Research on the Plane Multiple Cracks Stress Intensity Factors Based on Stochastic Finite Element...

8/20/2019 Research on the Plane Multiple Cracks Stress Intensity Factors Based on Stochastic Finite Element Method

http://slidepdf.com/reader/full/research-on-the-plane-multiple-cracks-stress-intensity-factors-based-on-stochastic 1/5

Chinese

Journal of

Aeronautics

Chinese Journal of Aeronautics 22(2009) 257-261 www.elsevier.com/locate/cja

Research on the Plane Multiple Cracks Stress Intensity Factors

Based on Stochastic Finite Element Method

Xue Xiaofeng*, Feng Yunwen, Ying Zhongwei, Feng Yuansheng

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

Received 28 March 2008; accepted 10 June 2008

Abstract

Taylor stochastic finite element method (SFEM) is applied to analyze the uncertainty of plane multiple cracks stress intensity

factors (SIFs) considering the uncertainties of material properties, crack length, and load. The stochastic finite element model of plane multiple cracks are presented. In this model, crack tips are meshed with six-node triangular quarter-point elements; and

other area is meshed with six-node triangular elements. The partial derivatives of displacement and stiffness matrix with respect

to all the random variables obtained by Taylor SFEM are derived. Meanwhile, the mean and variance expressions of SIF under

uncertain factors are also derived. Parallel double-crack illustrative example of using the proposed method is given. The calcula-

tion results indicate that the uncertainty of SIF is influenced by the distance of the cracks, the smaller the distance of cracks is,

the greater the SIF uncertainty is, and the uncertainties of material elastic modulus, load and crack length markedly affect the

uncertainty of SIF; and the Poisson ratio of material has little influence. Among the variables, the elastic modulus has the great-

est effect on SIF uncertainty. The next is external load. The crack length has lower effect on SIF uncertainty than both the elastic

modulus and external load.

Keywords: multiple crack; uncertainty; stochastic finite element; stress intensity factor

1. Introduction*

At present, the study on multiple cracks issue is fo-

cused on fatigue multiple cracks propagation and re-

sidual strength under the influences of deterministic

factors[1-4]

. However, the uncertainty of fracture pa-

rameters for multiple crack structure factors has rarely

been studied. Researches show[5-6] that geometrical and

physical random variables of structure, such as load,

material property, and crack length, strongly influence

the fracture parameters of structure with single crack.Hence, the fracture parameters should be considered as

random variables, and then the result of research for

multiple cracks issue will be more reasonable. Because

of the complexity of the calculation of multiple crack

fracture parameters and the uncertainty of the interac-

tion of cracks, no report about the uncertainty of mul-

tiple crack has appeared yet currently. Meanwhile,

*Corresponding author. Tel.: +86-29-88460383.

E-mail address: [email protected]

Foundation items: National Natural Science Foundation of China

(10577015); Chinese Aeronautics Foundation (2006ZD53050,03B53008)

1000-9361© 2009 Elsevier Ltd.

doi: 10.1016/S1000-9361(08)60096-5

multiple cracks on aging aircraft often lead to severe

fatigue damages[7-9]. Stochastic finite element method

(SFEM) is used to explore the uncertainty of plane

multiple cracks and the corresponding influence fac-

tors.

2. Analysis of Influence Factors with Uncer-

tainties for Multiple Cracks

The stress, displacement, and stress intensity factors

(SIFs) of crack tip should be considered as randomvariables during establishing the fracture mechanics

model of multiple cracks structure. The uncertainty of

fracture parameters is coming from geometrical and

physical uncertainties of structure with cracks. Geo-

metrical uncertainty includes structure pattern, crack

pattern, and length; Physical uncertainty includes ma-

terial properties, load, and so on. Considering all the

parameters as random variables, the fracture parameter

calculation of multiple cracks using analytic method is

very difficult, even difficult to be realized. Hence, the

numerical method should be used. In recent 20 years,

SFEM as a numerical method is extensively applied to

dealing with structure reliability and nonlinearity is-

sues because the random variables can be considered

by SFEM[10-16]

. In recent years, SFEM is graduallyOpen access under CC BY-NC-ND license.

http://creativecommons.org/licenses/by-nc-nd/4.0/







· 258 · Xue Xiaofeng et al. / Chinese Journal of Aeronautics 22(2009) 257-261 No.3

applied to fracture reliability analysis, but it is still

rarely used to solve the multiple-crack problem[17-18].

