Two-dimensional (2D) materials hold great promise for future electronic, optical, thermal devices and beyond, underpinning which the predictability, stability and reliability of their mechanical behaviors are the fundamental prerequisites. Despite this, due to the layered crystal lattice structure, extremely high anisotropy and the independent deformation mechanism of out-of-plane bending, the proper homogenization for such materials still faces challenge. That is because the monolayer bending is of independent deformation mechanism distinct from the traditional bulk deformation which thereby brings couple stress to the bulk 2D materials, while the different interlayer constraints of bulk and surface layers bring surface layer effect. In this paper, by considering the two effects, a continuum mechanics framework for extremely anisotropic 2D materials (CM2D) is proposed, without necessities of ad hoc experiments for the unclassical parameters. Under the framework of the CM2D, beam-like deformation, plate-like deformation and indentation of 2D materials are studied to showcase its ability and applicability. An analytical expression of the effective bending rigidity is derived, which can be characterized by several dimensionless parameters. It is found that the overall bending deformations of 2D materials are controlled by the competition between the intralayer deformation mode and the interlayer shear deformation mode. Besides, the size-dependent modulus is also identified on the indentation of 2D materials at the pure elastic deformation regime, distinct from the size effect caused by plasticity. In addition, we discussed the effects of monolayer bending and surface layer on the mechanical behaviors of 2D materials. Our work not only provides guidance for the studies and applications of 2D materials, but also serves as a good example with well-defined physical meanings for the strain gradient, high-order moduli and couple stress in high-order continuum mechanics theories.

中文翻译：

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整合单层弯曲和表面层效应的二维材料的均质化


二维 （2D） 材料在未来的电子、光学、热器件等领域具有巨大的前景，其机械行为的可预测性、稳定性和可靠性是其基本先决条件。尽管如此，由于层状晶格结构、极高的各向异性和独立的面外弯曲变形机制，此类材料的适当均质化仍然面临挑战。这是因为单层弯曲具有独立的变形机制，不同于传统的体变形，从而给体 2D 材料带来耦合应力，而体层和表面层的不同层间约束带来了表面层效应。在本文中，通过考虑这两种效应，提出了一个用于极端各向异性二维材料 （CM2D） 的连续介质力学框架，而无需对非经典参数进行临时实验。在 CM2D 框架下，研究了二维材料的梁状变形、板状变形和压痕，以展示其能力和适用性。推导出有效弯曲刚度的解析表达式，该表达式可以用几个无量纲参数来表征。研究发现，二维材料的整体弯曲变形受层内变形模式和层间剪切变形模式竞争的控制。此外，在纯弹性变形状态下，二维材料的压痕也确定了尺寸依赖性模量，这与塑性引起的尺寸效应不同。此外，我们还讨论了单层弯曲和表面层对 2D 材料力学行为的影响。 我们的工作不仅为二维材料的研究和应用提供了指导，而且为高阶连续介质力学理论中的应变梯度、高阶模量和耦合应力提供了一个很好的例子，具有明确的物理意义。