Effects of Crystalline Anisotropy and Indenter Size on Nanoindentation by Multiscale Simulation
© The Author(s) 2009
Received: 20 October 2009
Accepted: 19 November 2009
Published: 27 November 2009
Nanoindentation processes in single crystal Ag thin film under different crystallographic orientations and various indenter widths are simulated by the quasicontinuum method. The nanoindentation deformation processes under influences of crystalline anisotropy and indenter size are investigated about hardness, load distribution, critical load for first dislocation emission and strain energy under the indenter. The simulation results are compared with previous experimental results and Rice-Thomson (R-T) dislocation model solution. It is shown that entirely different dislocation activities are presented under the effect of crystalline anisotropy during nanoindentation. The sharp load drops in the load–displacement curves are caused by the different dislocation activities. Both crystalline anisotropy and indenter size are found to have distinct effect on hardness, contact stress distribution, critical load for first dislocation emission and strain energy under the indenter. The above quantities are decreased at the indenter into Ag thin film along the crystal orientation with more favorable slip directions that easy trigger slip systems; whereas those will increase at the indenter into Ag thin film along the crystal orientation with less or without favorable slip directions that hard trigger slip systems. The results are shown to be in good agreement with experimental results and R-T dislocation model solution.
KeywordsMultiscale Quasicontinuum method Nanoindentation Anisotropy Size effect
Nanostructural materials have been the subject of intensive research in recent years due to its unique mechanical properties [1, 2]. Research has shown that these unique mechanical properties are closely related to internal structure of nanomaterials and deformation mechanism . Nanoindentation is a complicated contact problem [4–6], which has become a standard technique for evaluating the mechanical properties of thin film. During nanoindentation, it can be strongly influenced by crystalline anisotropy [7, 8] and indentation size effect (ISE) [9, 10], besides test equipment, surface roughness [11, 12], substrate effects , grain boundaries effects , pre-existing defects [15, 16] and indenter geometry [17, 18]. Groenou et al.  investigated slip patterns on a single crystal MnZn ferrite workpiece by spherical indentation on (100), (011) and (111) crystallographic planes. Khan et al.  studied the deformation behavior and hardness in indentation of a single crystal MgO using Vickers and spherical indenter, and showed that crystal orientation has a distinct influence on the indentation crack patterns. Recently, a number of experiments [21–23] have pointed out the well-known indentation size effect for metals, in which the hardness is observed to increase with decreasing indentation size.
However, it is difficult to directly examine the dislocation activities by experiments. One of the available methods is the atomic scale simulations to identify microscopic mechanisms, and to put insights into microscopic behavior such as molecular dynamics (MD) that can effectively simulate the dynamic behavior of nanomaterials. Due to the computational intensity of the problem, many of MD simulations are limited to relatively small spatial sizes or very high loading rates, and inevitably include boundary effects. For example, Tsuru and Shibutani  performed the atomistic simulation of nanoindentation via MD and investigated anisotropic effect in elastic and incipient plastic behavior under nanoindentation. However, atomic model studied in this simulation is only 20 nm thick and 18 nm wide, it is hard to exclude boundary effect and especially to study size effect. Multiscale atomic simulation is in general more flexible and more efficient than fully atomic simulations when dealing with complicated dislocation configurations in relatively large atomic systems. The quasicontinuum (QC) method [25, 26] is one of the successful multiscale atomic simulations, which has been capable of implementing large-scale simulations without losing calculation precision. Nowadays, this novel method has been successfully applied to investigate the dislocation nucleation and interactions [27, 28], nanoindentation [29–32] and fracture . Using the QC method, Wang et al. [34, 35] investigated the size effect of the indenter and the microscopic mechanisms of dislocation deformation during nanoindentation. Dupont and Sansoz  examined the mechanical behavior and underlying mechanisms of surface plasticity in nanocrystalline Al with a grain diameter of 7 nm deformed under wedge-like cylindrical contact. Recently, Jin et al.  studied the dislocation nucleation and interactions at the onset of plasticity for the nanoindentation of the (001) surface of a single crystal Al using the QC method. They found that the sharp load drops in the load–displacement curves are associated with deformation twinning and collective dislocation activities. However, Jin et al.  is not enough to deeply discuss collective dislocation activities due to the shallow indentation depth (the maximum indentation depth is only 1.6 nm).
