# Minimal model of an active solid deviates from equilibrium mechanics

Mario Sandoval<EMAIL_ADDRESS>Department of Physics, Complex Systems,
Universidad Autonoma Metropolitana-Iztapalapa, Mexico City 09340, Mexico.

###### Abstract

In this work, the mechanical response of an one-dimensional active solid
–defined as a network of active stochastic particles interacting by nonlinear
hard springs– subject to an external deformation force, is numerically studied
and rationalized with a minimal model. It is found that an active solid made
of linear springs and subject to an external stress, presents an average
deformation which is independent of the system’s activity. However, when the
active solid is made of nonlinear hard springs, the solid’s average
deformation decreases with respect to a passive system under the same
conditions, and as a function of activity and rotational noise in the system.
The latter result may shed light on new ways to creating an active
metamaterial, which could tune its stiffness by moving either its activity or
rotational noise.

##  I INTRODUCTION

The materials science community Sigalas and Economou (1992, 1993); Kushwaha
_et al._ (1993); Liu  _et al._ (2000); Bukhari and Barry (2020a) is already
designing and characterizing metamaterials or metastructures defined as
synthetic materials with deliberately designed structures. These
metastructures are found to present exotic behavior such as the ability to
prevent wave propagation through them Sigalas and Economou (1992, 1993); Liu
_et al._ (2000) (the so-called bandgap), negative Young’s modulus Huang and
Sun (2012), a negative index of refraction Shalaev  _et al._ (2005), or a
negative effective mass density Huang and Sun (2012). After the inclusion of
nonlinearities, those metastructures are seen to present asymmetric wave
traveling phenomenon Liang, Yuan, and Cheng (2009); Zivieri  _et al._ (2019);
Fiore  _et al._ (2020) and the development of gap solitons Kivshar and
Flytzanis (1992); Nadkarni, Daraio, and Kochmann (2014). Quite recently, and
by exploiting 3D printing techniques researchers have designed metamaterials
with programmable response to uniaxial compression Florijn, Coulais, and van
Hecke (2014), a negative Poisson’s ratio Babaee  _et al._ (2013) or even
materials with a nearly constant stiffness at very low densities Zheng  _et
al._ (2014). Metamaterials are also being used for vibration suppression and
energy harvesting Hu, Tang, and Das (2018); Li  _et al._ (2017); Tol,
Degertekin, and Erturk (2016); Bukhari and Barry (2020b); Hu  _et al._ (2016);
Zivieri  _et al._ (2019); Fiore  _et al._ (2020).

In addition, the physics community has also spotted a different way to render
a metamaterial a different behavior. The key is to take this material out of
equilibrium by imposing a propulsion or active force to each of its building
components Ferrante  _et al._ (2013); Menzel, Ohta, and Löwen (2014); Lowen
(2018); Kaiser  _et al._ (2015); Barberis and Peruani (2019), thus creating an
active metamaterial. Explicit recent works on elucidating a change of a
material’s behavior due to its non-equilibrium state are: Souslov Souslov,
Anton and van Zuiden, Benjamin C and Bartolo, Denis and Vitelli, Vincenzo
(2017) et al. who built a lattice made of annular shapes filled with an active
fluid. Using numerical simulations, they observed the propagation of
unidirectional sound waves through the edges of their system domain. Wang Wang
_et al._ (2018) et al. and Trainiti Trainiti  _et al._ (2019) et al. proposed
a one-dimensional crystal whose elastic interactions are time-dependent. This
time-dependence breaks time-reversal symmetry, thus creating a directional
bandgap in the dispersion equation. Brandenbourger Brandenbourger  _et al._
(2019) et al. used a chain of springs with broken spatial-symmetry, that
together with actuation and local sensing, observed the emergence of
asymmetric standing waves with unidirectional amplification. As it can be
noticed, scientists have already spotted self-propulsion as a means of
creating novel active metamaterials.

Figure 1: (a) A chain of active particles with hexagonal shape about to
collide against a wall Brandenbourger  _et al._ (2021). This chain was seen to
rebound differently when its activity was turned off or on. (b) By only
keeping the main ingredients from (a), we build the active structure under
stress (red arrow) shown in (b). Thus, this work proposes to give an
analytical understanding of possible activity effects on the deformation of
this simpler active metastructure.

