# The Extreme Mechanics of Viscoelastic Metamaterials

David M.J. Dykstra  Shahram Janbaz and Corentin Coulais  Institute of Physics
University of Amsterdam
Amsterdam, 1098 XH
Netherlands

###### Abstract

Mechanical metamaterials made of flexible building blocks can exhibit a
plethora of extreme mechanical responses, such as negative elastic constants,
shape-changes, programmability and memory. To date, dissipation has largely
remained overlooked for such flexible metamaterials. As a matter of fact,
extensive care has often been devoted in the constitutive materials’ choice to
avoid strong dissipative effects. However, in an increasing number of
scenarios, where metamaterials are loaded dynamically, dissipation can not be
ignored. In this review, we show that the interplay between mechanical
instabilities and viscoelasticity can be crucial and can be harnessed to
obtain new functionalities. We first show that this interplay is key to
understanding the dynamical behaviour of flexible dissipative metamaterials
that use buckling and snapping as functional mechanisms. We further discuss
the new opportunities that spatial patterning of viscoelastic properties offer
for the design of mechanical metamaterials with properties that depend on
loading rate.

##  1 Introduction

Mechanical metamaterials exhibit exotic mechanical responses. Static responses
of interest span a wide range of tunable behaviours, such as auxetic [1, 2,
3], programmable [4, 5], shape-changing [6, 7], non-reciprocal [8] to chiral
responses [9], often by harnessing nonlinear mechanics and snap-through
instabilities [4, 5, 10, 11, 12]. Interesting dynamical responses include
shock absorption [13, 14, 15, 16], soliton propagation [17, 18] and transition
waves [19, 20].

In many of these dynamical responses, dissipation plays a key role.
Specifically, dissipation plays a key role in vibration transmission. As such,
dissipative vibration transmissions has been studied extensively in acoustic
metamaterials featuring wave propagation and band gap structures [21, 22, 23,
24, 25, 26, 27, 28, 29, 30] as well vibration and shock control [31, 32, 33,
34, 35, 36]. However, although the effect of the constitutive materials’
dissipation on mechanical instabilities has been well studied [37, 38, 39, 40,
41, 42, 43, 44], the role of dissipation in the instabilities that underpin
the extreme mechanics of mechanical metamaterials remains poorly understood
(Fig 1). Yet, there is growing evidence that dissipation can play a key role
in the response of mechanical metamaterials that are based on buckling [45,
46]. In addition, a growing body of work has recently demonstrated that a wide
variety of additional functionalities can be obtained by leveraging
dissipation in mechanical metamaterials [47, 46, 48, 49].

In this short review, we discuss the interplay between mechanical
instabilities and viscoelasticity and how this interplay can be harnessed to
offer new functionalities that depend on loading rate.

Figure 1: Viscoelastic Metamaterials. Viscoelasticity affects the dynamics of
buckling and can host new responses and functionalities.

##  2 Delayed Snap-through Buckling by Dissipation

Mechanical metamaterials often harness mechanical instabilities, including
snap-through instabilities [50]. In particular, multistable buckling paths can
be harnessed to switch between different responses [50]. As such, we need to
understand how dynamics and dissipation affect mechanical (snap-through)
instabilities and multistability before we can develop new types of
dissipative metamaterials.

On the one hand, while quasi-static buckling analyses do not consider any
timescale, even purely elastic snap-through instabilities show a very distinct
time dependence, transitioning from a very slow state of storing elastic
energy to a fast snap-through motion [51]. This snap-through motion is what
allows for instance the Venus flytrap or the hummingbird to catch their prey
[52, 53]. Such snap-through motion can further be tuned by controlling the
loading rate [54].

However, most solid materials are not purely elastic. Most solid materials are
viscoelastic instead, and exhibit both viscous and elastic characteristics
when undergoing deformation. Viscous materials, like liquids, start flowing
when a stress is applied. Purely elastic materials return to their original
state once the stress is removed without dissipating any energy. Viscoelastic
materials in turn exhibit dissipation and a time-dependent stress-strain
response. Specifically, viscoelastic materials stiffen with increasing strain
rates. Basic models used for viscoelastic solids are discussed in the
Materials & Methods.

