SUMMARY A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright 2011 John Wiley & Sons, Ltd.

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

Contact between deformable bodies is a dif- cult problem in the analysis of engineering systems A new approach to contact has been implemented us- ing the Material Point Method for solid mechanics, Bar- denhagen, Brackbill, and Sulsky (2000a) Here two im- provements to the algorithm are described The rst is to include the normal traction in the contact logic to more appropriately determine the free separation crite- rion The second is to provide numerical stability by scaling the contact impulse when computational grid in- formation is suspect, a condition which can be expected to occur occasionally as material bodies move through the computational grid The modications described preserve important properties of the original algorithm, namely conservation of momentum, and the use of global quantities which obviate the need for neighbor searches and result in the computational cost scaling linearly with the number of contacting bodies The algorithm is demonstrated on several examples Deformable body so- lutions compare favorably with several problems which, for rigid bodies, have analytical solutions A much more demanding simulation of stress propagation through ide- alized granular material, for which high delity data has been obtained, is examined in detail Excellent quali- tative agreement is found for a variety of contact con- ditions Important material parameters needed for more quantitative comparisons are identied

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An Improved Contact Algorithm for the Material Point Method and Application to Stress Propagation in Granular Material

A new algorithm is described which extends the material point method (MPM) to allow explicit cracks within the model material Conventional MPM enforces velocity and displacement continuity through its background grid This approach is incompatible with cracks which are displacement and velocity discontinuities By allowing multiple velocity fields at special nodes near cracks, the new method (called CRAMP) can model cracks The results provide an “exact” MPM analysis for cracks Comparison to finite element analysis and to experiments show it gets good results for crack problems The intersection of crack surfaces is prevented by implementing a crack contact scheme Crack contact can be modeled using stick or sliding with friction All results are two dimensional, but the methods can be extended to three dimensional problems keyword: Material point method, cracks, fracture, numerical methods, contact

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Material Point Method Calculations with Explicit Cracks

A numerical model for studying the dynamic evolution of landslides is presented. The numerical model is based on the Generalized Interpolation Material Point Method. A simplified slope with a house placed on top is analysed. An elasto-plastic material model based on the Mohr–Coulomb yield criterion is employed for the soil. The slide is triggered for the initially stable slope by removing the cohesion of the soil and the slide is followed from the triggering until a state of static equilibrium is again reached. Parameter studies, in which the angle of internal friction of the soil and the degree of discretisation are varied, are presented.

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Modelling of Landslides with the Material-Point Method

The inherent no-slip contact constraint in the standard material point method (MPM) creates a greater penetration resistance. Therefore, the standard MPM was not able to treat the problems involving impact and penetration very well. To overcome these deficiencies, two contact methods for MPM are presented and implemented in our 3D explicit MPM code, MPM3D. In MPM, the impenetrability condition may not satisfied on the redefined regular grid at the beginning of each time step, even if it has been imposed on the deformed grid at the end of last time step. The impenetrability condition between bodies is only imposed on the deformed grid in the first contact method, while it is imposed both on the deformed grid and redefined regular grid in the second contact method. Furthermore, three methods are proposed for impact and penetration simulation to determine the surface normal vectors that satisfy the collinearity conditions at the contact surface. The contact algorithms are verified by modeling the collision of two elastic rings and sphere rolling problems, and then applied to the simulation of penetration of steel ball and perforation of thick plate with a particle failure model. In the simulation of elastic ring collision, the first contact algorithm introduces significant disturbance into the total energy, but the second contact algorithm can obtain the stable solution by using much larger time step. It seems that both contact algorithms give good results for other problems, such as the sphere rolling and the projectile penetration. Copyright © 2010 John Wiley & Sons, Ltd.

S.G. Bardenhagen

Papers

Energy Conservation Error in the Material Point Method for Solid Mechanics