Motivated by the latter efforts on rendering a material new properties and by
recent active materials experimental observations, such as the peculiar
bouncing over a wall of a closed chain of active robots interacting by mobile
active arms Brandenbourger  _et al._ (2021) (see cartoon in Fig. 1(a) for
illustration), and its bouncing dependence on activity; or a network of active
robots interacting by means of springs and generating a synchronized dynamics
Baconnier  _et al._ (2021); or a larger structure made of several closed
active chains as its main unit (see Fig. 1(b) for a possible realization), and
observed to deform differently when activity in the structure was turned on
Brandenbourger  _et al._ (2021); I report on numerical and analytical results
obtained after analyzing an active solid made of particles interacting by
nonlinear hard springs described by constants $k$ and $\mu$. This simple one-
dimensional model retains the main ingredients (interaction and self-
propulsion) of the previous experiments Brandenbourger  _et al._ (2021);
Baconnier  _et al._ (2021), and can be used to evaluate the effect of those
ingredients on the active solid’s mechanical response to a deformation force
(stress). Briefly, it is found that the latter active solid deforms less on
average, as the activity in the system grows. The results of this model may
help to understand the mentioned experimental observations and could be
generalized to higher dimensions. It is interesting to mention that when
seeking a model able to provide insight on possible self-propulsive effects on
a material’s response, it was observed that a linear interaction between
building blocks was not able to do so; instead, nonlinear interactions Fermi,
Pasta, and Ulam (1955) proved to be a valuable tool to accomplish that. It is
worth mentioning that a nonlinear mechanical response of ephitelial monolayers
to stress, has been experimentally spotted Trepat  _et al._ (2007); Harris
_et al._ (2012). Numerical and analytical efforts highlighting the nonlinear
response of a tissue-model have also been reported Merkel  _et al._ (2019);
Huang  _et al._ (2022). Moreover, internal active forces as the responsible
for stiffening a biopolymer network Koenderink  _et al._ (2009), as well as
the responsible for a phase transition from a ductile to a brittle state in
ephitelial tissue of Trichoplax Adhaerens, and for a bistable fission process
(asexual reproduction) have already been experimentally identified Prakash,
Bull, and Prakash (2021). Therefore, this active nonlinear model extended to
higher dimensions may also be suitable for analyzing the dynamics of living
tissues. Note that although active forces have been already identified to play
an important role on the dynamics of artificial active metamaterials and
ephitelial tissue, a minimal model able to encompass elasticity and active
forces, and to systematically produce a change in the system’s elastic or
dynamic properties –after tuning certain parameters– has not been proposed.
This study is an attempt to fill this gap.

The present study is organized as follows: Section II introduces the model.
Section III studies the deformation –due to an external force– of active
solids made of self-propelled particles either linearly or nonlinearly
interacting and as a function of activity in the system, rotational
diffusivity, and strength of nonlinearities. An energy analysis of the system
is carried out in Section IV. Section VI discusses available experiments on
self-propelled robots and presents some orders of magnitude for relevant
parameters such as propulsion force, dissipation coefficient, and angular
velocity. Due to the similarity of the orders of magnitude between the used
numerical parameters in Section III and the experiments, a possible experiment
is proposed in this section. A simple deterministic model to rationalize the
present findings is introduced and discussed in Section V. Conclusions and
future directions are offered in Sec. VII, and two appendices complementing
this work are also added.

##  II Physical model

Figure 2: Schematic of the studied problem. One-dimensional active solid
defined as a network of $N-1$ active stochastic particles (blue circles) of
mass $M$, interacting by nonlinear springs. Note that the $N$ particle in this
model is passive (black circle) and is subject to a deformation constant force
$\mathbf{F}$ (red arrow). In the picture, $N-1=2$. Due to the dimensionality
of the system, their independent random propulsion forces $\mathbf{F}_{0}$
have to be projected, that is, $F_{0}\cos(\varphi_{n}(t))$. These projections
are indicated as orange arrows. The original equilibrium distance between
particles is $a$, and no steric interactions are considered.

Let us analyze a one-dimensional active solid defined as a network of $N-1$
active stochastic particles of mass $M$, and interacting either by linear or
nonlinear springs characterized by constants $k$ and $\mu$; where $k>0$ is the
linear restitutive constant, and $\mu>0$ is the degree of nonlinearity. In
this system, the last particle in the network, $N$, (see Fig. 2) is passive
and its right hand side is not subjected to a spring, but it is exposed to a
deforming force (${\bf F}$) either stretching or compressing the system. The
active particles, originally separated by their equilibrium distance $a$, are
assumed to stochastically rotate along their azimuthal direction (with origin
at their center) $\varphi_{n}(t)$, and towards which a propulsion force ${\bf
F}_{0}=(F_{0}\cos\varphi_{n},F_{0}\sin\varphi_{n})$ is acting. The latter is
just an ABP (Active Brownian particles) model. Due to the one-dimensionality
of this system, that may be accomplished by setting smooth walls in an
experiment (see Section VI and supplementary materials where an experiment is
proposed), the propulsion forces will only be present through their
projections along the $x$-axis, that is, $F_{0}\cos(\varphi_{n})$. See the
inset of Fig. 2 for further clarification. These active forces will be
constantly driving the system out-of-equilibrium. Note that dimensionality
does not change the effect of active forces on a system’s effective diffusion.
This has been proved by Lowen et al. ten Hagen, van Teeffelen, and Lowen
(2009) who showed that in an one-dimensional active system, and even if the
magnitude of the propulsion force varies according to $\cos(\varphi_{n}(t))$;
it induces the same effective diffusion as in higher dimensions, and for which
the magnitude of the propulsion force is kept constant. Thus we expect to have
a similar behavior of our active solid at higher dimensions.

