---
author:
- Julien Chopin
- Moumita Dasgupta
- Arshad Kudrolli
title: Dynamic wrinkling and strengthening of an elastic filament in a viscous fluid
---

Slender structures embedded in complex fluids which buckle and fold as a result of mechanical compression are commonly found as in F-actin and microtubules in cell mechanics [@Gardel2004; @Chaudhuri2007; @Jiang2008] flagella in swimming organisms [@Powers2010; @Goldstein2006; @Son2013], fibers in paper processing [@Lindner2012], and the earth’s crust in orogenesis [@Biot1961]. A classical result dating back to Euler states that a thin sheet or filament will buckle under axial loading above a critical strain which is proportional to the square of the mode number and the square of the ratio of its thickness to length [@timoshenko1940strength]. While buckling typically occurs in the fundamental mode corresponding to the lowest strain, higher modes can occur depending on the constraints along the filament which may be static or dynamic in nature [@Chopin2013; @Chopin2015; @miller2015buckling; @lagrange2016wrinkling; @Audoly2005; @Vermorel2007; @gladden2005dynamic]. Although theoretical analysis of the problem are numerous, there are few experimental systems allowing close comparison with predictions. Traditional analysis to the wrinkling observed in elas- tic filaments consider linear stability analysis with instantaneous loading which can be an oversimplification in many situations  [@Biot1961; @Huang2001; @Jiang2008; @Li2008; @kodio2016lubricated]. For example, the buckling of membranes, the motion of living systems or actuated membrane involves the change of distance between material points in response to a stimuli (stress, light, pH, temperature) whose dynamics is set externally. Additionally, in-plane and out-of-plane modes experience dynamics with a timescale highly dependent on the surrounding environment as in air [@gladden2005dynamic], intra-cellular medium [@Brangwynne2007], or a viscous fluid [@wiggins1998trapping]. In this context, the interplay between the dynamics of an external stimuli and the dynamics of the elastic modes have been largely overlooked. Here, we examine a model system that reveals the emergence of a new timescale for the growth of wrinkles that explicitly depends on the loading rate. The precise understanding of this interplay is important for the development of fast reacting metamaterials and the tuning of the rheological response of polymers.

Experiments were performed with an elastic filament clamped at both ends and immersed in a container filled with a viscous fluid. The filament has a length $L = 92\,$mm, width $W = 3.0\,$mm and thickness $h = 0.30\,$mm and is composed of vinylpolysiloxane with a Young’s modulus $E= 1\,$MPa, and Poisson’s ratio $\nu \approx 0.5$. The bending and stretching moduli are given by $B = Eh^3W/12$ and $K = EhW$ respectively. An anisotropic filament cross section was chosen to have a well defined plane in which the filament buckles, simplifying data acquisition and analysis. One clamped end of the filament is attached to a motorized translating stage which can be moved through a displacement $u$ which varies between $u_0 < 0$ when the filament is under tension, and $u_f > 0$ when the filament is under compression with speed $V_e$ in the range $5\times10^{-3}$mm/s – $10\,$mm/s. The fluid composed of a glycerol-water mixture is prepared so that the filament is neutrally buoyant and has a dynamic viscosity $\eta= 0.9\,$Pas at 25$^{\circ}$C. The viscous drag coefficient is given by $\mu \approx 4\pi \nu /\log(L/W)$ [@batchelor1970slender; @Powers2010]. Unless otherwise stated, distances and time are normalized by $L$ and a time scale $\tau = \mu L^2/K=3\times10^{-2}$s given by a balance between viscous drag and elastic forces. With this rescaling, the non-dimensional speed $V = \tau V_e /L$ ranges between $2\times 10^{-6}$ and $3\times 10^{-3}$.

