# Designing a braille reader using the
snap buckling of bistable magnetic shells

Arefeh Abbasi1 Tian Chen2 Bastien F.G. Aymon1 Pedro M. Reis1∗ 1Flexible
Structures Laboratory, Institute of Mechanical Engineering,
École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
2Architected Intelligent Matter Laboratory, Department of Mechanical
Engineering, University of Houston, Houston, TX, 77024, U.S.A.
∗Email<EMAIL_ADDRESS>
###### Abstract

A design concept is introduced for the building block, a dot, of programmable
braille readers utilizing bistable shell buckling, magnetic actuation, and
pneumatic loading. The design process is guided by Finite Element simulations,
which are initially validated through precision experiments conducted on a
scaled-up, single-shell model system. Then, the simulations are leveraged to
systematically explore the design space, adhering to the standardized
geometric and physical specifications of braille systems. The findings
demonstrate the feasibility of selecting design parameters that satisfy both
geometric requirements and blocking forces under moderate magnetic fields,
facilitated by pneumatic loading to switch between the two stable states. The
advantages of the proposed design include the reversible bistability of the
actuators and fast state-switching via a transient magnetic field. While the
study is focused on experimentally validated numerical simulations, several
manufacturing challenges that need to be resolved for future physical
implementations are identified.

Keywords: Shell buckling; Bistability; Magneto-rheological elastomers; Braille
reader.

Shell buckling, Bistability, Magneto-rheological elastomers, Braille reader

## 1 Introduction

Worldwide, 285 million people experience visual impairments, including 39
million living with blindness [1]. These impairments present challenges in
navigating and interacting with the world, influencing various aspects of
daily life, such as access to printed or digital content. Braille code, a
tactile writing system, facilitates these interactions, mapping symbols (e.g.,
letters, numbers, and punctuation) into arrays of cells, each comprising a
3$\times$2 matrix of dots. Each dot can independently be raised or flat, and
words are then formed by assembling a series of such cells. Braille users
typically read by tracing their fingertips across rows of these cells, whose
dimensions are optimized to allow the index finger pad to cover the entire
cell and discern each dot. While static embossed paper is the traditional
medium for braille, modern assisting devices, such as refreshable braille
displays (RBDs), enable dynamic reading and writing [2, 3]. RBDs provide
access to written content using arrays of morphable physical dots, whose
configuration adjusts dynamically to represent different sequences of braille
symbols in accordance with international braille standards [4, 5, 6, 2].

In recent years, technological advancements have stimulated the development of
RBD devices driven by a variety of actuation mechanisms, including
piezoelectrics [7, 8, 9, 10], electromagnetics [11, 12, 13], or
thermopneumatics [14]. These devices often use advanced materials such as
electroactive polymers [15, 16, 17, 18], shape memory alloys [19, 20, 21], or
dielectric elastomers [22, 23]. Piezoelectric actuators have been favored for
commercial RBDs due to their fast refresh rates, low power consumption, and
reliability, albeit at a relatively high cost [2, 7, 8]. Electromagnetic
linear actuators tend to have a low ratio between output force and operating
velocity, requiring complex packaging and a considerable force to hold a
raised dot [24, 25]. RBD devices utilizing shape memory alloys require
intricate heating and cooling processes, posing practical implementation
challenges [9, 21, 19]. Dielectric elastomers have been gaining traction for
lightweight tactile displays, offering high actuator density and a wide range
of motion, with performance comparable to previous technologies, but in a more
compact form [22, 23]. However, their required large driving voltages can be
impractical for some applications.

To date, most technological solutions for RBDs cannot offer sufficiently high-
quality performance, especially regarding the balance between fast shape-
changing dynamics and low power consumption [26]. Additionally, the mainstream
adoption of advanced tactile displays is hindered by the lack of compact,
large-area actuator arrays that can stimulate multiple sensory receptors while
adhering to high user-safety standards. Existing solutions tend to be costly
and require complex manufacturing processes. Despite ongoing efforts [27],
designing RBD devices that are simple, compact, low-cost, large-scale, user-
friendly, and reliable remains a formidable challenge.

