Annex 5 Discretization of the FdV variational principle for XNEM

Annex 5    Page 235 

Annex 5

Discretization of the FdV variational principle for XNEM

Content

A5.1. Discretization of the FdV variational principle 236 A5.2. Euler equations deduced from the FdV variational principle 250

A5.3. Calculation of { } K

_ε ^L

253

A5.4. Calculation of { } ^γ

^{0 L}

²⁵⁴

Annex 5    Page 236 

A5.1. Discretization of the FdV variational principle

∫

∫

∫

∫

∫ _⎥ ^{⎥ − − + −}

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂ Σ ∂

+

= Π

u

t

S

i i i S

i i A

i i A

i ij j j ij i

A

ij ) dA F u dA T u dS r ( u~ u ) dS

X u X ( u dA

) (

W ε ε

2

1

6

0

5 4 3 2

1

+ Π + Π + Π + Π + Π =

Π

=

Π δ δ δ δ δ δ

δ (A5.1)

∑ ∫ −

=

= N

I A ij ij ij I

I

dA ) (

1 1

σ Σ δε

Π

δ (A5.2)

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂

= ∂

= N

I I

A i

j j

ij i

) dA

X u X

( u

1 I

2

δ δ 2 Σ 1 Π

δ (A5.3)

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂

= N

I I

A ij

i j j

ij i ) ε dA

X u X

( u δ

δ

1

I

3 2

Σ 1

Π (A5.4)

∑ ∫

=

−

=

Π

^N

I

I A

i

i

u dA

F

1 I

4

δ

δ (A5.5)

∑ ∫

−

=

= _K

t

S i i K

M

K

T δ u dS

Π δ

5 1

(A5.6)

∑ ∫ ∫

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − −

=

Π

^u

K K

M K

K i S

i S

K i i

i ( u~ u ) dS r u dS r

1

6

δ δ

δ (A5.7)

Development of δ Π

₁

∑ ∫

∑ ∫

∑ ∫

₌

^{− Σ +}

_∈

^{− Σ +}

_∈

^{− Σ}

= Π

C L

H J O

I L A

L ij ij ij

J A

J ij ij ij N

I A

I ij ij

ij

dA dA dA

λ λ

δε σ

δε σ

δε σ

δ ( ) ( ) ( )

1

1

(A5.8)

Let :

[ ]

[ ] { } [ ] { }

[ ] { } [ ] { }

⎥ ⎥

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢

⎣

⎡

=

∫

∫

∫

∫

L L

L L

A

L L L L

A

L L L L

A

L L L L

A

L L L L

L

dA H D H dA

H D H

dA H D H dA

H D H HDH

2 2

1 2

2 1

1 1

(A5.9)

[ ] { } { }

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡ ∫ ∫

L

L A

L L A

L L

L

H dA H dA

IH

_{1 2}

; { } [ ] { }

[ ] { }

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

= ∫

∫

L L

A

L L L L

A

L L L L

L

dA D

H

dA D

H

H τ

τ

τ

2 1

(A5.10)

{ } ⎭ ⎬ ⎫

⎩ ⎨

= ⎧

Σ Σ

Σ L

L L

K K K

2

1

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_L^L

L

K K K

2 1 ε

ε ε

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_L

U L L U

U

K

K K

2

1

(A5.11)

Annex 5    Page 237 

The rotation into the global frame of the last term of (A5.8) gives:

{ }

{ } [ ] { } [ ] { }

{ }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

∑ ∫ ∫

∫

∫

∑ ∫ ∫

∫

∫

∑ ∫ ∫ ∫

∑ ∫

∑ ∫

∑ ∫

λ

∈ ε

Σ Σ

λ

∈ ε

Σ Σ

λ

∈ Σ Σ

λ

∈ Σ Σ ε ε

λ

∈ λ

∈

δ

⎥ ⎥

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢

⎣

⎡

−

−

− τ

+

δ

⎥ ⎥

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢

⎣

⎡

−

−

− τ

+

γ

⎥ δ

⎥

⎦

⎤

⎢ ⎢

⎣

⎡ τ − − −

=

δ + δ

+ γ δ

−

−

− τ

=

γ δ

− τ

= δε

Σ

− σ

C

L L

L L

C

L L

L L

C L L L

C L

C L

C L

L

L

A A

L L L L

L L L L L

L

A L L L L

A

L L L L

L

L

A A

L L L L

L L L L L

L

A L L L L

A

L L L L

L

L

A A

L L L L

L L L L A

L L

L A

L L L L

L L L L

L L L L

L

L A

L L L L

L A

I ij ij ij

K dA H D H K

dA H D H K

dA H D P dA H D

K dA H D H K

dA H D H K

dA H D P dA H D

dA H K

dA H K

P A dA

dA ) H D K H

D K (

) H K H

K P

(

dA )