The filament is imaged with a $1200 \times 1200$ pixel camera at 1500fps by placing a diffused light source on the opposite side of the container which allows a good contrast for subsequent image analysis. We then obtain the filament deflection $w$ to within $0.15 h$ as a function of the non-dimensionalized coordinate along the longitudinal direction $x$ and time $t$. We calculate the bending content $\kappa(t) = h^2 \langle (w_{,xx}(x,t))^2 \rangle $, where $\langle ..\rangle$ denotes average over the length of the filament, and the averaged strain normalized by $L$ given by $$\begin{aligned}
\gamma(t) =  -u(t) + 1/2 \int_0^1 w_{,x}^2(x,t) dx\,,
\label{eq:gamma}\end{aligned}$$ This expression is obtained by integrating the 1D nonlinear strain $\gamma(x,t) = u_{,x}(x,t) + 1/2 (w_{,x}^2(x,t))$, where $u(x,t)$ is the local in-plane axial displacement [@landau1986course]. The evolution of the stretching and bending energies stored in the filament can be then quantified by $\gamma(t)$ and $\kappa(t)$, respectively.

Fig. \[Fig1\](a) shows the measured profile at various times while the filament is loaded as denoted by the bar on the left. The filament is observed to undergo a wrinkling instability during the loading phase ($t<t_f = u_f/V$) with a wavelength which is approximately a third of the length of the ribbon. Shortly thereafter, the ribbon is observed to undergo spontaneous coarsening till the wavelength reaches the filament length, and the shape corresponds to the fundamental buckling mode calculated using time-independent Euler analysis [^1]. We obtain the evolution of $\gamma$ and the bending content $\kappa$, and plot $u(t)$, $\gamma(t)$, and $\kappa(t)$ in Fig. \[Fig1\](b) focusing on a time interval over which most of the initial buckling and coarsening occurs. At the onset of instability, $\gamma(t)$ and $\kappa(t)$ are observed to increase rapidly showing the increase of the bending energy and the relaxation of the compression. From the plot, we observe that the rate of change of $\gamma(t)$ and $\kappa(t)$ starts to decrease well before compression is stopped. Although $\gamma(t)$ is found to vanish much faster than $\kappa(t)$, stretching is expected to play an important role at onset. We can then anticipate that a wrinkling growth model is likely to include small but finite extensibility and bending rigidity.

We obtain the root mean square amplitude $A(t) = \sqrt{\langle w^2(x,t) \rangle}$ from the measured $w(x,t)$ to understand the growth of the buckling modes. Fig. \[Fig1\](c) shows a plot of $A(t)$ as a function of time scaled with a parameter $\alpha$. This parameter is chosen so that the initial growth of $A(t)$ collapses onto a single curve before a time when its rate of increase starts to decrease. Plotting $\alpha$ as a function of $V$ in log-log scale in Fig. \[Fig1\](d), we find that the data can be described by the function $\alpha \sim V^n$ with $n = 0.7 \pm 0.08$. This form is observed to be significantly different from a linear scaling if the dependence on time was captured simply by $V$.

To develop a model of this non-trivial scaling, we next consider the spatio-temporal variation of the normal deflection $w(x,t)$ and the axial strain averaged over the length given by $\gamma(t)$. We assume that axial force balance develops rapidly in our system because of the homogeneous development of wrinkles along the length of the filament in contrast with observations with very viscous fluids which show wrinkling localized near the moving ends [@Gosselin2014]. Under these conditions, the axial tension in the filament $T$ just depends on time and is given by $T(t) = K \gamma(t)$. Then, neglecting inertia and balancing drag, bending and stretching forces, the non-dimensional equation of evolution for $w(x,t)$ is given by $$\begin{aligned}
\dot{w} &=& -\frac{B}{K}w_{,xxxx} + \gamma(t) w_{,xx}\,.
\label{eq:deflec}\end{aligned}$$ The corresponding boundary conditions are $w(0,t) = w(1,t)= 0$, $w_{,x}(0,t)=  w_{,x}(1,t) =0$, $u(1,t) = 0$, and $$\begin{aligned}
u(t) &\equiv& u(0,t) = 
\begin{cases}
    Vt,& \text{if } t<{u_f}/{V}.\\
    u_f,              & \text{otherwise}.
\end{cases}\, 
\label{eq:BC}\end{aligned}$$

This results in a series of ordinary differential equations (ODEs) for the evolution of the deflection. Further, differentiating the strain given in Eq. \[eq:gamma\], an additional equation for the evolution of the strain is obtained [@kodio2016lubricated]. Then, by setting the initial conditions for the deflection with white noise, the equations are integrated forward in time using the ODE solvers in MATLAB [@Note1].

