Stability of the Plane Stressed State of the Graphene Sheet Based on the Moment-Membrane Theory of Elastic Plates

Two-dimensional nanomaterials (graphene, carbon nanotube) are high-strength and ultra-light materials that have several promising areas of application. From theoretical and applied perspectives, it is relevant to study various problems of their statics, stability, vibrations, and calculations of the required mechanical characteristics based on the corresponding continuum theory of the deformation behavior of two-dimensional nanomaterials.

In this work, based on the moment-membrane theory of elastic plates, which is interpreted as the continuum theory of the deformation behavior of graphene, stability problems of a freely supported graphene sheet (rectangular plate) are studied. The sheet is uniformly compressed in one direction, compressed in two directions, and subjected to shear stresses in its plane. The stability problem of uniformly compressed graphene sheets, freely supported on two opposite sides and having different boundary conditions on the other two sides, is also considered.

When solving stability problems of the graphene sheet (rectangular plate), the Euler method is applied, considering a form of equilibrium that is slightly deviated from the initial (moment-free) position (buckled plate). Differential equilibrium equations and boundary conditions are formulated for this shape. The critical load value is determined from the solution of these boundary problems, i.e., the load value at which the initial flat form of the plate becomes unstable. All solutions are accompanied by numerical results: tables or diagrams providing the critical load values for each particular case.