Page 252 - The Combined Finite-Discrete Element Method

STRAIN SOFTENING BASED SMEARED FRACTURE MODEL 235

element size, and with a finer mesh the localisation zone width is smaller and, in addition,
localisation zones tend to follow preferred directions (along the finite element edges
or diagonals) dictated by the mesh. According to how these problems are approached,
finite element based discretisations of problems involving localisation can be classified as
follows:
• Sub-h concept is intended for situations where the localisation bandwidth b is signifi-
cantly smaller than the element size h. It is realised by implementing a localisation zone
into a finite element. The strain field therefore incorporates a localisation band (rather
than just a discontinuity), which can be triggered in a state of homogeneous strain. The
embedding of localisation zones is usually carried out for low order elements, such as
the four node quadrilateral and the three node constant-strain triangle.
• Super-h concept is implemented when the localisation band is larger than the element
size. The bandwidth is usually uniquely determined by the field equations, such as
a non-local strain-softening formulation where the stress at any point is related to the
weighted strain within a finite volume about that point, leading to the bandwidth arising
from the governing field equations.

• Iso-h concept is characterised by the size of finite elements being equal to the width
of the localisation zone. The constitutive law is usually modified to include the char-
acteristic length of the discretisation grid, such as the element size or area around the
integration point, in order to deal with mesh sensitivity.
Localisation is closely related to smeared crack models, where the localisation zone
(crack band) is usually assumed to propagate into the next finite element when the stress
in that element reaches a strength limit. In this way, the propagation of the zone is
influenced and largely determined by the zone width, because the narrower the zone the
larger the stresses ahead. If the crack bandwidth is specified as a material property within
a non-local continuum framework, there is nothing wrong with this approach. However,
when it is determined by the finite element size as discussed above, additional criteria for
band propagation need to be introduced.
The smeared fracture model implemented in the combined finite-discrete element
method also uses the concept of localisation band propagation, and in essence belongs to
the iso-h group of strain-softening models. The underlying assumptions of the model are:
• the localisation (tensile fracture) occurs on the finite element integration point level;

• the size of the overall model is significantly larger than the size of the finite elements
employed (h);

• the strain energy accumulated before the peak stress is reached within an area associated
with the integration point undergoing softening can be neglected;

• the plasticity model is assumed to be isotropic, i.e. the accumulated effective plastic
strain is monitored in the principal directions only. If, after the strength limit is reached,
a full breakage does not occur (stress state on the softening branch), the effective plastic
strain is treated as a scalar state variable for the next state of deformation, which will
be valid for any new rotated principal direction.