Fracture Mechanics and Crack Growth (Recho/Fracture Mechanics and Crack Growth) || Introduction to...

Chapter 5

Introduction to the Finite Element Analysisof Cracked Structures

Many analytical expressions can determine stress intensity factors as a functionof geometrical parameters and applied loading. These expressions are only valid forcertain geometries and for some given types of loads, within the limits of thevalidity specified. This limits their use because, even if a large number ofconfigurations have been calculated, it is unusual for the engineer to have thedesired geometry and loading. Before the finite element method (FEM) was aswidespread, the engineer had to approximate the stress intensity factor usinganalytical expressions. This chapter presents the principles of the finite elementanalysis of cracked structures. Two main categories of calculation methods arediscussed: local methods and energetic methods. This chapter also gives someinformation about the specific finite element computation of cracked structures anddiscusses the case of nonlinear behavior.

The asymptotic behavior of the linear elasticity solution is known for problemsin two-dimensional (2D) plane stress and plane strain.

To explain the solution in the vicinity of a crack, it is fitting to introduce thepolar coordinates at the tip of the crack (see Figure 5.1).

The stress solution in the x′, y′ axial system is given by equations [4.43]. Note

that the stresses show a singularity ofr1 and tend to infinity at the crack tip.

Equations [4.43] correspond to 2D problems (plane stress and plane strain).

Fracture Mechanics and Crack Growth Naman Recho© 2012 ISTE Ltd. Published 2012 by ISTE Ltd.

188 Fracture Mechanics and Crack Growth

Figure 5.1. Stress field around a crack tip

The displacement components relative to the fracture tip are also given in the x′,y′ axial system, under asymptotic form for a linear elastic (2D) medium forequations [4.44].

The stress intensity factors KI and KII, which relate the behavior around thefracture to the global geometry and the boundary conditions of the problem beingconsidered, are not determined by the previous asymptotic analysis.

Linear fracture mechanics is based on the stress intensity factors. It is thusimportant to elaborate a numerical model that is able to determine KI and KII fordifferent geometries and boundary conditions. The FEM treats different geometriesand boundary conditions, but there are some issues with applying this method tolinear fracture mechanics, such as:

– the standard finite element cannot represent a singularity (crack); and

– the finite element software produces incoherent results around a crack.

Thus, the following are needed:

– combinations of asymptotic solutions to describe the behavior near the cracktip; and

– a finite element solution to describe the linear elastic behavior.

In what follows, the reader will find a detailed example of such modeling. Itshould be noted that other techniques are used in this work that are in part related toapplications.

5.1. Modeling of a singular field close to the crack tip

The objective when developing a numerical calculation for cracking problems isto determine the stress and strain fields in the structure containing the crack, inparticular, to calculate KI and KII:

FE Analysis of Cracked Structures 189

– equations [4.43] and [4.44] determine the solution close to the crack tip; and

– finite element techniques determine the solution in all other places.

First, we determine the neighborhood of a crack in a circular radius ro (core),whose center is the crack tip. In this element, the solution is determined by equations[4.43] and [4.44] (asymptotic solution). From equation [4.44], the displacement fieldon the border of this element is given in the x′, y′ local system by the following:

( )

( )

( )

1

2

1

1 32 1 cos cos2 2 2 2

1 32 3 sin sin2 2 2 2

1 32 1 sin sin2 2 2 2

ox’ ox’

o

oy’ oy’

rν θ θu u k χπ

rν θ θk χπ

rν θ θu u k χπ

-

⎧ ⎫+⎪ ⎪⎡ ⎤= + − −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭⎧ ⎫+⎪ ⎪⎡ ⎤+ + +⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

⎧ ⎫+⎪ ⎪⎡ ⎤= + + −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

( )21 32 3 cos cos2 2 2 2

orν θ θk χπ

⎧ ⎫+⎪ ⎪⎡ ⎤− +⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

[5.1]

1

2

normalized stress intensity factor

I

II

Kk

E

Kk

E

⎫= ⎪⎪⎪⎬⎪= ⎪⎪⎭

where uox′ and uoy′ are the displacements of the crack tip in x′, y′.

The unknown parameters in equations [5.1] – 1 2 andox oy’k ,k ,u u – will be

determined by the finite element solution.

The circular element (the core-element) is connected to the standard finiteelements along its face by accepting the ux′, uy′ displacements as the imposeddisplacements.

The core has a certain number of nodes on its face. This number is a function ofthe standard elements that surround it. Assume that this number is equal to N1.


Notes on the compatibility between core-element and standard element

If using standard elements with straight segment sides, there will be a geometricincompatibility with the core circular element. This difficulty is overcomecompletely by the use of isoparametric finite elements or by specific elementsbetween the core and the standard part.

It should be noted that continuity in displacement between the core and standardcomponents is only achieved through the nodes surrounding the core.

It has been found that the error introduced due to the discontinuity of thedisplacement field is negligible for a sufficient number of standard elements and asufficiently small core radius.

Determination of the stress intensity factor

Let us assume that the stress intensity factor needs to be obtained from the stressfield in mode I fracture opening. Consider a volume element corresponding to θ = 0,in this case σ22, is written along the x axis as follows:

rKIπ

σθ 20

22 == [5.2]

If the value (calculated by finite element) of022 =θ

σ is used, and

rπσθ

20

22 ⋅=

is traced as a function of r, the evolution of KI as a function of r

will be obtained. The precision in the value of KI will therefore strictly depend on

the precision of the calculation of022 =θ

σ .

This will require the use of fine meshing around the fracture tip and specialelements at the crack tip. Figure 5.2 shows the KI ∼ r curve. To obtain KI, the valuesof σ22, calculated in the elements that are very close to the fracture tip, are avoided.This is useful as they are erroneous. The curvature is then extrapolated to estimateKI.


Figure 5.2. Evolution of factor KI, determined from thestress field as a function of radius r

In the case of the FEM with an assumed displacement field, the approximateddisplacement values are more reliable than the stresses and strains. By using θ = πin equations [5.1], the displacement is obtained based on y, which is known as v:

( )122

+== χπμπθrKv I

which represents the evolution of displacement v along the upper lip of the crack.

If the variable r is followed, KI is obtained by extrapolation up to:

rv πχμ πθ 2

1.2+=

where r = 0 (see Figure 5.3). It should be noted that in some cases extrapolation maynot be evident (see Figure 5.4), which leads to an inaccurate estimate of KI.

Figure 5.3. Evolution of factor KI, determined from the displacementfield as a function of core element radius r


To practically determine KI, Head [83] showed that the relationship between KIand r has the form of the curve in Figure 5.3. Head proposed retaining the value ofKI as the corresponding maximum value of the curve (r = ro), or the value of KIobtained by extrapolation. When comparing these two volumes, the error obtainedfor KI is less than 1% for an appropriate mesh.

Figure 5.4. Errors in estimating factor KI

5.1.1. Local method from a “core” element

The displacement field is written in the global system as follows:

cossin

sincos y'

αααα

y'x'y

x'x

uuu

uuu

+=

−=

where ux′ and uy′ are given in equation [5.1] as a function of α (see Figure 5.5)

If the nodes around the core are numbered in an anticlockwise direction, from 1to N1, there will therefore be (2N1) global displacement components, ui.

