Material stability analysis of particle methods
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Springer 2005
Material stability analysis of particle methods S.P. Xiao a and T. Belytschko b,∗ a Department of Mechanical and Industrial Engineering and Center for Computer-Aided Design,
The University of Iowa, 3131 Seamans Center, Iowa city, IA 52242, USA E-mail: [email protected] b Department of Mechanical Engineering, Northwestern University, 2145 N Sheridan Rd, Evanston, IL 60208, USA
Received 8 April 2003; accepted 20 December 2003 Communicated by Z. Wu and B.Y.C. Hon
Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with reasonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz–Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability, so that material instabilities can occur in stress states that should be stable. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material stability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations. Keywords: particle methods, stability, Lagrangian kernels AMS subject classification: 74S30
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Introduction
Particle methods are a class of meshfree methods. The original particle methods was based on kernel approximations [20]. In some cases, as shown by Belytschko et al. [4], kernel methods are closely related to the mesh-free methods that are based on field approximations. However, the kernel approximations used in particle methods are somewhat inaccurate because they cannot exactly reproduce linear functions. With corrected derivative methods, as developed by Randles and Libersky [23] and Krongauz and Belytschko [17], the kernel approximations are corrected so that they can reproduce derivatives of linear functions exactly. Alternatively, Liu et al. [18] has developed correction functions that enable particle methods to reproduce linear functions exactly; these ∗ Corresponding author.
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S.P. Xiao, T. Belytschko / Particle methods
corrected particle methods are identical to the moving least square (MLS) approximation with a linear basis as used in the element-free Galerkin (EFG) methods [6]. Material instabilities occur in nonassociative plasticity and softening materials where stress decreases with increasing strain. The literature on material instability goes back at least as far as Hadamard [13] who examined the question of what happens when the tangent modulus is negative. He identified the conditions for a vanishing propagation speed of an acceleration wave as a material instability. In 1962, Hill [14] mad
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