The above discussions motivate us to understand the nature of crystalline anisotropy and indenter size on the onset of plasticity during nanoindentation without boundary effect. The QC method is employed to study the nanoindentation deformation processes under different crystalline orientation and various indenter size, respectively. The organization of the remainder of the paper is as follows: First, the choice of the nanoindentation orientations and the QC method adopted are briefly introduced in Sect. “Nanoindentation Simulation”. Subsequently, a series of different dislocation activities and underlying deformation mechanisms of nanoindentation under different crystallographic orientations on single crystal Ag thin film are analyzed in Sect. “Results and Analysis”. Then, the influences of crystalline anisotropy and indenter size on hardness, contact stress distribution, critical load for first dislocation emission and strain energy under the indenter are discussed, and the simulation results are compared with experimental results and Rice-Thomson (R-T) dislocation model solution in Sect. “Discussion”. Finally, some concluding remarks are made in Sect. “Conclusions”.
Description of the Nanoindentation Model
The QC method proposed by Tadmor et al.  in 1996 is a remarkably successful multiscale simulation method. This method is a molecular static technique finding the solution of equilibrium atomic configurations by energy minimization, given externally imposed forces or displacements. The problem is modeled without explicitly representing every atom in the cell; the regions of small deformation gradients are treated as a continuum media by the finite element method. The major reasons that the QC method is selected in this investigation are that: (1) It can describe larger-scale atomic systems in which <1% of the total atoms are treated explicitly ; (2) Compared with a single scale method, such as MD, the QC method is able to save considerable computing time while maintaining sufficient accuracy ; (3) The issues with artificial boundary conditions encountered in MD can be avoided ; (4) The automatic adaptation mesh criterion is adopted to reduce the degrees of freedom and computational demand without losing atomistic detail in regions where it is required . More details on the QC method can be found in Refs.  and .
Choice of the Nanoindentation Orientation
Coordinate systems are taken as X,Y and Z for surface nanoindentation. The indenter is pushed into the surface with the indenter sides parallel to the (111) slip plane of Ag thin film. Along this orientation, there are two favorable slip directions, namely, slip direction (5) and (12) as shown in Fig. 2a.
Coordinate systems are taken as X,Y and Z for (001) surface nanoindentation. The indenter is pushed into the (001) surface with a 35.7° angle between the indenter sides and the (111) slip plane of Ag thin film. Along this orientation, there are eight favorable slip directions, namely, slip direction (1), (2), (3), (4), (7), (8), (9) and (10) as shown in Fig. 2a.
Coordinate systems are taken as X ,Y and Z for (111) surface nanoindentation. The indenter is pushed into the (111) surface with the indenter sides perpendicular to the (111) slip plane of Ag thin film. Along this orientation, there are no favorable slip directions, as shown in Fig. 2a.
Results and Analysis
Nanoindentation on Surface
For the surface nanoindentation, the slip systems can be activated under a smaller external load and two parallel slip planes yield around the indenter tip. Subsequently, one can clearly observe the dislocation nucleation, emission and dissociation. The obvious discontinuity of load–displacement curve is the result of these dislocation activities.
