SPH (smoothed particle hydrodynamics) is a gridless Lagrangian techniq
ue which is appealing as a possible alternative to numerical technique
s currently used to analyze large deformation events. Recent tests of
the standard SPH method using the cubic B-spline kernel indicated the
possibility of an instability in the tensile regime, even though no su
ch difficulties were observed in compression. A von Neumann stability
analysis of the SPH algorithm has been carried out which identifies th
e criterion for stability or instability in terms of the stress state
and the second derivative of the kernel function. The analysis explain
s the observation that the method is unstable in tension while apparen
tly stable in compression but shows that it is possible to construct k
ernel functions which are stable in tension and unstable in compressio
n. The instability is shown to result from an effective stress with a
negative modulus (imaginary sound speed) being produced by the interac
tion between the constitutive relation and the kernel function and is
not caused by the numerical time integration algorithm. That is, chang
es in the effective stress act to amplify, rather than reduce, perturb
ations in the strain. The analysis and the stability criterion provide
insight into possible methods for removing the instability. (C) 1995
Academic Press, Inc.