Contact probing of stretched membranes and adhesive interactions: graphene and other two-dimensional materials

Abstract

Contact probing is the preferable method for studying mechanical properties of thin two-dimensional (2D) materials. These studies are based on analysis of experimental force–displacement curves obtained by loading of a stretched membrane by a probe of an atomic force microscope or a nanoindenter. Both non-adhesive and adhesive contact interactions between such a probe and a 2D membrane are studied. As an example of the 2D materials, we consider a graphene crystal monolayer whose discrete structure is modelled as a 2D isotropic elastic membrane. Initially, for contact between a punch and the stretched circular membrane, we formulate and solve problems that are analogies to the Hertz-type and Boussinesq frictionless contact problems. A general statement for the slope of the force–displacement curve is formulated and proved. Then analogies to the JKR (Johnson, Kendall and Roberts) and the Boussinesq–Kendall contact problems in the presence of adhesive interactions are formulated. General nonlinear relations among the actual force, displacements and contact radius between a sticky membrane and an arbitrary axisymmetric indenter are derived. The dimensionless form of the equations for power-law shaped indenters has been analysed, and the explicit expressions are derived for the values of the pull-off force and corresponding critical contact radius.

Adhesion of an elastic sphere on a tensioned membrane

“…which is identical to the results of Borodich and Galanov [10] where adhesion of a rigid sphere on a membrane of ignored bending rigidity is studied.…”

Indentation of thin elastic films glued to rigid substrate: Asymptotic solutions and effects of adhesion

“…Borodich noted that if one derives an expression for the slope of the force-displacement curve of an appropriate problem of non-adhesive contact then the JKR theory may be extended to the problem of adhesive contact for an axisymmetric convex indenter of arbitrary profile. These problems were solved for transversely isotropic elastic materials [37], prestressed materials [34], a two-dimensional stretched membrane [38], isotropic frictional indenter [33], and a single thin isotropic layer [20]. Here, we extend the above ideas to problem of adhesive indentation into a bilayer elastic coating.…”

“…Using (38) it is possible to show that for an arbitrary convex body of revolution f (r), f (0) = 0, the JKR theory leads to the following expressions…”

An ordinary state-based peridynamic model for the fracture of zigzag graphene sheets

“…For example, they can be used in display screens of mobile devices [1], to design ultra-capacitors with better performance than batteries [2] and for water desalination [3]. Especially in engineering, they are incorporated in composite materials as reinforcements [4,5], which makes it of fundamental importance to understand its physical properties and behaviours beforehand because the mechanical performance of composites is affected significantly by the reinforcements. As a common phenomenon, failure may occur in the form of generation and growth of discontinuities such as cracks in such structures when they are under certain loading conditions.…”