COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
3D Crack Propagation Analysis Using Free Mesh Method
H. Osaki 1*, H. Matsubara 2, G. Yagawa 1
1
2
Center for Computational Mechanic Research (CCMR), Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo,
112-8611, Japan
Japan Atomic Energy Agency
Email: [email protected]
Abstract The Free Mesh Method (FMM) is a kind of meshless method in the sense that the user can obtain the solution
without the consciousness of meshing process. Also, the solution gained by using the FMM is equivalent to that of the
FEM. However, the computational fracture mechanics requires frequent re-meshing as the crack propagates. Therefore,
the FMM is considered to be useful in the crack propagation analysis. In this paper, in order to show the effectiveness
of the FMM in crack propagation analysis, we studied the 2D and 3D crack propagation problems with the FMM. The
crack propagation is simulated here according to the Erdogan-Sih Criterion and the Paris law.
Key words: Finite Element Method (FEM), Free Mesh Method(FMM), 3D Crack Propagation
INTRODUCTION
The FEM [1] is the numerical analysis technique widely used in the engineering field. The FEM has many superior
characteristics, including high adaptability to complicated shape. However, its weak point is that great labor is needed
for mesh generation. There is though the Meshless Method as a method of solving this problem. This is a technique
only using node information from among the nodes and element information which are needed by the FEM. The FMM
is in one of the Meshless Methods. The FMM temporarily creates a local element stiffness matrix during the process of
analysis using the shape function of the FEM. Then, the process which adds a local element stiffness matrix to all
stiffness matrixes for every central node is created. Therefore, a researcher can improve their analysis, without being
conscious of mesh generation, and can achieve a solution equivalent to the FEM. From the above, it is thought that the
FMM is highly compatible with computational fracture mechanics. It is because computational fracture mechanics
constantly requires re-meshing near a crack tip while a crack propagates. However, a 3D crack analysis which follows
the progress arbitrarily has not yet been developed. Thus, in this paper it is shown that the FMM is effective in a 3D
complicated crack problem. As an analysis model, we used a 3D specimen model.
THE FREE MESH METHOD
1. Basic algorithm Initially the basic algorithm of the FMM is shown in Fig. 1. In the usual FEM, nodal point
information and its connection information are needed as input data, and domain integration is performed for every
element. For such a algorithm, the FMM does not need the connection information of the nodal point as input data, but
only the nodal point information. The creation of a global stiffness matrix and a right-hand vector can be used to calculate
a stiffness matrix for every nodal point by not calculating every element. Under the present circumstances, a local element
division (local mesh) is temporarily created around a central node. In this way, the usual FEM demands the element
division in a domain as input data before analysis, whereas the FMM has a form which includes a part of the pre-process
in analysis processing. Therefore, the stiffness matrix finally created will become completely the same, and a solution
equivalent to the FEM can be obtained (The turn that a difference with the usual FEM makes a stiffness matrix is not
every element, and it is to be every nodal point.) (The meaning of this sentence has been confused. It is very difficult to
understand what you are trying to say. Thus, I could not edit this sentence.) Through this procedural change, the parallel
computing of the global stiffness matrix can be carried out for every central node. In the calculation of each central node,
only the information on the satellite node which is near the central node is needed. (Referring to the data has locality, the
features, such as being suitable for parallelization – cannot be edited as sentence is too confusing).
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・Input data(nodal information)
・Assemble stiffness matrix/right hand vector
for each node(center node)
-select satellite nodes
-construct local elements
-assemble rows in the stiffness matix/ right hand vector concerning the central node
・Consider boundary conditions
・Solve the equation
Figure 1: Basic algorithm of Free Mesh Method
2. Local elements Next, the local mesh generation of the FMM is shown in Fig. 2. The domain which contains
sufficient node for the surroundings which focuseson one central node in the node group distributed over the analysis
domain, is defined. The node which goes inside this domain, is called a candidate node point. The satellite node which
constitutes the element (local mesh) which surrounds a central node out of a candidate node point is selected. The local
element group called a satellite element using the satellite node group is then generated. In this way, surrounding the
obtained satellite element, only the ingredient corresponding to the central node is calculated among the element
stiffness matrix in the FEM. The ingredient of the global stiffness matrix corresponding to a central node is calculated
by adding the ingredient of all satellite elements. This operation is performed to all nodes and simultaneous linear
equations are calculated.
