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. 2023 May 9;13:7473. doi: 10.1038/s41598-023-34025-x

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© The Author(s) 2023

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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Results of root penetration numerical analysis using the contact finite element method. (a) Root displacement, soil principal stress and soil displacement of the simulations for two β-values by changing soil strength E (kPa). The units of color codes are cm (left), MPa (middle), and cm (right). From the initial displacement (left), the resulting stress and displacement after deformation were calculated (right). (b) Schematic illustrations of root displacement and soil principal stress. (c) Schematic illustrations of soil displacement for low β and for high β.