Material point method for deteriorating inelastic structures

The material point method (MPM) is one of the latest developments in particle in cell methods (PIC). The structure is discretized into a number of material points that hold all the state variables of the system [1] such as stress, strain, velocity, displacement etc. These properties are then mapped to a temporary background grid and the governing equations are solved. The momentum conservation equations (together with energy and mass conservation considerations) are solved at the grid nodes. The state variables of the particles are then updated by transferring the solutions from the grid nodes back to the material points. Since the background grid is used only to solve the governing equations at the end of each computational step it can be reset to its undistorted form and thus mesh distortion and element entanglement are avoided. In this work an explicit MPM accounting for elastoplastic material behavior with degradations is proposed. The stress tensor is decomposed into an elastic and a hysteretic – plastic part [5] where the hysteretic part of the stresses evolves according to a Bouc-Wen type hysteretic rule [2]. The inelastic constitutive material law provides a smooth transition from the elastic to the inelastic regime and accounts for the different phases during elastic loading, unloading, yielding and stiffness and strength degradation. Heaviside type functions are introduced that act as switches, incorporate the yield criterion and the terms for stiffness and strength degradation as in the Bouc-Wen model of hysteresis [2]. The resulting constitutive law relates stresses and strains with the use of the tangent modulus of elasticity, which now includes the Heaviside functions and gathers all of the governing inelastic degrading behavior

Solid deformation by material point method

Solid materials are responsible for many interesting phenomena. There are various types of them, such as deformable objects and granular materials. In this paper, we present an MPM based framework to simulate the wide range of solid materials. In this framework, solid mechanics is based on the elastoplastic model following small deformation theory. We use von Mises criterion for deformable objects, and the Drucker–Prager model with nonassociated plastic flow rules for granular materials. As a result, we can simulate different kinds of deformation of deformable objects and sloping failure for granular materials

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