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\begin{document}

\date{\today}

\title{Distortion of a poly-crystalline Al bar in a vice fixture: Molecular dynamics analysis of grain movement and rotation}


\author{Vardan Hoviki Vardanyan}
\affiliation{%
Physics Department and Research Center OPTIMAS,
University Kaiserslautern,
Erwin-Schr{\"o}dinger-Stra{\ss}e, D-67663 Kaiserslautern, Germany}

\author{Barbara S. Linke}
\affiliation{%
University of California Davis, Mechanical and Aerospace Engineering Department, 
1 Shields Ave, Davis, CA 95616, USA}


\author{Herbert M.~Urbassek}
\email{Corresponding author:  Email: urbassek@rhrk.uni-kl.de}
\homepage{http://www.physik.uni-kl.de/urbassek/}
\affiliation{%
Physics Department and Research Center OPTIMAS,
University Kaiserslautern,
Erwin-Schr{\"o}dinger-Stra{\ss}e, D-67663 Kaiserslautern, Germany}

\begin{abstract}

\noindent{\bf Abstract}\\
The deformation of a nano-sized polycrystalline Al bar under the action of vice plates is studied using molecular dynamics simulation. Two grain sizes are considered; deformation in the fine-grained sample is mainly caused by grain-boundary processes (sliding and rotation), while dislocation plasticity dominates in the coarse-grained sample. We show that the sample distortion is reflected by the center-of-mass motion of the grains. Grain rotation is responsible for surface roughening after the loading process. While the plastic deformation is caused by  the loading process, grain rearrangements under load release also contribute considerably to the final sample distortion.


\end{abstract}

\keywords{ nanofabrication, distortion, plasticity, 
grain boundary,
grain rotation,
molecular dynamics
}


\maketitle

\section{Introduction}

The plastic deformation of polycrystalline metal samples is a topic of long-standing interest in materials science \cite{Die86}. It is of immediate importance in the field of mechanical engineering, where plastic deformation processes accompany more or less every treatment of a workpiece, leading both to residual stresses and to workpiece distortions \cite{WKD*19}. The origin of such distortions is the topic of actual scientific research \cite{YW13,ACSS15,KYP*19,ALL20,FCL20}. 

While machining processes on the microscale are commonly used in ultra-precision engineering \cite{ACW15}, nanometer-sized mechanisms are out of reach for most applications. However, in the fields of photolithography, chemical etching, laser machining, atomic force lithography, or focused ion beam lithography,  nanofabrication techniques are already being used \cite{Bhu17}. It is therefore of interest to widen our understanding of plastic processes by including processes occurring in nanofabrication.  Here, atomistic simulation  methods based on molecular dynamics simulation, have proven useful \cite{SHB*10,GTK*17}. The behavior of samples under homogeneous strain has been investigated best \cite{STJ98,SW06,SNK05,HS09,LWYG09,ZSOB17,ZLSY17,ZFZ*20}. 
Also several machining processes have been studied, including nanoindentation \cite{HCF15,RBGU17}, scratching \cite{ABKU17} and nanocutting  \cite{GU14} of surfaces. Several of the results obtained also allowed extension to larger spatial scales, i.e., to $\mu$m-sized samples, for instance the generation of dislocations \cite{RBGU17} or the effect of the tip geometry on the plastic processes \cite{ABKU17}.

On the nanoscale, distortion and the generation of residual stresses are based on the plastic processes occurring in the material under mechanical loading. In many metals, plasticity is dominated by dislocations which are activated during loading \cite{Die86,MC10}. In nano-crystalline material, in addition grain-boundary-based processes -- in particular grain-boundary sliding and rotation -- contribute  to the plastic deformation \cite{FMM08,VAB*08,ZRRZ08,RJB*14}. 
The balance of these processes for describing distortion has to our knowledge not been investigated up to now. 

In the present paper, we will study the  mechanical deformation of an Al bar which is compressed in an inhomogeneous way by vice plates  \cite{GZLU18}, see \qfig{f1}a. Besides the plastic deformation processes, we investigate in particular the processes occurring on the individual grains, i.e., grain sliding and rotation, and the eventual distortion of the bar, which is the externally measurable result of the plastic processes occurring inside the bar.  



\section{Method}

We study an Al block, see \qfig{f1}a, with  sizes of 80 nm $\times$ 40 nm $\times$ 60 nm  in  $x$, $y$ and $z$ directions, consisting of approximately $12\times10^6$ atoms. The block is subject to periodic boundary conditions in $y$ direction; the other   surfaces are free. The bottom half of the $x$--$z$ surfaces are held  by rigid vice plates. In the course of the simulation, these plates move inward along the $x$ direction with a velocity of 10 m/s for 200 ps, compressing the Al bar from 80 to 78 nm; then the fixtures are held for 200 ps to allow the sample to relax; and finally the vices are released by moving with 10 m/s to their initial positions.

