# Effect of prestress on phononic band gaps induced by inertial amplification

M. Miniaci  CNRS, Centrale Lille, ISEN, Univ. Lille, Univ. Valenciennes, UMR
8520 - IEMN, F-59000 Lille, France  M. Mazzotti  Department of Mechanical
Engineering, CU Boulder, 1111 Engineering Drive, UCB 427 Boulder, CO 80309,
USA  A. Amendola  Department of Civil Engineering, University of Salerno, Via
Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy  F. Fraternali  Department
of Civil Engineering, University of Salerno, Via Giovanni Paolo II, 132, 84084
Fisciano (SA), Italy<EMAIL_ADDRESS>
††preprint: Submitted to

Phononic crystals and elastic metamaterials have recently received significant
attention due to their potential for unconventional wave control. Despite this
interest, one outstanding issue is that their band diagram is typically fixed,
once the structure designed. To overcome this limitation, periodic structures
with adaptive elastic properties have recently been proposed for Bragg- and
local resonance-driven structures.
In this context, we report about the effect of an applied external mechanical
load on a periodic structure exhibiting band gaps induced by inertial
amplification mechanism. If compared to the cases of Bragg scattering and
ordinary local resonant metamaterials, we observe here a more remarkable curve
shift, modulated through large but fully reversible compression(stretch) of
the unit cell, eventually triggering significant (up to two times)
enlargement(reduction) of the width of a specific band gap. An important
up(down)-shift of some dispersion branches over specific wavenumber values is
also observed, showing that this selective variation may lead to negative
group velocities over larger(smaller) wavenumber ranges. In addition, the
possibility for a non-monotone trend of the lower limit of the first BG under
the same type of external applied prestrain is found and explained through an
analytical model, which unequivocally proves that this behaviour derives from
the different unit cell effective mass and stiffness variations as the
prestrain level increases. These peculiarities derive from the hinge-like
behaviour of some regions of the unit cell, which is typical of structures
exhibiting the inertial amplification mechanism.
The effect of the prestress on the dispersion diagram is investigated through
the development of a 2-step calculation method: first, an Updated Lagrangian
scheme, including a static geometrically nonlinear analysis of a
representative unit cell undergoing the action of an applied external load is
derived, and then the Floquet-Bloch decomposition is applied to the linearized
equations of the acousto-elasticity for the unit cell in the deformed
configuration.
Finally, the most evident consequence on the dispersion curves of the
application of an external prestress, i.e. the band gap shift with respect to
the unloaded structure, is demonstrated through nonlinear transient numerical
simulations, clearly proving the capability of the structure to switch from a
pass- to a stop-band behaviour over the same frequency range.
The results presented herein provide insights in the behaviour of band gaps
induced by inertial amplification, and suggest new opportunities for real-time
tunable wave manipulation.

##  I Introduction

Phononic crystals and elastic metamaterials have proved to be powerful
platforms to achieve a plethora of unconventional vibrational behaviours
deriving from their peculiar dispersion diagrams, characterized by frequency
band gaps (BGs) and pass-bands, i.e. frequency regions where the propagation
of waves is inhibited or supported, respectively [1]. This allowed to achieve
frequency selective wave filtering [2], guiding [3, 4, 5] and splitting [6,
7], stimulating the conception of pioneering applications, such as ultra-
sensitive devices [8], large scale metamaterials for vibration shielding [9,
10, 11], sub-wavelength imaging [12], elastic wave cloaking / lensing [13, 14,
15], etc.
In this context, an important issue is that phononic crystal and elastic
metamaterial operational frequencies are typically fixed once the structure
has been designed and fabricated. In contrast, in the majority of practical
applications, including the ones previously mentioned, it may be desirable to
dynamically or adaptively tune the BGs (or some of the dispersion branches) in
terms of frequency, also after the structure has been fabricated. To overcome
this limitation, periodic systems with adaptive elastic properties have
recently been proposed. For instance, BG tuning has been achieved by means of
piezoelectric materials [16, 17, 18], temperature variation [19, 20, 21, 22],
magneto- and light-based approaches [23, 24, 25, 26, 27], as well as by the
application of external mechanical loads [28]. Among them, the latter approach
can be easily implemented by imposing controlled displacements into specific
portions of the structure.
In this context, Bigoni et al. [29] proposed for the first time the prestress
as a practical way to reversibly alter the dispersion diagram of a periodic
structure, including the possibility of shifting the BG position. The
feasibility of the approach was confirmed formulating a theoretical model for
an orthotropic, prestressed (compressible) elastic layer vibrating on an
elastic half space and assuming long-wave asymptotics for the solution. Gei et
al. [30] relaxed the hypothesis of perfect periodicity and investigated the
effect of the prestress in quasi-periodic structures in the case of flexural
vibrations. Amendola et al. [31] studied the band structure of tensegrity
mass-spring chains, and the possibility to tune the dispersion relation of
such systems by suitably varying local and global prestress variables, given
their remarkable softening / stiffening response under axial or compressive
loading [32, 33].
Periodic elastomeric structures, thanks to their capacity of repetitively
undergoing large strain deformations in a fully reversible manner, brought to
BG nucleation / annihilation mechanisms triggered by the application of
external loads radically changing the unit cell geometry [34]. Deformations in
the linear and nonlinear regimes, as well as diverse geometrical topologies
have been explored [35, 36, 37, 38].
Finally, it has been observed that also mechanical instabilities induced, for
instance, by the application of external loads, may alter the propagation of
elastic waves. Slesarenko et al. induced instabilities in soft composite
materials achieving a significant decrease of the group velocity (up to going
negative) of the transverse waves under specific micropolar conditions [39].
However, the majority of the aforementioned investigations are limited to the
context of (i) Bragg- or (ii) ordinary resonance-induced BGs, whereas in the
present paper, we report about the effects of an applied prestress on (iii)
periodic structures exhibiting BGs induced by inertial amplification
mechanisms. While in (i) the wave scattering derives from assemblages of
periodic unit cells, requiring thus the wavelength of the incoming wave to be
comparable to the structural periodicity [40], and in (ii) the dynamic
behaviour is mainly governed by the eigenfrequencies of resonators included in
the structure [41], in (iii) large inertial forces are generated by amplifying
the motion of a mass, which in turn increases the inertia of the overall
system and lowers its resonance frequency, allowing thus for sub-wavelength
and broadband BGs, while keeping the structure lightweight [42, 43].
If compared to the cases of Bragg scattering and ordinary local resonant
metamaterials [44], we show that the inertial amplification allows for a more
remarkable curve shift, modulated through large but fully reversible
compression(stretch) of the unit cell, eventually triggering significant
enlargement(reduction) of a specific band gap width. We also observe a
selective up(down)-shift of some dispersion branches, leading to negative
group velocities over specific wavenumber ranges. In addition, a non-monotone
trend of the lower limit of the first BG under the same type of external
applied prestrain is found and explained through an analytical model, which
shows that this behaviour derives from the different unit cell effective mass
and stiffness variations as the prestrain level increases. Examining the
deformed geometries consequent the application of the prestress highlights how
the remarkable band shift is due to the hinge-like behaviour of some regions
of the unit cell, typical of the inertial amplification mechanism and
responsible for a large deformation state, i.e. change of the unit cell
geometry. We consider the static deformation induced by the prestress to be in
the linear elastic regime so to have a complete reversibility of the phenomena
(tunability). The analysis is performed in terms of small amplitude motions
superimposed on a deformed state once the desired load has been applied.
The paper is organized as follows: in section II, the 2-step Updated
Lagrangian scheme, including (i) a static geometrically nonlinear analysis of
a representative unit cell undergoing the action of an applied external load
and (ii) the Floquet-Bloch decomposition applied to the linearized equations
of the acousto-elasticity for the unit cell in the deformed configuration, is
presented. Section III provides evidence for the dispersion band alteration
induced by the application of the prestress in a periodic structure exhibiting
BG induced by inertial amplification mechanism. Parametric dispersion curves
for different values of prestress are presented and compared to the original
structure (i.e., without the application of any prestress). Afterwards, a
2-step nonlinear transient numerical simulation confirms the BG shift induced
by the prestress in a finite structure, proving its ability to switch from a
pass- to a stop-band behaviour over the same frequency range. The prestress is
easily implemented by applying prescribed displacement at specific edges of
the structure prior to the wave propagation. The full tunability of the
structure is guaranteed by the possibility of readily applying and removing
the imposed deformations. Finally, section IV summarizes the main results of
the present research and provides future perspectives, shedding light on the
possibility of investigating the effect of an additional state of prestress,
internal, cross-linking the fields of metamaterials and tensegrity structures.

