Indentation Modulus

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Indentation modulus

Definition of the indentation modulus

The indentation modulus E_IT (in MPa) is determined in the micro load range of the hardness test using a method described in detail in ISO 14577 [1] from the initial rise of the unloading curve of the load (F)–indentation depth (h) diagram (see: instrumented hardness testing – method & material parameters) at the maximum load F_max (dF/dh|_Fmax). In the nanolast range of hardness testing, the method according to Oliver and Pharr is usually used [2]. According to ISO 14577, it is taken into account that the mechanical resistances of the test specimen and diamond indenter are connected in parallel (see Figure and Eq. 1)

${\frac {1}{E_{G}}}={\frac {1\,-\,\nu ^{2}}{E_{IT}}}\,+\,{\frac {1-\nu _{D}^{2}}{E_{D}}}$

(1)

This results in E_IT:

$E_{IT}={\frac {1-\nu ^{2}}{{\frac {1}{E_{G}}}\,-\,{\frac {1-\nu _{D}^{2}}{E_{D}}}}}$

(2)

from which the following expression can be derived using the relationships described in detail in ISO 14577:

$E_{IT}={\frac {1-\nu ^{2}}{0,5{\sqrt {\frac {24,5}{\pi }}}\cdot \left({\frac {dh}{dF}}\right)_{F_{max}}\cdot \left(4h_{t}-3F_{max}\left({\frac {dh}{dF}}\right)_{F_{max}}\right)-8,73\cdot 10^{-13}Pa^{-1}}}$

(3)

The Poisson's ratio required to calculate E_IT is known for most materials and is relatively independent of temperature. The value 8.73 x 10^-13 Pa^-1 is the effective compliance of diamond.

Application limits

When using the indentation modulus E_IT, it is important to note that although E_IT is fundamentally equivalent to a modulus of elasticity, it only describes the stiffness behaviour very locally and under triaxial loading. This results in a difference in value to the characteristic values of the modulus of elasticity, which were determined using conventional plastic testing methods, such as the uniaxial tensile or compression test or the three- or four-point bending test. The indentation modulus can therefore not be used for dimensioning purposes.

See also

• Elastic modulus
• Instrumented hardness testing – method & material parameters
• vickers hardness
• Ultrasound – elastic parameters
• Indentation fracture mechanics

References

[1]

ISO 14577: Metallic Materials – Instrumented Indentation Test for Hardness and Materials Parameters Part 1 (2024-08): Test Method (Draft) Part 2 (2024-08): Verification and Calibration of Testing Machines (Draft) Part 3 (2024-07): Calibration of Reference Blocks (Draft) Part 4 (2016-11): Test Method for Metallic and Non-metallic Coatings Part 5 (2022-10): Linear Elastic Dynamic Instrumented Indentation Testing (DIIT) Part 6 (2025-11): Instrumented Indentation Test at Elevated Temperature

[2]

Oliver, W. C., Pharr, G. M.: An Improved Technique for Determining Hardness and Elastic Modulus using Load and Displacement Sensing Indentation. J. of Materials Research 7 (1992) 1564–1583