Parametric excitation of wrinkles in elastic sheets on elastic and viscoelastic substrates
EEPJ manuscript No. (will be inserted by the editor)
Parametric excitation of wrinkles in elastic sheets on elastic andviscoelastic substrates
Haim Diamant
School of Chemistry, and Center for Physics and Chemistry of Living Systems, Tel Aviv University, Tel Aviv 6997801, IsraelFebruary 11, 2021
Abstract.
Thin elastic sheets supported on compliant media form wrinkles under lateral compression. Sincethe lateral pressure is coupled to the sheet’s deformation, varying it periodically in time creates a parametricexcitation. 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. Wefind distinctive behaviors as a function of excitation amplitude and frequency, including (a) a differentdependence 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 sufficientlylarge pressures. In the case of a viscoelastic substrate, resonant wrinkling requires crossing a threshold ofexcitation amplitude. The frequencies for observing these phenomena in relevant experimental systems areof the order of a kilohertz and above. We discuss various experimental implications of the results.
Wrinkling is one the common deformation patterns whichthin elastic sheets form when subjected to lateral compres-sion [1,2]. In many cases wrinkles appear when the sheetis supported on a softer substrate, a scenario which is rele-vant to a range of applications (e.g., coatings, paints) andnaturally occurring structures (e.g., skin and tissue lin-ings). Studies have been directed more recently at active wrinkling [4,5,6,7,8]. The interplay between the topogra-phy of supported thin sheets and their delamination offthe support [9,10,11,12,13] suggests active wrinkling asan anti-fouling strategy adopted by Nature and mimickedin man-made systems [3,5,6,8].Studies of active wrinkling have considered static orquasi-static wrinkles, arising from mechanical equilibriumat pressures exceeding the static flat-to-wrinkle transition.The dynamic effects, when considered [5,8], are due tolow-frequency (below 1 Hz) actuations, where the wrinklesfollow the external stimulus quasi-statically. Works goingbeyond the quasi-static limit addressed the time evolu-tion of the flat-to-wrinkle transition in sheets supportedon viscous [14,15] and viscoelastic [16] media.The present work investigates a different regime, wherenon-equilibrium inertial effects take the supported sheetout of plane at pressures lower that the critical pressureof static wrinkling, through a mechanism of parametricresonance [17]. Parametric resonance suggests itself natu-rally for compressed sheets, because the actuating pressureproduces a force that depends on the sheet’s out-of-planedeformation.Let us examine heuristically the relevant scales of thesuggested phenomenon. The wavelength of static wrinkles is determined by a competition between the rigidities ofthe sheet and the supporting medium [1,2]. For exam-ple, in the case of a semi-infinite elastic medium of shearmodulus G , the wavelength is of order λ c ∼ ( B/G ) / ,where B is the sheet’s bending rigidity [18]. In termsof the Young moduli of the sheet and medium, E s and E m , it can be rewritten as λ c ∼ h ( E s /E m ) / , where h is the thickness of the sheet. Thus, because of the small1 / ω . This introduces two additionallengths, G/ ( ρω ) and [ B/ ( ρω )] / , depending on whetherthe dominant restoring force comes from the medium orthe sheet. Here, ρ = ρ s h is the sheet’s effective 2D massdensity arising from its mass density ρ s and thickness h .We expect (and show below in detail) that the most un-stable resonant mode should have a wavelength compa-rable to the static λ c . Hence, we equate those two dy-namic length scales with λ c , obtaining ω ∼ G/ ( ρλ c ) and ω ∼ B/ ( ρλ ). Looking for a lower bound for the rele-vant frequencies, we take large but relevant length scales, λ c ∼ h ∼ . G ∼ Pa. This implies B ∼ Gλ ∼ − J. Theresulting lower frequency bound (taking ρ s ∼ kg/m )is 10 –10 Hz. These values probably lie outside the rangeof natural scenarios, but they are well within experimentalfeasibility.In sect. 2 we present the model and the general equa-tions of motion which are common to the more specificcases that follow. Section 3 presents results for a sheetsupported on two types of substrate — an elastic substrate(sect. 3.1) and a viscoelastic one (sect. 3.2). In sect. 4 we a r X i v : . [ c ond - m a t . s o f t ] F e b Haim Diamant: Parametric resonance of supported sheets P ( t ) xyzu ( x,t ) Fig. 1.
Schematic view of the system. summarize the predictions for experiments and describepotential extensions of the model.
