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One essential problem related to instrumented indentation is the effect of pileup, which could introduce significant errors on the measured hardness and elastic modulus. In this work, we have assessed some critical issues associated with instrumented indentation by means of numerical simulation. Dimensional analysis is adopted to acquire the principal governing parameters of the process, such as the ratio of yield strength to elastic modulus

Instrumented indentation, particularly nanoindentation, has been widely used for characterizing the mechanical properties of materials at small scales [

The commonly used definition of hardness is as follows:

The elastic unloading curve of nanoindentation is approximately expressed in the power-law relation [

From (

Fit the unloading curve of the indentation; find values of

Find contact depth

Determine the projected contact area

Obtain hardness by (

Derive

Apart from geometric difference of indenters, there are several other factors that may influence the measurement of instrumented indentation. The thickness and properties of the test materials and the substrate effect have been studied systematically by many researchers [

In this paper, we will focus on the differences of indenter geometries and their effect on the indentation results by means of numerical simulation. Dimensional analysis will be adopted to acquire the principal governing parameters. The effect of pileup on the simulation results will also be given.

Three kinds of indenters, namely, conical indenter, Berkovich indenter, and Vickers indenter, will be discussed separately. The half angle of the conical indenter is set to be 70.3°, while the half angles of Berkovich indenter and Vickers indenter are chosen to be 65.3° and 68°, respectively. The relationship between the projected contact area

For a simple instrumented indentation test, the procedure is composed of two steps, that is, loading and unloading. During the loading process, the produced load

During unloading, the imposed load

Finite element analysis (FEA) of the indentation process has been carried out by using the commercial software Abaqus. The above-mentioned scaling relationships are carefully examined. These dimensionless functions, such as

As mentioned before, three kinds of indention, namely, conical, Vickers, and Berkovich, are simulated. All of the indenters have the same projected area function (

The indenters were taken as rigid body, while the tested materials were regarded as elastic-perfect-plastic. The yield strengths of the materials are set to range from 80 MPa to 8 GPa, which covers most of the engineering materials. Two elastic modulus values, that is, 100 GPa and 200 GPa, are considered. Poisson’s ratio is set to be 0.38, corresponding to a material of niobium that will be used for comparison with the experimental observations. Actually, the values of Poisson’s ratio taken from 0.2 to 0.4 will not lead to any significant difference [

A total number of 60,000 brick elements are meshed in the material and about 3,000 quadrilateral elements are meshed in the indenter. An example of the mesh of half of the conical indentation model is given in Figure

Finite element meshes of the numerical model of conical indentation (only half model is shown for better view). The image in the red box provides the details of the mesh in the region of contact.

Similar to experiments, the

Figure

Displacement fields at indentation depth of 3

The evolution of force

Force-displacement curves for different kinds of indenters. The solid lines are the power law fitting of the loading parts.

True projected area-displacement curves for three kinds of indenters.

The dimensionless functions, that is,

The variation of dimensionless function

The variation of dimensionless function

The variation of dimensionless function

The variation of dimensionless function

The projected contact area

Influence of

The derived elastic modulus of the material depends on the contact stiffness

Influence of

Figure

Effect of

The hardness is normalized by yield stress of the material and plotted against

Influence of

It should be noted that

Our experimental results [

Appearance of the indentation geometry by AFM at the center (a) and edge (b) of the HPTed niobium disk. It can be observed that, at the center of the disk, significant pileup effect is present (a2), while, at the edge, neither sink-in nor pileup effect is observed (b2) [

Three kinds of indentation, namely, conical, Berkovich, and Vickers, were comparatively studied by means of numerical simulations. The effect of pileup on the derived quantities was carefully examined. Dimensionless analysis was adopted to acquire the principal governing parameters of the process. The main findings are summarized in the following:

The dimensionless functions

When

The derived elastic modulus could be different if different indenters are adopted, even if the effect of pileup is considered. An approximate relationship of

The Tabor factor,

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the financial support by National Natural Science Foundation of China (nos. 11102168, 11102166, 11472227, 11202168, and 10932008), the 111 Project (no. B07050), and the Fundamental Research Funds for the Central Universities (no. 310201401JCQ01001).