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Extended Finite Element analysis of a stationary crack in hygrothermal isotropic media subjected to thermal shock

Mohammad Bagher Nazari

^*

, Hamid Rajaei

School of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

* P.O.B. 3618785545, Shahrood, Iran. [email protected]

A RTICLE I NFORMATION A BSTRACT

Original Research Paper Received 21 August 2016 Accepted 02 November 2016 Available Online 04 January 2017

In this paper, the extended Finite Element Method (XFEM) is implemented to compute the Stress Intensity Factors (SIFs) for rectangular media subjected to a hygrothermal loading. In governing hygrothermoelasticity equations, the cross coupling of temperature and moisture fields and temperature- dependent diffusion in some cases are considered. Furthermore, an interaction integral for hygrothermal loading is developed to compute the stress intensity factors. The non uniform mesh of isoparametric eight-nod rectangular element is used in XFEM to decrease the absolute error in SIFs computations. In order to validate numerical results, the SIF of mode I is obtained analytically. The coupled governing equations are firstly decoupled in terms of new variables and then solved by the separation of variable method. According to the results, the moisture concentration gradient has a significant effect on the SIFs so should be considered in the model. Until temperature reaches its steady state, the cross coupling of temperature and moisture synchronizes their time variation which affects on the time variation of SIF. At the beginning of thermal shock, the SIF for shorter cracks is not necessarily less than the longer ones. Also, the mode I SIF for longer and inclined cracks is smaller. On the other hand, considering the moisture concentration as a temperature function increases the time required to reach the moisture steady state.

Keywords:

Extended Finite Element Method (XFEM) Isotropic Materials

Stress Intensity Factors (SIFs) Hygrothermal Loading

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sin sin 2, sin cos2 -3

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Fig. 1 Enrichment nodes in XFEM. Square: crack tip enrichment.

Circle: Heaviside enrichment.

:

1

: .

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) ( )

( 7

=

) - 7 (

=

) - 7 (

= :

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1 0

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(1 ) 0

0 0 (1 2 )

. 2

) ( 1 )

( 11

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=

) - 11 (

_

+

^_

=

Fig. 2 Long enough substance with internal borders

and crack under hygrothermal loading

2

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= 0

1

Table 1 Initial and boundary conditions for temperature and humidity

( , 0) = ( , 0) = (0, ) =

(0, ) = ( , ) =

( , ) =

) ( 18 ]

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,

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C)

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(C + C + 2C )

+(C + C + 2C )

+2(C + C + 2C )

(C + C + C )

2(C + C + C )

(C + C + C )

2(C + C + C )

( 21 )

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(C + C + C )

2(C + C + C )

) ( 23

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(C + C + C )

2(C + C + C )

) ) ( 22 ( 23

) ( 24 ( ) =

Fig. 3 Conversion of the contour integral into an equivalent domain integral (EDI)

3

= + +

Crack

(5)

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) ( 18

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) - 28 (

( = 1, = 0) = 2

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^{( )}

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2 2 sin(2) + 1 + 2 cos (2)

) ( 30 ( , ) = 2 2 sin(2) + 1 2cos (2)

2 2 cos(2) 1 2 sin (2) ) ( 31

) - 31 (

= 2(1 + )

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= 3 4

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= 3

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) - 34 ( T ( , ) = ( , )

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( )

T ( , ) C ( , )

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= (C ( , ) T ( , ))

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C T T + C

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) ( 42 [(1 + )T ( , ) + (1 + )C ( , )]

T ( , ) + C ( , ) )

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=

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+ 2

sin( )

⁽ ⁾

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) ( 48

1

( , ) =

₁

T ( , ) + C ( , )

2

( , ) =

₂

T ( , ) + C ( , ) )

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¹

( , )

₂

( , )

(

₁

+

₂

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^{2 1}

( , ) +

_{1 2}

( , )

(

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+

₂

) - 6

- 2

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) - 50 ( , ) = 0 , ( , = , )

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= 0

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( 52

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+ C ( , )

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T ( , )

+ C ( , ) 3 ( )

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.

"

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) ( 56 ]

.[ 30

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= 2 1 ( , ) ( , )

( , )

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(7)

] [ 31

= 0.3

= 7.78×10

^-7

cm /hr

= 7.78×10

^-6

cm /hr

= 2.68 × 10 cm/cm %H o

= 31.3×10

^-6

cm/cm °C

= 0.5 g / cm °C

= 64.3 GPa

= 0.5 cm °C/g

60 ×240

"

" 4

. .

.

. I .

.

"

" 5

. 42 (MPa m) 1.67

. .

= 0 .1

= 0 .2

= 0 .3

"

6 7

" 8 .

-

. t = 0.02

t = 0.26

= 0.2

"

9 10

" 11 .

t=0.02 .