Here, we propose a novel design concept for braille dots, the building block
of braille readers. Our design leverages the buckling and bistability of thin
shells fabricated from hard magneto-rheological elastomers (h-MREs). Inspired
by the popular ‘pop it’ toy [28], these shells can be reversibly set in a
convex or concave state (Figure 1a). Each of these shells (dots) can then be
arranged in a 3$\times$2 matrix and programmed, on-demand, to form a braille
symbol. The dots have independent writing and reading phases under magnetic
and mechanical loading, respectively. During writing, a transient external
magnetic field can induce snap-through buckling to transition the shell
between its two stable states: from ON (bump) to OFF (dimple) or vice versa.
For reading, shells in the ON state must sustain a blocking force, in reaction
to the finger indentation, without snapping to the OFF state. Throughout, our
mechanics-based design process is centered on Finite Element Method (FEM)
simulations. We first validate these simulations against precision experiments
on a scaled-up (centimeter-scale) physical model of a braille dot. Then, we
study dots at their actual scale, ensuring adherence to standard braille
specifications [6], with a special focus on their geometry, elastic response,
and actuation. Although the primary driver of actuation is an external
magnetic field, it is supplemented with a transient pneumatic loading to aid
in widening the design space. Our numerical exploration of the design
parameters allows us to identify the regions that meet the various design
constraints, making a step toward a new class of programmable braille
displays.

## 2 Problem definition: braille reader design concept

The specifications for braille cells and dots are standardized by the World
Blind Union [6], and the relevant parameters are required to lie within the
following ranges: $\ell_{\mathrm{d}}\in[2.3,\,2.5]\,$mm for the dot-to-dot
spacing, $\ell_{\mathrm{c}}\in[6,\,7]\,$mm for the distance between two
distinct cells, and $\ell_{\mathrm{l}}\in[10,\,11]\,$mm for the distance
between two lines of words. Furthermore, each raised dot must feature a quasi-
hemispherical cap with base diameter $D\in[1.4,\,1.6]\,$mm and height
$h\in[0.4,\,0.9]\,$mm [29]. Finally, to sustain the normal indentation force
applied by the index finger during reading, each dot must be able to withstand
a minimum blocking force of $F>50$ mN [6].

Figure 1: Design, fabrication, and operation of a bistable Braille dot. (a)
Geometry: A word is formed by assembling a series of braille cells, each
comprising $3\times 2$ dots. “EPFL” is shown as an example. (b) Fabrication: A
bistable shell is fabricated by sandwiching a circular h-MRE plate between two
radially pre-stretched boundary annuli. This pre-stretch is then released to
buckle the plate into a shell. (c) Writing phase: an external magnetic field,
$\mathbf{B}^{\mathrm{a}}$, sets each shell in one of its two stable states,
either ON (bump) or OFF (dimple). (d) Reading phase: an index finger applies
an indentation force, $F$ on each dot.

Our objective is to design a programmable braille dot that adheres to the
aforementioned braille specification. We consider a bistable shell clamped at
its base (Figure 1a, b). The analysis is segmented into two phases: “Writing”
(Figure 1c) and “Reading” (Figure 1d). For writing, the ON-OFF switching is
done via magnetic actuation. Concurrently with this phase, we temporarily
depressurize the shell to lower the energy barrier required for buckling. By
contrast, during the reading phase, the shell is pressurized to increase its
rigidity. We seek to identify the key design variables and protocols required
for the fabrication and operation of our system. Next, we describe the
geometric considerations and the two operational phases.

The geometry of our model braille-dot (Figure 1a) comprises a shell of
diameter $D$=1.45 mm and height $h$=0.48 mm, in accordance with braille
standards. This shell is fabricated by the buckling of a radially compressed
circular plate (thickness $t$) made of h-MRE [30] when the in-plane pre-
stretch, $\lambda$, of two boundary annuli is released (Figure 1b), as
detailed in Section 5.1. One first goal of the design is to select appropriate
values of $t$ and $\lambda$ that, upon buckling of the plate, yield a shell
with the target value of $h$, satisfying braille requirements.

For the writing phase (Figure 1c), we will characterize the snap buckling of
the shells under loading by a uniform magnetic field,
$\textbf{B}^{\mathrm{a}}_{c}$, to switch between their two stable states. We
assume that each dot, which would eventually form the 3$\times$2 cell, can be
actuated independently. For the present study, we restrict our focus to the
operation of a single dot. The goal of this design phase is to identify the
critical magnetic field, $B^{\mathrm{a}}_{\mathrm{c}}$, required for snapping
under the limitation set by upper-bound of the linear regime of the B-H
hysteresis curve for the h-MRE material [31] (additional details are provided
in Section 5.1). Subsequently, we aim to determine the corresponding geometric
and fabrication parameters, $t$, and $\lambda$, that yield the desired snap-
buckling characteristics.

For the reading phase (Figure 1d), the braille dots must be designed such that
the user can tactilely discern the dots without altering their state. The
challenge lies in ensuring that a dot in the ON state can sustain the
indentation force mentioned above ($F\geq 50$ mN) without snapping to the OFF
state, thereby inadvertently erasing the braille pattern. This design phase
targets the determination of optimal geometric and fabrication parameters for
the dot, specifically its thickness $t$ and pre-stretch $\lambda$, to meet
this requirement.

