Annex 2 Discretization of the FdV variational principle for LEFM

(1)

Annex 2

Discretization of the FdV variational principle for LEFM

Content

A2.1. Discretization of the FdV variational principle 210

A2.2. Recasting in matrix form 215

(2)

A2.1. Discretization of the FdV variational principle

The FdV variational equation writes:

6

0

5 4 3 2

1

+ Π + Π + Π + Π + Π =

Π

=

Π δ δ δ δ δ δ

δ (A2.1)

∑ ∫

=

Π

^N

I A

I ij ij

I

dA

1

σ δε

δ (A2. 2)

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂

= N

I I

A ij

i j j

ij i

) δε dA

X u δ X

u ( δ δ

1 I

2

2 Σ 1

Π (A2. 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 (A2. 4)

∑ ∫

₌

−

=

Π

^N

I

I A

i

u dA

F

1 I

4

δ

δ (A2. 5)

∑ ∫

−

=

= _K

t

S i i K

M

K

T δ u dS

Π δ

5 1

(A2. 6)

∑ ∫ ∫

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ δ − − δ

= Π

δ

^u

K K

M K

K i S

i S

K i i

i

( u~ u ) dS r u dS r

1

6

(A2. 7)

From the assumption on the stresses and strains in LFMVC, we get successively:

{ }

J

{ }

O J

{ }

J

{ }

C

N J H

K H

K P

P = δ + δ

_Σ₁ ^Σ¹

+ δ

_Σ₂ ^Σ²

, = 1 ,

δ (A2.8)

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

C

J J J J

J O

N J H

D K H

D

K

₁ ¹

+

₂ ²

, = 1 , +

= δγ δ

_Σ ^Σ

δ

_Σ ^Σ

δγ (A2.9)

Development of δ Π

₁

Introducing (A2.8) and (A2.9) in (A2.2), we get:

Π

1

δ

{ } [ ] { } [ ] { } ∑ { }

∑ ∫ ∫ ∫

+

=

Σ Σ

Σ

+

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + +

=

^N

N I

I I I N

I A

I I I

I A

I I I

I A

I O I I

C C

I

A dA

H D K

dA H D K

dA

1 1

2 2

1 1

δε σ τ

δ τ

δ δγ

τ

∑ ∫ { } ∑ { } ∑ { }

+

=

= Σ

=

+ +

=

^N

N I

I I I N

I

I I N

I A

I O I I

C C

C

I

A e

K dA

1 1

1

δε σ δ

δγ

τ (A2.10)

with

{ }

^I ^I

K K K

⎭ ⎬

⎫

⎩ ⎨

= ⎧

Σ Σ Σ

2

1

; (A2.11)

(3)

{ }

[ ] { }

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

I I

A

I A

I

A

I I I

A

I I I

I

D

dA H

dA H dA

H D

dA H D e

e e

I I

I

τ

τ τ

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

⎭ =

⎬ ⎫

⎩ ⎨

= ⎧

∫

Σ Σ

2 1

2

1

(A2.12)

Let

[ ]

_C

A

I A

I I

I

I N

dA H

dA H IH

IH IH

I

1 ,

2 1

2

1

=

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

⎭ =

⎬ ⎫

⎩ ⎨

= ⎧

∫

Σ Σ

(A2.13)

Hence

{ } e

^I

= [ ] [ ] IH

^I

D

^I

{ } τ

^I

(A2.14) e

₁I

and e

₂^I

can be seen as a generalized strains in LFMVC n° I conjugated to the generalized stresses (or stress parameters) K

_Σ^I₁

and K

_Σ^I₂

Development of δ Π

₂

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂

= N

I I

A ij

i j j

ij i

) δε dA

X u δ X

u ( δ δ

1 I

2

2 Σ 1 Π

For I = 1 , N

_C

(LFMVC’s), we have:

∑ ∫

₌

_⎪⎩ ^⎪ ^⎨ ^⎧ _⎥ ^⎥ _⎪⎭ ^⎪ ^⎬ ^⎫

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

C

I

N

I I

A

I ij i

j j

I i

ij

) dA

Y v Y

( v P

1

2 1 δ δ δγ

∑ ∫ ∑ ∑

= = =

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ Φ + ∂

∂ Φ

=

^C

∂

I

N

I I

A

I ij N

J

N

J

J j i J J

i j I J

ij

v ) dA

v Y ( Y

P

1

2

1 1

1 δ δ δγ

[ ] { } ∑ ∫ { }

∑ ∑ ∫

=

= =

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

− ⎡

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ∂ Φ

=

^C

I C

I

N

I A

I I I

N

J N J I

I J A

I

dA v P dA

P

1

1 1

δγ δ

[ ] [ ] [ ] { }

∑ ∑ ∫ ∫ ∫

= =

Σ Σ

Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ∂ Φ + ∂ Φ + ∂ Φ

=

^N

J N J I

I J A

I I J A

I

dA K H dA K H dA v

P

C

I I

1 1 I

2 2

1 1

0

δ

{ } [ ] { } [ ] { }

[ ]

^I

N

I A

I I I I

O I J

I J

O

K H K H K D H K D H d A

C

P

I

∑ ∫

=

Σ Σ Σ Σ

Σ Σ

⎥⎦ ⎤ + +

⎢⎣ ⎡ + +

−

1

2 2

1 1

2 2 1

1

δγ δ δ

[ ] { }

∑ ∑

₌ ₌ ^Σ ^Σ

∫

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ + + ∂

=

^N

J N J I

I A

I J I IJ

I

w

IJ

K w P dA v

K

C

1 1 I

0 2

2 1

1

φ δ

(4)

{ } [ { } { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ⁺ ∫

^Σ ^Σ

⁺

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬

−

I I I

A

I I I

dA H K H

K A

P

1

2 2 1

1 0

0

δγ

{ } [ [ ] { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧

^Σ

∫ ⁺ ∫

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I I

N

I A

I I I I

A

I I I

dA K V K dA

e K

1

0

δ

[ ] [ ] [ ] [ ] { }

∑ ∑ ∫

= = Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂

=

^N

J N

I

J I I A

I J I

T , I IJ

C

I

u dA R P

R W K

1 1

0

φ δ

{ } [ { } { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧ ⁺ ∫

^Σ ^Σ

⁺

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I

N

I I

A

I I I

dA H K H

K A

P

1

2 2 1

1 0

0

δγ

{ } [ [ ] { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧

^Σ

∫ ⁺ ∫

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I I

N

I A

I I I I

A

I I I

dA K V K dA

e K

1

0

δ

(A2.15)

with

{ } [ ]

^I

{ }

^J

J

R u

v

v v =

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧

2

1

the displacements of node J expressed in the local reference frame attached to the LFMVC n° I

{ }

^J ^J

[ ] ^R

^I

{ } ^u

^J

v

v v δ

δ δ δ

2

1

=

⎭ ⎬

⎫

⎩ ⎨

= ⎧

[ ]

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

∂

∂ ∂

∂

=

∂

1 2

2 1

Φ Φ

0 Φ Φ 0 Φ

Y Y

J J

J

; [ ] _⎥

⎦

⎢ ⎤

⎣

⎡

= −

I I

I

cos sin

sin R cos

α α

{ } [ ] { }

I A

T J

IJ

H dA

w

I

, 1

1 Σ

∫ ^∂ ^Φ

= ; { } [ ] { }

I

A

T J

IJ

H dA

w

I

, 2

2 Σ

∫ ^∂ ^Φ

= (A2.16)

[ ] ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡

IJ IJ IJ

w W w

2

1

(A2.17)

[ ]