In a previous publication \cite{GZLU18}, this procedure was applied to a single-crystalline Al block where all surfaces were oriented in \{100\} directions. In the present work, we use poly-crystalline Al blocks. In order to understand the effect of grain size, we use two samples: a fine-grained sample consisting of 180 grains with an average size of 10 nm, and a coarse-grained sample with 18 grains of  22 nm size; these samples are depicted in \qfig{f1}b. They are generated  using  Voronoi tesselation \cite{Vor08,Hir15,PHSB16a}. For the preparation of relaxed samples, we proceed as follows.   First, we heat the samples to 500 K during 50 ps in order to relax the grain boundaries in the samples; 
then  we keep the temperature constant for another 50 ps, and finally  decrease the temperature to 300 K during 50 ps.
In a final step, the samples are relaxed for 250 ps at 300 K in an isothermal-isobaric NPT ensemble  such that all   components of the stress tensor decrease to zero. During the compression and release simulations, the atoms in a stripe of width 1 nm in the middle between the vice plates are subjected to a Nose-Hoover thermostat  to keep the temperature at 300 K. In addition, the lower part of this stripe with a height of 1 nm has been fixed in order to prevent rigid-body translation and rotation of the entire sample during the release phase.

The  vice plates have a thickness of 2 nm and a height of 34 nm. They consist of around 500,000 carbon atoms each, arranged in a rigid diamond-lattice structure. 
 


The Al atoms interact via the Mendelev \etal potential \cite{MKBA08} with each other.  The Al-carbon interaction is modeled by a purely repulsive potential obtained from  a Lennard-Jones potential cut off at its minimum \cite{GZLU18}. 
% \VVa{epsilon is 0.02495, sigma 3.7 and cutoff  4.151}.

The simulations are performed with the software LAMMPS \cite{Pli95} and the results are analyzed and visualized using OVITO \cite{Stu10}. 


\section{Results}

\subsection{Forces}  \label{s_forc}

The force on the vices during the pressing, holding and release phases of the simulation are shown in \qfig{forces}. We show  the forces acting on the vice plates on the left- and right-hand-side, in order to be able to discuss the asymmetry caused by grain sliding and rotation, see Sects.\ \ref{s_COM} and \ref{s_rot} below. 

During the pressing phase, initially the force increases in proportion to the distance moved by the plates; however, in the second half of the pressing phase, the forces saturate and even start decreasing. This happens since the yield stress of the material  has been reached. The average pressure reached can be estimated as $p=F/A$; with the force maximum $F=3$ $\mu$N and the area of the vice plates of $A=1200$ nm$^2$, we have $p=2.5$ GPa. This value may be compared to the hardness of single-crystalline Al in this potential of 6 GPA \cite{ZU18_ind}. The yield stress in nanocrystalline samples has been found to be smaller and is in the range of 1.6 to 3.5 GPa, varying with grain size \cite{SNK05,BDAS08,DS08,KGL*04,PWB17,MZ19}. We note that these values are considerably above the yield stress of macroscopic polycrystalline Al  due to the Hall-Petch effect.


The yield mechanisms in the coarse- and fine-grained samples are different \cite{Sch04a,KGL*04}, see discussion in \qsect{s_def} below, and hence the two samples have two different force maxima. 

%We checked in an additional simulation that the force maxima also depend on the pressing velocity. 
%\hmu{higher v: leads to ? literature results
%coarse: higher F only at high v: no time for disloc nucleation
%v= 1m/s  no difference}

During the holding phase, the force decreases as the material relaxes internally by dislocation- and grain-boundary-based mechanisms, see \qsect{s_def} below. Stress release is particularly pronounced for the coarse-grained sample, as here dislocations are more active; the dislocation activity is visible in the force evolution by the considerable fluctuations seen that originate from individual dislocation processes occurring in the sample. After around 100--150 ps of holding time, the forces have reached their equilibrium values. The final pressure reached in the coarse-grained sample is only  half of that in the pressing-phase and amounts to only 1.25 GPa. 

During  release of the vices, the force continuously decreases towards zero. The vices lose contact with the sample after a time of 500 ps, indicating that both samples have shrunk in $x$ direction by around 1 nm due to the plastic deformation in the entire compression and release operation.


We note that the strain rate used in our simulations amounts to  $2.5 \times 10^8$ s$^{-1}$. It has been shown \cite{SVTJ99,SJ03} that below strain rates of around $1 \times  10^9$ s$^{-1}$, elastic and plastic properties -- in particular the yield stress -- depend only negligibly on strain rate.  Above that critical strain rate, dislocation nucleation is prohibited.





