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

TY - JOUR

T1 - On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity

AU - Coombs, William M.

AU - Augarde, Charles

AU - Brown, Michael

AU - Brennan, Andrew

AU - Charlton, Tim

AU - Knappett, Jonathan

AU - Motlagh, Yousef

AU - Wang, Lei

N1 - This work was supported by the Engineering and Physical Sciences Research Council [grant numbers EP/N006054/1, EP/M017494/1 and EP/M000397/1]. The fifth author (Charlton, T.J.) was supported by theEngineering and Physical Sciences Research Council [grant number EP/K502832/1]. All data created during this research are openly available at http://doi.org/10.15128/r2pk02c977f.

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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.

AB - 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.

KW - Elasto-plasticity

KW - Finite deformation mechanics

KW - Generalised interpolation

KW - Lagrangian mechanics

KW - Material point method

UR - https://www.scopus.com/pages/publications/85072752983

U2 - 10.1016/j.cma.2019.112622

DO - 10.1016/j.cma.2019.112622

M3 - Article

SN - 0045-7825

VL - 358

SP - 1

EP - 32

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

M1 - 112622

ER -