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Lat. Am. j. solids struct. vol.13 número1

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(1)

! " # " "

! # !$ # % ! " &" " # "

" &

'

(

" )

' &

*

!

" #$% & '!%#&$

( " "

& "

( " " " &

) +,-. " ( +,,/ " 0

+,-+ +,,1 0 0 +,-.

2 3 +,-4 * ! #

( )( ! ( !

5 6

7 " 58 *

* 9 ) : ;-<.= ,<.+. /1, 58 * 5* ) ' > ? &

& 5 6

7 " 58 *

* 9 ) : ;-<.= ,<.+. /1, 58 * 5* ) ' > ? &

@AA : A-, -</,A-;1/ 1=+<-;/. B -4 -- +,-.

(2)

& ( "

φ & &" " &

C & &

' φ≈ ( :

" ! # + " " C

& : &

& : ! # ! #

" φ= π ≈ ( +,,, ( &

0 D $ & " " " E +,,< C

' " &

& φ≈ 5 +,,= "

C " &

" " D &

0 ( +,,+ F 5 5 ( +,,= F ) 5

( +,-- F G ( +,-4 3 " 5 -//=

" &

& " &

" &

9 & H & " 5

H & " 5 -//,

" " I

( G -//. 5 " -//= "

-//, & &" " " C '

+,,-& " : +,,4 9

" & "

" &" " :

" 3 9J5 +,,4 &

"

: &

&" K E

C 9 +,,< F 5 * 5 -/// F L 0 H +,,4 F 5

M +,,1 & "

$ B " : &

&" H +,,= & " 4

&" ( 6 3 +,-+ :

:

& &

9

B5% "

& " "

(3)

% " ! " &" " #

!$ # " "

& '

C ' ' &

&" B & "

& " " &" %

"

" " " '

" "

& & "

& "

9 & H & " 5

H & " 5 -/// "

" ' &

" &

" " &

" $

"

" & &" L +,,4

& & ' &

"

& ' ( & " &

"

5 " " &"

& " " "

& & &

" C " " &

( ( & +,,, F 0 ( 5

+,,+ F 5 " -//= F E C 9 +,,< +,,;

C "

3 5 -/1/ F ) +,,. F G +,,= F 9D5 +,-- F

G +,-+ F -//1 F *N 5 +,,. ( -/// "

( ' 5 +

5 4 &

D " 5 .

5 <

& "

(4)

: … α β… … &

(

α β… …

)

(

&" ⋅ &" = ⋅

* % & & % ! $#+'

9 " &" " & & &

( & ' &

+ 4 C "

& "

& & " & B5%

" " &"

-& " " +

B5% " φ ( "

" % $

&" ( $ "

" &

" "

φ

φ

" ( "

C

-5 +

new layer

previous layers

Jamming forces (DEM simulation)

Next layer

(repeat process)

(5)

" &

& 6 &

& ! #

& " : (

&

$

H & " 5 " (

" "

&

C & "

" ( " @

-$

" " " "

& "

& " F + :

& " F 4

" φ = + φ = 4

"

B5% F .

"

" & 5 <

& "

" " " ( &" "

: " & " &

& " &

( &

&

" ' % - &

, - % ' & -$ #!

C &

" "

& "

( " & &"

"

( &

5 '

(6)

H " &

= &" " &"

!H " &" " 5

- 6 & &

+ 6 " J " φ ' ' &

J

4 6 $

. 6 $

< 6 '

; C9B = = & & 9@

C9B = = & 9@

O ,

E H P 9@

6 " " B5%

% " $ 3 K 9 K 9 K 9

&" ω C + ( &

&" α

&" α

D D " & "

ω F : & :

& & (

" &

&"

D &" C D

(7)

= = ω

ɺɺ

ɺ

-D

= (

= + + + + +

"

A &

( & "

& " " "

= 4

-<

: & + (

: &"

= − − .

(8)

"

" (

&"

(

β β

)

α α = ≠ − − = = − − −

<

& αD βD

" Q R α β

α β

− =

− ;

D ( A

A & E 'D " L

0 B &

-/1-δ δ = ∗ ∗ ∗ = = +

ɺ 1 & & A ν ν== + − + − =

" Q R

ν ν " * "

(

)

δ = − − + /

& D δɺ

ξ δ

(9)

" D

( +,-+ ξ

& ∗

=

+

--C 4 A (

&"

"

& : A C 4

& )" & &

% & "

&" 3 +,-< " "

: "

" " 3 &D

F "

$ "

( &"

3 A &

=

=

=

τ

-+

(10)

& " τ τ τ τ − = − τ -4 " τ τ = − ⋅ = − ⋅ -.