The constraints associated with the fabrication protocol and the ensuing dot
geometry, coupled with the requirements for the reading and writing phases,
underscore the intricacies involved with designing our braille dot. We aim to
identify the feasible design parameter space of the system (specifically, $t$
and $\lambda$) that satisfies the constraints on $h$, $F$, and
$B^{\mathrm{a}}_{\mathrm{c}}$. This design exploration will be performed
solely using FEM simulations, which will be initially validated against
experiments in a scaled-up system.

## 3 Results and Discussion

### 3.1 Validation of the FEM simulations against experiments

We first validate the FEM simulations (technical details are provided in
Section 5.2) against experiments (see Section 5.1) on the scaled-up model
system, considering the results from the geometry characterization of the
fabricated shells, as well as from the reading and writing experiments.

During sample fabrication, the buckling of the plate, which yields a shell,
may produce undesirable wrinkling patterns [32]. To act as braille dots, ideal
shells should be smooth (i.e., free of these wrinkles). Toward identifying the
design space for these ideal shells, Figure 2(a) presents a phase diagram of
the thickness-diameter parameter space ($t,\,D$) for representative shells
fabricated with a pre-stretch of $\lambda=0.1$ (see Section 5.1, and 5.2).
There is excellent agreement between experiments (crosses) and FEM (circles),
serving as a first step in validating the simulations. Wrinkling is observed
for higher values of the slenderness ratio $D/t$. For the chosen pre-stretch
($\lambda=0.1$), the empirical phase boundary between ideal and wrinkled
shells is $t\approx 0.02D$ (dashed line).

Figure 2: Validation of FEM simulations against the scaled-up experiments. (a)
Phase diagram in the ($t,\,D$) parameter space. The experiments and
simulations correspond to the cross and circle symbols, respectively. The
smooth and wrinkled shells are represented by the blue and red symbols,
respectively; the empirical phase boundary between the two is represented by
the dashed line. The normalized (b) shell height, $h/t$, (c) critical magnetic
field, $B^{\mathrm{a}}_{\mathrm{c}}B^{\mathrm{r}}/(E\mu_{0})$, and (d)
blocking force, $FD/(Et^{3})$ are plotted as functions of the normalized base
diameter of the shell, $D/t$, for 10 different values of thickness, $t$. The
error bars of the experimental data represent the standard deviation of 6
independent measurements on the same specimen. The different values of $t$ are
color coded (see adjacent color bar). The solid lines and data symbols
correspond to FEM and experiments, respectively. The insets show the log-log
plots of the data. Throughout, the fabrication pre-stretch is $\lambda=0.1$.

First, focusing on the geometry characterization, Figure 2(b) plots the
normalized height, $h/t$, of the smooth shells (blue region in Figure 2a)
versus $D/t$. The FEM simulations (lines) and experimental data (symbols) are
obtained with a fabrication pre-stretch of $\lambda=0.1$ and thickness in the
range $t\in[0.18,\,1.08]\,$mm (see Section 5.1). The data collapses onto a
single curve, with excellent agreement between experiments and FEM, thereby
validating the FEM for the shell fabrication. Moreover, the inset of Figure
2(b) shows that the relation between $h$ and $D$ follows a power law,
consistently with the prediction from Föppl–Von Kármán theory for the buckling
of a circular plate under in-plane radial compression:
$h/t\sim\left(D/t\right)^{2/3}$ [33].

Toward validating the writing-phase simulations, in Figure 2(c), we plot the
normalized magnetic field required for snapping,
$B^{\mathrm{a}}_{\mathrm{c}}B^{\mathrm{r}}/(E\mu_{\mathrm{0}})$, as a function
of $D/t$. Naturally, increasingly slender shells require a lower magnetic
field for snapping, and the data collapse into a single curve, exhibiting a
power law with an exponent $\approx-1$. This scaling originates from the
balance between magnetic and elastic energies investigated in our previous
work [34, 35], which suggests
$B^{\mathrm{a}}_{\mathrm{c}}B^{\mathrm{r}}/(E\mu_{\mathrm{0}})\sim(D/t)^{-1}$;
a prediction in agreement with our present data (inset of Figure 2c).

In Figure 2(d), we present the results for the reading, plotting the
normalized blocking force, $FD/(Et^{3})$ (required to snap the shell), as a
function of $D/t$. Again the experiments (symbols) are in excellent agreement
with the FEM (lines). The data is consistent with a power law with exponent
$\approx 1.5$, which can be rationalized using well-established results for
the indentation of a spherical shell by a flat plate, causing mirror buckling
(to produce an inverted cap) of the shell [36, 37, 38, 39]. Balancing the
stretching and bending energies of the shell, the dimensionless indentation
force is expected to scale as $FD/(Et^{3})\sim(D/t)^{3/2}$, a prediction that
is consistent in our data (inset of Figure 2d). For completeness, the
dimensional version of the plots shown in Figure 2(b, c, d) are provided in
Supplementary Information.

