Effect Analysis of Gurson Model Parameters on Crack Extension of Pipeline

doi:10.4236/jamp.2013.16011

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Journal of Applied Mathematics and Physics, 2013, 1, 54-56

Published Online November 2013 (http://www.scirp.org/journal/jamp)

http://dx.doi.org/10.4236/jamp.2013.16011

Open Access JAMP

Effect Analysis of Gurson Model Parameters on Crack

Extension of Pipeline

Tieping Li, Xinlu Tian

Nuclear and Radiation Safety Center of MEP, Beijing, China

Email: [email protected], [email protected]

Received August 2013

ABSTRACT

Leakage-before-break technique is widely used in high energy pipeline of nuclear plant, for which crack stability of

pipeline under complex loading condition is a key issue, and crack growth resistance curve of pipeline material is the

important foundation for crack stability analysis. In this paper, ferritic steel A533B is studied, Gurson damage model is

used to simulate crack process of contact tension specimen under uni-tension, and effect of Gurson mod el parameter on

simulation result is discu ssed. The following results are found dur ing simulation: initial porosity factor

is the main

parameter, when it increases gradually, unstable crack extension will be observed; however, only initial J toughness is

affected by critical po rosity factor

; the minor parameter is loa d step control

, when it increases, stable and con-

vergent result is obtained. All results in this paper can be used to determine parameters in Gurson mode, which will be

foundation for cra c k extension anal y s i s of pi pe line.

Keywords: Leakage-Before-Break; Crack Growth Resistance Curve; Gurson Damage Model; Crack Extension

1. Introduction

In present, leakage-before-break (LBB) technique is

widely used for collant pipeline in most operating and

abuilding nuclear plants. In LBB technique, leakage can

be detected in pipeline before unstable crack extension

occurring in pipeline, and protective hardware and whip

restraints can be removed for convenience of inspection

operability, from which economic efficiency can be

guaranteed. Since austenitic steel with high toughness is

used for nuclear pipeline, elastic-plastic fracture is the

main failure mode. Elastic-plastic fracture is a complex

process, and fracture toughness

cannot be adapted

directly to fracture analysis in complex structure [1].

Combination of fracture test and material model is pro-

posed by some researchers since fracture mechanism is

required by fracture process simulation [2], material

model parameters can be deduced from combination re-

sult [3], which can be used to simulate fracture process in

any complex structure.

Gurson model is proved to simulate fracture process

well for metallic material from a lot of tests and theoretic

studies [4]. Contact tension specimen made of ferritic

steel A533B is studied in this paper, Gurson model is

combined to describe micro fracture process in A533B

material, and effect of Gurson model parameter on simu-

lation results is analyzed by finite element method.

2. Gurson Model

Metallic material is divided into two parts in Gurson

model: basement material and porosity. In itial porosity is

assumed as

, when material is loaded, bigger poro sity

is formed by connection of smaller porosity, and new

crack surface is obtained at the same time. Material fail-

ure can be judged by the following equation:

( )

2 cosh10

g qfqf

σσ

 

=+−+ =

 

 

(1)

In Equation (1),

is regarded as equivalent stress of

the whole material, m

is hydrostatic pressure,

equivalent stress of basement material,

is real-time

porosity rate of the whole material. When

is larger

than

, which is defined as the critic porosity, crack

extension will be found. In above equation, q1, q2, q3, is

aspect ratio of porosity, which can be deduced by

stress-strain curve of uni-tension test [5].

3. Simulation Analysis

3.1. Material Property

The stress-strain curve of ferritic steel A533B can be

described as the following relationship:

T. P. LI, X. L. TIAN

Open Access JAMP

σσ

εσ



=



(2)

In Equation (2),

is initial yielding stress of ma-

terial, N is strain hardening index, E is young’s modulus,

and for material of A533B, parameters defined in Equa-

tion (2) are fixed a s

400,2 10,0.1MPa EMPa N

= =×=

3.2. Finite Element Model

According to ASME fracture toughness test standard

E1820 [6], the geometric character of contact tension

specimen can be plotted as Figure 1: w is 50.8 mm as

width of specimen, B is 127 mm as depth of specimen,

and size of the remanent part can be deduced by the pro-

portion in Figure 1. Geometric size in Figure 1 is used

to build the finite element model.

