The completeness of Smooth Particle Hydrodynamics (SPH) and its modiÿcations is investigated. Completeness, or the reproducing conditions, in Galerkin approximations play the same role as consistency in ÿnite-di erence approximations. Several techniques which restore various levels of completeness by satisfying reproducing conditions on the approximation or the derivatives of the approximation are examined. A PetrovGalerkin formulation for a particle method is developed using approximations with corrected derivatives. It is compared to a normalized SPH formulation based on kernel approximations and a Galerkin method based on moving least-square approximations. It is shown that the major di erence is that in the SPH discretization, the function which plays the role of the test function is not integrable. Numerical results show that approximations which do not satisfy the completeness and integrability conditions fail to converge for linear elastostatics, so convergence is not expected in non-linear continuum mechanics. ? 1998 John Wiley & Sons, Ltd.

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In 2011, Holmquist and Johnson presented a model for glass subjected to large strains, high strain rates and high pressures. It was later shown that this model produced solutions that were severely mesh dependent, converging to a solution that was much too strong. This article presents an improved model for glass that uses a new approach to represent the interior and surface strength that is significantly less mesh dependent. This new formulation allows for the laboratory data to be accurately represented (including the high tensile strength observed in plate-impact spall experiments) and produces converged solutions that are in good agreement with ballistic data. The model also includes two new features: one that decouples the damage model from the strength model, providing more flexibility in defining the onset of permanent deformation; the other provides for a variable shear modulus that is dependent on the pressure. This article presents a review of the original model, a description of the improved model and a comparison of computed and experimental results for several sets of ballistic data. Of special interest are computed and experimental results for two impacts onto a single target, and the ability to compute the damage velocity in agreement with experiment data.This article is part of the themed issue 'Experimental testing and modelling of brittle materials at high strain rates'.

This article addresses some issues and solutions for ballistic impact computations. A discussion of the strengths and weaknesses of existing computational techniques is presented, and this is followed by a description of a new computational technique that is wellsuited for ballistic impact computations. This new approach uses both finite elements and meshless particles. The initial grid is composed entirely of finite elements. Then as the solution progresses, the highly strained finite elements are automatically converted into meshless particles. Generally, most of the grid remains as finite elements, and this allows for an accurate and efficient solution for the less distorted portion of the problem. Only the highly distorted regions of the problem are converted into meshless particles, and these meshless particles can accurately and robustly represent the high distortions that the finite elements are not able to represent. Several examples are provided to illustrate this approach. Included is the capability to compute the formation of Behind Armor Debris (BAD) and to track it through large distances.