Overall, we found excellent agreement between the experiments and the FEM
simulations of the scaled-up model system for the shell-fabrication protocol
and their geometric characterization, as well as for the reading and writing
phases.

### 3.2 Design of braille dots at the real scale.

We shift our attention from the scaled-up model system to investigate the
design of the real-scale braille dots. Leveraging the FEM simulations
validated above, and following the protocol details in Section 5.2, we explore
the design space for braille dots and determine their optimal fabrication and
operational conditions. Each dot must conform to the specifications laid out
in Section 2. Our objective is to determine the optimal ranges for the key
geometrical parameters (thickness, $t$, and fabrication pre-stretch,
$\lambda$) that simultaneously meet the acceptable design constraints for dot
height (obtained from fabrication) and meet feasible operational conditions
for the writing phase (critical magnetic field for actuation), and reading
phase (blocking force). Finally, we will identify the intersecting region of
these three design sub-spaces.

Figure 3: Design of the real-scale braille dot. The (a) dot height, $h$, (b)
critical magnetic field, $B^{\mathrm{a}}_{\mathrm{c}}$, and (c) blocking
force, $F$, are all plotted versus thickness, $t$, at different levels of pre-
stretching, $\lambda$=$[0.05,0.2]$ (increments of 0.05). The horizontal dashed
lines represent the design constraints of the corresponding quantities. (d)
Phase diagram in the ($B^{\mathrm{a}}_{\mathrm{c}}$, $F$) parameter space,
with the desired shaded region. (e) Phase diagram in the ($t$, $\lambda$)
parameter space for the geometry step and reading and writing phases, with the
feasible design space indicated by the rectangle region. All results were
obtained from FEM simulations. Figure 4: Improved design of real-scale braille
dots with pneumatic system. (a) Blocking force, $F$, and (b) magnetic field,
$B^{\mathrm{a}}_{\mathrm{c}}$, plotted as a function of the pressure
difference, $\Delta P$. The thickness was varied in the range
$t=[0.075-0.3]\,$mm (in increments of 0.025 mm) for a specific pre-stretch
$\lambda=0.15$. The horizontal dashed lines represent the limiting bounds
imposed by the braille standards. (c) Phase diagram in the
($B^{\mathrm{a}}_{\mathrm{c}}$, $F$) parameter space with the desired regions
of viable parameters (shaded region), for different levels of pressure
difference, $0<|\Delta P|<120$ kPa. All results were obtained from FEM
simulations.

First, we characterize the geometry of the dots obtained from the fabrication
step of the simulations. In Figure 3(a), we plot the dot height, $h$, versus
thickness, $t$, for different levels of $\lambda$. The color map represents
the various levels of $\lambda$, whose range is specified in Section 5.2. The
resulting $h(t)$ curves exhibit a non-monotonic trend, with overall values
(including the maximum of the curves) that increase with $\lambda$. This non-
monotonic behavior arises because for either very thick or thin plates, the
pre-stretch release (compression) leads to planar (radial) contraction rather
than increasing the out-of-plane deformation of the buckled plate (shell). The
horizontal dashed line represents the minimum dot height, $h\geq 0.48\,mm$,
required by braille specifications. Thus, to satisfy this requirement, we find
that the feasible range for the fabrication pre-stretch is $\lambda\geq 0.15$,
and the viable thickness range is $t\in[0.05,0.325]\,$mm.

Next, we consider the writing phase, which enables the braille dot (magnetic
shell) to switch between its two stable stages. In Figure 3(b), we plot the
critical amplitude of the magnetic field, $B^{\mathrm{a}}_{\mathrm{c}}$,
required to snap the dot as a function of $t$, for different fabrication pre-
stretches. We observe that, $B^{\mathrm{a}}_{\mathrm{c}}$ increases with
$\lambda$ (and thus for taller dots; cf. Figure 3a), also with a non-monotonic
dependence on $t$. To prevent the demagnetization of the shell due to high
magnetic fields [31], we established the upper limit
${B^{\mathrm{a}}_{\mathrm{c}}}\leq$500 mT, represented by the horizontal
dashed line in Figure 3b). Consequently, for actuation within this magnetic-
field limit, the allowed parameters are in the ranges
$t\in[0.025,\,0.325]\,$mm, and $\lambda\in[5,\,17.5]$.