The dynamics of each particle is described by its translational velocity
$v_{n}(t)$ and position $x_{n}(t)$, with respect to its equilibrium position.
Therefore, the motion of the $n-th$ particle under a hard spring symmetric
potential, $V(x)=\left(k/2\right)x^{2}+\left(1/4\right)\mu x^{4},$ and in
dimensionless variables
$\left(\widetilde{x}_{n},\widetilde{v}_{n},\widetilde{t}\right)$ reads

$\displaystyle d\widetilde{v}_{n}$ $\displaystyle=$
$\displaystyle\left\\{\left[\widetilde{F}_{0}\cos\left(\varphi_{n}\right)+\left(\widetilde{x}_{n+1}-\widetilde{x}_{n}\right)\right](1-\delta_{nN})\right.$
(1)
$\displaystyle-\left(\widetilde{x}_{n}-\widetilde{x}_{n-1}\right)-\alpha\widetilde{v}_{n}$
$\displaystyle+\lambda\left(\widetilde{x}_{n+1}-\widetilde{x}_{n}\right)^{3}(1-\delta_{nN})$
$\displaystyle\left.-\lambda\left(\widetilde{x}_{n}-\widetilde{x}_{n-1}\right)^{3}+\delta_{nN}\widetilde{F}\right\\}d\widetilde{t},$
$\displaystyle d\widetilde{x}_{n}$ $\displaystyle=$
$\displaystyle\widetilde{v}_{n}d\widetilde{t},$ (2) $\displaystyle
d\varphi_{n}$ $\displaystyle=$ $\displaystyle\sqrt{2\widetilde{D}_{R}}dW_{n},$
(3)

where
$\widetilde{x}_{n}=x_{n}/a,\widetilde{v}_{n}=v_{n}\sqrt{M/k}/a,\widetilde{t}=t/\sqrt{M/k}$,
$\delta_{ij}$ is the kronecker’s delta, $\widetilde{F}_{0}=F_{0}/ka$ is the
dimensionless magnitude of the imposed propulsion forces along the random
directions $\cos\left(\varphi_{n}\right)$, $W_{n}$ represent standard Wienner
processes with zero mean and standard deviation $\widetilde{t}$,
$\widetilde{D}_{R}={D}_{R}\sqrt{M/k}$, with ${D}_{R}$ as the strength of
rotational noise, which could be caused by a vibrating motor/surface Scholz
_et al._ (2018) such as in the Hexbugs-nano robots Tapia-Ignacio, Gutierrez-
Martinez, and Sandoval (2021); Baconnier  _et al._ (2021),
$\alpha=R_{T}/\sqrt{Mk}$, is a dimensionless number which accounts for
dissipation, and where $R_{T}$ is the friction coefficient, $\lambda=\mu
a^{2}/k$ is an introduced dimensionless parameter accounting for nonlinearity,
and $\widetilde{F}=F/ka$ is the magnitude of the imposed deformation force
along the $x-$direction, and in dimensionless units the solids’ original
equilibrium length is simply given by $L_{N}=N$. Note that Eq. (1) is subject
to the initial conditions $\widetilde{x}_{n}(0)=\widetilde{v}_{n}(0)=0$ for
$n=1,2...,N$, and we define $\widetilde{x}_{0}(t)=0$ for completeness.

Figure 3:  Average strain of five active solids made of
$N=\\{6,16,32,64,128\\}$ masses (either linearly or nonlinearly interacting)
as a function of stretching and compressing forces. The influence of activity
on the deformation of the system, when nonlinearity is present, can be
appreciated. Here, $\widetilde{D}_{R}=1$.  Figure 4:  Average strain of five
active solids made of $N=\\{6,16,32,64,128\\}$ particles, as a function of
their propulsion force. The nonlinear spring constant is set as $\lambda=20$,
whereas $\widetilde{D}_{R}=1$.

##  III Deformation

Let us start by numerically solving Eqs. (1)-(3) for the case of five active
solids made of $N=\\{6,16,32,64,128\\}$, respectively; and look for its
deformation. For the simulations, we choose a second-order Verlet algorithm, a
time step $\Delta\widetilde{t}=0.025$, a total time to reach the system’s
steady state of $\widetilde{T}=\\{50,300,1000,4000,10000\\}$, and perform from
50 to 500 realizations to calculate ensemble averages. Due to the number of
interacting particles, a vectorized code had to be implemented. It was also
noticed that the larger an active solid, the longer the time it takes to reach
a steady state. A typical scenario of this steady state can be seen in the
Appendix. In addition, the following set of parameters are considered:
$\alpha=1$, $\widetilde{F}_{0}=[0,0.7]$, $\widetilde{D}_{R}=[1,40]$,
$\widetilde{F}=[-0.6,0.6]$, and $\lambda=\\{0,5,10,20\\}$. It is worth
mentioning that the previous parameters were chosen in such a way that the
acting forces deform the solids in such a way that their constituent particles
never overlap, that is,
$\widetilde{x}_{n-1}<\widetilde{x}_{n}<\widetilde{x}_{n+1}$ for all time. The
same rule will be applied thereafter. The results are visualized in Fig. 3,
where the average strain ($\langle\Delta L_{N}\rangle/L_{N}$) of the mentioned
active solids versus applied force ($\widetilde{F}$) is shown. See also
supplementary material for visualization purposes. We start validating the
employed code by reproducing the linear deformation region or Hooke’s law.
Here, we set $\lambda=0$, and try two propulsion forces
$\widetilde{F}_{0}=\\{0,0.7\\}$. The results for passive solids
($\widetilde{F}_{0}=0$) made of $N=\\{6,16,32,64,128\\}$ are shown as blue
solid lines in Fig. 3.

