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# Micro and Nano Indentation, Oliver-Pharr Model

- Authors
- Name
- Universal Lab
- @universallab

**1. Definition**

Traditional material mechanical property testing techniques usually provide only macroscopic mechanical parameters of materials. They struggle to give quantitative explanations for the micro-mechanical behavior of materials under complex loading conditions, the evolution of service performance (such as thermal and electromagnetic properties), and the correlation and interaction mechanisms between these behaviors and structural evolution. Therefore, testing techniques for materials at small scales have emerged, with micro and nano indentation testing technology showing great potential.

Unlike the method of directly observing the residual indentation size in microhardness testing techniques, micro and nano indentation testing technology establishes the corresponding mathematical model based on the P-h curve shown below to solve for material mechanical performance parameters:

For the load-displacement curve in Figure 1, the commonly used analytical models is the Oliver-Pharr model . Before explaining these model, let's first understand some key parameters involved in Figure 1:

P(max): Maximum load;

S: Contact stiffness, which is the initial slope at the beginning of unloading;

h(f) : Residual indentation depth after complete unloading.

Curve Analysis: During the loading phase (the segment marked "loading" in the figure), the material undergoes elastic-plastic deformation. After reaching the maximum load P(max), unloading begins (the segment marked "unloading" in the figure). During the unloading phase, the elastic deformation involved in the loading phase recovers, while the plastic deformation does not. After the elastic deformation completely recovers, the indentation depth at this point is h(f) .

Note: The following analysis is based on the widely used Berkovich indenter.

**2. Oliver-Pharr Model**

**2.1 Calculation of Hardness H**

To calculate hardness, we need to understand the unloading process of the Berkovich indenter, as shown in the figure below:

In elastic-plastic materials, there is material pile-up at the contact edges, resulting in an indentation depth, denoted as h(s). During the unloading process, the indenter does not contact the material over this depth. From the dotted line marked by "indenter" in Figure 2, one can see the contact situation between the material and the indenter.

For an ideal geometric indenter, the first term of the polynomial is π.

**2.2 Calculation of Elastic Modulus E**

β is a constant related to the geometric shape of the indenter, with β=1.034 for the Berkovich indenter. ν is the Poisson's ratio of the test sample,E is the elastic modulus of the test sample,ν(i) and E(i) are the Poisson's ratio and elastic modulus of the indenter, respectively. For a diamond indenter, ν(i)=0.07 and E(i)=1140GPa. Using the above formulas, the elastic modulus E can be calculated.

Assumptions of the Oliver-Pharr Model:

Isotropic material;

Semi-infinite elastic half-space;

The material undergoes elastic-plastic deformation during loading and only elastic deformation recovery during unloading;

Material around the rigid indenter undergoes only indentation deformation;

Time-dependent deformations like creep and viscoelasticity are not considered.

**3. Summary**

This overview covers the two models, including the calculation of hardness and elastic modulus. To obtain other mechanical performance parameters, conduct in-depth analysis of the impact of each parameter, or investigate the influence of different indenter geometries and materials, further study and analysis are required.