A sample initial spatio-temporal growth and coarsening of the wrinkles observed in the simulations by plotting $w(x,t)$ in Fig. \[Fig2\](a) and $w(x,t)$ normalized by $A(t)$ in Fig. \[Fig2\](b) [@Note1]. One can observe that the amplitude of the wrinkles increases rapidly and the wavelength appears to decrease till $t \approx 10$ before coarsening starts to develop and amplitude increases more slowly. To quantify the variation in wavelength, the temporal evolution of the Fourier modes as a function of $k$ and $t$ are plotted in Fig. \[Fig2\](c). One observes that the peak in $k$ increases initially before decreasing confirming the trends noted in the evolution of $w(x,t)/A(t)$ in Fig. \[Fig2\](b). Comparing the three plots, one also observes that the rapid increase in amplitude occurs at a time similar to time $t_{max}$ at which the wavenumber reaches a maximum $k_{max}/2\pi = 7$.

We then measure the normalized stretching energy and bending energy $U_b =  \frac{1}{2} \frac{B}{K} \int (w_{,xx}(x,t))^2 dx$.

Fig. \[Fig2\](e) shows a plot of $A(t)$ in semi-log scale which is observed to first increase faster than exponential followed by a much lower growth rate for $t>t_{max}$. We find that the initial growth is accurately fitted by $A_{fit}(t) = A_0 \exp{(\alpha t)^3}$ as shown in Fig. \[Fig2\](e). Varying the compression speed $V$ in the range $10^{-8}$ and $10^{0}$, we measure the dependence of $t_{max}$, $k_{max}$, and $\alpha$ and plot the result in Fig. \[Fig2\](e)(inset), Fig. \[Fig3\](a), and Fig. \[Fig3\](b), respectively. Because the observed trends are captured by the dashed lines, we find that $\alpha \sim 1/t_{max} \sim V^n$ with $n = 0.68 \pm 0.05$ in very good agreement with the experiment. Further, the wavenumbers measured in the experiments and in the simulations are also in agreement with $k_{max} \sim V^{0.17}$. Thus, we conclude that the simplified Eq. \[eq:deflec\] captures the overall evolution of the wrinkling patterns observed in the experiments.

Now, Eq. \[eq:deflec\] in Fourier space is given at linear order by $$\begin{aligned}
\dot{\hat{w}}(k,t)=\left[u(t)k^2 -\frac{B}{K} k^4 \right] \hat{w}(k,t)\,,
\label{eqFourier}\end{aligned}$$ where $\hat{w}(k,t)$ is the Fourier transform of $w(x,t)$. The general solution of this equation is simply $\hat{w}(k,t) = w_0 \, e^{\Gamma(k,t)}$ where $w_0$ is proportional to the amplitude of the white noise used as initial condition in the numerics and $\Gamma(k,t) = k^2\phi(t)-B/K k^4t $ with $\phi(t) = \int_0^t u(t')dt'$. The standard deviation of $w(x,t)$ is then $$\begin{aligned}
A^2(t) &=& \frac{|w_0|^2}{2 \pi} \int_{-\infty}^{+\infty}e ^{2\Gamma(k,t)}dk\,, \end{aligned}$$ where we used the Parseval-Plancherel theorem. The integral can be evaluated using the Laplace method [@budd2000approximate]. When $\phi(t) > 0$, $\Gamma(k,t)$ reaches a maximum for $k^2  =k^{*2}(t) = K/(2B) \phi(t)/t$. Therefore, $$\begin{aligned}
A(t) &\sim&\frac{|w_0|/2}{\pi^{1/4}} \frac{e ^{\Gamma^*(t)}}{ \phi^{1/4}(t)} \,,
\label{At}\end{aligned}$$ where $\Gamma^*(t) = K/(4B) \phi^2(t)/t$. . Using the boundary conditions, we have $\phi(t<u_f/V) = Vt^2/2$ and $\phi(t>u_f/V) = -u_f^2/(2V)+tu_f$. Thus, $$\begin{aligned}
\Gamma^* (t) &=& \frac{Ku_f^2}{4B}
\begin{cases}
    \frac{V^2t^3}{4u_f^2}\,, & \text{if }  t<\frac{u_f}{V}\,.\\
    \frac{u_f^2}{4V^2 t} -\frac{u_f}{V} + t\,, & \text{if } t>\frac{u_f}{V}\,.
\end{cases}\, 
\label{BigGamma}\end{aligned}$$ Thus, we find that $A(t) \sim \exp\left[(\alpha t)^3 \right]$ during the loading phase. This super-exponential growth is consistent with the fits used to describe the experimental and numerical data, and where $$\begin{aligned}
\alpha = \left( \frac{K}{8B}\right)^{1/3}V^{2/3}\,.
\label{eq:alpha}\end{aligned}$$ This calculated scaling with $V$ is also in very good agreement with both the observed $n= 0.7 \pm 0.08$ in experiment in Fig. \[Fig1\](d), and $n = 0.68$ in simulations in Fig. \[Fig2\](e). . Further, the fastest mode has a wavenumber $$\begin{aligned}
k^{*2}(t) &\approx& \frac{Vt}{4B/K}\,.\end{aligned}$$ We cannot capture a time dependence of the wave vector near the initial growth because of the noise in the image detection, unlike $A(t)$ which is much cleaner. However, we can measure $k$ near coarsening, i.e. when non-linearities start to dominates. These non-linearities become dominant for large amplitude for $t>t_{max}$ where $$\begin{aligned}
t_{max} \sim 1/\alpha \sim V^{-2/3}.
\label{tmax}\end{aligned}$$ Then, at time $t = t_{max}$ we have $$\begin{aligned}
k_{max}^2 = \left(\frac{K^2V}{8B^2} \right)^{1/3}.
\label{EqKmax}\end{aligned}$$ The derived scaling of $k_{max} \propto V^{1/6}$ is consistent with the weak $V^{0.17}$ dependence seen in the numerics and the experiment shown in Fig. \[Fig3\](b) for large enough $V$. At the lowest $V$ investigated, one observes that $k_{max}$ is constant and corresponds to the fundamental mode as can be expected. The crossover speed $V^*$ between wrinkling and buckling dynamics is derived using Eq. \[EqKmax\] and $k_{max} = 2\pi$ leading to $V^* \approx 8 (2\pi)^6 (B/K)^2 \sim 10^{-7}$ consistent with numerical and experimental results. Thus, our model explains quantitatively the dependence of the growth rate and the wavevector with the loading speed.

