Page 248 - The Combined Finite-Discrete Element Method

7

Transition from Continua

to Discontinua

7.1 INTRODUCTION

Transition from continua to discontinua in the combined finite-discrete element method
occurs through fracture and fragmentation processes. A typical combined finite-discrete
element method based simulation, such as rock blasting, may start with a few discrete
elements and finish with a very large number of discrete elements.
Fracture in general occurs through alteration, damage, yielding or failure of microstruc-
tural elements of the material. To describe this complex, material-dependent phenomenon,
the alteration of stress and strain fields due to the presence of microstructural defects
and stress concentrations must be taken into account. Several approaches are available,
and these include global approaches, local approaches, smeared crack models and single
crack models.
Global approaches to fracture are based on the representation of the singularity of
the stress field at the crack tip. It was shown by Griffith that the failure of a brittle
elastic medium due to such singularity can be characterised by the energy release rate
G. The critical value of G = 2γ (where γ represents the surface energy) is a material
characteristic. The alternative formulation of the Griffith method is achieved through stress
intensity factors, which characterise the stress singularity on a semi-local basis in terms of
force, while the same singularity is characterised in terms of energy by contour integrals.
Local approaches to crack analysis usually employ a smeared crack approach, with a
single crack being replaced by a blunt crack band. This approach has been justified by the
fact that engineering materials show a reduction in the load-carrying capacity accompa-
nied by strain localisation after the maximum load-carrying capacity is reached. Beyond
the peak load (when the material gradually disintegrates), two types of failure mechanism
are observed, namely decohesion and frictional slip. In the first type of failure fracture,
zones are observed (cracks), while in the latter failure zones propagate along shear bands
(faults). Smeared crack models attempt to describe these processes through constitutive
laws, such as a strain softening constitutive law or damage mechanics based formulation.
However, standard continuum mechanics formulations incorporating softening fail, as the
underlying mathematical problem becomes ill-posed. As a result, the numerical solution

The Combined Finite-Discrete Element Method A. Munjiza
 2004 John Wiley & Sons, Ltd ISBN: 0-470-84199-0