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Vol.13 (2010 8 )

XFEM FEM

Implementation of XFEM into a general-purpose FEM code and simulation of fatigue crack propagation in actual structure

* ** *** **** *****

Kazuki SHIBANUMA, Hiroki AOI, Tomoaki UTSUNOMIYA, Masahiro SAKANO and Yoshihiro NATSUAKI

* -DC 615-8540 ** 615-8540

*** 615-8540 **** 564-8680 3-3-35

***** 550-0005 1-8-2 5

In order to evaluate the behavior of a fatigue crack in the local part of complex and large-scale civil engineering structures such as orthotropic steel deck, we develop a fatigue crack simulation code by implementation of PU-XFEM into the general-purpose FEM analysis software. We have adopted the multiple-nodes to increase the nodal degrees of freedom. This makes possible to implement the PU-XFEM approximation to various software without any changes of the definitions in the convention rules. The performance of the developed code is evaluated through verification of the numerical accuracies of fracture mechanics parameters using basic models. Then, the developed code is applied to the numerical simulations of the fatigue crack propagation in the orthotropic steel deck specimen using bulb rib, as an example. It is concluded that the developed PU-XFEM analysis code is useful for the quantitative evaluation of fatigue propagation in actual structure.

Key Words: XFEM, PU-XFEM, implementation, fatigue, crack propagation, actual steel structure

1.

1)

FEMFEM

(1)

(2)

2) (3)(1) (2)

FEM3), 4), 5)

eXtended FEM;

XFEM 6) FEM

XFEMFEM

FEM

XFEM

Blending Elements; BE7), 8)

PUFEM 9) XFEMPU-XFEM BE

10), 11) FEM

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応用力学論文集 Vol.13, pp.945-956　 （2010 年 8 月） 土木学会

12), 13) FEM

PU-XFEM

XFEM FEMXFEM

Giner 14) 2XFEM FEM

2

3XFEM FEM

FEM

ABAQUS 15) PU-XFEM3

ABAQUS

ABAQUS XFEM(1)

XFEM XFEM

BE (2)3

2. PU-XFEM

III III 3

-1

I IIIII

III

3

PU-XFEM

PU-XFEM 11)

PU-XFEM

(1)

Partition of Unity PU

(2)

PU

(3)

(2) (3)

4

-1

- 946 -

DRCRC

CC

: C : C' : J : C' and J

1 XFEM

(4)

(5)

(4) Heaviside 16)

(5)

17)

PU-XFEM

XFEM PU

(2) (3)

(6)

(7)

PU-XFEM

PU

(8)

(6) ~ (8)-2

PU-XFEM (6)

~ (8)

3 PU-XFEM

11) 3. FEM ABAQUS

FEMABAQUS 15) PU-XFEM

3

ABAQUS

PRE-PRO: FEM PU-XFEM

POST-PRO:

XFEM3DS:

3.1XFEM3DS

3.2PRE-PRO 3.3 POST-PRO

3.2 XFEMFEM

ABAQUS

ABAQUS

ABAQUSABAQUS User’s Manual 15)

-2 PU-XFEM

- 947 -

ABAQUS CAE( INPUT.inp )

Xn.dat

PRE-PRO*USER ELEMENT

XFEM

POST-PRO

ABAQUS CRACK.dat ( )

XFEM

(a)

(b), (f)

(c)

(e)

(h)

ABAQUS SOLVER(d)

3.1

-3 FEMABAQUS PRE-PRO ABAQUS

PU-XFEM POST-PRO

XFEM3DS

(a) ABAQUS CAEFEM

FEM

INPUT.inp (b) (a)

Crack.dat (c) (b) (f)

Crack.dat (a)INPUT.inp PU-XFEM

Xn.inp nPRE-PRO

(d) (c) PU-XFEMXn.inp FEM

Xn.dat (e) (d) Xn.dat

POST-PRO

-1 ABAQUS 15)

1 – 3 x, y, z - 4 - 6 x, y, z -

7 8 , 9 10 11 12 2 13 3 14 Etc.

(f) (e)

Crack.dat (g)

(c) ~ (f)

(h) TECPLOT 18)

DSn.dat STn.dat PU-XFEM FEM

ABAQUS

3.2 FEM PU-XFEM

FEM

PU-XFEM FEM XFEM

ABAQUS

*USER ELEMENT FEM XFEM

ABAQUS-1

XFEM

(1)(2)

PU-XFEM FEM

-3 ABAQUS PU-XFEM

XFEM3DS

- 948 -

-2 XFEM

(a) 4

4 24 (4 × 6)

(b)

- node + 4 + 12 (4 × 3) node + 4 + 12 (4 × 3)

-node + 1 + 3 (1 × 3) - node + 5 + 15 (5 × 3)

ABAQUSPRE-PRO

(1)

1ABAQUS

*USER ELEMENT PU-XFEM

36 3 3

24 4

2PU-XFEM

3

-2 PU-XFEM(6) ~ (8)

