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X-FEM for Three-Dimensional Dynamic Crack Growth

Kan Ni1, Patrick Hu2, Xiaochen Fan3 Advanced Dynamics Inc. Lexington, KY, 40511

Shaofan Li4 University of California Berkeley, CA 94720

In this paper, the extended finite element (X-FEM) methodology is studied in the modeling and simulation of 3D crack and 3D dynamic crack growth in linear materials. And a 3D X-FEM computer code is developed. The B-bar method to modify the shape function of the 3D brick element is adopted. Stress intensify factors of 3D crack are calculated in X-FEM. Several 2D and 3D crack problems are modeled by X-FEM. In the end, a 3D dynamics crack growth is simulated by using X-FEM. It is found that X-FEM is superior to the traditional FEM in the modeling and simulation of 3D dynamic crack growth.

I. Introduction ATIGUE crack initiation and propagation is one of the most catastrophic and common failure modes in aircraft, marine and offshore, and automotive structures, etc.. In US, the annual cost of fatigue amounted to about 4% of

the gross national product1. The design of durable structures for fatigue damage tolerance requires an understanding of fracture mechanics1-3, in which there are two key issues: (1) the calculation of crack tip parameters such as stress intensity factor (SIF), energy release rate (G), and J-integral; and (2) the prediction and simulation of crack propagation. For real aircraft structural components - due to the complexity of the geometry, boundary, and loading conditions - these issues can only be resolved numerically1. This paper studies a method of the extended finite element (X-FEM) for 3D crack analysis and 3D crack growth under dynamic loading.

II. State of the Art of Numerical Simulation of Crack Growth During the last decades, linear elastic fracture mechanics, especially for mode I cracking, has matured and has

been applied in practical engineering applications. Until now, all numerical simulations of crack growth can be divided into three major approaches - finite element methods (FEM), boundary element methods (BEM), and meshfree methods:

The first two approaches are plagued by the tedious, time-consuming, and expensive remeshing requirement2-4. While the FEM has been widely applied and is a powerful technique for stress analysis, the simulation of crack growth requires a full remeshing, and under large deformation, the mesh will distort. In BEM, only the crack surface needs to be remeshed. Recently, the symmetric Galerkin BEM (SGBEM) and the coupled FEM/BEM approach are two major technologies developed for fracture mechanics analysis, but remeshing is still needed. Therefore, other alternatives should be found to simulate crack propagation.

The third approach avoids remeshing. Three different meshfree methods have been developed related to crack growth: (1) the element-free Galerkin (EFG) method5, (2) the Material Point Method (MPM) 6, and (3) the extended finite element method (X-FEM) 7.

A. Element-Free Galerkin Methods Belytschko et al.5 developed the element-free Galerkin method (EFG), in which moving least-square interpolants

are used to construct the trail and test functions of the weak form variational principle. EFG requires only nodal data and the geometry description, which makes it very convenient for simulating arbitrary crack growth. Belytschko et

1 Principal Scientist, Correspoding Author, kan.ni.08@gmail.com, AIAA Senior Member. 2 President and Principal Scientist, AIAA Senior Member. 3 Research Scientist 4 Professor, Department of Civil and Environmental Engineering, University of California Berkeley.

F

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference<BR> 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-1990

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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al.8 first applied EFG in 2D fracture and static crack growth. Then Belytschko et al.9 presented an EFG method to calculate dynamic stress intensity factors and the modeling for dynamics crack propagation (2D). Later, Sukumar and Belytschko et al.10 proposed a coupled EFG and FEM method for 3D fracture mechanics analysis: domain integral methods were used to calculate 3D stress intensity factors.

These methods were then followed by a number of modified versions proposed by Belytschko et al.: (1) an enriched EFG for crack growth11-12; (2) an arbitrary Lagrangian-Eulerian formulation of EFG for crack growth13; (3) a EFG technique for modeling arbitrary 3D dynamics crack propagation in elastic bodies14; (4) a vector level set instead of the broken line method for 2D crack growth by EFG15; and (5) cracking particles methods for discrete cracks in brittle material16.

Meanwhile, various EFG methods were developed by other researchers. Rao and Rahman17-19 developed an enriched meshless method for 2D nonlinear fracture mechanics - they calculated J-integral and crack opening displacement (COD); Chen et al. 20 developed a meshless method for 2D elastic-plastic crack growth; Chen, et al. 21 developed an enriched EFG method for 3D fracture analysis; and Duflot22 developed an enriched EFG method for 3D elastic crack propagation.

