Localised material failure for large deformation problems

Classical continuum mechanics-based theories are unable to capture localised material failure. This is because they do
not include any information about the length scale (such as information on the crystalline structure or grain size) of a
material. Therefore when modelling problems involving localised failure (such as concentrated shear failure which is typical
in failure of geotechnical structures, such as landslide events) using numerical analysis techniques, such as the finite
element method, the failure zone does not converge to a finite size with mesh refinement. This means that the failure
load is highly dependent on the mesh size and it will not converge towards a steady value. The key way to overcome
this problem is to use high-order continuum mechanics theories that include information about the material length scale.
Options here include non-local methods, gradient theories and Cosserat (or couple-stress) theory [7].
The vast majority of numerical analyses in engineering are conducted using the finite-element method (FEM). However,
the conventional FEM suffers from a number of drawbacks, principally its inability to handle large deformations without
the computationally expensive task of re-meshing. This makes the simulation of such problems numerically tiresome.
The material point method (MPM) is very similar to the finite element method, with one key difference - the points that
represent the physical material (known as material points) are allowed to move, no longer being directly coupled to their
parent element. This allows material to deform through a regular background grid and avoids mesh distortion and the
computationally expensive task of re-meshing. The MPM was developed by Sulsky et al. in 1994 [10] as particle method
for history-dependent materials, making it ideal for modelling geotechnical materials undergoing large deformations. The
MPM does however have some drawbacks for certain applications. For example, due to the non-matching nature of the
physical boundaries and the mesh it is difficult to apply boundary conditions, and there are issues associated spurious locking
with certain types of material behaviour (researchers at Durham have pioneered solutions to these problems [1, 5, 6]).
To the best of the student and supervisor team's knowledge to date there have been only two papers that have looked at
combining non-local methods with the MPM [2, 8]. However, the adopted formulation results in a integral-type non-local
model that has difficulties when solving the resulting system of equations. They are also restricted to early version of
the material point method that suffers from instabilities as material points move between background grid elements.
This research project will follow a different approach and look to extend a more advanced version of the material point
method [3] to include Cosserat theory. The project will start by implementing the Cosserat finite element formulation
from [9] within an existing Durham University finite element code. The project will then extend the formulation such
that it can be used with the MPM and implemented in Durham's in-house code [4]. Once this has been achieved the
formulation will be extended to include large deformation mechanics and elasto-plasticity so that it can be applied to
challenging localisation problems in geotechnical engineering. This is a new and exciting area of research that will open
the door to the MPM to be used to understand the true nature of failure in geotechnical structures.