‘Turbulent’ shear flow of solids under high-pressure torsion

ABSTRACT

The term ‘solid-state turbulence’ may sound like an oxymoron, but in fact it is not. In this article, we demonstrate that ‘solid-state turbulence’ may emerge owing to a defining property of the solid state: the ability of a solid to retain its shape. We consider shear flow under high-pressure torsion of layers of solids with different flow stress and show that the stiffer ones may spontaneously decompose into a set of blocks. This effect is fundamental for the occurrence of ‘solid-state turbulence’ (SST). To visualize SST, we use a heuristic model based on discretization of a continuum into interacting ‘particles’. The outcomes of the numerical experiments conducted support the occurrence of pulsations of velocity and pressure in plastically deforming solids and the emergence of vortices characteristic of classical turbulence. This phenomenon may have important practical implications for solid-state mixing as an ecologically beneficial alternative to conventional metallurgical processing routes.

KEYWORDS:

• High-pressure torsion
• laminate
• shear flows
• mixing
• turbulence
• moveable cellular automata

1. Introduction

It is well-known that under certain conditions laminar flow of gases and fluids moving in an orderly manner becomes irregular, their velocity fields becoming chaotic and pulsatile. This is referred to as turbulence [1]. It is almost a platitude to say that turbulence is one of the greatest enigmas of Nature, but this is, indeed, an apt statement [2].

Turbulence is not restricted to gases and fluids, though, as some of its attributes are found for deforming solids. In 1953, Cottrell used the term ‘turbulence’ to describe a transition from the ‘laminar’ dislocation motion on a single slip system in stage I of strain hardening of metals to multiple slip when several slip systems get activated in stage II [3]. Irregularity of plastic flow of metals at micro and meso length scales associated with movement of crystal lattice defects was reported by many researchers, cf., e.g. [4–8]

As distinct from this view of plastic deformation at small length scale, traditional plasticity theory going back to the classical work by Tresca, cf. [9], considers macroscopic flow of metals as laminar. This has been convincingly supported by various technological processes involved in metal forming, such as rolling, extrusion, drawing, stamping, etc. However, recent studies of metal laminates and powders processed by high-pressure torsion (HPT) produced evidence of irregular, pulsatile flow of metals at macroscopic length scale. These processes involve shear flows of two or more metals giving rise to vortices at their interfaces, which lead to random oscillations of the flow velocity and vigorous mixing of the metals, cf., e.g. [10–13].

The subject of this article is shear flows of laminates leading to mixing-type motions [14], when closely spaced particles of a material depart from each other by large distances as a result of the motion. The topology of the flux is changed: as seen in Ref. [15], shear under pressure turns a singly connected solid layer of a laminate to a multiply connected one. This results in stirring of the constituents of the laminate. The described character of the flux is one of the five characteristic attributes of turbulence [16]. That is why we choose to refer to the phenomenon observed as ‘solid-state turbulence’ (SST).

Investigations of SST as a stochastic nonlinear phenomenon are of great scientific significance, as it has certain universality and can occur at different length scales: from laboratory experiments to processes in the Earth’s lithosphere [17]. The phenomenon is also of practical importance, as SST opens an avenue for solid-state mixing and physicochemical reactions under high pressure [18], whose potentialities in materials engineering cannot be overestimated [10–12]. Indeed, SST offers an ecologically advantageous alternative to conventional metallurgical processing.

Current literature contains numerous publications about materials obtained by HPT of laminates, see comprehensive reviews [11, 12], yet the SST phenomenon is practically unexplored. Just a handful of articles address the mechanisms of the loss of stability and fracture of layers [15, 19], computational modelling of plastic flow [13, 20], and a quantitative analysis of the quality of mixing of the components of a laminate [21]. Although the possibility of the occurrence of SST may be evident from these reports, it is fair to say that the scientific community is largely unaware of the existence of the phenomenon. At any rate, the mechanisms of SST are not understood, and it appears timely to address them. In this article, we propose and corroborate a hypothesis on the nature of SST and investigate some salient features of this phenomenon. To that end, a computational model was developed and applied.