P (

dA ) (

2 1

2 2 1

2 1

0 2 2

1 1

2 2 1

1 1

0 1 1

2 0 2 1

0 1

2 2

1 0 1

2 2 1

0 1

or, using the notations (A5.9,A5.10,A5.11):

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

[ ]

[ ] { }

∑

∑

∑ ∫

∑ ∫

∈ Σ

∈

∈ Σ

∈

−

− +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − −

= Σ

−

C C

C L

C L

L

L L L

L

L L T L L L

L

L L L

L A

L L L

L A

I ij ij ij

K HDH K

P D IH H

K

K IH P

A dA dA

λ ε

λ ε τ

λ λ

δ δ

τ δγ

δε σ

, 0 0 0

) (

(A5.12)

Recalling (II.34):

ij ij

ij

ε

ε σ W

∂

= ∂ ( )

(II.34)

On the other hand, in cells of type O, as a consequence of (II.34) and (VI.7), the constitutive stresses are constant.

Similarly, in cells of type H, as a consequence of (II.34) and (VI.10), the constitutive stresses are constant in parts A and B.

Consequently, the discretized form of δ Π

₁

is:

Annex 5    Page 238 

{ } { }

[ ]

{ } { }

[ ] [ { } { } ]

{ }

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

[ ]

∑

∑ ∫

∑

∑

∈ Σ

∈ Σ

∈

=

−

− +

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − −

+

Σ

− +

Σ

− +

Σ

−

= Π

C

C L

H O

L

L L L

L T L L L

L

L L L

L A

L L L

J

J B J B I B J J B

A J A J A J A N

I

I I I I

K HDH P

D IH H

K

K IH P

A dA

A A

A

λ ε τ

λ λ

δ

τ δγ

σ δε

σ δε

σ δε

δ

, 0 0 0

, ,

, , ,

, , , 1 1

(A5.13)

with the notations:

{ } ⎪

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

I I I I

12 22 11

Σ Σ Σ

Σ ; { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

I I I I

12 22 11

σ σ σ

σ ; { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

I I I I

12 22 11

ε 2

ε ε

ε (in global frame) (A5.14)

{ } ⎪

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

L L L L

12 22 11

γε γ γ

γ ; { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

L L L L

0 12 0 22 0 11 0

2 γ γ γ

γ ; { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

L L L L

P P P P

0 12 0 22 0 11

0

(in local frame) (A5.15)

Development of δ Π

₂

∑ ∫

∑ ∫ ∑ ∫

∑ ∫

∈

∈ ∈

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

+

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

⎥ +

⎥ ⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

+

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

= Π

C L

H J H J

O

I

L L

A i

j j

L i ij

J J

J

A i

j j

J i B ij J

A i

j j

J i A ij N

I

I

A i

j j

i I

ij

X dA u X

u

X dA u X

dA u X

u X

u X dA

u X

u

λ

λ λ

δ δ

δ δ δ δ

δ δ δ

) 2 (

1

) 2 (

) 1 2 (

1

) 2 (

1

, ,

1 2

(A5.16)

where the Σ

_ij^L

of the last term are given by (VI.15). Since the stresses in (VI.15) are given in the local frame (Y

1

,Y

2

)

^L

attached to the crack tip L, they must be rotated to be expressed in the global frame (X

1

,X

2

) because the displacements are expressed in this global frame.

Another approach consists in using the local frame (Y

1

,Y

2

)

^L

to express the displacements in the last term of (A5.13).

Let v

_i^L

be the components of the displacements in the local frame.

