COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
2006 Tsinghua University Press & Springer
The State-of-the-Art Methodology to Compute the Three-Dimensional
Stress Intensity Factors for Arbitrary Shaped Cracks in Complex
Shaped Structures
H. Okada1*, G. Yagawa2, H. Kawai3, K. Shijo4, D. Fujita4,Y. Kanda5, T. Fujisawa5, T. Iribe5
1
Department of Nano-structure and Advanced Materials, Kagoshima University, 1-21-40 Korimoto, Kagoshima,
890-0065 Japan
2
Center for Computational Mechanics Research, Toyo University, 2-36-5 Hakusan, Bunkyoku, Tokyo 112-8611, Japan
3
Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohokuku, Yokohama 233-0051, Japan
4
Technostar Co., Ltd., M21 Bldg., 2-2-5, Roppongi, Minatoku, Tokyo 106-0032, Japan
5
Prometech Software, Inc., Sangakurenkei-Plaza, 4F, 7-3-1 Hongo, Bunkyoku, Tokyo 113-0033, Japan
Email: [email protected]
Abstract In this paper, new developments in the field of fracture mechanics analyses are presented. New fracture
analysis systems are being developed under the collaboration of private industries (Technostar Co., Ltd. and
Prometech Software, Inc., Japan), academic institutes (Kagoshima University, Toyo University and Keio University,
Japan) and a government agency (Nuclear Energy Safety Organization, Japan). Time and labor, that fracture
mechanics analyses using the finite element method take, will drastically be reduced. The new systems are composed
of new VCCM (Virtual Crack Closure-Integral Method) for quadratic tetrahedral finite elements, the state-of-the art
finite element model generation programs and the finite element program that can perform large scale computations
efficiently. This paper describes the key technologies in the system.
Key words: fracture mechanics, stress intensity factor, virtual crack closure-integral method (VCCM), automatic
mesh generation, free mesh method (FMM)
INTRODUCTION
In present research, engineering softwares that can perform highly efficient and accurate fracture analyses based on the
finite element method (FEM) is being developed. The developed softwares will drastically shorten time and labor cost
to perform the fracture mechanics analyses.
Structural integrity problems have been recognized as the major concerns by engineers in power plant, civil
engineering, aerospace and ship-architectural industries. One of the scenario leading to structural failure is that (1)
cracks from in the structure, (2) they propagate through the structure under the applied stress and (3) unstable crack
propagation lading to catastrophic structural failure occurs. In this scenario, it is very important to detect the structural
defects by non-destructive inspection (NDI) techniques and to predict how the existing cracks propagate through the
structure. Thus, the remaining service life of damaged structure can be predicted.
The major tool to perform the residual life prediction analysis is the finite element method (FEM). Once FEM analysis
is performed for the damaged structure, fracture mechanics parameter such as the stress intensity factor is evaluated
and it characterizes the severity of crack tip deformation field. Rate and direction of crack propagation are predicted
based on the stress intensity factor. In order to carry out the crack propagation analysis, the finite element model needs
to be updated as the crack propagates though the structure.
Fracture mechanics analyses have historically been carried out by using hexahedral finite elements [1-3], because
many researchers and engineers have believed that the hexahedral finite elements are superior over tetrahedral
elements in terms of their accuracies. Algorithms to evaluate the fracture parameters have been developed for
hexahedral finite elements [4-6]. Thus, the use of hexahedral finite elements became a sort of standard way.
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To carry out an FEM analysis, one needs to generate an appropriate finite element model. Since there is a sever stress
concentration, very fine finite elements must be placed at the crack tip. For the other parts of the structure, the sizes of
the finite elements need to appropriately be varied with respect to the shape and the variation of stresses. However,
when a model with hexahedral finite elements for complex shaped structure with a crack is built, there are many
manual operations and therefore it is very time and labor consuming. On the other hand, though building a model with
tetrahedral finite elements seems to be less troublesome, methods to evaluate the fracture mechanics parameters are not
well established.