While the sudden compression regime is not reached in our experiment before coarsening develops, we can examine this regime by numerically solving Eq. \[eq:deflec\] by imposing an instantaneous compression $u_f$. Numerics confirms an exponential growth of the wrinkling amplitude with a growth rate $\beta$ which is shown in Fig. \[Fig4\] as a function of applied strain $u_f$. Interestingly, we find that the regime $\beta \sim u_f^2$ discussed is only valid when $u_f  \gg u_1$, the strain corresponding to the buckling in the fundamental mode. Near threshold ($u_f \approx u_1$), we find another regime characterized by $\beta$ linearly increasing with $u_f$. In this regime, the discreteness of the modes imposed by the boundary conditions plays a dominant role. When $u_1<u_f<u_2$, the only possible wave vector is $k = 2\pi$, yielding $\dot{w}(k,t)=4 \pi^2 (u_f - u_1) w(k,t)$, where $u_1 = 4 \pi^2 B/K$. Upon integration, we obtain an exponential growth with a growth rate linearly depending on $u_f-u_1$ as shown in Fig. \[Fig4\].

In conclusion, our study reveals that filaments subjected to a ramped loading show a super-exponential wrinkling growth in a regime where the filament is extensible which is qualitatively different than the growth derived when assuming a sudden loading. In particular, we find the wrinkles to grow as $\exp((\alpha t)^3)$ with a timescale $\alpha^{-1} \sim \beta^{-1/3} \dot{\gamma}^{2/3}$ which is a combination of the spontaneous timescale $\beta^{-1}$ of sudden compression and a timescale given by the loading rate $\dot{\gamma}$. Our analysis based on our model experimental system thus advances a new time scale which is important to the mechanics of filaments which are subjected to time-dependent confining stresses.

We thank Vincent Démery, Dominic Vella and Etienne Barthel for their help with the analytical and numerical analysis. This work was supported by the National Science Foundation under grant number DMR 1508186.

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[^1]: see Supplementary Documentation for a movie of the shape evolution, the calculation of the Euler buckling modes, and the comparison of the fundamental mode with the final shape shown of the filament.
