To understand the formation of roughness from plastic deformation, we used a simple model system: A periodic slab of material with two free surfaces that is biaxially compressed. In atomic scale models, the simplest realization is a pure single crystal, but we also studied a high-entropy alloy and a metallic glass to introduce different amounts of disorder. The atomic-scale calculations were carried out on systems in excess of 50 million atoms, with lateral dimensions on the order of 100 nm. Calculations of this size are at the limit of what is possible on present-day computers but were necessary to extract information on the statistical properties of the emerging roughness. These molecular simulations of biaxial compression showed the emergence of roughness for these three materials under various loading conditions. In particular, roughness emerged at atomically flat interfaces beyond the yield point of the material. The rough topography is imprinted at yield and is reinforced during subsequent deformation. Crucially, these simulations not only allow evaluation of the emerging surface topography, but they also allow a detailed inspection of the subsurface displacement field. We were able to show that the topography is nothing but a fingerprint of the structure of the larger subsurface deformation during flow of a solid body.
These atomic-scale simulations are like an experiment and do not intrinsically tell us why the specific self-affine structure of roughness emerged. However, they do allow insights into subsurface deformation fields that are difficult to obtain experimentally, and allowed us to connect this deformation to surface roughness. In order to gain further understanding about the origin of self-affinity, the proposal suggested mesoscale continuum models where the mechanism behind the individual plastic flow event can be controlled. Those continuum models require efficient solvers for large simulation domains, that we developed within this project. We carried out two-dimensional and three-dimensional calculations of this character. The marked difference between these is that in two dimensions, we see clear emergence of shear bands. The ensuing surface roughness does look self-affine, but with a Hurst exponent much smaller than 0.5. In contrast, the three-dimensional calculations do not emerge shear bands and the surface topography that emerges is not self-affine. The reason for the suppression of shear banding in three dimensions is that the percolation of deformation events is more difficult in three than in two dimensions.
These calculations allow a key insight into the formation of surface roughness: A discrete carrier of deformation (the shear band) is necessary for self-affinity to emerge. In the molecular calculations, this discrete carrier is either a dislocation (for crystals) or a shear transformation (for glasses). Conversely, a discrete microscopic event is intrinsically absent in most continuum models such as the one used by us. The deformation is assumed to be of a laminar nature, and this only leads to self-affine topographies and if the deformation self-organizes into shear bands. The fact that self-affine topographies are observed even on geological scales (the earth’s surface) could therefore also be related to the fact, that deformation of rocks also proceeds in shear bands - albeit at much larger (macroscopic) scales than the effects studied by us.
