Virtual Crack Extension Theory
The virtual crack extension method is a way to compute the energy release rate for a crack. The idea in the virtual crack extension method is to evaluate the change in total energy when the surface area of a crack is changed, but without actually solving for different geometries. Several different variants on this theme can be found in the literature.
The energy release rate, G, is defined as
where U is the total potential energy, and A is an increase in crack area. The total potential energy consists of the strain energy and the potential of the loads.
For a 2D case, the interpretation is simple, since the crack length, a, is a scalar. Then,
where d is the thickness.
For a crack in 3D, the energy release rate must be viewed as local quantity that varies along the crack front. There is no well defined crack length. Let s be a curve parameter along the crack front. Then
Here A(s) is some kind of local crack area. The new area that is created by a small crack growth must be located in the plane of the crack, and by extending the crack perpendicular to the crack front. If s is the tangent to the crack front, and n(s) is the local normal to the crack face, then the local crack extension direction is
Of course, it is not possible to create a new area of the crack in a pointwise fashion. It must have some, arbitrary, extension along the crack front. In a finite element context, the local crack extension is typically distributed over one or two elements.
In the current implementation, the built-in functionality for sensitivity analysis is used together with a deformed geometry. In a small cylindrical region around the crack tip, the sensitivity region, there is a prescribed deformation field. The prescribed deformation, dX inside the domain is
Here, r is the distance from the crack tip in the plane perpendicular to the crack tip. Rs is the radius of the sensitivity region, which can be given by the user. The control variable field for the sensitivity analysis is denoted c(s).
Figure 3-45: The deformed geometry with interpolated mesh. The yellow ring shows the limit of the sensitivity region. The mesh translation is largest at the crack tip and then decreases linearly through the sensitivity region.
The energy release rate is then determined as
where Ac is the area of the crack.
The size of the sensitivity region should not be too small. If it just encompasses the elements closest to the crack, the singular fields that cannot be represented well in the solution will have a too large influence on the sensitivity evaluation. On the other hand, it should be relatively small, so that it does not include, for example, exterior boundaries.
Computing Stress Intensity Factors From Energy Release Rate
Stress intensity factors can be computed from the energy release rate in exactly the same way as from a J-integral. The methods described in Computing Stress Intensity Factors From J-Integral are applied also in this case.