Figure 5.5. Nodes around the crack tip


12i xiu u +⎛ ⎞= ⎜ ⎟

⎝ ⎠for i non-pairs, and

2i yiu u ⎛ ⎞= ⎜ ⎟

⎝ ⎠for pairs.

These components are written in the following form:

1 1 2 2 3 4' 'i i i i iox oyu a k a k a u a u= + + +

with:

1

1

2

i 2

12

3i=4 u

etc.

i x

i y

i x

y

i = u ui = u u

i = u uu

⇒ =⇒ =

⇒ =⇒ =

[5.3]

where:

( )

1 2

1 2

121

2

cos sin with for non-pairs

sin cos with for pairs

x x i

ix x i /

u α u α θ= θ ia

u α u α θ θ i

+⎛ ⎞⎜ ⎟⎝ ⎠

⎧ −⎪

= ⎨⎪ + =⎩

( )

1 2

1 2

122

2

cos sin with for non-pairs

sin cos with for pairs

y y i

iy y i /

u α u α θ= θ ia

u α u α θ θ i

+⎛ ⎞⎜ ⎟⎝ ⎠

⎧ +⎪

= ⎨⎪ − =⎩

( )

123

2

cos with for non-pairs

sin with for pairs

i

ii /

α.......... θ = θ ia

α.......... θ θ i

+⎛ ⎞⎜ ⎟⎝ ⎠

⎧⎪= ⎨⎪ =⎩

( )

124

2

sin with for non-pairs

cos with for pairs

i

ii /

α.......... θ = θ ia

α.......... θ θ i

+⎛ ⎞⎜ ⎟⎝ ⎠

−⎧⎪= ⎨⎪ =⎩


as well as:

( )

( )

( )

( ) ⎥⎦⎤

⎢⎣⎡ +−+=

⎥⎦⎤

⎢⎣⎡ −++=

⎥⎦⎤

⎢⎣⎡ +++=

⎥⎦⎤

⎢⎣⎡ −−+=

23cos

2cos32

221

23sin

2sin12

221

23sin

2sin32

221

23cos

2cos12

221

2

2

1

1

θθχπrνu

θθχπrνu

θθχπrνu

θθχπrνu

oy

ox

oy

ox

In order to apply the minimum potential energy principle (see equation [2.101]),the strain energy is expressed as follows:

w (ε) = w (core) + w (Elts standard)

w (Elts standard) = { }[ ] { }iiEl

Ti δKδ

ts∑2

1

with:

– δι being the nodal displacement vector in element i;

–Elts∑ being a symbolical summation to express the assembly of all standard

elements in the structure;

– [K]i being stiffness matrix of the standard element (i);

–θ=π

intθ=-π

w (for a node around the core)or r

r o

W(core) r.dr.dθ=

=

= ∫ ∫ ; and

– Wint =12

mnε

ij ij mn mno

σ dε σ ε=∫ in linear elasticity

= ( )11 11 22 22 12 121 22σ ε σ ε σ ε+ + .


Replacing ijσ by equations [4.43] and ijε as function of ijσ , considering the

linear elastic behavior law (equation [2.32]), the following are obtained:

( ) ( ) ( )[ ]( ) ( ) ( )[ ] 2

22

21

2int

cos14cos3cos238

1

cos14coscos2381

kθνθθπrEν

kθνθθπrνEW

−−+−++

+−−++=

(in plane strain); and

( )[ ]( )[ ] 2

232

21

2int

cos9cos72coscos238

cos12coscos238

kθ)θν(θθπrE

kθ)ν(θθπrEW

−−++−+

+−−+=

(in plane stress).

Thus:

2 21 1 2 2( )W core B k B k= +

with:

1

1 5 2 planestrain2 4

5 planestress2 4

o

o

ν Er νB

Erν

⎧ +⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎪⎪⎝ ⎠ ⎝ ⎠⎨

⎛ ⎞⎪ −⎜ ⎟⎪ ⎝ ⎠⎩

2

1 9 2 planestrain2 4

7 planestress2 4

o

o

ν Er νB

Erν

⎧ +⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟⎪⎪⎝ ⎠ ⎝ ⎠⎨

⎛ ⎞⎪ +⎜ ⎟⎪ ⎝ ⎠⎩


Work of external forces

( ) { } { }RdsuTW T

sjjext .. δ=⋅= ∫

where {R} is the forces’ vector external to the surface.

{ }δ is the displacement vector on the surface, and therefore does not depend onk1, k2, uox’ or uoy’.

The minimum potential energy principle can therefore be written as follows:

{ } [ ] { } { } { }2 21 1 2 2

12 ts

T Tpot i ii

ElW B k B k δ K δ δ R= + + − ⋅∑ [5.4]

where:

– { }iδ depends on k1, k2, uox’ and uoy’ in the standard element nodessurrounding the (core element); and

– { } [ ]1 2

1 1 2 2 3 3 4 4 5 5 12 12depend on and donot depend on these parametersox' oy'

Ti

k ,k ,u u

δ u ,v ,u ,v ,u ,v ,u ,v ; u ,v ..........u ,v=

The potential energy is therefore minimized relative to displacement, as is thecase in finite element analyses (see section 2.6). The energy relative to k1, k2, Uox’and Uoy’ is also minimized.

Thus:

{ }{ } { } [ ] { } { }

{ } [ ] [ ]{ }∑

∑

==⇒

=⎥⎦

⎤⎢⎣

⎡−⇒⇒=⋅=

ts

ts

Elii

Elii

TT

potpot

δKδKR

RδKδdδdδ

Wd W 00

∂

∂

NOTE 5.1.– if the problem is a simple finite element problem (in a non-crackedmedium), k1 = 0 and k2 = 0:


{ } [ ] { } { } { }RδδKδW Tii

T

Elipot

ts−= ∑2

1

with [K] being the stiffness matrix of the assembly:

{ }[ ] { } { }

[ ] { } { }

[ ] { } { }

[ ] { } { }

[ ] { } { }000

000

020

002

000

222

2

111

1

=++=

=++=

=++=

=++=

=−++=

∑

∑

∑

∑

∑

oy'

ii

Eli

oy'

pot

ox'

ii

Eli

ox'

pot

ii

Eli

pot

ii

Eli

pot

iEl

iTpot

uδ

δKu

W

uδ

δ.Ku

W

kδ

δKkBk

W

kδ

δKkBk

W

RδKδ

W

ts

ts

ts

ts

ts

∂∂

∂

∂

∂∂

∂

∂

∂∂

∂

∂

∂∂

∂

∂∂

∂

[5.5]

These equations allow the determination of nodal displacements as well as thevalues of k1 and k2.

The core element is a half moon shaped for symmetrical problems in mode I anda full moon shaped for problems with combinations of modes I and II (seeFigure 5.6). The continuous geometry between the elements near the curved sidesand the conventional finite elements on straight sides is assured through the use ofisoparametric elements.

The total number of nodes existing on the ring depends on the number ofelements around it and on the formulation of these elements. These are determinedand prepared before the number of nodes. The geometric definition of the coreresults from a study by Gifford and Hilton [81] shows that the radius r must bebetween 2% and 3% of the crack length and the ratio r/h should be between 1/6 and1/10. These conditions provide results with good accuracy.