Figure 5: Average strain of an active solid made of $N=32$ particles as a
function of stress, and for several active forces. Here, the nonlinear spring
constant and rotational noise are set as $\lambda=20$ and
$\widetilde{D}_{R}=1$, respectively. Clearly, as activity increases the active
solid tends to a Hookean behavior.

Clearly, all the solids’ deformations collapse to one curve and the Hookean
behavior is well reproduced. This collapsing is attributed to the fact that we
are stressing the same material but of different lengths. We now analyze some
possible active forces implications on the active solids’ strain by setting
$\widetilde{F}_{0}=0.7$ and $\widetilde{D}_{R}=1$, while keeping a linear
interaction ($\lambda=0$). After performing the needed simulations, the
results are shown with the respective symbols in Fig. 3. Interestingly, all
the solids’ strains fall into the same blue line, therefore it can be
concluded that there is not any activity effect on the system since both
cases, $\widetilde{F}_{0}=\\{0,0.7\\}$, have the same Hookean behavior, and of
course this behavior is seen to be independent of the solids’ size. This
activity-independence result for linearly interacting masses has also been
observed by Kaiser et al. Kaiser  _et al._ (2015) when studying an active
Rouse model.

The next step is to add nonlinear effects by setting for example $\lambda=20$.
Thus, a first numerical experiment is to simulate the solids’ strain with null
activity ($\widetilde{F}_{0}=0$). The results for $N=\\{6,16,32,64,128\\}$ are
shown in Fig. 3 as black-solid lines. Once again all the passive solids’
deformation fall into the same curve, thus one apparently sees only one black-
solid line in Fig. 3, but in fact there are five curves. This was done for
clarity, otherwise Fig. 3 would have too much information. These curves
indicate that nonlinear interactions have modified the solids’ response to a
not Hookean behavior, and as expected, the solids irrespective of their
length, now show a symmetric hardening effect.

The interesting results start here. Let us now turn on activity and keep the
nonlinearity in the solids. Consider for example
$\lambda=20,\widetilde{D}_{R}=1$, and set activity and deformation forces,
respectively, as $\widetilde{F}_{0}=0.7$ and $\widetilde{F}=[-0.6,0.6]$. The
numerical results for all the active solids are shown in Fig. 3, where strain
versus external force are represented by the symbols shown in that figure.
Notice how the behavior of the five active solids ($N=\\{6,16,32,64,128\\}$)
is the same and is independent of the system’s size. This time, it is seen
that activity affects the systems’ mechanical response, since on average, a
hardening effect in the systems has appeared. This novel observation may
originate similar behaviors as the recent experiments dealing with a hexagonal
array of six active masses colliding against a wall Brandenbourger  _et al._
(2021), or the deformation of a structure (made with the mentioned hexagonal
array as its building block) due to the impact of a mass through it
Brandenbourger  _et al._ (2021). Notice as well from the same figure, that the
Hookean region is longer, that is, our system hardens but nearly behaves
Hookean. Only after higher stresses, one can see again the nonlinearity in the
system.

Figure 6: Average strain of four active solids made of
$N=\\{6,16,32,64,128\\}$ particles as a function of rotational noise. Here,
the nonlinear spring constant and active force are set as $\lambda=20$ and
$\widetilde{F}_{0}=0.5$, respectively. For reference, the strain of a passive
system is also shown as red-dashed lines.  Figure 7:  Average strain of a
chain of active particles as a function of deformation force, propulsion
force, and rotational noise; and for three nonlinear springs, namely
$\lambda=\\{5,10,20\\}$. (a) $\widetilde{D}_{R}=1$. (b) $\widetilde{F}=0.3$
and $\widetilde{D}_{R}=1$. (c) $\widetilde{F}_{0}=0.5$ and
$\widetilde{F}=0.3$.

Once a self-propulsive effect has been spotted, the next step is to understand
the dependence of deformation on the magnitude of the propulsion force. This
is achieved by setting for example $\widetilde{F}=0.3$, $\widetilde{D}_{R}=1$,
$\lambda=20$, and vary the self-propulsion force $\widetilde{F}_{0}=[0,0.7]$.
Under the latter parameters, five active solids are deformed, and their
resulting mechanical behavior (average strain vs deformation force) is plotted
in Fig. 4. It can be observed that on average the solids’ deformation
decreases as activity increases, and this behavior is similar for all of them
but this time size effects seem to appear. Notice for example how the pink
squares ($N=6$) are always above the cyan triangles ($N=128$). For a better
accuracy, the offered measurements in Fig. 4 also indicate their respective
one standard deviation error. From this plot one can also see that the maximum
decrement of the strain occurs at the highest proposed propulsion force
$\widetilde{F}_{0}=0.7$; however, bear in mind that higher propulsion forces
may not satisfy, the no overlap condition,
$\widetilde{x}_{n-1}<\widetilde{x}_{n}<\widetilde{x}_{n+1}$.

