Study of Nanoindentation Using FEM Atomic Model

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1 Yeau-Ren Jeng Professor Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi, Taiwan Chung-Ming Tan Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi, Taiwan; Department of Mechanical Engineering, WuFeng Institute of Technology, Chia-Yi, Taiwan Study of Nanoindentation Using FEM Atomic Model This paper adopts an atomic-scale model based on the nonlinear finite element formulation to analyze the stress and strain induced in a very thin film during the nanoindentation process. The deformation evolution during the nanoindentation process is evaluated using the quasi-static method, thereby greatly reducing the required computation time. The finite element simulation results indicate that the microscopic plastic deformation in the thin film is caused by instability of its crystalline structure, and that the magnitude of the nanohardness varies with the maximum indentation depth and the geometry of the indenter. DOI: / Keywords: Nanoindentatoin, Finite Element Methods, Nanohardness 1 Introduction It is known that the mechanical properties of a material may vary locally if they are measured at the nanoscale. The mechanical properties adopted for thin films, e.g., hardness and Young s modulus, are generally those obtained from the bulk material since it is difficult to derive these values experimentally at the nanoscale. It is necessary, therefore, to develop techniques to determine the in situ mechanical behavior of nanoscale materials if science and technology at this scale is to progress. Nanoindentation provides a means of determining mechanical properties through the analysis of the load-depth curve obtained from indentation at nanoscale dimensions 1 7. Nowadays, there are a number of commercially available nanoindenters with the ability to measure loads and displacements with resolutions of better than 1N and 1nm, respectively 8. Harrison et al. 9 employed molecular dynamics simulations to investigate the indentation of a hydrogen-terminated diamond crystal and a nonhydrogen-terminated diamond crystal with a hydrogen-terminated diamond tip. Kallman et al. 10 used nonequilibrium molecular dynamics to simulate the elastic-plastic deformation of silicon indented by tetrahedral indenters. In a study that incorporated both molecular dynamics and finite element methods, Hoover et al. 11 applied massively parallel low-cost computers to the simulation of plane-strain elastic-plastic flow in the indentation of ductile metals. Belak et al. 12 utilized a molecular-dynamics approach to simulate the nanoscale deformation of metallic and ceramic surfaces under point indentation and nanocutting. Scagnetti et al. 13 considered a two-dimensional indentation problem via a molecular-dynamics simulation method that used an atomistic model to calculate the corresponding stresses and strains. Yan et al. 14 conducted a three-dimensional molecular dynamics analysis of atomic-scale indentation, and showed that the energy consumed by irreversible deformation always exceeds that caused by heating. Richer et al. 15 performed experimental and numerical investigations into the compressive behavior of various carbon materials under nanoindentation. From the literature review presented above, it is clear that a nanoindentation is generally investigated using moleculardynamics simulations. This technique is very time consuming since it adopts a high-resolution time step of at least one picosecond. Therefore, the present study adopts an alternative approach which is based on the fact that in condensed matters, both atoms and molecules oscillate thermodynamically around their minimum-energy positions. Under this approach, the changes in Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division November 24, 2003; revised manuscript received March 11, Review conducted by: J. Tichy Editor. the minimum-energy positions are calculated incrementally during the indentation process. The objective of the present study is to provide a qualitative description of the stress and strain distributions within the thin film during the nanoindentation process. The study also addresses the deformation mechanisms of the elasticplastic flow and determines the influence of indentation depth, film thickness, and film length upon the deformation behavior of the thin film and upon the magnitude of hardness. 