Parametric excitation of wrinkles in elastic sheets on elastic and viscoelastic substrates
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EPJ manuscript No.
(will be inserted by the editor)
Parametric excitation of wrinkles in elastic sheets on elastic and
viscoelastic substrates
Haim Diamant
School of Chemistry, and Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, Israel
February 11, 2021
arXiv:2102.05129v1 [cond-mat.soft] 9 Feb 2021
Abstract. Thin elastic sheets supported on compliant media form wrinkles under lateral compression. Since
the lateral pressure is coupled to the sheet’s deformation, varying it periodically in time creates a parametric
excitation. We study the resulting parametric resonance of wrinkling modes in sheets supported on semi-
infinite elastic or viscoelastic media, at pressures smaller than the critical pressure of static wrinkling. We
find distinctive behaviors as a function of excitation amplitude and frequency, including (a) a different
dependence of the dynamic wrinkle wavelength on sheet thickness compared to the static wavelength; and
(b) a discontinuous decrease of the wrinkle wavelength upon increasing excitation frequency at sufficiently
large pressures. In the case of a viscoelastic substrate, resonant wrinkling requires crossing a threshold of
excitation amplitude. The frequencies for observing these phenomena in relevant experimental systems are
of the order of a kilohertz and above. We discuss various experimental implications of the results.
1 Introduction is determined by a competition between the rigidities of
the sheet and the supporting medium [1, 2]. For exam-
Wrinkling is one the common deformation patterns which ple, in the case of a semi-infinite elastic medium of shear
thin elastic sheets form when subjected to lateral compres- modulus G, the wavelength is of order λc ∼ (B/G)1/3 ,
sion [1, 2]. In many cases wrinkles appear when the sheet where B is the sheet’s bending rigidity [18]. In terms
is supported on a softer substrate, a scenario which is rele- of the Young moduli of the sheet and medium, Es and
vant to a range of applications (e.g., coatings, paints) and Em , it can be rewritten as λc ∼ h(Es /Em )1/3 , where h
naturally occurring structures (e.g., skin and tissue lin- is the thickness of the sheet. Thus, because of the small
ings). Studies have been directed more recently at active 1/3 power, wrinkles much wider than the sheet thickness
wrinkling [4, 5, 6, 7, 8]. The interplay between the topogra- require sheets that are orders of magnitude stiffer than
phy of supported thin sheets and their delamination off the medium. Let us now include the sheet’s inertia and
the support [9,10, 11, 12, 13] suggests active wrinkling as an actuation frequency ω. This introduces two additional
an anti-fouling strategy adopted by Nature and mimicked lengths, G/(ρω 2 ) and [B/(ρω 2 )]1/4 , depending on whether
in man-made systems [3, 5, 6, 8]. the dominant restoring force comes from the medium or
Studies of active wrinkling have considered static or the sheet. Here, ρ = ρs h is the sheet’s effective 2D mass
quasi-static wrinkles, arising from mechanical equilibrium density arising from its mass density ρs and thickness h.
at pressures exceeding the static flat-to-wrinkle transition. We expect (and show below in detail) that the most un-
The dynamic effects, when considered [5, 8], are due to stable resonant mode should have a wavelength compa-
low-frequency (below 1 Hz) actuations, where the wrinkles rable to the static λc . Hence, we equate those two dy-
follow the external stimulus quasi-statically. Works going namic length scales with λc , obtaining ω 2 ∼ G/(ρλc ) and
beyond the quasi-static limit addressed the time evolu- ω 2 ∼ B/(ρλ4c ). Looking for a lower bound for the rele-
tion of the flat-to-wrinkle transition in sheets supported vant frequencies, we take large but relevant length scales,
on viscous [14, 15] and viscoelastic [16] media. λc ∼ 1 mm and h ∼ 0.1 mm, and a very soft hydrogel sub-
The present work investigates a different regime, where strate, G ∼ 103 Pa. This implies B ∼ Gλ3c ∼ 10−6 J. The
non-equilibrium inertial effects take the supported sheet resulting lower frequency bound (taking ρs ∼ 103 kg/m3 )
out of plane at pressures lower that the critical pressure is 103 –104 Hz. These values probably lie outside the range
of static wrinkling, through a mechanism of parametric of natural scenarios, but they are well within experimental
resonance [17]. Parametric resonance suggests itself natu- feasibility.
