Current material point method (MPM) particle updates use a PIC approach, a FLIP approach, or a linear combination of PIC and FLIP. A PIC update filters velocity in each time step, which causes unwanted numerical diffusion, while FLIP eliminates that diffusion, but may retain too much noise. This paper develops a new particle update termed XPIC(m) (for eXtended PIC of order m) because it generalizes PIC updates. XPIC(1) is identical to current PIC methods, but higher orders of XPIC(m) address the over filtering and numerical diffusion of PIC, while still filtering out noise caused by the nontrivial null space of the extrapolation matrix used in MPM. As m → ∞, XPIC(m) converges to a modified FLIP update with orthogonal removal of null space noise. The frequency response and filtering properties of XPIC(m) are investigated and several numerical examples demonstrate its advantages over other update methods.

This paper describes implementation of anisotropic damage mechanics in the material point method. The approach was based on previously proposed, fourth-rank anisotropic damage tenors. For implementation, it was convenient to recast the stress update using a new damage strain partitioning tensor. This new tensor simplifies numerical implementation (a detailed algorithm is provided) and clarifies the connection between cracking strain and an implied physical crack with crack opening displacements. By using 2 softening laws and 3 damage parameters corresponding to 1 normal and 2 shear cracking strains, damage evolution can be directly connected to mixed tensile and shear fracture mechanics. Several examples illustrate interesting properties of robust anisotropic damage mechanics such as modeling of necking, multiple cracking in coatings, and compression failure. Direct comparisons between explicit crack modeling and damage mechanics in the same material point method code show that damage mechanics can quantitatively reproduce many features of explicit crack modeling. A caveat is that strengths and energies assigned to damage mechanics materials must be changed from measured material properties to apparent properties before damage mechanics can agree with fracture mechanics.

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Nanoindentation tests were carried out on the surface of polymer nanocomposites exhibiting either graded or homogeneous distributions of Fe 3 O 4 @silica core-shell nanoparticles in a photocurable polymeric matrix. The results reveal a complex interplay between graded morphology, indentation depth, and calculated modulus and hardness values, which was elucidated through numerical simulations. First, it was experimentally shown how for small (1 µm) indentations, large increases in modulus (up to +40%) and hardness (up to +93%) were obtained for graded composites with respect to their homogeneous counterparts, whereas at a larger indentation depth (20 µm), the modulus and hardness of the graded and homogeneous composites did not substantially differ from each other and from those of the pure polymer. Then, through a material point method approach, experimental nanoindentation tests were successfully simulated, confirming the importance of the indentation depth and of the associated plastic zone as key factors for a more accurate design of graded polymer nanocomposites whose mechanical properties are able to fulfill the requirements encountered during operational life.

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