Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

COUPLING OF PERIDYNAMIC THEORY AND THE FINITE ELEMENT METHOD 711
Figure 3. Definition of geometric parameters.
η = u − u between points x ′ and x, as shown in Figure 3. The stretch, s, is given as
s =
|ξ + η| − |ξ|
|ξ|
and it is the ratio of the change in distance to initial distance between points x ′ and x. Failure is included
in the material response through a history-dependent scalar-valued function µ (see [Silling and Bobaru
2005]), defined as
�
1 if s(t ′ , ξ) < s0 for all 0 < t
µ(ξ, t) =
′ < t,
(4)
0 otherwise,
in which s0 is the critical stretch for failure to occur, as shown in Figure 4. In the solution phase, the
displacements and stretches between pairs of material points are computed. When the stretch between
two points exceeds the critical stretch, s0, failure occurs and these two points cease to interact. As derived
in [Silling and Askari 2005], the critical stretch value can be related to the well-known fracture parameters
such as the energy release rate. Thus, damage in a material is simulated in a much more realistic manner
Figure 4. Model for bond failure.
(3)