Adding dissipation complicates the dynamics of instabilities. Viscoelastic
dissipation in instabilities has already been explored extensively since the
pioneering works of Kempner in 1954 for beam buckling [37] and Nachbar and
Huang in 1967 for snap-through instabilities [38]. As one might expect, they
showed that the buckling force increases with increasing loading rate, in line
with what one might expect with linear viscoelastic materials.

From a qualitative perspective, an equivalent quasi-static analysis can offer
significant insights, such as the Von Mises Truss set-up explored by Santer in
Fig. 2A [39]. Here, a rigid Von Mises truss is suspended by a vertical spring
$A$ and a horizontal spring $B$ on the top and right, with spring stiffness
$K_{A}$ and $K_{B}$ respectively. When pulling down the top of the Von Mises
truss, it can be argued that increasing the ratio of $K_{A}/K_{B}$ is
equivalent to increasing the strain rate in a viscoelastic analysis: spring A
experiences a much larger deflection rate than spring B when the Von Mises
truss is deflected downward, and since both springs are viscoelastic, the
instantaneous spring constant or spring A is effectively larger than that of
spring B. The relation between force $P$, deflection delta and $K_{A}/K_{B}$
is then captured in Fig. 2B [39]. Two major effects can be identified in Fig.
2B. First of all, as $K_{A}/K_{B}$ increases, not only does the buckling force
of the first limit point increase, but so does its deflection. Second, as
$K_{A}/K_{B}$ increases, the second limit points moves upward and stops
crossing 0 above a critical $K_{A}>K_{crit}$. At this point, the structure
loses its bistability and becomes monostable. In other words, a viscoelastic
Von Mises Truss can be bistable if it is loaded slowly but monostable when it
is loaded fast.

Figure 2: Pseudo-bistability controlled by viscoelasticity. (A) A single rigid
truss supported at both ends by elastic springs (Reproduced from [39] with
permission from Elsevier). Changing the ratio of the spring stiffnesses can be
considered equivalent to increasing the loading rate in a viscoelastic
analysis. The equilibrium paths in (B) (Reprinted from [39] with permission
from Elsevier) show that both monostable and bistable configurations are
possible. (C) A viscoelastic jumping popper shows pseudo-bistability
(Reproduced from [42] under CC BY 4.0). (D) A viscoelastic version of the Von
Mises Truss of (A), featuring Generalised Maxwell springs with 5 spring-
dashpots (Modified from [39] with permission from Elsevier). (E) Force-
Displacement curves due to moving at constant speed, with the colour
indicating the total loading time in seconds. Both very high and very low
speeds induce bistability, while intermediate speeds can generate
monostability.

The opposite effect can also occur. A viscoelastic structure can appear stable
in its snapped state after it is loaded. But after viscoelastic relaxation
occurs, the snapped state will lose its stability and the structure will snap
back into the initial state. This behaviour is in fact very common in
viscoelastic snap-through instabilities and has been referred to as temporary
bistability [39], pseudo-bistability [40, 41, 42], acquired bistability [43]
or metastability [40]. For instance, viscoelastic jumping poppers (Fig. 2C)
can appear bistable initially but may still snap back to the original state
due to viscoelasticity [42].

Figure 3: Delayed buckling by dissipation in mechanical metamaterials. (A) A
viscoelastic unit cell (a) after compression (b), showing initially bistable
behaviour (c) before snapping back after relaxation (d) (Copyright Wiley-VCH
GmbH. Reproduced from [55] with permission). (B) A laterally pre-confined
viscoelastic mechanical metamaterial for shock absorption experiences a snap-
through instability only when compressed slowly (Reproduced from [45] with
permission from ASME).