2 Computer Simulation A Atomistic Simulation Model. Figure 1 illustrates the copper atom configuration of the thin film considered in the present investigation. Note that a two-dimensional model is selected in order to simplify visualization. The arrangement of atoms can be viewed as one of a family of close-packed planes, 111, in face-centered cubic fcc monocrystalline copper. As can be seen from the three-dimensional illustration of the copper crystal presented in Fig. 1(b), the profile of the thin film can be considered as lying along an oblique plane. In this illustration, the x axis represents one of the families of close-packed directions, i.e., 110. During the indentation process, a rigid sharp diamond tip is impressed into the thin film along the y-axis direction, i.e., in the direction of the thin-film thickness. It is assumed that the hardness of the diamond tip far exceeds that of the thin copper film, and hence deformation of the tip can be neglected during the indentation process. This assumption implies that there is no change in the relative positions of the indenter carbon atoms during the simulation process. Unlike the molecular dynamics simulation approach, the present method cannot employ a periodic boundary condition to simulate the infinite domain condition. Rather, the current simulation adopts boundary conditions in which the atoms at the extremities in the x direction and at the base in the y direction are fully constrained. Furthermore, the interatomic potential energy is assumed to be given by the sum of the pairwise empirical potentials, which depend only on the distance between the atoms. The current simulation employs the pairwise Morse potential to model the interatomic pairwise potential of the copper atoms in the thin film. This potential has the following form: r ij Dexp2r ij r 0 2 expr ij r 0 (1) where r ij is the distance between atoms i and j, and D,, and r 0 are constants which are determined from the physical properties of the corresponding material. As shown in Fig. 2, this potential model produces repulsive forces over a short range, attractive forces over a medium range, and then decays smoothly to zero over a long range. In this research, we take five times the equilibrium distance 0.25 nm be- Journal of Tribology Copyright 2004 by ASME OCTOBER 2004, Vol. 126 Õ 767

2 r ij A exp2r ij r 0 (2) The relevant C-Cu and Cu-Cu constants are indicated in Fig. 2. B Simulation Method. The nanoindentation process is simulated by increasing the displacements of the atoms in the indenter downward until the indenter penetrates the thin film to a specified depth. The indenter is then retracted to its original position. During the indentation process, the copper atoms in the film always move to their minimum-energy positions under equilibrium conditions. Hence, the nonlinear finite element formulation can be employed to establish a computationally efficient procedure to model the indentation process 16. In this procedure, two arbitrary atoms, i and j, are regarded as two nodes, and their potential is considered to be one element. It is assumed that atom i is located at position (x i, y i ) with displacements u i and v i in the x and y directions, respectively. By defining the nodal displacement vector for the i and j atoms as u Ij and the corresponding external nodal force vector as F ij ( f i,g i, f j,g j ) T, the total pairwise potential energy can be expressed as E ij r ij u T ij F ij (3) where the atomic distance r ij is given by r ij x i u i x j u j 2 y i v i y j v j 2 1/2 (4) The differential of the atomic distance with respect to u ij can be expressed as: dr ij x i u i x j u j,y i v i y j v j,x i u i x j u j,y i v i y j v j du ij Bdu ij (5) Fig. 1 a Atomistic model used in present nanoindentation simulation and b Three-dimensional illustration of monocrystalline fcc copper thin film tween two neighboring copper atoms as the cutoff radius. Since the indenter is considered to be a rigid diamond tip, the interatomic potential of the diamond tip can be ignored. The potential between the carbon atoms of the indenter and the copper atoms of the thin surface is modeled by the Born-Mayer potential. This potential yields only an impulsive force, and is given by Fig. 2 Interatomic potential energy functions used in present simulation, and corresponding parameter values The principle of minimum work enforces the minimization of E ij with respect to u ij such that, E ij u ij r ijb T F ij 0 (6) Equation 6 expresses the element equilibrium equation, which represents the equilibrium of the forces acting on atoms i and j. The residual force, ij, can then be defined as ij r ijb T F ij (7) When the equilibrium equation is solved using an iterative procedure, it converges to zero with a tolerance of In order to solve this nonlinear equilibrium equation in this way, it is necessary to differentiate ij with respect to u ij, i.e., where and d ij d r ijb T T B 2 2 r ij K ij r ijdb T dr ij r ijdb T K ij K ij du ij K T ij du ij (8) T K ij B 2 2 r ij B (9) r ij BT u i, BT v i, BT u j, BT u j (10) Equation 9 can be solved by substituting Eq. 5 into the second line of Eq. 8. Subsequently, the conventional finite element formulation assembly procedure can be employed to assemble Eq. 8 in order to obtain the total system equation, i.e., dk T du (11) 768 Õ Vol. 126, OCTOBER 2004 Transactions of the ASME

