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Lat. Am. j. solids struct. vol.11 número4

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(1)
(2)

1

ˆ

2

u

C ij ij ijkl ij i ij j C C

V

C

dV

u

n dS

U

W

 

 

 

  

,

i

d ij

d

i

d

i ij

n dS

j

 

 

1

,

2

d

C

ij

d

i

C

d V

ij

C

ijkl

ij

dV

d

i

ij

n dS W

j P

(3)

ij

s

C

ijkl

j

n

s

ij

n

j

d

i

P

W

1

,

2

d T T

C

M u

U

C

W

d V

M CM dV

T u dS W

P

 

 

3

1

0

12

1

0

0

0

2 1

C

Et

4

P x y

( , )

w

D

 

4 4 4

4

4

2

2 2 4

w

w

w

w

x

x y

y

 

 

 

3 2

12 1

Et

D

( , )

P x y

(4)

 

nnx xx xx yy yxy

ny y y x xy

S

n Q

n Q

T

M

n M

n M

M

n M

n M

 

 

 

 

 

 

 

2 1 21 21 21 2 1 21

21 21

cos

sin

x

y

y

y

y

n

l

l

x

x

x

n

l

l

 



 

 

 



 

2 2 2 2 2 2 2 2 2

1

x y xy

w

w

M

D

x

y

w

w

M

D

y

x

w

M

D

x y

 

 

 

 

 

 

 

2 2 x y

Q

D

w

x

Q

D

w

(5)

4

0

c

w

Ñ

=

 

 

1 1 0

cos

sin

cos

1

sin

1

n n n n

c n n n n

n

w

a r

n

b r

n

c r

n

d r

n

 

2 2

0

Re

n

Im

n

c n n n n

n

w

a

r b

z

c

r d

z

2 2 2

r

=

x

+

y

z

=

x

+

iy

w

j

 

 

 

 

2 1 2 2 2 3 2 4

Re

Im

0,1, 2,

Re

Im

k k k k k k k k

w

r

z

w

r

z

k

w

z

w

z

     



0, 1, 2

k

=

3 2 4 4

4 8

0

2 3 3 3

2 2

5 9

1

3 2 4 2 2 4

2 6 10

2 2 2 3 3 3

3 7 11

0

2

2

0

;

1

;

2

2

3

6

3

4

4

w

x

xy

w

x

y

w

w

x y

y

w

x y

xy

w

x

y

k

k

k

w

xy

w

x

xy

w

x

x y

y

w

x

y

w

x y

y

w

x y

xy

 

 

 

 

 

 

 

 

 

2 2 2 2 2 2 2 2 1 2

1

i i x m i i y i xy i

w

w

x

y

M

w

w

M

M

D

a

a

(6)

 

 

 

2 2

2 2 2

2 2

1

2 2 2

2 2

1

1

x i y i

n

m

i i i

nx x y

i ny

i i i

y x

n

w

n

w

x

y

S

w

w

w

T

M

D

n

n

a

a

x

y

x y

M

w

w

w

n

n

x

y

x y

 

  

 

 

 

m

³

n

-

r

3

m

³

NDOF

-,

x

,

y

w q q

1 n

q

q

n2

q

s1

q

s2

1

(7)

3 1 2 3 2 21 3 3 2 3 4 21

2 3

4

1

8

2 3

4

1

8

N

s

s

N

s

s

s

l

N

s

s

N

s

s

s

l

   

    

  

    



 

 

5 6

1

2

1

2

N

s

N

s

  

  



w%

q%

s

q%

n

1

1

1 2 3 4

2

2

n

n

w

w

N

N

N

N

w

 

 

 

 

 

 

 

 

 

2 2 1 1 21 21 2 2 2 2 21

2

3

1

1

3

2

1

2

4

3

1

1

3

2

1

2

4

n n

n

w

s

w

s

w

s

s

s

l

s

l

s

w

s

s

(8)

(

)

1

(

)

2

1

1

1

1

2

2

s

s

s

s

s

q

%

=

-

q

+

+

q

x

-

y

cos

sin

1, 2

cos

sin

ni xi yi

si yi xi

i

 

 

 

 



 



x

q%

q

%

y

cos

sin

sin

cos

x n s

y n s

 

 

 

 

 

 



21 21

21 21

cos

sin

y

l

x

l

 

(9)

1

1

11 12 13 14 15 16

1

21 22 23 24 25 26

2

31 32 33 34 35 36

2 2 x y x y x y

w

w

N

N

N

N

N

N

N

N

N

N

N

N

w

N

N

N

N

N

N

 

 

 

  

  

 

 

 

 

 

 

 

 

 

 

 

 

    

      

1

2

T T T

d

C

a

F

a

a

G

D

f

D

  

 

e

    

T

F

t

C

d

 

 

e

   

T

G

t

N d

{ }

D

{ }

f

 

  

  