##  II Floquet-Bloch Analysis of a Prestressed Phononic Crystal

In this work, the band structure diagrams are computed using a Floquet-Bloch
finite element method formulated within an Updated Lagrangian scheme. As
schematically presented in Fig. 1, the procedure consists of two main
computational steps, namely (i) a nonlinear static analysis involving large
strains and displacements, and (ii) a so called small-on-large dynamic
analysis, in which small vibrations are superimposed on the statically
deformed unit cell. The main computational aspects of the two steps are
outlined in the following.

###  II.1 Static analysis

Following the application of a static volume load $\mathbf{f}_{V0}$, or
surface load $\mathbf{f}_{S0}$, the unit cell $\Omega_{0}$ identified by the
position vector $\mathbf{x}$ and represented by the lattice vector
$\mathbf{r}_{0}=\\{r_{01},0\\}^{\rm{T}}$ and its reciprocal vector
$\mathbf{g}_{0}=\\{g_{01},0\\}^{\rm{T}}=\\{r_{01}/2\pi,0\\}^{\rm{T}}$,
undergoes a displacement $\mathbf{u}_{0}$ that results in a change of
configuration from the undeformed state $C_{0}$ to the static deformed state
$C$ (see Fig. 1). The deformed domain and matrix of the lattice vectors in the
(deformed) configuration $C$ are identified by $\Omega$ and
$\mathbf{r}=\\{r_{1},0\\}^{\rm{T}}$, respectively. The relation between
$\mathbf{r}_{0}$ and $\mathbf{r}$ can be expressed as
$\mathbf{r}=\mathbf{F}_{L}\mathbf{r}_{0}$, where $\mathbf{F}_{L}$ defines the
affine component of the deformation gradient such that
$\mathbf{F}_{L}=\mathbf{F}^{-1}\mathbf{F}_{P}$, being
$\mathbf{F}=\nabla_{\mathbf{x}}\mathbf{u}_{0}+\mathbf{I}$ the deformation
gradient, $\mathbf{F}_{P}$ a periodic non-affine deformation [45] and
$\nabla_{\mathbf{x}}=\\{\partial/\partial x_{1},\partial/\partial
x_{2}\\}^{\rm{T}}$. The equilibrium equations with respect to the undeformed
configuration $C_{0}$ can be derived from the variational statement:

$\int_{\Omega}\left(\mathbf{S}(\mathbf{x}):\delta\mathbf{E}(\mathbf{x})-\mathbf{f}_{V0}\cdot\delta\mathbf{u}_{0}\right)da=\int_{\partial\Omega}\mathbf{f}_{s0}\cdot\delta\mathbf{u}_{0}ds,$
(1)

subjected to the Dirichlet boundary conditions:

$\mathbf{u}_{0}(\mathbf{x}+\mathbf{r}_{0})=\mathbf{u}_{0}(\mathbf{x}),$ (2)

in which $\mathbf{E}=\frac{1}{2}(\mathbf{F}^{\rm{T}})\mathbf{F}-\mathbf{I}$ is
the Green-Lagrange strain tensor and $\mathbf{S}=\mathbf{D}_{0}:\mathbf{E}$
the second Piola-Kirchhoff stress tensor. The tensor of tangential elastic
moduli, $\mathbf{D}_{0}$, is expressed by
$\mathbf{D}_{0}=4\partial^{2}\psi/(\partial\mathbf{C}\partial\mathbf{C})$,
being $\mathbf{C}=\mathbf{F}^{\rm{T}}\mathbf{F}$ the right Cauchy-Green
deformation tensor and $\psi$ the elastic energy density. The material of the
unit cell in $C_{0}$ is specified by the density $\rho_{0}$ while, assuming a
hyperelastic material behavior described by the Murnaghan’s model [46, 47, 48,
49], the elastic energy density can be defined as:

$\begin{split}\psi=&\frac{1}{2}\left(\lambda+2\mu\right)I_{1}^{2}(\mathbf{E})-2\mu
I_{2}(\mathbf{E})+\frac{1}{3}\left(l+2m\right)I_{1}^{3}(\mathbf{E})\\\
&-2mI_{1}(\mathbf{E})I_{2}(\mathbf{E})+nI_{3}(\mathbf{E}),\end{split}$ (3)

in which $\lambda$ and $\mu$ denote the first and second Lamé parameters,
respectively, ($l$, $m$, $n$) the third order Murnaghan parameters, and
$I_{1}(\mathbf{E})$, $I_{2}(\mathbf{E})$ and $I_{3}(\mathbf{E})$ the first,
second and third invariants of the Green-Lagrange strain tensor, respectively.
The application of a standard Galerkin approach to Eq. (1) results in the
generalized system of equations:

$\left[\bm{\Gamma}_{0}^{\mathrm{T}}\mathbf{K}(\mathbf{Q}_{0})\bm{\Gamma}_{0}\right]\mathbf{Q}_{0}(\mathbf{X})=\mathbf{P}_{0}(\mathbf{X}),$
(4)

where $\mathbf{K}(\mathbf{Q}_{0})$ is the static stiffness matrix,
$\mathbf{P}_{0}$ the global vector of nodal forces, $\mathbf{Q}_{0}$ the
global vector of independent nodal displacements and $\bm{\Gamma}_{0}$ the
mapping operator resulting from Eq. (2) and realizing the condition
$\mathbf{U}_{0}=\bm{\Gamma}_{0}\mathbf{Q}_{0}$. $\mathbf{U}_{0}$ indicates the
full vector of nodal displacements. In this work, the solution of Eq. (4) is
carried out using Comsol Multiphysics 5.3 [50].
Following the Updated Lagrangian scheme, once the displacements
$\mathbf{Q}_{0}$ are obtained, the reference configuration is updated from
$C_{0}$ to $C$ by calculating the corresponding nodal coordinates
$\mathbf{x}=\mathbf{x}_{0}+\bm{\Gamma}_{0}\mathbf{Q}_{0}(\mathbf{x}_{0})$. The
updated material properties in $C$ are given by
$\rho=\rho_{0}(\det\mathbf{F})^{-1}$ and
$D_{ijkl}=(\text{det}\mathbf{F})^{-1}F_{iI}F_{jJ}F_{kK}F_{lL}(D_{0})_{IJKL}$,
while the Cauchy stress tensor is obtained from the relation
$\bm{\sigma}=(\det\mathbf{F})^{-1}\mathbf{F}\mathbf{S}\mathbf{F}^{\mathrm{T}}$.
The geometry of the unit cell in the configuration $C$ is then re-meshed and
used as the basis for the linear dynamic analysis described in the next
section.