[ ] { } [ ] { }

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

=

∫

Σ Σ

I I

A

I I

A

I I

A

I I

A

I I

I

dA H D H dA

H D H

dA H D H dA

H D H V

2 2

1 2

2 1

1 1

(A2.18) X

2

X

1

) ( X

_c₁

, X

_c₂

r θ Y

₁

Y

2

α

(5)

{ }

[ ] { }

[ ] { } ^{[ ]}

^I

{ }

^I

A

I A

I

A

I I I

A

I I I

I

D P

dA H

dA H dA

H D P

dA H D P e

e e

I I

I

I 0

2 1

2 0

1 0

0 2 0 0 1

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

⎭ =

⎬ ⎫

⎩ ⎨

= ⎧

∫

Σ Σ

(A2.19)

For I = N

_C

+ 1 , N (OVC’s), we have:

∑ ∫

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

+

= N N

I I

A ij

i j j

ij i

C I

dA δε X )

u δ X

u ( δ

1

2 Σ 1

∑ ∫ ∑

+

= +

=

Σ

⎥⎦ −

⎢⎣ ⎤

⎡ +

Σ

=

^N

N I

I I ij I ij I

N

I C

j I i i I j I

ij

C

C I

A dC

) u N u N (

1

2 1 δ δ δε (A2.20)

Taking account of = ∑

= N J

iJ J

i

δ u

u δ

1

Φ as well as of the symmetry of the Σ

_ij

, this becomes :

∑ ∫

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

+

= N N

I I

A ij

i j j

ij i

C I

dA δε X )

u δ X

u ( δ

1

2 Σ 1

[ ] { } ∑ { }

∑ ∑

∑

+ = = + = + = = +

=

Σ

⎥ −

⎦

⎢ ⎤

⎣ Σ ⎡

= Σ

⎥ −

⎦

⎢ ⎤

⎣ Σ ⎡

=

^N

N I

I I I

N

N I

N

J

J T , I IJ

N

N I

I I ij I ij N

J

J j IJ i N

N I

I ij

C C

A u

A A

u A

1

1 1

1

δε δ

(A2.21)

with

{ }

^J ^J

u u u

⎭ ⎬

⎫

⎩ ⎨

= ⎧

2 1

δ

δ δ ; { }

I I

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

12 22 11

2 δε δε δε

δε ; { }

I I

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧ Σ Σ Σ

= Σ

12 22 11

Development of δ Π

₃

For I = 1 , N

_C

(LFMVC’s), we have:

∑ ∫

₌

_⎪⎩ ^⎪ ^⎨ ^⎧ _⎥ ^⎥ _⎪⎭ ^⎪ ^⎬ ^⎫

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

C

I

N

I

I A

I ij i j j I i

ij

) dA

Y v Y ( v P

1

2 1 γ

δ

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

∑ ∑

₌ ₌ ^Σ

∫

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂

=

^N

J N

I

I J I A

I J J

I T , I IJ

C

I

dA u R P

u R W K

1 1

0

φ

δ δ

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

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧ ⁺ ∫

^Σ ^Σ

⁺

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I

N

I

I A

I I I I

I I

dA H D K H

D K P

A P

1

2 2

1 1

0 0

0

γ δ

δ

{ } [ [ ] { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧

^Σ

⁺ ∫

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I

N

I A

I I I I

I

e K V K dA

K

1

δ

^γ

(A2.22)

(6)

{ } { }

{ } ^{[ ]}

^I

{ }

^I

A I

I

IH

dA H

e e e

I

I 0

0 2

0 1

γ 2 γ

γ 1

γ

γ γ

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

∫

⎭ =

⎬ ⎫

⎩ ⎨

= ⎧ (A2.23)

For I = N

_C

+ 1 , N (OVC’s), we have:

∑ ∫

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

+

= N N

I I

A ij

i j j ij i

C I

dA ε X ) u X ( u δ

1

2 Σ 1 ⎥ − ∑

⎦

⎢ ⎤

⎣

⎡ ∑

= ∑

+

=

= +

=

N N

I I I

I ij ij N

J

Jj iIJ N

N I

ijI

C C

A ε δ u

A δ

1 1

1

Σ Σ

[ ] { } ∑ { }

∑ ∑

+

= +

= =

Σ

⎥ −

⎦

⎢ ⎤

⎣ Σ ⎡

=

^N

N I

I I I N

N I

N

J

J T , I IJ

C C

A u

A

1

1 1

ε δ

δ (A2.24)

with

{ }

I I

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧ Σ Σ Σ

= Σ

12 22 11

δ δ δ

δ

Development of δ Π

₄

∑ ∑ ∫ ∑ { }

∑ ∫

= = = =

−

= Φ

−

=

−

=

Π

^N

I

N

J

J J I

A J i N

J J i N

I

I A

i

F ~

u dA

F u dA

u F

I

I 1 1 1

1

4

δ δ δ

δ (A2.25)

Development of δ Π

₅

∑ { }

∑ ∑ ∫

∫ ⁼ ⁻

₌ ₌

^Φ ⁼ ⁻

₌

−

=

Π

^N

J

J J M

K S

K J i N

J J i S

i

T ~

u dS

T u dS

u

T

^t

K

t 1 1 1

5

δ δ δ

δ (A2.26)

Development of δ Π

₆

∑ ⎥

⎦

⎢ ⎤

⎣

⎡ ∫ − − ∫

=

u

K K

M

K KS i K

S i i K i

K

i

( u~ u ) dS t u dS

t

6 1

δ δ

Π δ

We logically assume that none of the edges of the LFMVC’s is subjected to imposed displacements. So, they do not belong to S

_u

.

Consequently, the result of the calculation of δ Π

₆

is the same as in [5]:

Π

6

δ = ∑ ∑ ∑ ∑ { }

= =

⎭ ⎬ ⎫

⎩ ⎨

− ⎧

⎭ ⎬

⎫

⎩ ⎨

⎧ −

^N

J

M K K

KJ M J

K

N

J

KJ J i K

i K

i

U ~ u B u B t

t

^u

u

1 1

δ δ

_∑ { } { } _∑ ^{{ }} _∑ _∑ ^{}

= =

⎭ ⎬ ⎫

⎩ ⎨

− ⎧

⎭ ⎬

⎫

⎩ ⎨

⎧ −

=

^N

J

M K K

KJ J

M

K

N J J K KJ

K

U ~ B u u B t

t

^u

u

1 1

δ

δ (A2.27)

(7)

A2.2. Recasting in matrix form

In matrix form, these results can be summarized as follows:

{ } ∑ { } ∑ { }

∑ ∫

= Σ = +

=

+ +

=

Π

^N

N I

I I I N

I

I I N

I A

I O I I

C C

C

I

A e

K dA

1 1

1

τ δγ δ σ δε

δ (A2.28)

[ ] [ ] [ ] [ ] { }

∑ ∑

₌ ₌ ^Σ

∫

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂

=

Π

^N

J N

I

J I I A

I J I

T , I IJ

C

I

u dA R P

R W K

1 1

0

2

φ δ

δ (A2.29)

{ } [ { } { } ]

∑ ∫

=

Σ Σ Σ

Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + +

−

^C

I

N

I

I A

I I I

dA H K H

K A

P

1

2 2 1

1 0

0

δγ

{ } [ [ ] { } ]

∑ ∫

= Σ Σ Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ +

−

^C

I

N

I A

I I I I

I I

dA K V K e

K

1

0

δ

[ ] { } ∑ { }

∑ ∑

+

= +

= =

Σ

⎥ −

⎦

⎢ ⎤

⎣ Σ ⎡

+

^N

N

I I

I I N

N I

N

J

J T , I IJ

C C

A u

A

1

1 1

δε

δ (A2.30)