 
\subsection{Deformation mechanism} \label{s_def}

The deformation mechanisms of nanocrystalline materials are known to depend on grain size. In particular, it has been found \cite{SJ03} for the case of fcc Cu that above a grain size of around 15 nm, deformation is governed by dislocation   plasticity, which occurs inside grains, and below that grain size by grain-boundary processes, such as grain-boundary sliding and  rotation. Dislocations nucleate in grains as a result of stress concentration in grain boundaries and relief of this stress in the  grain interior for sufficiently large grains.
While the quoted study  \cite{SJ03} was on Cu, the grain sizes chosen by us -- 10 and 22 nm -- lie clearly below and above the critical grain size, such that we expect to find these distinct behaviors also in our study on Al. 

In order to quantify the deformation processes both inside the grains and in the grain boundaries, we use the von-Mises shear strain (VMSS) that is readily evaluated for each atom in the sample \cite{GGR*20}. Shear strain is the prime agent to nucleate and move dislocations, as well as to shape grain boundaries. 

%More interestingly Brink \etal \cite{BA18} argued that the grain boundary phase in fine-grained NC metals behaves like a metallic glass. \VV{ok -- but what does this mean for us here?}
 
 Focussing on the grain-boundary activity first, 
 \qfig{s_activity} shows the fraction  of grain-boundary atoms with  VMSS $>0.1$. The strain at the end of each phase (pressing, holding, and release) is calculated with respect to the configuration at the beginning of the phase.
We see that in the fine-grained sample, in each phase more than half of the grain-boundary atoms have been considerably sheared, evidencing a high occurrence of grain-boundary mediated plastic deformation. The fraction of grain-boundary atoms that have been similarly sheared is considerably smaller for the coarse-grained sample. These numbers show that the above distinction made for nanocrystalline Cu also holds in our case of machined Al. Interestingly, the fraction of sheared grain-boundary atoms is largest under the release step. i.e., when the sample is allowed to return to a pressure-free state. 

Apart from these fractions, the snapshots shown in \qfig{ss_disloc} demonstrate the occurrence of dislocation-based and grain-boundary-mediated processes in greater detail. In this view on the surface of the sample, dislocations inside the grains are marked and grain boundaries are colored according to whether their shear exceeded VMSS $=0.1$. First, it is clear that the number and length of dislocations is considerably higher in the coarse-grained sample. 
Second, the region where dislocations are nucleated extends towards the upper end of the vice plates and even above in the coarse-grained sample, while they are concentrated towards the lower surface in the fine-grained sample. Not surprisingly, the largest amount of dislocation nucleation is found in the pressing phase, where a previously well-relaxed sample is quickly strained out of equilibrium. However, even the holding and release phases allow to produce novel dislocations or let already existing dislocations migrate. 
These findings corroborate our conclusion that dislocation plasticity is more readily found in the coarse-grained sample.

The role of grain-boundary activity can also be analyzed in \qfig{ss_disloc}. After pressing the samples, almost all grain boundaries in the bottom half of the sample are found to be strongly sheared,  VMSS $>0.1$. Interestingly, the grains close to the vice plates are not as much sheared as those in the middle of the samples; this is due to the fact that close to the plates, the stress field is mostly uniaxial, and more shear is expected at the upper ends of the plates and in the middle of the sample. Again grain-boundary activities continue during the holding and the release phases, but are more prominent in the fine-grained sample.

Besides shear strain, also hydrostatic pressure is an important indicator of the stress fields generated in the sample. It is shown in \qfig{hydro_pressure} in atomic resolution; pressure data are averaged over 0.5 ps to get rid of atomic vibrations. The plots  give  the same view as \qfig{ss_disloc}. As was noted above, the averaged (uniaxial) stress in $x$ direction amounts to 2.5 GPa  in the bottom half of the sample. The figure shows that  even in the bottom half, the hydrostatic pressure is  far from a homogeneous distribution; this is caused  by the differing  orientations of the grains. 
%and the difference between the reactions of the grain interiors to the grain boundaries to the applied compressive pressure. 
While at the end of the pressing phase, the grain interiors are all compressively strained, some regions -- in particular where a high dislocation density has been produced -- are under tensile strain. Grain boundaries act as sources of high stress; some low-angle grain boundaries can be resolved in this figure as a chain of dislocations with their alternating compressive and tensile stress fields. Other grain boundaries are under  primarily compressive  stress. 
%These grain boundaries with their open atomic structure thus act as stress concentrators.

During the holding phase, pressures equilibrate somewhat in the grains, as can be seen by a stronger mix of compressive and tensile stressed regions; this goes in line with the reduction of the forces needed to keep the vice plates in position,  \qfig{forces}. After release, this equilibration process has further proceeded, since now the global pressure vanishes. Interestingly, stress release at grain boundaries is faster in the case of the fine-grained system.