###  II.2 Dynamic Analysis Using the Floquet-Bloch Decomposition

Following the small-on-large analysis approach [51, 52, 47, 53, 54], in which
$C$ is the new reference configuration, the position vector for the unit cell
in the dynamic deformed configuration $C^{\prime}$ (Fig. 1) is approximated as
$\mathbf{x}^{\prime}\approx\mathbf{x}$ while, from the application of the
Floquet-Bloch theorem, any small harmonic perturbation
$\mathbf{u}(\mathbf{x})$ can be expressed as [55]:

$\mathbf{u}(\mathbf{x})=\tilde{\mathbf{u}}(\mathbf{x})\text{exp}(\text{i}kx)\text{exp}(-\text{i}\omega
t),$ (5)

in which $\tilde{\mathbf{u}}(\mathbf{x})$ is a $\Omega$-periodic displacement
amplitude, $t$ denotes the time, $\omega$ the angular frequency, and
$k\in\Lambda$ the Bloch wavenumber, being $\Lambda$ the reciprocal unit cell
defined in $C$ by the reciprocal lattice vector
$\mathbf{g}=\mathbf{F}_{L}^{-\rm{T}}\mathbf{g}_{0}$.
By defining the $k$-shifted gradient of a generic $\Omega$-periodic vector
field $\tilde{\phi}(\mathbf{x})$ as:

$\nabla_{k}\tilde{\phi}(\mathbf{x})=\nabla_{\mathbf{x}}\tilde{\phi}(\mathbf{x})+\mathrm{i}k\tilde{\phi}(\mathbf{x})\otimes(\mathbf{r}\|\mathbf{r}\|^{-1}),$
(6)

the solution of the elastodynamic problem for free vibrations of the unit cell
in $C$ subjected to an initial stress $\bm{\sigma}_{0}$ can be obtained from
the variational statement:

$\begin{split}-\omega^{2}&\int_{\Omega}\rho(\mathbf{x})\delta\tilde{\mathbf{u}}^{*}(\mathbf{x})\cdot\tilde{\mathbf{u}}(\mathbf{x})\mathrm{d}\Omega\\\
+&\int_{\Omega}\delta\tilde{\mathbf{e}}_{k}^{*}(\mathbf{x},\vartheta):\mathbf{D}(\mathbf{x}):\tilde{\mathbf{e}}_{k}(\mathbf{x})\mathrm{d}\Omega\\\
+&\int_{\Omega}\bm{\sigma}_{0}(\mathbf{x}):\left[\left(\nabla_{k}\delta\tilde{\mathbf{u}}^{*}(\mathbf{x})\right)^{\mathrm{T}}\nabla_{k}\tilde{\mathbf{u}}(\mathbf{x})\right]\mathrm{d}\Omega=0,\end{split}$
(7)

subjected to the Dirichlet boundary condition:

$\tilde{\mathbf{u}}(\mathbf{x}+\mathbf{r})=\tilde{\mathbf{u}}(\mathbf{x})\quad\text{on}\;\partial\Omega,$
(8)

in which $(\cdot)^{*}$ stands for the conjugate of a complex vector or tensor
field,
$\tilde{\mathbf{e}}_{k}(\mathbf{x})=\frac{1}{2}[\nabla_{k}\tilde{\mathbf{u}}(\mathbf{x})+(\nabla_{k}\tilde{\mathbf{u}}(\mathbf{x}))^{\mathrm{T}}]$
follows from Eq. (6) and denotes the linearized Green-Lagrange strain tensor.
The finite element discretization of Eqs. (7) and (8) proceeds by first
generating a new mesh for the deformed geometry of the unit cell in $C$ and
then applying a Galerkin approach [56]. As a results, the following
generalized linear eigenvalue problem is obtained:

$\left\\{\bm{\Gamma}^{\mathrm{T}}\left[k^{2}\mathbf{K}_{3}+\mathrm{i}k\left(\mathbf{K}_{2}-\mathbf{K}_{2}^{\mathrm{T}}\right)+\mathbf{K}_{1}-\omega^{2}\mathbf{M}\right]\bm{\Gamma}\right\\}\tilde{\mathbf{Q}}(\omega)=\mathbf{0},$
(9)

in which $\bm{\Gamma}$ is a mapping operator implementing the Dirichlet
boundary condition in Eq. (8), such that
$\tilde{\mathbf{U}}=\bm{\Gamma}\tilde{\mathbf{Q}}$, where $\tilde{\mathbf{U}}$
is the global vector of nodal displacement amplitudes and $\tilde{\mathbf{Q}}$
a subvector of $\tilde{\mathbf{U}}$ collecting only its independent
components. The expressions of $\mathbf{K}_{3}$, $\mathbf{K}_{2}$,
$\mathbf{K}_{1}$ and $\mathbf{M}$ are given in the Appendix A, for the sake of
brevity.
The band diagrams of the phononic structure can be computed from the
eigenvalue problem in Eq. (9) (i) by extracting the Bloch wavenumbers
$k(\omega)$ for any fixed frequency $\omega$, or (ii) by computing the natural
frequencies $\omega(k)$ of the system for any fixed Bloch wavenumber $k$.
Since the focus of the present research does not require the knowledge of the
spatial attenuation, the latter approach has been used, resulting
computationally more convenient, as it does not require the transformation of
the system into the state space.

##  III Results

To study the effect of an external mechanical load on a periodic structure
exhibiting BGs induced by inertial amplification, the numerical method
presented in the previous section is here applied to the representative unit
cell reported in Fig. 2A. The unit cell consists of an epoxy matrix exhibiting
elongated cross-like holes, and it was proposed for the first time by Acar and
Yilmaz [43]. Its geometrical and material parameters are reported in Tables 1,
2.

Table 1: Material constants for the epoxy [44]. |  $\rho_{0}$ [kg$/$m3] |  $\lambda$ [GPa] |  $\mu$ [GPa] |  $l$ [GPa] |  $m$ [GPa] |  $n$ [GPa]
---|---|---|---|---|---|---
Epoxy | 1540 | 2.59 | 1.34 | -18.94 | -13.36 | -9.81
Table 2: Geometrical parameters of the unit cell presented in Fig. 2A. All the given parameters are in [mm]. $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $d_{1}$ | $d_{2}$ | $d_{3}$ | $d_{4}$
---|---|---|---|---|---|---|---
3.2 | 0.4 | 3.2 | 0.4 | 5.2 | 2.0 | 20.0 | 4.0