The principal difficulty is how to calculate the interac-

tion of cracks[19-20]

. The procedure of calculating the

fracture parameter uncertainty with SFEM is as fol-

lows: meshing the interacting area of cracks prop-erly; calculating the mean and variance of each

node displacement; introducing the mean and vari-

ance into the finite element formulas of fracture me-

chanics.

3. FEM Calculation of Plane Multiple Crack

SIFs

In the finite element analysis of fracture mechanics,

meshing the crack tip with 12 singular elements is the

current method as shown in Fig.1(a). For the arbitrary

plane crack shown in Fig.1(b), the displacement of

crack tip in global coordinate system should be trans-

formed into crack tip coordinate system.

sin cos1,2,3,4,5

cos sin

i i i

i i i

u u vi

v u v

(1)

Fig.1 Meshing of general plane crack tip and coordinate

transformation of node displacement at crack tip.At crack tip node, the relationship between SIF and

displacement is:

I 1

2

(1 )( 1)

E K v

k r

II 1

2

(1 )( 1)

E K u

k r

At crack tip node 1, r is 0, and SIF cannot be solved

directly by the above mentioned formulas but can be

solved by the extrapolation of nearby nodes. The pro-

posed calculation formulas of SIFs are utilizing the

displacements of 4 nodes on the crack plane of singu-

lar element as shown below:

T T

I 1 5 4 2 3 2, , K M v v M v v (2)

T T

II 1 5 4 2 3 2, , K M u u M u u (3)

where1

1

2 1 2

2(1, 1)

2( )(1 )( 1)

r E M

r r k r

22

2 1 1

2(1, 1)

2( )(1 )( 1)

r E M

r r k r

(3 ) / (1 ) plane stress

3 4 plane strain

v vk

v

iu and iv are the x' and y' direction displacements of

node i at crack tip, respectively.

The mean and variance of SIFs are:

T T

I 1 5 4 2 3 2[ ] , , E K M v ' v ' M v ' v ' (4)

5 5T

I 1 1

4 4

3 3T

2 2

2 2

Var[ ] Cov ,

Cov ,

i j

i j

i j

i j

K M v v M

M v v M

(5)

T T

II 1 5 4 2 3 2[ ] , , E K M u ' u ' M u ' u ' (6)

5 5T

II 1 1

4 43 3

T2 2

2 2

Var[ ] Cov ,

Cov ,

i j

i j

i j

i j

K M u u M

M u u M

(7)

where E [ ] is the mean, Var[ ] the variance, and Cov( )

the covariance, iu ' and iv ' are the x' and y' direction

displacement means of node i at crack tip, respectively,

and the mean and variance of which can be obtained

by SFEM. For the crack tip of plane multiple cracks,

the mean and variance of SIFs can be obtained by

executing the calculation mentioned above.

4. SFEM Model of Plane Multiple Cracks

Taylor SFEM is a kind of widely used SFEM



No.3 Xue Xiaofeng et al. / Chinese Journal of Aeronautics 22(2009) 257-261 · 259 ·

method for its simple principle and good precision[21]

.

The uncertainties of parameters involved in the plane

multiple cracks SFEM model in this article are uncer-

tainty of elastic modulus E , Poisson ratio , load P ,

and cracks length [a1 a2 ··· an], respectively.

Six-node triangular quarter-point element is used tomesh each crack tip, and six-node triangular element is

used to mesh other area.

4.1. Taylor SFEM

IfT

1 2[ ]nY Y Y Y is the random variable array

and Y j is the jth primary random variable, then accord-

ing to Taylor SFEM, the first order Taylor expansion

of displacement array u at its mean point is

2

1 1

1[ ] Cov( , )

2

n n

i j

i j i j

E Y Y Y Y

Y Y

uu u (8)

1 1

Var[ ] Cov( , )n n

i j

i j i j

Y Y Y Y

Y Y Y Y

u uu (9)

4.2 Calculation of the mean and variance of multi-

ple cracks structure node displacement consi-

dering interaction of cracks

The derivative of multiple cracks structure node

displacement array with respect to each random vari-

able is expressed as follows:

1( ) j j jY Y Y

u F K K u (10)

where load array F is derived according to the bound-

ary conditions and element properties, and the deriva-

tive of F with respect to each random variable can be

easily derived, / Y K is very important for the plane

fracture mechanics SFEM model, and the calculation

of / jY K should be performed for both the general

and singular elements.