Integrating by part, the last term of (A5.16) becomes:

Annex 5    Page 239 

L A

L L i j

L ij L

C

L i L ij L j

L A

L i

L j L

j L L i

ij L

A i

j j

L i ij

dA Y v

dC P v P M

Y dA v Y

P v X dA

u X

u

L L

L L

∫

∫

∫

∫

∂ + ∂

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂

= ∂

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

δ δ

δ δ δ δ

) 2 (

) 1 2 (

1

(A5.17)

where C

_L

is the contour of the cell and M

^L_j

are the components of the outward normal to C

_L

expressed in the local frame (Y

1

,Y

2

)

^L

However, since { } ^P

^L

⁼ { } { } ^P

^{O L}

^{+ P}

^{Σ L}

^where { } ^P

^{O L}

is constant and { } ^P

^{Σ L}

is an exact solution of the Theory of Elasticity, which satisfy the equilibrium equations, we have, in absence of body force,

= 0

∂

∂

L j

L ij

Y P

so that the last term of (A5.17) vanishes.

The other terms of (A5.16) are also integrated by parts. This gives:

V H

2 2 2

2

= Π + Π + Π

Π δ

^Φ

δ δ

δ (A5.18)

with

∑ ∫

=

=

O

I

N

I I

C I i

I j

ij

N u dC

1

Φ2

Σ δ

Π

δ (A5.19)

∑ ∫ ∑ ∫

∈ ∈

Σ + Σ

= Π

H AJ H BI

J J

J B C

i J B j J B ij J

A C

i J A j J A ij

H

N u dC N u dC

λ λ

δ δ

δ

₂ ^{, ,} _, ^{, ,} _,

, ,

(A5.20)

L C

i L ij L j L

V

M P v dC

C L

∑ ∫

∈

=

Π δ

δ

2 λ

(A5.21)

The assumed displacements are given by (VI.22):

J i J J

N

J

J i J

i

u C X a

u = ∑ Φ + ∑ Φ

Λ

∈

=

) (

1

Their variations are given in the global frame ( X

₁

, X

₂

) by:

C i i

i

u u

u δ δ

δ =

^Φ

+ (A5.22)

with

∑

= Φ

=

^N

Φ

J

J i J

i

u

u

1

δ

δ (A5.23-a)

J i J J

C

i

C X a

u δ

δ = ∑ Φ

Λ

∈

)

( (A5.23-b)

They can also be expressed in a local frame ( Y

₁

, Y

₂

) by:

C i i

i

v v

v δ δ

δ =

^Φ

+ (A5.24)

with

Annex 5    Page 240 

∑

= Φ

=

^N

Φ

J

J i J

i

v

v

1

δ

δ (A5.25-a)

J i J J

C

i

C X b

v δ

δ = ∑ Φ

Λ

∈

)

( (A5.25-b)

For the calculation of δ Π

^Φ₂

, we have:

[ ]

^IJ

{ }

^I

N I

N J J N

I I

C I i

j I

ij

N u dC

^O

u A

O

I

Σ δ

δ Σ

1 1

1

Φ

= ∑ ∑

∑ ∫

= =

=

(A5.26-a)

[ ] _C ^IJ { } ^I

N I

J J

N

I I

C

i C I j

ij I N u dC

^O

a A

O

I

Σ

=

Σ ∑ ∑

∑ ∫

=1 =1 ∈Λ

δ

δ (A5.26-b)

where

{ }

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

=

I I I I

12 22 11

Σ Σ Σ

Σ ; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_I^I

I

u u u

2

1

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_I^I

I

a a a

2

1

; (A5.27)

[ ] _⎥

⎦

⎢ ⎤

⎣

= ⎡

^IJ _{IJ IJ}^IJ

IJ

A A

A A A

1 2

2 1

0

0 ; [ ]

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡

_IJ

C IJ C

IJ C IJ

IJ C

C

A A

A A A

1 2

2 1

0

0 (A5.28)

= ∫

C

I

I J I

IJ j

j N dC

A Φ ; ⁼ ∫ ^Φ

CI

I J I

j IJ

j

C

N C X dC

A ( ) (A5.29)

Finally, δ Π

^Φ₂

becomes:

[ ] { } ^N [ ] _C ^IJ { } ^I

I

J J

I N IJ

I N J J

A a A

u

^O

O

δ Σ + δ Σ

= Π

δ ∑ ∑ ∑ ∑

= ∈Λ

= = Φ

1

1 1

2

(A5.30)

H

Π

2

δ is calculated in the same way:

[ ] { } [ ] { }

[ ] { } [ _{C , B} ] ^IJ { } ^{B I,}

I

J J

I , A A IJ

, C I

J J

I , B B IJ

I N J J I,

A A IJ

I N J J H

A a A

a

A u A

u

H H

H H

Σ δ

+ Σ δ

+

Σ δ

+ Σ δ

= Π δ

∑ ∑

∑ ∑

∑ ∑

∑ ∑

λ

∈ ∈Λ λ

∈ ∈Λ

λ

∈ =

λ

∈ =1 1

2

(A5.31)

where the subcripts A and B refer to the 2 parts of the cell of type H.

