List of Symbols
Step 3: Step 3: Derive the element characteristic (stiffness and mass) matrices and characteristic (load) vector. The strain can be expressed as
3.6 Experimental Validation of FEM based Nanoindentation Simulation
72
infinite value of Young’s modulus for the diamond as well. Thus, the second term in the Equation (3.44) will become zero [Yu et al. (2003)]. Then by using Equation (3.44), the elastic modulus of SiC sample was computed as
1
𝐸𝑟
=
1−𝐸𝑖2𝑖
+
1−𝐸𝑠2𝑠
1
3.67253×1011
= 0 +
1−(0.23)𝐸 2𝑠
𝐸𝑠 = 348 GPa (3.49)
Thus, the Young’s modulus (Es) for the silicon carbide sample is found as 348 GPa.
B. Hardness
The hardness value of silicon carbide can be calculated by using Equation (3.45). It is given as,
Hardness, 𝐻 =𝑃𝑚𝑎𝑥𝐴 =9.348790.00012210−15 = 13.04 𝐺𝑃𝑎 (3.50) After determining the required output parameters, the results were validated with the published experimental results. These are presented in the following section.
73 displacement plots of experimental results and numerical simulations for 5, 10, 12, 20, 25 and 50 nm depth indentations. From the figure, it is observed that numerical results fairly match with the experimental results. With these encouraging results, further comparisons have been carried out by using Oliver and Pharr analytical procedure as explained in section 3.5.
Figure 3.13 Load vs. displacement: numerical and experimental results for SiC
The computed values of Young’s modulus and hardness values are listed in Table 3.7. It also shows peak loads, projected area, calculated Young’s modulus and hardness for various levels of indentation depth. From the table, it is observed that, peak loads predicted in simulations are slightly higher than that of experimental peak loads. This is possibly due to consideration of various assumptions such as diamond indenter was considered as perfectly TH-2306_10610325
74
rigid body and spherical and workpiece and indenter both assumed to be isotropic and homogeneous. The simulations were found to be predicting well with overall prediction error of 19.9% considering all responses together. Also, the computed values of projected area are found to be similar with experimental results with maximum prediction error of 2.63%. The numerical model is capable of predicting the Young’s modulus and hardness with 32% and 24.7% of absolute prediction error respectively.
Table 3.7 Comparison of experimental and simulated Young’s modulus and hardness at different indentation depth for SiC
Peak displacement Peak Load
(μN)
Projected area (nm2)
Young's Modulus (GPa)
Hardness (GPa)
Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%)
5 nm 103.3 121.9 15.3 9978.2 9348.8 −6.7 233 348 33.1 10 13 23.0 10 nm 276.2 336.9 18.0 18055.9 18541.7 2.6 275 358 23.1 15 18 15.8 12 nm 312.2 434.5 28.2 22422.1 22174.6 −1.1 270 351 23.0 14 20 29.7 20 nm 654.1 889.5 26.5 35791.4 36455.9 1.8 246 376 34.7 18 24 25.8 25 nm 821.8 1215.1 32.4 45967.9 45177.4 −1.7 255 397 35.7 18 27 34.2 50 nm 2415.2 3065.9 21.2 84811.4 86427.8 1.9 265 459 42.4 28 35 19.8
Average prediction error 23.6 2.63 32.0 24.7
3.6.2 Experimental Validation of Nanoindentation of Silicon
Nanoindentation simulations were also carried out for silicon as the workpiece. To validate the developed finite element based nanoindentation model, the experimental results given by Rao et al. (2007) were considered. All the process conditions in the simulation were set similar to that of experimental studies carried out by Rao et al. (2007). The workpiece geometry, boundary condition, element type and mesh model used in the present simulation were similar to that for SiC (section 3.4.2) except the material model and process conditions.
Numerical simulations were carried out and results were extracted in terms of load vs.
displacement values. Figure 3.14 shows the plots of experimental and numerical simulation results for indentation depths 45 nm, 75 nm, 95 nm, 115 nm, 130 nm and 140 nm. It can be seen that the numerical results are in well agreement with the experimental results.
TH-2306_10610325
75 Figure 3.14 Load vs. displacement: numerical and experimental results for Si
After the numerical simulations, using Oliver and Pharr analytical procedure explained in section 3.5, the Young’s modulus and hardness values were calculated. These are listed in Table 3.8. It also depicts the comparison of experimental results with the numerical results in terms of peak loads, projected area, calculated Young’s modulus and hardness for various levels of indentation depths. It can be observed that, the peak load for simulations are higher than the experimental peak load. However, computations of projected area, Young’s modulus and hardness were found to be agreeing very well with the experimental results with the overall prediction error of 10.9%. Computation of projected area is found to be quite accurate with 1.3% as average prediction error. The average prediction errors for Young’s modulus and hardness were noted to be 9.2% and 16.5% respectively.
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From the Figure 3.14, it is also noticed that the peak loads are higher for simulation results at lower depths and gradually become lower at higher depth (140 nm) as compared to that of experimental results. Furthermore it is to be noted that the our results are found to be in-line with the established values of Young’s modulus and hardness values for silicon 190 GPa and 12 GPa respectively [Reddy (2008)]. The variations in the results between experimental results and numerical simulations may be due to the consideration of indenter as rigid body and employing the work material as isotropic and homogeneous.
Table 3.8 Comparison of experimental and simulated Young’s modulus and hardness at different indentation depth for Si
Peak displacement Peak Load
(μN) Projected area
(nm2)
Young's Modulus
(GPa)
Hardness (GPa)
Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%) Experiment Simulation Prediction error (%)
45 nm 1004.2 1275.2 27 81157.7 78492.9 −3 139 155 11 12 16 31 75 nm 2000.4 2604.4 30 121056.3 123750.0 2 176 169 −4 17 21 27 95 nm 3000.0 3566.8 19 149824.4 150778.6 1 196 179 −9 20 24 18 115 nm 4001.7 4568.5 14 173974.9 175292.9 1 193 188 −3 23 26 13 130 nm 4975.3 5296.1 6.4 192637.2 192028.6 −0 211 193 −9 26 28 7 140 nm 6026.0 5826.6 −3 203447.6 202400.0 −1 243 198 −18 30 29 −3
Average prediction error 16.6 1.3 9.2 16.5
After successful validation of the present numerical model for Si and SiC, it was thought appropriate to carry out a detail study on understanding of ductile to brittle transition (DBT) of brittle materials using the developed numerical model. In what follows the study on the DBT is presented in the next section.