However, while a quasi-static analysis can show some agreement with
viscoelastic analyses, it can also differ substantially. To compare, we model
the same Von Mises Truss of Fig. 2A but by fully modelling the viscoelasticity
of the springs (Fig. 2D) so that they faithfully describe the behaviour of a
3D printed rubber, which has also been explored in depth analytically by Gomez
et al. [42]. We do so with the Viscoelastic Solver in Abaqus 2021 (See
Section5), using a three element truss model (T2D2). Specifically, we replace
springs A and B in Fig. 2A by viscoelastic springs using a viscoelastic 5-term
Prony series, which is equivalent to a Generalised Maxwell-Model with 5
spring-dashpots [56] (see Materials & Methods). For the strength and time
scales of the Prony series, we assume that the springs are made from Stratasys
Agilus30 material, a rubber-like material that is used in polyjet additive
manufacturing and is commonly used as a prototypical example of a highly
viscoelastic solid [45, 57, 58, 49, 59, 60]. The strength and time scales of
the Prony series have been adapted from [45] and can be found in the Materials
& Methods. We employ $w/h=10$ and $K_{A}/K_{B}=0.05$. We then move the truss
down with a constant rate and track the force-displacement response at various
loading rates in Fig. 3E.

When going from small loading rates in yellow to intermediate loading rates in
green and cyan, we observe a change from bistability (force goes through zero
three times) to monostability (force crosses zero only once). Also, the
reaction force $P$ increases over the entire loading regime when increasing
the loading rate. Both of these phenomena were also captured by the quasi-
static analysis in Fig. 2B. However, when we further increase the loading
rate, we again identify bistability, which is not captured by a simple quasi-
static analysis. Relaxation from this state could induce another state of
intermediate monostability instead, so the same structure could also capture
pseudo-bistability. This shows that it is not self-evident to predict in what
way viscoelasticity affects the pseudo-stability of a buckling structure.
Moreover, while the quasi-static analysis in Fig. 2B suggests that the first
limit point will shift to the right when increasing the equivalent loading
rate, the viscoelastic analysis in Fig. 2E shows the opposite effect. This
shows that it is not trivial to identify a priori whether viscoelasticity will
increase or decrease the buckling deflection either. In both of these cases,
increased viscoelasticity, as identified by a higher loading rate, has led to
an increase in buckling force.

Yet, the opposite can also be true. Stein-Moldavo et al. recently showed that
the buckling pressure of spherical shells could also drop due to viscoelastic
effects [44]. When a negative pressure was kept stable just below the elastic
buckling limit, a viscoelastic shell would still buckle after relaxation. This
is because the relaxation induces additional geometric imperfections, which in
turn leads to a reduction in the buckling pressure. However, depending on the
direction of the imperfections, the result could also act opposing and make
the system more bistable [61].

In summary, the interplay between instabilities, imperfections and
viscoelasticity all contribute to shaping how snap-through buckling is
affected by viscoelasticity.

In turn, this knowledge of viscoelasticity-induced pseudo-stability can be
used to understand how the behaviour of mechanical metamaterials is affected
by viscoelasticity. First of all, Li et al. have shown that viscoelastic
dissipation can delay buckling in viscoelastic metamaterials, which in turn
can delay a shift from non-auxetic to auxetic in mechanical metamaterials
[62]. Moreover, Berwind et al. showed in Fig. 3A that pseudo-stability can
occur in mechanical metamaterials just as it could for simpler visco-elastic
snap-through instabilities [55]. Pseudo-bistability can even be harnessed to
program a morphing pattern of viscoelastic snap-through shells [63]. On the
other hand, Dykstra et al. showed in Fig. 3B that viscoelasticity can prevent
a mechanical metamaterial featuring snap-through instabilities from snapping
altogether under large strain rates [45]. From a shock damping perspective,
this means that a higher strain rate could actually induce poorer shock
damping performance when no snap-through occurred. The strain at which this
transition occurred was then tailored by shape-changing. This showed the very
careful interrelation between geometry and dissipation in viscoelastic
metamaterials.

In summary, delayed snap-through and modification of stability by
viscoelasticity can occur because different parts of the structure or the
metamaterial can experience different strain rates. As a result, these
different parts experience different levels of stiffening induced by
viscoelasticity and this modifies the energy landscape. In the second section,
we will show that suitable spatial patterning of the viscoelastic properties
allows to further control how the energy landscape depends on strain rate.

##  3 Dissipation to Seed Dynamic Imperfections

Flexible metamaterials based on buckling and snap-through instabilities
exhibit complex energy potentials—with mountains and valleys. We argue in this
section that viscoelastic dissipation can be used to introduce a new stable
bifurcation state [64]. This in turn is an efficient way to control how
metamaterials navigate such landscapes, based on the loading rate (Fig. 4A)
[47]. As such, the deformation pathways and ensuing shape-changes and
effective properties of the metamaterials can depend dramatically on whether
the metamaterials are loaded fast or slow.

