ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

In the Element-Free Galerkin (EFG) method [16], only nodal data and boundary
descriptions are required to formulate the discrete Galerkin equations. Basically, the
main difference between the EFG method and FEM is how the shape functions are
constructed. The construction of the shape functions in the EFG method is based on a
connectivity-free technique known as the moving least square (MLS) approximation
[17]. One of the advantages of MLS approximants is that their continuity is related to
the continuity of the weight function. A background-cell structure is employed for
evaluating the integral expression. More information about meshfree methods can also
be found in many literatures [18, 19, 20].
In this research, crack propagation is simulated by cohesive modeling (Fig. 2a) that
captures the nonlinear behavior of the process zone, which cannot be modeled by linear
elastic fracture mechanics [21]. It represents the physical processes occurring in the
vicinity of a propagating crack by a simplified traction-displacement relationship. Two
properties of the softening curve are most important: the tensile strength (σc) and the
energy release rate (Gc), which are introduced in the cohesive crack model as constant
material parameters. Several types for this softening function have been developed. So
that, a linear traction-displacement curve was employed in current work and the
cohesive model parameters, energy release rate and critical stress are obtained from the
experimental results reported in the literature for human cortical bone [8] – see table 1.
Table 1
Elastic and cohesive properties of the osteonal cortical bone tissue in transverse direction.
Model
Osteon
Interstitial matrix
Cement line
Elastic modulus
(GPa) [11,12]
Poisson’s
ratio[1]
Normal strength
(MPa) [8]
Fracture toughness
(N/mm) [8]
In EFG cohesive fracture methods, in contrast to cohesive element method, there are no
interface elements involved. It uses cohesive zone to simulate the fracture surface and δu
is the displacement jump across the crack surface. The representation of crack in the
meshfree method is depicted by the so-called “visibility” criterion [16, 22]. Since the
growth of crack surfaces can be reflected without changing a connectivity of nodes in
such a method, consequently it is possible to simulate crack propagation without
remeshing and it is an excellent choice for solving these problems. The presence of the
crack, a surface of discontinuity, only affects how nodes influence the displacement at a
given point. A visibility criterion [16] can be used to select these points and
constructing shape functions near a crack. In this method, the crack boundary is
considered to be opaque (Fig. 2b). Nodes that are on the opposite side of the crack are
excluded in the approximation of the displacement field, during the construction of
weight functions [19].
(a)
12
15
6
Cohesive traction
Crack
Fracture process zone
0.3
0.3
0.4
43.5
43.5
0.71
0.71
Fig. 2. (a) Cohesive crack model, (b) Domain of influence near a crack for the visibility criterion[23]
(b)
Domain of influence of node I
Crack