0

d

C

F

a

G

D

a

  

 

 

   

0

d

T

C

G

a

f

D

 

 

    

1

(10)

    

K

D

f

       

T 1

K

G

F

G

1

C

0

C

(

)

(

4

100

)

c

(11)

(

)

(

4

100

)

c

w

qa

D

(

)

(

2

)

c

(12)

-7 -4 -1 2

0 2 4 6 8 10 12 14 16 18

Mesh N×N

E

r

r

o

r

%

DKT

DKQ MITC4 DKMT DKMQ ARS-Q12 THS QHS

-12 0 12 24

0 2 4 6 8 10 12 14 16 18

Mesh N×N

E

r

r

o

r

%

(13)

-36 -24 -12 0 12 24

0 2 4 6 8 10

Mesh N×N

E

r

r

o

r

%

DKT DKQ REC4 ACM T6/3 THS QHS

-16 -8 0 8 16

0 2 4 6 8 10

Mesh N×N

E

r

r

o

r

%

(14)
(15)

(

)

(

4

10

)

c

w

qa

D

(

)

(

2

10

)

c

w

P a

D

-10

-8

-6

-4

-2

0

2

4

0

2

4

6

8

10

Number of radius division

E

r

r

o

r

%

DKT

DKQ

MITC4

DKMQ RDKTM

ARS-Q12

(16)

-24

-20

-16

-12

-8

-4

0

4

8

12

0

2

4

6

8

10

Number of radius division

E

r

r

o

r

%

DKT

DKQ

MITC4

DKMT

DKMQ

RDKTM

ARS-Q12

THS

(17)

60

o

-70 -60 -50 -40 -30 -20 -10 0 10

0 2 4 6 8 10

Number of radius division

E

r

r

o

r

%

T6/3 MITC7 AST6 THS QHS

-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10

0 2 4 6 8 10

Number of radius division

E

r

r

o

r

%

(18)

(

)

(

4

100

)

c

w

qa

D

-60 -50 -40 -30 -20 -10 0 10

0 2 4 6 8 10 12 14

Mesh N×N

E

r

r

o

r

%

(19)

(

)

(

4

1000

)

c

w

qa

D

-30 0 30 60 90

0 4 8 12 16 20 24 28 32

Mesh N×N

E

r

r

o

r

%

(20)

(

)

(

4 3

)

c

w

qa

Et

(21)
(22)
(23)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

(

)

)

(

)

(

)

(

)

3 11 2 3 12 21 2 3 13 21 3 14 2 3 15 21 2 3 16 21 2 2

21 21 21

2 2 2

22 21 21 21

2 2

23 21 21 21

2 2

24 21 21

2

25 21

2

3

4

1

8

1

8

2

3

4

1

8

1

8

3

1

2

1

2

1

3

4

3

1

4

3

1

2

1

2

1

3

N

s

s

N

s

s

s

y

N

s

s

s

x

N

s

s

N

s

s

s

y

N

s

s

s

x

N

s

y

l

N

s

x

s y

l

N

s

x y

l

N

s

y

l

N

s

x

=

-

+

=

-

-

+

= -

-

-

+

=

+

-=

-

-

+

+

= -

-

-

+

+

=

-

+

=

-

+

-

+

+

= -

-

+

= -

-

+

=

+

(

+

(

-

+

)

)

(

)

(

)

(

)

(

) (

(

)

)

(

)

(

)

(

) (

(

)

)

2 2 21 21 2 2

26 21 21 21

2 2

31 21 21

2 2

32 21 21 21

2 2 2

33 21 21 21

2 2

34 21 21

2 2

35 21 21 21

2 2 2

36 21 21 21

4

3

1

4

3

1

2

3

1

4

1

1

3

2

4

3

1

2

3

1

4

1

1

3

2

4

s y

l

N

s

x y

l

N

s

x

l

N

s

x y

l

N

s

s x

y

l

N

s

x

l

N

s

x y

l

N

s

s x

y

l

= -

-

+

= -

-

+

= -

-

+

=

-

+

+

-=

-

+

= -

-

+

=

+

-

+

+

x

h

 

 

4 4 1 1

,

,

,

i i i i

i i

x

N

 

x

y

N

 

y

 

i

N



1

1

1

1, 2, 3, 4

4

i i i

N





i

(

x h

i

,

i

)

 

1 1

 

 

 

1 1

,

,

T

F

h S

 

C

S

 

J d d

 

 

(24)

J

 

4

1

i i

i i

i i i

i i

N

N

x

y

x

y

J

x

y

N

N

x

y

ij

 

1

 

1

,

,

2

2

2

ij

j j i j j i

l

s

s

x

 

x

x

x

y

 

y

y

y

d

 

ds

     

1

 

 

 

1

2

ij

T

T T ij T

ij ij

l h

G

h S

A

N d

S s

A

N

s

ds

 

 

 

 

1

1

1

1

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