1 1 1

1.1 3 1.1.1 (...) 3 1.1.2 Love 8

1.2 11 1.3 17 1.4 20 1.5 22 1.6 , 24 1.7 Tresca von Mises 27 1.8 Mohr-Coulomb 30 1.9 : Mohr 32

.

. (2008) , . 1 2

1. , 2009

. , Dr-Ing., ..

.. 144, 190-02, http://geolab.mechan.ntua.gr/, I.Vardoulakis@mechan.ntua.gr

. (2008) , . 1 3

1.1 1.1.1 (...) 1

. 1-1:

, 2,3. , 4 (REV). 30 5,6 (. 1-1). . , n

m , ( , )c m n , ( , )m n , nc mck kn n= ,

c . , nc mck kF F= , (. 1-2). () (. 1-3). . , .

1 Bardet, J.-P. and Vardoulakis (2001). The asymmetry of stress in granular media. Int. J. Solids Struct.,38, 353-367. 2 Christoffersen, J., M. M. Mehrabadi, S. Nemat-Nasser (1981), A micromechanical description of granular material behavior, Journal of Applied Mechanics, ASME, Vol. 48, pp. 339-344. 3 Rothenberg, L., and A. P. S. Selvadurai (1981). Micromechanical definition of the Cauchy stress tensor for particulate media, In: Mechanics of Structured Media (edited by A.P.S. Selvadurai), Elsevier, pp. 469-486. 4 . Representative Elementary Volume (REV). 5 . Discrete Element Method (DEM) 6 Cundall P.A., Strack O.D.L. (1979). A discrete numerical model for granular assemblies, Gotechnique 29, No 1, 47-65.

. (2008) , . 1 4

. 1-2: ( )REV .

. 1-3: :

. 1-4:

. (2008) , . 1 5

( )REV , N , ( )REV . ( )REV . ( )REV ,

{ } { }1, , , , 1, , ,a NB a N = = (1.1) ( )REV M ,

{ } { }1, , , , 1, , , ,s MC e e e s M= = (1.2) I C ( )REV , E C ( )REV :

{ }{ }

1

1

, , {1, , }

, { 1, }

,

M

M M

I e e M

E e e M M

I E C I E

+

= == = + = =

(1.3)

a aI a aE a ( )REV ( )REV , a aC ,

,

,

a a a aa B

a aa B a B

C C C I E

I I E E

= = = =

(1.4)

a b { },a b a b c a bE E I I e B = = (1.5) , . ,

0a

aci

c CF

= (1.6)

( ) 0a

c a acijk j j k

c Cx x F

= (1.7)

aix cix

.

. (2008) , . 1 6

a ( ) aiu

ai (. 1-5).

. 1-5: .

(1.6) (1.7) aiu ai ( )REV ,

( )( ) 0a

ac a c a ac ai i ijk j j k i

a c CF u x x F

+ = (1.8)

aC . ,

c ac bci i iF F F= = (1.9) ...

( , ) ( ,int)D ext DW W = (1.10) ( , )D extW ( ,int)DW 7: ) ,

( , )D ext e ei ie

W F u

= (1.11) eiu e

eiF ) :

( ,int)D c ci ic

W F u

= (1.12) ciu c a b (. 1-6),

7 D (. discrete)

. (2008) , . 1 7

( ) ( )( )c b a b c b a c ai i i ijk j k k j k ku u u x x x x = + (1.13)

. 1-6:

. :

a ai i ij ju a b x = + +" (1.14)

a ai i ij jx = + +" (1.15) ,i ija b ,i ij , :

( ) ( ) ( ) ( )( )c b a b a b c b a c ai ij j j j ijk k k jl ijk l k k l k ku b x x x x x x x x x x = + +" (1.16)

( )

( ) ( )e a a e aei i ijk j k k

ae e ae ae e aei ij j ijk j k k ijk jl l k k

u u x x

a b x x x x x x

= + += + + + +

""

(1.17)

aekx a a e .

:

( ,int) ( ) ( )D c b a c b aij i j j j ijk i k kc c

W b F x x F x x

= + " (1.18) ( , ) ( )D ext e e ae e e aei i ij i j j ijk i k k

e e eW a F b F x F x x

= + + + " (1.19)

..., . (1.10) :

. (2008) , . 1 8

0 , 0 , 0

0

eij i i i i

ee

ie

b a F a

F

= = = =

(1.20)

. (1.20) ( )REV . ,

0 , 0 , ( )

( )

c b a e aei i ij i j j ij i j ij

c ec b a e ae

i j j i jc e

a b F x x b F x b

F x x F x

= = = =

(1.21)

0 , 0 , ( ) ( )

( ) ( )

c b a e e aei ij j ijk i k k j ijk i k k j

c ec b a e e ae

ijk i k k ijk i k kc e

a b F x x F x x

F x x F x x

= = = =

(1.22)

( )e aek kx x

( )e aek k gx x O R = (1.23)

(1.22) ,

( ) 0c b aijk i k kc

F x x

= (1.24) 1.1.2 Love 8 Cauchy . ( )REV , ,

0ij k REVi

x Vx = (1.25)

ij i j k REVn t x V = (1.26) ( )REV 9: . (1.25) kx ( )REV , ,

8 Froiio, F., Tomassetti, G. and Vardoulakis, I. (2006). Mechanics of granular materials: the discrete and the continuum descriptions juxtaposed, Int. J. Solids Structures, 43, 76847720. 9 L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Vol.7, sect. 2, p.7, Pergamon Press, 1959.

