CIRP Encyclopedia of Production Engineering

Living Edition
| Editors: The International Academy for Production Engineering, Sami Chatti, Tullio Tolio


  • Nikolaos MichailidisEmail author
  • Konstantinos-Dionysios Bouzakis
  • Ludger Koenders
  • Konrad Herrmann
Living reference work entry


Indentation Depth Indentation Hardness Indentation Modulus Test Force Indentation Force 
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Nanoindentation is a method for testing the hardness and related mechanical properties of materials, facilitated by high-precision instrumentation in the nanometer scale, as well as analytical and computational algorithms for result evaluation.

Theory and Application


The origin of the indentation method goes back to Martens at the end of the nineteenth century (Martens 1898). Later, the force–indentation depth curve and a method for deriving hardness and elastic modulus were described (Ternovskij et al. 1973). In the early 1980s, the development of a high-resolution nanoindentation followed (Newey et al. 1982). In the 1990s, refinements to the instrumentation and methods for extracting mechanical properties from the test contributed to the establishing of nanoindentation as an important tool in materials research (Oliver and Pharr 1992 and Field and Swain 1993). Since then, further developments of the technique and its use in a range of materials have been performed. This is highlighted by several special focus series, the most notable being the Journal of Materials Research series in the years 1999, 2004, and 2009. The CIRP Annals in 2010 published a keynote paper on nanoindentation (Lucca et al. 2010). The standardization of the instrumented indentation test started in the 1990s in Germany (DIN 50359-1 to -3), establishing the basis for the development of the international standard ISO 14577-1 to -3, published in 2002. To address the peculiarities of indentation of thin films and coatings, the ISO 14577-4 was developed. In 2008, the ISO/TR 29381 was published, allowing for the evaluation of tensile properties of metallic materials by instrumented indentation.


Method and Instrumentation

During an instrumented indentation test, the applied test force and the indentation depth of the indenter are continuously measured. ISO 14577-1 defines three application ranges:
  • Nano-range: h max ≤ 200 nm

  • Micro-range h max > 200 nm and F max < 2 N

  • Macro-range 2 N ≤ F max ≤ 30 kN

Typical indenter materials include diamond, tungsten carbide, and sapphire. Indenter geometries are pyramidal with square base (Vickers), pyramidal with triangular base (Berkovich and cube corner), and spherical.

Either the force or the displacement of the indenter is controlled by the indentation instrument. The testing cycle is composed of a linear loading to the maximum force, a hold at the maximum force to allow time-dependent plasticity or creep effects to diminish, and an unloading to 10 % of the maximum force, a hold at this force to measure the thermal drift of the instrument followed by a final unloading. Figure 1 shows an example of a test cycle. Figure 2 shows the results of an indentation test where the test force is plotted as a function of the indentation depth based on the test cycle of Fig. 1. Curve (a) represents the application of the test force, (b) the hold at maximum test force, (c) the first part of the unloading, (d) the hold at 10 % of the maximum test force, and (e) the final unloading.
Fig. 1

Typical test force versus test time cycle for indentation

Fig. 2

Test force versus indentation depth plot

Nanoindentation instruments mainly differ by the principle of the force measurement system (nN resolution), where electromagnetic, capacitive, and inductive systems are applied, and of the displacement measurement system (sub-nm resolution) which can be inductive or capacitive. Furthermore, the use of a small ac signal that is superimposed on the force signal to measure the contact stiffness throughout the complete cycle for calculating the indentation modulus is also a trend. Systems have been developed which enable the rapid application of the test force, thus resulting in a nano-impact test. Indentation has also been combined with transmission electron microscopy (TEM). Nano-scratch tests were rendered possible by including another axis of motion. Most of the known nanoindentation instrument manufacturers are presented in Bhushan and Palacio (2012). A further development was testing at high temperatures, offering more realistic predictions about the materials performance close to operation conditions. A characteristic example of Fig. 3a shows nanoindentation measurements on a TiAlN film, at temperatures varying up to 400 °C. The h max and the h p values are fluctuating over the temperature (see Fig. 3b) as discussed in Bouzakis et al. (2010).
Fig. 3

(a) Nanoindentation diagrams of a TiAlN coating and (b) h max and h p at various temperatures

Indenter Tip Calibration Procedures

A prerequisite for high accuracy in nanoindentation result evaluation is the precise knowledge of the indenter and its tip geometry, especially when results attained using different indenters or even devices need to be correlated (Oliver and Pharr 1992; Alcala et al. 1998; Herrmann et al. 2010). In nanometer scale, indenter deviations from the ideal geometry due to manufacturing imperfections or operation wear significantly affect the result accuracy (ISO 14577 2002; Bouzakis et al. 2001, 2002a, b). Therefore, analytical–empirical and FEM-based methods have been developed to overcome this problem.