3 Similarly, Eq. 7 can be assembled to obtain the equilibrium equation of the total system, i.e., i j r ij B T F ij f int F ext 0 (12) In terms of the finite element formulation, Eq. 11 represents the tangent stiffness equation, while the terms f int and F ext in Eq. 12 denote the internal force vector and the external force vector, respectively. The present simulation adopts the Newton-Raphson iterative technique to solve Eq. 12 via the following displacement control scheme. First, it is assumed that the external force vector F retains a specified form during the iteration process, i.e., F i F i1 i Fˆ, i1,2,... (13) where Fˆ is the reference load vector. If Eq. 13 is substituted into Eq. 11, the iterative tangent stiffness equation becomes: K T du i i Fˆ d i, i1,2,... (14) The iterative displacement increment can be written in a similar form, i.e., du i i u a i du b i, i1,2,... (15) where: K T u i a Fˆ K T du i b d i, i1,2,... (16) The displacement control scheme is so called because the qth component of the incremental displacement vector is maintained as a constant during the iteration process, i.e., Fig. 3 a Load versus indentation depth curve of complete nanoindentation cycle, and b Configuration of deformed thin film after completion of nanoindentation cycle. i i i i u aq du bq du q du i q du q, i1 0, i1 (17) Fig. 4 Flooded contour subplots of hydrostatic stressõstrain and deviatoric stressõstrain evaluated at maximum indentation depth Journal of Tribology OCTOBER 2004, Vol. 126 Õ 769

4 Fig. 5 Flooded contour subplots of hydrostatic stressõstrain and deviatoric stressõstrain evaluated after completion of nanoindentation cycle The iterative solution strategy described above yields the complete equilibrium path of the nanoindentation. During the complex deformation process associated with the nanoindentation process, the atomic structure of the thin film experiences structural instabilities which cause the iterative scheme to diverge. To overcome this divergence, it is necessary to add a suitable constant to each diagonal element of the tangent-stiffness matrix in order that the nonpositive eigenvalues of the tangent stiffness matrix are shifted to a positive value. C Stress Calculation. In the atomistic simulation, the interatomic potential energy function determines the interactive forces between the atoms. Once the forces acting at each atom have been determined, the stress tensor at the atomic site can be determined 17 in the following form: N i km 1 V i ji f k ij r m ij (18) Fig. 6 Load versus indentation depth curves and configuration of deformed thin films after completion of nanoindentation cycle for six different maximum indentation depths 770 Õ Vol. 126, OCTOBER 2004 Transactions of the ASME

5 Table 1 Maximum loads, projected areas, and magnitudes of hardness calculated using two common hardness definitions for six different maximum indentation depths Indentation depth layer Max load nn The projected area of residual cavity (nm 2 ) The projected contact area at max.load (nm 2 ) Hardness Gpa Nanohardness Gpa where i refers to the atom in question, j refers to the neighboring ij atom, r m is the displacement vector from atom i to atom j, N is the number of nearest-neighboring atoms, and V i is the volume of the atom in question. D Strain Calculation. Although the displacements in the atomistic model can only be defined at the point of each atom, in the continuum model, the displacement can also be defined at a point between two atoms. This implies that the displacement at one point in the continuum model can be defined via some form of interpolation from the displacements of the surrounding atoms 18. Hence, in the continuum model, the displacement ū(r p )at point r p (x p,y p ) is defined by ūr p q wr p,r q ur q (19) where r p (x p,y p ) and u(r q ) are the position and displacement, respectively, of the qth atom in the atomistic model, and w(r p,r q ) is the weight function, which should satisfy q wr p,r q 1 wr p,r q wr q,r p (20) The weight function used in the present study has the following form: wr p,r q C pq exp r pr q (21) where the constant, C pq, is chosen so as to satisfy the conditions of Eq. 20, and the constant,, determines the range of averaging. By redefining the x and y components of the displacement of the qth atom by u(r q ) and v(r q ), respectively, the three strain components can be expressed as xr p x p wr p,r q ur q (22) q Fig. 7 a Load versus indentation depth curves, and b Deviatoric strains in thin film calculated at the maximum indentation depth for one can in which the constraint boundary condition at the base of the thin film is released, and in a second case where the thickness of the thin film is doubled Journal of Tribology OCTOBER 2004, Vol. 126 Õ 771