We consider a thin elastic sheet attached to the surfaceof a (visco)elastic medium. The sheet, lying at rest on the z = 0 plane, is assumed to be incompressible, infinite,and made of a much stiffer material than the supportingmedium. The medium occupies the region z ∈ ( −∞ , x axis,by a time-dependent actuating pressure (force per unitlength) P ( t ). It can deform on the xz plane from z = 0to z = u ( x, t ). See fig. 1. We assume | ∂ x u | (cid:28) u ( x, t ) to a two-dimensional one, u ( x, y, t ), issimple, and we restrict the discussion to 1D for brevity.We account for the elasticity of the sheet and its inertia.For the supporting medium we neglect inertia, i.e., we as-sume that its time-dependent response, if it exists, arisessolely from viscoelasticity. For the wrinkling phenomenaaddressed here, this assumption implies that the veloc-ity of bending modes along the sheet is smaller than thevelocity of sound modes in the medium. Both sheet and medium respond to the surface deforma-tion u ( x, t ). The sheet experiences a restoring normal forceper unit area due to bending and the lateral compression, F s ( x, t ) = − Bu (cid:48)(cid:48)(cid:48)(cid:48) − P ( t ) u (cid:48)(cid:48) , (1)where a prime denotes an x -derivative. We take the actu-ating pressure to be P ( t ) = P + P cos( ω t ) , (2)with actuation frequency ω .The normal force per unit area which the medium ex-periences at its surface is given by the general linear re-sponse, F m ( x, t ) = (cid:90) t −∞ dt (cid:48) (cid:90) ∞−∞ dx (cid:48) K ( x − x (cid:48) , t − t (cid:48) ) u ( x (cid:48) , t (cid:48) ) . (3) The kernel K ( x, t ) encodes the effect of the medium’s spa-tial and temporal response on normal stresses at its sur-face. In Fourier space,˜ K ( q, ω ) ≡ (cid:82) ∞−∞ dt (cid:82) ∞−∞ dxe iqx − 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 mediumis relevant. This limit allows the approximation,˜ K ( q, ω ) (cid:39) K ( q ) + iωK ( q ) . (4)Explicit expressions for K ( q ) and K ( q ) will be given inthe following sections.The equation of motion of the sheet’s deformation is ρ ¨ u = F s − F m , (5)where ρ is the sheet’s mass per unit area, and a dot de-notes a time derivative. Using eqs. (1)–(5) while applyinga spatial Fourier transform, ˜ f ( q, t ) ≡ (cid:82) ∞−∞ dxe iqx f ( x, t ),turns the equation of motion into ρ ¨˜ u + K ( q ) ˙˜ u + [ Bq − P ( t ) q + K ( q )]˜ u = 0 . (6)The transformation ˜ v ≡ ˜ ue − [ K / (2 ρ )] t (7)eliminates the friction term, yielding ρ ¨˜ v + [ Bq − P ( t ) q + K − K / (4 ρ )]˜ v = 0 . (8)We rewrite eq. (8) as¨˜ v + ω [1 + a cos((2 ω + (cid:15) ) t )]˜ v = 0 , (9)where ω ( q ) ≡ ρ (cid:18) Bq − P q + K − K ρ (cid:19) ,a ( q ) ≡ − P q ω , (10) (cid:15) ( q ) ≡ ω − ω ( q ) . The problem has been transformed into an analogouschain of independent, parametrically actuated oscillators,with intrinsic frequencies ω ( q ), actuation amplitudes a ( q ),and detuning parameters (cid:15) ( q ). We see in eq. (10) that in-creasing the static pressure P weakens the ‘spring con-stant’ ω . For the analogy to work we must have ω ( q ) > , (11)and ‘oscillators’ (modes) q which do not satisfy it are over-damped. Further, from the known solution to the classicalproblem of parametric resonance [17], we infer the condi-tion for instability (i.e., exponentially growing amplitude˜ u ( q, t )), to leading order in the actuation a , Γ ( q ) ≡ a ω − K ρ > . (12) aim Diamant: Parametric resonance of supported sheets 3 This is the squared rate of amplitude growth. The fastestgrowing mode is the one which maximizes Γ ( q ). The al-lowed detuning for each ‘oscillator’ q , i.e., the actuationfrequency range providing resonance, is obtained from theinequality (cid:15) ( q ) < Γ ( q ). To simplify the discussion, wewill assume perfect tuning, (cid:15) = 0 , ω = 2 ω ( q ) . (13)Thus, by “unstable band” we will refer simply to the setof tuned ‘oscillators’ (i.e., range of q ) for which Γ ( q ) > The kernel K ( x − x (cid:48) ) gives the nonlocal normal force den-sity, acting at a