The final snapshots in \qfig{hydro_pressure} show the stresses remaining in the sample after the entire process; they constitute the residual stresses. A comparison with \qfig{ss_disloc} demonstrates that some of the stresses inside the grains may be attributed to the dislocations that have been produced there; other sources of high pressure are in the vicinity of triple junctions of grain boundaries. However, the inhomogeneity of the pressure distribution -- which is under vanishing global pressure -- is astonishing, since pressures vary between $-1$ and $+1$ GPa. 


%\hmu{hold: fg $>$ cg : change of hardening mechanism. cg disloc nucleation and large grain movement
%	disloc weaken the system 
%	larger fluctuations  because of dislocs 
%fg less GB movement and higher shear strain in GB less disloc mechanism is changing}

%The hardness of Al in this potential amounts to 6 GPA \cite{ZU18_ind}.

\subsection{Surface roughness} \label{s_surf}


The snapshots, \qfig{ss_disloc} and \qfig{hydro_pressure}, show already strong deformations of the sample surface. During pressing and holding, the upper surface becomes curved in a convex way; this curvature disappears after release. The lower surface becomes rough, and this roughness does not disappear. This is due to the different grain orientations and the various grain rotations (and sliding processes). Another interesting feature is the left-right asymmetry, which develops for the coarse-grained sample in the pressing phase and remains through the holding and release phases and will be discussed in \qsect{s_COM} and \ref{s_rot} below. 
%This asymmetry is caused by the fact that vice plates proceed with a fixed velocity; accoringly the forces acting after each time step on the two plates may vary from each other. While the center-of-mass of the sample remains fixed
%Indeed we find that the forces 

%we see atomic planes
%we see dislocaitons in red
%why GBs in 

We analyze the roughness developing in the bottom surface in greater detail in \qfig{surface} which shows the height distribution of the bottom surface profile for the coarse-grained  and fine-grained samples. The average   and the standard deviation of the distributions are assembled in \qtable{T1}. We use the standard deviation of the height profile as a measure of surface roughness.

%Originally, the bottom was on the level of $-304.42$ \AA\ with a spread of 0.60 \AA\ for the coarse-grained (fine-grained) sample. 

During pressing, the standard deviation increases by a factor of 4.5 (7) with respect to the original value for the coarse-grained (fine-grained) sample. During holding, the roughness increases even further, but reduces somewhat in the release phase. The reason for this surface roughness is grain sliding and rotation as will be discussed in the next two sections; these processes cannot be made undone during the release phase. The height profile of the coarse-grained sample, \qfig{surface}a, is far from a Gaussian profile as it exhibits two peaks  in the height spectrum, which are caused by individual grains popping out of the original surface; this feature is even enhanced during the holding phase. There are 42 grains in the fine-grained sample at the bottom surface, but only 13 grains in the coarse-grained sample. Therefore, 
the multitude of grains contributing to the surface in the fine-grained sample randomize the height profile and lead to a distribution that is closer to a Gaussian. 


\subsection{Center-of-mass grain translation}  \label{s_COM}

In order to describe the translational motion (known as sliding) of grains, we study their  center-of-mass    displacements, $\vec{d}$, \cite{FMM08}. Since there is no standard analysis  available for this purpose, 
%and surpasses for our atom number of 12 million atoms the capabilities of the software OVITO, 
we proceed as follows. 
We cut 7 slabs extending in the $x$--$z$ plane with a thickness of 2 nm in the $y$ direction, which are evenly distributed on the $y$ axis. In these slabs it is possible to analyze the center-of-mass translation of the grains   in the $x$--$z$ plane, as we can neglect any motion in $y$ direction. We present the resulting translational motions as histograms in 
\qfig{COM}. This figure shows the displacements during the pressing, holding and release phases. In each phase, the displacement between the beginning and the end of the phase are shown. 
The total displacement, $d_{\rm abs}= |\vec{d}|$, and the components $d_x$ and $d_z$ along  the 
 $x$ and $z$ axes are shown.

Grain translation during the pressing phase, \qfig{COM}a,  is best described by the component parallel to the force axis, $d_x$. The distributions shows two peaks at $-20$ \AA\ and $+20$ \AA, corresponding to grains located close to the vice plates, which are forced to move the entire distance covered by the nearby vice. The majority of the grains, however, suffer only little motion, since they are located in the middle of the sample, between the two vices. Apart from this horizontal motion, a component $d_z$ in vertical direction is seen. The majority of the grains are moved upwards and thus contribute to the curvature of the upper surface seen in  \qfig{surface}. In the pressing phase, no large difference between the coarse-grained and the fine-grained samples shows up.

Note that in a purely elastic compression,  grain centers initially positioned at $x$ would move by a distance $(1-\epsilon)x$, where $\epsilon=0.05$ is the applied uniaxial strain. The $d_x$ data in \qfig{COM}a would thus follow a rectangular distribution of constant height. Our simulation results thus demonstrate the large deviation of plastic grain flow from this elastic reference case.