Figure 5.6. Isoparametric finite elements around a core:(a) singular element in mode I; and (b) singular elements in modes I and II

5.1.2. Local methods from enhanced elements

The enhanced element is composed of n nodes, where one of the angular nodescorresponds to the crack tip and with the displacement field is augmented by theterms of the local solution around the crack tip:


u =α1 +α2x +α3y +α4 xy +α5x 2+α6y 2 αn ym

+K I f1(r,θ) +K II g1(r,θ)

v = β1 + β2x + β3y + β4 xy + β5x 2 + β6y 2 βn ym +K I f2(r,θ) +KII g2(r,θ)

[5.6a] and [5.6b]

where m is the order of the polynomial being used.

Equation [5.6a] is written in the following matrix form:

( )[ ]{ } 11 gKfKyx,Pu III ++= α [5.7]

with:

[ ] [ ] { } { }TnmyyP αααααααα ....,,,,,,,xxy,y,x,,1 6,5432122 =⋅⋅⋅⋅=

The nodal displacement vector u:

{ } [ ]{ } { } { }11 gKfKC IIIe ++= αδ

with:

[ ]

1

,1

1

22

222

222222

121

211111

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅⋅

⋅⋅⋅⋅⋅⋅⋅

=

mmmmmmmm

m

m

y,, y, xy, x, y, x

y,y, xy, x, yx,

y,,y, xy, x, y, x

C

Solving this equation for values of {α}:

{ } [ ] { } [ ] { } [ ] { }111

11 gKCfKCC IIIe −−− −−= δα

{α} is replaced in equation [5.7]:

[ ][ ] [ ][ ] { } [ ][ ] { } { } { }1111

111 gKfKgKCPfKCPCPu IIIIII ++−−= −−−

Let:

[ ] [ ][ ] 1−= CPN


The following:

[ ]{ } [ ] { } [ ] { } { } { }1111 gKfKgKNfKNNu IIIIIIe ++−−= δ

or is obtained:

( ) ⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡−+= ∑∑∑

=

=

=

=

mi

iiiII

mi

iiiIii gN(x,y)gKfNx,yfKuNu(x,y)

111

111

where the indices on f1i and g1i indicate that f1 and g1 are evaluated at node “i”.

Similarly, for v:

( ) ( ) ( )∑ ∑∑ ⎥⎦

⎤⎢⎣

⎡−+⎥

⎦

⎤⎢⎣

⎡−+=

=

=

=

=

mi

iiiII

mi

iiiIii gNx,ygKfNx,yfKvNx,yv

122

122

The last two equations represent displacement fields u and v in the enhancedelement.

The rest of the calculation follows the steps developed from equation [4.234] tocalculate KI and KII.

5.2. Energetic methods

The second category of methods is based on the concept of energy release rates.This rate represents the change in potential energy with respect to the crack length inthe plane problems. The finite element potential energy is therefore written as:

{ } [ ] { } { } { }12 ts

ext

T Tpot i i i iEl

W(ε) W

W δ K δ δ R= −∑[5.8]

Thus, the displacement, strain and stress fields have been evaluated using finiteelements for a crack length of (a) and (a+Δa), see Figure 5.7. (Wpot) is obtained forboth problems. The energy release rate is written as:

aW

aW

G potpot

ΔΔ

−=−=∂

∂[5.9]


Figure 5.7. The evaluation of mechanical fields for a crack length of a and a+Δa

5.2.1. Finite variation methods

From [5.8] and [5.9]:

{ } { } { } { } { } daGdW

daa

Wd

WdW i

i

potpoti

potpot ⋅−⋅=⋅⋅+= δ

δ∂∂

∂∂

δδ∂

∂

{ } { } { } { } { } { } { }

{ } { } { }

{ } { } { } { } { } { } { } { }

1 1 12 2 2

12

T T T TdW d K K d d R dKpot i i i i i i i i i

T dRi iT T T TdW d K d R dK dRpot i i i i i i i i

δ δ δ δ δ δ

δ δ

δ δ δ δ δ δ

= + − +⎡ ⎤ ⎡ ⎤ ⎡ ⎤∑ ∑ ∑⎣ ⎦ ⎣ ⎦ ⎣ ⎦

−

= − + −⎡ ⎤ ⎡ ⎤∑ ∑⎣ ⎦ ⎣ ⎦

[5.10]

where {dR} and {dKi} are the variation in nodal external forces and the stiffnessmatrix, respectively.

For the case where the two structures possess the same crack length, {dKi} = 0.When the applied load is constant for both structures, {dR} = 0, we have:

{ } [ ]{ } { } { }∑ −= RdKddW Tiii

Tipot δδδ


During equilibrium:

[ ]{ } { }RKdW iipot =⇒= ∑ δ0 [5.11]

For the case where the two structures contain a crack of a and a+Δa,respectively, we have:

{ } [ ]{ } { } { }aR

aKG

aW T

iiT

ipot

∂∂δδ

∂∂δ

∂∂

+=−= ∑21 [5.12]

Special case

In the case of a crack not affecting the applied boundary conditions during its

propagation, the work of external forces is constant. That is, δ{ }T.∂R{ }∂a

is

constant with the extension of the crack from (a) to (a+Δa), thus:

{ } [ ]{ }∑−= iiT

i aKG δ∂∂δ

21 [5.13]

Only the variation in the stiffness matrix needs to be calculated between twostructures with the same geometry and boundary conditions, one with a crack that is(a) long and the other with a crack that is (a+Δa) long.

From a practical perspective, identical meshing is carried out that is onlydifferent in the area at the crack tip (see Figure 5.8).

Figure 5.8. Meshing of two structures made by changing the crack tip zone


5.2.2. Contour integrals

The energy released may be written in contour integral form. Assuming a 2Dmedium of unit thickness and considering equations [5.8] and [5.9] (see Figure 5.9),we get:

( )[ ] ( ) ∫∫ ⋅⋅−=−−=−=FS

FiiV

extpot d SuT

ad V+εw

aWεW

aaW

G∂∂

∂∂

∂∂

∂∂

Figure 5.9. Contour integral Γ1, Γ2

Assuming a crack along the x axis, and transforming the volume integral to asurface integral, we obtain:

( )F

ii F

S

uG J w ε d y+T d S

x∂∂

= = ⋅ ⋅∫ [5.14]

Suppose that Γ is a continuous contour from the lower lip to the upper lip of thecrack. SF represents the trace of this contour in the x,y plane. The J-integral isreferred to as the Rice integral [10].

Bui [11] defined a dual integral of the J-integral, named as the I-integral. In thesame way, the I-integral is equal to the complementary energy release rate relative tothe crack length:

( ) ui Su+yd dxTwIG i

Su

⋅⋅−== ∫ ∂∂σ

[5.15]


where:

( ) 12ij ij ij ijw dσ ε σ ε σ= =∫ in linear elasticity;

Su also represents the Γ contour trace in the x,y plane.

It can be shown that G = J = I and that the two integrals I and J are independentof the Γ contour in finite elements.

All quantities (displacements, strains and stresses) that are required to determinethe J-integral are easily obtained in a finite element analysis. On the other hand,these quantities are not affected by the error estimates induced by the vicinity of thecrack tip.

For convenience, the J-integral is determined from the rectangular contour(see Figure 5.10).