In the calculation of contour integrals of the type ∫ •

I ,

CA

dC

A,I

or ∫ •

I ,

CB

dC

B,I

, the contours

include the crack lips (figure A5.1).

Annex 5    Page 241 

Figure A5.1. Cell of type H

For δ Π

^V₂

, we have:

L C

L j L ij C i i L

L C

L j L ij i L

V

v P M dC v v P M dC

C L C L

∑ ∫

∑ ∫ ^{= +}

=

Π

^Φ

∈

∈

)

2

δ ( δ δ

δ

λ λ

(A5.32) with P

_ij^L

given by:

{ } P

^L

= { } { } { } P

^{O L}

+ P

^{Σ L}

= P

^{O L}

+ K

_Σ^L₁

{ } H

₁ ^L

+ K

_Σ^L₂

{ } H

₂ ^L

, L ∈ λ

_C

(A5.33) The 1

^st

term of δ Π

^V₂

writes:

L C

L j L ij J J i L

N

J L

C

L j L ij i L

dC M P v

dC M P v

C L C L

∑∑ ∫

∑ ∫ ^{= Φ}

∈ =

Φ

∈

δ δ

λ

λ 1

(A5.34) Let

[ ] _⎥

⎦

⎢ ⎤

⎣

= ⎡

^L _{L L}^L

L

M M

M M M

1 2

2 1

0

0 (A5.35)

[ ]

^JL _J ^L _{L L}^L _J

[ ] M

^L

M

M M

M ⎥ = Φ

⎦

⎢ ⎤

⎣ Φ ⎡

=

1 2

2 1

0

η 0 ; [ ] ⁼ ∫ [ ]

CL

L JL

JL

dC

W η (A5.36)

{ } ^{= Φ} [ ] { } { } ⁼ ∫ { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎦

⎢ ⎤

⎣ Φ ⎡

=

CL

L JL JL

L L J L

L L

L L

J

JL

M H V dC

H H H M M

M M

1 1

1 3

1 2 1 1 1

1 2

2 1

1

;

0

0 η

η (A5.37)

{ } ^{= Φ} [ ] { } { } ⁼ ∫ { }

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎦

⎢ ⎤

⎣ Φ ⎡

=

CL

L JL JL

L L L J

L L L

J

JL

M H V dC

H H H M M

M M

2 2

2 3

2 2 2 1 2

1 2

2 1

2

;

0

0 η

η (A5.38)

X

1

X

2

Contour of part A

Contour

of part B

Annex 5    Page 242 

[ ] ^V

^JL

= [ { } { } ^V

1^JL

^V

2^JL

] ; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_J^J_L^L

L J

v v v

_,

2 ,

, 1

(A5.39)

Then, the term δ v

_i^J,L

Φ

_J

P

_ij^L

M

^L_j

can be written in matrix form

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

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

[ ]

[ ] { } { } { }

[

^{JL L L JL L JL}

]

L J

L L J L L

L J L L

L J L J

L L L L

L L J L J

L L J L J L

L J

L J L J L j L ij J L J i

K K

P v

H M K

H M K

P M v

H K H K P M v

P M v

P P P M

M M v M

v M

P v

1 2 1

1 , 0

2 2

1 1

, 0

2 2 1

1 , 0

,

12 22 11

1 2

2 , 1

2 , 1 ,

0 0

η η

η δ δ δ

δ δ

δ δ

Σ Σ

Σ Σ

Σ Σ

+ +

=

Φ + Φ

+ Φ

=

+ +

Φ

=

Φ

⎪ =

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎦

⎢ ⎤

⎣ Φ ⎡

= Φ

(A5.40)

so that:

[ ] { } { } { }

[ ]

[ ] { } [ ] { }

[

^{JL L JL L}

]

L J L

N

J

L JL L JL

JL L L J L

N

J L

C

L j L ij L i L

K V P

W v

V K V

K P

W v dC

M P v

C L C C

Σ

∈ =

Σ Σ

∈ =

Φ

∈

+

=

+ +

=

∑ ∑

∫ ∑ ∑

∑

, 0 1

2 2 1 1

, 0 1

,

δ

δ δ

λ

λ

λ

(A5.41)