Considerable differences are observed during the holding phase, \qfig{COM}b. Here, grains in the coarse-grained sample are shifted to the right hand side, $d_x > 0$, with an amplitude of 5 \AA. This shift corresponds to the left-right asymmetry  shown in  \qfig{ss_disloc} and \qfig{hydro_pressure}.  The fine-grained sample also drifts further in horizontal direction in order to release the compressive strain present in the lower part of the sample; grain movements in the coarse-grained sample are larger in size, but show as much upwards as downwards motion.  

The left-right asymmetry showing up here is caused by the individual grain movements. In the coarse-grained sample, an analysis of grain motion shows a relative movement of the upper part of the sample towards the right, accompanied by a downward motion of the grains in the vicinity of the right-hand vice plate. The fine-grained sample does not show a corresponding asymmetry, since here, grain motion is more random.

Also during the release phase, considerable grain movements are observed, \qfig{COM}c. The curvature of the upper part of the sample is reduced, leading to a downward grain motion, $d_z<0$. While the grain motion in $x$ direction is symmetrical for the fine-grained sample, the coarse-grained sample tends to reduce the asymmetric motion that occurred during the holding phase, featuring an average $d_x < 0$. 
%\qfig{COM}c shows displacements distribution during \textit{releasing}.

The asymmetry observed in the grain motion for the coarse-grained sample also shows up when considering the forces acting on the vices, \qfig{forces}. As the vices are compressed with constant velocity, individual grain movements oppose the vice motion with different forces. Interestingly, this asymmetry is particularly pronounced in the holding stage, but is also visible in the pressing and release stages. 

We conclude that grain motion is mainly governed by the movement of the vices and by the upwards expansion of the sample. The coarse-grained sample shows more individuality in grain motions such that even an asymmetry with respect to the behavior of the left- and right-hand side builds up. In contrast, the grain motion of the fine-grained sample is more randomized and does not show this asymmetry. 



 
\subsection{Grain rotation} \label{s_rot}

In the course of pressing, grains are not only translated, but also rotated. We calculate the rotation angle by using a quaternion formulation to identify rotation,  which transforms the original orientation of each grain to the final orientation after pressing \cite{Bar08b,EC16,KZC*20}. 
 

\qfig{f_rot} shows the distribution of the rotation angles during the pressing phase. Quite large angles up to $8.5^\circ$ are observed with an average value of $1.29^\circ$ ($1.40^\circ$ )  for the coarse-grained (fine-grained) sample. The angles are approximately comparable for the 2 samples; however, the largest angles occur for the fine-grained sample. The effect of the grain rotation is the tilting of the grains at the lower sample surface visible in \qfig{ss_disloc} and \qfig{hydro_pressure} which leads to the sample roughness discussed in \qsect{s_surf}.
During the holding and release phase, rotation angles were $< 0.01^\circ$, indicating that grains do not rotate back towards their initial orientations. This demonstrates that sizable rotations only occur under immediate sample loading.


\subsection{Sample distortion} \label{s_dist}

The length changes of the sample after release from the fixture are called distortions. We discuss here these length changes in all phases during and after the loading process.
We measure these changes in vertical direction at 3 positions A, B, C at the left, middle and right of the sample, and at 3 positions in horizontal direction (D, E, F) in the bottom, middle, and top, see \qfig{distortion}a. In detail, due to the atomistic roughness of our samples,  
for the example of position A, we measure the average height of the sample in a slab of width 1 nm extending at a distance of 3 nm from the lateral surface in the $z$--$y$ plane; this averaging process is performed because of the roughness of our samples due to their nano-crystalline structure, cf.\  \qsect{s_surf}.

%\VV{where exactly are these positions? D for instance: at the very bottom of the sample?  at the front, in the middle or in the back? or averaged? \\ Note: D should show exactly $-5$ \%: because the sample shrank from 80 to 76 nm. Is this the case in (c), (d)? What about (b)?}

%3 nm from surface 



\qfig{distortion} shows the distortions measured as fractional changes of the lengths with respect to their initial values. 
The sample shrank horizontally under the compressive loading by the vice plates. These induce a compressive strain of $-5$ \% which exactly equals the distortion measured at D in the pressing and holding phase. After release, the sample does not return to its original length, since the deformation was plastic; the final distortion amounts to $-3.8$ ($-2.7$) \% for the coarse-grained (fine-grained) sample. Horizontal distortions in the middle of the sample at position E are considerably reduced, and at the sample top, position F, the sample expands in all phases. This expansion is caused by material flow from the lower half of the sample to the upper half.