Nanoindentation on (001) Surface
When the indentation depth reaches 1.2 nm, two new slip planes on the lower left-hand side of the wedge are activated and then glide along the and directions, respectively, as seen in Fig. 6c. Subsequently, a new dislocation lock is produced under the lower left-hand side of the indenter. In order to pass this new dislocation lock, the two slip planes compete and suppress each other. The slip plane on the lower left-hand side of the wedge passes by the dislocation lock again and extends to a further distance along the direction. Simultaneously, a second twinning plane is produced due to the appearance of cross-slip. Corresponding to the appearance of cross-slip, a third abrupt drop in the load–displacement curve is displayed. Figure 6d (corresponding to point H in Fig. 5) shows the von Mises distribution at the indentation depth of 1.80 nm. The severe lattice distortion is caused by the strong driving force of indenter. Furthermore, the orderly arrangement of atoms beneath the indenter is broken, which changed from order to disorder. As the lattice distortion becomes more severe, the number of distinct slip planes increases and these slip planes intertwine with one another. The glide directions are changed due to the severe lattice distortion, which no longer merely extend along the direction, but glide along the and directions by turns. In the meantime, both the deformation twinning and cross-slip are emerged. However, in Jin’s  numerical results, the cross-slip of dislocations was not be observed due to the shallow indentation depth. One can find that the nature of plastic deformation on (001) surface nanoindentation is collective dislocation activities, such as deformation twinning and cross-slip. The dislocation morphologies obtained by QC simulation are found to be quite similar to the experimental results , although the crystal material is not the same.
For the (001) surface nanoindentation, the slip systems can be activated under a very small external load. A dislocation lock is noted to form during the glide process. These slip planes in order to pass the dislocation lock compete with each other and eventually extend along the predominant slip direction. During the nanoindentation, one can clearly observe the dislocation lock, deformation twinning and cross-slip. The load drop in the load–displacement curve is the result of the complex collective dislocation activities.
Nanoindentation on (111) Surface
With the increase of indentation depth, the load–displacement curve experiences load drop several times, as the AB, CD and EF segments shown in Fig. 7, in which the amplitude of the first load drop is largest among these load drops in above three load–displacement curves. The atomic structure beneath the indenter at points B, D and F are plotted in Fig. 8b–d, respectively. It can be seen from Fig. 8b that the slip systems cannot be directly activated along this nanonindentation orientation without favorable slip directions. The crystal lattice beneath the indenter undergoes severe distortion and rotation under the action of external load. Subsequently, two new favorable slip directions along the  and the directions, respectively, are formed and then the glide starts. Figure 8c gives the atomic structure beneath the indenter at the indentation depth of 1.44 nm (corresponding to point D in Fig. 7). It is noted that if the dislocations glide along the direction are hampered, the slip direction will be reversed (from the to the  direction) that results in the release of strain energy and a second load drops at CD segment in Fig. 7. With the aggravation of lattice distortion, a lot of strain energy is stored in the lattice, but these strain energy cannot be released at once because the glides along ,  and  directions are fully hampered. Therefore, new slip planes along the direction will be produced (as shown in Fig. 8d) in order that the strain energy can be rapidly released, which corresponds to the third abrupt drop of the load–displacement curve at EF segment in Fig. 7.
For the (111) surface nanoindentation, the slip surfaces cannot be directly activated. The crystal lattice beneath the indenter at the beginning of nanoindentation undergoes severe distortion and rotation under the action of external load, then the favorable slip directions can be formed and the glide starts. A lot of strain energy is stored in the lattice due to the severe lattice distortion and rotation. The load–displacement curve represents abrupt drop when this strain energy is rapidly released.
A significant difference exists in the hardnesses for nanoindentation on various crystallographic surfaces, which can be attributed to the anisotropy of lattice orientation. The slip systems for the (001) and the surfaces nanoindentation can be activated under a smaller external load with favorable slip directions (as shown in Fig. 2), thus the calculated hardness values are lower. Whereas the slip systems for the (111) surface cannot be directly activated at the outset of nanoindentation without favorable slip directions (as shown in Fig. 2), the calculated hardness values are the highest because the crystal lattice beneath the indenter first undergoes severe distortion and forms the favorable slip directions and subsequently starts their glide under the action of intense external load.
Contact Stress Distribution Under the Indenter
In order to elucidate the mechanisms of dislocation nucleation and interaction, we adapt the criterion derived by Rice and Thomson [49, 50] for dislocation nucleation. It is assumed that dislocation nucleation occurs when the distance between the dislocation and indenter is equal to the dislocation core radius. There are two forces acting on this dislocation, in which one is the Peach-Koehler force (FL) due to the indenter stress field driving the dislocation into the bulk, and the other is the image force (FI) pulling the dislocation to the surface. The competition between the two forces imply that if the dislocation is too close to the surface, it will be attracted to the surface, while if the dislocation is at a sufficient depth, it will propagate into the solid.
where P is the force per unit length exerted by the indenter, ν is Poisson’s ratio of the thin film. Here, P in Eq. (4) can be obtained from the QC simulations and a can be determined from the nanoindentation models in Fig. 1. Using Eqs. (4) and (5), we can get the contact stress distribution under the indenters.