Liu and Li23 developed reproducing kernel particle method (RKPM). In addition, RKPM meshfree method has been applied to ductile fracture and ductile crack growth (Li24 2005).

B. Material Point Method - MPM Sulsky et al. 6,25-26 developed the extension of the particle-in-cell method27 from fluid mechanics to solid

mechanics and established the Material Point Method (MPM). MPM is a particle-based method that is essentially “meshfree.” With MPM, each material is discretized into a mesh of Lagrangian “material points” The material points carry the state variables such as mass, velocity, stress, and strain with them. An Eulerian background mesh is used as a computational pad to assist in the updating of the material state. This methodology can be implemented easily in massive parallelization due to the structured Cartesian background mesh. However, only a few literatures addressed MPM in crack growth28-29 and most of them are 2D cracks.

C. X-FEM As proposed by Belytschko7, the X-FEM for crack growth was based on the Partition of Unity method. In X-

FEM, the singular and discontinuous displacement field along the crack is approximated by a set of enriched shape functions. The enrichment includes the asymptotic function near crack tip and a Heaviside jump function across crack surfaces. In the X-FEM mesh, the crack is not geometrically modeled; thus, the mesh does not need to conform to the crack growth path. As a result, remeshing is avoided while the merit of FEM is retained. Compared to meshfree methods, the X-FEM is much more easily implemented into current commercial FEM packages for the simulation of crack growth.

X-FEM was first applied in 2D and 3D quasi-static elastic crack growth7,32. Thereafter, the level-set method was combined with X-FEM to track the growth of a crack and to locate the crack tip31. After that, some new X-FEM enrichment techniques were proposed for both dynamic crack growth33 and elastic-plastic fracture34. X-FEM has been applied to a wide range of applications, and has shown greater efficiency in the modeling of 3D crack growth compared to meshfree methods.

In summary, on one side, both FEM and BEM have a significant disadvantage in the simulation of crack growth, namely, they are plagued by the time-consuming, cumbersome, inaccurate, and expensive re-meshing requirements. On the other side, meshfree methods such as the element free Galerkin method (EFGM)5 have been proposed, in which tedious remeshing is avoided. However, these meshfree methods are slower than the FEM for stress analysis in the absence of a crack. Therefore, an optimum and economical solution would involve the development of a novel method that not only avoids dealing with re-meshing but also can take the advantage of existing FEM. The extended finite element method (X-FEM)7, recently developed Belytschko, retains the merits of FEM in regions without cracks and uses the Heaviside jump function to represent the strong discontinuities of the displacement field along cracks. To model the surface evolution of fatigue cracks in three dimensions, the X-FEM can be combined with the level-set method31, which can accurately capture evolving fatigue crack surfaces with high fidelity, and without tedious and expensive re-meshing. Therefore, in this paper, we studied X-FEM in the modeling and simulation of 3D crack and 3D crack growth under dynamics loading.

III. 3D X-FEM For the linear elastic crack problem, the classical X-FEM enriched displacement field approximation, as

proposed by Belytschko et al.7, is as follows:

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4

1

( ) ( )h li i j j k k l

i I j J k K l

N N H N F

u u b x c x (1)

where iu are the traditional FE displacement components of node i , iN is the shape function associated with node i , jb represents the displacement jump across the crack, ( )H x is the Heaviside jump function along the crack, l

kc are the additional degrees of freedom associated with the crack tip enrichment function, lF , I is the set of all nodes in the domain, J is the set of nodes whose shape function is cut by the crack, and K is the set of nodes whose shape function contains the crack tip. The crack tip enrichment function lF , is defined by the asymptotic crack tip displacement in the classical linear elastic fracture as follows:

( ) cos , sin , sin sin( ), cos sin( )2 2 2 2

F x r r r r

(2)

where r and are the usual crack tip polar coordinates. Once an enrichment strategy has been established, the stiffness of enriched FE elements will be added to the global stiffness matrix. In order to assure that the stiffness of enriched elements is accurate, a numerical integration will be performed for those enriched elements that are cut by the crack. Then, the displacement, stress, and strain of all nodes will be calculated in a manner that is similar to the standard FEM method.

