A general adhesion model of stretched membranes


Thin membranes find applications in various fields. Over the past decades, the focus has been to enhance their mechanical properties and overall performance. In particular, their adhesion mechanics has drawn significant research attention. This can be attributed to the critical impact of adhesive forces, such as altering the load-displacement curves. Therefore, developing appropriate adhesive contact models is highly desirable for improving the mechanical properties, reliability, and functionality of thin membranes. In practical scenarios involving thin membranes such as micro/nano-electromechanical systems and biological tissues, the knowledge of membrane adhesion helps distinguish various physical mechanisms.

The adhesion of two elastic spheres has been extensively studied, resulting in improved models like the Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT) models. Unfortunately, these models fail to predict the adhesion behavior of thin membrane accurately due to the difference between the elastic response of solid spheres and membranes subjected to external loadings. Interestingly, based on the previous findings on the similarities between the adhesion of solid spheres and that of membranes, the framework of the exiting adhesive models can be used to develop adhesion mechanics of thin membranes. This has provided more opportunities for advanced studies on the membrane elastic energy and the adhesion of spherical/cylindrical particles in contact with membranes.

Several models have been proposed to describe the interaction between contact surfaces. Among them are the Lennard-Jones potential that requires solving the self-consistent equations, and the Maugis-Dugdale model based on uniform adhesive stress. Additionally, the Maugis-Dugdale model has exhibited great potential for modeling the adhesive interaction of membranes. Despite the good progress, the contact of thin membranes due to general adhesive interactions is not fully explored. To this note, Mr. Weike Yuan and Professor Gangfeng Wang from Xi’an Jiaotong University developed a theoretical model for studying the adhesive interactions between the stretched circular membrane and a rigid spherical indenter. Their research work is currently published in the International Journal for Solids and Structures.

In their approach, the adhesive traction between the tensioned membrane and the rigid spherical indenter was described using the cohesive force law adopted from the Maugis-Dugdale model. Based on the constant value within the finite length, it was easy to approximate the adhesive stress between the contact surfaces. The mutual relationships between different parameters: contact displacement, contact radius, and net external force were obtained analytically. To accurately characterize the adhesion properties of the thin membranes via indention experiments, it was necessary to express the critical contact state at zero external force or that achieving the pull-off force.

Results demonstrated a general contact solution that led to a complete transition from the DMT-type limit to the JKR-type limit. The dimensionless adhesion parameter and the normalized membrane radius were used to regulate the general contact response. An increase in the adhesion parameter results in the contact response closer to the JKR-type limit. Furthermore, the analytical equations of the contact parameters could facilitate further study of thin membranes through indentation experiments.

In a nutshell, an analysis of the asymmetric contact between the circular clamped membrane subjected to constant tension and a rigid sphere in the presence of uniformly distributed adhesive stress was reported in the study. Based on the results, the expressions of the pull-off force and the contact displacement and contact radius at zero external force were successfully derived analytically as functions of the adhesion parameter. The explicit equations provide more opportunities for characterizing the adhesion properties of thin membranes via indentation experiments. In a statement to Advances in Engineering, the authors said their study would advance the knowledge of adhesion mechanics of thin membranes and their associated practical applications.


Yuan, W., & Wang, G. (2021). Adhesion between a rigid sphere and a stretched membrane using the Dugdale modelInternational Journal of Solids and Structures, 208-209, 214-220.

Go To International Journal of Solids and Structures

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