Critical Load for First Dislocation Emission
The critical load for first dislocation emission is an important parameter in nanoindentation, which not only reflects the intrinsic property of dislocation nucleation in the thin film, but also infers the elastic-to-plastic transition mechanism during nanoindentation.
Where , the parameter KR-T in the R-T dislocation model depends on the dislocation core radius rc. From Eq. (10), one can conclude that the critical load for first dislocation emission is in direct proportion with the square root of the indenter half-width.
The obvious discontinuity in load–displacement curves is closely related to the entirely different dislocation activities triggered by nanoindentation on various surfaces, which correspond to the dislocation nucleation and emission for nanoindentation on surface, deformation twinning and cross-slip for nanoindentation on (001) surface, and lattice distortion and rotation for nanoindentation on (111) surface. The results obtained by QC simulation show quite similar dislocation morphologies to the experimental results.
A distinct effect of crystalline anisotropy and indenter size are found on hardness. At the nanoindentation along the crystal orientation with more favorable slip directions, the corresponding hardness will decrease; whereas that will increase at the nanoindentation along the crystal orientation with less or without favorable slip directions. The calculated hardnesses of the thin film agree well with the experimental data.
The contact stress distribution under the indenter is very sensitive to indenter size and crystalline anisotropy. Along the orientation that the slip systems are easy to be activated, both the normal and shear contact stress are small; on the contrary, they are large along the orientation that the slip systems are hard to be activated.
Crystalline anisotropy and indenter size play an important role in measuring the critical load for first dislocation emission. At the same indenter size, the nanoindentation on (111) surface requires the largest critical load, while the nanoindentation on (001) surface requires the lowest. At the same crystallographic surface, the critical loads are increased as indenter widths are increased. In addition, they are in direct proportion to the square root of the indenters’ half-width, which is in good agreement with R-T dislocation model solution.
The strain energy of nanoindentation on (111) surface is the largest, while that of nanoindentation on (001) surface is the lowest. Although these strain energy curves for nanoindentation on various crystallographic surfaces exhibit difference, they show a similar increasing tendency. At the same crystallographic surface and indenter width, there is a corresponding relationship between load–displacement curve and strain energy.