The horizontal narrowing of the sample is accompanied by an expansion in vertical direction. This expansion is more or less uniform for the fine-grained sample, but shows a considerable left-right asymmetry for the coarse-grained sample, which was already observed and discussed previously, \qsect{s_COM}.

%The difference of distortions  at $A$ and $C$ is the result of grain sliding in the vicinity of the  right vice plate.
%\VV{how do you know?}
%Thus the  $-10$ \AA\ displacements   of COM in coarse-grained sample during releasing \qfig{COM}c is  result of  grain sliding near right vice.



We conclude that sample distortions after release are stronger for the coarse-grained than for the fine-grained sample. This is caused by the easier grain movement of the large grains, as evidenced in \qsect{s_COM}. 



\section{Summary}

We monitored the plastic deformation processes occurring in a polycrystalline Al bar under external inhomogeneous loading, representing vice plates. The processes occurring in the pressing, the holding and the release phases were analyzed and gave the following results.

\begin{enumerate}
\item Fine-grained material (grain size of 10 nm) showed only little dislocation activity inside the grains during plastic deformation, while dislocation activity dominates the deformation of coarse-grained material (grain size of 18 nm). 
This result is in agreement with previous studies of the deformation of nano-crystalline fcc metals  \cite{STJ98,SJ03,Sch04a,SW06,LWYG09}.

\item We followed the grain motion (sliding) by monitoring their center-of-mass trajectories. The results show strong deviations from a simple affine compaction of the material; rather, grains move in a strongly inhomogeneous way.

\item As a consequence of the nanometric size of the investigated Al bar, grain motions are not symmetric with respect to the bar center, but exhibit a left-right asymmetry. It is caused by grain sliding and rotation and occurs pronouncedly in the coarse-grained sample, while it averages out in the fine-grained sample. It may be expected that this asymmetry will decrease for larger samples.

\item The grains also suffer considerable rotations with an average rotation angle of $1.3^\circ$ --$1.4^\circ$. These grain rotations only occurred in the pressing phase but not during holding or releasing.

\item These grain rotations lead to a considerable roughening of the bottom surface of the sample, which does not vanish after releasing the vice fixtures.

\item The residual stresses in the material resulting from dislocation activity and grain-boundary processes build up in the pressing phase and do not relax during the holding and release phases. 

\item Due to the plastic deformations occurring, the sample shows a distortion after the process. The final distortion at the position where the vice plates acted amounts to around half the applied strain, but decreases in other areas. In the direction perpendicular to the applied strain, the sample grows. The coarse-grained sample shows higher distortion values than the fine-grained sample because of longer-ranged grain translation.

\end{enumerate}

Our results show that the final sample distortion is based on plastic processes that are mediated  both by dislocation glide  and grain-boundary sliding and rotation. In future work, the results presented here should be extended to other material classes. Besides bcc metals, such as Fe, the distortion processes in alloys would be particularly relevant.
 

%\cite{HHF10,HCF15,BA18, ZLSY17, FMM08, LWYG09, STJ98, KGL*04,SNK05, YW13,Sch04a,SW06,KYP*19,ZFZ*20}


\section*{Declarations}

\subsection*{Funding}
We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- project number 252408385 --  IRTG 2057.

\subsection*{Conflict of interest}
The authors declare that they have no conlict of interest.

\subsection*{Availability of data and material}
This study is based only on data obtained using the methods described in this paper.
%The study uses no material other than that described in this paper

\subsection*{Code availability}
The simulations were performed by the  open-source code LAMMPS available under http://lammps.sandia.gov/ \cite{Pli95}.


\begin{acknowledgments}
Simulations were performed at the High Performance Cluster Elwetritsch (RHRK, TU Kaiserslautern, Germany). 
\end{acknowledgments}