Figure 5.10. Rectangular contour

From equation [5.14], the following can be written for the chosen contour:

( ) dxxu

xvdy

xv

xuwJ ∫∫ ⎥⎦

⎤⎢⎣⎡ −+⎥⎦

⎤⎢⎣⎡ −−=

2

11222

10 1211 ∂

∂σ∂∂σ

∂∂σ

∂∂σε

{ } ∫∫ ⎥⎦⎤

⎢⎣⎡ ++⎥⎦

⎤⎢⎣⎡ −−+

4

32212

3

21211 dx

xv

xudy

xv

xuw

∂∂σ

∂∂σ

∂∂σ

∂∂σε


( )∫ ⎥⎦⎤

⎢⎣⎡ −−+

412115 dy

xv

xuw

∂∂σ

∂∂σε [5.16]

In practice, the integration contour Γ is defined by a group of nodes in the finiteelement mesh. The numerical integration is done on each side of the elements thatconstitute the contour (by the Gauss or trapezium method).

A finite element code provides the values of stresses and the gradient ofdisplacements in each element at each of its nodes: for a given contour, it is possibleto calculate three values of the J-integral depending on whether it uses quantitiesfrom the components interior to the contour (Jint value), exterior to the contour(Jext value) or the arithmetic average quantities at each node of the contour (Jmvalue), see Figure 5.11. These three quantities must be very close to each other,which allows the justification of the integral as being independent of the contour.

Figure 5.11. Rectangular contour’s zones

5.2.3. Other integral/decoupling modes

In the case of a planar problem in linear elasticity, we can show that:

I = J = G =2 2

*I IIK KE+

with E* = E in plane stresses and E* =1Eν−in plane strains.


Note that determining the energy release rate, directly or through integrals I andJ, presents a major difficulty due to the combination of the two failure modes KI andKII. The integrals T and A essentially allow us to separate these two modes [187][188] [189] [190]. Integrals T and A have the same values and can be expressed as abilinear form of KI and KII. These two integrals are a family of dual integrals to Jand G.

5.2.3.1. T-integral

For a solid V with a fracture on the x-axis, let C(ui, εij, σij) be the field of singulardisplacements, strains and stresses, and consider KI and KII to be the stress intensityfactors in an elastic medium in plane strain.

Assume another auxiliary singular field C * ui*,εij*,σ ij *( ) with stress intensity

factors and* *I IIK K . C* corresponds to the solution of any elasticity problem on

the same solid V. The T-integral is defined as being independent of the Γ contour:

dsunuT ijijiji ⎥⎦⎤

⎢⎣⎡ ⋅−⋅= ∫

Γ

*1,j

*1, 2

1n21 σσ

The T-integral is scalar product of field C and field C*, which are written in thefollowing bilinear form relative to the stress intensity factors:

( )**'1

IIIIII KKKKE

T +=

where E' = E in plane stress, and E' = 21Eν−

in plane strains.

By successively taking the auxiliary field corresponding to ( )* , 0IK and to

( )*II0,K , ( )**

1III KK

ET = and ( )*

*1

IIIIII KKE

T = are obtained, which allow the

calculation of KI and KII.


5.2.3.2. A-integral

Another version of the J-integral on a V domain is given by Ohtsuka andDestuynder et al. [191], [192], [193]:

dVXuXuJV

jkkiijkkijij∫⎭⎬⎫

⎩⎨⎧ ⋅+⋅−= ,,2

1 σσ

where Xi is an arbitrary, scalar, continuous and differentiable field, with:

– X1 = X2 = 0 at the exterior of domain V;

– X1 = 1, X2 = 0 around the crack tip; and

– Xi . ni = 0 on the lips of the crack.

An appropriate choice of X allows the calculation of the previous integrals on aD ring that does not include the crack tip. The expression for the A-integral is thusobtained (see Figure 5.12):

dV,21

21 *

1,*1, j

Diijiji uuA φσσ∫ ⎥⎦⎤

⎢⎣⎡ ⋅+−=

where:

– ( )* *,i iju σ is an auxiliary field;

– φ is a scalar, arbitrary and continuously differentiable field in the D ring, withthe boundary conditions on Γ1 and Γ2:

φ = 0, x ∈ Γ1, and

φ = 0, x ∈ Γ2.

Figure 5.12. Zone of the contour


The equivalence between the T and A integrals can be demonstrated bytransforming the surface integral of A to the contour integral. Similarly:

( )'1 **

IIIIII KKKKE

A +=

5.3. Nonlinear behavior

The solution to nonlinear problems is much more difficult to obtain than thesolution to linear problems. It depends heavily on the choice of behavior law, theincrement chosen and the mesh. The user’s experience and understanding of themodels used are decisive.

The following discusses the different concepts needed to establish an iterativeprocedure for finite element calculation of the J-integral.

Modeling the nonlinear behavior of the material in the field of structuralmechanics can be done by using global approaches based on the J-integral or COD(crack opening displacement) opening the crack lips.

Given the variety of elastic–plastic constitutive equations and methods fortreating large deformations, we will limit ourselves in this chapter to the case of anelastic–plastic behavior law with multilinear or power forms and updating of thegeometry at large deformations.

Equation [5.14] provides the expression for the J-integral, the crack beingparallel to the x axis (see Figure 5.9). This integral may be used in the elastic–plasticcase where w(ε) is expressed in an elastic–plastic medium as follows:

( )0

ij

ij ijw dε

ε σ ε= ∫

The J-integral represents a measure of the amplitude of the energy release rateduring fracture propagation (dx).

Compared to the critical energy release rate Gc, which corresponds to the surfaceenergy (2γ), which is necessary for the separation of the crack lips in a givenmaterial, this integral may constitute the basis for an elastic–plastic fracture criterionof the specimen.


5.3.1. Case of a power law

For a hardened elastic–plastic material, the HRR (Hutchinson, Rice andRosengreen) models [17], [194] show that the measurement of the J-integral may,on its own, characterize the stress and strain fields in plastic zone spread in a crackfront. Considering a hardened material with behavior following a Ramberg-Osgood type power law:

( ) npC εσ =

where σ is the equivalent stress, pε is the equivalent plastic strain, C is a materialconstant and n is the hardening parameter. Stress and strain fields in the plastic zoneat the crack tip are given by the following:

( )θσσijij nn

nn

n rICJC ~1. 1/

1/

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛= +

+

( )θεεijij n

n

n rICJ ~1. 1/1

1/1

⋅⋅⎟⎟⎠

⎞⎜⎜⎝

⎛= +

+

where In is a function of n and ( ) and ( )ij ijσ θ ε θ are the normalized functions of θ.

5.3.2. Case of a multilinear law

Hilton and Hutchinson [20] defined an elastic–plastic stress intensity factor Kpwhich, similar to KI or KII in elasticity (see section 5.1), depends on the geometry ofthe specimen and the boundary conditions, but also on the elastic–plastic mechanicalproperties of the material. The behavior law of the material considered is multilinearand written as follows:

( ) ( ) ( )1122

11

−−⋅⋅⋅+−+−+= mm

yd EEEEσσασσασσασε

where 1 and m mm m m

m

EΔε Δσσ σ σ αΔσ−−

< ≤ = , and ydσ represents the elastic

limit.

The number of segments (elastic segment and plastic segments defining thebehavior law) is greater than or equal to m. The last plastic segment is parallel to theε axis, in other words αN= 0 (see Figure 5.13).