This work was supported by the National Natural Science Foundation of China (Grant No. 10576010). The authors would like to thank Tadmor E.B. and Miller R. for helpful comments and suggestions during the multiscale simulations.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
- Siegel RW, Fougere GE: Nanostruct. Mater.. 1995, 6: 205. COI number [1:CAS:528:DyaK2MXmtleqs7c%3D] 10.1016/0965-9773(95)00044-5View ArticleGoogle Scholar
- Wang Y, Chen M, Zhou F, Ma E: Nature. 2002, 419: 912. COI number [1:CAS:528:DC%2BD38Xot1Klur8%3D]; Bibcode number [2002Natur.419..912W] 10.1038/nature01133View ArticleGoogle Scholar
- Valiev R: Nat. Mater.. 2004, 3: 511. COI number [1:CAS:528:DC%2BD2cXmtVOhs7k%3D]; Bibcode number [2004NatMa...3..511V] 10.1038/nmat1180View ArticleGoogle Scholar
- Oliver WC, Pharr GM: J. Mater. Res.. 1992, 7: 1564. COI number [1:CAS:528:DyaK38XktlWqtb4%3D]; Bibcode number [1992JMatR...7.1564O] 10.1557/JMR.1992.1564View ArticleGoogle Scholar
- Bhushan B, Koinkar VN: Appl. Phys. Lett.. 1994, 64: 1653. COI number [1:CAS:528:DyaK2cXivVKhtLw%3D]; Bibcode number [1994ApPhL..64.1653B] 10.1063/1.111949View ArticleGoogle Scholar
- Chen J, Bull SJ: Thin Solid Films. 2006, 494: 1. COI number [1:CAS:528:DC%2BD2MXht1aqsbnP]; Bibcode number [2006TSF...494....1C] 10.1016/j.tsf.2005.08.176View ArticleGoogle Scholar
- Viassak JJ, Nix WD: J. Mech. Phys. Solids. 1994, 42: 1223. Bibcode number [1994JMPSo..42.1223V] Bibcode number [1994JMPSo..42.1223V] 10.1016/0022-5096(94)90033-7View ArticleGoogle Scholar
- Kiely JD, Houston JE: Phys. Rev. B. 1998, 57: 12588. COI number [1:CAS:528:DyaK1cXjtFSmtbo%3D]; Bibcode number [1998PhRvB..5712588K] 10.1103/PhysRevB.57.12588View ArticleGoogle Scholar
- Kim JY, Kang SK, Greer JR, Kwon D: Acta Mater.. 2008, 56: 3338. COI number [1:CAS:528:DC%2BD1cXovFyhtL4%3D] 10.1016/j.actamat.2008.02.049View ArticleGoogle Scholar
- Shim S, Bei H, George EP, Pharr GM: Scr. Mater.. 2008, 59: 1095. COI number [1:CAS:528:DC%2BD1cXhtFSqs7rJ] 10.1016/j.scriptamat.2008.07.026View ArticleGoogle Scholar
- Zimmerman JA, Kelchner CL, Klein PA, Hamilton JC, Foiles SM: Phys. Rev. Lett.. 2001, 87: 165507. COI number [1:STN:280:DC%2BD3MnjslGnsQ%3D%3D]; Bibcode number [2001PhRvL..87p5507Z] 10.1103/PhysRevLett.87.165507View ArticleGoogle Scholar
- Shankar RM: Appl. Phys. Lett.. 2007, 90: 171924. Bibcode number [2007ApPhL..90R1924S] Bibcode number [2007ApPhL..90R1924S] 10.1063/1.2733027View ArticleGoogle Scholar
- Nowak R, Li CL, Maruno S: J. Mater. Res.. 1997, 12: 64. COI number [1:CAS:528:DyaK2sXisFyjsLo%3D]; Bibcode number [1997JMatR..12...64N] 10.1557/JMR.1997.0011View ArticleGoogle Scholar
- Soifer YM, Verdyan A, Kazakevich M, Rabkin E: Scr. Mater.. 2002, 47: 799. COI number [1:CAS:528:DC%2BD38XnvFOgs7g%3D] 10.1016/S1359-6462(02)00284-1View ArticleGoogle Scholar
- Jarausch KF, Kiely JD, Houston JE, Russell PE: J. Mater. Res.. 2000, 15: 1693. COI number [1:CAS:528:DC%2BD3cXlsFOltbo%3D]; Bibcode number [2000JMatR..15.1693J] 10.1557/JMR.2000.0244View ArticleGoogle Scholar
