Page 253 - The Combined Finite-Discrete Element Method

236 TRANSITION FROM CONTINUA TO DISCONTINUA

To deal with mesh size sensitivity, the local softening material law is formulated in terms
of the fracture energy release rate in tension, G f , and the local control length h:

4A
h = (7.8)
π
where A is the area associated with the Gauss integration point considered. To avoid
limitations to the upper limit of the element size to be used, which arises from the diffi-
culties in numerically capturing the so-called snap back in the constitutive law (softening
slope return), the fracture energy is assumed to control only the post-peak behaviour, i.e.
after the peak stress f t is reached. The local softening slope for each Gauss-point is then
obtained from the energy balance

2
f t 2 1 f √
t

A = 2γh = G f h;using h from (7.8) yields E = πA (7.9)
2E 4 G f
This modification of the constitutive law resolves the problem of sensitivity of the fracture
energy release rate to the mesh size. However, the sensitivity of crack initiation to element
size remains. This is because the crack is replaced by a localisation band which is equal to
the element size. A further consequence of this is also the sensitivity to mesh orientation.
With a deformable discrete element, discretised into finite elements, a critical state of stress
(or strain) is reached when an element separates into two or more discrete elements, or
a discrete element changes its boundary (if the failure is only partial). At the stage when
the strength of material in some Gauss-points is reduced to zero, a crack is assumed
to open. The direction of the crack coincides with the direction of the greater principal
plastic stretch. A re-meshing of finite elements within every discrete element is therefore
performed, and when breakage occurs new boundaries are created.
As mentioned above, the constitutive law taking into account the mesh size and fracture
energy of rock is employed to deal with the sensitivity of the model to the mesh size in
terms of energy dissipated through the fracture process. The convergence of the solution
in terms of energy dissipation is best illustrated by an example (Figure 7.4), showing a
strain-softening bar with both ends moving in opposite directions with a constant velocity
v = 0.02 m/s. The bar is of length l = 100 m and rectangular cross-section of h = 5m and
b = 0.2 m. The material of the bar is a linear softening material with a fracture energy
2 2
G f = 2γ = 0.05 kNm/m , modulus of elasticity E = 900 kN/m , tensile strength f t =
2
3
1kNm/m and density ρ = 1000 kg/m . The combined finite-discrete element simulation
of the problem using coarser and finer meshes both resulted in the bar being broken in
two parts that move further away. The total energy of the bar (i.e. kinetic energy plus

Figure 7.4 Failure sequence of a strain softening bar together with finite element meshes
employed.