In this research we aim at developing softwares that can automate the fracture mechanics analysis by coupling the 1)
automatic mesh generation software, 2) finite element program for large scale analysis and 3) a new virtual crack
closure-integral method (VCCM) (see references [6] and [7] for conventional VCCM for three- and two-dimensional
crack problems) for evaluating the stress intensity factor based on tetrahedral finite elements. Among them the VCCM
technique is the key technology. In this paper, the VCCM for quadratic tetrahedral finite element is described first and
the-state-of-the-art mesh generation software called VENUS [8] and the use of free mesh method (FMM) [9] are
discussed.
VIRTUAL CRACK CLOSURE-INTEGRAL METHOD (VCCM) FOR SECOND ORDER
TETRAHEDRAL FINITE ELEMENT
In this section VCCM for second order tetrahedral finite element is briefly discussed by following Okada et al. [10].
VCCM was first proposed by Rybicki and Kanninen [7] and was extended to three-dimensional cases (Shivakumar et
al. [6]). Shivakumar et al. [6] added a thickness to two-dimensional finite element model in their extension to the
three-dimensional case. Therefore, they implicitly assumed the use of hexahedral finite elements. Okada and
Kamibeppu [11] and Okada et al. [10] proposed VCCM formulations that enables us to use linear and quadratic
tetrahedral finite elements, respectively.
A structure with a crack is modeled by tetrahedral finite elements, as shown in Figure 1 (a). At the vicinity of the crack
front, the faces of the finite elements are placed in the plane of the crack, as shown in Figures 1 (b) and (c). The faces
of the finite elements are required to have the same width ∆ across the crack front. This needs to be satisfied locally.
We discuss about the case hat the faces of the elements are arranged as shown in Figure 1 (b). Figure 1 (b) shows the
case that the faces of the element across the crack front are placed such that the faces in the ligament side are translated
across the crack front. This is an ideal case in present VCCM formulation. However, the case that the arrangement of
the element faces is somewhat irregular as shown in Figure 1 (c) can be dealt with, although such cases are not
discussed in this paper for simplicity. The energy to open an element face area S1 , when the crack propagates a length
∆ , can be computed by using the asymptotic solution of stresses and opening displacements. The area S1 is illustrated
in Figure 2 (a).
(b) Arrangement of element faces at the crack front
(Regular arrangement)
(a) A crack model with tetrahedral finite elements
(c) Arrangement of element faces at the crack front
(Non-regular arrangement)
Figure 1: An example of crack model with tetrahedral finite elements and typical arrangements of finite
element faces at the crack front.
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where
takes E and
for the cases of plane stress and of plane strain, respectively. E and ν are the
at a point P on S1 is multiplied by
Young’s modulus and the Poisson’s ratio. In Eq. (1), the cohesive stress
and the integration is carried out over S1.
the crack tip opening displacement at its corresponding point P’ on
Eq. (1) leads to:
For the case that is shown in Fig. 2(b), the same procedures lead to:
where S2 is the area of an element face as shown in Fig. 2(b). From Eqs. (2) and (3), the following expression can be
established for the energy release rate.
On the other hand, the energies
and
can be computed approximately by using the nodal forces and the
and
,
crack tip opening displacements at nodes. They are illustrated in Fig. 3. We denote them to be
respectively. They are computed by:
To compute the energy
for the area S1, five pairs of nodal forces and opening displacements are used. For
, three pairs are used.
and
are replaced by
and
, respectively, and the energy release rates
The energies
and
can be expressed, by:
VCCM computes the energy release rate by using nodal reaction forces and crack tip opening displacements at
nodes.
Since the cohesive stress at the element faces are not known, we compute the nodal reaction force using a group of
element immediately above the plane of crack as shown in Fig. 4(a). The nodal reaction forces are computed from
the stresses inside these elements, by:
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(a) Typical arrangemate-1
(b) Typical arrangemate-2
Figure 2: Paris of finite element faces for the computation of VCCM
(a) Typical arrangemate-1
(b) Typical arrangemate-2
Figure 3: Paris of nodal reaction forces and nodal crack opening displacements
where [B ] is the displacement-strain matrix (B-matrix) and {σ } is the vector of stresses. Ω CFE
IEL is the volume of
{ }
element IEL and NELCFE is the total number of elements that are used to compute the nodal reaction forces. F FE is
{ }
FE
the vector of evaluated nodal reaction forces. In this way each component of F
contains influences from
neighboring finite elements. Therefore, the nodal reaction forces computed by equation (7) have to appropriately be
partitioned, when they are used in the VCCM calculation of equation (6). To do so we introduce partitioning constants
that are determined based on the ratio of the areas of neighboring finite element faces.