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

∑ ∑ ∫

= = Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂

=

Π

^N

J N

I

I J I A

I J J

I T , I IJ

C

I

dA u R P

u R W K

1 1

0

3

δ δ φ

δ

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

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧ ⁺ ∫

^Σ ^Σ

⁺

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I

N

I I

A

I I I I

I I

dA H D K H

D K P

A P

1

2 2

1 1

0 0

0

γ δ

δ

{ } [ [ ] { } ]

∑

₌

_⎪⎩ ^⎪ ^⎨ ^⎧

^Σ

⁺ ∫

^Σ ^Σ

_⎪⎭ ^⎪ ^⎬ ^⎫

−

^C

I

N

I A

I I I I

I I

dA K V K e

K

1

δ

^γ

[ ] { } ∑ { }

∑ ∑

+

= +

= =

Σ

⎥ −

⎦

⎢ ⎤

⎣ Σ ⎡

+

^N

N I

I I I N

N I

N

J

J T , I IJ

C C

A u

A

1

1 1

ε δ

δ (A2.31)

∑ { }

=

−

=

Π

^N

J

J J

F ~ u

1

4

δ

δ (A2.32)

∑ { }

=

−

=

Π

^N

J

J J

T ~ u

1

5

δ

δ (A2.33)

{ } { } ^{{ }} _∑ _∑ ^{{ }}

∑ ∑

= =

⎭ ⎬ ⎫

⎩ ⎨

− ⎧

⎭ ⎬

⎫

⎩ ⎨

⎧ −

=

Π

^N

J

M K K

KJ J

M

K

N J J K KJ

K

U ~ B u u B t

t

^u

u

1 1

6

δ δ

δ (A2.34)

Substitution in δ Π = 0 gives:

= Π

δ ∑ { } { } { } [ ] [ ] { } ∑ [ ] [ ] { }

= Σ Σ =

⎭ ⎬

⎫

⎩ ⎨

⎧ − − − + +

NC

I

J N I

J

T , IJ I

T , I I I

I I

I

e e e ( V V ) K W R u

K

1 1

0 γ

δ

(8)

+ _∑ { { } { } }

+

=

Σ

−

N I

I I

C

A A

1

σ δε

+ ∑ ∑ [ ] { } { }

+

= =

⎭ ⎬ ⎫

⎩ ⎨

⎧ −

N

Σ

N I

I N I

J

J T , I IJ

C

A u

A

1 1

ε

δ + ∑ { } { } ∑ { }

= =

⎭ ⎬ ⎫

⎩ ⎨

⎧ −

Mu

K

N J J K KJ

K

U ~ B u

t

1 1

δ

+ [ ] [ ] { } [ ] [ ]

⎪⎩

⎪ ⎨

⎧

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂

∑ ∫

∑

= Σ

=

C

I

N

I

I A

J I T , I I

IJ T , N I

J

R W K R P dA

u

1

0 1

φ δ

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

⎭ ⎬

− ⎫

−

− Σ

+ ∑ ∑

= +

=

M K K J KJ N J

N I

I

IJ

F ~ T ~ B t

A

^u

C 1 1

_∑ _∫ [ { } { } { } { } ]

=

Σ Σ Σ

Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ − − −

+

^C

I

N

I A

I I

I I I I

dA H K H

K P

1

1 2 1 1 0

0

τ

δγ

_∑ _∑ _∫ [ ] [ ] { } { } _∫ [ [ ] { } [ ] { } ]

= =

Σ Σ

Σ

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ ∂ − − +

+

^C

I I

N

I

N

J A

I I I

I A

I J I I J

dA H D K H

D K A

dA u R P

1 1

1 2

1 1

0

φ γ

δ

= 0 (A2.35)

with [ ] [ ] V

^I

+ V

^I^,^T

= 2 [ ] V

^I

since [ ] V

^I

is symmetric.

From (A2.35), all the equations (V.27) to (V.33) in chapter V can be deduced.