###  III.1 Band structure analysis

As discussed in Section II, the first step for the band structure calculation
is to extract the deformation induced in the unit cell by an initial state of
stress / strain applied to the structure. To do this, two different sets of
loading conditions are applied to the structure in the form of a normal
displacement
$\mathbf{u}_{0}(\mathbf{x}_{0})\cdot\mathbf{n}_{0}(\mathbf{x}_{0})$ prescribed
to the $x_{02}$-parallel faces of the unit cell (highlighted in purple in Fig.
2A). The out-of-plane degrees of freedom of the unit cell are blocked, so to
prevent any possible bending deformation during the application of the loading
conditions. The first loading set induces a state of compression and varies
from $0$ (no prestrained condition) to $-360$ $\mu$m (maximum compression
condition) with a step of
$\Delta\mathbf{u}_{0}(\mathbf{x}_{0})\cdot\mathbf{n}_{0}(\mathbf{x}_{0})=-20$
$\mu$m, while the second set induces a tensile state and varies from $0$ (no
prestrained condition) to $+130$ $\mu$m (maximum traction condition) with a
step of
$\Delta\mathbf{u}_{0}(\mathbf{x}_{0})\cdot\mathbf{n}_{0}(\mathbf{x}_{0})=+10$
$\mu$m. For the sake of brevity, only the deformation of the unit cell under
the conditions of maximum compression / traction are reported in Figs. 2B,C,
respectively. Examining the induced deformation, it is possible to infer that
the stress, and thus the deformation, is mainly localized into the hinge-like
regions, responsible for the activation of the inertial amplification
mechanism [43]. The analysis also shows that the maximum Von Mises stress
level reached in the structure is of 33.1 and 21.2 MPa, for the two types of
loading condition, respectively. Given these values of maximum stress, and
applying the von Mises yield criterion [57], the minimum mono-axial tensile
strength required for our material to have a safety factor $\geq 1$ is
$\sigma_{min}=33.1\cdot\sqrt{(}3)=56$ MPa (if we consider a state of pure
shear solicitation), which is an acceptable value for epoxy [58]. Considering
that the aforementioned value is obtained in the most strict condition of pure
shear (which is not the case in the system under consideration), an elastic
behaviour of the material, and thus of the band diagram, over the full range
$[-360,+130]$ $\mu$m, modulated by the intensity of the applied prestrain, is
guaranteed, allowing for a full reversibility of the undeformed configuration
(original dispersion diagram), once the load is removed. For the sake of
completeness, since the small on large theory is considered here, the dynamic
component of the stress is negligible with respect to the static prestress.
For this reason it can be assumed that the safety factor does not change
between the static and dynamic configuration.
Once the static analysis performed for the aforementioned sets of prestrain,
the deformed geometries are assigned as the input unit cells to calculate the
dispersion diagrams exploiting the Floquet-Bloch theory (see section II). The
out-of-plane displacement of the structure is kept blocked also in this phase,
in order to limit the dispersion analysis to waves belonging to the
$x_{1}-x_{2}$ plane. The band structures are computed considering the unit
cell to infinitely duplicate in a periodic linear array, and assuming the
epoxy in its linear elastic regime (the hypothesis of small displacements is
now applied). The unit cell domain is meshed by means of 8-node hexagonal
elements of maximum size $L_{FE}=0.2$ mm, which is found to provide accurate
eigensolutions up to the frequency of interest [59]. It should be noted that,
since the applied prestrain applied to each face of the cell induces isotropic
deformation in the $x_{1}-x_{2}$ plane in both compression and traction cases,
the deformation gradient $\mathbf{F}$ and its affine component
$\mathbf{F}_{L}$ are diagonal and, as a consequence, the orientation of the
reciprocal lattice vectors $\mathbf{g}_{1}$ in the deformed configurations
does not change with respect to that in the undeformed configuration. This
implies that the orientation of the Bloch wavevector also remains unchanged
between the undeformed and deformed configurations (see Fig. 1D). Therefore,
the resulting eigenvalue problem
$(\mathbf{K}-\omega^{2}\mathbf{M})\mathbf{u}=\mathbf{0}$ is solved by varying
the non-dimensional wavevector $\textbf{k}^{*}$ along the irreducible path
$\left[\Gamma-X\right]$, with $\Gamma\equiv(0,0)$ and $X\equiv\pi/a,0)$, being
$a=\left(2\cdot\sum_{i=1}^{3}d_{i}+d_{4}\right)$ the lattice parameter.
Figure 3 reports parametric plots of the dispersion diagrams of the unit cell
as a function of the external prestrain intensity inducing compression (Fig.
3A) or traction (Fig. 3B) states in the structure. The dispersion curves are
color-coded on the base of the level of prestrain applied at the boundaries of
the unit cell in the pre-loading phase. Specifically, the color bar of Fig. 3A
varies gradually from $-360$ $\mu$m (blue: maximum compression state) to $0$
(green: unprestrained condition), respectively. Analysing the band diagrams,
it is possible to observe that increasing the compressive state in the unit
cell induces (i) a general up-shift (curves shading into dark blue) of the
dispersion branches, as well as (ii) group velocity inversion in the third
band when $k^{*}$ gets close to the high symmetry point $X$, as indicated by
the black arrow. The band inversion is more evident comparing the dispersion
curves singularly plotted for the $-360$ $\mu$m and $0$ $\mu$m prestrain
cases, as reported in Fig. S1A and Fig. S1B. Further inspection of the the
dispersion diagram allows to infer that the unprestrained unit cell (green
curves) allows for the opening of two BGs in the $[0-3000]$ Hz frequency
range, going from $170$ to $470$ Hz and $703$ to $1180$ Hz, respectively (see
Fig. 4A at $0$ $\mu$m imposed displacement). The upper and lower bounds for
both the first and second BGs shift in frequency for increasing values of the
prestrain (see Fig. 4A). However, although the global width of the first BG
experiences a limited variation, both its upper and lower bounds undergo a
remarkable frequency shift (they both almost triple their frequency). On the
contrary, the lower bound of the second BG is rather stable in frequency for
increasing values of the prestrain, while its upper limit experiences a
considerable frequency up-shift. This is responsible for an overall
enlargement of the BG width (almost doubling with respect to the unprestressed
case). Finally, it is possible to notice that the lower bound of the second BG
experiences a sort of inflection point when no displacement is imposed. The
different behaviours of the two BGs highlights a selective nature of the
prestress in altering the dispersion diagram.
The aforementioned effects partially apply also for the case of unit cell
subjected to an external traction pre-loading, i.e. both curve shifting and
group velocity inversion are observed (Fig. 3B). However, in this case, group
velocity inversion involves more bands (the third and fourth ones) and occurs
at lower reduced wavenumber values ($k^{*}\simeq\frac{\pi}{2a}$), as
highlighted by the black arrows in Fig. 3B (refer to Fig. S1B and Fig. S1C for
a direct comparison of the dispersion curves singularly plotted for the $0$
$\mu$m and $+130$ $\mu$m prestrain cases). This implies that, differently from
the compression case, when an external state of traction is induced in the
pre-loading phase, only few dispersion curves clearly shift towards higher
frequencies (the first, the second, the fifth and sixth bands) over the full
range of $k^{*}$ (from $0$ to $\pi/a$), whereas some others (the third and the
fourth ones) exhibit both a down- and an up-shift over a wide range of
$k^{*}$. As a consequence, while the first BG is almost kept unaltered in
terms of frequency width and slightly shifts towards higher frequencies (see
Fig. 4B), the second BG experiences a remarkable width decrease (up to 3 times
less the original BG).
At this point, it is worth to point out here the different dynamic behaviour
of inertially amplified elastic metamaterials with respect to the ordinary
ones when subjected to an external state of prestress that emerges from the
present research. Focusing the attention on the case of prestrain inducing
tensile deformation in the structure, in ordinary PCs the BGs tend to decrease
its frequency regime, being this effect mainly driven by the geometrical
changes of the unit cell rather then by the effective stiffness and mass
alteration introduced by the prestress (see for instance Fig. 3A of ref. [44]
and relevant literature on ordinary PCs presented in section I). On the
contrary, in the case of inertially amplified elastic metamaterials, where the
displacement mechanism is used to amplify the effective inertia of small
masses [60], the state of external solicitation represents an important means
to control the effective stiffness / effective mass ratio of the structure.
This additional degree of freedom may lead to both an increase or decrease of
the lower limit of the first BG frequency regime, under the same type of
applied prestrain (see for instance Figs. 3B and 4B, where the lower limit of
the first BG increases although a prestress condition inducing tensile
solicitation in the structure is applied).
To gain further mechanical insights about this peculiar behaviour, a thorough
explanation on the change of the effective stiffness / effective mass ratio of
the structure as a function of the applied prestrain is provided with the help
of the model reported in Fig. 5, and proposed for the first time by the
research group of Yilmaz [61, 62, 63, 64]. Figure 5A shows the rigid link
equivalent model, which assumes pin joints at the middle points of the
flexural hinges. Lumped parameters $m$, $m_{a}$ and $k_{eff}$ are reported
along with the principal geometrical parameters $\vartheta$ (initial link
angle) and displacements ($\delta_{x}$ and $\delta_{y}$) consequent the
application of a prestrain inducing tensile solicitation in the system.
Already in the undeformed configuration, an angle $\vartheta$ exists between
the equivalent rigid links (lines in red) and the ground.

$f_{p}=\left(\sqrt{\frac{k_{eff}}{m_{eff}+m}}\right)/(2\pi)=\left(\sqrt{\frac{k_{eff}}{m_{a}(cot^{2}(\vartheta)+1)/4+m}}\right)/(2\pi)$
(10)

