Localised material failure for large deformation problems

经典的基于连续介质力学的理论无法捕捉局部材料失效。这是因为它们不包括任何关于材料长度尺度的信息（如关于晶体结构或晶粒尺寸的信息）。因此，当使用数值分析技术（如有限元法）模拟涉及局部破坏的问题时（如岩土结构破坏中典型的集中剪切破坏，如滑坡事件），破坏区域不会收敛到网格细化的有限尺寸。这意味着失效载荷高度依赖于网格大小，它不会收敛到一个稳定的值。克服这一问题的关键方法是使用包含材料长度尺度信息的高阶连续介质力学理论。这里的选择包括非局部方法、梯度理论和Cosserat（或偶应力）理论[7]。工程中绝大多数的数值分析都是采用有限元方法进行的。然而，传统的有限元法有许多缺点，主要是它无法处理大的变形，而不需要计算昂贵的重新网格划分任务。这使得这类问题的数值模拟令人厌烦。材料点法（MPM）与有限元法非常相似，有一个关键的区别-代表物理材料的点（称为材料点）被允许移动，不再直接耦合到它们的父元素。这允许材料通过一个规则的背景网格变形，避免网格失真和计算昂贵的重新网格划分任务。MPM是由Sulsky等人于1994年开发的，作为历史相关材料的粒子方法，使其成为模拟大变形岩土材料的理想方法。然而，对于某些应用程序，mpm确实有一些缺点。例如，由于物理边界和网格的不匹配性质，很难应用边界条件，并且存在与某些类型的材料行为相关的虚假锁定问题（杜伦大学的研究人员率先解决了这些问题[1,5,6]）。据学生和导师团队所知，迄今为止只有两篇论文将非局部方法与MPM相结合[2,8]。然而，所采用的公式导致了一个积分型非局部模型，在求解所得到的方程组时存在困难。它们也仅限于早期版本的材料点方法，因为材料点在背景网格元素之间移动时存在不稳定性。本研究项目将采用不同的方法，并寻求扩展材料点法[3]的更高级版本，以包括Cosserat理论。该项目将首先在杜伦大学现有的有限元代码中实施b[9]中的Cosserat有限元公式。然后，该项目将扩展配方，使其可以与MPM一起使用，并在Durham的内部代码[4]中实现。一旦实现了这一点，该公式将扩展到包括大变形力学和弹塑性，以便它可以应用于岩土工程中具有挑战性的局部问题。这是一个令人兴奋的新研究领域，它将为MPM打开大门，用于理解岩土工程结构破坏的真实本质。

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.