Page 249 - The Combined Finite-Discrete Element Method

232 TRANSITION FROM CONTINUA TO DISCONTINUA

predicts a vanishing energy dissipation upon spatial discretisation refinement. A mathe-
matically well-posed problem is obtained by using an enriched continuum formulation
(such as a Micro-polar Cosserat) or higher-order constitutive law (such as a non-local
constitutive law, where the higher order gradients of the deformation field are included in
the formulation). A relatively straightforward alternative utilising a fracture energy based
softening plasticity framework has also been successfully adopted in the past, where a
mesh size dependent softening modulus ensures objective energy dissipation.
The local approaches to crack analysis based on a single crack concept are usually
based on the Dugdale model or Barenblatt model. The Dugdale model is a relatively
simple nonlinear model for a crack with a plastic zone at its tip, where the zone of
plastically strained material is replaced by a zone of weakened bonds between the crack
walls. As the crack walls separate the bond stress reaches maximum. At the point when
the separation reaches a critical value, the bonding stress drops to zero.
The main tasks in describing fracture in the combined finite-discrete element method
are to predict crack initiation, predict crack propagation, perform the necessary remesh-
ing, transfer variables from the old to the new mesh and replace the released internal
forces with equivalent contact forces. Robustness, accuracy, simplicity and CPU require-
ments of the fracture algorithms implemented are of major importance, and both single
and smeared crack models have been employed in the past. In the rest of this chapter,
two most widely employed fracture models are described together with numerical exper-
iments demonstrating the complexity of the combined finite-discrete element simulations
involving complex fracture and fragmentation patterns. Fracture and fragmentation is still
intensively researched field in the combined finite-discrete element method and Computa-
tional Mechanics of Discontinua in general, and currently available simulation techniques
are far from optimum.

7.2 STRAIN SOFTENING BASED SMEARED FRACTURE MODEL

In experimental tests of rock and rock-like materials, a gradual load decrease with an
increase in displacements is observed. The phenomenon occurs under uniaxial tension as
well as under uniaxial pressure and triaxial stress states. Figure 7.1 shows a typical stress-
displacement diagram for a rock specimen under uniaxial tension. Due to stress decreasing
with increasing strain, pre-failure strains are highly localised in a narrow band, which
eventually results in a discontinuity in the form of a crack. The phenomenon of decreasing
stress in a localisation band area with increasing strains is called ‘strain-softening’.

P s
P t f t
L
P 1
E′
L + d E
1
d e t e

Figure 7.1 A typical stress-displacement diagram for rock under uniaxial tension and idealised
stress-strain diagram in the localisation zone.