Analytical–Empirical Methods
Direct and indirect measurement methods are applied for determining the indenter area function. In the direct method, usually atomic force microscope (AFM) is employed for attaining a high-resolution shape characterization of the indenter (see Fig. 4a). Characteristic indenter tip form deviations, as well as an example of AFM measurement of a Berkovich indenter tip, conducted according to ISO 14577-2 standard, are illustrated in Fig. 4b (Bouzakis et al. 2013). Results at indentation depths less than roughly 25 nm can be significantly affected by the tip geometry, due to the manufacturing nano-imperfections of the indenter tip in the area A (Bouzakis et al. 2001; ISO 14577 2002; Bouzakis and Michailidis2007). In the indirect methods, nanoindentations are performed on reference materials with determined properties. A typical method counts on a material of known Martens hardness as a function of the test force, while a further method involves testing with a material of known Young’s modulus.
Fig. 4

Characteristic indenter tip form deviations and AFM measurement of a Berkovich nanoindenter

FEM-Supported Methods
The evolution of FEM-based algorithms in the evaluation of nanoindentation results offers advanced capabilities in determining the exact contact between the indenter and the test piece, thus allowing the accurate calculation of the material hardness and stress–strain curve. A precondition for that is an effective evaluation approach of the exact indenter tip geometry. Figure 5a presents an FEM model of nanoindentation process (Bouzakis et al. 2001). For describing analytically the indenter tip nano-deviations from the ideal sharp tip geometry, the pyramid is replaced by an equivalent cone possessing the same projected area versus the indentation depth h i (see Fig. 5b) (Bouzakis et al. 2002a). The indenter tip geometry of the equivalent cone is described in Fig. 5c. Due to the high indenter tip radius in this small indenter tip area, this curve can be substituted by a line described via the parameters height t h and width b (see Fig. 5c). The approximation of the indenter tip as a sphere with radius up to 200 nm may lead to substantial errors in the evaluation of nanoindentation results (Bouzakis and Michailidis 2007).
Fig. 5

(a) FEM model of nanoindentation. (b) Replacement of a Berkovich indenter pyramid by an equivalent cone. (c) Indenter tip geometry of the equivalent cone

Although diamond pyramids possess very high elasticity modulus and hardness, their elastic deformation during penetration can be considerably high, depending on the properties of the test material, causing significant changes in the contact surface (Bouzakis and Michailidis 2006).

A fast method for estimating indenter tip nano- and micro-geometry, based on a combination of nanoindentations on Si(100) with FEM-supported calculations of Martens hardness, is reported in Bouzakis et al. (2010), offering also the option to determine indenter tip geometry changes (see Fig. 6) over its operation time, due to wear. The worn tip is blunted. In this way, when this indenter is applied, the maximum indentation depth at the same indentation force is smaller compared to that one of the new indenter. This fact could be misleading, interpreted as if the material hardness was higher. At the right part of Fig. 6, the new indenter geometry (dashed line) and the worn one (solid line) are displayed, with the indenter wear volume amounting to 160 ∙ 103 nm3.
Fig. 6

Application example for determining the tip geometry of an unworn and worn indenter

Effect of Surface Roughness on the Nanoindentation Accuracy

A further crucial parameter affecting the accuracy of the nanoindentation results is the surface roughness, through the different contact area of the indenter with the test piece when indenting on roughness peak or valley. When indenting on relatively rough specimens, as experienced in most surfaces of technical materials, an indentation depth scatter may occur as a consequence of the surface topography.

The distribution of the number of measurements versus the maximum indentation depth, for three specimens with different roughness specifications, is demonstrated in Fig. 7a. The registered maximum indentation depths on the polished specimen lie closer to the maximum occurring penetration depth mean value, leading to a smaller scattering of the measurements in comparison to the honed and the ground specimen. However, after a certain number of measurements depending on the surface roughness, the moving average of the maximum indentation depths h max in all three investigated cases are converging (see Fig. 7b). The scatter of the moving average of h max per indentation has been stabilized after the first 15 measurements in the case of the polished specimen, while the honed and the ground specimens require more than 20 and 25 measurements, respectively, to obtain a stable moving average of h max.
Fig. 7

Statistical analysis of the nanoindentation measurements for specimens of different roughness

Determination of Material Properties (Stress–Strain Law, Hardness, etc.)

Analytical–Empirical Methods
Analytical–empirical methods are basically applied for the determination of bulk materials’ hardness and Young’s modulus based on the nanoindentation force–depth diagram. The Martens hardness HM and indentation hardness H IT are the most commonly used parameters. Martens hardness is defined as the test force F divided by the surface area A s(h) of the indenter penetrating beyond the zero point of the contact and is expressed in N/mm2 by the following equation:
$$ HM=\frac{F}{A_{\mathrm{s}}(h)} $$

An improved method for the determination of the Martens hardness is offered for homogeneous materials by calculating the slope of the increasing force/indentation depth curve, to avoid the determination of the zero point.