Figure 2: Local elements
THE CRACK PROPAGATION METHOD
1. The crack progress method Here, the crack progress method used in this paper is next described. The target
analysis model is a piece model of a 3-dimensional examination which contains a complicated crack as shown in Fig. 3. A
complicated crack was made into the crack shape which then becomes symmetrical with a y-axis, and is shown in
Fig. 4. In addition, the analysis domain was made into 1/2of the whole domain in relation to the object nature of a
domain.
In this research, analysis which limited the 3D complicated crack to the crack propagation within a field, is performed.
Here, the crack propagation within a field progresses only on the complicated crack side which is shown in the progress
direction of a crack in Fig. 3. In the direction of the outside of a field, it is assumed that it (need to specify what ‘it’ is
here) is what is not progressing. A conceptual figure is shown in Fig. 5. The amount of crack progress of each node of
the crack front tip shown in Fig. 5 is the amount of crack progress computed from the K value (which asked at least for
it being strange by displacement method – with this last part, it is very difficult to understand the meaning). According
to the displacement method, the amount of crack progress of each node of a crack front tip is only strange. The amount
type of crack progress is expressed by the formula (1).
da = c × K
2 .0
× dt (1)
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y
y
x
Figure 3: Complicated Crack in a 3D body
Figure 4: Crack Geometry
However, da is the amount of crack progress (mm). ｃ is a parameter in the amount anticipation type of crack progress,
while K is the stress intensity factor ( MPa m ). dt expresses the time increment in the amount of the anticipation type
of crack progress. In addition, since the purpose of this research is to examine the validity to the 3D crack problem of
the FMM, the amount of crack progress of the direction of z has been disregarded.
δa
y
x
δa
Figure 5: Incremental Crack Propagation
THE COMPLICATED CRACK MESH CREATION METHOD
In this research, complicated crack mesh creation was performed according to the complicated crack mesh creation
method using the B-spline curve. The B-spline curve defines some control points, and can draw a smooth free form
curve. In this research, crack propagation analysis which is characterized by the arbitrary crack of a complicated form
is performed. Here, the complicated crack mesh creation method concept figure is shown in Fig. 6 to Fig. 8 using
B-spline curve from each node coordinates of the crack front tip used in this research. In Fig. 6, the B-spline is used
from the node coordinates of each crack front tip, thus, enabling the curve of a crack front tip to be drawn. Next, the
node inside a crack is created from the node coordinates of each crack front tip, and a curve is drawn using the B-spline.
This is shown in Fig. 7. Fig. 8 is a figure in which the patch mesh from the node at the tip of the complicated crack is
created.
y
y
x
x
Figure 6: Spline Curve Fitting (1)
Figure 7: Spline Curve Fitting (2)
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y
x
Figure 8: Spline Curve Fitting (3)
THE EXAMPLE OF NUMERICAL ANALYSIS
As an example of numerical analysis, tensile analysis of the model using a 3D examination which has the complicated
crack shown in Fig. 3, was performed. Young's modulus set 200Gpa(s) and a poisson ratio to 0.3. The initial
complicated crack y-axis assumes that the 6.0(mm) first stage complicated crack x-axis to be 2.0 (mm). The results of
having performed tensile analysis are shown in Fig. 9 to Fig. 11 using this model of examination. Fig. 9 expresses the
initial crack. In Fig. 10, it turns out that a crack progresses on the complicated crack side and that the crack form is
changing to a great degree. In Fig. 11, the fracture of the model under examination takes place as a crack progresses
further. Therefore, if the FMM is used in 3D crack analysis, progress will be simulated on a complicated crack side as
are shown in Fig. 10 and Fig. 11, and an initial crack piles up a step.
Figure 9: Mesh for Propagating Crack (1)
Figure 10: Mesh for Propagating Crack (2)
Figure 11: Mesh for Propagating Crack (3)
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CONCLUSION
In this research a crack progress simulation of a 3D examination model with a complicated crack was performed in
order to demonstrate that the FMM is effective in 3D crack analysis. In the example given of numerical analysis, the
simulation of the crack progress was successfully achieved and the validity of the FMM was able to be proven. The
next step is to further develop this research in order to greater understand the progress problem aside from a crack 3D
complicated crack side.
REFERENCES
1.
Yagawa G, Yamada T. Free Mesh Method: A New Meshless Finite Element Method. Computational Mechanics,
1996; 18: 383-386.
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