rally for compressed sheets, because the actuating pressure In sect. 2 we present the model and the general equa-
produces a force that depends on the sheet’s out-of-plane tions of motion which are common to the more specific
deformation. cases that follow. Section 3 presents results for a sheet
Let us examine heuristically the relevant scales of the supported on two types of substrate — an elastic substrate
suggested phenomenon. The wavelength of static wrinkles (sect. 3.1) and a viscoelastic one (sect. 3.2). In sect. 4 we2 Haim Diamant: Parametric resonance of supported sheets
z The kernel K(x, t) encodes the effect of the medium’s spa-
P(t) u(x,t) y
x tial and temporal response on normal stresses at its sur-
face. In Fourier
R ∞ space,
R∞
K̃(q, ω) ≡ −∞ dt −∞ dxeiqx−iωt K(x, t) is a complex func-
tion related to the medium’s viscoelasticity. We will as-
sume that the actuation frequency is sufficiently small,
such that only the lowest relaxation rate of the medium
Fig. 1. Schematic view of the system. is relevant. This limit allows the approximation,
K̃(q, ω) ' K1 (q) + iωK2 (q). (4)
summarize the predictions for experiments and describe
potential extensions of the model. Explicit expressions for K1 (q) and K2 (q) will be given in
the following sections.
The equation of motion of the sheet’s deformation is
2 Model ρü = Fs − Fm , (5)
2.1 The system where ρ is the sheet’s mass per unit area, and a dot de-
notes a time derivative. Using eqs. (1)–(5)
R ∞ while applying
We consider a thin elastic sheet attached to the surface a spatial Fourier transform, f˜(q, t) ≡ −∞ dxeiqx f (x, t),
of a (visco)elastic medium. The sheet, lying at rest on the
turns the equation of motion into
z = 0 plane, is assumed to be incompressible, infinite,
and made of a much stiffer material than the supporting ¨ + K2 (q)ũ˙ + [Bq 4 − P (t)q 2 + K1 (q)]ũ = 0.
ρũ (6)
medium. The medium occupies the region z ∈ (−∞, 0).
The sheet is compressed unidirectionally, along the x axis, The transformation
by a time-dependent actuating pressure (force per unit
length) P (t). It can deform on the xz plane from z = 0 ṽ ≡ ũe−[K2 /(2ρ)]t (7)
to z = u(x, t). See fig. 1. We assume |∂x u| 1 and con-
struct the leading-order (linear) model. Within this ap- eliminates the friction term, yielding
proximation the extension from a one-dimensional surface
deformation u(x, t) to a two-dimensional one, u(x, y, t), is ρṽ¨ + [Bq 4 − P (t)q 2 + K1 − K22 /(4ρ)]ṽ = 0. (8)
simple, and we restrict the discussion to 1D for brevity.
We account for the elasticity of the sheet and its inertia. We rewrite eq. (8) as
For the supporting medium we neglect inertia, i.e., we as-
sume that its time-dependent response, if it exists, arises ṽ¨ + ω02 [1 + a cos((2ω0 + )t)]ṽ = 0, (9)
solely from viscoelasticity. For the wrinkling phenomena
where
addressed here, this assumption implies that the veloc-
ity of bending modes along the sheet is smaller than the K2
1
velocity of sound modes in the medium. ω02 (q) ≡Bq − P0 q + K1 − 2
4 2
,
ρ 4ρ
2
P1 q
a(q) ≡ − 2 , (10)
2.2 Equations of motion ω0
(q) ≡ ω1 − 2ω0 (q).