Figure 5.13. Multilinear elastic–plastic behavior law

Hutchinson [17] provides an asymptotic solution to determine the displacementfield in a singular field around a crack tip:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ +−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ +−+=

23sin

231

2sin

21372

4

23cos

231

2cos

27352

40

θανθανπ

θανθανπ

NNp

NNp

rEK

v

rEK

uu

[5.17]

in plane stress, and:

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ −+−=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ −+−+=

23sin

231

2sin8

21372

4

23cos

231

2cos8

27352

40

θανθβανπ

θανθβανπ

NNp

NNp

rEK

v

rEK

uu

in plane strain.

u0 is the displacement at the crack tip based on the local x axis through the crack,and v0 is zero, considering the symmetry relative to the local system axis.

β is equal to

2

21

N

N

αν

α

⎛ ⎞+⎜ ⎟⎝ ⎠

+

Nα represents the slope of the last segment of the multilinear behavior law

when Nα = 0⇒ 2β ν= .


The strain field is determined by updating the displacement field. The updatedLagrangian formulation is used to integrate the strain field in the displacement fieldfor each step of the calculation. The coordinate system allows the relationshipbetween the updated strain variation and updated displacement variation field to be

written. The updated stress deviation fieldC

ijσ (in the sense of Cauchy) is related

to the updated strain variation field klε as follows:

C

ij ijkl klDσ ε=

The indices take values from 1 to 3. The repeated indices signify summations.Dijkl represents the constants of the behavior law for each segment (corresponding toeach deviation). The Cauchy stress tensor is not invariant in relation to rotations.

In the case of small strains, the strain field may be explicitly determined. Thestress field is therefore determined from the behavior law.

The knowledge of displacement, strain and stress fields as a function of Kpallows the J-integral to be determined:

( )2

1 pN

KJ

Eα= + in plane stress; and [5.18]

22 231 1,5 12 4

pN

KJ

Eγ γν γ α γ

α⎡ ⎤⎛ ⎞

= + − ⋅ + − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦in plane strain with:

N

N

α

ανγ

+

⎟⎠⎞

⎜⎝⎛ +

=1

2

With regards to the setting up of an iterative procedure by finite elements, whichallows the calculation of the J-integral, an assumed displacement field is used. Thisis enhanced by the local field (see section 5.1.2) expressed in Kp (see equation[5.17]):

]~),(),(~[),(

]~),(),(~[),(121

121

121

121

iii ipi

ii i

iii ipi

ii i

vtsNtsvKvtsNv

utsNtsuKutsNu

∑∑

∑∑==

==

==

==

−+=

−+=


Here s and t are the local coordinates associated with finite elements (consideredas isoparametric with 12 nodes here), Ni represents the terms of the serendipfunction. ( , )u s t and ( , )v s t are the terms in the local solution in plane strain orstress (see equations [5.17]).

The J-integral is therefore determined from the equations [5.18] after Kp hasbeen determined.

This is only one particular way to evaluate the J-integral. This integral may, infact, be calculated from the choice of various contours, as described in section 5.2.2(see Figure 5.9).

5.3.3. Relationship between COD and the J-integral

The COD may be determined directly from the crack lip displacements of thefracture, calculated by FEM.

It can also be shown that in linear elasticity that a relationship exists betweenCOD and the J-integral, written in the following form:

e

J5.0σ

=COD

with eσ being the elastic limit. COD is considered here to be half of δ in equation

[4.147], where2IKJE

= (see Figure 4.33).

Several simplified models were used to determine the COD by incorporating theplastic zone at the crack tip [3]. In the case of a 2D plate containing a central crackthat is (a) long, subject to a stress σ in mode I, the COD based on the Dugdale-Barenblatt model considered in [16] is written as follows:

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

e

e aE

COD

σσπ

σπ

.2

cos

1ln...8.21


In the case of a “generalized” plastic deformation, Rice has shown that theproperties of the J-integral remain unchanged using the theory of confined plasticity(Hencky equations) and that the formalism is identical to that of the nonlinearelasticity. In this case, although the integral may no longer be considered a variationof potential energy, it can nevertheless be regarded as an energy balance betweentwo specimens with notches of adjacent lengths.

Finally, note that in generalized plasticity described by a power behavior lawwith n as an exponent (plastic coefficient), Mc Meeking [195] showed that we couldconnect the COD to the J-Integral:

CODMJ e ..σ=

or at the critical moment where the crack propagates:

cec CODMJ ..σ=

COD = CODc is the crack propagation criteria. M is a factor between 1 and 3 formetals at ambient temperature. McMeeking proposed the following expression forM:

( ) ( )( )n

eEn

nn

M−⎟⎠⎞

⎜⎝⎛ ++

+=

..1.1

32

154.01 σν

5.4. Specific finite elements for the calculation of cracked structures

Several finite elements incorporating the singular fields of stresses and strainshave been developed to improve the performance of the numerical results obtainedfrom a finite element analysis of a cracked structure. These are mainly used near thecrack tip, where the stress gradient is very high since it tends to infinity when r tendsto zero. The finite elements reduce the number of elements needed for goodaccuracy of the results in terms of KI, KII, J, G, etc.

5.4.1. Barsoum elements [19] and Pu and Hussain [20]

Finite elements enable an approximation of a displacement field and strain usinginterpolation functions. In the case of fracture mechanics, the Westergaard solution

gives a strain field inr1 , where r is the distance to the crack tip. To better

represent this field of fracture mechanics, Barsoum [196] proposed to move the


middle nodes of a quadratic element of six or eight nodes to the side quarter. Otherauthors [81], [190] propose elements with nine, 12 or 17 nodes, completely coveringthe crack tip.

In an isoparametric finite element with eight nodes in a plane linear elasticmedium, Barsoum showed that the nodes aside (t) are found at t/4 of the crack tip

(see Figure 5.14), and the strain field is affected by a singularity ofr1 .

Figure 5.14. Finite elements with eight nodes, one of them is situated at t/4

Pu and Hussain [197] showed that the same effect exists for isoparametricelements with 12 nodes, by placing the intermediate nodes at t/9 and 4t/9 from thecrack tip. As for an elastic–plastic medium, the singularity of the strain and stress

fields is not more thanr1 , but depends on the mechanical characteristics of the

material. Senzley [198] showed that the use of intermediate nodes at t/4 producesgood results for the J-integral.

It can also be noted that the use of the elastic–plastic multilinear behavior law

(see section 5.3.2) conserves the singularity ofr1 of the strain and stress fields.

5.4.2. Verification of the strain field form

The crack tip is composed of four elements of rectangular form (seeFigure 5.15).


Figure 5.15. Eight noded elements and movement ofthe middle nodes to a side

Nodes 2 and 8 are placed at 1/4 of the distance d between 1 and 3 and between 1and 7, respectively:

( ) ( ) ( )1

1 8

8

, , ... , . ...

x

x N N

x

ξ η ξ η ξ η=⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

If the origin is considered at point 1, the serendip functions for the eight nodedelements are on the X1 axis (η = -1) [190]:

( )

( )

1

2

2

3

11

2

1

11

2

N

N

N

ξ ξ

ξ

ξ ξ

= − −

= −

= +

Thus, by using x1 = 0, x2 = d/4 and x3 = d, the following may be written:

( )214

2 1

dx

x

d

ξ

ξ

= +

= −


The derivative of ξ may thus be calculated:

1

x xd

ξ∂=

∂

This allows the determination of strain based on x:

1x x xx

u u u

x x xd

ξε

ξ ξ∂ ∂ ∂∂

= = =∂ ∂ ∂ ∂

for the displacement:

( ) ( ) ( )2

1 2 3

1 11 1 . 1

2 2x x x xu u u uξ ξ ξ ξ ξ ξ= − − + − + +

Hence the strain expression, with the term1

x, representing the strain field

singularity around the crack tip is:

1 2 3

1 1 12 .