& !τ# & ( &"

" &"

=

=

× -<

=

C 4 &

" # ! : " "

& &" & A

3 G +,-. &"

! E H # % - 5 +

( " " % &

$ ! #

" (

& &" &

! # ! # "

$ K M -//. ( &"

& ( &

"

. %# #$% %#&$ ! /& &0'%#&$ &/ % - % 1 ' & -$ #!

( " D " &"

- & +

(11)

( -; : &" ' ' @ θ θ θ θ + +   ≈ + + −   ≈ + + −

-1 θ

≤ ≤ θ = :

% - 5 +F θ= &

F θ= ' )" -1 -;

θ θ θ θ   + = + + + −   + = + + + − ω ω -=

9 &" " &

+

! : " D "

- ' @

= = ! = ! ω = ! α =

+ * @ E H ≤ 9

3 D @

(

)

(

)

= = ω α ω α

H @ C9B = 9

7 " @

+ = + + = + ω ω 7 @ + = + + = + + α ω

7 : @

(12)

+ + + = + + =

α ω -/ ( ' ' : -/ θ θ θ θ + +   ≈ + + −   ≈ + + −

ω ω ω

+,

)" +, -/

θ θ θ θ   + = + + + −   + = + + −

α ω ω

+-: -= +- =

" + " &

ω θ≠ &

" -= + +

" + - 9

" θ= & &" θ = -=

+- & " % " ' %

+ &

2 3 0

" & + 4

& % " $

& & % " &

" C

" " J

(

φ= ++

( "

" " E : 5

(13)

&

& &

E " "

& " " & ++

& & & " " (

: @

" O -,,, A 4F

" *

A @ = KA + ν = F

A @ ξ= F

3 " A @ = F

" $ @ = A +F

@ = K A = F

K @ = α =α =β =β = F

" " @ =ω =

( ' @ = F

$ " "

@ =

(" & @ = +

= ∼ 4

' & & "

" " " ( &"

" & " & " &

1 " & " &"

& " "

" & & & δ δɺ 1 & "

E =

A ( &" E 'D

(14)

∗ ∗

 

 

⇒ ≤

 

 

+4

" " D

& A

2(" * !

B ! = =

= = : ( "

= " φ =

&" % - ( C .

(

( & - &

( C < (

&

-& &

C . <

& :

& :

: & " "

5 - φ ≈ 9 &" &

&

9 : "

& &" " "

& > & &"

&

" " % "

& ( &

& " & "

(

& φD '

(15)
(16)
(17)

* 9K 9CC - , =<. , =-4 + , =<. , =4= 4 , =<. ,

=.-. , =;4 , =4,

< , =<. , =4=

; , =;. , =41

1 , =<4 , =.,

= , =;4 , =4/

/ , =/+ , =.;

-, , =;+ , =.;

, =;- , =41 , ,-+ , ,-, , ,-. , ,-+

% & * φ & : <

-2(* * #

E : &

% " D 6 &

= σ = ( &

σ

= − = = + σ = //S

& ( & C ;

& C 1 ( & + '

&

" & " :

( ( :

:

& " " : :

& " ( &"

" " % &

(18)
(19)
(20)

* B 9K B 9CC

- , =+; , =,4

+ , =4, , =+1

4 , =+< , =-. . , =-/ , =-< < , =++ , =-. ; , =+< , =-/

1 , =+; , =-4

= , =++ , =-,

/ , =+1 , =,;

-, , =+; , =,;

, =+< , =-4 , ,,4 , ,,1 , ,,. , ,,/

% & * φ & : < +

2(, , ) !

( : < - " % "

& ' &

=

!

=

= 3 = =

( " = "

φ = ( &

C = & C / ( & 4 '

&

9 & & : @

" 9 & ' @

" " &

& :

& : ( : &" &

&" : '

&

' > : < - >

& F &

& %

( & 4 : & "

4 5 - φ

" φ ≈ %

& " & "

(21)
(22)
(23)

* B 9K B 9CC - , <;, , <., + , <1; , <4= 4 , <1+ , <4/ . , ;,= , <.+ < , <;. , <4. ; , <</ , <.-1 , <;4 , <4/ = , <;, , <., / , <;. , <4/ -, , <;- , <4; , <;/ , <4/ , ,-< , ,,+ , ,+; , ,,.