Both sheet and medium respond to the surface deforma-
tion u(x, t). The sheet experiences a restoring normal force The problem has been transformed into an analogous
per unit area due to bending and the lateral compression, chain of independent, parametrically actuated oscillators,
with intrinsic frequencies ω0 (q), actuation amplitudes a(q),
Fs (x, t) = −Bu0000 − P (t)u00 , (1) and detuning parameters (q). We see in eq. (10) that in-
creasing the static pressure P0 weakens the ‘spring con-
where a prime denotes an x-derivative. We take the actu- stant’ ω02 . For the analogy to work we must have
ating pressure to be
ω02 (q) > 0, (11)
P (t) = P0 + P1 cos(ω1 t), (2)
and ‘oscillators’ (modes) q which do not satisfy it are over-
with actuation frequency ω1 . damped. Further, from the known solution to the classical
The normal force per unit area which the medium ex- problem of parametric resonance [17], we infer the condi-
periences at its surface is given by the general linear re- tion for instability (i.e., exponentially growing amplitude
sponse, ũ(q, t)), to leading order in the actuation a,
Z t Z ∞
0 1 2 2 K22
Fm (x, t) = dt dx0 K(x − x0 , t − t0 )u(x0 , t0 ). (3) Γ 2 (q) ≡ a ω0 − 2 > 0. (12)
−∞ −∞ 4 ρHaim Diamant: Parametric resonance of supported sheets 3
This is the squared rate of amplitude growth. The fastest For P0 = 0 (uncompressed sheet) the most unstable wave-
growing mode is the one which maximizes Γ (q). The al- length is indefinitely small. Thus wrinkles of well-defined
lowed detuning for each ‘oscillator’ q, i.e., the actuation finite wavelength require a finite static pressure, P0 > 0.
frequency range providing resonance, is obtained from the The wavelength of these dynamic wrinkles is shorter than
inequality 2 (q) < Γ 2 (q). To simplify the discussion, we the static one by a factor of P0 /P0c . As P0 is increased, the
will assume perfect tuning, wrinkles’ growth rate and their wavelength increase, un-
til, at the static wrinkling transition, the fastest-growing
= 0, ω1 = 2ω0 (q). (13) mode converges to the critical static one, qf → qc = 1, and
its growth rate diverges. The actuation frequency produc-
Thus, by “unstable band” we will refer simply to the set ing the fastest growth is obtained from eqs. (13), (15), and
of tuned ‘oscillators’ (i.e., range of q) for which Γ 2 (q) > 0. (18), as
√ (P 3 − P 3 )1/2
3 Results ω1f = 2 3 0c 2 0 , (19)
P0
3.1 Elastic substrate
independent of the actuation amplitude P1 .
The kernel K(x − x0 ) gives the nonlocal normal force den- The most natural control parameters, however, are the
sity, acting at a point on the medium’s surface, in response actuation frequency and amplitude, and the static pres-
to a normal surface displacement elsewhere. For a semi- sure. Given P0 , the choice of ω1 selects a dynamic wrinkle
infinite elastic medium it corresponds to the inverse of the wavenumber, q1 (ω1 , P0 ), according to eqs. (13) and (15).
Boussinesq problem [19]. In q space inverting the solution This wavenumber is not equal to qf in general, and is in-
to this problem is immediate, yielding dependent of P1 . Figure 2 shows the selected wavenumber
as a function of ω1 for several values of P0 between 0 and
G P0c . The figure shows also the asymptotes of q1 for small
K1 = q, K2 = 0, (14)
1−ν and large ω1 , which are both independent of P0 ,
where G is the medium’s shear modulus, and ν its Poisson
ω12 /8,
ratio. ω1 1
q1 (ω1 , P0 ) ' (20)
To make the expressions concise, we hereafter use B (ω1 /2)1/2 , ω1 1.
as the unit of energy, (B/Ĝ)1/3 as the unit of length, and
(ρ/B)1/2 (B/Ĝ)2/3 as the unit of time, where Switching for a moment back to dimensional parameters,
Ĝ ≡ G/(2(1 − ν)). This allows us to set B = Ĝ = ρ = 1. the two asymptotes become q1 ∼ (ρ/G)ω12 and
1/2
The 2D pressure is then measured in units of B 1/3 Ĝ2/3 . q1 ∼ (ρ/B)1/4 ω1 , revealing the different physical mech-
(In sect. 4.1 we will rewrite the most relevant expressions anisms in the two limits. At low frequencies the restoring
in dimensional form.) mechanism is the substrate’s elasticity, whereas at high
Substituting eq. (14) in eqs. (10) and (12), we obtain frequencies it is the sheet’s bending rigidity. Note that
these asymptotes agree with our heuristic arguments in
ω02 (q) = q(q 3 − P0 q + 2), (15) sect. 1.
