structural geology, propagation mechanics and - Stanford School of ...

and
ts = G·ds + H·dn (2)
where tn and ts are column vectors containing the normal and shear traction boundary
conditions prescribed for the midpoint of each successive element 1 through N; dn and ds
are column vectors representing the unknown components of displacement discontinuity,
Dn and Ds., for each successive element 1 through N; and E, F,G and H are N x N
matrices containing the influence coefficients that relate displacement on one element to
traction on another. For example, the component Eij of E gives the normal traction on the
ith element due to a unit normal displacement occurring on the jth element.
Analytical functions relate the displacement discontinuity on each element to the
stress field it generates in the surrounding elastic medium (Crouch and Starfield, 1983).
The complete state of stress at any point within the medium can be determined by
superimposing the contributions from each anticrack boundary element and the remote
values. Thus, once Dn and Ds have been determined for all N elements representing a
given band geometry, the code uses them to determine the magnitude and location of
σθθ max along a circle of specified radius centered on the model band. If this value exceeds
the prescribed threshold limit for propagation, an additional displacement discontinuity
element is appended in the appropriate direction, creating a new system of 2(N+1)
equations to be solved. By iterating this process, propagation is simulated.
As currently written, the code allows for any combination of traction and
displacement discontinuity boundary conditions to be specified for each element. If only
boundary displacements are specified, then the problem is fully constrained and the
solution is already in hand. If one or both displacement discontinuities are not specified,
compensating traction boundary conditions are determined during the solution step based
on elastic modulii that must be prescribed for each element. Other schemes for applying
boundary conditions on the elements without assuming linear elastic behavior (e.g.
nonlinear elasticity, plasticity and frictional sliding) would allow for the prescription of a
greater variety of potentially more realistic internal CB behaviors. These improvements
are currently being developed and implemented.
Finally we re-emphasize that, while the code treats positive Dn as a material
interpenetration across the elements, actual CBs represent porosity loss across a band of
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