. (2008) , . 1 9

( )0

0REV REV REV

REV REV

REV REV

ij kij kk ij

i i iV V V

ij k i ij kiV V

j k kjV V

x xx dV dV dV

x x x

x n dS dV

t x dS dV

= = =

=

(1.27)

1

REV

ij ijREV V

dVV

= (1.28) . (1.22) (1.27)

( )REV REV

ae e b a ck i ki k i k k i

e cV V

x t dS dV x F x x F

= = (1.29) (REV) 10,

1 ( )b a cij i i jcREV

x x FV

(1.30)

1 c cij i jcREV

FV

A (1.31)

b ai i ix x= A (1.32) . , .(1.31), Love11 . . (1.24) (1.31) ()

0 0c cijk i k ijk kic

F

A (1.33)

312 23 213 32 23 321: 0j = + ... (1.34)

10 e.g. information stemming from a DEM simulation of the mechanical description of a granular medium. 11 A.E.H. Love, A Treatise of the Mathematical Theory of Elasticity, Cambridge University Press, 1927.

. (2008) , . 1 10

ij ji (1.35) Love ,

( )12 c c c cij i j j icREV F FV + A A (1.36)

. 1-7:

1-1: (. 1-7)

Love , . (1.36), : , , [ / ]F kN m , . 1-7. ( a ) , (1) (6). ( 1-1) Love :

( )621

12

c c c cij i j j i

cS S = + A AA (1.37)

(a) :

[ ] 0 1/ 41/ 4 0

F = A (1.38)

. (2008) , . 1 11

1.2

. 1-8:

Cauchy 1 2 3( , , )O x x x (. 1-8.),

[ ] 11 12 1321 22 2331 32 33

= (1.39)

12,

13ij kk ij ij

s = + (1.40)

11 22 33kk = + + (1.41)

( )11 11 22 33 12 121 2 , , . . .3s s = = (1.42) . (;).

1 2 3( , , )O x x x

[ ] [ ]1 12 23 3

0 0 0 00 0 , 0 00 0 0 0

ss s

s

= = (1.43)

12 . deviator

. (2008) , . 1 12

[ ]ijA (;)

3 20 0ij ij A A AA I II III = + = (1.44) AI , AII AIII 3 3[ ] XA , [ ] [ ]ijA A= ( 1,2,3)i i = : 11 22 33 1 2 3AI A A A = + + = + + (1.45)

22 23 11 1311 12 1 2 2 3 3 132 33 31 3321 22

A

A A A AA AII

A A A AA A = + + = + + (1.46)

11 12 13

21 22 23 1 2 3

31 32 33

A

A A AIII A A A

A A A = = (1.47)

13.

. 1-9: Haigh-Westergaard

, , Haigh-Westergaard (. 1-9).

13 Pettofrezzo, A.J., Matrices and Transformations, Dover, 1966

. (2008) , . 1 13

1

2

3

OP

= (1.48)

(). , , 1 2 3 = = , (), , , , . 1 2 3 0 + + = . 1 ,

1 1 2 3kkI = = + + (1.49) ,

1 1 11 , . . .3

s I = + (1.50) 1 ,

1 1 2 3 0s kkJ s s s s= = + + = (1.51) ( 1,2,3)i i = ( 1,2,3)is i = . (;)

3 2 3 0s ss J s J = (1.52) 2 3 ,

( )2 2 22 1 2 31 12 2s ij jiJ s s s s s= = + + (1.53) ( )3 3 33 1 2 31 13 3s ij jk kiJ s s s s s s= = + + (1.54) , , . (1.52) s :

2 2 21 2 32 2 2

cos , cos , cos3 33 3 3

s s ss s s

J J Js s s = = = + (1.55) (. 1-10)

. (2008) , . 1 14

2 3 1 0

2 1 3 0

1 2 3 0

1 3 2 0

3 1 2 0

3 2 1

(1) : 0 / 3 ( ) :(2) : / 3 2 / 3 ( ) : 2 / 3(3) : 2 / 3 ( ) : 2 / 3(4) : 4 / 3 ( ) : 4 / 3(5) : 4 / 3 5 / 3 ( ) : 4 / 3(6) : 5 / 3 2 ( )

s s s

s s s

s s s

s s s

s s s

s

s s ss s s

s s ss s s

s s ss s s

= = + = +

= + = + 0: 2s s = +

(1.56)

. 1-10: . (1.52)