P12 q 3 At P0 = P0∗ = 3/21/3 ' 2.38 and ω1 = ω1∗ = 31/2 /21/3 '
Γ 2 (q) = . (16)
4(q 3 − P0 q + 2) 1.37, the selected wavenumber, which is at this point q1∗ =
2−2/3 ' 0.630, bifurcates into three. The bifurcation en-
Static wrinkling appears when ω0 = 0. This occurs at the tails anomalous dynamics. At the bifurcation point we
critical pressure and wavenumber have dω0 /dq = 0, implying that an excitation with P0∗ and
ω1∗ at one edge of the sheet will not propagate through the
P0c = 3, qc = 1, (17) sheet. For P0 > P0∗ and ω1 < ω1∗ , we find from eq. (12) that
the largest of the three solutions for q1 (ω1 , P0 ) grows the
in agreement with earlier results [18].
fastest. Thus, for P0 > P0∗ , as the excitation frequency
For P0 < P0c we have ω02 (q) > 0 and Γ 2 (q) > 0 for
ω1 is gradually increased from 0, the observed wrinkle
all q regardless of P1 . Thus all wrinkling modes q are os-
wavenumber will undergo a discontinuous jump. For in-
cillatory and will resonate if excited by ω1 = 2ω0 (q). The
creasingly larger static pressure P0 , the jump occurs at
resonance does not require the actuation amplitude to ex-
lower and lower frequencies, until, at P0 = P0c , the sys-
ceed a finite threshold, P1c = 0; the growth rate simply
tem selects q1 = qc at zero frequency (see fig. 2). This is
increases linearly with P1 [eq. (16)]. This is due to the
how the static-wrinkling limit is reproduced from the dy-
absence of damping (K2 = 0).
namic one. Note that this entire behavior is independent
Maximizing eq. (16) gives the fastest-growing mode
of P1 ; hence, the discontinuous transition is present also
and its growth rate as
for an arbitrarily weak actuation.
√
P0c 3 3 Figure 3 presents 2D maps of the growth rate Γ as a
qf = > 1, Γf = 3
P1 (P0c − P03 )−1/2 . (18) function of P1 and ω1 for P0 = 0 and P0 = P0c /2.
P0 24 Haim Diamant: Parametric resonance of supported sheets
2.5 10
1.25
2.0
8
1.5 1.00
q1
1.0 6
ω1
0.75
0.5 4
0.0 0.50
0 2 4 6 8 10 12 2
ω1 0.25
Fig. 2. Wrinkle wavenumber as a function of actuation fre- 0
quency for an elastic substrate. Different curves correspond to 0.0 0.5 1.0 1.5 2.0 2.5 3.0
different values of static pressure P0 (from right to left): 0, 1.5, 0
P0∗ = 3/21/3 , 2.8, and P0c = 3. Solid circles indicate the fastest- P1
growing modes for the corresponding pressure. Dashed lines 10
show the asymptotes given in eq. (20). For P0 > P0∗ there are 1.5
three solutions for q1 , the largest of which growing the fastest,
implying a discontinuous jump in the observed wavenumber as
8
ω1 is ramped up. The empty circle marks the bifurcation point.
All parameters are normalized (see text).
