A phase-field model for fracture in beams from asymptotic results in 2D elasticity
Giovanni Corsi, Antonino Favata, Stefano Vidoli
download PDFAbstract. We propose a derivation of a damage model in slender structures, focusing on the particular case of a rod. The peculiarity of the model is that it takes into account the changes in rigidity of the body, distinguishing between bending, traction and the possible mixed interactions between the two. The approach is based on a matched asymptotic expansion, taking the recent work of Baldelli et al [1] as starting point. Choosing the slenderness of the rod as small parameter for the asymptotic expansion, we determine the first order at which a correction occurs with respect to the Saint-Venant solution of the elastic problem, due to the presence of a crack. The results highlight that the presence of a defect affects in different ways the bending and traction rigidities of the rod, and that a coupling between the two deformation modes might occur, depending on the geometry of the crack. Moreover, the derivation allows to explicitly calculate the coefficients of this correction, for any given depth of the crack, by means of a simple numerical procedure. Application to the classic three-point bending problem is considered in order to highlight the predictive capabilities of the model. These results suggest ways in which state of the art phase-field models (e.g. [2]) for damage could be refined. This work goes in the direction of developing phase-field models suitable for application to slender structures, where the use of reduced dimensional models has proved promising [3].
Keywords
Asymptotic Approach, Fracture Mechanics, Slender Structures
Published online 3/17/2022, 6 pages
Copyright © 2023 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA
Citation: Giovanni Corsi, Antonino Favata, Stefano Vidoli, A phase-field model for fracture in beams from asymptotic results in 2D elasticity, Materials Research Proceedings, Vol. 26, pp 115-120, 2023
DOI: https://doi.org/10.21741/9781644902431-19
The article was published as article 19 of the book Theoretical and Applied Mechanics
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
References
[1] Baldelli, A.L., Marigo, JJ. Pideri, C., Analysis of Boundary Layer Effects Due to Usual Boundary Conditions or Geometrical Defects in Elastic Plates Under Bending: An Improvement of the Love-Kirchhoff Model. J Elast 143, 31–84 (2021). https://doi.org/10.1007/s10659-020-09804-6
[2] Gerasimov, T., De Lorenzis, Laura, “On penalization in variational phase-field models of brittle fracture.” Computer Methods in Applied Mechanics and Engineering 354 (2019): 990-102 6. https://doi.org/10.1016/j.cma.2019.05.038
[3] Brunetti M., Freddi F., Sacco E., “Layered Phase Field Approach to Shells”. In: Carcate rra A., Paolone A., Graziani G. (eds) Proceedings of XXIV AIMETA Conference 2019. AIMETA 2019. Lecture Notes in Mechanical Engineering. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41057-5_36
[4] Brunetti, M., Vincenti, A., & Vidoli, S. (2016). A class of morphing shell structures satisfying clamped boundary conditions. International Journal of Solids and Structures, 82, 47-55. https://doi.org/10.1016/j.ijsolstr.2015.12.017
[5] M. Brunetti, A. Favata, S. Vidoli (2020), Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability, Proceedings of the Royal Society of London A, 476: 20200455. https://doi.org/10.1098/rspa.2020.0455
[6] Corsi, G., De Simone, A., Maurini, C., & Vidoli, S. (2019). A neutrally stable shell in a Stokes flow: a rotational Taylor’s sheet. Proceedings of the Royal Society A, 475(2227), 20190178. https://doi.org/10.1098/rspa.2019.0178
[7] Corsi, G. (2021). Asymptotic approach to a rotational Taylor swimming sheet. Comptes Rendus. Mécanique, 349(1), 103-116. https://doi.org/10.5802/crmeca.75
[8] Kiendl, J., Ambati, M., De Lorenzis, L., Gomez, H., & Reali, A. (2016). Phase-field description of brittle fracture in plates and shells. Computer Methods in Applied Mechanics and Engineering, 312, 374-394. https://doi.org/10.1016/j.cma.2016.09.011
[9] Proserpio, D., Ambati, M., De Lorenzis, L., & Kiendl, J. (2020). A framework for efficient isogeometric computations of phase-field brittle fracture in multipatch shell structures. Computer Methods in Applied Mechanics and Engineering, 372, 113363. https://doi.org/10.1016/j.cma.2020.113363
[10] Proserpio, D., Ambati, M., De Lorenzis, L., & Kiendl, J. (2021). Phase-field simulation of ductile fracture in shell structures. Computer Methods in Applied Mechanics and Engineering, 385, 114019. https://doi.org/10.1016/j.cma.2021.114019
[11] Bessoud, A. L., Krasucki, F., & Serpilli, M. (2011). Asymptotic analysis of shell-like inclusions with high rigidity. Journal of Elasticity, 103(2), 153-172. https://doi.org/10.1007/s10659-010-9278-1
[12] Abdelmoula, R., & Marigo, J. J. (2000). The effective behavior of a fiber bridged crack. Journal of the Mechanics and Physics of Solids, 48(11), 2419-2444. https://doi.org/10.1016/S0022-5096(00)00003-X
[13] Geymonat, G., Krasucki, F., Hendili, S., & Vidrascu, M. (2011). The matched asymptotic expansion for the computation of the effective behavior of an elastic structure with a thin layer of holes. International Journal for Multiscale Computational Engineering, 9(5). https://doi.org/10.1615/IntJMultCompEng.2011002619
[14] Marigo, J. J., & Pideri, C. (2011). The effective behavior of elastic bodies containing microcracks or microholes localized on a surface. International Journal of Damage Mechanics, 20(8), 1151-1177. https://doi.org/10.1177/1056789511406914
