# A simple model for the dynamics of adhesive failure

Original Entry: Ian Burgess Fall 2009

## References

D. Vella, L. Mahadevan, "A simple model for the dynamics of adhesive failure" Langmuir 22, 163 (2006).

## Summary

This paper describes a simple model of adhesive failure used to describe the physical processes that result in bimodal nature of stress/strain curves in experiments. They consider two parallel plates attached to each other by an array of $N(t)$ nonlinear springs, which dissociate as the plates are pulled apart at a constant velocity. They describe the dissociation rate by the equation:

$k(t)=k_{0}\exp\left(\frac{\gamma vt}{bL(1-vt/L)}\right)$

where $\gamma$ is a lengthscale associated with the adhesive's bond potential, $k_{0}$ is the dissociation rate when no force is applied, v is the velocity at which the plates are being pulled apart, and b and L are lengths associated with the polymer chain in the adhesive. $N(t)$ is then described by the equation,

$\frac{\mathrm{d}N}{\mathrm{d}t}=-k(t)N$

The authors show that under certain conditions, even in this simple model, that a plot (such as the one shown above) of applied force on the plates required to maintain the constant velocity against time (or separation of the plates) can have one or two maxima (bimodal) depending on the material fitting parameters and the separation velocity chosen. They also extend this model to a continuum and show that the same property exists.