Crack Paths 2012

The 4th International Conference on “Crack Paths”

1 () () ! R o u m u m u D D f ¦ 1 ( ) ( )

D D [ [ [ [ [ D

(9)

In order to exactly reproduce nth order polynomials function, the following conditions need to be

satisfied;

1

¯®mmo

(10)

0 D

,...,2,1 n

[[D

Or in summary:

(11)

m ,...o,2,1,0 ; D G [ D D

n

If a correction function including n+1 unknown coefficient is defined, n+1 equation of 11 can be

satisfied simultaneously. The correction function is defined by:

¦ nxxC0,DDD[[E[[

(12)

It can be also express in matrix form:

(13)

C x x [ [ [ [ 7 P β

Where PT(ξ - x) is a set of basic functions and including n+1 components and β(ξ) is a set of

unknown coefficient. Substituting Equation 13 into Equation 11 and helping Equation 8 leads to:

[ [

D [ I

G

D

D

dx

³

x

,

x

0,1,2,...,

n

a

o

..

E [ E [ ½

:

1 0,1,2,...,

D

D

1 o

x x [ [

°

°

I [

x d x

[

G

n

x

°

° ¾

n a

o

D

D

®

³

oD

°

°

° ¯

E [ °

¿

n

D

:

(14)

From the Equation 14 the unknown coefficient sets of βi(ξ) is obtained. The Equation 14 can be

also rewritten as Equation 16.

D [ [ I [ : ³

(15)

D

m

x

x dx

a

(16)

0 [ [ Μ β P

Momentmatrix M can be shown like:

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