Figure 4: Dissipation to seed dynamic imperfections. (A) Viscoelasticity and
loading rate can be used to navigate an energy landscape with multiple valleys
(Reproduced from from [47] under CC BY 4.0). (B) Elastic-viscoelastic bi-beams
can switch buckling direction when changing loading rate (Reproduced from [48]
under CC BY-NC 4.0) and (C) can allow switching between auxetic and non-
auxetic when turned into a mechanical metamaterial (Reproduced from [48] under
CC BY-NC 4.0). (D) Loading rate can also be used to switch modes altogether in
metamaterials with a distinct number of global modes and both elastic and
viscoelastic hinges (Reproduced from [49] under CC BY-NC-ND 4.0). (E) An
origami sheet featuring a higher number of modes and both elastic and
viscoelastic hinges can also experiences alternative modes at intermediate
loading rates (Reproduced from from [47] under CC BY 4.0).

The control over such deformation pathways can be achieved by spatially
patterning the viscoelasticity of the flexible building blocks. The simplest
example is perhaps that of Euler beam buckling. In such a case, the patterning
of the viscoelastic relaxation strength allows to induce an imperfection that
depends on strain rate [48] (Fig. 4B). In practice, take a column split in two
parts: the left (right) half of the column is made of a material with a small
(large) viscoelastic relaxation strength. When the column is loaded along its
axis, it will buckle to the left (right) when loaded slower (faster) than the
typical relaxation timescale of the more viscoelastic material. This happens
due to a shift in the neutral axis. When loaded slowly (fast), the
viscoelastic material is much softer (stiffer), so the neutral axis shifts
left (right) of the point of load application and the beam therefore buckles
to the left (right). When sufficiently compressed, nonlinear deformation
prevents snap-back, implying a permanently stable bifurcation. This idea can
be applied to construct flexible structures that exhibit switching between
auxetic and non-auxetic behaviour (Fig. 4C) or show an apparent negative
viscoelasticity—where the apparent stiffness is lower for larger loading rate
and larger for lower loading rates [48].

The section above shows that the direction of buckling of a beam or a hinge
can be controlled by spatially patterning its viscoelastic properties. In this
case there is a single deformation mode (the out-of-plane bending) and one
controls its direction by inducing a strain-rate dependent imperfection.

In the following, we also discuss a different approach, which consists of
using structures with more than one deformation mode and where the spatial
patterning allows to selectively actuate the mode of choice based on loading
rate. Architectures that have been used for such an approach are oligomodal
metamaterials (Fig. 4D) [49] or origami with multiple folding branches (Fig.
4E) [47]. The idea is the following: a metamaterial has a distinct number of
global deformation modes (or folding branches for origami), such as the
metamaterial visualised in Fig. 4D which has two modes. One mode only actuates
viscoelastic hinges, all of which act soft during slow loading of the
structure. The other mode also actuates elastic hinges, which are much softer
than the viscoelastic hinges during fast loading of the structure. As a
result, fast loading will energetically favour deforming the mode featuring
elastic components. Where there are more modes available, it is then also
possible to obtain additional deformation modes at intermediate loading rates
(Fig. 4E) [47].

However, using only viscoelasticity is often not enough to navigate the energy
landscape effectively and to allow for a switchable actuation of the mode of
choice [49]. In some cases, some modes tend to have a lower energy barrier and
as such can be preferentially actuated regardless of the rate of actuation or
of the mismatch of viscoelasticity. Imperfections—traditionally considered a
hazard in most mechanical systems—tailored to the preferred mode, can be used
to tweak the energy landscape and hence to alleviate this problem. In quasi-
static metamaterial designs they have allowed branching into specific chosen
bifurcation paths [57, 65], while they can enable switching between modes when
combined with viscoelasticity [49]. This means that imperfections can also be
programmed specifically to change the loading rate at which mode switching
occurs.

