On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity

Coombs, WM; Augarde, CE; Brennan, AJ; Brown, MJ; Charlton, TJ; Knappett, JA; Ghaffari Motlagh, Y; Wang, L

Abstract

The material point method is ideally suited to modelling problems involving large deformations where conventional mesh-based methods would struggle. However, total and updated Lagrangian approaches are unsuitable and non-ideal, respectively, in terms formulating equilibrium for the method. This is due to the basis functions, and particularly the derivatives of the basis functions, of material point methods normally being dened based on an unformed, and sometimes regular, background mesh. It is possible to map the basis function spatial derivatives using the deformation at a material point but this introduces additional algorithm complexity and computational expense. This paper presents a new Lagrangian statement of equilibrium which is ideal for material point methods as it satises equilibrium on the undeformed background mesh at the start of a load step. The formulation is implemented using a quasi-static implicit algorithm which includes the derivation of the consistent tangent to achieve optimum convergence of the global equilibrium iterations. The method is applied to a number of large deformation elasto-plastic problems, with a specic focus of the convergence of the method towards analytical solutions with the standard, generalised interpolation and CPDI2 material point methods. For the generalised interpolation method, dierent domain updating methods are investigated and it is shown that all of the current methods are degenerative under certain simple deformation elds. A new domain updating approach is proposed that overcomes these issues. The proposed material point method framework can be applied to all existing material point methods and adopted for implicit and explicit analysis, however its advantages are mainly associated with the former.

Citation

Coombs, W., Augarde, C., Brennan, A., Brown, M., Charlton, T., Knappett, J., Ghaffari Motlagh, Y., & Wang, L. (2020). On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity. Computer Methods in Applied Mechanics and Engineering, 358, Article 112622. https://doi.org/10.1016/j.cma.2019.112622

• Journal Article Type: Article
• Acceptance Date: Sep 5, 2019
• Online Publication Date: Sep 28, 2019
• Publication Date: Jan 1, 2020
• Deposit Date: Sep 10, 2019
• Publicly Available Date: Oct 8, 2019
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 358
• Article Number: 112622
• DOI: https://doi.org/10.1016/j.cma.2019.112622
• Public URL: https://durham-repository.worktribe.com/output/1322478