2.1. Smoothed-Particle Hydrodynamics (SPH)

SPH is a meshless method to approximate the solution of partial differential equations. The family of meshless methods is, besides many other applications, of particular interest for the prediction of fracture and fragmentation at high strain rates in metals. Meshless methods can deal with large deformations as well as propagation, bifurcation, and joining of cracks. In contrast, mesh-based methods, such as the Finite Element Method (FEM), need additional modeling techniques, such as node-splitting or erosion criteria, to represent cracks that occur during natural fragmentation. Furthermore, large deformations decrease the accuracy of standard element formulations and increase the run time. Initially, the SPH method was developed for astrophysics problems by Lucy [23] and Gingold [24]. Later, Libersky and Petschek et al. [25,26] extended SPH to deal with materials with strength. The discretized conservation equations used in this paper are

where ρa denotes the density, va the velocity, Ea the internal energy, ma the mass, and σa the stress of a particle a. Wab is the so-called kernel function W evaluated at the distance ||xa−xb|| between two particles a and b, fb are the body forces and ∇a is the gradient with respect to xa. The standard SPH formulation presented in Section 2.1 suffers from a numerical instability [27]. Therefore, total Lagrangian descriptions that overcome the problem have been developed in the recent years [28,29]. However, these kinds of formulations are only applicable to moderate strains [30]. Thus, a total Lagrangian description poses numerical difficulties for our application. Eulerian methods, on the other hand, are able to deal with large deformations and fracture [31,32]. We assured by using a Monaghan bond viscosity [33] that tensile instability is not influencing the solution in our application (compare Figure 1).

Cracks are visible only when the fracture model is enabled. Otherwise, the ring stays intact until the particles loose contact.

Our SPH solver integrates these equations with a central difference scheme in time [4,34].

In the following subsections, we present the constitutive models to describe σ in (1).