Large deformation analysis using a quasi-static material point method. - Beuth, L.; Benz, T.; Vermeer, Pieter A.; Więckowski, Zdzisław (Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej, 2008)
The Finite Element Method (FEM) has become the standard tool for the analysis of a wide range of solid mechanics problems. However, the underlying structure of a classical updated Lagrangian FEM is not well suited for the treatment of large deformation problems, since excessive mesh distortions can lead to numerical difficulties. The Material Point Method (MPM) represents an approach in which material points moving through a fixed finite element grid are used to simulate large deformations. As the method makes use of moving material points, it can also be classifed as a point-based or meshless method. With no mesh distortions, it is an ideal tool for the analysis of large deformation problems. MPM has its origin in fluid mechanics and has only recently been applied to solid mechanics problems. It has been used successfully for impact analyses where bodies penetrate each other and for silo discharging problems. All existing MPM codes found in literature are dynamic codes with explicit time integration and only recently implicit time integration. In this study a quasi-static MPM formulation and implementation are presented. The paper starts with the description of the quasi-static governing equations and the numerical discretisation. Afterwards, the calculation process of the quasi-static MPM is explained, followed by the presentation of some geotechnical boundary value problems which have been solved with the newly developed quasi-static MPM code. The benchmark problems consist of an oedometer test and a slope. For validation, the results are compared with analytical solutions and FEM results, respectively.
Keywords: meshless methods; Material Point Method; large deformations; metoda elementów skończonych; metoda quasi-statyczna; metoda punktów materialnych; metoda MPM; odkształcenia