6 Fig. 8 Load versus indentation depth curves and deviatoric strains in thin film calculated after completion of nanoindentation cycle for one case in which the constraint boundary conditions at either end of the film are released, and in a second case where the length of the thin film is doubled xy r p x p q 3 Results and Discussion yr p y p q wr p,r q vr q (23) wr p,r q ur q y p q wr p,r q vr q (24) A Evolution of Deformation Under Nanoindentation The present study simulates the entire nanoindentation process using the finite element atomistic model. In this simulation, a sharp rigid diamond tip is initially positioned 4.4 Å above the surface of the thin copper film. The indenter is then moved downward incrementally to indent the thin film. Once the indenter has reached the specified indentation depth, it is retracted from the thin film and returned to its original position. At each incremental displacement of the indenter, the force equilibrium equations of the atoms as determined by their interatomic potentials are nonlinear functions of their incremental displacements. These equations are solved using the Newton-Raphson iterative technique. Generally, nanoindentation proceeds under quasi-static conditions. For a molecular simulation, the indenter usually penetrates the substrate in an unrealistic speed. Although this is difficult to represent in a molecular dynamics simulation due to its high timeresolution requirements, the present finite element approach is quasi-static, and hence the indenter can be considered to indent the thin film at a very low speed. Figure 3a indicates the relationship between the load, i.e., the force experienced by the indenter, and the indentation depth over the complete nanoindentation cycle. Meanwhile, Fig. 3b presents the configuration of the indented thin film. Note that for reasons of clarity, atoms within the same layer of the thin film prior to deformation are connected by a continuous line. As can be seen, slips orientated at approximately 60 to the horizontal direction are propagated from the region of the thin-film surface around the residual cavity. This is to be expected since slip generally occurs along the closed-packed directions. Furthermore, it is observed that the atoms in the region just below the apex of the residual cavity exhibit a disordered arrangement. These observations indicate that at least two plastic deformation mechanisms take place during the nanoindentation process. The slips occur as the result of the nucleation and propagation of dislocations, while the disordered atomic arrangement below the residual cavity is caused by the phenomenon of amorphization. By neglecting the thermal vibrations of the atoms, the present study transforms the molecular dynamics formulation into a static finite element structural problem. Therefore, the stability of the crystalline structure can be monitored by the nonpositiveness of the tangent stiffness matrix, K t, of the present finite element formulation. The irreversible plastic deformations observed in the simulation, i.e., slips and amorphization, can be considered to be the consequence of changes in the crystalline structure caused by instabilities induced by high localized stresses 19. From Fig. 3a, it can be concluded that localized instability of the thin-film crystal structure occurs initially in the simulation steps between Point A and Point B, i.e., because the rate of load increase in this region is seen to be somewhat less than in the previous indentation steps. A similar phenomenon is also observed in the steps between Point C and Point D, and between Point E and Point F. Since these structural instabilities cause the region of plastic deformation within the thin film to enlarge, there is no increase in load over these simulation steps. This phenomenon resembles that of the yielding of ductile metals in continuum mechanics. The simulation results also indicate the presence of strain hardening, i.e., the stresses induced in the thin film increase as yielding occurs. Equations can be used to calculate the stresses and strains induced in the thin film from the simulated positions of the atoms as the indenter reaches its maximum depth and is then retracted to its original position. The hydrostatic stress/strain and the deviatoric stress/ strain are presented in the contour subplots of Figs. 4 and 5 at the position of maximum indentation and after completion of the nanoindentation cycle, respectively. A comparison of the two sets of figures suggests that the compression of the thin film is almost fully recovered after the indentation load is removed. However, it is also noted that a large part of the distortion of the thin film appears to be permanent. Hence, the change of the crystalline structure caused by the high localized stresses tends to result in distortion of the thin film. Furthermore, it can be seen that the nanoindentation process induces high residual stresses, which by definition remain within the thin film once the indenting load is removed. B Parametric Study. Initially, the maximum indentation depth is varied in order to examine its influence on the magnitude 772 Õ Vol. 126, OCTOBER 2004 Transactions of the ASME