= 0.2 2.6%

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= 0.3×b 3.16%

. 8%

= 0.2 0.001

) 1%

5

.

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.

"

" 11 .

.

= 0.2

Fig. 4 Eight-node network elements with edge crack and enriched nods

4

_t*

Fig. 5 Coupled and uncoupled maximum SIF in an isotropic plate with an edge crack under hygrothermal loading

5

t*

Fig. 6 time variation of SIF for a crack with length of 0.1 × 0.1 × 6

0.01 0.02 0.03 0.04 0.05

10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 10⁶ 10⁷

Uncoupled Coupled

0 0.2 0.4 0.6 0.8 1

-9E+07 -6E+07 -3E+07 0 3E+07

Analytical a = 0.1× b XFEM a = 0.1× b

(Pam)(Pam)

(8)

_t*

Fig. 7 time variation of SIF for a crack with length of 0.2 × b 0.2 × b 7

t = 0.02 .

t = 0.26

.

"

" 9 T < 0 .

t = 0.26

) - 3 .

t = 0.26

T = 0

. .

.

) ( 56 .

. t = 0.02

( C < 0) ( C > 0)

.

. .

T > 0 )

- 3

. -

) ( - 56 .

. - C

) (- 8 -

) (- 26

t*

Fig. 8 time variation of SIF for a crack with length of . × 0.3 × b 8

b (cm)

Fig. 9 Dimensionless moisture in an isotropic plate with an edge crack under hygrothermal loading

9

_{b (cm)}

Fig. 10 Dimensionless temperature in an isotropic plate with an edge crack under hygrothermal loading

10

= 0.2 60×240

0 0.2 0.4 0.6 0.8 1

-6E+07 -4E+07 -2E+07 0 2E+07 4E+07

Analytical a = 0.2× b XFEM a = 0.2× b

0 0.2 0.4 0.6 0.8 1

-6E+07 -4E+07 -2E+07 0 2E+07 4E+07

Analytical a = 0.3 × b XFEM a = 0.3 × b

0 0.2 0.4 0.6 0.8 1

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

t* = 0.02 t* = 0.26

0 0.2 0.4 0.6 0.8 1

-1 -0.8 -0.6 -0.4 -0.2 0

t* = 0.02 t* = 0.26

(Pam)

(Pam)C*T*

(9)

Fig. 11 Stress in an isotropic plate with an edge crack under hygrothermal loading

11

75×290

"

12

"

.

- 7 - 2

) ( 13

.

= 21 ° C

= 0 °C .

.

= 0.1 .

-

] .

[ 10

.

= 64.3 GPa

= 0.33

= 1.53×10

³

cm /hr

= 31.3×10

^-6

cm/cm ° C

= 1.25×10

⁴

cal/gr mol

= 2.68 × 10

³

cm/cm %H

₂

O

= 0.5 cm ° C/g

= 0.5 g / cm ° C .

) ( 32

_t*

Fig. 12 The first mode SIF for different meshes

12

60×240

)

( 14

= 0.2×b 30

45 60

"

" 15 .

"

" 16 .

) ( 4

)

"

" 17 . .

"

" 15 .

Fig. 13 Long enough substance with internal bordersand angled crack under hygrothermal loading

13

_

Fig. 14 Eight-node elements with edge crack and enriched nods

14

0 0.2 0.4 0.6 0.8 1

-1.5E+08 -1E+08 -5E+07 0 5E+07

1E+08 t* = 0.02

t* = 0.26

0 0.2 0.4 0.6 0.8 1

-6E+07 -4E+07 -2E+07 0 2E+07 4E+07

75 × 290 60 × 240 b (cm)

(Pa)(Pam)

(10)

_t*

Fig. 15 The first mode SIF, in an isotropic plate with an edge crack under hygrothermal loadingwith different crack angles

15

_t*

Fig. 16 The second mode SIF, in an isotropic plate with an edge crack under hygrothermal loading with different crack angles

16

. I 2

.

"

" 16

-8

. .

-

Temperature (K)

Fig. 17 Moisture diffusion coefficient changes according to temperature

17

2

Table 2 Time and amount of maximum stress intensity factors for different angles

(Pa m)

0° 0.033

1.4277 × 10

30° 0.025

1.2251 × 10

45° 0.02

1.1854 × 10

60° 0.0181

6.5134 × 10

.

-

- I

. II

: -9

( , ) = + + + ( )

= 0.46 + 3.06 + 0.84(1 ) + 0.66 (1 )

= 3.52

= 6.17 28.22 + 34.54 14.39

(1 )

^.

5.88(1 ) 2.64 (1 )

= 6.63 + 25.16 31.04 + 14.41 + 2(1 )

^.

+ 5.04(1 ) + 1.98 (1 )

=

-10

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