In summary, we have discussed how spatial patterning of viscoelastic
relaxation strength allows to dynamically tune buckling imperfections or
relative mode stiffness. We have discussed a few examples where this approach
has been used to create metamaterials with shape-changes that depend on
loading rate. In the next section, we will show that stored energy combined
with viscoelasticity can also induce extreme mechanical properties.

##  4 Extreme Mechanics due to Negative Stiffness

Traditionally, the mechanical properties of a composite material typically
cannot surpass those of its constituents [66, 67]. For example, for the
Young’s modulus, upper and lower bounds for laminates are provided by the
Voigt [68] and Reuss [69] bounds, while upper and lower bounds for isotropic
composite materials are provided by the Hashin-Shtrikman bounds [70] (Fig.
5A). A similar compromise is present for damping. As a result, a combination
of high damping and high stiffness seems to be mutually exclusive [67].

However, these bounds all assume that the constituent materials have positive
definite elastic moduli, also known as positive stiffness [66, 71, 72, 73,
67]. In other words, they assume that the constituents do no not contain
stored energy. However, while a freestanding material made out of a single
material must display positive stiffness, confined systems can store energy.
For example, post-buckled systems can display negative stiffness by releasing
mechanical energy [67]. Viscoelasticity in turn stabilises this state of
negative stiffness [74, 75, 76, 71, 77, 72, 67]. In turn, the negative
stiffness in these post-buckled systems can be harnessed to generate extreme
damping [78, 79, 67].

Furthermore, confined material unit cells can also display negative stiffness.
In a notable example, Jaglinski et al. showed in Fig. 5B that a BaTiO3 unit
cell could transition from positive to negative stiffness when it is cooled
without allowing deformation [71]. When the normalised temperature $T_{N}$
becomes negative, the Landau energy [80] as function of strain $\epsilon$
becomes non-positive definite, implying negative stiffness (Fig. 5B). While
such a negative stiffness unit cell would be unstable in an unconfined
configuration, Jaglinski et al. embedded such unit cells in a tin matrix,
creating a mechanical metamaterial featuring negative stiffness inclusions
[71]. As a result, they were able to generate extreme damping in cyclic
loading (Fig. 5C) [71]. In turn, it even allowed them to generate a
viscoelastic modulus greater than diamond, the hardest natural material known
to humankind (Fig. 5C) [71].

Other authors have shown similar results, both theoretically and
experimentally: negative stiffness stabilised by viscoelasticity can generate
extreme mechanical properties, including extreme mechanical damping, extreme
viscoelastic modulus and extreme thermal expansion [74, 75, 76, 71, 77, 72,
67]. This shows the great potential of using negative stiffness in
viscoelastic metamaterials.

Figure 5: Negative Stiffness Inducing Extreme Dynamic Properties. (A) Young’s
modulus bounds for composite materials made out of constituents with positive
definite elasticity. The Young’s modulus for laminates is bound between the
Voigt [68] (blue dashed line and inset) and Reuss [69] (red solid line and
inset) bounds. The Young’s modulus for isotropic materials is bound between
the upper (green dotted line) and lower (cyan dash-dotted line) Hashin-
Shtrikman [70] bounds. The two constituents have a stiffness ratio of 10 and
Poisson’s ratio $\nu=0.3$. $E_{n}$ is the composite Young’s modulus normalised
with the Young’s modulus of the stiff constituent. $f$ is the volume fraction
of the stiff constituent. (B) Landau energy function of strain $\epsilon$ and
normalised temperature $T_{N}=(\alpha\gamma/\beta^{2})(T–T_{1})–0.25$, with
unit cells of BaTiO3 in cubic and tetragonal phases. $\alpha$, $\gamma$, and
$\beta$ are constants that depend on the material (Reproduced from [71] with
permission from AAAS). (C) Young’s modulus $|E^{*}|$ and viscoelastic damping
$tan\delta$ of tin-BaTiO3 composite metamaterial, showing extremely high
modulus over a range of temperatures. $\delta$ is the phase angle between
stress and strain. Young’s moduli of the constituents are BaTiO3, 100 GPa; and
tin, 50 GPa (Reproduced from [71] with permission from AAAS).

