Probe–suspended membrane resonance model for obtaining out-of-plane shear modulus and prestress in graphene

The out-of-plane shear modulus of two-dimensional (2D) materials is critical for understanding their strain-modulated electronic, optical, and tribological behavior. However, existing experimental methods are limited and often yield inconsistent results due to substrate interference. Here, we introduce a contact-resonance-based methodology utilizing a probe–suspended membrane system to simultaneously measure both the out-of-plane shear modulus and in-plane prestress of 2D material membranes. The membrane's load–deflection response is modeled via a novel cubic equation fitted from numerical simulations. The probe–membrane dynamic system is modeled as a tilted cantilever with normal and lateral springs at its tip, yielding a frequency equation that links resonance frequency to membrane stiffness. Experimental measurements on suspended multilayer graphene under varying loads yield an out-of-plane shear modulus of 4.16 ± 0.30 GPa, consistent with theoretical predictions. The in-plane prestress is observed to decrease with membrane thickness. Pressure-blister tests corroborate that van der Waals attraction from the hole sidewalls is a primary origin of this prestress. This work establishes a precise, nondestructive method for characterizing 2D material mechanical properties with minimal substrate influence, offering fundamental insights for the design and performance evaluation of 2D material-based nanodevices and flexible systems.