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An inverse problem in film/substrate indentation: extracting both the Young’s modulus and thickness of films

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Abstract

In an indentation test, the effective Young’s modulus of a film/substrate bilayer heterostructure varies with the indentation depth, a phenomenon known as the substrate effect. In previous studies investigating this, only the Young’s modulus of the film was unknown. Once the effective Young’s modulus of a film/substrate structure is determined at a given contact depth, the Young’s modulus of the film can be uniquely determined, i.e., there is a one-to-one relation between the Young’s modulus of the film and the film/substrate effective Young’s modulus. However, at times it is extremely challenging or even impossible to measure the film thickness. Furthermore, the precise definition of the layer/film thickness for a two-dimensional material can be problematic. In the current study, therefore, the thickness of the film and its Young’s modulus are treated as two unknowns that must be determined. Unlike the case with one unknown, there are infinite combinations of film thickness and Young’s modulus which can yield the same effective Young’s modulus for the film/substrate. An inverse problem is formulated and solved to extract the Young’s modulus and thickness of the film from the indentation depth-load curve. The accuracy and robustness of the inverse problem-solving method are also demonstrated.

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (11772335, 21622304, 61674045, and 21203038) and by the Ministry of Science and Technology (MOST) of China (2016YFA0200700). Z.H. Cheng was supported by the Distinguished Technical Talents Project and the Youth Innovation Promotion Association of Chinese Academy of Sciences.

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Authors and Affiliations

  1. State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Yin Zhang & Feifei Gao

  2. School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yin Zhang & Feifei Gao

  3. Department of Physics and Beijing Key Laboratory of Optoelectronic Functional Materials and Micro-Nano Devices, Renmin University of China, Beijing, 100872, China

    Zhiyue Zheng & Zhihai Cheng

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  1. Yin Zhang
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  2. Feifei Gao
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  3. Zhiyue Zheng
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  4. Zhihai Cheng
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Correspondence to Yin Zhang.

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Zhang, Y., Gao, F., Zheng, Z. et al. An inverse problem in film/substrate indentation: extracting both the Young’s modulus and thickness of films. Acta Mech. Sin. 34, 1061–1071 (2018). https://doi.org/10.1007/s10409-018-0778-8

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