HPl Sect-01-09

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HPl Sect-01-09

Text of HPl Sect-01-09

  • 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)