In a similar way, we have, for the 2

^nd

term of δ Π

^V₂

:

L C

L j L ij J L

J i

L J

L C

L j L ij L C i L

dC M P X C b dC

M P v

C L C L

∑ ∑ ∫

∑ ∫ ^{= Φ}

∈ ∈Λ

∈

)

,

(

,

δ

δ

λ λ

(A5.42) which can be written in the following matrix form:

[ ] { } { } { }

[ ]

[ ] { } [ ] { }

[

C ^{JL L}

]

JL L C L J

L J

JL C JL L

C L L

JL C L J

L J

L C

L j L ij L i L

K V P

W b

V K V

K P

W b dC

M P v

C L C C

Σ

∈ ∈Λ

Σ

∈ ∈Λ Σ Φ

∈

+

=

+ +

=

∑ ∑

∫ ∑ ∑

∑

, 0

2 2 1 1

, 0 ,

δ

δ δ

λ

λ

λ

(A5.43)

with

[ ]

_C ^JL _J ^L

^{C X}

_J

[ ] ^M

^L

M M

M X M

C ⎥ = Φ

⎦

⎢ ⎤

⎣ Φ ⎡

= ( )

0 ) 0

(

1 2

2

η

1

; [ ] ⁼ ∫ [ ]

CL

L JL JL

C

dC

W η (A5.44)

{ }

_J

[ ]

^L

{ }

^L

L L

J JL

C

C X M H

H H H M

M M X M

C

₁

3 1

2 1

1 1

1 2

2 1

1

( )

0 ) 0

( = Φ

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎦

⎢ ⎤

⎣ Φ ⎡

η = (A5.45)

Annex 5    Page 243 

{ }

_J

[ ]

^L

{ }

^L

L L

J JL

C

C X M H

H H H M M

M X M

C

₂

3 2 2 2 1 2

1 2

2 1

2

( )

0 ) 0

( = Φ

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎦

⎢ ⎤

⎣ Φ ⎡

η = (A5.46)

{ } ⁼ ∫ { }

CL

L JL C JL

C

dC

V

₁

η

₁

; { } ⁼ ∫ { }

CL

L JL C JL

C

dC

V

₂

η

₂

; [ ] [ { } { }

JL C ^JL

]

C JL

C

V V

V =

_{1 2}

(A5.47)

{ } ⎭ ⎬ ⎫

⎩ ⎨

= ⎧

_J^J_L^L

L J

b b b

_,

2 , , 1

(A5.48) Finally,

[ ] { } [ ] { }

[ ]

[ ] { } [ ] { }

[

C ^{JL L}

]

JL L C L J

L J

L L JL

L JL J L

N

J V

K V P

W b

K V P

W v

C C

Σ

∈ ∈Λ

Σ

∈ =

+ +

+

= Π

∑∑

∑∑

, 0 , 0 1

2

δ δ δ

λ λ

(A5.49)

Summarizing, we get:

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ]

[ ] { } [ ] { }

[

C ^{JL L}

]

JL L C L J

L J

L L JL

L JL J L

N

J

I B IJ B C I

J J

I A IJ A C I

J J

I B IJ B I

N J J I

A IJ A I

N J J

I IJ C N

I

J J

I N IJ

I N J J

K V P

W b

K V P

W v

A a A

a

A u A

u

A a A

u

C C H H

O O

Σ

∈ ∈Λ

Σ

∈ =

Λ

∈ ∈Λ

∈ ∈Λ

Λ

∈ =

∈ =

= ∈Λ

= =

+ +

+ +

Σ +

Σ +

Σ +

Σ +

Σ +

Σ

= Π

∑∑

∑∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

, 0 , 0 1

, ,

, ,

, 1

, 1

1

1 1

2

δ δ

δ δ

δ δ

δ δ

δ

λ λ λ λ

(A5.50)

Development of δ Π

₃

∑ ∫

∑ ∫

₌

_⎥ ^{⎥ −}

₌

^Σ

⎦

⎤

⎢ ⎢

⎣

⎡

∂ + ∂

∂ Σ ∂

=

Π

^N

I

I A

ij ij N

I

I

A i

j j

ij i

dA dA

X u X

u

I

I 1

1

3

( )

2

1 δ ε

δ δ

The first term is obtained by a calculation that is perfectly similar to that of δ Π