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\bibitem [{\citenamefont {Dieter}(1986)}]{Die86}%
  \BibitemOpen
  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {George~Ellwood}\
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\bibitem [{\citenamefont {Weber}\ \emph {et~al.}(2019)\citenamefont {Weber},
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\bibitem [{\citenamefont {D’Alvise}\ \emph {et~al.}(2015)\citenamefont
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  {\enquote {\bibinfo {title} {Ideal maximum strengths and defect-induced
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  {\enquote {\bibinfo {title} {Review article: Case studies in future trends of
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\bibitem [{\citenamefont {Schi{\o}tz}\ \emph {et~al.}(1998)\citenamefont
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\bibitem [{\citenamefont {Shimokawa}\ \emph {et~al.}(2005)\citenamefont
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\bibitem [{\citenamefont {Zhou}\ \emph {et~al.}(2017)\citenamefont {Zhou},
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  \bibinfo {author} {\bibfnamefont {Yijun}\ \bibnamefont {Yao}},\ }\bibfield
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\bibitem [{\citenamefont {Zhou}\ \emph {et~al.}(2020)\citenamefont {Zhou},
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  {Miyagi}, \citenamefont {Dong}, \citenamefont {Sheng}, \citenamefont {Wang},
  \citenamefont {Li}, \citenamefont {Ma}, \citenamefont {Zhang}, \citenamefont
  {Yan}, \citenamefont {Tamura}, \citenamefont {Kunz}, \citenamefont {Lutker},
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  {author} {\bibfnamefont {Xiaoxu}\ \bibnamefont {Huang}}, \ and\ \bibinfo
  {author} {\bibfnamefont {Bin}\ \bibnamefont {Chen}},\ }\bibfield  {title}
  {\enquote {\bibinfo {title} {High-pressure strengthening in ultrafine-grained
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\bibitem [{\citenamefont {Huang}\ \emph {et~al.}(2015)\citenamefont {Huang},
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\bibitem [{\citenamefont {Ruestes}\ \emph {et~al.}(2017)\citenamefont
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  {Gao}}, \ and\ \bibinfo {author} {\bibfnamefont {Herbert~M}\ \bibnamefont
  {Urbassek}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Molecular
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  \bibinfo {editor} {\bibfnamefont {Atul}\ \bibnamefont {Tiwari}}\ and\
  \bibinfo {editor} {\bibfnamefont {Sridhar}\ \bibnamefont {Natarajan}}}\
  (\bibinfo  {publisher} {Wiley},\ \bibinfo {address} {Chichester, UK},\
  \bibinfo {year} {2017})\ Chap.~\bibinfo {chapter} {14}, pp.\ \bibinfo {pages}
  {313--345}\BibitemShut {NoStop}%
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\bibitem [{\citenamefont {Farkas}\ \emph {et~al.}(2008)\citenamefont {Farkas},
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  \BibitemOpen
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  }\bibfield  {title} {\enquote {\bibinfo {title} {Strain-driven grain boundary
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  {493}},\ \bibinfo {pages} {33--40} (\bibinfo {year} {2008})},\ \bibinfo
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\bibitem [{\citenamefont {Vo}\ \emph {et~al.}(2008)\citenamefont {Vo},
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  {author} {\bibfnamefont {A.}~\bibnamefont {Caro}},\ }\bibfield  {title}
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  }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Phys. Rev. B}\
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\bibitem [{\citenamefont {Romero}\ \emph {et~al.}(2014)\citenamefont {Romero},
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  {Coarse graining and localized plasticity between sliding nanocrystalline
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  \citenamefont {Zhang}, \citenamefont {Linke},\ and\ \citenamefont
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  \bibnamefont {Linke}}, \ and\ \bibinfo {author} {\bibfnamefont {Herbert~M.}\
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  {Molecular dynamics simulations of single grain pure aluminum in a vice
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  {Prakash}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Hummel}},
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  }\bibfield  {title} {\enquote {\bibinfo {title} {Analysis of semi-empirical
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  {D.~E.}\ \bibnamefont {Wolf}},\ }\bibfield  {title} {\enquote {\bibinfo
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  {Schi{\o}tz}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Atomic-scale
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  }\bibfield  {title} {\enquote {\bibinfo {title} {Atomic-scale simulations of
  the mechanical deformation of nanocrystalline metals},}\ }\href@noop {}
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  {1999})}\BibitemShut {NoStop}%
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  \bibnamefont {Jacobsen}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {A
  maximum in the strength of nanocrystalline copper},}\ }\href {\doibase
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  \BibitemOpen
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  \bibinfo {author} {\bibfnamefont {Garritt~J.}\ \bibnamefont {Tucker}},\
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  {school} {RWTH Aachen} (\bibinfo {year} {2008})\BibitemShut {NoStop}%
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  {El~Shawish}}\ and\ \bibinfo {author} {\bibfnamefont {Leon}\ \bibnamefont
  {Cizelj}},\ }\bibfield  {title} {\enquote {\bibinfo {title} {Numerical
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  {\bibinfo  {journal} {Acta Materialia}\ }\textbf {\bibinfo {volume} {189}},\
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%\bibliography{./bibo/STRING.bib,./bibo/all.bib,./bibo/PUBL.bib,./bibo/Glasses.bib}
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\newpage \clearpage
%\clearpage

\begin{table*}[h]
\setlength{\tabcolsep}{1em}
\begin{center}
\caption{ Average (avg) and standard deviation (std) of the surface profiles,  \qfig{surface}. Data are in \AA\ units.
}
\begin{tabular}{r|ll|ll|ll|ll}
\hline
sample  & \multicolumn{2}{c|}{initial} & \multicolumn{2}{c|}{pressing} & \multicolumn{2}{c|}{holding} & \multicolumn{2}{c}{release} \\ 
                & avg   &   std    &   avg    &  std   &  level \AA\   &   std   &   avg    & std  \\
\hline
coarse-grained  & $-304.42$   &   0.60   &  $-305.85$   & 4.15   & $-305.30$   &  7.12   &  $-306.61$   &  5.94    \\ 
fine-grained    & $-305.05$   &   0.60   &  $-305.13$   & 2.62   & $-303.97$   &  3.21   &  $-303.31$   &  2.63    \\ 
\hline
\end{tabular}
\label{T1}
\end{center}
\end{table*}


