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
3.7 Study of Ductile to Brittle Transition (DBT) using Nanoindentation
3.7.1 Determination of DBT
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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.
77 Simulations were carried out on silicon and silicon carbide materials using rigid diamond indenter. Diamond indenter was given the downward movement with velocity of 0.1 mm/s.
The indentation pressure below the indenter tip was observed closely until the failure occurs.
The indentation depth and pressure were recorded for the comparison with the experimental ductile to brittle transition depths.
Figure 3.15 Indentation depth (a) before fracture and (b) after fracture for silicon carbide Figure 3.15 shows the von-Mises stress field for 4H-SiC specimen before and after the occurrence of fracture. From Figure 3.15 (a), it can be noted that a maximum von-Mises stress of about 38 GPa occurs when the indenter’s depth is 375 nm. This value is noted when the step time is 375 μs and ODB frame is 15. This pressure (~38 GPa) is sufficient to change the phase (> hardness of SiC, i.e., 25‒35 GPa [Noreyan et al. (2005), Reddy (2008), Patten and Jacob (2008)]) of the material and the depth can be comparable to the ductile to brittle transition depth [Patten et al. (2005)]. Just after the 15th ODB frame, i.e. at 16th frame and step time 400 μs, there appeared a crack/fracture (Figure 3.15 (b)) marked with red circle. The von-Mises stress also suddenly increased to 49 GPa from 38 GPa. The average stress was calculated along the contact surface between indenter and specimen (shown with blue dots in Figure 3.15 (a)). It is to be noted that von-Mises stresses plotted in the Figure 3.16 are the average values of stresses generated at nodes 1 to 6 as shown in Figure 3.15 (a).
Figure 3.16 shows the plot of average stress versus indentation depth. It reveals that the stress increases linearly up to some depth (~120nm) and when the pressure/stress reaches the threshold value for dislocation initiation, micro-cracks start forming underneath the tip.
Because of the movement of dislocations, the stresses are absorbed to some extent which is depicted in the Figure 3.16. When all the dislocation movements are completed, a linear stress of 30-35 GPa is obtained which is near to the phase changing pressure. Similar observations have also been presented by Szlufarska et al. (2004, 2005) and Mishra and Szlufarska (2009) TH-2306_10610325
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on their nanoindentation simulation using molecular dynamics. Just after the fracture (375 nm depth), stresses of the selected nodes were drastically reduced.
Figure 3.16 Average stress (nodes 1 to 6) vs. indentation depth
Similar to the numerical simulation of SiC, simulations were carried out to determine the DBT for silicon (isotropic) sample. It was found out that the fracture occurs at ~16.6 GPa, where the cubic diamond structure of silicon transforms to metallic -silicon (Figure 3.17 (b)). Here, the peak displacement was noted as ~ 91 nm. During silicon cutting, generally, the machining pressure in the ductile regime is higher than 10 GPa, which is sufficiently high for phase transformation of silicon [Yan et al. (2009a)].
Figure 3.17 Indentation depth (a) before fracture and (b) after fracture for silicon According to Yan et al. (2005b), dislocation phenomenon of silicon is still not fully understood. Literature suggests that dislocations do not move during conventional mechanical TH-2306_10610325
79 testing at temperatures below 450˚C [Alexander and Haasen (1969), Ray et al. (1971)]. Under extreme conditions such as indentation loading, where localized stresses can approach the theoretical shear strength of the material, dislocation structures have been observed in silicon.
pure edge dislocations generally do not form in silicon because, at room temperature, the dislocations are relatively immobile in silicon [Yan et al. (2009b)]. From the Figure 3.18, it is observed that the movements of dislocations are not clearly identifiable. Only a small portion of stresses has been absorbed at room temperature during the indentation process.
Figure 3.18 Average stress generated at the selected nodes vs. indentation depth
From the above two analyses, the DBT values for SiC and Si were found to be around 375 nm (Figure 3.16) and 91 nm (Figure 3.18) respectively. These are the values at which the first micro crack initiates in the workpiece. The validation of these findings is presented in the following section.