Another way of observing activity effects in the current active solids is to
plot the solids’ average strain versus stress for several active forces. Since
this plot has already proved to be size’s independent, I choose $N=32$
particles, $\widetilde{F}_{0}=\\{0,0.3,0.5,0.7\\}$, $\lambda=20$,
$\widetilde{D}_{R}=1$, and $\widetilde{F}=[0,0.6]$. The results are shown in
Fig. 5. A very interesting result from this plot is the fact that as activity
increases, the strain tends to behave Hookean, that is, activity seems to
dominate nonlinear effects at least for a given region. One can also notice
that when the deformation force increases, self-propulsion effects tend to
disappear. See for example the blue solid line at high stresses. Clearly, it
tends to the black-solid passive behavior. This can be understood if one
exaggerates the situation and visualizes a large stress applied to the system.
This stress surely will deform the system no matter if activity is present or
not. Finally, the effect of the strength of rotational noise on the
deformation of the five active solids ($N=\\{6,16,32,64,128\\}$) is analyzed.
By fixing the propulsion and stretching forces as $\widetilde{F}_{0}=0.5$ and
$\widetilde{F}=0.3$, respectively, and varying $\widetilde{D}_{R}=[1,40]$
together with $\lambda=20$; we obtain Fig. 6. This plot indicates that the
average strain’s behavior of the solids (see their respective symbols in Fig.
6) is very similar, and increases as the rotational noise grows. For a better
precision, error bars indicating a standard deviation are also indicated. It
is also appreciated that rotational noise may be used as tune parameter, since
it enables the system to behave either as a passive solid for high
$\widetilde{D}_{R}$ (see the horizontal red-dashed line asymptote, indicating
the passive behavior, $\tilde{F}_{0}=0$) or as a solid that deforms less for
low $\widetilde{D}_{R}$. The strain’s behavior in the limit
$\widetilde{D}_{R}\rightarrow 0$ is discussed in the Appendix section.
Basically, the strain seems to asymptotically reach a minimum value as
rotational noise tends to zero.

Let us now analyze the effect of nonlinearity ($\lambda$), on the strain in
our system when varying deformation force, activity, and rotational noise. For
this experiment, the case $N=6$ will be considered thereafter since the other
solids have shown to behave very similar. In a first experiment, it is set
$\widetilde{D}_{R}=3$, $\widetilde{F}_{0}=\\{0,0.5\\}$,
$\widetilde{F}=[-0.3,0.3]$ and $\lambda=\\{5,10,20\\}$. The results are shown
in Fig. 7(a). It can be appreciated that larger nonlinear coefficients do not
seem to change previous scenarios (see Fig. 3) but that these coefficients
only harden the system as usual. In the same figure, the behavior of a passive
solid linearly interacting and under the same conditions, appears in solid
blue lines as a reference. In a second experiment, the previous numerical
parameters are also used but this time it is fixed $\widetilde{F}=0.3$ and
vary $\widetilde{F}_{0}=[0,0.7]$. The numerical results are shown in Fig. 7(b)
which indicates that for the three considered values of $\lambda$, the average
strain in the system behaves in the same manner. Finally, a last experiment
using the same previous parameters but fixing $\widetilde{F}=0.3$ and
$\widetilde{F}_{0}=0.5$, and varying $\widetilde{D}_{R}=[0.5,40]$ is shown in
Fig. 7(c). Once again, nonlinearity does not change the behavior shown in Fig,
6, it only hardens the system as usual. Horizontal asymptotes indicating the
respective passive strains are also shown as dashed lines.

Figure 8: (a) Log-Log plot of the average non-Hookean, Hookean elastic energy,
and average kinetic energy of a chain of active particles as a function of
propulsion force either linearly
($\lambda=0,\widetilde{D}_{R}=3,\widetilde{F}=0.3$ ) or nonlinearly
($\lambda=20,\widetilde{D}_{R}=3,\widetilde{F}=0.3$) interacting. (b) The same
as in (a) but as a function of $\widetilde{D}_{R}$.

##  IV Energetics

The next goal is to understand the behavior of the dimensionless Hookean
energy ($\widetilde{U}_{H}$), non-Hookean elastic energy
($\widetilde{U}_{NH}$), and kinetic energy ($\widetilde{K}$) as a function of
propulsion force and rotational noise in the system. These energies are
defined respectively, as

$\displaystyle\widetilde{U}_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{n=1}^{N}\left[-\left(\widetilde{x}_{n+1}-\widetilde{x}_{n}\right)^{2}(1-\delta_{nN})\right.$
(4)
$\displaystyle\left.+\left(\widetilde{x}_{n}-\widetilde{x}_{n-1}\right)^{2}\right],$
$\displaystyle\widetilde{U}_{NH}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4}\sum\limits_{n=1}^{N}\left[-\left(\widetilde{x}_{n+1}-\widetilde{x}_{n}\right)^{4}(1-\delta_{nN})\right.$
(5)
$\displaystyle\left.+\left(\widetilde{x}_{n}-\widetilde{x}_{n-1}\right)^{4}\right],$
$\displaystyle\widetilde{K}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum\limits_{n=1}^{N}\widetilde{v}_{n}^{2}.$ (6)

A first numerical experiment fixes $\lambda=\\{0,20\\},\widetilde{D}_{R}=3$,
$\widetilde{F}=0.3$, performs $1000$ realizations for ensemble averaging, and
vary $\widetilde{F}_{0}=\\{0,0.1,0.2,0.3,0.4,0.5\\}$. The results are shown in
Fig. 8(a). It can be seen that for an active solid made either of nonlinear or
linear springs, their kinetic energy increases, almost identically, as a
function of the propulsion force. However, for a solid made of nonlinear
springs, its Hookean elastic energy decreases as its propulsion force
increases. The opposite occurs for a solid made of linear springs and whose
elastic energy is larger compared to the nonlinear solid. In addition, Fig.
8(a) illustrates that the non-Hookean elastic energy also decreases as
propulsion in the system increases. These results suggest that as propulsion
in the active solid increases, a solid made of nonlinear springs will deform
less on average compared to a system made of linear springs under the same
conditions.