6 1.0
ω1
3.2 Viscoelastic substrate
For a viscoelastic medium the response is generalized by 4
replacing the modulus G and Poisson ratio ν with frequency-
dependent complex functions, G̃(ω) and ν̃(ω). In prac- 0.5
tice, viscoelastic media such as polymer networks usually 2
contain a host liquid, which makes them virtually incom-
pressible. Hence, ν̃(ω) ' 1/2 for any ω. Applying the low-
frequency approximation of eq. (4), we generalize eq. (14) 0
to 0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
P1
K̃ = K1 + iωK2 , K1 = 2Gq, K2 = 2ηq, (21)
Fig. 3. Density plots of wrinkle growth rate as a function of
where G = Re(G̃) and η = Im(G̃)/ω are the low-frequency excitation amplitude and frequency for an elastic substrate.
shear modulus and shear viscosity of the substrate. We use The static pressure values are P0 = 0 (upper panel) and
the same units of energy, length, and time as in sect. 3.1, P0 = P0c /2 = 3/2 (lower panel). The dashed line in the lower
making B, G, and ρ all equal to unity. The viscosity η is panel shows the excitation frequency that produces the fastest-
measured then in units of ρ1/2 B 1/6 G1/3 . growing mode (which for an elastic substrate is independent of
P1 ). All parameters are normalized (see text).
Substituting eq. (21) in eqs. (10) and (12), we obtain
ω02 (q) = q[q 3 − (P0 + η 2 )q + 2], (22)
minimum amplitude of actuation. The right-hand side of
P12 q
2 2
2
Γ (q) = (23) eq. (23) has the asymptotes −4η q in both limits of small
− 4η q 2 .
2
4(q 3 − (P0 + η 2 )q + 2) and large q. Hence, the resonant band of modes, when it
exists, must be of finite width and centered around a fi-
The viscous component leads to several essential changes nite q. Real positive solutions to the equation Γ 2 (q) = 0
compared to the elastic case. First, only for P0 < P0c − appear for
η 2 are all the modes oscillatory (ω02 > 0). Thus, before
reaching the static wrinkling transition,
√ increasingly more
√ P1 > P1c (P0 , η) = 4η(3 − η 2 − P0 )1/2 , (24)
modes become damped. If η > P0c = 3, the substrate
is too viscous to allow underdamped excitation (unless and the wavenumber at the threshold is q = qc = 1. Thus,
we ‘strengthen the springs’ by stretching the sheet with crossing the resonance threshold leads to finite-wavelength
P0 < 0). dynamic wrinkles with wavelength similar to that of the
Another change brought about by viscosity is that static wrinkles. Figure 4 shows the appearance of the un-
parametric resonance of the oscillatory modes requires a stable band for P1 > P1c .Haim Diamant: Parametric resonance of supported sheets 5
0.1 2.5
0.0 2.0
-0.1 1.5
2
q1
Γ
-0.2
1.0
-0.3
0.5
-0.4
0.0 0.5 1.0 1.5 2.0 0.0
q 0 2 4 6 8 10 12
Fig. 4. Wrinkle growth rate squared as a function of wrin-
ω1
kle wavenumber for an uncompressed sheet (P0 = 0) on a Fig. 5. Wrinkle wavenumber as a function of actuation fre-
viscoelastic substrate. Different curves correspond to different quency for a viscoelastic substrate. The viscosity is η = 0.2.
excitation amplitudes P1 (bottom to top): 0.8P1c , P1c , and Different curves correspond to different values of static pressure
1.2P1c . Parametric resonance (Γ 2 > 0) of finite-wavelength P0 (from right to left): 0, P0∗ = 2.34, 2.8, P0c − η 2 = 2.96, and
wrinkles occurs for P1 > P1c . The viscosity is η = 0.2, for P0c = 3. Dashed lines show the asymptotes given in eq. (20).
which (and P0 = 0) P1c ' 1.38. All parameters are normalized For P0 > P0∗ there are three solutions for q1 , the largest of
(see text). the three growing the fastest, implying a discontinuous jump
in the observed wavenumber as ω1 is ramped up. The empty
circle marks the bifurcation point. For P0 > 2.96 (leftmost,
Another difference from the elastic-substrate case is brown curve) a band of modes are damped. All parameters are
that the rate of amplitude growth is not proportional to normalized (see text).