For 3D X-FEM, B matrix of the shape function for 3D brick finite element is modified by the B-bar method35. Herein we adopt B-bar method too. The volumetric average in the deformation gradient operator is

, ,

1i x i xv

N N dvv

(3)

Then define

, , ,( ) / 3i x i x i xN N N (4)

The modified B-matrix for the unenriched nodes becomes as

, , , ,

, , , ,

, , , ,,

, ,

, ,

, ,

0

0

0

i x i x i y i z

i x i y i y i z

i x i y i z i zi con

i y i x

i z i y

i z i x

N N N N

N N N N

N N N NBN N

N N

N N

(5)

For the jump-enriched nodes, the extra terms of the B-matrix are , ,i jump i conB HB , and the B-matrix is:

, ,j j con j jumpB B B (6)

For the tip enriched nodes the extra terms of the B-matrix are:

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, , , , , , ,

, , , , , , ,

, , , , , , ,, ,

, , ,

( ) ( )

( ) ( )

( ) ( )

i x i x j i i j x i y j i j y i z j i j z

i x j i j x i y i y j i i j y i z j i j z

i x j i j x i y j i j y i z i z j i i j zi j tip

i y j i j y i x j

N N F N N F N F N F N F N F

N F N F N N F N N F N F N F

N F N F N F N F N N F N N FBN F N F N F

,

, , , ,

, , , ,

0

0

0

i j x

i z j i j z i y j i j y

i z j i j z i x j i j x

N F

N F N F N F N F

N F N F N F N F

(7)

where

( , ) { sin , cos , sin sin , sin cos }2 2 2 2jF r r r r r (8)

2 2( , , ) ( , , )r , 1 ( , , )tan ( )

( , , )

(9)

The level set fields { , } in the X-FEM domain are interpolated by the shape functions:

( , , ) ( , , )i iiN (10)

Finally the B-matrix for tip-enriched nodes can be written as:

, ,1, ,2, ,3, ,4,T T con T tip T tip T tip T tipB B B B B B (11)

where T  denotes the nodes with tip enrichment.

To obtain the derivatives of the branch functions, the chain rule is used.

1

, , ,1

, , ,

, , ,

j x j j

j y j j

j z j j

x y z

F F Fx y z

F F J F

F F Fx y z

(12)

, , ,,1 1

, , ,,

, , ,

jj r

jj

j

F rF

J F J rF

F r

(13)

where

1,

1sin

22rF

r

, 1, cos

2 2

rF

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2,

1cos

22rF

r

, 2, sin

2 2

rF

(14)

3,

1sin sin

22rF

r

, 3,

1( cos sin sin cos )2 2 2

F r

4,

1cos sin

22rF

r

, 4,

1( sin sin cos cos )

2 2 2F r

therefore

, , , ,, ,1 1

, , , ,, ,

, , , ,

rr

J r Jr

r

(15)

where

,r r

, ,r r

, , 2r , , 2r

(16)

, ,i iiN , , ,i ii

N , , ,i iiN

, ,i iiN , , ,i ii

N , , ,i iiN (17)

Therefore

, ,, , 21

, , , ,

2, , , ,

i x i i x ii i

i y i i y ii i

i z i i z ii i

N Nrr rJ r N N

r N N r r

(18)

Finally, from the enriched B matrix, we can obtain the element stiffness matrix in 3D X-FEM.

In addition, we adopt the slicing schemes proposed by Sukuma36 for numerical integration of completely or partially cut elements. In X-FEM, for an element being by a crack surface, its volume is divided into sub-regions as shown in Figure 1 bellow, in which black lines are cracks, and tetrahedrons constituted by red lines are sub-elements.

Figure 1. Partition a 3-dimensional finite element by a crack plane

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IV. SIF of 3D Crack In the last three decades, various 3D FEM methodologies have been developed to calculate the stress intensity factor, energy release rate, and J-integral for a crack. These methods include the displacement extrapolation method to calculate K, the quarter point singular elements to calculate K, the virtual crack model to calculate the energy release rate G, and domain integral methods to calculate the J-integral, etc.3.

In this paper, we adopt the relatively simple method to calculate the stress intensity factors (SIF) for mode I 3D crack. The relationship between the KI and crack tip opening displacement is

2

2

4(1 )

EK u

r

(19)

where u is displacement jumps at a point on the crack tip surface:

1( ( , , ) ( , , ))

2 i iiu N H x y z H x y z b (20)

( , , )H x y z and ( , , )H x y z are coinciding points associated with to the positive and negative surfaces of

a crack, respectively:

( , , ) 1H x y z , ( , , ) 1H x y z (21)

Therefore

i iiu N b (22)

where ib is the value of the Heaviside enrichment DOF.

For increasing K precision, a least square method is employed:

2

1 1 1 1

2

1 1

( )

N N N N

i i i i ii i i i

N N

i ii i

r K r r KK

r N K

(23)

where N is the number of points on the crack plane at which stress intensity factors are computed.