7 Fig. 9 a Load versus indentation depth curves for three different indenter angles, and corresponding deformed configuration of thin films after completion of nanoindentation cycle, b 75 deg, c 90 deg, and d 105 deg of nanohardness. The corresponding simulation results are presented in Fig. 6, which presents the load versus depth curves and corresponding residual cavities after unloading for six different maximum indentation depths. The hydrostatic stress is the first invariant of the stress tensor and represents the physical meaning of pressure. We determine the contact area by means of the hydrostatic stress that vanishes at the boundary on the surface of the contact area. Meanwhile, Table 1 provides the corresponding hardness and nanohardness results. Although an inspection of the results reveals that there is no obvious tendency in the correlation between the hardness and the indentation depth, it is clear that the maximum load, the area of the residual cavity, and the contact area at the peak load all depend strongly on the evolution of the deformation at the nanoscale dimensions considered in the present study. A frequently quoted rule for the indentation of thin films is that the maximum indentation depth should be less than 10% of the total film thickness if its hardness is to be measured independently of that of the substrate. To verify the validity of this rule, two further simulations are conducted, namely one in which the constraint of the atoms at the base of the thin film is removed, and another in which the film thickness is doubled. The corresponding simulation results are presented in Fig. 7. In the first simulation, the maximum indentation depth is specified to be one-quarter of the film thickness. Hence, doubling the film thickness causes the indentation depth to become just one-eighth of the film thickness. As can be seen in Fig. 7b, there is very little difference in the distortions of the film in the two cases. Furthermore, the loadversus-depth curves are very similar. Therefore, it may be concluded that the indentation rule is rather conservative. Two similar simulations are also conducted to determine the influence of the length of the film. In the first simulation, the constraint of the atoms at either extremity of the film length is removed, while in the second, the length of the film is doubled. Figure 8 presents the corresponding load-versus-depth curves and the distortion strains induced in the film at the maximum indentation depth. A comparison of the simulation results of Fig. 8 with those of the original simulation indicates that the length of the original thin film model is sufficiently large. These simulation results confirm that neither releasing the boundary conditions imposed at either end of the film, nor increasing the film length, has an influence upon the nanoindentation simulation results. To examine the influence of indenter bluntness, the complete nanoindentation cycle is simulated using indenters with three different apex angles. The corresponding load-versus-depth curves and residual configurations of the indented thin films are presented in Fig. 9. Meanwhile, the corresponding hardness and nanohardness values are provided in Table 2. These results indicate that the hardness at an indenter angle of 105 deg exceeds that of either the 75 deg or 90 deg case. But the effect of the indenter angle on the material hardness between indenter angles 90 deg and 75 deg is not obvious. This phenomenon can be explained by reference to the load-versus-depth curves of Fig. 9a, which reveal that for a constant indentation depth, the load for an indenter Table 2 Maximum loads, projected areas, and magnitudes of hardness calculated using two common hardness definitions for three different indenter angles Indenter angle deg Max load nn The projected area of residual cavity (nm 2 ) The projected contact area at max.load (nm 2 ) Hardness Gpa Nanohardness Gpa Journal of Tribology OCTOBER 2004, Vol. 126 Õ 773