point on the medium’s surface, in responseto a normal surface displacement elsewhere. For a semi-infinite elastic medium it corresponds to the inverse of theBoussinesq problem [19]. In q space inverting the solutionto this problem is immediate, yielding K = G − ν q, K = 0 , (14)where G is the medium’s shear modulus, and ν its Poissonratio.To make the expressions concise, we hereafter use B as the unit of energy, ( B/ ˆ G ) / as the unit of length, and( ρ/B ) / ( B/ ˆ G ) / as the unit of time, whereˆ G ≡ G/ (2(1 − ν )). This allows us to set B = ˆ G = ρ = 1.The 2D pressure is then measured in units of B / ˆ G / .(In sect. 4.1 we will rewrite the most relevant expressionsin dimensional form.)Substituting eq. (14) in eqs. (10) and (12), we obtain ω ( q ) = q ( q − P q + 2) , (15) Γ ( q ) = P q q − P q + 2) . (16)Static wrinkling appears when ω = 0. This occurs at thecritical pressure and wavenumber P = 3 , q c = 1 , (17)in agreement with earlier results [18].For P < P we have ω ( q ) > Γ ( q ) > q regardless of P . Thus all wrinkling modes q are os-cillatory and will resonate if excited by ω = 2 ω ( q ). Theresonance does not require the actuation amplitude to ex-ceed a finite threshold, P = 0; the growth rate simplyincreases linearly with P [eq. (16)]. This is due to theabsence of damping ( K = 0).Maximizing eq. (16) gives the fastest-growing modeand its growth rate as q f = P P > , Γ f = 3 √ P ( P − P ) − / . (18) For P = 0 (uncompressed sheet) the most unstable wave-length is indefinitely small. Thus wrinkles of well-definedfinite wavelength require a finite static pressure, P > P /P . As P is increased, thewrinkles’ growth rate and their wavelength increase, un-til, at the static wrinkling transition, the fastest-growingmode converges to the critical static one, q f → q c = 1, andits growth rate diverges. The actuation frequency produc-ing the fastest growth is obtained from eqs. (13), (15), and(18), as ω = 2 √ P − P ) / P , (19)independent of the actuation amplitude P .The most natural control parameters, however, are theactuation frequency and amplitude, and the static pres-sure. Given P , the choice of ω selects a dynamic wrinklewavenumber, q ( ω , P ), according to eqs. (13) and (15).This wavenumber is not equal to q f in general, and is in-dependent of P . Figure 2 shows the selected wavenumberas a function of ω for several values of P between 0 and P . The figure shows also the asymptotes of q for smalland large ω , which are both independent of P , q ( ω , P ) (cid:39) (cid:26) ω / , ω (cid:28) ω / / , ω (cid:29) . (20)Switching for a moment back to dimensional parameters,the two asymptotes become q ∼ ( ρ/G ) ω and q ∼ ( ρ/B ) / ω / , revealing the different physical mech-anisms in the two limits. At low frequencies the restoringmechanism is the substrate’s elasticity, whereas at highfrequencies it is the sheet’s bending rigidity. Note thatthese asymptotes agree with our heuristic arguments insect. 1.At P = P ∗ = 3 / / (cid:39) .
38 and ω = ω ∗ = 3 / / / (cid:39) .
37, the selected wavenumber, which is at this point q ∗ =2 − / (cid:39) . dω /dq = 0, implying that an excitation with P ∗ and ω ∗ at one edge of the sheet will not propagate through thesheet. For P > P ∗ and ω < ω ∗ , we find from eq. (12) thatthe largest of the three solutions for q ( ω , P ) grows thefastest. Thus, for P > P ∗ , as the excitation frequency ω is gradually increased from 0, the observed wrinklewavenumber will undergo a discontinuous jump. For in-creasingly larger static pressure P , the jump occurs atlower and lower frequencies, until, at P = P , the sys-tem selects q = q c at zero frequency (see fig. 2). This ishow the static-wrinkling limit is reproduced from the dy-namic one. Note that this entire behavior is independentof P ; hence, the discontinuous transition is present alsofor an arbitrarily weak actuation.Figure 3 presents 2D maps of the growth rate Γ as afunction of P and ω for P = 0 and P = P / Haim Diamant: Parametric resonance of supported sheets ω q Fig. 2.