The dependence of the average non-Hookean, Hookean elastic potential energy,
and average kinetic energy of a chain of active particles
($\lambda=20,\widetilde{F}=0.3$) as a function of rotational diffusivity is
shown in Fig. 8(b). Clearly, as rotational noise grows, $\langle
U_{NH}\rangle$ and $\langle U_{H}\rangle$ increase, meaning that the system is
tending to a passive behavior. This time, the kinetic energy decreases. The
decrease of energy could be originated by the fact that for high
$\widetilde{D}_{R}$, the propulsion force quickly changes direction, thus
avoiding an increase in the velocity of the system.

##  V Rationalization

Figure 9:  Comparison between the predicted theoretical deformation (red solid
lines) given by Eq. (11), and the numerical solution (blue circles) of Eq. (7)
with $\varepsilon=\left\\{0.2,0.17\right\\},\beta=2,f=\left[1,5\right]$, and
$\widetilde{\Omega}=1.5$.

So far we have seen the effect of rotational noise and activity on the average
deformation of an active solid. Briefly, we have concluded that as activity in
the system increases, its average deformation decreases; and that as the
rotational noise increases, the average deformation in the system tends to the
deformation of a passive system. The question that in this section will be
addressed is: Can we provide some theoretical explanation to the previous
numerical experiments? If we carefully analyze Eqs. (1) and (3), we can
extract some sufficient ingredients of its dynamics; namely, an active
stochastic force, a deformation force and a nonlinear interaction. Notice that
the nonlinear ingredient was numerically observed to be crucial for the
deformation of the system to become a function of activity. Let us keep those
ingredients, and analyze the dynamics of the building block of an active solid
with a minimal model. This consists of a driven single mass, attached to a
nonlinear spring and subject to an external constant force. In addition, and
in order to make some theoretical progress, let us consider a weakly nonlinear
spring ($\varepsilon\widetilde{x}^{3}$) such that $\varepsilon<<1$; with
weakly dissipation ($\alpha=\varepsilon\beta$); weakly deterministic driven
($\widetilde{F}_{0}=\varepsilon f\cos(\widetilde{\Omega}\tilde{t})$) and
deformation forces ($\widetilde{F}=\varepsilon f_{D}$); henceforth, our
building block satisfies the so-called Duffing equation with an extra
deformation force, expressly

$\frac{d^{2}\widetilde{x}}{{d\tilde{t}}^{2}}+\widetilde{x}+\varepsilon\beta\frac{d\widetilde{x}}{d\tilde{t}}{+}\varepsilon\widetilde{x}^{3}=\varepsilon
f\cos(\widetilde{\Omega}\tilde{t})+\varepsilon f_{D}\mathbf{,}$ (7)

where the dimensionless angular frequency of the driven force
($\widetilde{\Omega}=\Omega/\sqrt{k/M}$) near resonance satisfies
$\widetilde{\Omega}=1+\varepsilon\omega$, with $\omega$ a driver dimensionless
angular frequency of order one. Note that the justification for the
implementation of a deterministic driven force in this model, is given in
Section VI. Following the method of strained coordinates, a slow variable,
$\tau=\varepsilon\tilde{t}$, will also be introduced. Seeking a perturbation
solution to Eq. (7) of the form
$\widetilde{x}(\tilde{t},\tau)=\widetilde{x}_{0}(\tilde{t},\tau)+\varepsilon\widetilde{x}_{1}(\tilde{t},\tau)+\varepsilon^{2}\widetilde{x}_{2}(\tilde{t},\tau)+O(\varepsilon^{3})$
and substituting it into Eq. (7), it is possible to prove (after neglecting
terms of $O(\varepsilon^{3}),$ and defining the operators
$L[]=\partial^{2}/\partial\tilde{t}^{2}+1$ and
$P[]=-2\partial^{2}/\partial\tilde{t}\partial\tau-\beta\partial/\partial\tilde{t}$)
that the following coupled system arises

$\displaystyle L[\widetilde{x}_{0}]$ $\displaystyle=$ $\displaystyle 0,$ (8)
$\displaystyle L[\widetilde{x}_{1}]$ $\displaystyle=$ $\displaystyle
P[\widetilde{x}_{0}]-\widetilde{x}_{0}^{3}+f\cos\left(\tilde{t}+\omega\tau\right)+f_{D},$
(9) $\displaystyle L[\widetilde{x}_{2}]$ $\displaystyle=$ $\displaystyle
P[\widetilde{x}_{1}]-\frac{\partial^{2}\widetilde{x}_{0}}{\partial\tau^{2}}-\beta\frac{\partial\widetilde{x}_{0}}{\partial\tau}-3\widetilde{x}_{0}^{2}\widetilde{x}_{1}.$
(10)