Finally, we turn our attention to the reading phase. In Figure 3(c), we plot
the blocking force, $F$, as a function of $t$, using the same ranges of the
other parameters which are specified in Section 5.2. Increasing $\lambda$
leads to an overall increase of the $F(t)$ curves, much like the writing
phase, which is also non-monotonic. Combining these results with the $h(t)$
data in Figure 3(a) implies that taller dots require a higher indentation
force for inversion, presumably due to their geometry-induced rigidity [40,
41]. According to braille specifications, the blocking force of the dot must
be at least $F\geq 50\,$mN (horizontal dashed line in Figure 3c), which is
limited by the potential snapping of the shell due to the touch by a
fingertip. Under this constraint, we determine that the ranges of feasible
parameters for this reading phase are $t\in[0.075,\,0.325]\,$mm, and
$\lambda\geq 12.5$.

Combining the above results for the viable ranges of the parameter space
$(t,\,\lambda)$ dictated by the geometry characterization, writing, and
reading phases, we present the intersection of these three design phases in
Figure 3(d). At each level of $\lambda$ (colored symbols), we plot
$B^{\mathrm{a}}_{\mathrm{c}}$ as a function of $F$ for all the thickness
values. An overall correlation emerges between $B^{\mathrm{a}}_{\mathrm{c}}$
and $F$. In the plot, the design constraints on $B^{\mathrm{a}}_{\mathrm{c}}$
and $F$ mentioned above are represented by the shaded region, which only
intersects with a few of the explored designs ($\lambda\geq 0.125$). The
design constraints require a sufficiently high blocking force while ensuring a
sufficiently low magnetic field, a trade-off that is challenging to achieve in
our system.

In Figure 3(e), we present an alternative version of the overlap of all of the
design constraints explored above, now in the final target design parameter
space $(t,\,\lambda)$. Each separate shaded/textured region relates to the
individual viable bounds obtained above for the geometric characterization,
writing, and reading phases. Meeting all the constraints and ensuring braille
standards requires an overlap of these three regions; i.e., the domain
enclosed by the dashed rectangle with $t\in[0.1,0.325]\,$mm and
$\lambda\in[0.15,0.175]$.

### 3.3 Design improvement using a pneumatic system

The feasible design space identified from the results in the previous section
is rather limited, making a flexible fabrication process challenging. The
design requires the shell to be able to snap during the writing phase yet
remain resistant to snapping during the reading phase. To address these
conflicting limitations without altering the geometry, we propose the
incorporation of an additional pneumatic loading system. This pneumatic
component modulates the energy barrier for buckling in both the reading and
writing phases; the pressure difference, $\Delta P$, between the inside and
outside of the dot is positive for the reading phase and negative for the
writing phase. The implementation details of this pneumatic loading in the FEM
simulations are provided in Section 5.2. Hereon, we focus on shells fabricated
with a pre-stretch of $\lambda=0.15$, which was deemed practical from the
parameter exploration presented above.

For the writing phase, the dot is depressurized to reduce the critical
magnetic field $B^{\mathrm{a}}_{\mathrm{c}}$ required for snapping, thereby
facilitating the switching of the dot. In Figure 4(a), we plot
$B^{\mathrm{a}}_{\mathrm{c}}$ versus $\Delta P$, for different values of $t$.
The results show that applying pneumatic loading substantially reduces the
magnetic field $B^{\mathrm{a}}_{\mathrm{c}}$. The
$B^{\mathrm{a}}_{\mathrm{c}}(\Delta P)$ curves are linear, with a slope that
varies with $t$. For example, depending on $t$, a dot depressurized by $\Delta
P\lesssim-60\,$kPa can lower the critical magnetic field for snapping by as
much as 50%, compared to the zero-pressure case. The horizontal dashed line
represents the maximum acceptable magnetic field.

For the reading phase, in contrast to the writing phase, the dot is
pressurized to enhance the blocking force, $F$, and better resist indentation.
The dot is first pressurized and then loaded at its pole. In Figure 4(b), we
present the dependence of the blocking force $F$ on the applied $\Delta P$ for
different thickness values, $t$ (see adjacent color bar). In the explored
range of parameters, the $F(\Delta P)$ response is linear, as expected from
previous work [40, 41], and no buckling occurs. For example, the resistance of
the dot to indentation force can be increased by up to 100$\%$ (at
$t=0.2\,$mm) compared to the zero-pressure case. We find that all curves are
now well above the $50\,$mN limit imposed by braille standards.

Finally, in Figure 4(c), we combine data from both phases with pneumatic
loading (Figure 4a and 4b) and plot $B^{\mathrm{a}}_{\mathrm{c}}$ versus $F$.
The shaded region indicates the viable range of parameter space; with a
pneumatic load of $|\Delta P|\geq 40\,$kPa, we achieve successful reading and
writing operations across the full thickness range. The limit values of
$|\Delta P|$ could be further tuned by varying the fabrication pre-stretch
$\lambda$, but we leave a more systematic exploration for future work. Our
results demonstrate that using this additional pneumatic component, the design
space of the system is significantly expanded compared to the zero-pressure
case explored in Section 3.2.

