Determination of plastic yield stress from spherical indentation slope curve

The indentation slope curve from a spherical indentation on elastic-plastic materials is examined. By comparing it with that of an linear elastic material of the same elastic properties, we found that the start point of plastic yielding for an elastic-plastic material can be easily located from the indentation slope curve. Based on this analysis, a simple but effective method is proposed to measure the plastic yield stress of very small samples from a spherical nano-indentation slope curve.     

Characterization of stress-strain relationships of elastoplastic materials: An improved method with conical and pyramidal indenters

The load-displacement curve in indentation is widely used to extract elastoplastic properties of materials. It is believed that such a measurement is non-unique and a full stress-strain curve cannot be obtained with a sharp indenter or even plural and spherical indenters. By introducing a ratio of the additional residual area to the area of a profile indenter, we proposed a new set of dimensionless functions. Based on these functions and finite element simulations, analytical expressions were derived between indentation data and elastoplastic properties. It is shown that this method can effectively distinguish highly elastic and plastic solids (Cheng and Cheng, 1999) and mystical materials (Chen et al., 2007), which provides a useful guideline for properly using the indentation technique to measure elastoplastic properties of materials with conical and pyramidal indenters.     

基于纳米压痕及区间优化的Al 2024-T3铝合金本构参数识别研究

纳米压痕实验由于试样准备简便、使用范围广等优势,在材料力学测试领域得到了广泛关注。本研究建立了一种考虑纳米压痕实验不确定性的Al 2024-T3铝合金材料塑性参数识别方法。首先,针对Al 2024-T3铝合金开展了纳米压痕实验,获取了载荷-位移曲线,由于材料存在不均匀性,所以实验曲线存在不确定性。基于超参数优化的人工神经网络,建立了材料性能参数与压痕响应加载曲线的关联。基于区间优化理论,引入压痕实验曲线的不确定性,以压痕实验曲线加载曲率为不确定性量,提出了基于双层嵌套遗传算法的材料参数反分析识别区间优化模型,并进行了参数识别反问题的求解。该方法的优势在于能够考虑到实验测量的不确定性,识别结果更可信。所建立方法的有效性在Al 2024-T3铝合金塑性参数识别中得到了验证,识别误差分别为：屈服应力-0.87%,硬化指数2.76%。该识别方法可用于小尺寸试样局部力学性能的检测领域。     

Measuring mechanical properties of micro- and nano-fibers embedded in an elastic substrate: Theoretical framework and experiment

We propose to measure the elastoplastic properties of micro- and nano-fibers by a normal indentation technique in which the vertically aligned fibers are embedded in an elastic matrix. Measurements are taken at two different indentation depths, which represent different levels of the matrix effects and lead to the establishment of two independent equations that correlate the fiber/matrix properties with the indentation responses. Effective reverse analysis algorithms are proposed, and by following which the desired fiber properties can be determined from a sharp indentation test. Comprehensive analysis is also carried out to verify the effectiveness and error sensitivity of the presented method. The extracted material properties agree well with those measured from the parallel experiments on human hair and glass fibers.     

Determination of deformation and failure properties of ductile materials by means of the small punch test and neural networks

This paper describes an approach to identify plastic deformation and failure properties of ductile materials. The experimental method of the small punch test is used to determine the material response under loading. The resulting load displacement curve is transferred to a neural network, which was trained using load displacement curves generated by finite element simulations of the small punch test and the corresponding material parameters. The simulated material behavior of the specimen is based on the ductile elastoplastic damage theory of Gurson, Tvergaard and Needleman. During a training process the neural network generates an approximated function for the inverse problem relating the material parameters to the shape of the load displacement curve of the small punch test. This technique was tested for three different materials (ductile steels). The identified parameters are verified by testing and simulating notched tensile specimens.     

