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

William M. Coombs (Lead / Corresponding author), Charles Augarde, Michael Brown, Andrew Brennan, Tim Charlton, Jonathan Knappett, Yousef Motlagh, Lei Wang

Research output: Contribution to journalArticle

2 Citations (Scopus)
76 Downloads (Pure)


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 of 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 defined 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 satisfies 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 specific 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, different domain updating methods are investigated and it is shown that all of the current methods are degenerative under certain simple deformation fields. 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.

Original languageEnglish
Article number112622
Pages (from-to)1-32
Number of pages32
JournalComputer Methods in Applied Mechanics and Engineering
Early online date28 Sep 2019
Publication statusPublished - 1 Jan 2020


  • Elasto-plasticity
  • Finite deformation mechanics
  • Generalised interpolation
  • Lagrangian mechanics
  • Material point method

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  • Projects

    Research Output

    • 2 Citations
    • 6 Conference contribution
    • 3 Article
    • 1 Book
    • 1 Paper

    ISSPEA 2019: 1st International Symposium on Screw Piles for Energy Applications

    Davidson, C. (ed.), Brown, M. (ed.), Knappett, J. (ed.), Brennan, A. (ed.), Augarde, C. (ed.), Coombs, W. (ed.), Wang, L. (ed.), Richards, D. (ed.), White, D. (ed.) & Blake, A. (ed.), 27 May 2019, Dundee: University of Dundee. 121 p.

    Research output: Book/ReportBook

    Open Access
    419 Downloads (Pure)

    On implementation aspects of implicit MPM for 3D analysis

    Wang, L., Cortis, M., Coombs, W. M., Augarde, C., Brown, M., Knappett, J., Brennan, A., Davidson, C., Richards, D. & Blake, A., 10 Jan 2019, 2nd International Conference on the Material Point Method (MPM2019). Anura3D, p. 97-102 6 p.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

  • On the use of domain-based material point methods for problems involving large distortion

    Wang, L., Coombs, W. M., Augarde, C. E., Cortis, M., Charlton, T. J., Brown, M., Knappett, J., Brennan, A., Davidson, C., Richards, D. & Blake, A., 1 Oct 2019, In : Computer Methods in Applied Mechanics and Engineering. 355, p. 1003-1025 23 p.

    Research output: Contribution to journalArticle

    Open Access
  • 2 Citations (Scopus)
    83 Downloads (Pure)

    Student Theses

    Screw piles as offshore foundations : Numerical and physical modelling

    Author: Al-Baghdadi, T., 2018

    Supervisor: Brown, M. (Supervisor) & Knappett, J. (Supervisor)

    Student thesis: Doctoral ThesisDoctor of Philosophy


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