₂

. For the second term, we have:

∑ ∫

∑ ∫

∑ ∫

∑ ∫

= = ∈ ∈

Σ +

Σ +

Σ

= Σ

C L

H J O

I

I L A

L ij ij

J A

J ij ij N

I A

I ij ij N

I

I A

ij

ij

dA dA dA dA

λ λ

ε δ ε

δ ε

δ ε

δ

1 1

(A5.51)

∑ { }

∑ ∫

= =

Σ

=

Σ

^O

O

I

N

I

I I I N

I A

I ij

ij

dA A

1 1

δ ε ε

δ (A5.52)

Annex 5    Page 244 

{ } { }

{ }

∑ ∫ ∑

∈

∈

Σ +

Σ

= Σ

H

H J J

J J B

B J B J J A

A J A

J A

J ij

ij

dA A A

λ λ

δ ε δ

ε ε

δ

_, ^{, ,} _, ^{, ,}

(A5.53)

{ }

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

∑ ∫

∑ ∫

∑ ∫

∑ ∫

∈ Σ Σ

∈

∈

∈

+ +

+ +

=

=

= Σ

C L

C L

C L

C L

L A

L L L

L L L L

L L L L

L A

L L L

L A

L ij ij

L A

L ij ij

dA H K H

K P

D H K H

K

dA P dA

P dA

λ ε ε

λ λ

λ

δ δ

δ γ

δ γ γ

δ ε

δ

2 2 1

1 0

2 2 1

1 0

(A5.54)

Using the notations [ HDH ]

^L

and [ ] IH

^L

introduced in (A5.9) and (A5.10), we get

{ }

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

∑

∑

∑

∑ ∫ ∑

∈ Σ

∈ Σ

∈

∈

∈

+

+ +

= Σ

C

C C

C

C L

L

L L L

L

L L L L

L L L

L L

L L L

L A

L ij ij

K HDH K

K IH D P K

IH P A

dA

λ ε

λ ε

λ λ λ

δ

δ δ

γ

δ γ ε

δ

0 0

0 0

(A5.55)

Finally, we obtain:

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ] { } [ ] { }

[ ]

[ ] { } [ ] { }

[ ]

{ } { { } { } }

{ }

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

∑

∑

∑

∑

∑

∑

∑∑

∑∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∈ Σ

∈ Σ

∈

∈

∈

=

Σ

∈ ∈Λ

Σ

∈ =

Λ

∈ ∈Λ

∈ ∈Λ

Λ

∈ =

∈ =

= ∈Λ

= =

−

−

−

−

Σ +

Σ

− Σ

−

+ +

+ +

Σ +

Σ +

Σ +

Σ +

Σ +

Σ

= Π

C

C C

C

H O

C C H H

O O

L

L L L

L

L L L L

L L L

L L

L L L

J

J J B B J B J J A A J A N

I

I I I

L JL C JL L

C L J

L J

L L JL

L JL J L

N

J

I B IJ B C I

J J

I A IJ A C I

J J

I B IJ B I

N J J I

A IJ A I

N J J

I IJ C N

I

J J

I N IJ

I N J J

K HDH K

K IH D P K

IH P A

A A

A

K V

P W

b

K V P

W v

A a A

a

A u A

u

A a A

u

λ ε

λ ε

λ λ

λ λ

λ λ λ

δ

δ δ

γ

δ γ

δ ε δ

ε δ

ε

δ δ

δ δ

δ δ

δ δ

δ δ

δ

0 0

0 0

, , , , ,

, 1

, 0 , 0 1

, ,

, ,

, 1

, 1

1

1 1

3

(A5.56)

Annex 5    Page 245 

Development of δ Π

₄

∑ ∫

∑ ∫

∑ ∫

= =

Φ

=

−

−

=

−

=

Π

^N

I

I A

C i i N

I

I A

i i N

I

I A

i

i

u dA F u dA F u dA

F

I I

I 1 1

1

4

δ δ δ

δ (A5.57)

∑ ∫ ∑

∑ ∫ ∑

= ∈Λ

= =

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ Φ

⎥ −

⎦

⎢ ⎤

⎣

⎡ Φ

−

=

Π

^N

I

I A

J i J J

i N

I

I A

N

J

J i J

i

u dA F C X a dA

F

I

I 1

1 1

4

δ ) ( ) δ

δ (A5.58)

