Abstract Elastic modulus of multiple layer two-dimensional (2D) materials is typically measured in indentation testing via existing indentation bending model of a beam/plate, in which the sample is assumed to be a doubly clamped beam/plate under a central point load. However, a paradox is commonly found in indentation testing of multiple layer 2D materials: the measured indentation load-displacement curve has a much longer linear regime than the corresponding theoretical counterpart; in addition, the elastic modulus determined is significantly lower than its theoretical counterpart. Based on the theoretical and numerical analyses, the above paradox can be well explained via the separation of the adhesive boundary of 2D materials in indentation tests, resulting in that the freestanding portion of sample behaves like a pure bending beam/plate, instead of a doubly clamped beam/plate subjected to a central point load. These results agree with the reported experimental results very well, and thus, the indentation response of multiple layer 2D materials should be analyzed via the pure bending model instead of the existing indentation bending model of beam/plate.

中文翻译：

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基于现有压痕弯曲模型的多层二维材料力学性能表征的悖论

摘要 多层二维 (2D) 材料的弹性模量通常在压痕测试中通过梁/板的现有压痕弯曲模型进行测量，其中假设样品是中心点载荷下的双夹紧梁/板。然而，在多层 2D 材料的压痕测试中经常发现一个悖论：测得的压痕载荷-位移曲线的线性范围比相应的理论对应曲线长得多；此外，确定的弹性模量明显低于其理论对应值。基于理论和数值分析，上述悖论可以通过压痕测试中二维材料粘合边界的分离得到很好的解释，导致样品的独立部分表现为纯弯曲梁/板，而不是承受中心点载荷的双夹紧梁/板。这些结果与报道的实验结果非常吻合，因此，应通过纯弯曲模型而不是现有的梁/板压痕弯曲模型来分析多层二维材料的压痕响应。