A mathematical description of mixing-type motion cannot be based on a traditional continuum approach, which presumes smoothness of the velocity field [15]. In such a case theoretical studies resort to discretization of a continuum, cf., e.g. [22], or generalized continuum descriptions, as in Ref. [7]. In the present communication we employ a computational model of SST based on a particular version of the discretization approach.

2. The model of solid flow

We start by outlining qualitative considerations and on that basis put forward a hypothesis regarding the physical cause of SST and then establish requirements on an adequate computational model. We showed earlier [15] that for simple shear the shear rate in the bulk of the laminate is inverse in the flow stress. A corollary is that the harder layers inhibit simple shear relative to the softer ones. According to Ref. [23], this will lead to the occurrence of rotational modes in a laminar flux, which will distort it. In our view, this is the root cause of SST.

Our aim was to employ a model of plastic flow in solids that would be robust and simple, yet capable of capturing the above principal factor. We are obviously dealing here with a particular case of a general problem: how do meso- and macroscopic properties (including the elastic and plastic ones) and the associated geometrical patterns derive from dynamic processes described at a deeper theoretical level? Suitable modelling tools are provided by discretization of a continuum and tracing the dynamics of ‘particles’ representing it. For example, the model proposed in Ref. [24] in terms of the dynamics of discrete ‘particles’ yields a broad range of mass and energy transport regimes – from the solid to the gaseous state – for the simplest case of a repulsive interaction potential between the ‘particles’. This approach provides control of the correlation radius between the elements of an effective continuum. As shown below, the flow stress of a model material represented by a set of ‘particles’ can be governed by tuning the correlation radius. To avoid confusion, it should be stated from the outset that the use of the notion of ‘particles’ here and below does not imply that individual atoms are considered. Recently, а significant progress with understanding structural lubricity in soft and hard matter systems was achieved by using a similar modelling approach [25]. At present, it is still not possible to advance to the satisfactory length and time scales by using classical molecular dynamics that operates with atoms or molecules. That is why modelling approaches in which representative larger-scale objects whose motion in its entirety reflects the pertinent aspects of the behaviour of the system have come to the fore. These include, for example, the method of moveable cellular automata [26], smoothed particle hydrodynamics [27], and the discrete automata method [28, 29]. Not only do these approaches show a qualitative accord between modelling and experiment, but they also enable a good quantitative agreement for various systems considered. However, they are computationally costly. Besides, visualization and processing of the results of the computations are very involved [30].

Our aim in the present work was primarily the extraction of qualitative information from the numerical simulations, with a convenient access to direct visualization for various scenarios and over large periods of time. As a method of choice, we used the isotropic ‘moveable automata’ technique (cf., e.g. [31]), which involves a weak long-range attractive interaction between particles in addition to a short-range repulsive one. The form of the interparticle interactions used ensures the existence of a minimum of the overall potential and the attendant equilibrium state. Such a system does not require ‘walls’ created by the boundary conditions to form compact fragments of a crystalline lattice. Even for the case when the average density of the material is lower than that required for contiguous space filling, the particles show a tendency to aggregation. The freedom of the solid to deform plastically is still retained, as the lattice is subdivided in fragments capable of moving over distances far greater than the lattice constant. In doing so they can rotate and deform elastically nearly as a whole, or fuse with other fragments that may have been distant initially. The ability to rupture enables the lattice to engulf particles of other substances that were separated from it. Naturally, the combinations of attraction and repulsion are different for the sub-systems of a two-component or, generally, multicomponent system, as well as between the subsystems. It can be shown that a strong mutual repulsion promotes separation of subsystems, whereas an attraction enhances the tendency to mixing. We used this observation in connection with modelling of the behaviour of materials with different stiffness that show some (but not exceedingly strong) proclivity for separation. The details of the model used are given in the Supplementary Information.