After eliminating secular terms in Eqs. (8)-(10), their remaining
inhomogeneous terms will be harmonic, subharmonics, and constants terms. The
constant terms precisely provide the information for the net deformation in
the system as a function of stretching/compressing forces. This deformation
can be explicitly calculated, namely

$\Delta L=\varepsilon f_{D}-\varepsilon^{2}\frac{3A^{2}f_{D}}{2},$ (11)

where $A$ represents the steady amplitude of Eq. (7). After the elimination of
the secular terms, this amplitude is found to satisfy Kevorkian and Cole
(1981)

$\beta^{2}A^{2}+\left(2\omega A-\frac{3A^{3}}{4}\right)^{2}=f^{2}.$ (12)

The latter expression couples the steady amplitude and the propulsion force.
Thus Eq. (11) indicates that only to $O(\varepsilon^{2})$ and due to
nonlinearity, activity will influence the system’s deformation. To prove that
our analytical expression, Eq. (11), correctly describes such a deformation in
our building block; we numerically solve Eq. (7) under the following
parameters:
$\varepsilon=\left\\{0.2,0.17\right\\},\beta=2,f=\left[1,5\right]$, and
$\widetilde{\Omega}=1.5$. Note that the previous parameters were chosen in
such a way that the acting forces in the building block do not deform it
beyond a third of its original equilibrium distance $a$. The results can be
seen in Fig. 9, where the deformation in the building block as a function of
self-propulsion is illustrated. The predicted theoretical deformation using
Eq. (11) is shown as a red line, whereas the numerical solution of Eq. (7) is
shown in blue circles. This figure shows an excellent agreement between theory
and simulations, hence one can argue that Eq. (11) contains sufficient
ingredients to understand the dependence of strain on activity in our building
block. In addition, one can see that this figure possesses a very similar
behavior as Fig. 4 and Fig. 7(b), which allow us to have some physical insight
on the behavior of those plots.

##  VI Possible experiment

At this stage, theoretical and numerical analysis have been offered but, Is it
possible to build an experiment? What are the minimum components to make an
active solid? To explicitly see this, let us take some macroscopic
experimental values reported by Lowen et. al Scholz  _et al._ (2018). Their
experiment considers a vibrobot which as it will be seen, may be a candidate
for experimentally realizing our active solid. Using a specific set of their
data for a given vibrobot, namely, $M=0.004kg$, $U=0.09m/s$ (propulsion
speed), $\Omega=0.7s^{-1}$ (intrinsic angular velocity), $D_{B}=8\times
10^{-5}m^{2}/s$, $D_{R}=2.59s^{-1}$, and $R_{T}/M=6.46s^{-1}$, and assuming a
strong trap constant (see Tapia-Ignacio, Gutierrez-Martinez, and Sandoval
(2021) for the strong trap condition) of $k=0.1kgs^{-2}$, a deformation force
$f_{D}=1N$, and a separation equilibrium distance between robots of $a=0.15m$;
leads to the first row of Table 1, where orders of magnitude for some relevant
experimental parameters are indicated.

| $\alpha$ | $\widetilde{D}_{R}$ | $\widetilde{\Omega}$ | $\widetilde{F}_{0}$
---|---|---|---|---
Vibrobot | 1.4 | 0.54 | 0.14 | 0.17
Hexbug | 1.9 | 0.02 | 0.17 | 0.14
Table 1: Some relevant experimental dimensionless quantities, for a vibrobot
Scholz  _et al._ (2018) and a hexbug-nano robot Tapia-Ignacio, Gutierrez-
Martinez, and Sandoval (2021), appearing in Eqs. (1) and (13).

In addition, Sandoval et. al Tapia-Ignacio, Gutierrez-Martinez, and Sandoval
(2021) considered the dynamics of trapped hexbug-nano robots (another possible
candidate for the experimental realization of the current system). They
emphasized that hexbugs also posses an intrinsic torque which makes those
robots to rotate with an angular velocity of around $\Omega=1.9s^{-1}$. Thus
by using the parameters describing those robots namely, $U=0.13m/s$
(propulsion speed), $M=0.0072kg$, and $S_{T}=0.0027$ (Stokes translational
number), where $S_{T}=\tau_{M}/\tau_{R}$ with $\tau_{M}=M/R_{T}$, and
$\tau_{R}=1/D_{R}$; and assuming a strong trap constant of $k=0.9kgs^{-2}$
together with $D_{B}=8\times 10^{-5}m^{2}/s$, $a=0.15m$, and $f_{D}=1N$; one
gets the second row of Table 1. Interestingly, this table shows that the
dimensionless experimental quantities fall into the numerical parameters used
during the simulations. This directly establises a connection between theory,
simulations and experiments. Notice as well that both of the mentioned teams
propose an internal degree of freedom ($\varphi$) whose dynamics in
dimensionless variables reads

$d\varphi=\tilde{\Omega}d\widetilde{t}+\sqrt{2\widetilde{D}_{R}}dW,$ (13)

where $W$ represents a standard Wienner process. Therefore, based on Table 1,
one could approximate the rotational dynamics of a hexbug-nano simply as
$d\varphi=\tilde{\Omega}d\widetilde{t}$, hence the projected active force,
$\varepsilon fcos(\varphi)$, would become $\varepsilon fcos(\tilde{\Omega})$,
thus justifying the proposed driven force in Eq. (7). Moreover, the discussed
order of magnitude for hexbugs can be visualized in the supplementary material
video: Home-made-experiment.mov, where two hexbugs are practically rotating
clockwise and with a very small angular noise. A snapshot of the mentioned
video is offered in Fig. 10. Here, springs taken from pens, two glasses as
corrals, two-hexbugs-nano, two walls, and a pulling force (hand of the author)
are shown to conform an active solid deformation experiment. These would be
the minimum components for a more sophisticated experiment.