- Smith R, Christopher D, Kenny SD, Richter A, Wolf B: Phys. Rev. B. 2003, 67: 245405. Bibcode number [2003PhRvB..67x5405S] Bibcode number [2003PhRvB..67x5405S] 10.1103/PhysRevB.67.245405View ArticleGoogle Scholar
- Mirshams RA, Pothapragada RM: Acta Mater.. 2006, 54: 1123. COI number [1:CAS:528:DC%2BD28XnvVGjsQ%3D%3D] 10.1016/j.actamat.2005.10.048View ArticleGoogle Scholar
- Qin J, Huang Y, Hwang KC, Song J, Pharr GM: Acta Mater.. 2007, 55: 6127. COI number [1:CAS:528:DC%2BD2sXhtFGkt7fF] 10.1016/j.actamat.2007.07.016View ArticleGoogle Scholar
- Groenou V, Broese A, Kadijk SE: Acta Metall.. 1989, 37: 2613. 10.1016/0001-6160(89)90294-0View ArticleGoogle Scholar
- Khan MY, Brown LM, Chaudhri MM: J. Phys. D Appl. Phys.. 1992, 25: A257. COI number [1:CAS:528:DyaK38XhtFajs74%3D]; Bibcode number [1992JPhD...25A.257K] 10.1088/0022-3727/25/1A/039View ArticleGoogle Scholar
- Nix WD: Metall. Trans. A. 1989, 20A: 2217. COI number [1:CAS:528:DyaK3cXmtVOmuw%3D%3D]; Bibcode number [1989MTA....20.2217N]View ArticleGoogle Scholar
- Feng G, Nix WD: Scr. Mater.. 2004, 51: 599. COI number [1:CAS:528:DC%2BD2cXltlyhsro%3D] 10.1016/j.scriptamat.2004.05.034View ArticleGoogle Scholar
- Durst K, Backes B, Goken M: Scr. Mater.. 2005, 52: 1093. COI number [1:CAS:528:DC%2BD2MXisFaqtb0%3D] 10.1016/j.scriptamat.2005.02.009View ArticleGoogle Scholar
- Tsuru T, Shibutani Y: Phys. Rev. B. 2007, 75: 035415. Bibcode number [2007PhRvB..75c5415T] Bibcode number [2007PhRvB..75c5415T] 10.1103/PhysRevB.75.035415View ArticleGoogle Scholar
- Tadmor EB, Ortiz M, Phillips R: Phys. Mag. A. 1996, 73: 1529. Bibcode number [1996PMagA..73.1529T] Bibcode number [1996PMagA..73.1529T] 10.1080/01418619608243000View ArticleGoogle Scholar
- Shenoy VB, Miller R, Tadmor EB, Rodney D, Phillips R, Ortiz M: J. Mech. Phys. Solids. 1999, 47: 611. Bibcode number [1999JMPSo..47..611S] Bibcode number [1999JMPSo..47..611S] 10.1016/S0022-5096(98)00051-9View ArticleGoogle Scholar
- Shenoy VB, Phillips R, Tadmor EB: J. Mech. Phys. Solids. 2000, 48: 649. COI number [1:CAS:528:DC%2BD3cXisFSntL4%3D]; Bibcode number [2000JMPSo..48..649S] 10.1016/S0022-5096(99)00055-1View ArticleGoogle Scholar
- Ni YS, Wang HT: Chin. Q. Mech.. 2005, 26: 366. [in Chinese] [in Chinese]Google Scholar
- Tadmor EB, Miller R, Phillips R: J. Mater. Res.. 1999, 14: 2233. COI number [1:CAS:528:DyaK1MXjs1yltb0%3D]; Bibcode number [1999JMatR..14.2233T] 10.1557/JMR.1999.0300View ArticleGoogle Scholar
- Li JW, Jiang WG: Acta Metall. Sin.. 2007, 43: 851. COI number [1:CAS:528:DC%2BD2sXhtFCns7fM] [in Chinese]Google Scholar
- Jiang WG, Li JW, Su JJ, Tang JL: Acta Mech. Solida Sin.. 2007, 28: 375. COI number [1:CAS:528:DC%2BD2sXjs1WrsLY%3D] [in Chinese]Google Scholar
- Ni YS, Wang HT: Chin. J. Mech. Eng.. 2007, 43: 101. [in Chinese] [in Chinese] 10.3901/JME.2007.08.101View ArticleGoogle Scholar
- Miller R, Ortiz M, Phillips R, Shenoy V, Tadmor EB: Eng. Fract. Mech.. 1998, 61: 427. 10.1016/S0013-7944(98)00047-2View ArticleGoogle Scholar