In solving the equations of motion, Supplementary Information Equation (S3), we confined ourselves to problems where the thermal energy $k_{B}T$, where T is the absolute temperature and k_B is the Boltzmann constant, is negligibly small compared to other energy terms involved. It can be proven [24], that in the limit of high kinetic energy density, E_kin >> C_ij, where C_ij is a stiffness parameter associated with the magnitude of the attractive interaction between particles i and j, the system described by Supplementary Information Equations (S1) and (S2) behaves like a gas of nearly free-flying particles. In the opposite limit case, E_kin << C_ij, a crystal lattice is formed spontaneously. The lattice constant $a$ is determined by the equilibrium distance between the particles. The nodes of the lattice oscillate around the bottoms of the potential valleys formed by the neighbouring particles. This oscillatory motion can be described in terms of a harmonic Hamiltonian slightly perturbed by non-linear terms. In this sense, the system deforms elastically. While in the gaseous state the system is isotropic on average, in the solid state its lattice has a hexagonal symmetry if the interaction potential between the particles is isotropic. In both limit cases, the system exhibits a regular dynamic behaviour. However, due to a difference in symmetry, there is a transition between the two states via a mixed disordered state, which, in the presence of dissipation, represents a viscous fluid. Thus, the minimalist model considered accounts for the occurrence of all three aggregate states of the system.

3. General features of solid-state flow as revealed by computational experiments

Based on the model of solid-state flow outlined above and detailed in Supplementary Information S1, we conducted computational experiments and visualized their results. As the system consists of discrete moveable particles, the following computational procedure was used for our numerical experiments. A certain number of particles were placed on a rectangle with the sides $\left[0,L_x\right]$ and $\left[0,L_y\right]$ at random at near-zero temperature. Due to their interactions described in Section 2, the particles formed a near-equilibrium structure consisting of a set of differently oriented domains. The system was then engaged in shear flow by the horizontal movement of the upper and lower plates in the opposite directions. For laminar flow, the velocity distribution along the $y$-axis of particles trying to follow the plates would become a smooth monotonic function. This is not possible, however, if the solid is sub-divided into domains. It will first deform elastically, the lattice nodes following the displacement of the plates smoothly. This is reflected in a smooth distribution of the horizontal velocity over the ordinate axis, $v_x\left(y\right)$. Once a critical shear strain is exceeded, the bonds between some of the nearest neighbours start breaking. Other particles now become neighbours. Although formally all particles of the system interact with each other, in reality a particle is enslaved by its immediate neighbourhood and tends to follow it. This means that locally a fragment of the lattice moves as a whole swapping crystallographic planes only at contacts between domains.

A typical picture of the particle movement of the kind described revealed by computer experiments is visualized in the video clip Supplementary Movie_01.avi. The red and blue circles in the spatial distribution map correspond to positive and negative projections of the velocity on the abscissa, respectively. Contrast spots and bands give a clear picture of instantaneous velocity bursts in proximity to the lines separating the regions where movement of the particles occurs with nearly the same velocity. These are the lines where switching of nodes between crystallographic planes occurs. These are also the lines with the greatest level of energy dissipation.

In addition to the velocity distributions, it is also instructive to use the Voronoi and Delaunay diagrams. A Voronoi diagram for a set of points (particles) on a plane is a tessellation of the plane into regions where each point is closer to all points within that region than to any point of the set outside of it. A Delaunay diagram is a triangulation of the same set of points such that for any triangle all points, except those at its apexes, are situated outside of its circumcircle. The Voronoi diagram permits identification of the symmetry of the neighbourhood of any point, while Delaunay triangulation, which is dual to the Voronoi mesh, enables visual construction of a lattice at any step of a numerical experiment. Figure 1 displays a typical instantaneous snapshot of a system represented by the Voronoi and Delaunay constructions. One can readily recognize a subdivision of the system into domains and chains of defects with fivefold and sevenfold symmetry axes located at domain boundaries (coloured blue and orange, respectively). A blow-up in the insert presents an enlarged view of the Delaunay lattice where some of the defects are highlighted by pentagons for clarity.

Figure 1. A typical snapshot of a ‘crystalline’ system of particles by means of the Voronoi and the Delaunay diagrams. The enlarged picture in the insert facilitates the visualization of the constructions described in the text.

It should be noted that such geometrical constructions are helpful both for qualitative presentation and for quantitative evaluation of the numerical simulation results, e.g. in the form of uniquely determined histograms for particle distributions.

Calculations show (see Supplementary Information S2) that under shear of a collective of particles the spatial distribution of the scattering power (Figure S1) is highly nonuniform, and that the external force required for the shear undergoes irregular oscillations (Figure S2).