where $m_{a}(cot^{2}(\vartheta)+1)/4$ defines the effective mass of the
system.
When a tensile prestress is applied, $\vartheta$ decreases to $\vartheta_{u}$,
and the effective mass increases. If the stiffness $k_{eff}$ is left
unaltered, from Eq. (10) it clearly emerges that the lower limit of the first
BG would decrease due to increase of the effective mass. However, if the
effective stiffness $k_{eff}$ increases more than the effective mass $m_{eff}$
(the edge mass $m$ does not depend on $\vartheta$), then the lower limit of
the first BG may also increase (this reasoning also applies to the case of
prestrain inducing a compression state of solicitation and it may lead to
analogous considerations).
To verify the correctness of the numerical approach proposed in this paper, a
comparison of the analytical values of $f_{p}$ deriving from Eq. (10) and the
numerical solutions is performed and reported in Figs. 5B-D. Figure 5B reports
the parametric plot of the dispersion curves of the unit cell when an external
prestrain ($[0-40]$ $\mu$m range) is applied inducing a tensile state in the
structure. The $[100-220]$ Hz frequency range is considered. The same
polarization reported in Fig. 3 applies. The values of the frequency of the
longitudinal branch responsible for the lower limit of the first BG are
reported at the $X$ high symmetry point in correspondence of the black arrows.
Left panel of Fig. 5C reports the direct comparison of the analytical solution
(triangular markers connected by the black line) deriving from Eq. (10) and
the numerical results (magenta square markers). The values of $k_{eff}$
introduced in the analytical model are directly calculated from the deformed
numerical configuration according to standard homogenization procedures [65,
66]. Right panel of Fig. 5C shows the variation of the effective stiffness
$\Delta k_{eff}$ and of the effective mass $\Delta m_{eff}$. This explains why
the lower limit of the first BG increases, as shown in Figs. 3B and 4B, even
if a tensile prestress is applied to the structure. Indeed, the effective mass
increase $\Delta m_{eff}$ (Fig. 5D) is far lower than the stiffness increase
$\Delta k_{eff}$ (this is due to the chosen geometrical characteristics of the
unit cell, including the out-of-plane dimension $h=t_{2}$ limiting the value
of $m_{a}$). As a consequence, it is possible to conclude that in this case
the BG alteration is mainly stiffness driven.
Finally, to clearly show the enhanced potential of inertial amplification PCs
with respect to the ordinary ones when an external prestrain condition is
applied, additional calculations by changing some of the geometrical and
mechanical parameters of the unit cell are performed. Inspired by the
analytical model reported in Fig. 5A, a configuration of the unit cell
allowing for the lower limit of the first BG to exhibit a non-monotone trend
has been found and reported in Fig. 6. It is possible to observe that the
lower limit of the first BG now first decreases, i.e., the increase of the
effective mass is dominant - the red curve is above the blue one at low values
of prestrain (yellow rectangle) and then increases, i.e., the increase in
effective stiffness is dominant - the red curve is below the blue one at
higher values of prestrain (green rectangle). This has been obtained by
changing some of the model parameters as follows: $d_{3}=18.5$ mm,
$t_{3}=18.5$ mm, $m_{a}=1$ kg$/$m3 and multiplying by a factor of $10$ the
Lamè and Murnaghan material parameters.
These results suggest that a deformation of the unit cell geometry induced by
a compressive / tensile prestress state, already in the elastic regime, can
lead to significant changes in the passband and BG behaviours of a periodic
structure, especially if the BG nucleation mechanism is lead by IA.

###  III.2 Transient Calculation

The confirmation of the above mentioned BG tunability is here verified through
a nonlinear transient numerical simulation of wave propagation conducted on a
finite waveguide comprising 50 unit cells disposed in the $x_{1}$-direction,
as shown in Fig. 7A. The idea, here, is to pre-load some unit cells of the
waveguide before exciting elastic waves at one of its edge, in order to
locally and reversibly change the dispersion diagram and confirm the ability
of the waveguide to switch from a pass- to a stop-band behaviour over a
specific frequency range.
After having pre-loaded the array, so to uniformly reach the prestrain
condition of $-360$ $\mu$m in 6 unit cells (Fig. 7B), elastic waves are
excited at the left edge of the waveguide by means of an imposed displacement
of $1$ $\mu$m in the $x_{1}$-direction (red arrow in Fig. 7A). Two input
signals are considered: (i) a triangular-like excitation (top-left panel of
Fig. 7C) and (ii) a Hanning modulated 11 sine cycles centered at $1400$ Hz
(top-right panel of Fig. 7C), exhibiting a rather broadband and narrowband
frequency content, respectively (bottom panels of Fig. 7C). Such pulses have
been chosen according to the band structure shifts reported in Fig. 3 and to
highlight the tunable filtering capabilities of the designed waveguide under
prestressed conditions. In both excitation cases, $20$ ms long time transient
explicit simulations have been performed in order to allow multiple wave
reflections to take place at both the edge of the waveguide and of the
prestressed unit cells.
In the case of excitation (i), time transient displacements in the
$x_{1}$-direction are recorded at the two acquisition points R1 and R2 (Fig.
7B), taken equidistant from the prestressed regions and chosen respectively
before (B/f) and after (A/t) the prestressed portion of the waveguide. After
acquisition, signals are Fourier transformed and compared to highlight the
differences of the two responses in terms of frequency content. Figure 7D
reports the displacement along the $x_{1}$-direction at points R1 and R2 (top
panel), as well as their energy content in the frequency domain (lower panel).
The frequencies of the computed BGs as a function of the applied prestrain are
also highlighted as a shaded region, where the color scale refers to the level
of imposed displacement in the pre-loading phase. Examining the energy content
it emerges that the Fourier transform of the signal acquired before the
prestrained region (B/f - red line), where the diagram reported in Fig. S1B
applies, presents component in the $1000-1600$ Hz frequency range, whereas the
signal registered after the prestrained region (A/t - black line) has no
frequency components in this frequency range. On the contrary, for frequencies
above $\simeq 1600$ Hz, the amplitudes of the frequency contents are
comparable. This is in agreement with the two different dispersion diagrams
and clearly confirms the possibility of the waveguide to to switch from a
pass- to a stop-band behaviour over the $1000-1600$ Hz frequency range by
readily applying and removing the imposed deformations, respectively.
When the second type of excitation (i.e., the Hanning modulated 11 sine cycles
centered at $1400$ Hz) is applied to the left edge of the waveguide, the
switch potential is even more evident. Indeed, reconstructing the full wave
field displacements at specific time instants, it is clearly visible that the
pulse is fully supported in the first portion of the waveguide (where no
prestress is applied), whereas when the prestrained region begins the wave is
strongly reflected back.

##  IV Conclusions and future perspectives

In conclusion, in this work the effect of the application of an external
prestress on the dispersion diagram has been investigated. The geometrical
deformation of the unit cell consequent the action of an applied external load
has been determined through a static geometrically nonlinear analysis,
representing the first step of a so called Updated Lagrangian scheme of
calculation. A Floquet-Bloch decomposition has then been applied (second step)
to the linearized equations of the acousto-elasticity for the unit cell in the
deformed configuration.
The effect of the prestress on the the original band structure (i.e., unit
cell without any pre-loading phase) has been demonstrated through parametric
Floquet-Bloch analysis and further confirmed by nonlinear transient numerical
simulations, proving the capability of the structure to switch from a pass- to
a stop-band behaviour in the same frequency range ($[1000-1600]$ Hz).
The results presented herein provide insights in the behaviour of band gaps
induced by inertial amplification, and suggest new opportunities for real-time
tunable wave manipulation.
Future investigations will concern the extension of the present study in the
direction of a tensegrity-inspired redesign [67, 68] of the unit cell reported
in Fig. 1. A self-similar [69] reorganization of the structural elements will
allow the construction of a tensegrity architecture. The design will derive
from the D-bar tensegrity systems [67, 68], equipped with longitudinal and
transverse cables. The idea will be to recursively divide the longitudinal
span of the cell into segments of equal length and replace each new segment
with a smaller scale unit with equal shape [67]. Self-equilibrated tensile
forces in the cables and compressive forces in the struts will give rise to an
internal state of prestress. Cables and struts will present an offset in the
transverse direction in order to prevent material overlapping. The application
of a self-equilibrated system of forces composed of tensile forces in the
cables and compressive forces in the struts will give rise to an internal
state of prestress, which can superimpose to the external state of prestress,
analysed in the present work. Such initial states of stress will contribute to
the geometric term of the stiffness matrix of the structure [67], and allow
for an extra degree of freedom for the optimal tuning of the dispersion
relation of the system [31].

References

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Acknowledgements MiM has received funding from the European Union’s Horizon
2020 FET Open (”BOHEME”) under grant agreement No. 863179.

Data availability. The data that support the plots within this paper and other
findings of this study are available from the corresponding author upon
request.

Figure 1: Graphic representation of the Updated Lagrangian scheme. (A) The
undeformed configuration $C_{0}$, i.e. the initial unit cell (delimited by
dashed lines) used to calculate the displacement and stress fields introduced
by the external mechanical load. (B) The static deformed configuration $C$,
resulting from the application of the external mechanical load. (C) The
dynamic configuration $C^{\prime}$ undergoing a harmonic motion. (D) The
reference systems in the direct and reciprocal spaces.

