Lattice Defects

Stacking faults are colored in red, partial dislocations and the surfaces are colored in gray.  The  nucleation of plasticity is homogenous. An amorphous blob of defective particles is nucleated in the region of the maximal stress. Later a stacking fault is nucleated and cross glide sets in.


View on the Whole Crystal

The formation of a screw dislocation can be seen on the top surface. Obviously it is not easy to observe the plastic deformation.

Plasticity under Nanoindenter in (111) Copper

Simulation result from a MD simulation on (111) Copper modelled with the EAM potential of Mishin. All particles in ideal lattice positions are omitted and the color code refers to the von Mises stress field.

Plasticity Under Spherical Indentation


Plasticity is the nonreversible reaction of a material under stress. Its physics is dominated by material transport (glide) and therefore, the stress-strain relation is not unique anymore. For an ideal single crystal, this means that the crystalline structure is disturbed locally and defects are generated. These defects can be classified in surface and volume defects. In experiments, surface defects are more directly accessible than volume defects. The latter can easily be investigated using atomistic simulations.
The following atomistic pictures have been obtained from a molecular-dynamics simulation of the indentation of a 8nm diameter sphere into a Cu (100) crystallite with a velocity of 12.8 m/s. The scales on the frames indicate the indentation time and the indentation depth.

Stress Distribution

We will consider an ideal spherical indenter which is moved down into a fcc substrate in (100) direction. This process will induce a stress field. In continuum theory the region of maximal stress is predicted to be under the indenter. Consistently plasticity will also set in there.
Predicted Stress Distribution [1]
As plasticity means that some particles will move and others not, an inhomogenous stress is essential. Therefore, ther shear stress is relevant. A scalar measure for the stress relevant for plasticity is the von Mises stress
It follows the symmetry of the crystalline structure.
Von Mises Stress Distribution (measured at the end of the elastic regime)

Onset of Plasticity

As discussed above, plasticity will set in not at the surface but in the volume of the substrate, where the theoretical maximal shear stress is exceeded.

Onset of Plasticity
After a primary defect cluster has been nucleated, partial dislocation loops propagate both into the volume and up to the surface. They can be observed as small cracks in the surface under the indenter. The inner of the partials has a hcp structure.
Propagating Partials
Crack under the Indenter

Gliding and Material Transport

Under the applied stress material transport will occur. This can be imagined similar to a carpet through which a fold is moving: the fold will propagate a piece of the carpet forwards. The same happens to the crystal.
Gliding of the Free Surface

Cross Glide and Prismatic Loops

Energy is not only transferred into the elastic deformation but also into plasticity. If a specific energy is reached cross glide occurs; it can be seen when the moving partials change their direction. This process ends in the nucleation of a prismatic dislocation loop.
Cross Glide
Prismatic Dislocation Loop
The Mises stress of this prismatic dislocation suggests that a prismatic loop transports encapsulated shear stress into the material.
Mises Stress of a Prismatic Dislocation Loop


  1. Schematic drawing after Fischer-Cripps, Nanoindentation (2004)

G. Ziegenhain – 2.11.2007

Atomistic Simulations of Nanoindentation

Introduction to Nanoindentation

Indentation experiments in which a hard tip is indented into a substrate allow research on basic material properties. Indentation in general is an important technique for determining the elastic properties of materials Tabor (1996,2000). Besides the reduced elastic modulus

$\displaystyle E_\mathrm{r}:=E/(1-\nu^2)$

Oliver and Pharr (1992); Hertz (1882); Chaudhri (2000) one can determine the contact hardness

$\displaystyle H:=F_\mathrm{ind}/A_\mathrm{contact}$

Fischer-Cripps (2004) of materials.

Figure 1: Force-Depth Curve – (100) Indentation into Copper
Image ForceDepth

Figure 2: Contact Hardness – (100) Indentation into Copper
Image ContactHardness

For small length scales Gane and Bowden (1968) the material properties differ from the macroscopic expectations. Particularly the force-depth curve has characteristic dips (the so-called pop-ins) at the elastic-plastic transition Göken and Kempf (2001). This and the overestimated theoretical shear stress$ \tau_\mathrm{th}\gg\tau_\mathrm{exp}$ Frenkel (1926) suggest that atomistic effects play an important role for the macroscopic length scale material behavior; namely these atomistic effects are dislocations Phillips (2001); Hull and Bacon (1992). The indented surface itself is also of interest: pile-up effects and regeneration are only two important topics.
Therefore an understanding of the atomistic plasticity is of great importance for material modeling on larger length scales and for understanding the material properties (hardness, elasticity) at all. Apart from that this the current length scale of electronic devices has already reached the nm-scale and atomistic effects itself are becoming important for industrial production.

