A continuum consistent discrete particle method for continuum–discontinuum transitions and complex fracture problems

Abstract

In numerical simulations where complex fracture behavior plays a prominent role in the material’s mechanical behavior, particle methods are an attractive computational tool since they adequately accommodate arbitrary discontinuities. However, existing particle methods are either limited in their constitutive flexibility, like the Discrete Element Method (DEM), or prone to instabilities, like Smoothed Particle Hydrodynamics (SPH) and Peridynamics. In this paper we present an alternative particle formulation, referred to as the Continuum Bond Method (CBM). The method has the same constitutive flexibility as conventional continuum methods like the Finite Element Method (FEM), while still being able to incorporate arbitrary discontinuities as in particle methods like DEM, SPH and Peridynamics. In CBM, the continuum body is divided into a series of material points where each material point carries a fraction of the body’s mass. A triangulation procedure establishes the bonds between the particles that interact with each other. The deformation gradient tensor is determined via a volume weighted averaging procedure over the volumes spanned by pairs of nearest neighboring particles. The obtained approximation of the continuum deformation field on the particles allows for a straightforward implementation of continuum constitutive laws. To assess this property in CBM, simulation outcomes for an elastic nonlinear plastic tensile bar are compared to FEM and SPH results. While the stress–strain curves obtained by FEM, CBM and SPH coincide quite accurately, it is found that the local plastic strains obtained by CBM are much closer to the FEM reference solution than the SPH results. The ability of CBM to account for arbitrary discontinuities is demonstrated via a series of dynamic fracture simulations. It is shown that, without the need of additional crack tracking routines, CBM can account for fracture instability phenomena like branches. In conclusion, CBM is suitable for the implementation of continuum constitutive behavior while maintaining the advantageous discontinuous fracture properties of particle methods.