Localised material failure for large deformation problems

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Classical continuum mechanics-based theories are unable to capture localised material failure. This is because they donot include any information about the length scale (such as information on the crystalline structure or grain size) of amaterial. Therefore when modelling problems involving localised failure (such as concentrated shear failure which is typicalin failure of geotechnical structures, such as landslide events) using numerical analysis techniques, such as the finiteelement method, the failure zone does not converge to a finite size with mesh refinement. This means that the failureload is highly dependent on the mesh size and it will not converge towards a steady value. The key way to overcomethis 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 withoutthe 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 thatrepresent the physical material (known as material points) are allowed to move, no longer being directly coupled to theirparent element. This allows material to deform through a regular background grid and avoids mesh distortion and thecomputationally expensive task of re-meshing. The MPM was developed by Sulsky et al. in 1994 [10] as particle methodfor history-dependent materials, making it ideal for modelling geotechnical materials undergoing large deformations. TheMPM does however have some drawbacks for certain applications. For example, due to the non-matching nature of thephysical boundaries and the mesh it is difficult to apply boundary conditions, and there are issues associated spurious lockingwith 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 atcombining non-local methods with the MPM [2, 8]. However, the adopted formulation results in a integral-type non-localmodel that has difficulties when solving the resulting system of equations. They are also restricted to early version ofthe 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 pointmethod [3] to include Cosserat theory. The project will start by implementing the Cosserat finite element formulationfrom [9] within an existing Durham University finite element code. The project will then extend the formulation suchthat it can be used with the MPM and implemented in Durham's in-house code [4]. Once this has been achieved theformulation will be extended to include large deformation mechanics and elasto-plasticity so that it can be applied tochallenging localisation problems in geotechnical engineering. This is a new and exciting area of research that will openthe door to the MPM to be used to understand the true nature of failure in geotechnical structures.