P1 [see eq. (23)]. As a result, the fastest-growing mode de-
pends now on P1 (follow the maxima in fig. 4). Finally, un- 6
like the elastic case, in the case of a viscoelastic substrate 1.0
we can get parametric excitation of the finite-wavelength
wrinkles even for an uncompressed sheet, P0 = 0 (see 5
again fig. 4). 0.8
Considering the excitation frequency ω1 = 2ω0 as a 4
control parameter, we get, as in sect. 3.1, a selected wavenum-
ber, q1 (ω1 , P0 , η), from eq. (22). Since qf depends in the 0.6
ω1
present case on P1 while q1 does not, the fastest-growing 3
mode does not belong in general to the set of selected
wavenumbers. In other words, one should tune P1 to get 2 0.4
q1 = qf (see fig. 6 below). Figure 5 shows the selected
wavenumber as a function of ω1 for several values of P0
between 0 and P0c . The asymptotes for small and large 1
ω1 remain as in eq. (20). Also here, the solutions bifurcate 0.2
∗ 1/3 2
above a certain static pressure, P0 = 3/2 − η , imply- 0
ing a discontinuous jump in the wrinkle wavenumber as 0.0 0.5 1.0 1.5 2.0 2.5 3.0
ω1 is increased. The bifurcation point is as in the elastic 0
case, q1∗ = 2−2/3 and ω1∗ = 31/2 /21/3 . For P0 > P0c − η 2 a P 1
band of modes becomes damped (with imaginary ω0 ) as Fig. 6. Density plot of wrinkle growth rate as a function of
manifested by the leftmost curve in fig. 5. excitation amplitude and frequency for an uncompressed sheet
Figure 6 shows a 2D map of the growth rate Γ as a (P0 = 0) on a viscoelastic substrate. The viscosity is η = 0.2.
function of the excitation parameters P1 and ω1 for an The dash-dotted line shows the threshold amplitude P1c . The
uncompressed sheet (P0 = 0). Unlike the elastic-substrate dashed line shows the excitation frequency that produces the
case (fig. 3), here the threshold for parametric resonance fastest-growing mode for each amplitude. All parameters are
makes the unstable region bounded. normalized (see text).
4 Discussion
4.1 Summary of experimental predictions
Let us summarize the results which seem most relevant
experimentally, and give them in dimensional form.6 Haim Diamant: Parametric resonance of supported sheets
In the case of an elastic substrate, one can first com- where qc is the static wrinkle wavenumber. Thus the thresh-
press the sheet until static wrinkling is reached. The mea- old of resonance may be used as a sensitive probe of the
sured critical pressure and static wrinkle wavenumber are viscous component η. As in the elastic case, at low and
related to the bending modulus of the sheet and the elastic high excitation frequencies the asymptotic dependence of
moduli of the substrate as the dynamic wrinkle wavenumber q1 on ω1 is given in
2/3 1/3 eqs. (26). The remark concerning the dependence on sheet
G G
P0c = 3B 1/3 , qc = . thickness in the elastic case holds here as well.
2(1 − ν) 2(1 − ν)B In the viscoelastic case, too, the value of P0 separates
(25) the behaviors when ramping up ω1 into two cases: a con-
This allows a measurement of B and G/(1 − ν). tinuous decrease of wavelength for low pressure and a dis-
For a finite P0 < P0c , and ramping up the actuation continuous one at high pressure. The transition now is at
frequency ω1 from zero, dynamic wrinkles should form
for any actuation amplitude. At low frequency the wrin- P0∗ = 0.794P0c − η 2 /ρ, (30)
kle wavenumber q1 increases (wavelength λ1 = 2π/q1 de-
creases) quadratically with ω1 , providing another sensitive probe of η.
