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. 2021 Nov 22;11:22668. doi: 10.1038/s41598-021-01036-5

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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(a,b) FEM analysis was performed using ABAQUS choosing a finite element quad-mesh composed by 64,000 elements (a) Normal displacement shown for PAA6 (b). The simulation was performed to calculate both the elastic modulus and the normal displacement exerted on the hydrogel by the indenting sphere. (c) The evolution of correction factor g(δ/h,Rh/D²) with strain (δ/h) for different values of the finite diameter and also given by the empirical fit.