Wrinkle wavenumber as a function of actuation fre-quency for an elastic substrate. Different curves correspond todifferent values of static pressure P (from right to left): 0, 1 . P ∗ = 3 / / , 2 .
8, and P = 3. Solid circles indicate the fastest-growing modes for the corresponding pressure. Dashed linesshow the asymptotes given in eq. (20). For P > P ∗ there arethree solutions for q , the largest of which growing the fastest,implying a discontinuous jump in the observed wavenumber as ω is ramped up. The empty circle marks the bifurcation point.All parameters are normalized (see text). For a viscoelastic medium the response is generalized byreplacing the modulus G and Poisson ratio ν with frequency-dependent complex functions, ˜ G ( ω ) and ˜ ν ( ω ). In prac-tice, viscoelastic media such as polymer networks usuallycontain a host liquid, which makes them virtually incom-pressible. Hence, ˜ ν ( ω ) (cid:39) / ω . Applying the low-frequency approximation of eq. (4), we generalize eq. (14)to ˜ K = K + iωK , K = 2 Gq, K = 2 ηq, (21)where G = Re( ˜ G ) and η = Im( ˜ G ) /ω are the low-frequencyshear modulus and shear viscosity of the substrate. We usethe same units of energy, length, and time as in sect. 3.1,making B , G , and ρ all equal to unity. The viscosity η ismeasured then in units of ρ / B / G / .Substituting eq. (21) in eqs. (10) and (12), we obtain ω ( q ) = q [ q − ( P + η ) q + 2] , (22) Γ ( q ) = (cid:20) P q q − ( P + η ) q + 2) − η (cid:21) q . (23)The viscous component leads to several essential changescompared to the elastic case. First, only for P < P − η are all the modes oscillatory ( ω > η > √ P = √
3, the substrateis too viscous to allow underdamped excitation (unlesswe ‘strengthen the springs’ by stretching the sheet with P < P ω P ω Fig. 3.
Density plots of wrinkle growth rate as a function ofexcitation amplitude and frequency for an elastic substrate.The static pressure values are P = 0 (upper panel) and P = P / / P ). All parameters are normalized (see text). minimum amplitude of actuation. The right-hand side ofeq. (23) has the asymptotes − η q in both limits of smalland large q . Hence, the resonant band of modes, when itexists, must be of finite width and centered around a fi-nite q . Real positive solutions to the equation Γ ( q ) = 0appear for P > P ( P , η ) = 4 η (3 − η − P ) / , (24)and the wavenumber at the threshold is q = q c = 1. Thus,crossing the resonance threshold leads to finite-wavelengthdynamic wrinkles with wavelength similar to that of thestatic wrinkles. Figure 4 shows the appearance of the un-stable band for P > P . aim Diamant: Parametric resonance of supported sheets 5 - - - - q Γ Fig. 4.
Wrinkle growth rate squared as a function of wrin-kle wavenumber for an uncompressed sheet ( P = 0) on aviscoelastic substrate. Different curves correspond to differentexcitation amplitudes P (bottom to top): 0 . P , P , and1 . P . Parametric resonance ( Γ >
0) of finite-wavelengthwrinkles occurs for P > P . The viscosity is η = 0 .
2, forwhich (and P = 0) P (cid:39) .
38. All parameters are normalized(see text).
Another difference from the elastic-substrate case isthat the rate of amplitude growth is not proportional to P [see eq. (23)]. As a result, the fastest-growing mode de-pends now on P (follow the maxima in fig. 4). Finally, un-like the elastic case, in the case of a viscoelastic substratewe can get parametric excitation of the finite-wavelengthwrinkles even for an uncompressed sheet, P = 0 (seeagain fig. 4).Considering the excitation frequency ω = 2 ω as acontrol parameter, we get, as in sect. 3.1, a selected wavenum-ber, q ( ω , P , η ), from eq. (22). Since q f depends in thepresent case on P while q does not, the fastest-growingmode does not belong in general to the set of selectedwavenumbers. In other words, one should tune P to get q = q f (see fig. 6 below). Figure 5 shows the selectedwavenumber as a function of ω for several values of P between 0 and P . The asymptotes for small and large ω remain as in eq. (20). Also here, the solutions bifurcateabove a certain static pressure, P ∗ = 3 / / − η , imply-ing a discontinuous jump in the wrinkle wavenumber as ω is increased. The bifurcation point is as in the elasticcase, q ∗ = 2 − / and ω ∗ = 3 / / / . For P > P − η aband of modes becomes damped (with imaginary ω ) asmanifested by the leftmost curve in fig. 5.Figure 6 shows a 2D map of the growth rate Γ as afunction of the excitation parameters P and ω for anuncompressed sheet ( P = 0). Unlike the elastic-substratecase (fig. 3), here the threshold for parametric resonancemakes the unstable region bounded. Let us summarize the results which seem most relevantexperimentally, and give them in dimensional form. ω q Fig. 5.
