Abstract
The so-called indentation size effect (ISE) observed mainly in nanoindentation measurements with prismatic tips, is theoretically modeled in this article with the use of gradient theory. It is shown that the ISE, i.e. the dependence of the calculated hardness value on the indentation depth, is rather an artifact of the geometry of the tip used, than a phenomenon related to the material tested. The model predictions are compared with nanoindentation measurements of Al specimens.
1 Introduction
As materials science advances towards smaller scales of observation, more advanced measuring techniques have been established and utilized to quantify material characteristics and properties at these scales, since measuring mechanical properties at smaller scales cannot be done using conventional uniaxial tensile testing. One such technique for measuring material “hardness”, i.e. the material resistance to plastic deformation, has been developed by indenting a material with a Vickers or a Berkovich tip. Indentation hardness tests comprise the majority of experiments used to determine material hardness, and can be divided into two classes: microindentation (with loads typically up to 2 N/0.45 lbf) and macroindentation tests.
Hardness is not considered to be a fundamental material property. Instead, it represents an arbitrary quantity used to provide a relative idea of material properties [1]. As such, it can only offer a comparative idea of the material’s resistance to plastic deformation, since different hardness techniques include different scales. In addition, the so-called indentation size effect (ISE) (e.g. [2]), i.e. the dependence of the indentation hardness value on the indentation depth, is another factor for not considering hardness as a material property.
Although the ISE has been studied by many researchers (e.g. [2], [3], [4], [5], [6], [7]), herein we consider the ISE to be nothing more than an artifact of the indentation experiment when using prismatic tips (Vickers, Berkovich), i.e. that the dependence of the calculated values of indentation hardness on the depth of indentation comes from the geometry of the tip and has nothing to do with the indented material. In addition, it is our suggestion that the material hardness should be measured after the yield strength has been reached, since only after this point the whole resistance to plastic deformation can be measured. Before the yield point the indentation hardness measurements produce overestimated values due to the elastic recovery.
The main motivation for this work is that the indentation hardness calculated from indentation tests performed with prismatic tips should not be depending on the way that the indentation experiment is performed, i.e. the maximum depth at which the hardness is calculated. Herein the indentation test is considered to be a combination of an “effective compression” experiment along with an influence coming from the tip geometry. Thus, the hardness should be calculated after this geometrical influence is subtracted. To this end a gradient model similar to the ones introduced by Aifantis and co-workers on gradient elasticity [8] and gradient plasticity [9], [10] is used, in order to approximate the effect of geometry and subtract it, thus being left only with the effective compression.
The present article is organized as follows: in Section 2 a description of the gradient model is given, while in Section 3 the theoretical model is applied to Al specimens subjected to indentation in order to extract one value for the hardness, independent of the indentation depth, and compare it with the hardness value calculated by other tests.
2 Theoretical modeling
2.1 Formulation of the model
Since the hardness calculation should be conducted after the yield point, we start from a variation of a 2D gradient model for plasticity [9], [10] where the second stress (instead of strain) gradient is introduced into the flow stress equation for the z component of the stress as
with ε being the plastic strain and c the so-called gradient coefficient, which has dimensions of length square, thus introducing an internal length into the formulation.
Since in this work we are interesting in applying Eq. (1) for metals, we assume Hollomon’s relation [11] for the post-yield behavior, i.e. κ(ε)=kεn, with k the hardening modulus and n the hardening exponent.
Next, we calculate the strain in the z direction for the material in contact with the faces of the indenter tip, assuming that the deformation of the material near the tip follows the tip’s geometry, which is known for a specific tip. In Figure 1 the geometry of a Vickers tip is shown.
From the Vicker’s tip geometry it is easy to calculate the strain in the z direction by dividing the indentation depth at each point with coordinates (x, y) under the tip by the height W of the tip as
For Vickers indentation the second gradient of κ(ε), evaluated at the center point of the tip for evaluating its maximum value, i.e. for (x→0, y→0), is given by
Combining Eqs. (1) – (3) we have
It is assumed herein that since indentation hardness is considered to actually be the value of the stress applied at the maximum indentation depth, as it is calculated by dividing the load at maximum indentation depth over the projected area of indentation, Eq. (2) should also hold for hardness. In this case the stress σz is replaced by the hardness H calculated at different indentation depths, while the “homogeneous” post-yield stress kεn is replaced by the hardness H0 of the material that should not depend on how the experiment is made, i.e. the maximum indentation depth, and Eq. (4) takes the form
If instead of using the Hollomon’s model for the post-yield behavior one uses the Voce model [12] of the form
with Rsat denoting the saturation stress, while a, b are the coefficients of the initial hardening state and the rate at which the size of the yield surface changes as plastic straining develops, respectively, then following the same approach Eq. (5) would take the form
In the following sections both models given by Eq. (5) and Eq. (7) will be used for modeling indentation size effect data in various materials.
It is important to clarify at this point that if the height of the tip W (usually equal to 2 µm) as well as the parameters of the Hollomon/Voce models are known, and the fitting parameter c′ is calculated by fitting, then it is very simple to determine the gradient coefficient c.