Figure 2: Schematic representation of the undeformed unit cell and its
deformed conditions under isotropic compression / traction. (A) Schematic
representation of the unit cell exhibiting inertial amplification mechanism,
positioned at $\mathbf{x}_{0}$ with respect to the original reference systems
$x_{0i}$, with i = 1,2,3. The structure is made of epoxy and it was proposed
for the first time by Yilmaz et al. in ref. [43]. (B, C) Deformed
configuration of the unit cell under isotropic
$\mathbf{u}_{0}(\mathbf{x}_{0})\cdot\mathbf{n}_{0}(\mathbf{n}_{0})=130\mu$m
traction($-360\mu$m compression). The colors denote the Von Mises stress in
MPa. A maximum stress of 21.2 (28.6 33.1) MPa is observed. Deformations are in
1:1 scale.

Figure 3: Band diagrams of the unit cell under different values of applied
compression / traction. (A) Parametric plot of the the real part of the
reduced wavenumber $k^{*}$ along the $\Gamma-X$ irreducible path as a function
of the applied prestrain inducing a compressive state in the structure. The
dispersion curves are color-coded on the basis of the applied prestrain level
at the boundaries of the unit cell in the pre-loading phase. The polarization
factor color bar varies gradually from $-360$ $\mu$m (dark blue: maximum
compression) to $0$ (green: unprestrained structure). A general up-shift trend
of the dispersion curves is observable (curves fading into dark blue) and a
group velocity inversion (highlighted by the black arrow) occurs for the third
band in proximity of the high-symmetry point $X$. (B) Parametric plot of the
the real part of the reduced wavenumber $k^{*}$ along the $\Gamma-X$
irreducible path as a function of the applied prestrain inducing a tensile
state in the structure. The dispersion curves are color-coded on the basis of
the applied prestrain level at the boundaries of the unit cell in the pre-
loading phase. The polarization factor color bar varies gradually from $130$
$\mu$m (dark red) to $0$ (green: unprestrain structure). A general down-shift
trend of the dispersion curves could be observed (curves fading into dark
red). Group velocity inversions occur, in this case, in more bands
(highlighted by the black arrows) and extend over a larger region of $k^{*}$.

Figure 4: Upper and lower bounds for the first and second BGs as a function of
the applied prestrain inducing a compressive or tensile state in the
structure. (A) In the case of increasing compressive prestrain, the first BG
(black dashed lines delimited by red square markers) experiences a limited
width variation, although both its upper and lower bounds are subjected to a
remarkable frequency shift (towards higher frequencies). On the contrary, the
lower bound of the second BG (black dashed lines delimited by blue square
markers) is rather stable in frequency, while its upper bound still
experiences a strong shift towards higher frequencies. This is responsible for
an important global BG width enlargement (twice the initial value). (B) In the
case of prestrain inducing a state of traction in the unit cell, a similar
behaviour is observed for the first BG, whereas an important width reduction
can be deduced for the second BG (up to three times the initial value), as the
initial imposed solicitation is increased to $130$ $\mu$m.

Figure 5: Change of the effective stiffness / effective mass ratio of the
structure as a function of the applied prestrain: numerical - analytical
comparison. (A) Schematic representation of the rigid link equivalent model of
the examined PC with inertial amplification mechanism [62]. Lumped parameters
$m$, $m_{a}$ and $k_{eff}$ are reported along with the principal geometrical
parameters $\vartheta$ (initial link angle) and displacements ($\delta_{x}$
and $\delta_{y}$) consequent the application of a prestrain inducing tensile
solicitation. (B) Parametric plot of the the real part of the reduced
wavenumber $k^{*}$ along the $\Gamma-X$ irreducible path as a function of the
applied prestrain inducing a tensile state in the structure in the $[100-220]$
Hz frequency range. The same polarization reported in Fig. 3 applies. The
values of the frequency of the longitudinal branch responsible for the lower
limit of the first BG are reported at the $X$ high symmetry point in
correspondence of the black arrows. (C) Comparison of the analytical solution
(triangular markers connected by the black line) deriving from Eq. 1 and the
numerical values (magenta square markers) is reported in the left panel. On
the right panel, the variation of the effective stiffness $\Delta k_{eff}$ and
of the effective mass $\Delta m_{eff}$ explaining the reason of the lower
frequency increase of the BG even if a tensile prestress is applied to the
structure. Indeed, the effective mass increase is far lower than the stiffness
increase. (D) Magnification of the change of the effective mass showing its
increase.

Figure 6: Additional degree of tunability of inertially amplified elastic
metamaterials under external prestrain solicitation state. (Left panel) Lower
limit frequency of the first BG as a function of the applied prestrain. (Right
panel) The variation of the normalized effective stiffness and of the
normalized effective mass explaining the reason of the decrease / increase of
the lower limit frequency of the first BG even if the same type of prestress
(tension) is applied to the structure. Yellow and green rectangles determine
the mass and stiffness driven regions, respectively.

Figure 7: Switch from a pass- to a stop-band behaviour of the waveguide. (A)
Schematic representation of the finite structure implemented for the nonlinear
transient simulation. The model comprises 50 unit cells arranged in the
$x_{01}$-direction. (B) Before exciting the propagation of elastic waves, an
external load is applied over $6$ internal unit cells of the array, so to
locally induce the uniform prestrained condition of $-360$ $\mu$m. (C)
Triangular-like (left panel) and 11 sine cycles centered at $1400$ Hz Hanning
modulated (right panel) excitations. (D) Signals recorded before (B/f) and
after (A/t) the $6$ prestressed unit cells (top panel) and their Fourier
Transform (low panel), for the triangular-like excitation signal. The Fourier
content clearly shows how the introduction of the prestress alters the
frequency response function of the structure. The tunable BG is reported as a
color-coded rectangle. The green region of the rectangle is the BG
corresponding to the unprestrained unit cells (up to $1180$ Hz), whereas the
one fading to dark blue corresponds to the BG extension (up to $1680$ Hz)
induced by the introduction of the external compressive prestrain (up to
$-360$ $\mu$m: dark blue). The comparison of the two signals proves that the
same waveguide is capable to support or to inhibit waves having the same
frequency content. (E) Snapshots at different time steps, for the narrowband
excitation signal, showing how the unprestrained waveguide is capable of
supporting the propagation of the wave, whereas the stressed region reflect
back the majority of the waveform.

## Appendix A

The different operators in Eq. (9) are expressed by:

$\displaystyle\mathbf{M}$
$\displaystyle=\bigcup_{e}\int_{\Omega_{e}}\mathbf{N}^{\mathrm{T}}(\mathbf{x})\rho(\mathbf{x})\mathbf{N}(\mathbf{x})da,$
(11) $\displaystyle\mathbf{K}_{1}$
$\displaystyle=\bigcup_{e}\int_{\Omega_{e}}\mathbf{N}^{\mathrm{T}}(\mathbf{x})\left[\mathbf{B}^{\mathrm{T}}\mathbf{D}(\mathbf{x})\mathbf{B}+\mathbf{B}_{0}^{\mathrm{T}}\bm{\Sigma}_{0}(\mathbf{x})\mathbf{B}_{0}\right]\mathbf{N}(\mathbf{x})da$
(12) $\displaystyle\mathbf{K}_{2}$
$\displaystyle=\bigcup_{e}\int_{\Omega_{e}}\mathbf{N}^{\mathrm{T}}(\mathbf{x})\left[\mathbf{B}^{\mathrm{T}}\mathbf{D}(\mathbf{x})\mathbf{H}+\mathbf{B}_{0}^{\mathrm{T}}\bm{\Sigma}_{0}(\mathbf{x})\mathbf{H}_{0}\right]\mathbf{N}(\mathbf{x})da,$
(13) $\displaystyle\mathbf{K}_{3}$
$\displaystyle=\bigcup_{e}\int_{\Omega_{e}}\mathbf{N}^{\mathrm{T}}(\mathbf{x})\left[\mathbf{H}^{\mathrm{T}}\mathbf{D}(\mathbf{x})\mathbf{H}+\mathbf{H}_{0}^{\mathrm{T}}\bm{\Sigma}_{0}(\mathbf{x})\mathbf{H}_{0}\right]\mathbf{N}(\mathbf{x})\mathrm{d}a,$
(14)

where $\Omega_{e}$ denotes the domain of the $e$-th finite element of the
mesh, $\bigcup_{e}(\cdot)$ the standard direct stiffness assembling procedure,
$\mathbf{N}(\mathbf{x})$ the matrix of shape functions for the $e$-th element,
and $\bm{\Sigma}_{0}(\mathbf{x})$ is a block-diagonal matrix of the form:

