Bending of Multilayer van der Waals Materials

TLDR

Out‑of‑plane deformation patterns such as buckling, wrinkling, scrolling, and folding in multilayer van der Waals materials have recently attracted attention, yet the key parameter governing these deformations—bending rigidity—remains controversial. This study directly measures the bending rigidity of multilayer graphene, MoS₂, and hBN by employing pressurized bubbles. By varying sample thickness and bubble deflection, the authors observe platelike responses and extract both Young’s modulus and bending rigidity using a nonlinear plate theory. The measured Young’s moduli agree with literature (E_graphene > E_hBN > E_MoS₂), but the bending rigidity follows the opposite trend (D_graphene < D_hBN < D_MoS₂) for comparable thickness, contradicting classical plate theory and indicating interlayer shear effects.

Abstract

Out-of-plane deformation patterns, such as buckling, wrinkling, scrolling, and folding, formed by multilayer van der Waals materials have recently seen a surge of interest. One crucial parameter governing these deformations is bending rigidity, on which significant controversy still exists despite extensive research for more than a decade. Here, we report direct measurements of bending rigidity of multilayer graphene, molybdenum disulfide (MoS_{2}), and hexagonal boron nitride (hBN) based on pressurized bubbles. By controlling the sample thickness and bubbling deflection, we observe platelike responses of the multilayers and extract both their Young's modulus and bending rigidity following a nonlinear plate theory. The measured Young's moduli show good agreement with those reported in the literature (E_{graphene}>E_{hBN}>E_{MoS_{2}}), but the bending rigidity follows an opposite trend, D_{graphene}<D_{hBN}<D_{MoS_{2}} for multilayers with comparable thickness, in contrast to the classical plate theory, which is attributed to the interlayer shear effect in the van der Waals materials.