8 apex angle of 105 exceeds that of smaller indenter apex angles of 75 deg and 90 deg as a result of the relatively larger contact area. However, an indenter angle of 105 deg induce less plastic deformation of the thin film during the indentation cycle than smaller indenter angles of 75 deg and 90 deg. 4 Conclusions The current simulation of the nanoindentation process for thin films adopts a nonlinear finite element formulation which ignores the thermal vibrations of atoms in the condensed matter, and hence enables a more computationally efficient approach for modeling the deformation evolution of the thin film than the conventional molecular dynamics simulation method. Simulations have also been conducted to examine the influence of maximum indentation depth, film thickness, and film length upon the deformation behavior of the thin film and upon the magnitude of its hardness. The major findings of the present study may be summarized as follows: 1. The current approach is less time consuming than the conventional molecular dynamics simulation method. 2. The current static finite element structural analysis indicates that plastic deformation of the thin film is the consequence of the instability of the crystalline structure of the film. Furthermore, the occurrence of this instability can be monitored by the nonpositiveness of the tangent stiffness matrix of the crystalline structure. 3. Two types of plastic deformation mechanism are evident in the present simulation results, namely slips and amorphization. Slips are caused by the nucleation and propagation of dislocations, while amorphization causes the arrangement of the atoms to become disordered. 4. The quantity of nanohardness varies with the maximum indentation depth and the geometry of the indenter, but there is no obvious correlation between them. 5. The frequently adopted rule of thumb that the maximum indentation depth should be less than 10% of the total film thickness has been shown to be rather conservative. Acknowledgments The authors gratefully acknowledge the support by AFOSR under Contract No. F P-0378 and the National Science Council of Taiwan under Grants No. NSC E and NSC M References 1 Adhihetty, I., Hay, J., Chen, W., and Padmanabhan, P., 1998, Thin Film Mechanical Properties Through Nano-Indentation, Mater. Res. Soc. Symp. Proc., 522, pp Lucas, B. N., Oliver, W. C., and Swindeman, J. E., 1998, The Dynamics of Frequency-Specific, Depth-Sensing Indentation Test, Mater. Res. Soc. Symp. Proc., 522, pp Hay, J. L., O Hern, M. E., and Oliver, W. C., 1998, The Importance of Contact Radius for Substrate-Independent Property Measurement of Thin Film, Mater. Res. Soc. Symp. Proc., 522, pp Hay, J. C., and Pharr, G. M., 1998, Experiment Investigation of the Sneddon Solution and an Improved Solution for the Analysis of Nanoindentation Data, Mater. Res. Soc. Symp. Proc., 522, pp Lugscheider, E., Barimani, C., and Lake, M., 1998, Mechanical Properties of TiC,N and TiN Thin Films on Cutting Tools Measured by Nanoindentation, Mater. Res. Soc. Symp. Proc., 522, pp Lu, W., and Komvopoulos, K., 2001, Nanotribological and Nanomechanical Properties of Ultrathin Amorphous Carbon Films Synthesized by Radio Frequency Sputtering, ASME J. Tribol., 123, pp Klapperich, C., Komvopoulos, K., and Pruitt, L., 2001, Nanomechanical Properties of Polymers Determined From Nanoindentation Experiments, ASME J. Tribol., 123, pp Bhushan, B., 1995, Handbook of Micro/Nano Tribology, CRC Press, Boca Raton, FL. 9 Harrison, J. A., White, C. T., Colton, R. J., and Brenner, D. W., 1992, Nanoscale Investigation of Indentation, Adhesion and Fracture of Diamond 111 Surfaces, Surf. Sci., 271, pp Kallman, J. S., Hoover, W. G., Hoover, C. G., Groot, A. J. De, Lee, S. M., and Wooten, F., 1993, Molecular Dynamics of Siilicon Indentation, Phys. Rev. B, 47, pp Hoover, W. G., Groot, A. J. De, and Hoover, C. G., 1992, Massively Parallel Computer Simulation of Plane-Strain Elastic-Plastic Flow via Nonequilibrium Molecular Dynamics and Lagrangian continuum Mechanics, Comput. Phys., 6, pp Belek, J., Boercker, D. B., and Stowers, I. F., 1993, Simulation of Nanometer-Scale Deformation of Metallic and Ceramic Surfaces, MRS Bull., 185, pp Scagnetti, P. A., Nagem, R. J., Sandri, G. V. H., and Bifano, T. G., 1996, Stress and Strain Analysis in Molecular Dynamics Simulation of Solids, ASME J. Appl. Mech., 63, pp Yan, W., and Komvopoulos, K., 1998, Three-Dimensional Molecular Dynamics Analysis of Atomic-Scale Indentation, ASME J. Tribol., 120, pp Richter, A., Ries, R., Smith, R., Henkel, M., and Wolf, B., 2000, Nanoindentation of Diamond, Graphite and Fullerene Films, Diamond Relat. Mater., 9, pp Inamura, T., Suzuki, H., and Takezawa, N., 1991, Cutting Experiments in a Computer Using Atomic Models of a Copper Crystal and a Diamond Tool, Int. J. Jpn. Soc. Precis. Eng., 25, pp Horstemeyer, M. F., Baskes, M. I., Godfrey, A., and Hughes, D. A., 2002, A Large Deformation Atomistic Study Examining Crystal Orientation Effects on the Stress-Strain Relationship, Int. J. Plast., 18, pp Inamura, T., Takezawa, N., and Kumaki, Y., 1993, Mechanics and Energy Dissipation in Nanoscale Cutting, CIRP Ann., 42, pp Jeng, Y-R., and Tan, C-M., 2004, Theoretical Study of Dislocation Emission Around a Nanoindentation Using a Static Atomistic Model, Phys. Rev. B, 69, pp Õ Vol. 126, OCTOBER 2004 Transactions of the ASME

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