Mixing instabilities during shearing of metals

Severe plastic deformation of solids is relevant to many materials processing techniques as well as tribological events such as wear. It results in microstructural refinement, redistribution of phases, and ultimately even mixing. However, mostly due to inability to experimentally capture the dynamics of deformation, the underlying physical mechanisms remain elusive. Here, we introduce a strategy that reveals details of morphological evolution upon shearing up to ultrahigh strains. Our experiments on metallic multilayers find that mechanically stronger layers either fold in a quasi-regular manner and subsequently evolve into periodic vortices, or delaminate into finer layers before mixing takes place. Numerical simulations performed by treating the phases as nonlinear viscous fluids reproduce the experimental findings and reveal the origin for emergence of a wealth of morphologies in deforming solids. They show that the same instability that causes kilometer-thick rock layers to fold on geological timescales is acting here at micrometer level.

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Supplementary Note 1: Feasibility of Kelvin-Helmholtz instability in solids
Our aim here is to evaluate the feasibility of Kelvin-Helmholtz (KH) instability in solids under our experimental conditions. Here we do the estimations for Cu, which is the phase with apparent KH instability features in Al/Cu system, but the calculation is similar for other phases. Since KH instability is caused by inertial forces, it requires the fluid to be turbulent. A flow becomes turbulent as its corresponding Reynolds number exceeds a critical value. The Reynolds number Re is defined as =8920 kg m -3 , and assuming an effective viscosity as low as η eff =10 4 Pa·s, implies that the material should flow at a velocity of v >500 km s -1 , to be in the turbulent regime. For comparison, the maximum shear velocity in our experimental setup is achieved at r = 5 mm, which for the rotational velocity of 1 rpm used in our experiments is equal to 2πr / 60 ≈ 500 µm s -1 . Hence, KH instability cannot be accounted as a viable mechanism for producing the vortices observed in Al/Cu multilayer.

Supplementary Note 2: Effective viscosity contrast and vortex formation
The numerical simulations suggest that the effective viscosity contrast between the weak and strong layers has an influence on whether vortices form or not. In a power-law viscous material, the effective viscosity depends on the local strain rate. Therefore, it cannot be directly estimated from the background strain rate. Instead, it is useful to consider a 1D-setup under simple shear with a single high viscosity layer of thickness h that is embedded in a lower viscous matrix in a system with overall thickness H ( Supplementary Fig. 6a).
In 1D, the governing equations are: whereas the boundary conditions are where ! γ BG is the applied background strain rate. From force balance it follows: If ! γ mat is the strain rate in the matrix and ! γ lay the strain rate in the layer, and η 0,mat , η 0,lay = VCη 0,mat the corresponding viscosity prefactors in the matrix and layer, respectively, it follows from force balance that The effective viscosity contrast between the layer and the matrix is η eff,lay η eff,mat = 2η 0,lay ! γ lay Simulations with a single layer show that for lower viscosity contrasts, vortex-like structures still form but in a kinematic manner, as a result of the applied simple shear (Supplementary Fig. 7). For a larger effective viscosity contrast, a folding instability occurs before developing to a larger vortex-like structure ( Supplementary Fig. 7).
Folding of a single layer with a power-law material embedded in a power-law matrix under pure shear is known to have the following dominant wavelength expression 3 : where H layer is the thickness of the layer, which is ~40 µm for the simulations shown in Supplementary Fig. 7. For the case that shows intermittent folding in this figure ( VC =5, n =3), we obtain λ dom =197 µm , which is in good agreement with the observed fold wavelength.