Evaluation the effect of aspect ratio for Young’s modulus of nanobelt using finite element method

The traditional nanoindentation method provides experimental data for the calibrating mechanical parameters of nanobelt through semi-empirical formulae. In this paper, a technique to identify Young’s modulus of nanobelts with different aspect ratios is introduced combining finite element method (FEM) and nanoindentation test. For the nanobelt on the substrate, the power function relationship is used to describe the loading curve of the nanobelt indentation behavior. The loading curve exponent of the power function which is the fitting parameter can reflect the influence of aspect ratio of nanobelt on Young’s modulus of nanobelts as well as the maximum indentation load. In the forward analysis, considering the substrate effect and the size effect, the numerical loading responses are simulated at the appropriate penetration depth, and then the dimensionless equations between the parameters characterizing the indentation loading curve and the properties of nanobelt/substrate system can be established via extensive FEM simulation. In the reverse analysis, the nanoindentation tests were performed on ZnO and ZnS nanobelts, and the experimental indentation loading curves can be fitted as power function. The maximum indentation loads and the loading curve exponents are extracted from two experimental loading curves, and then they are substituted into the dimensionless equations to solve the Young’s moduli of ZnO and ZnS nanobelts. The results show the Young’s moduli solved are close to previous values, indicating that the Young’s moduli are reasonable. This developed method is effective to identify the Young’s modulus of nanobelt and it can be applied to identify the Young’s modulus of other nanobelts in practice.     

Elastoplastic contact/impact of rigidly supported composites

A general elastoplastic contact/impact model of a spherical object and a supported transversely isotropic composite layer or a half-space is presented. The main feature of the model is a contact law that is developed based on elastic–plastic and fully plastic indentation theory. For an impact event, the model parameters can easily be obtained analytically, computationally using finite elements (FE), and from experiments. The model results are compared to those from a nonlinear FE model developed in ABAQUS, and with limited experimental data showing excellent agreement. The model has also been tested for low-velocity impact, which compared very well with the FE results using ABAQUS/Explicit. The model is simple, yet it successfully captures a fairly complicated contact behavior.     

Characterization of viscoelastic properties of molybdenum disulphide filled polyamide by indentation

In this paper, the creep behavior of molybdenum disulphide (MoS₂) filled polyamide 66 composite was investigated through sharp indentation at room temperature. Two types of indentation creep test, the 3-step indentation test, and the 5-step indentation test were considered in order to explore whether the measured creep response is mainly viscoelastic or includes a significant contribution from the plastic deformation developed during the loading phase. The experimental indentation creep data were analyzed within an analytical framework based on the hereditary integral operator for the ramp creep and a viscoelastic–plastic (VEP) model in order to determine the indentation creep compliance function including the short- and long-time modulus. The equivalent shear modulus calculated from the creep compliance function was compared to the indentation plane strain modulus derived from the initial slope of the unloading curve in order to investigate the validity of the Oliver and Pharr method.     

Simulation of instrumented indentation and material characterization

Extensive large deformation finite element analyses were carried out to investigate the response of elasto-plastic materials obeying power law strain-hardening during the loading and unloading process of instrumented sharp indentation. The functional forms of the relationships between the characteristics of the load–indentation curve and the material properties of elasto-plastic materials were examined. The governing equations relating the curvature of the loading curve to the elasto-plastic material properties were formulated based on cavity expansion analogy. Two simple and robust algorithms were proposed for forward and reverse analyses. The numerical results obtained are in good agreement with published values. The uniqueness of the results from the reverse analysis algorithm was also addressed. By considering the load–displacement curve of Al 6061-T651, it was demonstrated that a one-to-one relationship between the elasto-plastic material properties and the load–displacement curve does not always exist.     

Scaling functions in conical indentation of elastic-plastic solids

Summary The finite element method was used to simulate the conical indentation of elastic-plastic solids with work hardening. The ratio of the initial yield strength to the Young’s modulus Y/E ranged from 0 to 0.02. Based on the calculation results, two sets of scaling functions for non-dimensional hardness H/K and indenter penetration h are presented in the paper, which have closed simple mathematical form and can be used easily for engineering application. Using the present scaling functions, indentation hardness and indentation loading curves can be easily obtained for a given set of material properties. Meanwhile one can use these scaling functions to obtain material parameters by an instrumented indentation load-displacement curve for loading and unloading if Young’s modulus E and Poisson’s ratio ν are known.     

Finer-Scale Extraction of Viscoelastic Properties from Nanoindentation Characterised by Viscoelastic–Plastic Response

Abstract:  Motivated by recent progress in viscoelastic indentation analysis, the identification of viscoelastic properties from materials exhibiting elastic, viscous and plastic material behaviour by means of nanoindentation is dealt with in this paper. Based on existing solutions for pure viscoelastic material behaviour, two methods allowing us to consider the effect of plastic deformation are presented: (i) the so-called double-indentation technique, with the second indentation characterised by pure viscoelastic material response and (ii) the use of spherical indenter geometries instead of commonly used pyramidal indenters avoiding plastic deformation at all. Both methods are applied to three different polymers, giving access to the model parameters of the fractional dash-pot which is used to describe the viscoelastic behaviour. The results obtained are compared with results from standard (single) indentation tests using a Berkovich indenter. Moreover, the influence of the maximum load, determining the amount of plastic material response, on the identified model parameters is investigated. Finally, the creep-compliance functions identified by nanoindentation are compared with the respective macroscopic creep-compliance functions obtained from bending-beam rheometer tests.     

