Dynamics of stick-slip in peeling of an adhesive tape

We investigate the dynamics of peeling of an adhesive tape subjected to a constant pull speed. We derive the equations of motion for the angular speed of the roller tape, the peel angle and the pull force used in earlier investigations using a Lagrangian. Due to the constraint between the pull force, peel angle and the peel force, it falls into the category of differential-algebraic equations requiring an appropriate algorithm for its numerical solution. Using such a scheme, we show that stick-slip jumps emerge in a purely dynamical manner. Our detailed numerical study shows that these set of equations exhibit rich dynamics hitherto not reported. In particular, our analysis shows that inertia has considerable influence on the nature of the dynamics. Following studies in the Portevin–Le Chatelier effect, we suggest a phenomenological peel force function which includes the influence of the pull speed. This reproduces the decreasing nature of the rupture force with the pull speed observed in experiments. This rich dynamics is made transparent by using a set of approximations valid in different regimes of the parameter space. The approximate solutions capture major features of the exact numerical solutions and also produce reasonably accurate values for the various quantities of interest.

Figure 1

Schematic plot of experimental setup.

Figure 2

(a) Plots of $f(v,V)$ as a function of $v$ ($x$ axis in log scale) for $V=1$ (solid curve), $V=2$ (dashed curve), $V=4$ (dashed and dotted curve), $V=6$ (dotted curve); see Eq. (14). (b) Experimental strain energy release rate, $G\left(v\right)$ curve as in Ref. 1. [Units of $f(v,V)$ is in N, $G\left(v\right)$ in $\mathrm{J}∕{\mathrm{m}}^2$, and $v,V$ are in $\mathrm{m}∕\mathrm{s}$.]

Figure 3

(a) A typical phase space trajectory in the $v\text{−}F$ plane for $V=0.4,I={10}^{-5}$. The corresponding $f(v,V)$ is shown by a solid curve. (b) A phase space trajectory in the $v\text{−}F$ plane for $V=1.0$ and $I={10}^{-5}$. (c) A plot of $\alpha \left(t\right)$ for $V=1$ and $I={10}^{-5}$. (d) A plot of $F\left(t\right)$ (period 4) for $V=2$ and $I={10}^{-5}$. (Units of $v,V$ are in $\mathrm{m}∕\mathrm{s},F$ in N, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 4

(a) Phase space trajectory in the $v\text{−}F$ plane for a single cycle for $I={10}^{-2}$ and $V=1$. The corresponding $f(v,V)$ is shown by a thick solid curve. (b) Corresponding plots of $\alpha \left(t\right)$, (c) the pull force $F\left(t\right)$ (period 8), and (d) the peel velocity $v\left(t\right)$. (Units of $v,V$ are in $\mathrm{m}∕\mathrm{s},F$ in N, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 5

(a) A plot of $F\left(t\right)$ for $V=4$ and $I={10}^{-2}$. (b) Corresponding plot of $\alpha \left(t\right)$. The inset shows an expanded plot of decreasing trend of $\alpha \left(t\right)$. (c) Corresponding plots of phase space trajectory that reflects the chaotic nature and (d) the peel velocity $v\left(t\right)$. ($v,V$ are in $\mathrm{m}∕\mathrm{s},F$ in N, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 6

The plot shows the mean force drop $\overline{\Delta F}$ as a function of the pull speed $V$, for two distinct values of $I$. The dashed line corresponds to $I={10}^{-2}$ while the dotted line corresponds to $I={10}^{-5}$. ($v,V$ are in $\mathrm{m}∕\mathrm{s}$, $F$ in N, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 7

A phase plot of canard type of solution in $v\text{−}F$ plane for $V=0.4$ and $I={10}^{-3}$. ($v,V$ are in $\mathrm{m}∕\mathrm{s}$, $F$ in N, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 8

Comparison of approximate solution (continuous) with the numerically exact solution (dotted) of $v\left(t\right)$ for $I={10}^{-5}$ and $V=1$ for the $AB$ branch. The inset shows a similar comparison of $v\left(t\right)$ on the $CD$ branch. ($v,V$ are in $\mathrm{m}∕\mathrm{s},I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)

Figure 9

Comparison of the approximate solution (continuous line) for $v\left(t\right)$ with numerically exact solution (dotted line) for $V=1,I={10}^{-2}$. ($v,V$ are in $\mathrm{m}∕\mathrm{s}$, $I$ in $\mathrm{kg}{\mathrm{m}}^2$, and $t$ in s.)