2 2x

x x x x

uu u u

xdε ξ ξ ξ

ξ∂

= − + − + +∂

⎡⎛ ⎞ ⎛ ⎞ ⎤⎜ ⎟ ⎜ ⎟⎢ ⎥⎣⎝ ⎠ ⎝ ⎠ ⎦

Barsoum [196] thus verified in a cracked plate containing a central through crackwhere the change of the nodes to the quarter of the side actually provides thetheoretical results.

5.5. Study of a finite elements program in a 2D linear elastic medium

This is a program of finite elements that is especially designed for solvingproblems of fracture mechanics, i.e. the study of a cracked plate. It can be used forsolving problems of continuum mechanics without cracks in two-dimensions. Itsoriginality lies in the use of two sets of specific elements:

– The first set contains a single circular element type (denoted the core), which isincorporated as an additional unknown, and the stress intensity factors anddisplacements at the crack tip. This element is only used at the crack tip;


– The second set contains two types of “enriched” elements, and isoparametricquadrilateral element with 12 nodes (called QUAD-12) and an isoparametrictriangular element with nine nodes (called TRI-9). The term “enriched” refers to thefact that the global displacement field, on the finite element near the crack front, isenhanced by consideration of the local displacement field in the vicinity of thecrack. These elements are used at the crack tip, the rest of the structure beingmodeled using conventional elements (QUAD-12) and (TRI-9). These elements aredescribed below.

The 2D linear elastic version described here allows the resolution of anyproblem, whether cracked or not, in a state of plane stress or strain, according to thetheory of linear elasticity. In addition to the stresses and strains close to the cracktip, the program provides the stress intensity factors, which can then be introducedinto any instability criterion or law of propagation of fatigue fractures.

5.5.1. Definition and formulation of the conventional QUAD-12 element

The introduction and treatment of this element have been performed in detail byZienkiewicz [80]. The displacement field is chosen as a super-cubic1 function of xand y:

2 2 3 2 21 2 3 4 5 6 7 8 9

3 32 310 11 12

2 2 3 213 14 15 16 17 18 19 20

2 3 32 321 22 23 24

u a a x a y a x a xy a y a x a x y a xy

a y a x y a y x

v a a x a y a x a xy a y a x a x y

a xy a y a x y a y x

= + + + + + + + +

+ + +

= + + + + + + +

+ + + +

[5.19]

where u and v are the two components of displacement { }eΔ based on the (x,y)axes.

Considering equation [2.134], the following is obtained:

[ ] ]2 2 3 2 2 3 3 31P ,x, y,x ,xy, y ,x ,x y, xy , y ,x y,xy⎡= ⎣

1 This function is known as “super-cubic” as it contains two terms relative to x3y and y3x, ofthe order greater than the complete cubic function (Pascal’s triangle).


Let us take element QUAD-12 placed relative to the (x,y) axes, as shown inFigure 5.16.

Figure 5.16. Quadratic finite element with 12 nodes

From equation [2.135]:

[ ]

31 1 1 1

32 2 2 2

1

1

1

x y ....... .......y x

x y ...... ......y x

C⋅ ⋅ ⋅ ⋅ ⋅

= ⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅

312 12 12 12x y ..........y x

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

[5.20]

and considering equation [2.137], the following results are obtained for each nodefrom 1 to 12:

[ ] [ ][ ]1

1 2 12,N P C N N ........N−

⎡ ⎤= = ⎣ ⎦ [5.21]


with:

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )

2 21

22

23

2 24

25

26

2 27

28

29

10

1 1 1 10 9329 1 1 1 3329 1 1 1 3321 1 1 10 9329 1 1 1 3329 1 1 1 3321 1 1 10 9329 1 1 1 3329 1 1 1 3321 1 132

N y x x y

N y x x

N y x x

N y x x y

N x y y

N x y y

N x y x y

N y x x

N y x x

N y x

⎡ ⎤= − − − + +⎢ ⎥⎣ ⎦

= − − −

= − − +

⎡ ⎤= − + − + +⎢ ⎥⎣ ⎦

= + − −

= + − +

⎡ ⎤= + + − + +⎢ ⎥⎣ ⎦

= + − +

= + − −

= + − − ( )( ) ( ) ( )

( ) ( ) ( )

2 2

211

212

10 9

9 1 1 1 3329 1 1 1 332

x y

N x y y

N x y y

⎡ ⎤+ +⎢ ⎥⎣ ⎦

= − − +

= − − −

where x and y take the values ± 1 and ± 1/3 based on the position of each node in theelement. It is to be noted that the value of Ni is equal to 1 in node “i” and equal to 0in all other nodes.

The displacement field may then be written as:

( )

( )

12

112

1

i ii

i ii

u N x,y u

v N x, y v

=

=

=

=

∑

∑[5.22]


The geometric field is written as:

( )

( )

12

112

1

i ii

i ii

X N x, y X

Y N x, y Y

=

=

=

=

∑

∑[5.23]

Xi and Yi are the coordinates of the nodes; x and y are the coordinates of anypoint inside the element.

The geometry of this element varies in the same way as the displacement, whichis known as an isoparametric2 element.

5.5.2. Definition and formulation of the conventional TRI-9 element

This element is triangular with nine nodes, four being on the sides. Its serendipfunction is determined as a special case of the QUAD-12 element, keeping the sameexpression for N2, N3 … up to N9, and taking N1 (T) in the triangle as(N1 + N10 + N11 + N12) in the QUAD-12 (see Figure 5.17):

( ) ( )( )11 1 3 1 3 116

N (T) x x x= − + − [5.24]

Figure 5.17. Triangular element with nine nodes

2 If the geometry varies less than the displacement field, the element is known assubparametric; otherwise, the element is known as superparametric.


Using the isoparametric triangular element with 10 nodes (see Figure 5.18), witha complete cubic displacement field we obtain:

2 2 31 2 3 4 5 6 7

2 2 38 9 10

2 2 311 12 13 14 15 16 17

2 2 318 19 20

u a a x a y a x a xy a y a x

a x y a xy a y

v a a x a y a x a xy a y a x

a x y a xy a y

= + + + + + +

+ + +

= + + + + + +

+ + +

[5.25]

Figure 5.18. Triangular element with 10 nodes

It can be deduced that in the element TRI-9, the displacement field remains cubicby replacing N1 by N10 + N11 + N12 in the QUAD-12.

A problem arises when the complete cubic field associated with node 1 of theelement TRI-9 (see equation [5.25]) is not the same as the cubic field defined by theremaining nodes (see equation [5.19]: there are two more terms in x3y and y3x).Thus, the displacement field is not identical to at each node of element TRI-9. Thenumerical consequences of this problem are, however, minimal.

5.5.3. Definition of the singular element or core around the crack front

This specific element is a half-disc for symmetrical problems in mode I (seeFigure 5.6a) and a complete disc for problems involving combinations of modes Iand II (see Figure 5.6b). The geometric continuity between the elements on curvedsides and the conventional element QUAD-12 on straight sides is assured by the useof isoparametric elements.