4. Solid-state turbulence as revealed by computational experiments

Building upon the knowledge of the basic properties of the model discussed above, we can now turn to considering turbulence in a composite system comprised by materials with different stiffness. We construct the initial system as a layer of a stiff material sandwiched between two layers of a more compliant material. The latter are set in motion by two rigid plates shifted horizontally in the opposite directions with the same constant velocity, $V=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$ Figure 2 shows the results of numerical simulations.

Figure 2. ‘Turbulent’ flow of a layer of a stiff material sandwiched between two layers of a softer, more compliant material. Shear deformation is imposed by the movement of the upper confining plate to the left, and he lower one to the right. (a) and (c) show the morphology of the stiff material for shear strains equal to 5 and 10 (in relative units), respectively. The width of each layer is taken to be unity. In these units, the length of the zone considered is 10. The particles comprising the stiff and the pliant materials are represented by large and small circles, respectively. The magnitude of the velocity along the horizontal axis is visualized in (a) and (c) using the colours of the standard ‘jet’ colour code, in which red corresponds to the largest and blue to the smallest magnitude of a variable. The ‘comet tails’ in the subplots (b) and (d) reproduce short traces of the individual trajectories of particles preceding the moment at which the shown snapshot was taken. Different colours in these two subplots are used just for better visual separation of the individual trajectories and do not represent any numerical characteristic of the particle motion.

As could be expected from preliminary numerical experiments, the outer layers transmit the motion into the bulk of the sandwich relatively quickly, the stiff inner layer attaining a velocity of a substantial magnitude. The corresponding variation of the velocity with the depth is smooth and almost linear in the pliable outer layers and abrupt in the hard inner layer, Figure 2. An animation, Supplementary Movie_02.avi, clearly illustrates that at this initial stage the movement of the external plates is transmitted only to the more compliant sub-system, the stiff one being practically indifferent to this movement. Since the sandwich structure was created naturally, through spontaneous ordering of the layers of two different kinds before shear deformation, the interfaces between the layers exhibit some roughness. A phase separation between the layers is furnished by mutual repulsion between the particles of the two kinds considered to be the same as that between the particles of the same kind, combined with a smaller (or even non-existent) attraction. The smoothness of an interface and the separation distance between the layers of different kinds (or the opposite – the interdiffusion of the two particle species) are controlled by the magnitude of the attractive interaction.

To be specific, here we confine ourselves to an intermediate case when a tendency to phase separation does exist but is not excessive. Such a case is shown in Figure 2(c). Domains of different orientation are easily recognized. One can discern differently oriented domains within the stiff material and ordered (albeit more loosely) domains of the more compliant material lining the boundaries of the stiff layer without penetrating them.

A qualitative analysis of the deformation process at a later stage characterized by rupturing of the inner layer appears to be a good representation of the reality. The development of the process with time leads to deformation of the inner layer and the attendant smoothening of the distribution profiles of force and velocity integrated over the horizontal axis. Stress (or force) concentrations, as well as the associated waves of velocity, in proximity to the nascent discontinuities visualized in colour are readily recognisable.

Details of the entire process can be observed in the video Supplementary Movie_02.avi.

Smoothening of the extrema in the distributions of the force and especially velocity (where they disappeared altogether) with time is obvious. A characteristic velocity distribution within the islands indicates their practically independent solid-body rotation within the overall laminar flow. Computer experiments showed that, all other conditions being the same, the average block size grows with an increase in the correlation radius, as does the yield stress. The growth of the stress (or force) with C₁₁ is illustrated in Figure S3.

It should be noted that the volume of the layers gets slightly increased upon shear deformation, which is a result of the emergence of defects in particle packing. To explore the influence of this effect, we conducted numerical experiments with constant distance between the plates and with varying distance. No substantial difference between the results was found. All numerical experiments reported in the article were conducted for fixed distance between the plates.

It should be emphasized that solid-state turbulence promotes mixing of the two materials of which the sandwich structure is composed.

5. Discussion

The results of the numerical experiments conducted to model SST allow us to draw some general conclusions and establish the salient properties of free shear turbulence in solids.

As shown in Sections 3 and 4, long-range order in the system gives rise to a blocky structure of both materials of a ‘sandwich’. In the shear flow, the blocks obstruct the shear, which leads to the occurrence of SST.

