Overdamped stress relaxation in buckled rods

We present a comprehensive theoretical analysis of the stress relaxation in a multiply but weakly buckled incompressible rod in a viscous solvent. For the bulk, two interesting parameter regimes of generic self-similar intermediate asymptotics are distinguished, which give rise to approximate and exact power-law solutions, respectively. For the case of open boundary conditions the corresponding nontrivial boundary-layer scenarios are derived by a multiple-scale perturbation (“adiabatic”) method. Our results compare well with—and provide the theoretical explanation for—previous results from numerical simulations, and they suggest directions for further fruitful numerical and experimental investigations.

Figure 1

A typical scenario of a deterministically relaxing buckled rod. Initially the rod is wrinkled on small wavelengths. In the course of time undulations are pushed out at the free ends and the typical wavelength of the undulations grows.

Figure 2

Relaxation in the bulk. The situation is essentially the same as for a longitudinally confined rod. The pressure $f\left(t\right)$ exerted onto the confining walls exhibits the power-law decay Eq. (4).

Figure 3

The parameterization of the contour $\mathbf{r}\left(s\right)={({\mathbf{r}}_{\perp },s-r_{\Vert })}^T$ by transverse and longitudinal displacement variables $r_{\perp }$ and $r_{\Vert }$, respectively. Note that the displacements vanish for the straight contour.

Figure 4

The sensitive dependence of $A(q,\tau )$, given by Eq. (31b), on the parameter $\alpha =\tau Q^4$. For $\alpha =2$ the amplification factor takes the form of a pronounced peak (a) whereas for $\alpha =0.15$ it resembles a step function (b).

Figure 5

The fraction ${\varrho }_n\left(\tau \right)={\varrho }_n^{0}A(q_n,\tau )$ of contour length stored in mode $n$ versus the corresponding wave number $q_n=n\pi ∕l$ for three successive times ${\tau }_3>{\tau }_2>{\tau }_1$ and the particular initial condition ${\varrho }_{n⩽N}^0=\mathrm{const}$ and ${\varrho }_{n>N}^0=0$. The location of the maximum $Q\left(\tau \right)$ of $A(q,\tau )$ has been obtained upon solving the implicit equation (34c) numerically. As explained in the main text the dark shaded areas $\Delta {\varrho }_{-}$ and $\Delta {\varrho }_{+}$ can be interpreted as stored length that has been destroyed and generated during the relaxation, respectively. The global constraint of fixed stored length requires their sum to vanish identically, Eq. (37).

Figure 6

Typical tension relaxation of a rod with initial conditions satisfying Eq. (40) from a numerical solution of Eq. (34c). As in the example of Fig. 5 we chose the particular initial condition ${\varrho }_{n⩽N}^0=\mathrm{const}$ and ${\varrho }_{n>N}^0=0$. For two values of $N$ the graph displays the tension $\phi$ versus time $\tau$ in units of critical force ${\phi }_1={(\pi ∕l)}^2$ and the typical relaxation time ${\tau }_0=l^4$, respectively. In the case $N=100$ it is seen that the intermediate asymptotic power law $\phi \propto {\tau }^{1∕2}$ is valid in the time window ${\tau }_N⪡\tau ⪡{\tau }_0$, where ${\tau }_N\equiv N^{-4}{\tau }_0$ is the relaxation time of the highest excited mode. The extreme case $N={10}^{20}$ illustrates the asymptotic behavior of the staircase regime for large $N$.

Figure 7

The amplitude $\sqrt{{\alpha }_{\beta }}$ in the power law Eq. (53) for the line tension $\phi$ in the bulk of a relaxing rod with power-law initial conditions ${\varrho }^0\left(q\right)\propto q^{-\beta }$. For $\beta <1$ (type I initial conditions) an upper cutoff is required to keep the stored length finite. The relaxation then proceeds via the cascading of a localized peak in mode space, as depicted in Fig. 4a and explained in Sec. 4C. For $\beta >3$, the rod is essentially in the ground state from the beginning. The interval $1<\beta <3$ of type II initial conditions comprises the exact similarity solutions derived in Sec. 4D. Thermal initial conditions correspond to ${\alpha }_{\beta =2}\approx 0.146$.

Figure 8

The situation in mode space after time $\tau$ for representative initial conditions of type I ($\beta =0$, $\alpha ⪢1$, cutoff $q_N⪢Q$ as in Fig. 5) and type II (thermal initial conditions $\beta =2$) with the same total stored length ${\varrho }^0$.

Figure 9

The length $\lambda$ over which the tension increases toward the bulk value is much larger than the characteristic length $Q_{\infty }^{-1}$ of the transverse undulations in the bulk, as expressed by the inequality in Eq. (60). The slowly varying part ${\phi }_l\left(s\right)$ of the tension is obtained upon averaging over the coarse-graining scale $l$ that satisfies the condition in Eq. (61).

Figure 10

Type II: Stress profile in the boundary layer, given by the scaling function ${\chi }_{\beta }(s∕{\lambda }_{\beta }\left(\tau \right))$, Eq. (74). The case $\beta =2$ corresponds to thermalized initial conditions.

Figure 11

Type I: Stress profile in the boundary layer, given by the scaling function $\chi \mathbf{(}s∕{\lambda }_{\beta }\left(\tau \right)\mathbf{)}$, Eq. (89). The displayed curves correspond to $\beta =-1$ and ${\alpha }_{\infty }=5,10,15,20,25$. For increasing ${\alpha }_{\infty }⪢1$, $\chi \left(\xi \right)$ approaches its limiting form $\chi (\xi <1)=\xi$ and $\chi (\xi >1)=1$. The asymptotic behavior is independent of $\beta <1$.

Figure 12

The exponents ${\gamma }_{\beta }^{<}$ and ${\gamma }_{\beta }^{>}$ (grey) determine the growth $\delta R_{\Vert }\propto {\tau }^{{\gamma }_{\beta }}$ for the end-to-end distance on short $(\tau ⪡{\tau }_f)$ and long times $(\tau ⪢{\tau }_f)$, respectively. The exponent ${\delta }_{\beta }={\gamma }_{\beta }^{<}-{\gamma }_{\beta }^{>}$ (black) characterizes the growth ${\lambda }_{\beta }\sim {\tau }^{{\delta }_{\beta }}$ of the width of the boundary layer.