Atomistic insights into metal hardening


For millennia, humans have exploited the natural property of metals to get stronger or harden when mechanically deformed. Ultimately rooted in the motion of dislocations, mechanisms of metal hardening have remained in the cross-hairs of physical metallurgists for over a century. Here, we performed atomistic simulations at the limits of supercomputing that are sufficiently large to be statistically representative of macroscopic crystal plasticity yet fully resolved to examine the origins of metal hardening at its most fundamental level of atomic motion. We demonstrate that the notorious staged (inflection) hardening of metals is a direct consequence of crystal rotation under uniaxial straining. At odds with widely divergent and contradictory views in the literature, we observe that basic mechanisms of dislocation behaviour are the same across all stages of metal hardening.

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Fig. 1: Stress–strain response of an single aluminium crystal subjected to tensile straining along seven different initial orientations of the straining axis.
Fig. 2: Slip crystallography of fcc single crystals.
Fig. 3: Stress–strain response, dislocation densities and Schmid factors in 12 slip systems of aluminium.

Data availability

All figures in the main text and Supplementary Information, as well as the data used to produce the figures, are available at In addition, large arrays of data (around 400 GB in total) used for producing plots and Supplementary Videos 1–5 have been retained and are available from the corresponding author on reasonable request.

Code availability

The open-source computer code LAMMPS used in this study is developed and maintained at the Sandia National Laboratories. LAMMPS is available at


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We acknowledge discussions with W. Cai, E. Tadmor and D. Karls and editorial suggestions from D. Bulatova. This work was funded by the NNSA ASC Program and Technische Universität Darmstadt and was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under contract number W-7405-Eng-48. Computing support came from the DOE INCITE programme and LLNL Computing Grand Challenge programme. The simulations were performed on Mira and Vulcan supercomputers at the Argonne Laboratory Computational Facility and Livermore Computing Facility, respectively. We dedicate this paper to the memory of A. Argon.

Author information




L.A.Z.-R. and R.F. ran atomistic simulations. L.A.Z.-R. produced three atomistic videos. A.S. developed methods for in silico computational microscopy and visualization. T.O. optimized run-time efficiency and data management of ultra-large-scale simulations. N.B. developed algorithms for initialization of the atomistic simulations. N.R.B. performed finite-element simulations and produced the dog bone video. V.V.B. developed the concept, planned the research and generated starting configurations for the MD simulations. All authors analysed the simulation results and wrote the paper.

Corresponding author

Correspondence to Vasily V. Bulatov.

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Supplementary information

Supplementary Information

Supplementary Discussion 1–5, Figs. 1–4 and Tables 1 and 2.

Supplementary Video 1

Simulated response of single-crystal aluminium to tensile straining along the x axis initially oriented along the [001] direction of the cubic lattice. Two animated sequences at the top depict continuous elongation of the simulated crystal along the x axis and simultaneous reduction of its lateral dimensions due to the Poisson effect. The sequence at the top left shows emergence and coarsening of slip traces on fictitious surfaces of the simulated crystal. The sequence at the top right shows dislocation motion and multiplication resulting in the development of a dense dislocation network. Shown at the bottom left is the stress–strain response of this crystal under tensile straining, which is distinctly parabolic (no inflection). At the bottom right is the orientation trajectory of the tensile axis expressed in a frame tied to the cubic lattice of the strained crystal. The axis remains within 2 angular degrees from its initial [001] orientation during straining. All four animated sequences are synchronized. Axis orientations of the Laboratory (specimen) frame are shown at the far left.

Supplementary Video 2

Simulated response of single-crystal aluminium to tensile straining along the x axis initially oriented along the [101] direction of the cubic lattice. The meaning of the four synchronized sequences is as explained in the caption to Supplementary Video 1. In this case an initially orthorhombic crystal not only elongates along its straining axis x, but also changes its shape and becomes distinctly triclinic. Although not immediately obvious from the changing orientations of the slip traces in the upper left sequence, axis rotation from its initial [101] orientation at the corner of the standard triangle towards [112] as well as axis overshoot, are both clearly shown at the bottom right. The stress–strain response shown at the bottom left is a typical staged (inflection) hardening.

Supplementary Video 3

Simulated response of single-crystal aluminium to tensile straining along the x axis initially oriented along the [213] direction of the cubic lattice. The meaning of the four synchronized sequences is as explained in the caption to Supplementary Video 1. As shown at the bottom right, the axis rotates along the triangle edge towards the [112] orientation. The stress–strain response shown at the bottom left shows a distinct inflection.

Supplementary Video 4

Deformation of a dog bone specimen subjected to uniaxial straining in tension. Left, evolution of the stress tensor in the dog bone under tensile straining. Colours other than white indicate a misalignment between the first principal axis of the stress tensor (corresponding to its maximum eigenvalue) and the tensile axis. Right, six components of the stress tensor averaged over the middle portion of the specimen. The position of the vertical bar along the strain axis is synchronized with specimen deformation shown on left. All stress components but σxx remain close to zero along the entire straining path.

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Zepeda-Ruiz, L.A., Stukowski, A., Oppelstrup, T. et al. Atomistic insights into metal hardening. Nat. Mater. (2020).

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