Software Warp3d is used here for model creation and

element C3D8 is selected. Since plastic deformation is

included in simulation, crack tip singularity is not consi-

dered in this paper. Plot in Figure 2 is the half model of

contact tension specimen, one-layer element with Gurson

model property is assigned to the crack face, and the

elastic-plastic material relationsh ip in Equation (2) is still

suitable for the remanent part. In this paper, some para-

meters such as aspect ratio and Gurson element height

are fixed in simulation as

12 3

1.46, 0.931, 2.131qq q= ==

and

200Dm

, where

is the height of element

with Gurson model property.

The boundary condition is: all node freedom in depth

direction is fixed; node freedom in vertical direction for

symmetric surface except crack face is fixed; in order to

remove rigid displacement of the whole model, one of

the nodes far from loading position and crack surface is

Figure 1. Schematic plot of contact tension specimen.

Figure 2. Finite element model of contact tension specimen.

fixed in horizontal direction. Displacement loading me-

thod is used.

3.3. Calculation Results

Software Warp3d is used to simulate fracture process of

contact tension specimen. Effects of initial porosity

critic porosity

and load control parameter

em-

bedded in Warp3d on fracture process are studied in si-

mulation.

1) Effect of load control parameter

value

For the simulation of this part, initial crack length ratio

/ 0.3aw=

, initial porosity rate is

0.005f=

, critic

porosity ratio is

0.2

and only load control para-

meter is changeable. It is shown in Figure 3 that, curve

value vs. displacement loading is affected little by

load control parameter, and the specimen is perfect now

with no crack extension. As displacement

∆

increasing,

curves are distinct with different

: smaller

value

is related with stable crack extension, such as there is a

platform in curve corresponding to

0.005

; curves

with larger

are related with unstable crack extension

since there is a downtrend in these curves.

2) Effect of critic porosity

For the simulation of this part, initial crack length ratio

is defined as

/ 0.5aw=

, load control parameter is de-

fined as

0.005

, initial porosity is defined as

0.003f=

, and only critic porosity

is changeable.

It is shown in Figure 4 that almost no effect of

result can be observed.

3) Effect of initial porosity

For the simulation of this part，initial crack length ratio

is defined as

/ 0.5aw=

, load control parameter is de-

fined as

0.005

, critic porosity is defined as

0.2

, and only initial porosity

is changeable. It

is shown in Figure 5 that curves of

value vs. dis-

placement loading are coincident with smaller loading.

As loading

∆

increases, yielding is found around crack

tip since smaller

(

0.001f=

) corresponding to

more perfect material, and there is an uptrend for

value because yielding zone is expanding. As

in-

creases, there is a downtrend in curve, since material

with larger porosity rate is fragile, and relaxative stress

can be observed around crack tip with crack surface ex-

tension.

T. P. LI, X. L. TIAN

Open Access JAMP

012345

100

150

200

α=0.005

α=0.01

α=0.02

α=0.05

J(N/m)

∆(mm)

Figure 3. Effect of load-control parameter on calculation

results.

010 20 30 40 50

100

150

200

250

J(N/m)

fN=0. 15

f N=0. 20

f N=0. 25

P(KN)

Figure 4. Effect of critic porosity on calculation.

012345

100

150

200

250

300

350

∆(mm)

J(N/m)

f 0=0.001

f 0=0.003

f 0=0.004

f 0=0.005

Figure 5. Effect of initial porosity on calculation results.

4. Conclusions

The fracture process of contact tension specimen made of

ferritic steel A533B is simulated by finite element me-

thod combining with Gurson damage model, and effect

of parameters in Gurson model on simulation result is

studied, from which the following conclusions can be

obtained:

1) The initial porosity rate is the leading role for simu-

lation result, when

increases, there is an unstable

trend for crack extension, and only

value in initial

loading period is affected by critic porosity rate

2) With decreasing load control parameter of

, sta-

ble and convergent crack extension can be observed.

5. Acknowledgements

This work was financially supported by Natural Key

Projects of China (Grant No. 2011ZX06002-010).

REFERENCES

[1] Z. R. Huang and Y. K. You, “Analysis of the Load-Dis-

placement Relationship to Determine the JR Resistance

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[2] T. L. Anderson, “Fracture Mechanics: Fundamentals and

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http://dx.doi.org/10.1016/0022-5096(94)00064-C

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