Indentation hardness is expressed as the maximum applied force F max, divided by the projected (cross-sectional) contact area A p of the indenter with the test piece:
$$ {H}_{\mathrm{IT}}=\frac{F_{\max }}{A_{\mathrm{p}}} $$
Young’s modulus of the test material can be approached by the indentation modulus E IT, which can be calculated from the slope of the tangent:
$$ {E}_{\mathrm{IT}}=\frac{1-{\left({v}_{\mathrm{s}}\right)}^2}{\frac{1}{E_{\mathrm{r}}}-\frac{1-{\left({v}_{\mathrm{i}}\right)}^2}{E_{\mathrm{i}}}} $$
$$ {E}_{\mathrm{r}}=\frac{\sqrt{\uppi}}{2C\sqrt{A_{\mathrm{p}}}} $$
where ν s is Poisson’s ratio of the test piece, ν i is Poisson’s ratio of the indenter (for diamond 0.07), E r is the reduced modulus of the indentation contact, E i is the modulus of the indenter (for diamond 1.14 ∙ 106 N/mm2), C is the compliance of the contact (dh/dF of the test force removal curve, evaluated at maximum test force), and A p is the projected contact area, the value of the indenter area function at the contact depth.

The analytical–empirical methods are fast and simple, offering adequate approximation of material mechanical properties. However, they have limited accuracy: (i) in small indentation depths, where the real indenter area function is required; (ii) in materials with high Young’s modulus, since the indenter is considered rigid; and (iii) when materials have graded properties, like thin films. Moreover, significant differences between Young’s modulus and E IT may occur, when either pileup or sink-in is present.

FEM-Supported Methods
The geometry of the contact area between the diamond indenter and the specimen during the indentation decisively affects the resulting material hardness. The FEM model simulating the nanoindentation procedure and the actual indenter tip geometry, introduced in Bouzakis et al. (2001, 2002a, 2002b), can be applied to calculate the contact geometry during loading and the shape of the surface impression after unloading. In this way, all specimen’s hardness values, i.e., Martens HM, indentation hardness H IT, and Vickers, can be determined (Qian et al. 2005). The course of the Vickers hardness and indentation hardness versus the indentation force and depth, for a tested hardened steel, is presented in Fig. 8a. Both hardness magnitudes have a decreasing tendency, as the applied indentation force and depth increase. In the case of the Vickers hardness, the FEM-calculated values versus the indentation force were validated by SEM observations of the corresponding remaining impressions. The deviation between the Vickers and indentation hardness is higher in the nanoindentation range (h ≤ 200 nm). These tendencies are explained in Bouzakis et al. (2005). Characteristic examples applying FEM-based algorithms for determining the stress–strain properties and hardness of various materials are presented in the table of Fig. 8b. Corresponding stress–strain curves have been FEM calculated for reference materials based on round robin nanoindentations (Herrmann et al. 2010), showing a satisfactory convergence (Bouzakis and Michailidis 2007).
Fig. 8

(a) Vickers and indentation hardness distribution versus the indentation force and depth for a hardened steel. (b) Stress–strain characteristics and hardness of various materials

Further algorithms for determining stress–strain characteristics of materials are based on the representative stress and strain (Ahn and Kwon 2001), an inverse analysis by FEM (Dao et al. 2001), and on neural networks (Huber and Tyulyukovskiy 2004). FEM-supported methods offer advanced capabilities in determining all kinds of mechanical properties of materials, such as Young’s modulus, yield, and rupture strength.

Key Applications

Characteristic applications of nanoindentation in production engineering for estimating mechanical properties can be found in coatings on cutting and forming tools and on anti-wear and anti-corrosion surfaces/surfaces of manufactured products for determining their internal stresses.



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Copyright information

© CIRP 2014

Authors and Affiliations

  • Nikolaos Michailidis
    • 1
    Email author
  • Konstantinos-Dionysios Bouzakis
    • 2
  • Ludger Koenders
    • 3
  • Konrad Herrmann
    • 3
  1. 1.Physical Metallurgy Laboratory & Fraunhofer Project Center Coatings in Manufacturing (PCCM) / Mechanical Engineering DepartmentAristoteles University of ThessalonikiThessalonikiGreece
  2. 2.Laboratory for Machine Tools and Manufacturing Engineering & Fraunhofer Project Center Coatings in Manufacturing (PCCM) / Mechanical Engineering DepartmentAristoteles University of ThessalonikiThessaloniki, GreeceGermany
  3. 3.Surface MetrologyPhysikalisch-Technische Bundesanstalt (PTB)BraunschweigGermany