Molecular dynamics simulation of nano indentation

MOLECULAR DYNAMICS SIMULATION OF NANO-INDENTATION DAI LING (B Eng SJTU) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 ACKNOWLEDGEMENTS First and foremost, I am sincerely grateful to my supervisor, Dr Vincent, Tan Beng Chye who has patiently guided me throughout the project Also I am sincerely grateful to Dr Tay Tong Earn who has always given me advice on each stage of research progress Discussion with them are always fruitful and, more importantly, encouraging Their advice will always be much appreciated Also great thanks to my seniors, Deng mu and Serena Tan They have given me great help to overcome difficulties during the past two years Especially thanks to Deng mu who has offered me great support on the research direction, methodology and on some research resources Also thanks to officers of Impact Lab, Alvin and Joe who have offered me an excellent environment and computer resources to research work Thanks to Dr Yang Liming and Dr Yuan jianming who have given to great support with their research knowledge The thank you list also includes the staff and student in Impact lab – Simon, Norman whose assistance have made things easier I SUMMARY Simulation is an effective method to study the mechanical properties of materials Three kinds of simulations of nano-indentation on copper were carried out: Molecular Dynamics simulation (MD), Finite Element (FE) simulation and hybrid simulation of Molecular dynamics and Finite Element The molecular dynamics simulations predicted mechanical properties of copper: The Young’s modulus was calculated from the unloading curve and the yielding stress was obtained from the indenting stress Similar results of the mechanical properties were obtained from tensile simulations and a potential based XMD program Those results agree with previous works using atomistic simulations However, such results differ greatly from bulk material properties because the modeled specimen is a perfect single cubic When equivalent molecular properties were used in the Finite Element simulation similar quantitative properties such as force-indentation depth relation can be recovered but other aspects, like deformed shape and stress distribution, continued to show obvious disagreements due to the different theoretical basis of the two kinds of simulation methods At last, a hybrid MD-FE simulation of nano-indentation was carried out With proper definition on the MD-FE handshaking area of the model, the model was built up and similar force-displacement curve as MD simulation were obtained Then a comparison of stress distribution among three simulations (FE with MD deduced property, MD and hybrid MD-FE II simulations) was carried out There is roughly agreement among the three models during loading process But the FE simulation shows great difference from the other two on the unloading stage III LIST OF FIGURES Figure 2.1 Force versus indenting depth curve for copper thin films Figure 2.2 Relative variations in the resistance to indentation Figure 2.3 Schematic illustration of radial cracking induced by Vickers indentation Figure 2.4A Atomic configurations corresponding to characteristics states of the force response Figure 2.4B Force vs tip-substrate distance for an FCC copper substrate indented by a rigid FCC copper-like tip 11 Figure 2.5 Equivalent Von Mises stress distribution at onset of plastic deformation 12 Figure 2.6 Force-displacement curves and snapshots from the simulation with indentation into the specimen crystal 14 Figure 2.7 Top view chart corresponding to figure 2.6 15 Figure 2.8 Plane view of the deformed region in the Fe {100},{111} grain boundary substrate showing only first layer 16 Figure 2.9 Stress distribution during the nano-indentation simulation 18 Figure 2.10 Strain distribution during the nano-indentation simulation 18 Figure 2.11 Nano-indentation load versus depth response curves for O-implanted Al sample 19 Figure 2.12 Crack propagation of hybrid FE-MD-TB Model 20 Figure 2.13 Illustration of FE/MD handshaking 21 Figure 2.14 The distance vs time history of the two crack tips 22 Figure 2.15 The stress waves propagating through the slab using a finely tuned gray scale 22 Figure 2.16 Snapshots of a projectile impact on a silicon crystal 23 Figure 2.17 The geometry of the silicon micro-resonator 24 IV Figure 2.18 Partition of the micro-resonator system into MD and FE regions Figure 2.19 A plot of the Young’s modulus as function of the device size for a perfect crystal at two temperatures Figure 3.1 25 26 (a) FCC cubic structure view (b) Four atoms in one single FCC cubic (c) 3d view of diamond structure 31-32 Figure 3.2 Front view of the initial state of the model 33 Figure 3.3 Periodic boundary condition 35 Figure 3.4 Force-displacement curve of Morse potential 38 Figure 3.5 Force vs