When the energy release rate is computed by equation (6), the solutions may become erroneous. We use a group of
finite element faces to compute the energy release rate. In Figure 4 (b), a VCCM calculation involving five element
faces is illustrated. The same discussion as in equations (1)~(6) can be carried out for the group of element faces and
the energy release rate is expressed by:
(
)
G I S1J −2 ~ S1J + 2 =
{ (
)
( )
( )
( )
(
K I2 2 δW S1J −2 + δW S 2J −1 + δW S1J + δW S 2J +1 + δW S1J + 2
=
E′
3S1J −2 + S 2J −1 + 3S1J + S 2J +1 + 3S1J + 2
(a) Group of elements for the evaluation of nodal
reaction forces
)}
(8)
(b) Group of element faces for the evaluation of
energy release rate
Figure 4: Group of finite elements and finite element faces to compute the nodal reaction forces and to calculate the
energy release rate, respectively
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The stress intensity factor is then calculated by the following relationship.
K I = G I E′
(9)
PRELIMINARY NUMERICAL EXAMPLE OF VCCM CALCULATION
In this section, some results of our preliminary analyses to examine the accuracy of proposed VCCM are presented.
The problems of embedded circular/elliptical crack are solved by FEM and the stress intensity factors are computed by
VCCM. The stress intensity factors are compared with their analytical solutions for embedded circular/elliptical crack
in an infinite solid. The problem and a typical finite element mesh discretization are presented in Figure 5. Due to the
symmetry of the problem, only 1/8 region was analyzed by FEM. The total numbers of elements and nodes are 27663
and 40286. The size ∆ of crack front elements are a 76 and b 76 in the directions of a and b axes of the elliptical
crack. The number of element faces that were used in VCCM calculation, i.e., equation (8), is nine.
The results are presented in Figure 6 for the aspect ratio b a 0.2, 0.4, 0.6 and 1.0 (circular crack). In all the cases, the
solutions of FEM coupled with VCCM are very close to the analytical ones. It is noted that the results presented in
Figure 6 are normalized by σ πa Q(a b ) where Q(a b ) is the shape factor [1]. Analyses are also carried out for the
problems of semi-circular/elliptical in a plate subject to tension and the results are compared with the Raju-Newman
solutions [12]. The solutions of proposed method are very close to those of Raju-Newman [12].
DEVELOPMENT OF FRACTURE MECHANICS ANALYSIS SYSTEMS
Present VCCM for tetrahedral finite elements enables us to develop automatic fracture mechanics analysis system. In
present investigation, two kinds of advanced methods are being developed. One includes an advanced mesh generation
software in the system VENUS (a product of Technostar Co., Ltd., Japan) [8] and the other is free mesh method
(FMM) [9]. The outlines of the methodologies and their preliminary results are presented.
1. Use of Advanced Mesh Generation Software A fracture analysis system is being developed by integrating an
advanced FEM preprocessing software VENUS [8] (a product of Technostar Co., Ltd., Japan), FEM and VCCM
programs. VENUS has a powerful tool called 3D-mouse that freely cuts and displays the sections of solids. Analysts
(b) Typical finite element model for the problems
of embedded circular/elliptical cracks
(a) The problem of embedded circular/elliptical crack
Figure 5: An illustration for the problems of embedded circular/elliptical cracks and their typical finite element
mesh discretization
(a) Aspect ratio b/a =0.2
(b) Aspect ratio b/a =0.4
(c) Aspect ratio b/a =0.6
(d) Aspect ratio b/a =1.0
Figure 6: The stress-intensity factors computed by present VCCM and the theoretical solutions
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can see and check mesh arrangements and the distributions of stresses and strains inside solid parts by moving and
rotating the 3D-mouse. An example of displaying the sections of complex shaped model is shown in Figure 7.