Depth sensing indentation of linear viscoelastic-plastic solids: A simple method to determine creep compliance

Viscoelastic solids may deform plastically under indentation. This leads to an overestimation of the creep compliance when the analytical solution for indentation of linear viscoelastic materials [Lu H, Wang B, Ma J, Huang G, Viswanathan H. Measurement of creep compliance of solid polymers by nanoindentation. Mech Time-Depend Mater 2003;7(3-4):189-207] is used for its determination. Using finite element analysis, in this work it is shown that the plastic and viscoelastic deformation processes occur simultaneously, even during holding of a constant indentation load. A simple procedure to separate the viscoelastic response from the plastic response is proposed, which involves spherical indentations at different loads. To illustrate the proposed method, indentation tests are conducted on a polymer, polymethylmethacrylate (PMMA), and its viscoelastic properties are determined.     

Modeling the nanoindentation of elastoplastic materials with nonlinear adaptive springs (NASs)

In this paper, the /spl beta/-material concept helps to elaborate and explore a new model for the indentation cycle of elastoplastic materials. The proposed approach takes into account the nonlinear behavior of homogeneous and isotropic materials. It uses the idea of a nonlinear adaptive spring (NAS) with changing properties according to the depth of penetration to accurately reproduce the material behavior in loading and unloading stages. The properties of the adopted NAS are included in its own stiffness function /spl kappa/ appearing in the form of an infinite sum of which the convergence and some properties are discussed in detail. This new model, which allows the indentation cycle to be reproduced whatever the penetration depth, permits at the same time a direct calculation of the involved energy terms. It also provides the possibility to perform separate analysis of the plastic energy, which allows distinguishing between different types of the material behavior and a better understanding of its nature. A validation is accomplished by applying the method to three different materials.     

Finite element analysis of frictionless contact between a sinusoidal asperity and a rigid plane: Elastic and initially plastic deformations

A series of finite element simulations of frictionless contact deformations between a sinusoidal asperity and a rigid flat are presented. Explicit expressions of critical variables at plastic inception including interference, contact radius, depth of first yielding, and pressures are obtained from curve fitting of simulation results as a function of material and geometrical parameters. It is found Hertz solution is not applicable to the critical contact variables at plastic inception for sinusoidal contact, although contact responses of initially plastic deformation follow the same trend as that of purely elastic deformation. The contact pressure at incipient plasticity, which is defined as yield strength, is dependent on Poisson’s ratio, yield stress, and geometrical parameters, but independent of elastic modulus. It is not yield stress, but yield strength that correlates with indentation hardness. The results yield the insight into the specification of material properties to realize elastic contact. A larger ratio of yield stress to elastic modulus is beneficial to sustain a larger load before plastic deformation.     

Evaluation of the tensile properties of a material through spherical indentation: definition of an average representative strain and a confidence domain

In the present article, a new method for the determination of the hardening law using the load displacement curve, F–h, of a spherical indentation test is developed. This method is based on the study of the error between an experimental indentation curve and a number of finite elements simulation curves. For the smaller values of these errors, the error distribution shape is a valley, which is defined with an analytic equation. Except for the fact that the identified hardening law is a Hollomon type, no assumption was made for the proposed identification method. A new representative strain of the spherical indentation, called “average representative strain,” ε _aR was defined in the proposed article. In the bottom of the valley, all the stress–strain curves that intersect at a point of abscissa ε _aR lead to very similar indentation curves. Thus, the average representative strain indicates the part of the hardening law that is the better identified from spherical indentation test. The results show that a unique material parameter set (yield stress σ _y, strain hardening exponent n) is identified when using a single spherical indentation curve. However, for the experimental cases, the experimental imprecision and the material heterogeneity lead to different indentation curves, which makes the uniqueness of solution impossible. Therefore, the identified solution is not a single curve but a domain that is called “solution domain” in the yield stress–work hardening exponent diagram, and “confidence domain” in the stress–strain diagram. The confidence domain gives clear answers to the question of uniqueness of the solution and on the sensitivity of the indentation test to the identified hardening laws parameters.     

