Gradient elasticity with the material point method
Charlton Tim  1@  , Will Coombs  1@  , Charles Augarde  1@  
1 : Durham University
South Road, Durham, DH13LE -  United Kingdom

The Material Point Method (MPM) is a method that allows solid mechanics problems
with large deformation and non-linearity to be modelled using particles at which state variables
are stored and tracked. Calculations are then carried out on a regular background grid to which
state variables are mapped from the particles. There have been a selection of extensions to the
MPM, for example, a problem in the original method that arises when a material point crosses
the boundary between one background grid cell and another is addressed by the Generalised
Interpolation Material Point (GIMP) method [1]. The GIMP method attempts to alleviate this
problem by modifying the particle characteristic functions so that particles have an influence
on nodes other than those associated with the element it is inside.
An area yet to be studied in as much depth in MPM is that conventional analysis techniques
constructed in terms of stress and strain are unable to resolve structural instabilities such as
necking or shear banding. This is due to the fact that they do not contain any measure of the
length of the microstructure of the material analysed such as molecule size of grain structure.
Gradient elasticity theories provide extensions of the classical equations of elasticity with additional
higher-order terms [2]. This use of length scales makes it possible to model finite
thickness shear bands that is not possible with traditional methods. Much work has been done
on including the effect of microstructure on a linear elastic solid and has previously been combined
with the Finite Element Method and with the Particle In Cell Method. In this paper the
MPM will be developed to include gradient effects.

[1] S. Bardenhagen and E. Kober, “The generalized interpolation material point method,” Computer Modeling in
Engineering and Sciences, vol. 5, no. 6, pp. 477–496, 2004.
[2] H. Askes and E. C. Aifantis, “Gradient elasticity in statics and dynamics: an overview of formulations,
length scale identification procedures, finite element implementations

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