## 4 Conclusion

We have proposed a new design concept for a reversibly switchable braille dot
as a building block for refreshable braille displays. The proposed mechanism
uses bistable magnetic shells that can be snapped on-demand under an external
magnetic field (writing phase) while resisting buckling due to the indentation
by a fingertip (reading phase). An additional pressure-loading component
expands the available design space without modifying the dot geometry. First,
we performed experiments on a scaled-up model system to validate FEM
simulations. These simulations were then leveraged to systematically explore
the design space at realistic scales while meeting braille standards (geometry
and tactile sensitivity) with reasonable magnetic field strengths and
temporary pneumatic loading. Our design boasts several advantages over
existing solutions, including bistability for self-stabilization, as well as
fast state-switching and pattern refreshment. This switching can be triggered
by a transient magnetic field rather than a continuous energy input. Finally,
a constant pneumatic input for the whole actuator enables the tuning of the
power input of the system.

While our design introduces promising advances, it is not without potential
limitations. Its complexity calls for advanced manufacturing and assembly
techniques. Miniaturized solenoids to generate the required magnetic field
under each dot would need to be developed. Furthermore, incorporating a hybrid
magnetic and pneumatic system could pose challenges in terms of size and
power, particularly for portable or battery-operated devices. Despite these
potential obstacles, we anticipate that future research and physical
implementation of this concept could make it possible to build a new class of
compact, user-friendly, and cost-effective braille readers.

## 5 Methods Section

### 5.1 Experiments with the scaled-up system

In this section, we detail the fabrication protocol, geometric
characterization, and testing (writing and reading phases) for the experiments
on our scaled-up model system. The data obtained from these experiments serve
to validate the FEM simulations in Section 3.1.

Fabrication: We have followed an established experimental procedure to prepare
the h-MRE material used to fabricate our specimens [42, 43, 30]. First, we
mixed Vinylpolysiloxane (VPS-32, Elite Double, Zhermack) with NdPrFeB
particles (MQFP-15-7-20065-089, Magnequench), with volume fraction
$c_{v}$=18.7$\%$. Then, an automated film applicator (ZAA 2300, Zehntner)
spread the VPS-NdPrFeB mixture into a thin film, which, upon curing, yielded a
thin elastic plate. By modulating the gap height of the film applicator, we
fabricated 10 plates with thicknesses in the range $t$=$[0.180,\,1.080]$ mm,
measured using an optical microscope (VHX-950F, Keyence). Post-curing, we cut
eight circular plates (Figure 5a) with diameters in the range
$D_{\mathrm{p}}$=$[25,\,60]\,$mm, in increments of 5 mm.

Magnetization: Then, various steps are involved in the magnetization of the
magnetic plate, as illustrated in the schematic diagrams in Figure 5(a)-(d).
The cut circular h-MRE plate of diameter $D_{\mathrm{p}}$ from the fabricated
plate (Figure 5a) does not possess any magnetic properties. Therefore, to
magnetize the plate, we folded it symmetrically into a semicircle along
$x$-axis (Figure 5b) and then into a quarter-circle along $y$-axis (Figure
5c). Third, we placed the folded quarter-circle plate in the impulse
magnetizer (IM-K-010020-A, flux density $\approx 4.4\,$T, Magnet-Physik Dr.
Steingroever GmbH) at an angle of $\alpha=45^{\circ}$ with respect to the
positive $y$ axis, aligned with the edge of the quarter circle (Figure 5d).
The magnetizer generates a magnetic field of strength $\mathbf{B}$, inducing a
permanent magnetic dipole in the NdPrFeB particles. Assuming a uniform
particle dispersion within the polymer matrix, the magnetization magnitude
computed from the volume average of the total magnetic moment of the
individual particles is $M={\mu_{0}}^{-1}B^{\mathrm{r}}=$134.4 kAm-1, where
$B^{\mathrm{r}}$ is the residual magnetic flux density and $\mu_{0}$ the
relative permeability of air. After unfolding, the magnetization pattern of
the circular plate is four-fold symmetric (Figure 5e). In each of the plate’s
four quarters, $k=\\{1,2,3,4\\}$, the magnetization is expected to be
$\mathbf{M}\approx M\hat{\mathbf{n}}_{k}$, pointing along the unit vector:

$\mathbf{\hat{n}}_{k}=-\cos\left(\alpha+(k-1)\pi/2\right)\hat{\mathbf{e}}_{x}-\sin\left(\alpha+(k-1)\pi/2\right)\hat{\mathbf{e}}_{y},$
(1)

where $\alpha=45^{\circ}$ is the orientation of the folded plate in the
magnetizer (see Figure 5d).

