Since this work is the first in-depth review of the NOSB PD in capturing crack kinematics, it makes a sound basis for future work. Although the methods of zero-energy control worked reasonably well, the need for the user to pick an optimum constant coefficient is not ideal. The need for the zero-energy control to flow from the fundamental equations needs to be further investigated.
Additionally, the issue of grid dependency needs to be addressed and in-vestigated further. An adaptive grid orientation scheme based on the stress state could be an option in mediating the dependency. Also, a classical frac-ture mechanics criterion can be used as one, not ideal, option of orientating the grid to the predicted crack path. Other types of particle arrangements can also be further investigated to address the dependency. All the crack propagation simulations in this work had an initial crack, so an investigation into crack nucleation is also warranted. Also, by using the cohesive analysis outlined in Section6, one should be able to derive the bond failure relation that leads to a desired failure law, thereby forming a link between PD CZM and cohesive laws available in the literature.
In terms of enhancements to the NOSB formulation, the list is long. At the top of the list would be the implementation of an adaptive grid scheme.
This would allow, for example, refinement around notches and other stress concentrating geometrical features. The computational savings would have been substantial for the penny shaped crack and notched specimen tests
mentioned in Chapter 4 had this been available. Some work has already begun on this task [60,68] for the bond-based methods. Also, it would be beneficial to have a robust adaptive load stepping scheme to prevent to many broken bonds per load step, as this usually has a negative effect on the crack-tip singularity. Additionally, the solver routines should take into account the fact that the stiffness matrix remains relatively the same during the bond breaking iteration loop. Thus, only matrix entries affected by broken bonds would be recomputed, saving time in the matrix assembly. And lastly, the standard list of features usually available in local continuum codes would be welcome, such as a finite strain formulation, a contact detection scheme for completely broken bond surfaces and a nonlinear solver for handling nonlinear material models and finite strains.
A Discretized shape tensor
The components of bond ξ between x and x0 in the x1, x2 and x3 directions will be denoted as ξx,x0, ξy,x0 and ξz,x0, respectively. Additionally, xj is the coordinate location of the point x, and the points within xj’s horizon are designated as the nth points. For example, if the jth node is 2 and the nth node is 4 then the components of ξ are ξx4= xn−xj = x4−x2, ξy4= y4−y2 Since K is symmetric, positive definite according to Lemma 3.1 [13], the inverse of K is also symmetric and is stored in a 6× 9 matrix
K (xj) =
B Discretized displacement gradient
Using the properties of the outer product,
u⊗v = uvT= the tensor products present in (2.9) and (2.13) can be written as
(xn− xj)⊗ (xn− xj) =
where u, v and w are the displacement components in the x, y and z direc-tions, respectively,
Substituting (B.2) and (B.3) into∇u (2.13),
∇u(xj) = storage convention for K(xj), (A.2), and isolating in (B.4) the displacement components in vector form, (2.16), the resulting in the 9× 3m matrix,
N =
ω (|ξ|) ξznVn for the first three columns and where the remaining trip-licate columns correspond to the displacement component vector for point xn in the horizon.
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