$\bm{\Sigma}_{0}(\mathbf{x})=\left[\begin{array}[]{ccc}\bm{\sigma}_{0}(\mathbf{x})&\mathbf{0}\\\
\mathbf{0}&\bm{\sigma}_{0}(\mathbf{x})\\\ \end{array}\right],$ (15)

while the different compatibility operators are expressed as:

$\displaystyle\mathbf{B}$ $\displaystyle=\frac{\partial}{\partial
x_{1}}\left[\begin{array}[]{cc}1&0\\\ 0&0\\\ 0&1\\\
\end{array}\right]+\frac{\partial}{\partial
x_{2}}\left[\begin{array}[]{cc}0&0\\\ 0&1\\\ 1&0\\\ \end{array}\right],$ (22)
$\displaystyle\mathbf{B}_{0}$ $\displaystyle=\frac{\partial}{\partial
x_{1}}\left[\begin{array}[]{cc}1&0\\\ 0&0\\\ 0&1\\\ 0&0\\\
\end{array}\right]+\frac{\partial}{\partial
x_{2}}\left[\begin{array}[]{cc}0&0\\\ 1&0\\\ 0&0\\\ 0&1\\\ \end{array}\right]$
(31) $\displaystyle\mathbf{H}$ $\displaystyle=\left[\begin{array}[]{cc}1&0\\\
0&0\\\ 0&1\\\
\end{array}\right],\quad\mathbf{H}_{0}=\left[\begin{array}[]{cc}1&0\\\ 0&0\\\
0&1\\\ 0&0\\\ \end{array}\right].$ (39)

## Appendix B

In this Appendix a direct comparison of the dispersion curves plotted for the
specific cases of $-360$ $\mu$m, $0$ $\mu$m and $+130$ $\mu$m prestrain loads
are reported as singular diagrams in Fig. S1.

Figure S1: Plots of dispersion curves for single states of prestress.
Comparison of the dispersion curves plotted singularly for the (A) $-360$
$\mu$m, (B) $0$ $\mu$m and (C) $+130$ $\mu$m prestrain cases.


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##  Appendix A Details of observations

Here we provide some further details regarding the observations and the data
reduction procedures.

###  A.1 JVLA

Table 23 summarises the JVLA observations, including the date of the
observation for each source, and the integration times in all bands.

Table 23: Summary of the JVLA observations. Columns: (1) source name; (2) date of observations; (3) integration time in each band. Source | date |  $T_{\mathrm{int}}$ (X / Ku / K / Ka / Q)
---|---|---
| (yyyy-mm-dd) | (s)
J1029+5556 | 2022-04-22 | 594 / 594 / 534 / 706 / 754
J1228+5018 | 2022-04-24 | 596 / 596 / 532 / 706 / 754
J1232+4957 | 2022-04-25 | 594 / 594 / 532 / 708 / 754
J1509+6137 | 2022-03-26 | 594 / 594 / 532 / 590 / 754
J1510+5547 | 2022-03-13 | 594 / 594 / 534 / 706 / 868
J1522+3934 | 2022-03-16 | 594 / 594 / 532 / 704 / 810
J1641+3454 | 2022-03-15 | 594 / 594 / 543 / 708 / 696

###  A.2 VLBA

Table 24 summarises the VLBA observations, including the name of the
calibrator source, its distance from the target as well as its VLBI scale flux
density at 15 GHz, and the phase referencing cycle time of the observation.

Table 24: Summary of the VLBA observations. Columns: (1) target source name; (2) phase reference calibrator; (3) distance between the target and the calibrator; (4) calibrator’s VLBI scale flux density at 15 GHz; (5) phase referencing cycle time. Source | Calibrator | Distance | $S_{\nu,\mathrm{cal}}$ | $T_{\mathrm{cycle}}$
---|---|---|---|---
|  | (deg) | (mJy) | (s)
J1029+5556 | J1035+5628 | 0.99 | 300 | 180
J1228+5018 | J1223+5037 | 0.87 | 62 | 160
J1232+4957 | J1223+5037 | 1.53 | 62 | 160
J1509+6137 | J1526+6110 | 2.05 | 38 | 220
J1510+5547 | J1510+5702 | 1.26 | 200 | 160
J1522+3934 | J1528+3816 | 1.82 | 68 | 200
J1641+3454 | J1635+3458 | 1.21 | 190 | 180

###  A.3 MRO

Measurements that are considered to be of poor quality, for example, due to
unfavourable weather conditions, are discarded semi-automatically.
Additionally faint detections are checked manually in the final data reduction
stage. Bad weather conditions or other environmental effects are taken into
account and also, for example, conspicuous but rare flux density increases
caused by aircraft in the telescope beam. The general flux density levels are
checked to be consistent with adjacent measurements (i.e. other sources
observed before and after the target source). In addition, we checked if the
observations of these sources could be contaminated by a bright radio source
falling into the reference beam of the MRO telescope. Using LOFAR, FIRST, and
VLASS data we concluded that whereas there are a few moderately bright sources
with flux densities around a few hundred mJy at low radio frequencies that
could be in the reference beam, all of them have steep spectra, and it is thus
very unlikely that any of them could affect these observations.

Due to the fairly high detection limit of the telescope (i.e. approx. 200 mJy
in optimal conditions, which is more than adequate for the bright AGN
monitoring programmes conducted at MRO), we typically only see the highest
tips of the flares in faint sources, whereas most of the lower level activity
remains below the detection threshold (e.g., Acciari et al. 2014). This is
also seen in the upper limits, the level of which can drastically change in
even a short time due to compromised weather conditions that can also
significantly raise the detection limit. The undetected source could be
actually fainter due to variability or the observing conditions could be worse
(or both), and it is therefore not detected. The upper limits describe the
largest flux density the source could have in the current conditions, but
still remain below the 4$\sigma$ detection limit and cannot therefore be used
for data analysis. Occasionally there are several high upper limit values
clustered around detections, which indicate that the source could indeed be
active, but for some reason it does not exceed the current detection limit,
for example, due to weather. However, it has been shown that the high activity
periods of NLS1 sources detected at MRO correspond to flare peaks in OVRO data
(Lähteenmäki et al. 2017), confirming that at least most of the time the two
telescopes are catching the same events.

###  A.4 OVRO

The AGN monitoring sample at OVRO mostly consists of bright blazar-type
objects, with the majority having a mean flux density $>60$ mJy (Richards et
al. 2014). Therefore in the schedules, each observation consists of four on-on
integrations, each 8 s long, resulting in a total integration time of 32 s.
Given the small number of on-on integrations, it is possible that atmospheric
fluctuations or pointing errors result in outliers in the light curves
(Richards et al. 2014). Moreover, the number of observations performed each
day is large (up to 500) so that it is possible that some, apparently high
signal-to-noise ratio observations, occur purely due to random fluctuations.
The data are processed with an automated pipeline, where manual editing is
done to flag obviously bad periods of data, and data are automatically flagged
based on large changes within the four on-on integrations and other
diagnostics (see Richards et al. 2011, for details). However, individual data
points are not typically manually inspected, as for the variability analysis
of bright blazars, the effect of outliers in the data is small (Richards et
al. 2014).

Because of the faintness of the NLS1 targets, we have done additional manual
checks to inspect the quality of the detections, which we describe here. We
note that in all the cases described below, the flux density of the spurious
detection has been less than 20 mJy and mostly $<10$ mJy, or the uncertainty
has been larger than usual so that similar observations in our blazar light
curves would not be as problematic.