\begin{figure*}[h]
  \begin{center}
  \subfigure[]{\includegraphics[angle=0,width=0.3\linewidth]{sim_setup.eps}}
  \subfigure[]{\includegraphics[width=0.52\textwidth]{sim_system.eps}}
  \end{center}
\caption{ a) Schematic setup of the simulation system. The vice plates compress the Al bar laterally. 
%The heights and widths of the bar are measured at several locations indicated. 
b) Two simulation samples containing 180 and 18 grains with average grains sizes of  10 nm and 22 nm,  respectively.} 
%1035.73 nm$^3$ and 10679.73 nm$^3$ 
\label{f1}
\end{figure*}

\begin{figure*}[h]
  \begin{center}
\includegraphics[width=0.5\textwidth]{forces.eps}
  \end{center}
\caption{Time evolution of the forces acting in $x$ direction on the vices.}
\label{forces}
\end{figure*}

\begin{figure*}[h]
  \begin{center}
\includegraphics[width=0.45\textwidth]{shear_activity.eps}
  \end{center}
\caption{Fraction of grain-boundary atoms with VMSS $>$ 0.1.
The reference configurations for holding and releasing are the end of the pressing and the holding phase, respectively.}
\label{s_activity}
\end{figure*}

\begin{figure*}[h]
  \begin{center}
\includegraphics[width=0.9\textwidth]{ss_disloc.eps}
  \end{center}
\caption{VMSS map of  coarse-grained and fine-grained samples, during pressing, holding and release. 
The reference configurations are the same as in \qfig{s_activity}. In this side view on the sample, dark blue represents atoms in   fcc lattice structure inside the  grains and light blue represents   atoms at grain boundaries.
White characterizes  grain-boundary atoms with VMSS $>$ 0.1, and red atoms have VMSS $>$ 0.1  and belong to the grain interior.
}
\label{ss_disloc}
\end{figure*}

\begin{figure*}[h]
  \begin{center}
\includegraphics[width=0.9\textwidth]{hydro_pressure.eps}
  \end{center}
\caption{Hydrostatic pressure. Negative values denote compressive pressure. 
}
\label{hydro_pressure}
\end{figure*}

\begin{figure*}[h]
\begin{center}
\subfigure[]{\includegraphics[angle=0,width=0.9\textwidth]{18_surface.eps}}\\
\subfigure[]{\includegraphics[angle=0,width=0.9\textwidth]{180_surface.eps}} \\

\end{center}
\caption{Height  profile of the bottom sample surface for the (a) coarse-grained sample and b) fine-grained sample. The vertical line indicates the original average surface position. Note the change in  scale between the initial and the deformed profiles. 
} 
\label{surface}
\end{figure*}



\begin{figure*}[h]
\begin{center}
\subfigure[]{\includegraphics[angle=0,width=0.9\textwidth]{pressing.eps}}\\
\subfigure[]{\includegraphics[angle=0,width=0.9\textwidth]{holding.eps}} \\
\subfigure[]{\includegraphics[angle=0,width=0.9\textwidth]{releasing.eps}}
\end{center}
\caption{Normalized distributions of  the center-of-mass  displacements of grains during the (a)  pressing, (b)  holding, and (c) release phase. 
} 
\label{COM}
\end{figure*}

\begin{figure*}[h]
\begin{center}
\includegraphics[angle=0,width=0.9\textwidth]{misorient_press.eps}
\end{center}
\caption{Distribution of grain rotation angles during the pressing phase. 
} 
\label{f_rot}
\end{figure*}

\begin{figure*}[h]
\begin{center}
\subfigure[]{\includegraphics[angle=0,width=0.45\linewidth]{distortion_setup.eps}}
%\subfigure[]{\includegraphics[angle=0,width=0.45\linewidth]{distortion_previous.eps}}
\\
\subfigure[]{\includegraphics[angle=0,width=0.45\linewidth]{distortion_18.eps}}
\subfigure[]{\includegraphics[angle=0,width=0.45\linewidth]{distortion_180.eps}}
\end{center}
\caption{(a) Schematics indicating the positions where vertical (A--C) and horizontal (F--D) sample distortions were measured. (b) and (c) show simulation results for  distortions for the coarse-grained (b) and fine-grained (c) sample in the pressing, holding and release phase. 
%Data for (b) taken from \qref{GZLU18}.
}
\label{distortion}
\end{figure*}






\end{document}