Stress concentration in plates under tension with local thinnings in elastoplastic deformations and creep

The results are given of a calculation-experimental investigation of the stress concentration in the zone of two-sided and one-sided craters in the form of spherical segments. The elastoplastic problem was solved by the method of variable parameters of elasticity with use of the finite element method. The experiments were made on plates of AL2 alloy with the use of short-base strain gauges. The relationship was obtained of the stress concentration factors to the parameters of thinning and loading at characteristic points. The picture is given of the successive transition of the elements into the plastic state. The simulation of creep based on the method of isochronous curves made it possible to obtain curves of the change in stress concentration factors with time for the characteristic points applicable to the steels used in turbine production from the data of experimental investigation of aluminum elastoplastic models.Khar'kov and Perm'. Translated from Problemy Prochnosti, No. 12, pp. 19–23, December, 1989.     

Material characterization by instrumented spherical indentation

It was illustrated by the author in the previous work that combinations between material properties and indentation parameters can be used as mixed parameters in dimensionless functions to capture the sharp indentation response of materials. These issues are further extended for spherical indentation in the present study. Instrumented spherical indentation was performed by a parametric finite element analysis for a wide range of materials with maximum indentation depth-indenter radius ratios rising from 0.01 to 0.3 to investigate several fundamental features within the frame work of limit analysis. Frictional effects are taken into account. Regarding dimensional analyses and using a Taylor series expansion, a new set of dimensionless functions is constructed for spherical indentation parameters and hardness associated to a 70.3° conical indenter. Based on formulated functions, a reverse analysis procedure is suggested to extract material properties and hardness from spherical indentation force-depth curves with respect to two different indentation depth-indenter radius ratios. Effects of indenter compliance on indentation parameters and reverse results are considered. The accuracy of the proposed method is studied and discussed by carrying out reverse and sensitivity analyses for 22 representative materials with rigid and deformable indenters.     

材料局部性能的球形压痕评价技术研究进展

简介了近几年采用球形压痕法在测试材料性能方面的最新发展和应用.首先概括了球形压痕理论的研究与发展,重点介绍了球形压痕法的应用研究.采用球形压痕法并结合有限元模拟可以获得材料的弹性模量、屈服强度、拉伸强度、应变硬化指数、断裂韧度和应变速率敏感度等性能.通过球形压痕法与通常测试方法比较的结果证实,球形压痕法可以有效评价材料的性能,随着研究和应用工作的不断深入,球形压痕技术有可能在不远的将来成为评价材料局部性能最方便、最简单且相当准确的方法.     

Determination of Stress-Strain State and Strength of Structural Elements on the Basis of Strain-Hardening Analysis

We outline the fundamentals of the method whereby the stress-strain state and strength of structural elements loaded along linear paths and small-curvature paths are determined by allowing for strain hardening of material of the structure or plastic indicators attached to it. The method is based on a model of hardening which assumes that during the deformation beyond an elastic range the loading surface separating elastic and elastoplastic deformation ranges changes its shape and shifts in the direction of the vector which connects its center and an image point on the loading path. It is assumed that the material in its initial state is isotropic and the hypotheses for the unified stress-strain curve and for the proportionality of stress and strain deviators are met.__________Translated from Problemy Prochnosti, No. 2, pp. 28 – 48, March – April, 2005.     

Study of the concept of representative strain and constraint factor introduced by Vickers indentation

The application of the concept of the representative strain is often used in the stress–strain curve determination from indentation test because it can significantly simplify the analysis of the indentation response. A new methodology for determining the representative strain for Vickers indentation is presented in this article. Following a procedure based on finite element simulations of indentation of elastoplastic materials, two representative strains are defined: the representative strain characteristic of the mean pressure and the representative strain characteristic of the Martens hardness or the indentation loading curvature. The results obtained from this methodology show that there is no universal value of representative strain independent of the mechanical parameters of materials indented by Vickers indentation. It is also shown that the representative strain, obtained by Vickers indentation is much lower when it is obtained from the relationship between the applied force and the penetration depth, F-h, rather than from the relationship between the applied force and the contact radius, F-a. The values of the calculated representative strains show that simultaneous measurement of relationships F-a and F-h make it possible to characterize the hardening law with two unknown parameters by Vickers indentation.