-2

-43

6 42

1 + 1 + 6 + 2 = 10 6 + 6 + 21 + 9 = 42

: C : C' : J : C' and J

3

: 1: 6

: 10: 42

4

1 2: 1

: 6

: 6: 21

: 2: 9

FEM

XFEM

FEM

3

ABAQUS

FEM

(2) PU-XFEM

XFEMABAQUS

*USER ELEMENT *MATRIX

(1)

ABAQUS PU-XFEMPRE-PRO

(i) ABAQUS CAEFEM INPUT.inp

-CRACK.dat

(ii)

-2(b)

-4 *USER ELEMENT XFEM

- 949 -

Crack Tip

Crack Tip

aac

(iii)

(iv) ABAQUS CAE

INPUT.inp (iii)*USER

ELEMENT *MATRIX

Xn.inp XFEM

Xn.inp ABAQUS

3.3

POST-PRO POST-PRO

XFEM 2

M19) M

M 19), 20) PU-XFEM M

11)

(9)

(9)

M

21)

(10)

22) I II

23)

(11)

(12)

MPa 24)

-5

(12) (10) (12)

4.

1PU-XFEM FEM

-5

- 950 -

= 1

L = 16

a = 3.5

W = 7

10), 11) PU-XFEM

PU-XFEM

4.1 I II

4.2

FEMABAQUS

11)

3

3

2 4.1

-6 I II

Model-1

(12)

19)

(13)

M (9)XFEM16)

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

1.0E-02 1.0E-01

Erro

r (%

)

Normalized Mesh Size: h/a

std. XFEM (Present)PU-XFEM (Present)std. XFEM (Reference )PU-XFEM (Reference )

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

1.0E-02 1.0E-01

Erro

r (%

)

Normalized Mesh Size: h/a

std. XFEM (Present)PU-XFEM (Present)std. XFEM (Reference )PU-XFEM (Reference )

-20.0

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

1.0E-02 1.0E-01

Erro

r (%

)

Normalized Mesh Size: h/a

std. XFEM (Present)PU-XFEM (Present)std. XFEM (Reference )PU-XFEM (Reference )

-6 Model-1

(a)

(b)

(c)

-7 Model-1

11)

11)

11)

11)

11)

11)

- 951 -

a

S = 4W

W

P

-7

-7(a) PU-XFEM

0.5%0.3 2.0%

11)

0.31.0%

1.0%

FEM25)

XFEM

0.3 5.5%3.0%

XFEMPU-XFEM

-7(b) PU-XFEM

0.5%11)

XFEM

4.0% -7(c)

PU-XFEM

% ~ .5%11)

% ~ 8.3%

XFEM PU-XFEM

PU-XFEM

ABAQUS XFEMXFEM

ABAQUSXFEM

4.2

-8 3Model-2

26)

(14)

-7.5

-5.0

-2.5

0.0

2.5

5.0

7.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Erro

r (%

)

Normalized Crack Length: a/W

PU-XFEM (Present)

PU-XFEM (Reference )

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0E+00 5.0E+11 1.0E+12 1.5E+12

Nor

mal

ized

Cra

ck L

engt

h: a

/W

Cycle of Loads: N

PU-XFEM (Present)

PU-XFEM (Reference )

-8 3 Model-2

-9 Model-2

-10 Model-2

11)

11)

- 952 -

0.2% 3720 3570

M 4.1

(9)

-9 -10

11)

-9

1.6% 0.9%

11)

PU-XFEM

5. 27)

-128)

-1128)

3 5

-13(b)(c) -12

= 33.05mm -12

Young 200GPa Poisson 0.3

-13

(c) (b) A

(a) (b) A

P

2000mm

622mm

3200mm

A

0 20 (mm)40

-11 -12

- 953 -

(a) a0 = 9.30mm (b) a = 15.15mm (c) a = 33.50mm

ABAQUS CAE-13 =

20mm

-13(b) A -13(c)10 1

= 3mm

XFEMFEM

-13(a)

1.80 × 106 cycle-16

CRACK.inp = 1.0 × 105 cycle =

9.30mm -14(a)

-143

-15

-162

24)

XFEM

= 9.3mm 0.07%ABAQUS

CPUCPU

CPU

0.0E+00

1.0E+01

2.0E+01

3.0E+01

8.0 16.0 32.0

K(M

Pam

)

Crack length : a (mm)

8.0

16.0

32.0

8.0 16.0 32.0 64.0 128.0 256.0

Cra

ck le

ngth

: a(m

m)

Cycle of loads : N ( 106 cycle)

Numerical SimulationFatigue Test

-14

[ 104 cycle] [kN]0 < N < 70 140

70 < N < 90 200 90 < N <110 280

110 < N <140 140140 < N <150 200150 < N 280

P

-16 -15

29)

- 954 -

XFEM FEM

6.

FEMABAQUS

(1) PU-XFEM

(2)

(3)

ABAQUS

XFEM

ABAQUSFEM

PU-XFEM

11)

FEMPU-XFEM

3

PU-XFEM FEM

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:

, 62 , I ,

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2010 3 9

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