indentation depth with different indenting speed 39 Figure 3.6 3D view of the initial model 43 Figure 3.7 Neighbor list array 50 Figure 3.8 The velocity distribution reaches Maxwell distribution 54 Figure 3.9 Total energy of the model reaches equilibrating status 55 Figure 3.10 Plan view of the indentation process 57 Figure 3.11 Force-distance curves of Cu-Cu and Cu-C interaction 58 Figure 3.12 Cross-section view slip-line in metal working 59 Figure 3.13 Force-displacement curve of the nano-indentation process 61 Figure 3.14 Plan view of the specimen model in [110] direction 62 Figure 3.15A Force-displacement curve with various indenting directions 63 Figure 3.15B Unloading force-displacement curve with various indenting directions 64 Figure 3.16 Indenting stress of the loading process 66 Figure 3.17 Plan view of the tensile MD model 67 Figure 3.18 Plan view of the tensile process until breaking 69 V Figure 3.19 Stress-strain curve of the tensile process 70 Figure 3.20 Stress-strain curve with various potentials 71 Figure 4.1 Plan view of 3D initial model of FE nano-indentation simulation 78 Figure 4.2 Strain characteristics during the indention process 80 Figure 4.3 Force-displacement curves from three types of simulation 82 Figure 5.1 Handshake region of hybrid MD-FE 83 Figure 5.2A 3D view of the FE-MD model 85 Figure 5.2B Front view of the initial model of the specimen 86 Figure 5.3 Nano-indentation process on a hybrid MD-FE model 87 Figure 5.4 Force-displacement curves of four simulations 88 Figure 5.5 Comparison of Von Mises stress of three simulations 90-91 VI LIST OF TABLES Table 2.1 Properties of materials used in indentation cracking measurement of fracture toughness Table 3.1 Parameters of Morse potential 37 Table 3.2 Values of parameters of the highest derivative order 48 Table 3.3 Values of Young’s modulus 61 Table 3.4 Values of Young’s modulus for comparison 65 Table 3.5 Young’s modulus of various potential 72 Table 3.6 Ultimate stress and strain 72 Table 3.7 Results for XMD program 76 Table 4.1 Values of Young’s modulus and Yielding stress 81 VII TABLE OF CONTENTS Page Acknowledgements I Summary II List of Figures IV List of Tables VII Chapter Introduction Chapter Literature Review 2.1 Nano-indentation Experiment 2.2 Molecular dynamics Simulations 2.3 Finite Element Simulations 15 2.4 Hybrid Simulation of Molecular Dynamics Chapter and Finite Element 20 Molecular Dynamics Simulations 27 3.1 Potential Functions 27 3.2 Nano-indentation Simulation 31 3.2.1 Initialization 31 3.2.2 Algorithm 43 VIII 3.2.3 Equilibration 51 3.2.4 Indentation 56 3.3 Tensile Simulation 67 3.4 XMD Simulation 74 Chapter Finite Element Simulation Chapter Hybrid Simulation of Molecular Dynamics and Finite Element 83 Chapter Conclusions and Recommendations References 77 93 95 IX Figure 5.2B Front view of the initial model of the specimen The dots represent the MD atoms The FE meshes are assigned around the MD region In the handshaking region, the mesh nodes are at the same size as MD atoms At regions far from MD atoms, the meshes become coarser which makes it possible to simulation models with larger size Apart from the handshake region, the MD atoms and FE nodes are just simulated as normal The simulation of nano-indentation on a hybrid MD-FE model of copper can be carried out The input properties of FE model are deduced from MD simulation The process of the indentation is shown in figure 5.3 86 (a) (b) (c) (d) Figure 5.3 Nano-indentation process on a hybrid MD-FE model: Chart (a) shows pop-up phenomenon which is also the case of MD simulation Chart (b) (c) represent the loading process and chart (c) is at the stage of deepest loading Chart (d) is the status after unloading, some copper atoms sticking to the indenter as expected Figure 5.3 shows similar view at MD region as MD simulation Also the deformation of FE meshes can be seen in chart (c) which means at this stage, the indenting stress has propagated to the FE region However, no slip lines are formed during loading 87 process while it is the case with MD simulation Probably the MD model is of small size in which the stationary boundary atoms prevented the bottom atoms to go downward under indenting stress and caused a compressed slip line at somewhere in the model The force-displacement curve is obtained as shown in figure 5.4 M F F M 0 D s im u la tio n E s im u la tio n w ith E s im u la tio n w ith D -F E M D d e d u c e d d a ta b u lk p r o p e r ty Force (nN) 0 0 -5 -.5 0 In d e n tin g D e p th (n m ) Figure 5.4 Force-displacement curves of four simulations: FE simulation with macro property, FE simulation with MD deduced properties, MD simulation and Hybrid MD-FE simulation According to figure 5.4, there is fairly good agreement among the three forcedisplacement curves of FE simulation with MD deduced properties, MD simulation and