Abstract
Material point methods suffer from volumetric locking when modelling near incompressible materials due to the combination of a low-order computational mesh and large numbers of material points per element. However, large numbers of material points per element are required to reduce integration errors due to non-optimum placement of integration points. This restricts the ability of current material point methods in modelling realistic material behaviour.This paper presents for the first time a method to overcome finite deformation volumetric locking in standard and generalised interpolation material point methods for near-incompressible non-linear solid mechanics. The method does not place any restriction on the form of constitutive model used and is straightforward to implement into existing implicit material point method codes. The performance of the method is demonstrated on a number of two and three-dimensional examples and its correct implementation confirmed through convergence studies towards analytical solutions and by obtaining the correct order of convergence within the global Newton–Raphson equilibrium iterations. In particular, the proposed formulation has been shown to remove the over-stiff volumetric behaviour of conventional material point methods and reduce stress oscillations. It is straightforward to extend this approach to other material point methods and the presented formulation can be incorporated into all existing material point methods available in the literature.
Highlights
Lagrangian mesh-based methods dominate engineering numerical computations in solid mechanics
The material point method was developed by Sulsky et al [33] as a solid mechanics extension to the fluid implicit particle method [2] which itself was developed from the particle-in-cell method [10]
This paper presents for the first time a method to overcome finite deformation volumetric locking in standard and generalised interpolation material point methods for nearincompressible non-linear solid mechanics
Summary
Lagrangian mesh-based methods dominate engineering numerical computations in solid mechanics. Material point methods typically use more materials points per element than would be adopted if the elements were integrated using Gauss-Legendre quadrature Combining this with the fact that material point methods generally use a low order background grid (bi-linear quadrilaterals or tri-linear hexahedrals are a common choice) means that the method is susceptible to volumetric locking (resulting in over-stiff behaviour) when modelling near-incompressible materials. This volumetric locking is caused by excessive constraints placed on an element’s deformation by the points used to integrate the stiffness of the element. The majority of the mathematical development in the paper is presented in tensor form using index notation, the notable exception is the numerics that are presented in matrix-vector form for ease of implementation
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