Figure 5: Fabrication and experimental apparatus for the scaled-up model. (a)
A circular magnetic plate (diameter of $D_{\mathrm{p}}$) is first (b) folded
along the $x$-axis and then (c) along the $y$-axis to form a quarter-circle
shape. (d) The folded plate is inserted into the pulse magnetizer at an angle
$\alpha=45^{\circ}$. The magnetizer generates a strong axial magnetic field,
B, which induces a magnetic moment, $\mathbf{m}$. (e) The circular h-MRE plate
was magnetized while folded, yielding the magnetization profile described by
Equation (1). (f) Two VPS annuli serve to constrain the plate boundary. (g)
Cylinders pre-stretched the annuli, which sandwich the h-MRE plate. (h) Upon
release of the pre-stretch, the plate buckles to form a shell. (i) Photograph
of the apparatus for the writing experiments. The shell is placed between two
Helmholtz coils and loaded by a uniform magnetic field along
$\hat{\mathbf{z}}$. (j) Photograph of the apparatus for the reading
experiments; a plate indents the shell, and the blocking force $F$ is
measured.

Empirically, we found that our chosen magnetization profile is more effective
in inducing snap buckling than simpler patterns (e.g., uniform magnetization
parallel or perpendicular to the plate mid-surface). Our choice aligns with
the anti-symmetric profile selected in recent snap-buckling studies of h-MRE
beams [43, 44]. We also acknowledge that our choice of the magnetization
profile is motivated by fabrication simplicity. However, we recognize the need
to conduct a more systematic exploration of other profiles [45] in future
research.

Having fabricated and magnetized the plates, we proceeded to produce shallow
shells through radial compression of the said plates. First, we fabricated two
VPS-32 annuli to act as the clamped boundary of the shell, each with inner and
outer diameters of $D=D_{\mathrm{p}}/2$ and $2D=D_{\mathrm{p}}$, respectively
(Figure 5f). Next, we stretched these annuli using two rigid cylinders of
diameter $D+\lambda D$ (Figure 5g), resulting in the radial pre-stretch of
$\lambda=\Delta D/D$ (Figure 5g). The plate was then sandwiched and bonded
between the annuli using the same VPS material. After curing, the cylinders
were removed, thereby relaxing the pre-stretched. Consequently, the plate
buckled out-of-plane due to the in-plane ($x$-$y$) radial compression,
yielding a shell, the braille dot (Figure 5h). The height, $h$, of this newly
formed shell was measured using an optical profilometer (VR-3200, Keyence
Corporation).

For the writing experiments (Figure 5i), the sample designated for testing was
placed within the region of the uniform magnetic field produced by a set of
Helmholtz coils [34, 42, 43]. The sample was clamped between two acrylic
(rigid) plates. Gravitational effects were minimized by orienting the shell’s
snapping direction (along $\pm\hat{\mathbf{z}}$) perpendicularly to gravity
(-g$\hat{\mathbf{y}}$). We determined the critical magnetic field
$B^{\mathrm{a}}_{\mathrm{c}}$ needed for snap-buckling, thereby writing the
desired state of the braille dot. To do so, we gradually increased the
magnetic flux density by increasing the current $I$ in the coils, in
increments of 0.05 A over 20 s intervals, until snap-through occurred [43].

For the reading experiments, we used the apparatus shown in Figure 5(j). The
specimen was mounted on an acrylic plate containing a hole to equalize the in-
out differential pressure. The shell equator was clamped using its thick
boundary annuli and mounted onto an acrylic plate using silicone glue. For the
intender, we used a rigid acrylic disk of diameter $D_{\mathrm{ind}}=0.8D$,
smaller than the inner diameter ($D$) of the thick annular boundary to prevent
contact with the perimeter during indentation. The indentation displacement
was imposed at a constant velocity, 0.06 mms-1. The reaction force, $f$,
exerted on the indenter was measured by a load cell (2530-5N, Instron). We
define the blocking force, $F$, as the maximum of $f$, beyond which the shell
undergoes snap-through buckling, altering the state of the dot.

### 5.2 Finite element modeling simulations

We conducted the FEM simulations using the commercial package ABAQUS/Standard,
undertaking two distinct series of simulations with the same protocol but
different parameters. First, we worked with the same parameters as the scaled-
up model system described in Section 5.1. The objective was to validate the
FEM simulations against experiments. For the second series of simulations, we
shifted to the realistic dimensions of the braille dots discussed in Section
2. In this case, the plate thickness was varied in the range
$t$=$[0.025,0.325]$ mm (increments of 0.025 mm), and the fabrication pre-
stretch of the magnetic plate in the range $\lambda=[0.05-0.2]$ (increments of
0.025).