% & * φ & : < 4

2(. , ) #

E : :

( D 6 &

= σ = = − σ =

σ

= + = //S & (

& C -, C -- &

( & . ' & 9

" E " :

"

: > " " ( : &"

& " "

& &" & '

& " 5

(24)
(25)
(26)

* B 9K B 9CC - , <1= , <.= + , <;4 , <.4 4 , <;; , <<+ . , <1- , <.1 < , <;1 , <.+ ; , <1. , <<+ 1 , <1= , <<4 = , <1< , <.+ / , <1= , <.; -, , <1= , <4= , <14 , <.; , ,,; , ,,< , ,-, , ,,/

% & $ * φ & : < .

4 !&$!0' #&$

% & "

" &"

" & 9 "

$

& & &"

( "

& " & & " &

" & & & 9

"

& & & & "

&

( &" 3K* 3 K 3 T ( U C%* 5*

(27)

) B 5 C E ( 5 +,--" " L 3 * " -4<@,<.-,.

) K +,,. " 3 I -@ C

5 ) B E (LB L " V 5 3

) 5 6 6 5 H L ) * C * 5 L 5 * B +,-. * " 3 * F - 4 @4+->4<;

3 ) G ( +,-. % " &

L K ) 4,@--4+>--<+

3 ) G ( +,-. & & " &

3 5 /= + @++->+.<

3 ) +,-. 3 & L ) '

5 5 9 -, -,,1A .,.4, ,-. ,+14 <

3 ) +,-< % " 3 * +@-,/>

-+<

3 " L 5 K -//= 5 * H 6 5 )

3 H 9J5 3 +,,4 % " & 6 <@-4< -.<

3 * % 5 9 H -/1/ % & 6W +/@.1>;<

3 ( 6 B $ B E E L H 5 +,,1 * + 4 & : : * * 5" 3 +,,1F +.@+/>44

% +,,; L * * 7 "

L -//1 5 * 6 @ % " 5 K

M

C M ( E 0 9 B L +,,4 C @ L K

<;@;// 1-4

C ' L % +,,- " ( 4 6 *

* " C H H

E ( 3 +,,< % 0 $ % -;+@-,;<>--=<

E 0 C M ( 9 B L +,,< 5 & *

( " -<<@44

.-L L 0 L H 5 +,,4 %

3 6 4,@<1/></1

L 0 H 0 0 B & % -/1- 5 " * B 5 H

% 4+.@4,- 4-4

L 0 H -/=< 3 3 & 7 " * 3 &

0 0 0 6 +,-. &

3 * -@-;/>-1;

0 % B ( 5 5 5 E +,,+ " $

* " B ;;@,.--,/

0 0 $ I 0 L +,-+ "

(28)

H L * $ ' % 6 2 2 L 6 B +,,= (

& " C A +-@1/1@=,<

H & " ) 5 C E -//, 6 L 5 * " ;,@<;-><=4

9D5 3 +,-- * & @ 6 L 6 --@../

.;.

K 5 M B -//. * " B <,@B+=

B4-*N ( 5 ( +,,. 3 6 " 5 )

5 % 5 C E ( 5 +,,= "

L * "

./@,.44,-5 5 M L +,,1 * & K * ( -1.@=+>/+

5 " B L -//= B $ L * " @ 3 -,@.-=< 5 3 * E % +,,= % $ K .<4 1-/< @ ;+/>;4+

5 * 5 -/// % &" 3 3 B

+/@-+,->-+,;

( M M 5 M +,,/

" " 5 3 L 3 5 +/ ; @-,+/>-,41

( 3 6 6 3 % E 3 +,-+ 5 7 ( : ) 6

L -4=@<;, <;;

( 5 ( ( & * 6 +,,, X * " B

H =. -, @+,;.>+,;1

( 3 -/// ( & @ 9 0

3 4 4 < -=1>-/; ) B

2 2 3 E 5 +,-4 K ' &

3 5

--.>--<@4<>.<

3 * +,-+ % 3 %

+,<>+,=@.; <=

G 6 ( 5 +,-4 * "

* " B ==@,<44-+

G E * G G M M B M M % ) +,,= " @

$ 3 5 ;4@<1+=><11,

G % -//. % & " L 3

* " --.@+/=>4-<

G ( +,-+ " " @ " 5 K

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