Molecular Dynamic Simulations of Nanoindentation

The big advantage of simulations in contrast to the experiment is a total control of the system. Therefore simulations are predicted to research the atomistic effects. By using molecular dynamics (MD) Frenkel and Smit (1996); Plimpton and Ziegenhain (2006); Allen and Tildesley (2002) the onset of plasticity has been investigated in various systems Smith et al. (2003); Mulliah et al. (2003); Lilleodden et al. (2003); Christopher et al. (2001). The elastic properties are treated in detail in Lilleodden et al. (2003) and preliminary simulations for anisotropical effects have been done Tsuru and Shibutani (2007). Alternatively one could choose finite-element simulations for the modeling Durst et al. (2002,2004); these operate inherently on larger scales and are therefore not reasonable for atomistic length scales. Nevertheless it is promising to exploit the concurrent length scales of the physical system by coupling both simulation methods McGee et al. (2006). This strategy will not be pursued in the present project.In our simulations we model the indenter as an external constraint using the potential proposed in Kelchner et al. (1998):

Figure 3: Lattice Defects under (100)-Indentation into Copper

Figure 4: Lattice Defects and Mises Stress under (100)-Indentation into Copper


M. Allen and D. Tildesley.
Computer Simulation of Liquids.
Claredon Press Oxford, 2002.
M. Chaudhri.
A note on a common mistake in the analysis of nanoindentation data.
J. Mater. Res., 16 (2): 336-339, 2000.
D. Christopher, R. Smith, and A. Richter.
Nanoindentation of carbon materials.
NIM B, 180: 117-124, 2001.
K. Durst, M. Göken, and G. Pharr.
Finite element simulation of spherical indentation in the elastic-plastic transition.
Z Metallkd, 93 (9): 857, 2002.
K. Durst, M. Göken, and H. Vehoff.
Finite element study for nanoindentation measurements on two-phase materials.
J Mater Res, 19 (1): 85, 2004.
A. Fischer-Cripps.
Springer, 2004.
D. Frenkel and B. Smit.
Understanding Molecular Simulation – From Algorithms to Applications.
Academic Press, 1996.
J. Frenkel.
Zur Theorie der Elastizitätsgrenze und der Festigkeit kristalliner Körper.
Zeitschrift für Physik A Hadrons and Nuclei, 37: 572, 1926.
N. Gane and F. Bowden.
Microdeformation of solids.
J Appl Phys, 39 (3): 1432, 1968.
M. Göken and M. Kempf.
Pop-ins in nanoindentations – the initial yield point.
Zeitschrift fuer Metallkunde, 92 (9): 1061, 2001.
H. Hertz.
Über die Berührung fester elastischer Körper.
J. reine angewandte Mechanik, 92: 156-171, 1882.
D. Hull and D. Bacon.
Introduction to Dislocations.
Pergamon, 1992.
C. Kelchner, S. Plimpton, and J. Hamilton.
Dislocation nucleation and defect structure during surface indentation.
Phys. Rev. B, 58 (17): 11085, 1998.
E. Lilleodden, J. Zimmerman, S. Foiles, and W. Nix.
Atomistic simulations of elastic deformation and dislocation nucleation during nanoindentation.
J Mech Phys Solids, 51: 901-920, 2003.
E. McGee, R. Smith, and S. Kenny.
Multiscale modelling of nanoindentation.
Z Metallk, 2006.
D. Mulliah, D. Christopher, S. Kenny, and R. Smith.
Nanoscratching of silver (100) with a diamond tip.
NIM B, 202: 294-299, 2003.
W. Oliver and G. Pharr.
An improved technique for determining hardness and elastic modulus using loas and displacement sensing indentation experiments.
J Mater Res, 7 (6): 1564, 1992.
R. Phillips.
Crystals, Defects and Microstructure.
Cambridge, 2001.
S. Plimpton and G. Ziegenhain.
Lammps – large-scale atomic/molecular massively parallel simulator; with modified analysis., 2006.
R. Smith, D. Christopher, and S. Kenny.
Defect generation and pileup of atoms during nanoindentation of fe single crystals.
PRB, 67: 245405-1, 2003.
D. Tabor.
Indentation hardness: fifty years on – a personal view.
Phil Mag A, 74 (5): 1207-1212, 1996.
D. Tabor.
The Hardness of Metals.
Oxford, 2000.
T. Tsuru and Y. Shibutani.
Anisotropic effects in elastic and incipient plastic deformation under (001), (110) and (111) nanoindentation of al and cu.
PRB, 75: 035415, 2007.

G. Ziegenhain – 24.10.2007