During the analysis, the crack must be explicitly described in the finite element model and to pro- vide accurate numerical results, significant mesh refinement must be performed around the lead- ing edge of the crack. Since no existing solution was available within the ANSYS framework, an
Numerical methods for 3D crack growth analysis 21
(a) (b)
Figure 1.9 Overview(a)and closeup(b)of the specific geometric configuration used to introduce the crack at the geometry level.
automatic meshing methodology has been developed. This new methodology is very general and allows to tackle complex non-planar 3D crack geometries, with arbitrary front shapes. Anyway for the explanation of the methodology proposed herein, the crack will be considered planar.
In this approach, the crack is first introduced at the geometry level, that is, before the meshing step, using a specific configuration shown on Figure 1.9. The crack front is represented by a spline, which is of a great versatility and allows to handle any type of complex crack front shape.
Along this spline, two concentric toruses are swept (in light orange on the figure). In addition, three bodies with a small, finite thickness are extruded at the location of the crack plane (in blue, brown and green on the figure). At this point, the crack is not existing as such, since these bodies of finite thickness are present at the place of the crack faces. The whole geometry is then meshed, as presented on Figure1.10. Obviously, the resulting mesh is still not representing a crack, but the extruded torus have guided the mesher to provide concentric layers of elements near the crack front. So here takes place the second step of the methodology. The closest elements to the crack tip, that is, the gray elements inside the smaller torus, are deleted and replaced by new wedge elements, joining precisely at the crack front. In addition, the brown, yellow and blue elements constituting the thin bodies at the crack plane location are also deleted. Then the nodes belonging to the newly created crack faces are moved toward the crack plane to fill the gap leaved by the elements deletion. The resulting mesh is shown on Figure1.11, as well as an example of von Mises stress obtained on this mesh solicited by an arbitrary loading. This specific methodology has been used to mesh all the test-cases presented in this typescript.
Remark 7 In addition to the introduced meshing methodology, an automated remeshing strat- egy has also been developed to test some ideas. It takes advantage of the scripting capabilities available within the ANSYS framework, combiningPython,JScriptandAPDLlanguages. In this method, based upon the front position provided by the crack growth law, the spline constituting the crack front is merely updated and the meshing methodology is performed again on the obtained updated geometry.
Hence as shown above, in the vicinity of the crack front, a specific radial pattern is used for the meshing. Each row of element around the tip is called a “contour”. This definition as well as the radial mesh pattern are illustrated on Figure1.12. The first mesh contour is constituted by quadratic wedges. Since in our finite element problems the material behavior is linear elastic, in those elements the midside nodes near the crack tip are placed at the quarter point, hence follow-
(a) (b)
(c)
Figure 1.10 Overview(a)and closeups(b)- (c)of the initial mesh of the pseudo-crack geometry.
(a) (b)
(c) (d)
Figure 1.11 Overview (a)and closeup (b)of the final crack mesh. Example of von Mises stress solution obtained on that mesh for an arbitrary solicitation(c)- (d).
Numerical methods for 3D crack growth analysis 23
First contour
Third contour
Figure 1.12 Definition of the element “contours” on a section of the specific radial mesh pattern at the crack tip.
105000 110000 115000 120000 125000 130000 135000 140000 145000 150000
0 20 40 60 80 100
K1(Pa√ m)
Crack front node number Integration contour number: 1
2 3 4
Figure 1.13 Mode 1 stress intensity factor computed on several integration contours around the crack front.
ing the Barsoum’s recommendations [BAR 77]. The other contours are meshed with conventional quadratic hexahedra.
All along the crack growth, the stress intensity factors must be computed and provided to the crack growth law. They are extracted from the stress and displacement fields by means of the so-called interaction integral (see section1.6.2). The accuracy of this type of domain in- tegral methods, even with coarse meshes, is now well known. Nevertheless, in the context of crack evolution analysis, the validity of the results must be ensured at each propagation step. For that purpose, an indicator based on the path independence property of those integrals is used.
Hence at each propagation step, the interaction integral is computed several times, using the dif- ferent contours of elements around the crack front as several different integration domains. Since these integrals exhibit some path independence properties, all the obtained stress intensity factors should be the same. In practice, those properties do not hold very close of the crack tip and for bad quality meshes. Thereby, checking the invariance of the obtained stress intensity factors over the integration domains provides an indicator of the quality of the numerical solution [BUI 78].
This method is used in an automated manner, without any intervention of the user. An example is given on Figure1.13. For a simple 3D test case, the mode I stress intensity factors obtained at each crack front node are plotted. They are computed using different integration domains. Here, the K1that has been computed on the first contour of elements is much lower than the others.
However, when the other contours are used to evaluate the interaction integral, the obtainedK1 are quite similar, hence exhibiting the contour independence property. This permits to conclude that the stress intensity factors computed on the second, third and fourth contours can be consid- ered of a good numerical quality. During a crack analysis solve, we would then choose to use the results of the fourth contour.
Figure 1.14 Typical fatigue crack growth behavior in metals.