2.2 Conversion between hardness measurements with different tip geometries
Since both tip shapes (Berkovich and Vickers) produce approximately 8% strain in the material during indentation, the hardness results obtained from tests using these two different tip geometries can be directly compared. However, care must be taken to correctly perform the conversion of the results for the Vickers hardness tests using micro-indentation. There is a definition change between the hardness measured using micro-indentation to the hardness used in nano-indentation. While both hardness values are calculated as the peak force divided by the area of contact, the definition of the contact area differs between the test techniques. For micro- indentation, the contact area is the area of the tip faces in contact with the sample, while for nano-indentation the contact area is defined as the projected area of contact between the sample and the tip. To make a comparison of the results between the two indentation techniques, the results must be converted to common definitions. Below, the Vickers Hardness (VH) measurement is converted to define the hardness as the mean contact pressure. Using the hardness definition and substituting the geometry for the Vickers tip, with the contact area as defined for micro-indentation gives
where P is the force in units of kgf/mm2 and d is the length of the diagonal in mm for the residual impression from the indentation test. As a result, the Vickers hardness is given by
where H has units of kgf/mm2. Since nanoindentation results are typically provided in MPa or GPa, units conversion (1 kgf/mm2=0.0098 GPa) gives
with H and VH having units of GPa.
3 Application to experimental data
In this Section the models proposed in Eqs. (5), (7) are used for interpreting nanoindentation measurements in Al specimens.
The nanoindentation hardness H of a single crystal Al specimen was measured by using a Nano Indenter G200 (Agilent Technologies) with a Berkovich tip. The hardness at infinite depth H0 (i.e. bulk-equivalent hardness) is estimated as H0=0.3 GPa and the strain hardening exponent is n=0.20 The experimental data for the Vickers hardness were modeled with Eq. (5) and Eq. (7), utilizing also the conversion listed in Eq. (10).
It can be seen from Figure 2 that our theoretical predictions can model sufficiently well the experimental measurements. The fitting parameters c′ for the case of Eq. (5) was c′=8.8 10−5 Pa/mn−2, while for the case of Eq. (7) the fitting parameters were c′=0.698 GPa and k=8.14·106m−1.
4 Discussion and future work
From the results presented in the previous section, it can be concluded that both models given by Eqs. (5) and (7) can be used to model the so-called indentation size effect (ISE), i.e. the dependence of the calculated hardness value on the indentation depth. This was accomplished by utilizing a stress gradient model with a constant hardness value and the corresponding effect of the tip geometry. The theoretical predictions showed that a voce-type relation assumed for the homogenous part of the stress under the indenter provides somewhat better results compared with its counterpart model assuming a Hollomon relation for the homogeneous stress. Although the model has been applied herein to Al specimen nanoindentation data, preliminary results indicated that it could be applied in other metallic specimen, a task that is left for a future publication.
Acknowledgment
The authors acknowledge very useful discussions with Professor E.C. Aifantis. The support of the Ministry of Education of Russia under MegaGrant No. 14.Z50.31.0039, is also gratefully acknowledged.
References
[1] Meyer MA, Chawla K. Mechanical Behavior of Materials, Prentice-Hall, Upper Saddle River, 1999.Search in Google Scholar
[2] Milman YV, Golubenko АА, Dub SN. Acta Mater. 2011, 59, 7480–7487.10.1016/j.actamat.2011.08.027Search in Google Scholar
[3] Mott BW. Micro-indentation Hardness Testing, Butterworths, London, 1956.Search in Google Scholar
[4] Stelmashenko NA, Walls MG, Brown LM, Milman YV. Acta Metall. Mater. 1993, 41, 2855–2865.10.1016/0956-7151(93)90100-7Search in Google Scholar
[5] De Guzman MS, Neubauer G, Flinn P, Nix WD. Mater. Res. Soc. Symp. Proc. 1993, 308, 613–618.10.1557/PROC-308-613Search in Google Scholar
[6] Ma Q, Clark DR. J. Mater. Res. 1995, 10, 853–863.10.1557/JMR.1995.0853Search in Google Scholar
[7] Swadener JG, George EP, Pharr GM. J. Mech. Phys. Solids 2002, 50, 681–694.10.1016/S0022-5096(01)00103-XSearch in Google Scholar
[8] Aifantis EC. Int. J. Engng. Sci. 1992, 30, 1279–1299.10.1016/0020-7225(92)90141-3Search in Google Scholar
[9] Aifantis EC. Trans. ASME J. Engng. Mat. Techn. 1984, 106, 326–330.10.1115/1.3225725Search in Google Scholar
[10] Aifantis EC. Int. J. Plasticity 1987, 3, 211–247.10.1016/0749-6419(87)90021-0Search in Google Scholar
[11] Hollomon JH. Trans. AIME 1945, 162, 268–290.Search in Google Scholar
[12] Voce E. J. Inst. Metals 1948, 74, 537–562.Search in Google Scholar
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