The partial derivative of displacement array with

respect to physical random variable is easily derived.

When we calculate the partial derivative of displace-

ment array with respect to each crack length, we

should consider the interaction of cracks. To do this,

we set u as the displacement array of all nodes includ-

ing the nodes at all crack tips and other areas. The

nodes at crack tip belong to the singular element. Be-

cause we are just interested in the uncertainty of nodes

displacement at the crack tip, the displacement array of

all the nodes u will be simplified as

1 i nu u u u (11)

The partial derivative of load array with respect tocrack length is generally equal to 0. / ia K can be

derived according to the element properties and node

coordinates. The partial derivative of node displace-

ment with respect to the length ai of crack i is:

1 1( )i i i ia a a a

u F K K K u K u (12)

From Eq.(11) and Eq.(12), we get:

1

1, 2, , ; , 1, 2, ,

ji n

i i i i ia a a a a

i n j i j n

uu uuu

(13)

where ui is displacement array of finite element

method (FEM) node at crack i tip, /i ia u is the par-

tial derivative of displacement array of FEM node at

crack i tip with respect to crack length ai, / ia u is

the partial derivative of displacement array of FEM

nodes at other crack tips with respect to crack length ai,which can be regarded as the interaction of the uncer-

tainty of the length of crack i with the uncertainty of

the nodes displacement at the tip of crack j. Calculat-

ing / j ia u by SFEM is equivalent to considering the

interaction of stochastic multiple cracks. After intro-

ducing the partial derivative of all the nodes displace-

ment array with respect to all the cracks into

Eqs.(8)-(9), we will obtain the mean and variance of

nodes displacement under the influence of the uncer-

tainty of all crack lengths. The covariance of any two

displacement components under the influence of all the

random variables is:

1 1

Cov( , ) Cov( , )n n

k l k l i j

i j i j

u uu u Y Y

Y Y

Y Y Y Y

(14)

The mean and variance of nodes displacement array

under the influence of all the random variables are

obtained through introducing the partial derivative of

all node displacement array with respect to all the

random variables into Eqs.(8)-(9). After introducing

the mean and variance of node displacement at each

crack tip into Eqs.(4)-(7), we can obtain the mean and

variance of each crack SIF.

5. Numerical Results

Considering the plane parallel double-crack shown

in Fig.2, crack lengths are a1 = a2 = a = 10 mm, the

elastic modulus of material is E = 72 GPa, the Poisson

ratio is = 0.31, and the load is P = 50 MPa. For sim-

plicity, the variation coefficients of all the random

variables (the ratio of mean square variance to mean)

are assumed as 0.01, and all the random variables are

mutually independent. Table 1 shows the mean and

variation coefficients of I and II SIFs at crack tips A, B,

C , and D under the influence of all the random vari-

ables in the cases of d = 10 mm, 8 mm, and 6 mm. Theinfluence of each random variable on the uncertainty

of SIFs is also shown in Table 1.



No.3 Xue Xiaofeng et al. / Chinese Journal of Aeronautics 22(2009) 257-261 · 261 ·

[2]

Wang L, Brust F W, Atluri S N. The elastic-plastic

finite element alternating method (EPFEAM) and the

prediction of fracture under WFD conditions in aircraft

structures. Part I: EPFEAM theory. Computational

Mechanics 1997; 19(5): 356-369.

[3]

Wang L, Brust F W, Atluri S N. The elastic-plastic

finite element alternating method (EPFEAM) and the

prediction of fracture under WFD conditions in aircraft

structures Part III: computational predictions of the

NIST multiple site damage experimental results. Com-

putational Mechanics 1997; 20(3): 199-212.