Let

I A i J IJ

i

F dA

F ~

I

∫ Φ

= ; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_IJ^IJ

IJ

F F F

2

~

1

~ ~

(A5.59)

I J A

i IJ

i

C

C X F dA

F

I

Φ

= ∫ ^{( )}

~

,

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_IJ

C IJ IJ C

C

F

F F

2

~

1

~ ~

(A5.60)

We get:

{ } { }

C ^IJ N

I J N J

I N IJ J

J

F a F

u ~ ~

1

1 1

4

∑∑ ∑∑

= ∈Λ

= =

−

−

=

Π δ δ

δ (A5.61)

Development of δ Π

₅

∑ ∫

∑ ∫

∑ ∫

= Φ

=

=

−

−

=

−

= Π

K t

K t

K t

S

K C i i M

S K

K i i M

S K

K i i M

K

dS u T dS

u T dS

u

T δ δ δ

δ

1 1

1

5

(A5.62)

Let

K J S

i KJ

i

T dS

T

K

Φ

= ∫

~ ; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_KJ^KJ

KJ

T T T

2

~

1

~ ~

(A5.63)

K J S

i KJ

i

C

C X T dS

T

K

Φ

= ∫ ^{( )}

~

,

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_KJ

C KJ KJ C

C

T

T T

2

~

1

~ ~

(A5.64)

We get:

{ } { }

C ^KJ M

K J M J

K N KJ J

J

T a T

u

^t

t

~ ~

1

1 1

5

∑ ∑ ∑∑

= ∈Λ

= =

−

−

=

Π δ δ

δ (A5.65)

Development of δ Π

₆

∑ ∫ ∫

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − −

=

Π

^u

K K

M K

K i S

i S

K i i

i ( u~ u ) dS r u dS r

1

6

δ δ

δ

Annex 5    Page 246 

∑ ∫ ∑ ∫

∑ ∫

= =

=

+ +

= Π

A u

B u

K i K

i O

u

K

M K

M

K S

) K K ( B ) K ( i B S

) K K ( A ) K ( i A M

K S

K K i

i u~ dS r u~ dS r u~ dS

r

1 1

1

6

δ δ δ

δ

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ Φ + Φ

δ

−

O u

K K

M

K J s

K J J

i N

J s

K J J

K i

i u dS a C ( X ) dS

r

1 1

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ Φ + Φ

δ

−

A u

K K

M

K J s

K J J

i N

J s

K J J

) i K (

i A u dS a C ( X ) dS

r

1 1

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ Φ + Φ

δ

−

B u

K K

M

K J s

K J J

i N

J s

K J J

) i K (

i B u dS a C ( X ) dS

r

1 1

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ δ Φ + δ Φ

−

O u

K K

M

K J s

K J J

i N

J s

K J J

K i

i u dS a C ( X ) dS

r

1 1

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ δ Φ + δ Φ

−

A u

K K

M

K J s

K J J

i N

J s

K J J

) i K (

i A u dS a C ( X ) dS

r

1 1

∑ ∑ ∫ ∑ ∫

= = ∈Λ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ δ Φ + δ Φ

−

B u

K K

M

K J s

K J J

i N

J s

K J J

) i K (

i B u dS a C ( X ) dS

r

1 1

(A5.66)

where A ( K ) and B ( K ) are the parts A and B of an edge K cut by the crack.

O

M

u

is the number of edges not cut by the crack on which displacements are imposed.

A

M

u

and M

_u^B

are the numbers of parts A and B of edges cut by the crack on which displacements are imposed.

Let

∫

=

SK

K i K

i

u dS

U ~ ~

; ⁼ ∫

K i

S

K A K

A

i

u dS

U ~

_{( )}

~

; = _∫

K

S iB(K) K )

K ( B

i

u~ dS

U ~

(A5.67)

{ } ⎭ ⎬ ⎫

⎩ ⎨

= ⎧

_K^K

K

U ~ U ~ U ~

2

1

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_A^A₍⁽_K^K₎⁾

) K ( A

U ~ U ~ U ~

2

1

; { }

⎭ ⎬

⎫

⎩ ⎨

= ⎧

_B^B₍⁽_K^K₎⁾

) K ( B

U ~ U ~ U ~

2

1

(A5.68)

= ∫

S

K

J K

KJ dS

B Φ ; = _∫

SK J K

CKJ

C ( X ) dS

B Φ (A5.69)

{ } ⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _K ^K

K

r r r

2

1

(A5.70)