##  5 Modelling Viscoelastic Metamaterials

Another issue that comes with designing nonlinear viscoelastic metamaterials,
is that of computational effort. While modelling quasi-static or linear
systems require relatively little computational effort, modelling nonlinear
dissipative systems often requires computationally expensive nonlinear
dynamics. In most cases, the response can fully and accurately be modelled
using nonlinear dynamic finite element methods. For relatively smooth dynamic
analysis, implicit finite element methods can be used. Implicit dynamic
analyses are analyses where relatively large time steps can be taken which
each have to converge individually within numerical accuracy to the exact
solution. We have provided the codes for two of such cases on Zenodo [81],
corresponding to the analyses of Fig. 2DE and Fig. 4BC, which have been solved
using the viscoelastic solver in Abaqus (Simulia). For more complicated
dynamic analyses, explicit analyses may be required. Explicit dynamic analyses
are analyses, which use very small time steps that do not need to converge to
the exact solution individually. Instead, such analyses may jump around the
exact solution from one step to another. While implicit dynamic analyses can
already be computationally expensive for a detailed mesh, explicit dynamics
can be orders of magnitude more expensive when modelling long model time
durations. Often, using such dynamic finite element solvers is not
computationally viable for more complicated systems, parametric studies or
optimisation. As such, reducing this computational effort is another key point
to advance the field of viscoelastic shape-changing metamaterials.

Several strategies have already been identified. For one, many dissipative
shape changes without inertial effects can be approximated with fully elastic
quasi-static analyses, by scaling local time-dependent material stiffnesses
with a fixed value. Such analyses can qualitatively match the key shape-
change, which has been done among other for the results in Fig. 2AB [39] and
Fig. 4D [49]. However, these analyses only give a qualitative description and
can fall short in obtaining the requested response [49]. On the other hand, a
simplified model employing full nonlinear dynamics, such as that of Fig. 2DE,
can also offer qualitative agreement but may still fall short quantitatively
[45]. In such cases, the challenge remains in finding a minimum viable model
without quantitative sacrifice. For systems that are overdamped or quasi-
static, a viscoelastic finite element solver which neglects the effects of
mass can prove viable (Fig. 7 [48]). Such solvers are available in both an
implicit and explicit variant. For these overdamped or quasi-static analyses,
neglecting the effects of mass can lead to multiple orders of magnitude
improvement in computational effort, without quantitative sacrifice. In cases
where mass is not negligible or where many unit cells have to be analysed
simultaneously, another option would be to model mechanical metamaterials as a
continuum. Statically, a conformal mapping approach has showed excellent
quantitative agreement in modelling mechanism based metamaterials [82]. More-
over, Glaesener et al. have even shown that viscoelastic truss-based
metamaterials can be modelled as time-dependent continua [83]. In doing so,
they managed to capture the full nonlinear dynamics of the metamaterials
accurately.

These works have shown that various strategies can be taken to model
viscoelastic metamaterials more efficiently. These strategies, and more
efficient efficient strategies still, will allow us to predict and design a
new generation of viscoelastic metamaterials.

##  6 Outlook

We have reviewed here how dissipative effects can be used to control the
extreme mechanics of metamaterials in the context of viscoelasticity. Further
progress could be achieved in this direction by further improvements in
additive manufacturing [84], material science [85] and computational design
methods. In particular, better control of multimaterial printing of materials
with strong contrast in their viscoelastic properties could help in taking the
ideas presented in this article further—one could imagine more complex
geometries or different ways to embed internal energy. Printing at the
microscale could help in designing a wider range of shape-changing
metamaterials with multiple functionalities. One could also harness viscosity
to turn fluids subjected to acoustic vibrations into mechanical metamaterials
[86].

One can also look beyond the case of viscoelasticity treated here to manage
the energy landscape of adaptive and interactive materials and materials
systems [87]. For instance, Che et al. [46] showed that temperature can be
used to expand the design space of the previously discussed pseudo-bistability
in mechanical metamaterials [40, 41, 42]. In particular, they showed that by
changing the temperature, they could control the amount of time it took to
snap-back. This is because the viscoelastic properties of rubbers are highly
temperature dependent. Further down the line, one could envision that recent
ideas on odd viscoelasticity [88, 89, 90, 91, 92, 93]—where the viscoelastic
tensors are not symmetrical and which have so far required active
elements—could be realised via suitable combinations of purely passive
viscoelastic elements.