[4]

Hsu C L, Lo J, Yu J, et al. Residual strength analysis

using CTOA criteria for fuselage structures containing

multiple site damage. Engineering Fracture Mechanics

2003; 70(3-4): 525-545.

[5] Zhao J P, Huang W L, Dai S H. The applications of 2D

elasto-plastic stochastic finite element method in the

field of fracture mechanics. International Journal of

Pressure Vessels and Piping 1998; 175(4): 281-286.

[6] He B L, Huo L X, Zhang Y F, et al. Reliability analysis

of welded structure with cracks based on three dimen-

sional stochastic FEM. Journal of Mechanical Strength

2000; 22(4): 291-293. [in Chinese]

[7] Galatolo R, Nilsson K F. An experimental and numeri-

cal analysis of residual strength of butt-joints panels

with multiple site damage. Engineering Fracture Me-

chanics 2001; 68(13): 1437-1461.

[8] Armentani E, Citarella R. DBEM and FEM analysis on

non-linear multiple crack propagation in an aeronautic

doubler-skin assembly. International Journal of Fatigue

2006; 28(5-6): 598-608.

[9] Kane A, Doquet V. Surface crack and cracks networks

in biaxial fatigue. Engineering Fracture Mechanics

2006; 73(2): 233-251.

[10]

Deodatis G, Graham-Brady L, Micaletti R. A hierarchy

of upper bounds on the response of stochastic systems

with large variation of their properties: random field

case. Probabilistic Engineering Mechanics 2003; 18(4):

365-375.

[11] Stefanou G, Papadrakakis M. Stochastic finite element

analysis of shells with combined random material and

geometric properties. Computer Methods in Applied

Mechanics and Engineering 2004; 193(2): 139-160.

[12] Sergey S, Roman K. Probabilistic method for the

analysis of widespread fatigue damage in structures.

International Journal of Fatigue 2005; 27(3): 223-234.

[13]

Xu H, Rahman S. Decomposition methods for struc-

tural reliability analysis. Probabilistic Engineering

Mechanics 2005; 20(3): 239-250.

[14]

Zhang J G. Reliability analysis method of stochastic

structures. Acta Aeronautica et Astronautica Sinica

2000; 21(3): 244-246. [in Chinese]

[15]

Cui H T, Ma H Q, Wen W D. Reliability-based shape

optimization of tenon/mortise in aeroengines with con-

tact. Chinese Journal of Aeronautics 2003; 16(4):

193-197.

[16]

Dai X J, Lin J H, Chen H R. Non-stationary random

vibration analysis of composite laminated structure at-

tached with frequency-dependant damping layer. Acta

Aeronautica et Astronautica Sinica 2007; 28(1):

100-105. [in Chinese]

[17] Xu X F. A multiscale stochastic finite element method

on elliptic problems involving uncertainties. Computer

Methods in Applied Mechanics and Engineering Issues

2007; 196(25-28): 2723-2736.

[18] Hung N D, de Saxce G, Kang C H. The computation of

2-D stress intensity factors using hybrid mongrel dis-

placement finite elements. Engineering Fracture Me-

chanics 1991; 38(2-3): 197-205.

[19] Denda M. Mixed mode I, II and III analysis of multiple

cracks in plane anisotropic solids by the BEM: a dis-

location and point force approach. Engineering Analy-

sis with Boundary Elements 2001; 25(4-5): 267-278.

[20] Denda M, Marante M E. Mixed mode BEM analysis of

multiple curvilinear cracks in the general anisotropic

solids by the crack tip singular element. International

Journal of Solids and Structures 2004; 41(5-6):

1473-1489.

[21] Meng G W, Li F, Sha L R, et al. Prediction of optimal

inspection time for structural fatigue life. International

Journal of Fatigue 2007; 29(8): 1516-1522.

Biographies:

Xue Xiaofeng Born in 1982, Ph.D. candidate. His major

subject is aircraft structure analysis and design.

E-mail: [email protected]

Feng Yunwen Born in 1968, professor. She is devoted to

the study of aircraft structural strength analysis.

E-mail: [email protected]

Research on the Plane Multiple Cracks Stress Intensity Factors Based on Stochastic Finite Element...

Documents

Transcript of Research on the Plane Multiple Cracks Stress Intensity Factors Based on Stochastic Finite Element...