A high-order material point method

The material point method (MPM) is a version of the particle-in-cell (PIC) which has substantial advantages over pure Lagrangian or Eulerian methods in numerical simulations of problems involving large deformations. Using MPM helps to avoid mesh distortion and tangling problems related to Lagrangian methods and the advection errors associated with Eulerian methods are avoided. In this paper a novel high-order material point method within an isogeomeric analysis (IGA) framework is developed. Utilizing high order basis functions enables more accurate determination of physical state variables e.g. stress. The smooth spline function spaces, B-splines, are used to eliminate the non-physical effects are caused by use of standard high-order finite element basis function i.e. based on Lagrange polynomials

An implicit high-order material point method.

The material point method (MPM) is a version of the particle-in-cell (PIC) which has substantial advantages over pure Lagrangian or Eulerian methods in numerical simulations of problems involving large deformations. The MPM helps to avoid mesh distortion and tangling problems related to Lagrangian methods and as well as the advection errors associated with Eulerian methods. Despite the MPM being promoted for its ability to solve large deformation problems the method suffers from instabilities when material points cross between elements. These instabilities are due to the lack of smoothness of the grid basis functions used for mapping information between the material points and the background grid. In this paper a novel high-order MPM is developed to eliminate the cell-crossing instability and improve the accuracy of the MPM method

On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity

The material point method is ideally suited to modelling problems involving large deformations where conventional mesh-based methods would struggle. However, total and updated Lagrangian approaches are unsuitable and non-ideal, respectively, in terms formulating equilibrium for the method. This is due to the basis functions, and particularly the derivatives of the basis functions, of material point methods normally being dened based on an unformed, and sometimes regular, background mesh. It is possible to map the basis function spatial derivatives using the deformation at a material point but this introduces additional algorithm complexity and computational expense. This paper presents a new Lagrangian statement of equilibrium which is ideal for material point methods as it satises equilibrium on the undeformed background mesh at the start of a load step. The formulation is implemented using a quasi-static implicit algorithm which includes the derivation of the consistent tangent to achieve optimum convergence of the global equilibrium iterations. The method is applied to a number of large deformation elasto-plastic problems, with a specic focus of the convergence of the method towards analytical solutions with the standard, generalised interpolation and CPDI2 material point methods. For the generalised interpolation method, dierent domain updating methods are investigated and it is shown that all of the current methods are degenerative under certain simple deformation elds. A new domain updating approach is proposed that overcomes these issues. The proposed material point method framework can be applied to all existing material point methods and adopted for implicit and explicit analysis, however its advantages are mainly associated with the former

Resilience of rail support systems: the use of plastic sockets

The use of such plastic inserts and other soft fastening solutions in pre-stressed concrete sleepers and crossing bearers has become widespread within the UK, Europe and the rest of the world. The stiffness of these inserts is typically an order of magnitude lower than that of the surrounding concrete and their inclusion leads to significant stress concentrations. This article quantifies the magnitude of the stress raiser and explores the potential of these elevated stress levels to lead to fracture development

NURBS plasticity: yield surface evolution and implicit stress integration for isotropic hardening

This paper extends the non-uniform rational basis spline (NURBS) plasticity framework of Coombs et al. (2016) to include isotropic hardening of the yield surfaces. The approach allows any smooth isotropic yield envelope to be represented by a NURBS surface. The key extension provided by this paper is that the yield surface can expand or contract through the movement of control points linked to the level of inelastic straining experienced by the material. The model is integrated using a fully implicit backward Euler algorithm that constrains the return path to the yield surface and allows the derivation of the algorithmic consistent tangent to ensure optimum convergence of the global equilibrium equations. This provides a powerful framework for elasto-plastic constitutive models where, unlike the majority of models presented in the literature, the underlying numerical algorithm (and implemented code) remains unchanged for different yield surfaces. The performance of the algorithm is demonstrated, and validated, using both material point and boundary values simulations including plane stress, plane strain and three-dimensional examples for different yield criteria

NURBS plasticity: non-associated plastic flow

This paper extends the non-uniform rational basis spline (NURBS) plasticity framework of Coombs et al. (2016) and Coombs and Ghaffari Motlagh (2017) to include non-associated plastic flow. The NURBS plasticity approach allows any smooth isotropic yield envelope to be represented by a NURBS surface whilst the numerical algorithm (and code) remains unchanged. This paper provides the full theoretical and algorithmic basis of the non-associated NURBS plasticity approach and demonstrates the predictive capability of the plasticity framework using both small and large deformation problems. Wherever possible errors associated with the constitutive formulation are specified analytically and if not numerical analyses provide this information. The rate equations within the plasticity framework are integrated using an efficient and stable implicit stress update algorithm which allows for the derivation of the algorithmic consistent tangent which ensures optimum convergence of the global out of balance force residual when used in boundary value simulations. The important extension provided by this paper is that the evolution of plastic strain is decoupled from the yield surface normal. This allows the framework to model more realistic material behaviour, particularly in the case of frictional plasticity models where an associated flow rule is known to significantly overestimate volumetric dilation leading to spurious results. This paper therefore opens the door for the NURBS plasticity formulation to be used for a far wider class of material behaviour than is currently possible

NURBS plasticity: yield surface representation and implicit stress integration for isotropic inelasticity

In numerical analysis the failure of engineering materials is controlled through specifying yield envelopes (or surfaces) that bound the allowable stress in the material. However, each surface is distinct and requires a specific equation describing the shape of the surface to be formulated in each case. These equations impact on the numerical implementation, specifically relating to stress integration, of the models and therefore a separate algorithm must be constructed for each model. This paper presents, for the first time, a way to construct yield surfaces using techniques from non-uniform rational basis spline (NURBS) surfaces, such that any isotropic convex yield envelope can be represented within the same framework. These surfaces are combined with an implicit backward-Euler-type stress integration algorithm to provide a flexible numerical framework for computational plasticity. The algorithm is inherently stable as the iterative process starts and remains on the yield surface throughout the stress integration. The performance of the algorithm is explored using both material point investigations and boundary value analyses demonstrating that the framework can be applied to a variety of plasticity models