The total number of nodes existing at the core depends on the number ofQUAD-12 elements that surround it. The geometric definition of this core elementresults from a study by Gifford [81], which shows that the radius r must be between2% and 3% of the crack length, and that the relation r/h must be between 1/6 and1/10. These conditions allow the results to be of a good precision.


5.5.4. Formulation and resolution by the core element method

In this solution, analytical expressions for the displacement, strain and stressfields are adopted around the crack tip.

The expressions are given as functions of the polar coordinate system.

Figure 5.19. Point around an inclined crack tip

The expression established by Sih and Liebowitz [82] is the one obtained for thecase of an inclined crack in a planar problem (see Figure 5.19) and is presented inthe following form (see equation [4.44] for a non-inclined crack):

( ) ( )( ) ( )μχαθμχαθ

μχαθμχαθ,,,,,,,,,,,,,,,,

221

111rgKrfKvvrgKrfKuu

IIo

IIo++=++=

[5.26]

where:

– u and v are the two displacement components at point M in thex,y coordinate;

– uo and vo are the displacements at the fracture extremities;

– χ = 3 – ν in plane strain;

– χ = (3 – ν)/(1+ν) in plane stress;

– α is the inclination angle of the fracture; and

– r and θ are polar coordinates of point M, where the displacements u and v arecalculated.


There are four unknowns in equations [5.26], which are reduced to two for thecase of mode I where vo = KII = 0.

Thus in the general case, KI, KII, uo and vo are to be determined from the FEM.

The equations required to determine the unknown come from the application ofthe minimum potential energy principle.

5.5.5. The evaluation of stress intensity factor (K) as a function of the radius (r)

Knowing u, v, uo and vo from the finite element solution, KI and KII can be foundas a function of r from equations [5.26]; the issue is then the choice of (r) in order toobtain more exact values of KI and KII.

Consider the following example: the case of plate with a symmetrical fracturerelative to the x axis under perpendicular forces with respect to the crack axis.Thus: α = 0.

Consider the conditions uo = vo = 0 (in other words, the crack tip does not move).

Assuming a circle of radius r around the crack tip, point M of the circle forwhich θ = πx presents the greatest displacement v following the y axis (see Figure5.20).

Figure 5.20. Displacement of the crack lips

From equations [5.1], we obtain 0oy'u = :

( )r

vKrKvu I

Iy

πχμχ

πμ2

121

22' ⋅⋅+

=⇒+==[5.27]


where v represents the displacement (following y) of point M.

It is then noticed that the stress intensity factor in mode I is a function of r andthe value of v, determined by the FEM.

5.6. Application to the calculation of the J-integral in mixed mode

Some decoupling methods allow the decomposition of energy magnitude intotwo parameters that are independently linked to mode I and mode II.

Thus:

I IIG J J J= = +

In linear elasticity, the plane stress state is:

2

'I

IKJE

= and2

'II

IIKJE

=

Consider a inclined crack with an angle of θ at coordinate (X ; Y), seeFigure 5.21.

Figure 5.21. Definition of the local coordinate


The contour integral J is written in the following schematic way (see equation[5.14]):

uJ Wdy T ds

xΓ

∂= −

∂∫

by considering the total energy, W, as the summation of the elastic strain energy andthe plastic strain energy:

elastic plasticW W W= +

At point M, vector T is expressed as follows: .i ij jT nσ= where:

.X XX XY X

Y YX YY Y

T nT n

σ σσ σ

⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ =⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎭ ⎣ ⎦ ⎩ ⎭⎩

or. .

. .X XX X XY Y

Y YX X YY Y

T n n

T n n

σ σ

σ σ

= +

= +

The quantity dl is related to dX and dY by:

.Xn dl dY=

.Yn dl dX= −

Calculating . .u

T dlx

∂

∂, we get:

. . . . .X YX Y

u uuT dl T T dlx x x

∂ ∂∂ ⎛ ⎞= +⎜ ⎟∂ ∂ ∂⎝ ⎠

Expressing the partial derivatives relative to x in the global domain:

cos sinx X Y

θ θ∂ ∂ ∂= +∂ ∂ ∂


Therefore:

. .u

T dlx

∂

∂

cos sin cos sin .X X Y YYX YY

u u u u dXX Y X Y

σ θ θ σ θ θ⎡ ∂ ∂ ∂ ∂ ⎤⎛ ⎞ ⎛ ⎞− + + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

Calculating W.dy: on one hand cos . sin .dy dY dXθ θ= − , and on the other theenergy is obtained by integrating the product of .dσ ε .

Therefore:

W dε

σ ε= ∫

For a linear elastic material:

1

2yx

XX XX YY YY XY

uuW

y xσ ε σ ε σ

∂∂= + + +

∂ ∂⎛ ⎧ ⎫⎞

⎨ ⎬⎜ ⎟⎝ ⎩ ⎭⎠

For a plastic material, the expression of .dσ ε must be integrated by parts for then steps of the calculation.

Thus:

( ) ( )( )( )

( )

1 1

1 1

01 1

1

1

2

i i i i

XX XX XX XXi n

i i i i

YY YY YY YYi

i i i i

i i X Y X YXY XY

W

u u u u

Y X Y X

σ σ ε ε

σ σ ε ε

σ σ

+ +

=+ +

=+ +

+

+ − +

= + − +

∂ ∂ ∂ ∂+ + − −

∂ ∂ ∂ ∂

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎝ ⎠⎠

∑

dYYu

Xu

Yu

Xu YY

XYXX

XX .sincossincos ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂+

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂+

∂∂= θθσθθσ


This calculation is realized at the points of integration on the circular contours Γ(see Figure 5.22).

Figure 5.22. View of the finite element mesh

5.6.1. Partitioning of J in JI and JII

The goal is now to determine the partition in JI and JII, which corresponds to thepartition between KI and KII in the case of linear elasticity.

Figure 5.23. Definition of M′

M′ is the symmetry of M relative to the x axis (see Figure 5.23).


Using:

( )

( )

'

1 1 1

'

2 2 2

1

21

2

I

I

u u u

u u u

= +

= −

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

and( )

( )

'

1 1 1

'

2 2 2

1

21

2

II

II

u u u

u u u

= −

= +

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

the field 1

2

I

I

u

u

⎛ ⎞⎜ ⎟⎝ ⎠

is relative to the JI component and field1

2

II

II

u

u

⎛ ⎞⎜ ⎟⎝ ⎠

to the JII component.

Thus:

II

I

uJ W dy T ds

xΓ

∂= −

∂∫ andII

II

II

uJ W dy T ds

xΓ

∂= −

∂∫

In practice, Jinv integral is defined from the displacement field'

1 1

'

2 2

inv

inv

u u

u u=

−

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

.

