Here are several useful references that I have found relating to the use of the Material Point Method in modeling flexible objects.

  1. A particle method for history dependent materials - Sandia Version
  2. A particle method for history dependent materials
  3. An Evaluation of the Material Point Method
  4. Analysis and Reduction of Quadrature Errors in the Material Point Method
  5. The Generalized Interpolation Material Point Method
  6. Improved Velocity Projection for the Material Point Method
  7. A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations
  8. Analysis of stress updates in the material-point method
  9. The material point method in soil mechanics problems
  10. Application of a particle-in-cell method to solid mechanics - PDF Not Free, Sorry
  11. A weighted least squares particle-in-cell method for solid mechanics
  12. Tutorial: A beginner’s introduction to the Material Point Method (MPM)
  13. Enriched Convected Particle Domain Interpolation (CPDI) Method for Analyzing Weak Discontinuities - Page 675
  14. Finite element method: an introduction
  15. Fracturing in Concrete via Lattice-Particle Method
  16. Implicit time integration for the material point method: Quantitative and algorithmic comparisons with the Finite element method
  17. Implicit dynamics in the material-point method - PDF Not Free, Sorry
  18. Material Point Method Calculations with Explicit Cracks, Fracture Parameters, and Crack Propagation
  19. Meshfree and Particle Methods and their Applications
  20. Particle Methods
  21. Decoupling and balancing of space and time errors in the material point method (MPM)

Other References:

  1. The classical FEM method and discretization methodology
  2. Stress and Strain Tensors – Deformation and Strain
  3. Stress in 3D
  4. Kinematics: The mathematics of deformation
  5. Velocity gradients and stress tensors