The initially flat, circular magnetic plate was modeled as a three-dimensional
solid body. Geometric nonlinearities were accounted for throughout the
analysis. Similarly to the experiments, the plate was segmented into four
quadrants, each with a magnetization oriented along $\mathbf{n}_{k}$ (cf
Section 5.1, Equation (1) and Figure 5e). The magnetic plate was discretized
using the user-defined 8-node brick element proposed by Zhao et al. [31] for
the modeling of hard-magnetic deformable solids under a uniform magnetic
field. We conducted a convergence study to determine the appropriate level of
mesh discretization, resulting in 6 elements in the thickness direction, 60
elements along the diameter, and 200 elements circumstantially. Mechanical
loads, both contact (indentation) and distributed (pressure), were applied via
a dummy mesh of C3D8R solid elements sharing the same nodes as the user
elements. The material was assumed to be an incompressible ($\nu\approx 0.5$)
Neo-Hookean solid with a bulk modulus 100 times higher than its shear modulus
($G=E/3$) and a Young’s modulus of $E=1.76$ MPa.

In both the scaled-up and real-scale simulations, we explored various
combinations of the parameters ($t,\,D,\,\lambda$) to investigate (1) geometry
of the fabricated dot (Figure 1b), (2) writing phase (Figure 1c), and (3)
reading phase (Figure 1d), as specified next.

(1) Geometry: To account for the possible emergence of higher-order modes
during plate buckling, we simulated the entire magnetized plate without any
symmetry assumptions (cf. Section 5.1). In order to break the symmetry on the
$x-y$ plane and induce buckling, a small out-of-plane displacement ($0.01t$)
was applied as an initial perturbation. Then, we specified the Dirichlet
boundary condition on each boundary node and applied compression by imposing
radially inward-directed displacements toward the center of the plate, a
process that led to the formation of the shell. The extent of compression was
set through $\lambda$.

(2) Writing: For the writing-phase simulations, having set the dot geometry in
step (1), we subjected the raised dot (ON state) to a uniform external
magnetic field, $B^{\mathrm{a}}=1\,$T. The field was applied with a slight
misalignment of $1^{\circ}$ about the -$\hat{\mathbf{z}}$ direction to trigger
asymmetrical buckling modes, thus providing a closer approximation of actual
experimental conditions. The magnitude of the magnetic field was then
increased linearly in the range of [0, 1] T. The magnetic field magnitude,
$B^{\mathrm{a}}_{\mathrm{c}}$, needed to snap the dot to the second stable
state, was determined from the magnetic-field increment at which the
displacement of the shell pole exhibited a sudden jump.

(3) Reading: For the reading-phase simulations, the indentation was simulated
using a rigid circular plate indenter that exerted controlled displacements
and discretized using rigid elements. The contact between the indenter and the
shell was assumed to be hard and frictionless, thereby preventing surface
penetration and sliding. To quantify the mechanical force needed to induce
snapping, the dots were subjected to a downward indentation load (along
$-\hat{\mathbf{z}}$) until reversal occurred, and the blocking (maximum)
force, $F$, was recorded.

(4) Pneumatic loading: When simulating the real-scale braille dots, we also
considered a constant pneumatic load, as discussed in Section 3.3. This
pressure loading served to widen the design space by stiffening the dot
against snapping during the reading phase (indentation) and reducing the
energy barrier during the writing phase. For each dot geometry, we first
measured the critical pressure, $P_{\mathrm{cr}}$, required to snap the shell
on its own, following the same procedure used to measure the critical magnetic
field for snapping [34, 46]. A constant positive (or negative) pressure
difference, within the range $|\Delta P|\in[0,P_{\mathrm{cr}})$ (in increments
of $1\,$kPa) was applied normally to the surface of the shell before
initiating the reading (or writing) simulation steps, respectively. Steps (2)
and (3) described above were repeated under this constant pneumatic loading.
Finally, the blocking force, $F$, and the critical magnetic field amplitude,
$B^{\mathrm{a}}_{\mathrm{c}}$, were recorded for each pair of parameters
($\Delta P,\,t$). For these simulations, we set the fabrication pre-stretch to
$\lambda=0.15$.

Acknowledgements

We are grateful to Dong Yan for fruitful discussions and Naïs Coq for
providing the illustration of Figure 1. A.A. acknowledges funding from the
Federal Commission for Scholarships for Foreign Students (FCS) through a Swiss
Government Excellence Scholarship (Grant No. 2019.0619).

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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