The receiver records both right- and left-hand circular polarisation
separately with the final observation being a weighted average of the two. We
can thus inspect the two values separately to verify that the source has been
detected in both polarisations (here we assume that the circular polarisation
of the objects is negligible, as is the case for most blazars at 15 GHz (e.g.,
Homan & Lister 2006). This made us reject two spurious detections in
J1232+4957. Additionally, we have inspected observations of other nearby
sources to see whether there are data that have been automatically flagged in
the pipeline close to the observation of the NLS1, indicative of potentially
poor observing conditions. This resulted in the rejection of one spurious
detection in J1641+3454.

In October 2021 we also changed the observing strategy of these NLS1 targets
so that they are observed twice in a row in the schedules. This way we can see
whether short-term atmospheric effects or bad conditions have resulted in
spurious detections if the two consecutive observations show a large
difference, as we would not expect large changes on a time scale of $\sim 1$
min. This resulted in the rejection of single spurious detections in
J1029+5556, J1509+6137, J1510+5547, and J1522+3934, all of which had moderate
S/N values of $\sim$4-9.

The remaining detections in the paper either show detections in two
consecutive observations (J1522+3934) or consistent flux densities in the
right- and left-hand circular polarisation and no apparent problems with
nearby targets (J1029+5556, J1522+3934, J1641+3454). However, we cannot fully
exclude additional, unknown effects in the observations before October 2021
when the sources were observed only once in a row, especially in J1029+5556
and J1641+3454 that do not show any other detections in the OVRO light curves.
J1522+3934 on the other hand seems more reliable because of its multiple
detections.

##  Appendix B Radio maps

The JVLA radio maps with likely detections are shown here. This includes X,
Ku, K, Ka, and Q band maps of J1228+5017 (Figs. 8-12), X, and Ku band maps of
J1232+4957 (Figs. 13-14), X, Ku, K, and Ka band maps of J1522+3934 (Figs.
15-18), and X, Ku, K, Ka, and Q band maps of J1641+3454 (Figs. 19-23).

Figure 8: JVLA X band radio map of J1228+5017, rms = 7$\mu$Jy beam-1, contour
levels at -3, 3, 6, 12 rms, beam size 1.20 $\times$ 0.91 kpc.  Figure 9: JVLA
Ku band radio map of J1228+5017, rms = 6$\mu$Jy beam-1, contour levels at -3,
3, 6, 12 rms, beam size 0.78 $\times$ 0.58 kpc.  Figure 10: JVLA K band radio
map of J1228+5017, rms = 9$\mu$Jy beam-1, contour levels at -3, 3, 6, 12 rms,
beam size 0.52 $\times$ 0.38 kpc.  Figure 11: JVLA Ka band radio map of
J1228+5017, rms = 12$\mu$Jy beam-1, contour levels at -3, 3, 6 rms, beam size
0.34 $\times$ 0.26 kpc.  Figure 12: JVLA Q band radio map of J1228+5017, rms =
31$\mu$Jy beam-1, contour levels at -3, 3 rms, beam size 0.24 $\times$ 0.20
kpc.  Figure 13: JVLA X band radio map of J1232+4957, rms = 7$\mu$Jy beam-1,
contour levels at -3, 3 rms, beam size 0.98 $\times$ 0.88 kpc.  Figure 14:
JVLA Ku band radio map of J1232+4957, rms = 5$\mu$Jy beam-1, contour levels at
-3, 3, 6 rms, beam size 0.65 $\times$ 0.57 kpc.  Figure 15: JVLA X band radio
map of J1522+3934, rms = 8$\mu$Jy beam-1, contour levels at -3, 3, 6, 12, 24
rms, beam size 0.36 $\times$ 0.31 kpc.  Figure 16: JVLA Ku band radio map of
J1522+3934, rms = 6$\mu$Jy beam-1, contour levels at -3, 3, 6, 12, 24 rms,
beam size 0.24 $\times$ 0.21 kpc.  Figure 17: JVLA K band radio map of
J1522+3934, rms = 9$\mu$Jy beam-1, contour levels at -3, 3, 6, 12 rms, beam
size 0.17 $\times$ 0.16 kpc.  Figure 18: JVLA Ka band radio map of J1522+3934,
rms = 13$\mu$Jy beam-1, contour levels at -3, 3, 6 rms, beam size 0.10
$\times$ 0.07 kpc.  Figure 19: JVLA X band radio map of J1641+3454, rms =
7$\mu$Jy beam-1, contour levels at -3, 3, 6, 12, 24 rms, beam size 0.73
$\times$ 0.59 kpc.  Figure 20: JVLA Ku band radio map of J1641+3454, rms =
5$\mu$Jy beam-1, contour levels at -3, 3, 6, 12, 24 rms, beam size 0.49
$\times$ 0.41 kpc.  Figure 21: JVLA K band radio map of J1641+3454, rms =
8$\mu$Jy beam-1, contour levels at -3, 3, 6, 12 rms, beam size 0.33 $\times$
0.27 kpc.  Figure 22: JVLA Ka band radio map of J1641+3454, rms = 11$\mu$Jy
beam-1, contour levels at -3, 3, 6 rms, beam size 0.29 $\times$ 0.18 kpc.
Figure 23: JVLA Q band radio map of J1641+3454, rms = 33$\mu$Jy beam-1,
contour levels at -3, 3 rms, beam size 0.20 $\times$ 0.14 kpc.

##  Appendix C Light curves

The light curves of our sources from the beginning of 2014 to mid-2022 are
shown here. Figs. 24-30 show light curves including low-resolution (MRO and
OVRO) and high-resolution (JVLA, VLBA, and VLASS) data. Due to the strongly
varying flux densities these plots are in logarithmic scale. The light curves
in Figs. 31-37 show only the MRO and OVRO data in linear scale, and include
also the 4$\sigma$ upper limits for both observatories.

Figure 24: Light curve of J1029+5556. Symbols and colours explained in the
figure. Filled symbols denote integrated flux densities and empty symbols mark
peak flux densities, except empty red symbols with downward arrows that are
used for upper limits. VLA 1 data from Berton et al. (2020b) and VLA 2 data
from this paper.  Figure 25: Light curve of J1228+5017. Symbols and colours as
in Fig. 24.  Figure 26: Light curve of J1232+4957. Symbols and colours as in
Fig. 24.  Figure 27: Light curve of J1509+6137. Symbols and colours as in Fig.
24.  Figure 28: Light curve of J1510+5547. Symbols and colours as in Fig. 24.
Figure 29: Light curve of J1522+3934. Symbols and colours as in Fig. 24.
Figure 30: Light curve of J1641+3454. Symbols and colours as in Fig. 24.
Figure 31: MRO and OVRO light curves of J1029+5556. Symbols and colours
explained in the figure. Symbols with downward arrows denote upper limits, for
the JVLA and the VLBA only the epochs of the observations are marked. Figure
32: MRO and OVRO light curves of J1228+5017. Symbols and colours explained in
the figure. Symbols with downward arrows denote upper limits, for the JVLA and
the VLBA only the epochs of the observations are marked. Figure 33: MRO and
OVRO light curves of J1232+4957. Symbols and colours explained in the figure.
Symbols with downward arrows denote upper limits, for the JVLA and the VLBA
only the epochs of the observations are marked. Figure 34: MRO and OVRO light
curves of J1509+6137. Symbols and colours explained in the figure. Symbols
with downward arrows denote upper limits, for the JVLA and the VLBA only the
epochs of the observations are marked. Figure 35: MRO and OVRO light curves of
J1510+5547. Symbols and colours explained in the figure. Symbols with downward
arrows denote upper limits, for the JVLA and the VLBA only the epochs of the
observations are marked. Figure 36: MRO and OVRO light curves of J1522+3934.
Symbols and colours explained in the figure. Symbols with downward arrows
denote upper limits, for the JVLA and the VLBA only the epochs of the
observations are marked. Figure 37: MRO and OVRO light curves of J1641+3454.
Symbols and colours explained in the figure. Symbols with downward arrows
denote upper limits, for the JVLA and the VLBA only the epochs of the
observations are marked.