Wrinkle wavenumber as a function of actuation fre-quency for a viscoelastic substrate. The viscosity is η = 0 . P (from right to left): 0, P ∗ = 2 .
34, 2 . P − η = 2 .
96, and P = 3. Dashed lines show the asymptotes given in eq. (20).For P > P ∗ there are three solutions for q , the largest ofthe three growing the fastest, implying a discontinuous jumpin the observed wavenumber as ω is ramped up. The emptycircle marks the bifurcation point. For P > .
96 (leftmost,brown curve) a band of modes are damped. All parameters arenormalized (see text). P ω Fig. 6.
Density plot of wrinkle growth rate as a function ofexcitation amplitude and frequency for an uncompressed sheet( P = 0) on a viscoelastic substrate. The viscosity is η = 0 . P . Thedashed line shows the excitation frequency that produces thefastest-growing mode for each amplitude. All parameters arenormalized (see text). Haim Diamant: Parametric resonance of supported sheets In the case of an elastic substrate, one can first com-press the sheet until static wrinkling is reached. The mea-sured critical pressure and static wrinkle wavenumber arerelated to the bending modulus of the sheet and the elasticmoduli of the substrate as P = 3 B / (cid:18) G − ν ) (cid:19) / , q c = (cid:18) G − ν ) B (cid:19) / . (25)This allows a measurement of B and G/ (1 − ν ).For a finite P < P , and ramping up the actuationfrequency ω from zero, dynamic wrinkles should formfor any actuation amplitude. At low frequency the wrin-kle wavenumber q increases (wavelength λ = 2 π/q de-creases) quadratically with ω , q (cid:39) ρ (1 − ν )4 G ω , (26a)and at high frequencies it increases as the square root of ω , q (cid:39) (cid:18) ρω B (cid:19) / . (26b)Since ρ ∼ h and B ∼ h , the two limits differ sharplyin their dependence on the sheet thickness. At low fre-quencies the wrinkle wavenumber increases with thicknessas ∼ h , and at high frequencies it decreases with h as ∼ h − / . Both these dependencies differ from that of thestatic wrinkles, q c ∼ h − ; see eq. (25).Depending on the value of P , two distinct behaviorsare expected as ω is increased. At small pressures, P
( ρP ) / . In the system described above this correspondsto 1–10 Pa s (i.e., 10 –10 times the viscosity of water). The theory presented here is linear. As a result, it providesthe properties of the instability but not the ultimate formof the sheet’s deformation. Whether the deformation sat-urates to periodic wrinkles of finite height or localizes intofolds [20] should be checked in a future nonlinear theoryor simulation.We have assumed a semi-infinite substrate. Over lengthscales comparable and larger than the substrate thicknessthe results will be modified. In the opposite limit, of a thinsubstrate compared to the wrinkle wavelength, the effectof the medium will turn into that of Winkler foundation[21], i.e., completely localized ( ˜ K independent of q ).Another approximation employed above is the low-frequency limit, in which the relaxation of the viscoelas-tic medium is dominated by a single relaxation time. Ac-tual viscoelastic media, particularly biological ones, havea much richer frequency dependence, which will affect theresponse to the parametric excitation. Conversely, para-metric resonance may be used to tap into the medium’srich temporal response based on an extended theory.Besides relaxation times, complex media have also char-acteristic lengths which affect their response [22,23]. The aim Diamant: Parametric resonance of supported sheets 7 present theory describes a way to sample various lengthscales (wavenumbers) by sweeping the parametric-excitationfrequency. Recently we have derived the solution to theanalogous Boussinesq problem for a viscoelastic structuredmedium, accounting for its intrinsic correlation length [24].Similar to the derivations in sects. 3.1 and 3.2, these re-sults can be used to address the parametric excitation ofa sheet supported on such a structured medium. References
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