Advances in Materials Science and Engineering

Volume 2018 (2018), Article ID 8316384, 9 pages

https://doi.org/10.1155/2018/8316384

## Representative Stress-Strain Curve by Spherical Indentation on Elastic-Plastic Materials

^{1}School of Applied Science, Taiyuan University of Science and Technology, Taiyuan 030024, China^{2}Departamento de Tecnología Química y Ambiental, Tecnología Química y Energética y Tecnología Mecánica, Escuela Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos, c/Tulipán s/n Móstoles, Madrid, Spain^{3}Departamento de Ciencia de Materiales, UPM, E.T.S.I. Caminos, Canales y Puertos, c/ Professor Aranguren s/n, 28040 Madrid, Spain^{4}School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China

Correspondence should be addressed to Chao Chang; moc.361@tsuyt_gnahc_oahc

Received 12 August 2017; Revised 4 December 2017; Accepted 25 December 2017; Published 26 February 2018

Academic Editor: Baozhong Sun

Copyright © 2018 Chao Chang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Tensile stress-strain curve of metallic materials can be determined by the representative stress-strain curve from the spherical indentation. Tabor empirically determined the stress constraint factor (stress CF), *ψ*, and strain constraint factor (strain CF), *β*, but the choice of value for *ψ* and *β* is still under discussion. In this study, a new insight into the relationship between constraint factors of stress and strain is analytically described based on the formation of Tabor’s equation. Experiment tests were performed to evaluate these constraint factors. From the results, representative stress-strain curves using a proposed strain constraint factor can fit better with nominal stress-strain curve than those using Tabor’s constraint factors.

#### 1. Introduction

The instrumented indentation technique consists of applying load to the sample by means of an indenter of known geometry, while the applied load and the penetration depth of the indenter are recorded simultaneously during a loading and unloading cycle. The load-penetration data can be used to determine the mechanical properties of the material without having to image the residual impression left on the material’s surface. Consequently, this technique can be applied at different scales, from macro to nano. The main mechanical properties measured by this technique are Young’s modulus, *E*, and hardness, *H*, by sharp indenter [1, 2]. This methodology could also measure material properties like yield stress, creep property, and fracture [3–6]. However, these properties are not sufficient to characterize a material. The present work focuses on the methodology to determine the stress-strain curve of metallic materials by the depth-sensing indentation technique using a spherical indenter. Sharp indenters like Berkovich or Vicker are characterized by inducing a constant strain on the indented materials. This deformation depends on the indenter angle. Consequently, if the complete stress-strain curve is needed, the sharp indenters are not the best option because they would give only a point of such curve. This is the reason why the interest for the spherical indenter has been recently grown. The experimental curve from spherical indenter provides more information, as this type of indenter has a smooth transition from elastic to elastic-plastic contact [7].

There were many methodologies for the extraction of flow stress with the spherical indentation technique. One category belongs to mathematical and numerical methods (e.g., dimensionless method and artificial neural network (ANN)). After running many finite element models within certain range of material properties, the indentation curves, contact stiffness, or the work of loading-unloading is obtained, which are verified by corresponding experimental indentation and establish a relation between the indentation characteristics and material properties [8–15]. However, these methods consume great computational effort. The unique solution of these methods is still under discussion [16, 17]. Moreover, the contamination in the experimental process like deficient indenter tip, roughness of specimen, or intrinsic noise of instrument machine is not considered in these methods.