- Wang HT, Qin ZD, Ni YS, Zhang W: Trans. Nonferr. Met. Soc. Chin.. 2008, 18: 1164. COI number [1:CAS:528:DC%2BD1cXht1KqurnN] 10.1016/S1003-6326(08)60199-5View ArticleGoogle Scholar
- Wang HT, Qin ZD, Ni YS, Zhang W: Acta Phys. Sin.. 2009, 58: 1057. COI number [1:CAS:528:DC%2BD1MXltV2qt70%3D] [in Chinese]Google Scholar
- Dupont V, Sansoz F: Acta Mater.. 2008, 56: 6013. COI number [1:CAS:528:DC%2BD1cXhsVSqtL3M] 10.1016/j.actamat.2008.08.014View ArticleGoogle Scholar
- Jin J, Shevlin SA, Guo ZX: Acta Mater.. 2008, 56: 4358. COI number [1:CAS:528:DC%2BD1cXhtVCqsbfM] 10.1016/j.actamat.2008.04.064View ArticleGoogle Scholar
- Prudhomme S, Bauman PT, Oden JT: Int. J. Multiscale Comput. Eng.. 2006, 4: 647. COI number [1:CAS:528:DC%2BD2sXmsVGmtLw%3D] 10.1615/IntJMultCompEng.v4.i5-6.60View ArticleGoogle Scholar
- Heino P, Hakkinen H, Kaski K: Europhys. Let.. 1998, 41: 273. COI number [1:CAS:528:DyaK1cXpvFCqsg%3D%3D]; Bibcode number [1998EL.....41..273H] 10.1209/epl/i1998-00142-5View ArticleGoogle Scholar
- Voter AF: The embedded atom method. In Intermetallic Compounds: Principles. Edited by: Westbrook JH, Fleischer RL. Wiley, New York; 1994.Google Scholar
- Liu Y, Wang B, Yoshino M, Roy S, Lu H, Komanduri R: J. Mech. Phys. Solids. 2005, 53: 2718. COI number [1:CAS:528:DC%2BD2MXhtVKqtbbM]; Bibcode number [2005JMPSo..53.2718L] 10.1016/j.jmps.2005.07.003View ArticleGoogle Scholar
- Chen P, Shen Y: Int. J. Solids Struct.. 2008, 45: 6001. COI number [1:CAS:528:DC%2BD1cXhsFWgsLjM] 10.1016/j.ijsolstr.2008.07.008View ArticleGoogle Scholar
- Hirth JP, Lothe J: Theory of Dislocations. Wiley, New York; 1982.Google Scholar
- Van Vliet KJ, Li J, Zhu T, Yip S, Suresh S: Phys. Rev. B. 2003, 67: 104105. Bibcode number [2003PhRvB..67j4105V] Bibcode number [2003PhRvB..67j4105V] 10.1103/PhysRevB.67.104105View ArticleGoogle Scholar
- Gouldstone A, Koh HJ, Zeng KY, Giannakopoulos AE, Suresh S: Acta Mater.. 2000, 48: 2277. COI number [1:CAS:528:DC%2BD3cXjt1Sisbk%3D] 10.1016/S1359-6454(00)00009-4View ArticleGoogle Scholar
- Yu HH, Shrotriya P, Gao YF, Kim KS: J. Mech. Phys. Solids. 2007, 55: 489. COI number [1:CAS:528:DC%2BD2sXosFGnuw%3D%3D]; Bibcode number [2007JMPSo..55..489Y] 10.1016/j.jmps.2006.09.003View ArticleGoogle Scholar
- Abu Al-Rub RK, Voyiadjis GZ: Int. J. Plast.. 2004, 20: 1139. 10.1016/j.ijplas.2003.10.007View ArticleGoogle Scholar
- Ma Q, Clarke DR: J. Mater. Res.. 1995, 10: 853. COI number [1:CAS:528:DyaK2MXks1ehtbk%3D]; Bibcode number [1995JMatR..10..853M] 10.1557/JMR.1995.0853View ArticleGoogle Scholar
- Rice JR, Thomson R: Phil. Mag.. 1974, 29: 73. COI number [1:CAS:528:DyaE2cXhsV2mur4%3D]; Bibcode number [1974PMag...29...73R] 10.1080/14786437408213555View ArticleGoogle Scholar
- Rice JR: J. Mech. Phys. Solids. 1992, 40: 239. COI number [1:CAS:528:DyaK3sXksFeqsrY%3D]; Bibcode number [1992JMPSo..40..239R] 10.1016/S0022-5096(05)80012-2View ArticleGoogle Scholar
- Johnson KL: Contact Mechanics. Cambridge University Press, Cambridge; 1985.View ArticleGoogle Scholar