Alternatively, dissipation can be harnessed from different mechanisms than
viscoelasticity. For instance, internal adhesive contacts can be used to
generate extreme dissipation and recovery after loading of nanofoams [94, 95].
Active dissipation on the other hand can be used to control waves [96, 88] or
achieve robotic functionalities [89] in mechanical metamaterials. Otherwise,
when combining elastomers with Low Melting Point Alloys, plasticity and heat
be used to generate reversible plastic deformation [97]. This generates a
response similar to what can be obtained using shape memory materials. One can
also think of a plethora of alternative ways to harness dissipation, including
for instance piezoelectric [98], magnetic [99, 100], photo-responsive [101] or
superelastic [102] materials or fluidic devices [103]. In an even broader
sense, nearly any type of time-dependent mechanical properties can be
exploited for 4D printing: 3D printing with a fourth dimension in time [104,
105]. 4D printing can be used among others to generate morphing or
reconfigurable structures, where viscoelastic creep and stress relaxation can
play a vital role [105].

With so many possibilities to harness dissipation in the design of mechanical
metamaterials, the next step is to go from generating new types of mechanical
behaviour to new functionalities to actual applications. Obvious applications
include using extreme vibration and shock damping due to negative stiffness in
e.g. aerospace, automotive or high-tech machinery [67]. However, viscoelastic
metamaterials also show potential in synthetic soft tissues and implants
[106]. Dissipative materials can also be beneficial to enrich the design of
machine-like devices. For example, the locomotion of soft robots can be
influenced by embedding dissipative components in their bodies [89]. For
instance, the control of motion can be reduced to the control of flow rate in
pneumatic soft actuators [107, 103]. For example, at larger scales,
dissipative mechanisms in wearable devices and soft exo-suits [108] can
promote the design of such soft systems by reducing the duty of active
elements and annexing memory.

## MATERIALS & METHODS

## Viscoelastic Material Models

The two most commonly used viscoelastic material models for viscoelastic
solids are the Kelvin-Voigt model (Fig. 6A) and the Generalised Maxwell model
(Fig. 6B) [109].

Table 1: Fitted viscoelastic material properties for Agilus 30, with $N$ = 1, $N$ = 3 and $N$ = 5 $N$ | $E_{0}$ | $E_{e}$ | $\beta_{1}$ | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{4}$ | $t_{1}$ | $t_{2}$ | $t_{3}$ | $t_{4}$ | $t_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---
| [MPa] | [MPa] |  |  |  |  |  | [s] | [s] | [s] | [s] | [s]
1 | 2.58 | 0.75 | 0.71 |  |  |  |  | 0.43 |  |  |  |
3 | 3.25 | 0.64 | 0.45 | 0.26 | 0.10 |  |  | 0.047 | 0.97 | 18 |  |
5 | 3.27 | 0.62 | 0.43 | 0.11 | 0.05 | 0.047 | 0.015 | 0.04 | 3.94 | 27.1 | 0.65 | 323

The Kelvin-Voigt model (Fig. 6A) consists of a spring, with a dashpot in
parallel and is typically used to describe solids featuring fluid or
frictional dissipation. The Kelvin-Voigt model is governed by the following
equation [109], where $\sigma$ is stress, $E$ is the Young’s modulus, $\eta$
is the dashpot stiffness, $\epsilon$ is strain and $t$ is time:

$\sigma=E\epsilon+\eta\frac{\partial\epsilon}{\partial t}.$ (1)

The Generalised Maxwell model (Fig. 6B) consists of a spring, with multiple
spring-dashpots in parallel and is typically used to describe material
dissipation. It is also referred to as the Maxwell-Wiechert model [56]. The
spring-dashpots are each known as Maxwell elements, each with a time scale
$\tau_{i}=\eta_{i}/E_{i}$. When only one Maxwell elements is used, the
Generalised Maxwell model reduces to the Standard Linear Solid [109].

