Simulation of Nanoindentation Technology Based on Rough Surface

Abstract

In order to study the influence of roughness on the nanoindentation test, finite element method was used to simulate the nanoindentation test process, and secondary indentation method was further proposed to suppress the influence of roughness on the indentation. After optimizing the indentation parameters, the relative error between the results of nanoindentation simulation for rough surface samples and smooth surface samples is stable within 5%. The secondary indentation method can effectively reduce the influence of roughness on nanoindentation test results.

1 Introduction

With the development of modern industry technology, the research of precision and ultra-precision machining technology is extensive, and the requirements for surface properties of materials are also widely, especially the mechanical properties of materials at micro and nano scale have attracted widespread attention [1,2,3]. The application and research of micro-nano structured materials require properties characterization and fracture analysis. Meanwhile, with the development of various surface treatment technologies, the scale of material properties is getting smaller [4,5,6], due to the size effect, there is a great difference between the mechanical properties of materials at nano scale and bulk materials, thus traditional mechanical properties characterization methods have been unable to accurately obtain the mechanical properties of materials at nano scale. Nano-indentation method is very suitable for characterizing the mechanical properties at nano scale that does not need to process materials into standard size [7,8,9]. By recording the indenter load and displacement data during the indentation process, based on the Oliver-Pharr theory, the mechanical properties of the material can be calculated, such as hardness, elastic modulus, strain hardening effect, creep, and so on [10,11,12]. However, the roughness of the sample will affect the evaluation of the geometric parameters of the residual indentation and directly affect the analysis of load–displacement data, resulting in poor reliability and repeatability of the test results [13, 14]. Therefore, the application of nano-indentation technology theory based on smooth surface will be limited. In order to reduce the cost and improve the test efficiency, this paper studied the influence of roughness on the nanoindentation test based on finite element simulation, and further proposed the secondary indentation method nanoindentation test technology to suppress the influence.

2 Nanoindentation Simulation by Secondary Indentation Method

2.1 Modeling of Rough Surface Samples

There are two main criteria for evaluating surface roughness: the arithmetic mean deviation method $R{\text{a}}$ and the maximum height method $Rz$. The arithmetic mean deviation $R{\text{a}}$ refers to the arithmetic mean value of the absolute value between the point on the contour line of the measurement direction and the reference line in the sampling length, calculated by the equations:

$$Ra = \frac{1}{l}\int\limits_{0}^{l} {\left| y \right|} dx$$

(1)

$$Ra = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {y_{i} } \right|}$$

(2)

$Rz$ refers to the distance between the peak line and the bottom line within the sampling length l, as shown in Fig. 1.

Fig. 1

Maximum height method to evaluate roughness

The arithmetic mean difference method uses the average of the height difference between the sampling point and the reference contour to define the roughness, which does not reflect the characteristics of the highest and lowest points of the surface. The definition of maximum height method reflects the characteristics of the maximum height difference of the surface, which is more beneficial to the determination of the indentation parameters.

In the simulation, the maximum height method $Rz$ is used as the standard for evaluating roughness, and ${Z}_{v}={Z}_{p}$ simplifies the rough sample surface into a surface composed of continuous identical peaks and bottoms. According to the definition of roughness, different roughness sample models were built and assembled into ABAQUS software. Pure copper is selected as the simulation material. Considering the actual machining accuracy and wear, the tip of the indenter was set a radius of 400 nm. In order to reduce the total number of meshes and shorten the calculation time, the mesh density near the contact area was encrypted by means of edge planting.

2.2 Secondary Indentation Nanoindentation Technology

The nanoindentation test process is divided into two steps, as shown in Fig. 2. Step 1: flat indenter is carried out on the rough surface to reduce roughness. Step 2: Conventional nanoindentation testing is performed in the residual indentation area.

Fig. 2

Nanoindentation test process using the secondary indentation method

2.3 Simulation of Secondary Indentation Method

In order to understand the plastic deformation and stress distribution of the first indentation, the flat indenter nanoindentation simulation was carried out, and the smooth surface nanoindentation simulation was used to analyze the influence of surface roughness and maximum indentation depth. The diameter of the flat indenter $D_{f}$ = 2 mm, and the samples with different surface roughness ($Rz$) were simulated with different maximum indentation depth ($h_{1\max }$).The simulation results show that under the same indentation depth, with the roughness increase, the relative error of load–displacement curve is greatly affected. When $Rz$ = 120 μm, $h_{1\max }$ = 400 μm, the relative error of maximum load is 6.32%, as show in Fig. 3.