Hybrid MD-FE simulation 88 A comparison of the stress distribution among FE (with MD deduced properties), MD and hybrid MD-FE simulations was undertaken Stresses in the MD region are calculated from equation 3.30 ( σ mn = Ns ∑[ i M i v im v in − Vi 2V i ∑ j ∂ φ(r ij ) rijm rijn ∂ rij rij ] ), Equation 3.30 gives stresses at the position of each atom which shows great fluctuation The stresses are averaged over a local domain of one lattice size during post-processing, i.e.: _ σ= N N ∑σ i =1 (5.2) i where the σi is the stress on each atom, N is the number of atoms in each local domain _ (normally each mesh contains atoms) and σ is the average stress After obtaining the average stresses in all –XYZ directions, equation 4.2 can be used to get the Von mises stress σm = [(σ − σ ) + (σ − σ ) + (σ − σ ) 2 ] (5.3) where σm is the Von mises stress and σ1-3 are normal stress vectors The distributions of Von mises stress of FE, MD and hybrid MD-FE simulation models are shown in figure 5.5 In the chart of MD-FE hybrid model, the FE nodes are shown as larger dots 89 (a) (b) 90 (c) (d) Figure 5.5 Comparison of Von mises stress of three simulations Chart (a)-(c) represent the loading process Chart (d) is the final view after unloading 91 According to figure 5.5, during the loading process, the three simulation models reach roughly agreement of Von mises stress distribution Chart (c) represents the status of maximum loading and shows that some MD atoms contribute more deformation at the top surface of the specimen In the hybrid MD-FE model, it can be seen that the stress has propagated to the FE region which means the handshaking region is able to transfer stress from MD to FE region For the unloading process, the FE model shows a little more complicate stress distribution than other two models The MD and hybrid MD-FE model show similar deformation shape that some atoms are piling on the top surface Therefore, the three models above can offer similar force and stress characteristics for a nano-indentation test However, the MD region shows more deformation through the piling up of atoms 92 Chapter Conclusions and Recommendations A series of nano-indentation simulations have been carried out in the report Based on the atomic potential, the MD simulation of nano-indentation was carried out to study some mechanical properties: The Young’s modulus was calculated from the unloading curve and the yielding stress was obtained from the indenting stress The values are much higher than bulk values because the modeled specimen is a perfect single cubic Similar results of the mechanical properties were obtained from tensile simulations and a potential based XMD program FE simulation of nano-indentation was also carried out via Abaqus software The FE simulation was carried out two times with two groups of mechanical properties: bulk values and values deduced from MD simulation There is great difference between the two force-displacement curves due to different input properties The curve of MD deduced properties agrees with that of MD simulation Therefore, it can be concluded that, with the MD deduced properties, the force reaction of indentation can be obtained from FE simulation and the computational cost of MD can be saved here Finally, a hybrid MD-FE simulation of nano-indentation was carried out The MD atoms are assigned at regions closer to the indenter to offer detail information at the atomic level; FE elements are located at peripheral regions to save computational cost With proper definition on the MD-FE handshaking area of the model, the simulation was carried 93 out and similar force-displacement curve as MD simulation were obtained Then a comparison of stress distribution among three simulations (FE with MD deduced property, MD and hybrid MD-FE simulations) was carried out There is rough agreement among the three models during loading process However, during unloading, the FE model shows significant difference from the other two models Furthermore, at the deep loading stage of the hybrid MD-FE model, the stress has propagated to the FE region Therefore, the handshake region has built up the bridge between the MD atoms and FE nodes Due to the limited 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SUMMARY Simulation is an effective method to study the mechanical properties of materials Three kinds of simulations of nano- indentation on copper were carried out: Molecular Dynamics simulation. .. Simulations 15 2.4 Hybrid Simulation of Molecular Dynamics Chapter and Finite Element 20 Molecular Dynamics Simulations 27 3.1 Potential Functions 27 3.2 Nano- indentation Simulation 31 3.2.1 Initialization... curves of four simulations 88 Figure 5.5 Comparison of Von Mises stress of three simulations 90-91 VI LIST OF TABLES Table 2.1 Properties of materials used in indentation cracking measurement of