A capability to add cracks at the surface of and the interior of the solid is now being developed. For example, a surface
crack model is presented in Figure 8. The angle and location can be given by the analyst using a graphical user
interface that is depicted in Figure 9.
2. Free Mesh Method (FMM) Free mesh method (FMM) has been proposed by Yagawa and Yamada [13]. Then, it
evolved and became a powerful analysis tool (see Yagawa [9]). A facture mechanics analysis system based on FMM is
now being developed by Prometech Software, Inc., Japan [14]. FMM is a node based finite element method and no
finite element model needs to be generated prior to the analysis. FMM computer program automatically generates
elements to perform stiffness matrix construction around each nodal point. Analysts do not have to consider about the
mesh at all and, therefore, FMM is considered to be a virtually meshless method. In their development, FMM based
finite element method is being customized so that the program can cope with crack problems.
As a demonstration, a surface crack propagation problem is solved. A plate with a semi-elliptical crack is subject to
tension. The rate of crack propagation is postulated to be governed by the stress intensity factor by the following
equation.
da
= CK m
dt
(10)
Here, the constants C and m are set to be 1.0 × 10 −7 and 2.0 , respectively. The duration of a crack propagation step is
(a) Displaying the arbitrary sections of large scale finite element model
(b) Displaying the stress
Figure 7: Capabilities of 3D-mouse that displays the model and the distributions of stresses of any arbitrary section
(a) Whole model
(b) Close-up of crack
(c) Crack face
Figure 8: Finite element model for a complex structure with a surface crack
Figure 9: A user friendly GUI (Graphical User Interface) of the system under development
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set to be ∆t = 2.0 × 10 4 . Thus, the amount of crack propagation during a step is expressed by:
∆a = CK m ∆t
(11)
It is noted that the values of the constants are postulated ones and their units are not specified here. The Young’s
modulus and Poison’s ratio are 200 GPa and 0.3, respectively.
The shape of the crack evolves as it propagates in the plate, as shown in Figure 10. The finite element meshes that are
generated by using the FMM procedures are shown in Figure 11. The same analyses were performed with linear and
quadratic tetrahedral elements with the displacement method [15] to evaluate the stress intensity factor. The amounts
of crack propagations and the stress intensity factors are compared in Figure 12. The results based on quadratic
tetrahedral elements with proposed VCCM and the displacement method are similar. However, the variation of the
stress intensity factor computed by the displacement method is less stable than that by proposed VCCM. The results
based on the linear and the quadratic elements are quite different although their methods to compute the stress intensity
factor are the same. Therefore, it is concluded that quadratic tetrahedral elements with should be used for
three-dimensional crack analyses.
By using the FMM procedures, the crack propagation analyses can fully be automated.
CONCLUDING REMARKS
In this paper, the development of new fracture mechanics analysis systems based on VCCM are presented. Unlike
traditional fracture mechanics analyses, they use quadratic tetrahedral finite elements.
The results of numerical analyses show that proposed VCCM accurately compute the stress intensity factor. The
capabilities of proposed fracture mechanics analysis system will be as follows.
(1) Three-dimensional crack propagation analyses will fully be automated.
(2) The stress intensity factor will accurately be computed.
(3) Time and labor to carry out the fracture mechanics analyses will drastically be reduced.
Therefore, the systems which are under development will serve as a practical tool to examine the structural integrities
of damaged structures.
Acknowledgements
(a) Plate with a
(b) Initial (a=2.0 mm)
(c) a = 4.10 mm
(d) a = 5.96 mm
(e) a = 7.97 mm
Figure 10: The configuration of a plate with a surface crack and the evolution of crack profile
(a) Initial (a=2.0 mm)
(b) a = 4.10 mm
(c) a = 5.96 mm
Figure 11: Evolution of finite element mesh
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(d) a = 7.97 mm
(a) The variations of crack depths with respect to time
(b) The variations of the stress intensity factors at the
deepest position of the crack
Figure 12: The variations of crack depths and of the stress intensity factors based on the displacement method
and on proposed VCCM
Parts of present developments, that Technostar Co., Ltd. and Prometech Software are conducting, are supported by Japan
Nuclear Energy Safety Organization (JNES). The authors would like to express their sincere gratitude to the support.
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