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
3.7.2 Validation of DBT with Published Experimental Results
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.
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3.8 In-situ Study of Phase Changing Pressure
In general, brittle materials undergo phase change when they applied with compressive hydrostatic stress [Strössner et al. (1987), Yoshida et al. (1993), Lu et al. (2008)]. In case of 3C-SiC, the material undergoes a phase transition into the Rocksalt (RS)-type structure at 60- 100 GPa or higher accompanied by a volume reduction of 20.3%. Upon pressure release, the RS-structured SiC transformed back to 3C-SiC phase below 35.0 GPa. 6H-SiC polytype remained stable up to about 90.0 GPa with an indication of a phase transition above this pressure [Yoshida et al. (1993)]. Sekine and Kobayashi’s shock compression experiments on 6H-SiC demonstrated a first-order phase transition with a volume reduction of about 15% into a six fold coordinated structure around 105 GPa [Sekine and Kobayashi (1997)].
Silicon material exhibits different transition phases during the application of pressure.
Published literature have demonstrated that silicon undergoes plastic deformation and phase transformation under the existence of high hydrostatic pressure [Zhao et al. (1986), Hu and Spain (1984), Yan (2004), Cai et al. (2007b), Needs and Mujica (1995), Minomura and Drickamer (1962), Clarke et al. (1988)]. Initially, the silicon will be in cubic (diamond) structure (-silicon) in the pressure range of 0 −11.3 GPa. A transition to the -Sn structure (II) and amorphous phase initiates at 11.2 ± 0.2 GPa and two phases (I + II) coexist at 12.5 ± 0.2 GPa. However, during unloading stage, this metallic phase is not stable at low pressure (~4 GPa) and changes to an amorphous phase or other metastable phases. At 13.2 ± 0.2 GPa a new phase (V) initiates, and the transition completes at 16.4 ± 0.5 GPa. This hexagonal phase continues to exist from 25 GPa to 40 GPa. Beyond this, metastable phases (VIII and IX) can be found. On release of pressure, the phase sequence is V → (V + II) (14.5 –11.0 GPa) → II
→ (II + III) (10.8 –8.5 GPa) → III, the last phase persisting to room pressure. The simulation results indicate a pressure of about 38 GPa at the peak displacement. This is greater than the hardness of the material, i.e. 26 GPa [Patten et al. (2007)], as shown in Figure 3.19.
Figure 3.19 Different pressure zones (a) before failure at 372.97 nm depth and (b) after failure at 400 nm depth for silicon carbide
TH-2306_10610325
81 In case of numerical simulation for silicon nanoindentation, the von-Mises stress of
~16 GPa was noted during the deepest indentation which is larger than the transition pressure for silicon from cubic (diamond) to -Sn metallic phase. This finding is found to be in line with the published results of hardness 10.79 GPa by Cai et al. (2007b), Zhang and Tanaka (1999) and Cheong and Zhang (2000). As per our simulations, the residual stresses on the surface were found to be 0.37 to 0.7 GPa for silicon. This is also in line with experimental findings of 0.3 to 1.5 GPa reported by Juliano and Penrose (2002). The different pressure zones appeared during nanoindentation process during loading and after unloading and their corresponding phases are shown in Figure 3.20.
Figure 3.20 Different pressure zones after (a) Loading and (b) Unloading during nanoindentation on silicon
During our numerical simulations, some irregular stress fields were observed along the element layer (considering a plane along the center of the indenter and perpendicular to workpiece surface). These stress fields are unusual, as no anisotropic properties and crystal orientations were considered in the model. The reason behind this fact may be that, just under the indenter tip, there exists an amorphous layer of few nanometers thickness. Under the amorphous layer, dislocations are present which trigger the plastic deformation of the workpiece. Although the amorphous layer cannot be clearly differentiated, the dislocations can be identified by the stress field as shown in red colored circles in Figure 3.21. Similar observations have been made by Jasinevicius et al. (2005), Nix and Gao (1998) in their experimental works. Nix and Gao (1998) mentioned that during indentation with rigid cone, geometrically necessary dislocations beneath the indenter are required to account for the permanent shape change at the surface. Jasinevicius et al. (2005) described the dislocations and subsurface micro-cracks beneath the amorphous layer during pyramidal micro indentation TH-2306_10610325
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and diamond turning. Overall the present work indicates the presented numerical modeling using FEM can be successfully utilized to predict the micro-cracks as well
Figure 3.21 Spherical nanoindentation of SiC showing irregular stress field
3.9 Numerical Study on the Effect of Residual Stress on DBT during Repetitive