Figure 10:  Possible experimental realization of an 1D active solid using
hexbugs-nano robots. Notice that although the robots freely spin and hence
propel towards any direction, only their active forces projections (and due to
the presence of walls) along the $x$-axis affect the active solid dynamics.

##  VII Conclusions

It has been shown that a mixture of self-propulsion and nonlinear
interactions; in particular, a hard spring symmetric potential, is sufficient
to render an active solid the ability to deform less on average –compared to a
passive one– when subject to a deformation force. The latter observation may
serve as another mechanism to creating novel active metamaterials (a material
with its own energy, that given its similarity with the movie ’Flubber’ Hughes
(1997), it could be called a flubber material). The dependence of the studied
solids’ strain on activity was explained by introducing a minimal model that
proved that nonlinearity is the responsible for this feature. The strain
behavior versus stress from all the analyzed active solids collapsed to a
single curve, meaning that the properties of our new material remain the same
irrespective of the length of the sample. It was also discovered –see Fig. 5–
that as an active solid under stress gains internal energy, it tends to behave
Hookean, that is, a linear behavior seems to dominate over a nonlinear one.
This switching between a Hookean and a nonlinear behavior has not been
observed before and may be related to the way adherent cells modify their
elastic properties according to their needs Koenderink  _et al._ (2009). In
addition, by using available experimental data Scholz  _et al._ (2018); Tapia-
Ignacio, Gutierrez-Martinez, and Sandoval (2021), it was possible to show that
the numerical parametes used in this work fall into real orders of magnitude
of an experiment, thus establishing a connection between theory, simulations
and experiments.

Although further investigations are needed to see if this ’flubber material’
has some practical applications, probably, those applications may be in a
close manner to classical metamaterials, which have potential impact on energy
absorbing devices (helmets, footwear), noise reduction, automobile parts,
prosthetic implants, among others. Future directions, and already in progress,
will be to evaluate the effect of nonlinearity and activity, on the speed of
waves propagating through this system, as well as heat transferring on these
structures. I believe that the present active solid can be experimentally
realized and its extension to higher dimensions, already in progress by the
author, may be significative for the physics and materials science
communities.

##  VIII Data availability statement

This manuscript does not have associated data

##  IX Acknowledgements

M. S. thanks Consejo Nacional de Ciencia y Tecnologia, CONACyT for support.
The author also thanks nature for giving him extra time.

Figure 11:  Strain of each particle in an active/passive solid made of $N=6$
masses reaching a steady state.

##  X Appendix

###  X.1 Reaching a steady state

In this appendix, the way the strain of each particle, composing either an
active or passive solid made of $N=6$ masses, reaches a steady state is
illustrated in Fig. 11. The numerical values for the simulations are
$\lambda=20,\widetilde{D}_{R}=3,\widetilde{F}=3,\widetilde{F}_{0}=0.5$,
$\Delta\widetilde{t}=0.02$, a total time of $\widetilde{T}=100$, and $1000$
realizations to calculate the respective ensemble averages. Dashed lines
represent the passive chain dynamics, whereas solid lines represent the active
counterpart. Clearly, self-propulsion decreases on average the strain in the
system.

###  X.2 Case $\widetilde{D}_{R}\rightarrow 0$

Figure 12 illustrates the convergence to a minimum deformation of an active
solid made of $N=16$ particles, as rotational noise tends to zero
($\widetilde{D}_{R}\rightarrow 0$). Here, the domain of the rotational
diffusivity varies along three orders of magnitude, namely,
$\widetilde{D}_{R}=[0.005,0.05,0.5,1]$. The parameters for the simulations are
500 realizations, $\widetilde{F}_{0}=0.5$, $\widetilde{F}=0.3$, and $T=2000$.
Note that the case $\widetilde{D}_{R}=0$ does not provide an absolute minimum
since our model assigns random angles to the active particles, thus depending
on the random distribution of active forces, the active solid may deform more
or less from the true minimum due to activity. Some trivial configurations
could also be proposed. For example, all the active forces having $\pi$
radians. This leads to a minimum deformation of the active solid when subject
to a stretching force. However, this is not a symmetric configuration when
subject to a compressing force. Other simpler configuration that has seen very
effective for a hardening effect, is to locate at each particle a 1D central
force field Koenderink  _et al._ (2009). Clearly, this force configuration
will stiffen the solid.

Figure 12: Convergence to a minimum deformation of an active solid made of
$N=16$ particles, as rotational noise tends to zero. Its domain in this plot
is $\widetilde{D}_{R}=[0.005,0.05,0.5,1]$.

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