Fig. 3

Simulation of indentation with different roughness, $h_{1\max }$ = 400 μm

On the other hand, the plastic deformation of the material is independent of the surface roughness. Figure 4 shows the residual stress distribution and the comparison of plastic deformation regions in indentation of flat indenters with different roughness. The residual stress is mainly distributed just below the indentation surface, and the stress and plastic deformation area gradually increase with the increasing of the indentation depth. The stress distribution is mainly concentrated in the area near the surface, while the lower region is basically the same as that in the plastic region. This means that at the same indentation depth, the stress–strain properties of the central region away from the surface of the sample are not affected by the roughness, which means that the stress–strain relationship at a representative point under the surface of the residual indentation is not affected by the surface roughness.

Fig. 4

Residual stress distribution and plastic deformation region by indentation of flat indenter

2.4 Optimization of Simulation Parameters

According to the definition of roughness by maximum height method, when the maximum indentation depth of step 1 is $h_{1\max } \ge 2Rz$, the residual indentation satisfies the conditions of Step 2. For the sample with roughness $Rz$ = 120 μm, the secondary indentation simulation was carried out. In Step 1, the diameter of the flat indenter $D_{f}$ = 2 mm, the maximum indentation depth $h_{1\max }$ = 240 μm, $h_{2\max }$ = 400 μm. As show in Fig. 5, the results of the secondary indentation method are different from the smooth surface with Berkovich indenter, and the relative error of the maximum load is 11.65%.

Fig. 5

Comparison of indentation simulation with conical indenter on smooth surface and with secondary indentation method on rough surface when $h_{2\max }$ = 400 μm

The Step 2 is carried out in the central area of residual indentation. However, when the depth of indentation increases, the contact radius between the edge of the Berkovich indenter and the surface of the residual indentation gradually rises. If the area of the residual indentation is not large enough, the stress distribution and plastic deformation in the surrounding area will affect the results. Therefore, the diameter $D_{f}$ of Step 1 is an important parameter of second indentation method, and further analysis is necessary.

Simulation using the secondary indentation method was carried out for the diameters of $D_{f}$ = 4, 8,10 mm respectively. Table 1 lists the relative error values between the maximum load of the second indentation method and smooth surface indentation test under different $D_{f}$. As show in Fig. 6, with the increase of $D_{f}$, the relative errors between the second indentation method and smooth surface decrease gradually, and tend to be stable after reaching a certain degree. This indicates that the area of the first stage residual indentation is large enough, and the influence of the residual stress and plastic deformation on the surface of the material residual indentation is significantly reduced.

Table 1 Maximum load and relative error values of the second indentation method and smooth surface indentation under different $D_{f}$

Fig. 6

Comparison between the indentation results of the second indentation method with different flat indenter diameters $D_{f}$ and those of the smooth surface ($h_{2\max }$ = 400 μm): a $D_{f}$ = 4 mm; b $D_{f}$ = 8 mm; c $D_{f}$ = 10 mm

Define geometric parameter $\gamma = h_{2\max } /D_{f}$. when $h_{2\max }$ = 400 μm, the results show that the relative error with the maximum indentation load on the smooth surface tends to be stable at $\gamma \le 5$%, and a similar trend is also present in the indentation simulation with the second indentation method at $h_{2\max }$ = 300 μm and $h_{2\max }$ = 500 μm, as show in Fig. 7. Which means the smaller geometric parameter $\gamma$ is beneficial to the secondary indentation method. The data in Table 2 further show that when $\gamma \le 5$%, the secondary indentation method can effectively reduce the influence of material surface roughness on the nanoindentation test results.

Fig. 7

Comparison of simulation results of the second indentation method and smooth surface indentation for different geometric parameters ($\gamma$ = 20, 10, 5, 2%): a =  300 μm; (b)$h_{2\max }$ = 500 μm

Table 2 Maximum load and relative error for different geometric parameters

3 Conclusion

This study aims at the problem of poor repeatability of traditional nanoindentation testing technology, based on the finite element simulation technology, the influence of roughness on the results of nanoindentation testing is studied, and the secondary indentation method is further proposed to suppress the influence of roughness on the results of indentation testing. The indentation parameter $\gamma = h_{2\max } /D_{f}$ was defined by the secondary indentation method, and multiple indentation simulations were carried out for different values of $\gamma$. According the simulation, when $\gamma \le 5$%, the relative error of the maximum load decreases from 15% to <5%, indicating that the secondary indentation method can effectively reduce the influence of material roughness on nanoindentation testing.