What features of a real experiment correspond to the fragmentation of solid flow into blocks? It can be said with certainty that these blocks are not microstructural entities of the material (such as grains or sub-grains). Indeed, the solid considered in the model is structureless. The emergence of a blocky structure is a result of long-range order, which ensures that the material retains its shape as long as the loads acting on it do not exceed a critical level. That is to say, the model accounts for the existence of a yield strength above which plastic flow is possible. In this respect, our approach is generalized, or even schematized. Unlike the modelling of mixing fluxes in terms of smoothed particle hydrodynamics (SPH) [32], it does not assume any specific rheology of the system considered. What is essential in our treatment of the problem is the ability of a solid to have a definite shape and to alter it only once a certain stress is attained. It can thus be concluded that SST relates to the defining property of the solid state that distinguishes it from the gaseous and liquid ones.

The characteristic block size and the spatial pattern of SST in general are dictated by the length scale of the order in the system and reflect the level of freedom of motion of the material points.

When the shear flow of a solid is constrained by rigid boundaries, segmentation of the blocks down to sizes much smaller than the width of the shear layer occurs. Viewed at macro scale, the solid-state flow appears to be laminar, but at smaller length scales disruption of the laminar flow becomes evident. If the shear flow of a solid layer occurs in a relatively compliant environment, the big blocks into which it gets subdivided are less prone to further fragmentation than in the case when the layer is confined by rigid walls. In the latter case, the turbulent character of the flow is manifested at a length scale prescribed by the thickness of the layer, and SST is pronounced at macro scale. This macroscopic turbulent flow leads to intensive mixing of the rigid and the compliant constituents of the sandwich.

Let us consider, from this viewpoint, the behaviour of solid fibres embedded in a soft matrix under extrusion combined with simple shear. This is the case, e.g. in such processes as twist extrusion, high-pressure torsion extrusion, shear extrusion, etc. In such conditions, simple shear occurs in a thin layer normal to the extrusion axis. On the interfaces of this layer, the hard material is in contact with a hard, not a soft one. According to the above results, in this case turbulence will take place only at the micro scale, the plastic flow at the macro scale being laminar. This agrees with the experiment reported in [33], where it was shown that under extrusion with shear, the markers embedded in a specimen assume the shape of flux tubes of laminar spiral flow.

6. Conclusion

In this article, we considered patterns observed under high-pressure torsion of stacks of different metals and qualified them as a footprint of solid-state turbulence. It was hypothesized that SST is a result of the fundamental property of the solid state: the ability of a solid to retain its shape and to change it only under sufficiently high stresses. To validate this hypothesis, we employed a discretized, multi-particle model of a shear flow of a solid, which spontaneously decomposes into a system of blocks by means of self-organization of the particles. The model is an oversimplification, in that it does not consider the chemical and phase composition of the solid. Neither does it reflect the real interaction of mobile particles representing the solid. Despite its minimalism, or perhaps thanks to it, this model made it possible to mimic the behaviour of the system at the temporal and spatial scales not accessible to molecular dynamics simulations. The existence of turbulence in plastically deforming solids was demonstrated and non-trivial results were obtained. The most important of them is the recognition that the main property of the solid state described above is sufficient for the occurrence of stochastic vortices in a solid-state flow. The effect of the type of the constraints put on the shear flow of a solid on the length scale of the turbulence patterns formed was also elucidated.

The model used in the numerical simulations is a heuristic one. It is simple and user-friendly and can readily be applied for future use aimed at testing new hypotheses relating to the nature of solid-state turbulence.

Acknowledgements

Funding received from The Volkswagen Foundation through the Cooperation Project Az. 97751 is gratefully appreciated. YB acknowledges support from Karlsruhe Institute of Technology through a visiting fellowship. He also acknowledges funding received from the European Federation of Academies of Sciences and Humanities within the framework of the ‘European Fund for Displaced Scientists’ (Grant reference number of EFDS-FL2-05).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Karlsruher Institut für Technologie, European Federation of Academies of Sciences and Humanities [grant number EFDS-FL2-05], and the Volkswagen Foundation [Cooperation Project Az. 97751].