To get a feeling for the relevant scales, we consider a
ρ(1 − ν) 2
q1 ' ω1 , (26a) specific system, motivated by the experimental system of
4G ref. [4]. It is made of a 1-mm-thick stiffer elastomeric sheet
and at high frequencies it increases as the square root of (Es ∼ 106 Pa), supported on a softer elastomeric medium
ω1 , (Em ∼ 104 Pa). These properties fit also a layer of skin
2 1/4
ρω1 covering a muscle tissue. The sheet’s bending modulus is
q1 ' . (26b) B ∼ 10−4 J. The resulting static wrinkle wavenumber
4B
[eq. (25)] is qc ∼ 1 mm−1 . To excite dynamic wrinkles
Since ρ ∼ h and B ∼ h3 , the two limits differ sharply of a similar wavenumber we need, according to eq. (26),
in their dependence on the sheet thickness. At low fre- an excitation frequency of order ω1 ∼ 104 s−1 . (We have
quencies the wrinkle wavenumber increases with thickness taken ρs ∼ 103 kg/m3 , yielding ρ ∼ 1 kg/m2 ). This is close
as ∼ h, and at high frequencies it decreases with h as to the relevant lower frequency bound obtained in sect. 1.
∼ h−1/2 . Both these dependencies differ from that of the As already noted there, such frequencies are probably too
static wrinkles, qc ∼ h−1 ; see eq. (25). high to be produced naturally but readily attainable in
Depending on the value of P0 , two distinct behaviors experiments.
are expected as ω1 is increased. At small pressures, P0 < The viscous component η, which would suppress dy-
P0∗ , the wrinkle wavelength decreases continuously with namic wrinkling altogether in the viscoelastic case, is η >
ω1 . For larger pressures, P0∗ < P0 < P0c , a discontinuous (ρP0c )1/2 . In the system described above this corresponds
drop in the wavelength is expected as a function of ω1 . to 1–10 Pa s (i.e., 103 –104 times the viscosity of water).
The transition occurs at
P0∗ = 0.794P0c . (27)
4.2 Model extensions
The transition in λ1 is a particularly strong, distinctive
prediction. The theory presented here is linear. As a result, it provides
The behavior in the case of a viscoelastic substrate is the properties of the instability but not the ultimate form
qualitatively different. Thus it might be used to obtain of the sheet’s deformation. Whether the deformation sat-
information on the viscoelastic properties of the support- urates to periodic wrinkles of finite height or localizes into
ing medium. The present theory is restricted, however, folds [20] should be checked in a future nonlinear theory
to sufficiently low frequencies, where the viscoelastic re- or simulation.
sponse is governed by a single (the longest) relaxation We have assumed a semi-infinite substrate. Over length
time, τ = η/G, i.e., the complex modulus is given by scales comparable and larger than the substrate thickness
G̃ = G+iωη; cf. sects. 2.2 and 3.2. The static measurement the results will be modified. In the opposite limit, of a thin
of P0c and qc are as in the elastic case above, except that substrate compared to the wrinkle wavelength, the effect
now G appearing in eq. (25) is the low-frequency shear of the medium will turn into that of Winkler foundation
modulus, G = Re(G̃), and ν = 1/2. [21], i.e., completely localized (K̃ independent of q).
A pre-condition for having dynamic wrinkles in the Another approximation employed above is the low-
viscoelastic case is that the viscous component is not too frequency limit, in which the relaxation of the viscoelas-
large, tic medium is dominated by a single relaxation time. Ac-
η < (ρP0c )1/2 . (28) tual viscoelastic media, particularly biological ones, have
a much richer frequency dependence, which will affect the
If this is met, one can obtain dynamic wrinkles even in an
response to the parametric excitation. Conversely, para-
uncompressed sheet (P0 = 0); yet, one needs an excitation
metric resonance may be used to tap into the medium’s
with a pressure amplitude that exceeds the threshold
rich temporal response based on an extended theory.
η2
4η Besides relaxation times, complex media have also char-
P1c = √ P0c − P0 − , (29) acteristic lengths which affect their response [22, 23]. The
qc Bρ ρHaim Diamant: Parametric resonance of supported sheets 7
present theory describes a way to sample various length 18. J. Groenewold, Wrinkling of plates coupled with soft elas-
scales (wavenumbers) by sweeping the parametric-excitation tic media, Physica A 298, 32–45 (2001).
frequency. Recently we have derived the solution to the 19. L. D. Landau, E. M. Lifshitz, Theory of Elasticity, 3rd Ed.
analogous Boussinesq problem for a viscoelastic structured (Butterworth-Heinemann, Oxford, 1986), sect. I.8.
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