When a material based on the Generalised Maxwell model is subjected to a
stress relaxation test, the time response of the Young’s modulus, $E(t)$, is
described by:

$E(t):=E_{0}\left(1-\sum_{n=1}^{N}\beta_{n}\left(1-e^{\frac{-t}{\tau_{n}}}\right)\right),$
(2)

with $E_{0}$ the peak Young’s modulus under instantaneous load, $\beta_{n}$
the dimensionless relaxation strength, $\tau_{n}$ the timescale of the
individual Maxwell elements [56], $N$ the number of Maxwell elements
considered and $E_{e}=E_{0}\left(1-\sum_{n=1}^{N}\beta_{n}\right)$ the fully
relaxed Young’s modulus. Eq. 2 is also known as a Prony series. Whether the
response at short time scales can be captured largely depends on the loading
rate of the test: the higher the loading rate, the shorter the time scales
that can be captured.

Figure 6: Viscoelastic material models.(A) Representation of the Kelvin-Voigt
model of viscoelasticity. (B) Representation of the Generalised Maxwell model.
(C) Relaxation test for a dogbone sample of Agilus 30 3D printed material.
Time-dependent Young’s modulus vs. relaxation time. The thick blue line
indicates the test results, the thin dashed orange, green and red curves
present fitted data with $N=1$, $N=3$ and $N=5$ respectively.

## Viscoelastic Properties of Agilus30

To determine the visco-hyperelastic properties of Agilus30, we performed
stress relaxation experiments, which have largely been recorded previously in
[45].

An Agilus 30 dogbone section with a length $L$ = 50 mm, depth $d$ = 5 mm and
width $w$ = 10 mm has been 3D printed using a Stratasys Objet500 Connex3
printer. The sample is stretched quickly at a strain rate of
$\dot{\epsilon}=0.4\,s^{-1}$, using a uniaxial testing device (Instron 5943)
to a strain $\epsilon=20\%$, after which the force is allowed to relax for one
hour. The data is measured with a frequency of 1000 Hz with $t=0\,s$ defined
at the point of highest load.

The parameters of Eqn. (2) are then fitted using a least-squares fit to the
test data, which was interpolated on a logarithmic time scale from 0.01 s to 1
hour. The relaxation test results are presented in Fig. 6C, including fits
with $N$=1, $N$ = 3 and $N$ = 5, with corresponding material properties in
Tab. 1.

## Switchable Bi-beam

Here, we provide an example of a model of the strain-rate dependent bi-beams
described in Fig. 4BC [48]. The code is available on Zenodo [81].

Bi-beams are made by attaching two similar flexible beams made from two
different materials. The left beam (Fig. 7A) is made of a stiff elastic
material. The right beam is highly viscoelastic and softer than the elastic
one in the relaxed state but stiffer at high strain rates. Such a bi-beam will
predictably buckle to either the left (Fig. 7B) or the right (Fig. 7B) side,
depending on the rate of the applied strain.

Nonlinear buckling analysis was performed using the non-linear visco solver in
Abaqus (Abaqus 2021. Simulia, USA) [48]: an implicit dynamic solver (see
Section 5), which ignores inertial effects. In order to discretise the
geometry of the bi-beam, we used 8-node biquadratic hybrid elements (CPE8H)
while 12 elements were seeded through the width of the bi-beam. The bi-beam is
clamped at its both ends while the top clamped nodes are moved longitudinally
in our simulations. A linear elastic material model ($E=1.5$ MPa) was used to
define the elastic properties of the left beam and the viscoelastic properties
of the right beam is according to the five term Prony series materials
parameters that is proposed for Agilus in Table 1. The bi-beam in our model
has a length and width of 80 mm and 20 mm respectively.

The code shared [81] can be used to model buckling of bi-beam (to the left) at
a strain rate $\dot{\varepsilon}=1\times 10^{-3}$ 1/s (Fig. 7B) and (to the
right) at a strain rate $\dot{\varepsilon}=1$ 1/s (Fig. 7C). It can also serve
as a general example on how to model viscoelastic metamaterials.

Figure 7: Bi-beam model. (A) Bi-beam geometry. (B) Buckling to the left at low
strain rates. (C) Buckling to the right at high strain rates.

## Data and Code Availability

The data and codes that support the figures within this paper are publicly
available on a Zenodo repository [81].

## Acknowledgements

We acknowledge funding from the European Research Council under grant
agreement 852587 and the Netherlands Organisation for Scientific Research
under grant agreement NWO TTW 17883.

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