V. Validation Examples

A. X-FEM modeling of a 2D Crack We first used X-FEM to model a 2D crack problem as shown in Figure 2.

Table 1. SIF in example 1

SIFs( 1/2Mpa m ) Exact solution 11 23 meshes

11 23 meshes*

23 47 meshes

23 47 meshes*

K1 34.0 32.75 33.72 33.54 34.03

K2 4.55 4.48 4.52 4.51 4.53

Remark: * means that there are two tiers of elements that have crack tip enrichments.

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Parameters: L=16m, w=7m, a=3.5m, τ=1MPa (stretched distributed loads), E=10000MPa, μ=0.3. The X-FEM calculation of SIF is presented in Table 1 compared with the theoretical values. It can be found they

agree with each very well.

(a) Mesh (b) Deformation

Figure 2. X-FEM modeling of a 2D crack

B. X-FEM Modeling of 2nd 2D Crack We used X-FEM to study another 2D crack problem as shown in Figure 3.

Parameters: w=10m, σ=1MPa, E=20000MPa, μ=0.3 The theoretical solution of SIF is

2K1=σ πacos (β) ; K2=σ πasin(β)cos(β)

The SIF calculated by X-FEM and comparison with theoretical values are present in Table 2. We also can see they agree with each other very well.

(a) Mesh (b) Deformation Figure 3. X-FEM modeling of 2nd 2D crack

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Table 2. SIF in example 2

oβ=30 a K1 K2

Theory 0.94 0.54 X-FEM

0.5 0.96 0.54

Theory 1.33 0.77 X-FEM

1.0 1.40 0.78

C. X-FEM Modeling of a 3D Crack We used X-FEM to model a 3D crack problem as shown in Figure 4.

Parameters: 1000E , 0.3 , 1 , t=1, meshes are 23×30×47.

The theoretical SIF is Y=K

a , the theory and X-FEM result is compared in Figure 5.

(a) Mesh (b) Deformation

Figure 4. X-FEM of a single edge-crack tension specimen.

2.00

2.25

2.50

2.75

3.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

a

YXFEM

Plane Strain

Figure 5. X-FEM calculation of 3D SIF

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VI. 3D X-FEM Simulation of Dynamic Crack Growth Simonsen and Tornqvist37 presented an experimental set, which can govern large scale ductile fracture under in-

plane bending and stretching in a thin shell structure. This experiment can measure the data that is easy to interpret the crack propagation process and calibrate with engineering material model. The experimental set-up and specimen are shown in Figure 6. The main values measured by experimental are:

P: The crosshead load from the testing machine; : The crosshead displacement from the testing machine.

Figure 6. Experiment set-up37

(a) (b)

(c) (d)

Figure 7. X-FEM simulation of elastic 3D dynamic crack growth

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Both 3D meshfree method and 3D X-FE modeling and simulation of this dynamic crack growth have been carried out. The result of meshfree method is reported in the other two AIAA SDM 2011 papers38,39. Herein we only present the 3D X-FEM modeling and simulation of dynamic crack growth for the linear elastic material in Figure 7. It is found that X-FEM can be easily and conveniently applied in the modeling and simulate of elastic crack growth under dynamics loading. However, for ductile crack of nonlinear material, X-FEM has to be combined with other technologies such cohesive zoo model.

VII. Conclusions In this paper, we have studied X-FEM for 2D and 3D crack analysis and have developed 2D and 3D X-FEM

computer codes. We applied X-FEM in a 3D dynamic crack growth. It can be concluded that: (1) X-FEM can provide accurate results for 2D and 3D fracture simulations for linear elastic material; (2) X-FEM can provide accurate simulations for dynamic loading and subsequent dynamic crack growth for

linear elastic material; (3) X-FEM is superior to the traditional FEM, because the tedious remeshing is no longer needed; (4) X-FEM can easily and conveniently simulate 3D elastic crack growth. However, for ductile crack of

nonlinear material, X-FEM needs to be combined with other technologies such as cohesive zoo model. (We have developed meshfree 3D RKPM meshfree methodology for ductile crack growth under dynamic impact

loading, and this method is reported in the other two papers38,39 of AIAA SDM 2011.)

Acknowledgement This study was supported by NAVY STTR Phase I contract N00014-10-M-0253, (Technical Monitor: Dr. Paul

Hess and Dr. Ken Nahshon ). This support is gratefully acknowledged.

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38Li, S., Ren, B., Ni, K., and Hu, P., “Meshfree Method for Ductile Fracture with Modified Gurson’s Model”, 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, April 4-7, 2011, Denver, Colorado, AIAA-2011-2109.

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