by the track stiffness for low train speeds or long flat lengths; for high speeds and/or short flat lengths the interaction is governed by the inertial properties of the wheel and the rail. For a given flat geometry, nonlinearities in the relationship between the impact magnitude and the train speed occur in the stiffness-dominated speed domain, whereas this relationship is approximately linear in the inertia-governed domain. In the latter domain, the impact magnitude is found to be linearly dependent upon the maximum trajectorial curvature or inversely linearly dependent on the minimum circumferential wheel tread curvature. The above relationships are valid for the subcritical speed regime, in which no contact loss occurs. Different contributions from the literature are compared with respect to the established relationship between impact magnitude and speed. Significant differences are found, due to insufficiently defined parameters and conditions. Conditions are derived for a consistent application of the so-called equivalent rail indentation in experiments with wheel flats, and the indirect strain registration method for measuring dynamic wheel–rail contact forces is reviewed.