Another category is to determine the material parameters from spherical indentation by defining the representative strain and stress. Regarding the experimental measurements, these methods are more convenient and can be used directly. Following the work of Meyer [18], Tabor defined the representative indentation strain, *ε*_{r}, at contact edge of the spherical indenter as , where the values of *ψ* and *β* empirically were determined as 2.8 and 0.2, respectively, based on the quantity of tensile tests on common metals [19]. Most of the researchers agreed with Tabor’s indentation strain but emerge with the controversy on the choice of values of *ψ* and *β* [5, 20–25]. Richmond et al. [21] predicted that the mean pressure was approximately equal to 3 times the yield stress and that the representative strain was approximately equal to 0.32 times the impression to ball diameter ratio. Herbert et al. [23] had used higher stress constraint factor of 3.7 to obtain the representative stress-strain curve for Al 6061-T6 by spherical indentation with a radius of 385 nm. Based on the work of Matthews [22], Tirupataiah and Sundararajan [26] derived an expression for the indentation stress factor, *ψ*, which was related with hardening exponent, *n*. Similarly, Yetna N’Jock et al. [27] determined the tensile property by spherical indentation by simply using the expression of mean pressure to stress ratio as a function of hardening exponent which was derived from the FEM works of Taljat et al. [28]. Using the representative strain under spherical indentation, Ahn and Kwon [29] developed a shear strain definition by differentiating the displacement in the depth direction, in which the ratio of mean pressure to representative stress was equal to 3 in the fully plastic period by conducting instrumented spherical indentation on the steel specimen. By carrying out extensive forward analyses in FEM, Xu and Chen [24] using the indentation strain by Ahn and Kwon [29] found the indentation stress constraint factors, *ψ*, depended almost linearly on hardening exponent, *n*. Additionally, the corresponding indentation strain constraint factor *β* depended on both *n* and the ratio of Young’s modulus to yield stress. Milman et al. [30] assumed that the fully plastic zone beneath indenter was incompressible and proposed a new representative strain which was related with contact radius and contact depth. Fu et al. [31] used the Milman representative strain [30] and proposed a novel iterative process to determine the tensile stress-strain curve. Recently, Kalidindi et al. [32, 33] argued that the definition of indentation strain as *a*/*R* lacks any physical interpretation as a measure of strain and proposed a new definition of the indentation strain consistent with the Hertz theory, which was evaluated from several FEM simulations as well as from the analysis of experimental measurements.

From the literature, there is not uniform agreement among investigations concerning the representative stress and strain equations and also the values for the factors involved in the expressions. The main purpose of the present investigation is to systematically study Tabor’s indentation strain and propose the possibility to develop an analytical procedure to extract the stress-strain curve using experimental data from spherical indentations, which would be comparable to that obtained from a uniaxial test (i.e., tensile test).

#### 2. Theoretical Background

##### 2.1. The Analytical Relationship between Stress and Strain Factors

In 1908, Meyer had found that for many materials, the mean pressure increased with *a*/*R* according to the simple power law [18]in which *p*_{m} is the mean pressure, *a*/*R* is the ratio of indentation radius to the ball radius, and *m* is the Meyer index.

In (1), *k* and *m* are constants. The Meyer equation was verified by further experimental studies [19, 34], and they suggested that the Meyer index, *m*, was related with the hardening exponent, *n*. Following Meyer’s work, Tabor had proposed the concept of representative strain or stress, by which the mean pressure in Meyer’s equation and the *a*/*R* ratio can be converted into the true stress-strain curve. He assumed that the mean pressure, *p*_{m}, at the fully plastic regime was proportional to the representative stress, *σ _{r}*, and the impression radius was proportional to the corresponding representative strain,

*ε*

_{r}. Consequently, the representative stress and strain can be expressed as [19]where

*β*is the indentation strain constraint factor,

*a*is the contact radius,

*R*is the radius of the indenter,

*ψ*is the indentation stress constraint factor, and

*F*is the indentation load.

Tabor determined the parameters *β* and *ψ* from empirically experimental data from spherical indentations under the fully plasticity regime. Generally, the indentation strain constraint factor *β* is considered to be equal to 0.2, and the indentation stress constraint factor *ψ* ranges from 2.8 to 3.2 [35]. It should be emphasized that the representative strain and stress defined by Tabor are an average value of the stress and strain states induced inside the material [36, 37]. The stress and strain constraint factors allow us to establish equivalence between the stress-strain indentation curve and the corresponding one obtained from a uniaxial test. The true stress-strain curve from uniaxial tensile or compression test can be expressed aswhere *E* is the elastic modulus, *σ _{y}* is the yield’ stress,

*K*is the strength coefficient, and

*n*is the strain-hardening exponent.

Substituting (2) into the Hollomon equation:which represents the relationship between load, *F*, and the ratio of contact radius to indenter radius according to the power law at the fully plastic regime. Transforming (5) into the natural logarithm as

According to Hertz equation under the elastic regime, the loading and contact radius can be expressed aswhere , *E*, *ν* and *E*_{i} and *ν*_{i} are Young’s modulus and Poisson’s ratio for the bulk material to be measured and for the indenter, respectively.

Transforming (7) into the natural logarithm

Most metals have an *n* value between 0.10 and 0.50, plotting (6) and (8) in the ln *a* versus ln *F* coordinate, as shown in Figure 1. Inversely extending the two lines, there should be an intersectional point “*e*” for different linear slopes.