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 journalArticlepeer-review

37 Citations (Scopus)
330 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 Sept 2019
Publication statusPublished - 1 Jan 2020


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

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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