Thus:

invinv

inv

uJ W dy T ds

xΓ

∂= −

∂∫ and ;2 2

inv invI II

J J J JJ J

+ −= =

It is to be noted that this transformation replaces the strain in M with the strain inM′ by changing the sign of the crossed derivatives. This is only valid for thex,y domain:

'

'

inv

x x

inv

y y

u u

x x

u u

y y

∂ ∂

∂ ∂=

∂ ∂

∂ ∂

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

and

'

'

inv

x y

invxy

u u

y x

uuyx

∂ ∂−

∂ ∂=

∂∂−

∂∂

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


5.7. Different meshing fracture monitoring techniques by finite elements

Many tools of crack propagation in finite element models were developed inconjunction with finite element codes. The idea is to be able to analyze the localbehavior (welding) from a single global model (e.g. vehicle) using the techniques ofstructural zoom to locally incorporate defects and study their evolution. The finiteelement computer code is the tool that is built around the design, the calculation ofstructure, the definition of quality (e.g. welding) and risk analysis (presence of adefect – how it can evolve). Some codes can process 2D cases, e.g. FRANC2D,FORGE2D, etc., and others have suggested the insertion of cracks in three-dimensional models, e.g. ZENCRACK, FRANC3D, etc.

The principle of these tools is often based on “crack boxes” connected to theglobal finite elements mesh by an intermediate zone. Some tools includepropagation: the crack is propagated in increments. Each increment and the directionto be followed are determined using a criterion. Then the area around the crack is re-meshed. The remeshing preserves a good quality of mesh around the crack tip andwe get a satisfactory calculation of stresses and stress intensity factors. Figure 5.24shows an example of the propagation of a crack in a plate by the FORGE2D toolused in reference [199]. Remeshing is observed for different lengths of crack.

Figure 5.24. Crack propagation in a pre-fractured plate

There are also tools to simulate crack propagation in three-dimensionalenvironments. The Zencrack tool [200] shows the use of crack boxes in a three-dimensional medium. An example of the result of this is shown in Figure 5.25. Theuse of crack boxes associated with a mesh of three-dimensional elements (Brickelements) is still a problem at initialization, in the way in which the connectionbetween the brick elements and the elements of the crack tip is made. The advantageof the crack box is that we can have a regular mesh for which a relatively smallnumber of elements are required to precisely determine the magnitude of fracture


mechanics, unlike the free mesh (see Figure 5.24), which has highly refined meshesat the crack tip.

In France, the Snecma has developed a crack introduction methodology in three-dimensional structures [201]. The crack box constitutes of three zones:

– the fracture front is surrounded by a regulated hexahedral meshing;

– at a distance the meshing is tetrahedral; and

– between these two zones, a transition zone is present that is meshed bypyramidal elements.

This method, when applied to a plate with a semi-elliptical crack under tension,allows results with a tolerance smaller than 5% relative to reference analyticalsolutions.

Figure 5.25. Various stages of propagation (Zencrack [200])

Other authors, such as Givoli [202], Murthy [203] and Schnöllmann [204], havedeveloped automatic meshing methodologies, which are also known as adaptivemethodologies, to represent crack propagation in three dimensions.


5.7.1. The eXtended finite element modeling method

The eXtended Finite Element Modeling (X-FEM) method is based on the FEM,but adds a term to the approximation of displacement, describing the supplementarydisplacement modes [205]. The FEM requires that the meshing represents thegeometry being studied, which poses problems in the modeling of discontinuousgeometries, such as holes and cracks. The X-FEM method allows the considerationof these singularities at the mesh level. Thus, the defect propagation can be treatedwithout modifying the mesh [206].

The approximation of the displacement field is enhanced by the local solution,and written as:

( )

( ) ( ) ( )En k

k k kl lk l

u M N M u a F M= +⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∑ ∑

where Fl are the enriched functions and akl is the additional degree of freedom fornode k.

Dolbow, Belytschko and Moës [207] proposed dividing the crack in three zones(see Figure 5.26) and adding an enrichment function following these zones:

– two zones, each constituting a crack tip (near the tip regions); and

– the third zone containing a crack but not the crack tips (the interior region).

The enhanced formulation is therefore written as:

∑ ∑∑∑ ∑∑∈ ==∈

⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛++=2

4

1

224

1

11

1)()()()(

Jj ijjij

ijji

l Jjjll

kkk MFcNMFcNbxHNuNMu

where coefficients bl and cji are the degrees of freedom corresponding to theenrichment functions introduced:

– H describes the discontinuity due to the crack – it is the Heaviside function;and

– F1 and F2 describe the asymptotic behavior at the crack tip.


Figure 5.26. Example of a crack in X-FEM

The nodes of elements at the crack tips correspond to J1 and J2. These nodes areenriched by asymptotic functions relative to the local solution. The Heavisidefunction, H, is applied to the other nodes through the crack length. The rest of thenodes in the far field are not affected by the presence of the crack.

5.7.2. Crack box technique (CBT)

Around the crack tip, the stress fields are determined by the so-called asymptoticanalyses. These allow the prediction of the critical load level leading to crackpropagation and the determination of the crack extension angle. This field is veryimportant around the crack tip, which requires a fine and regular finite elementmesh. This mesh must be able to move when the crack propagates and develops in away to optimize the number of elements. In addition to the clarification of ruin orcrack extension criteria, it is necessary to first specify the characteristics of the meshat the crack tip, and the methodology that allows its integration into the rest of themesh (see Figure [208]).

The methodology presented here requires the establishment of a transition zonebetween the crack box (CB) local mesh and the global mesh of the structure. Threezones are therefore considered (see Figure 5.27):

– Zone (A) CB, see Figure 5.28: this area has a special and regular mesh. At thecrack tip, the asymptotic solution dominates. In the case of elastic calculations, few


things are needed. The tip of the crack is modeled using quadratic quadrangularelements that are degenerated into triangles. More nodes at the “center” of theelements are placed at the quarter of the element to represent a strain field in r-0.5,with r being the distance to the crack tip. For the case of plastic calculations, moreelements are necessary to precisely calculate the J-integral. To introduce asingularity in r-1 for the case of perfectly plastic materials, the quadraticquadrangular elements are also used, but the nodes at the crack tip can moveindependently and the nodes at the “center” are not moved to the quarter of theelements. Finally, for the case of Ramberg-Osgood type materials (following theelastic–plastic behavior law), the last mesh considered allows a good approximationof the displacement field r-n/n+1 (where n is the plastic exponent). The displacednodes at the quarter of the elements may be used for small values of n.

Figure 5.27. Crack box in a mesh (zones A, B and C)

Figure 5.28. CB fine mesh (left); and CB coarse mesh (right)

Transition zone

Initial meshCrack box


– Zone (B) – transition zone: this is a mesh composed of linearly optimizedtriangular elements for purely elastic or quadratic calculation (in the plastic case orfor a better precision of elastic calculation, where the use of triangular elements isnot recommended). This mesh is obtained using the Delaunay triangulationprocedure, developed in NAG [209]. These elements connect the specific mesh ofthe crack tip to the rest of the ABAQUS model [210], which can be 2D in planestrain and stress, or to the shell model.

– Zone (C) – global mesh: this is a classical quadrangular (or triangular) mesh.It is to be noted that it is not necessary to remesh the integrality of the structureduring the fracture propagation.

The Crack Box Technique (CBT) presented uses the ABAQUS code and consistsof:

– meshing the three zones (A), (B) and (C) for the initial crack;

– realizing the finite element calculation to determine the crack extension angle;

– incrementing the crack length in the predetermined direction;

– creating the meshing at the crack tip and linking it to the rest of the structure;and

– realizing a new finite element calculation, and so on.

NOTE 5.2.– Zone (B) works as a contour around the crack tip, which gets closer tostatic condensation of the structure at this contour through an intermediate mesh.This technique is similar to the boundary integral approach, in which the contour isreplaced by a transition zone here [211].

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