Механика деформируемого кабеля: Монография

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  • 8

    1 -

    1.1.

    ,

    -, , -. , - [3]. - -, , [41].

    -. , , .

    - -, , - . [3] , . - , , [71, 31, 68, 23, 7, 11, 28].

    - , , - . .. [35,36], .. [25], .. [4], .. [8], .. [6] .

    , - . -

    9

    , - . - , - - . , , [36].

    - - -- . , - - -- . - (a rod with helical anisotropy) - [73], 1977 . - , -- . 55. - , . [74], - , - (helical coordinate system). . - . - , - , . , , , . - , - . -

  • 10

    , -.

    [41] , , -, , . - , , - . [43] -, , . . -, . .

    -- (, , ) - () .

    60- , , [56], . . , - . , - kE - E 4/9 7/8. , - , . .. [21] - . , - kE E 0 . - - . :

    11

    04cos =

    EEk

    - .. , - 5%. , - , - , - , .. - -. - , [51,52].

    .. [17] [18,27,47] - , . - : - , , - .

    , . -, , . -, - -.

    , , - . - - , . ( -) , .

    , - .

  • 12

    [69, 70]. - - , . -. , - , . . , , , - , .

    , - , - , . - .

    - - -, -. -- , - [38, 39, 41, 43] - -. -, . -, (), -, , - .

    - . , - - [ ]. , - -

    13

    . - . "" . - . "-" , - , . -, - . , , - [15, 16]. - , - . - , , -, "" [ - ], , -, - - -, .. - .

    , - , , [] . -. y - - += yy~ , - . - , - ~ .

  • 14

    . , >0, - :

    .~ yy , ,

    .

    , - , ( ) . - , . - [53].

    1.2 ,

    1.2.1 -

    -, - z , - - h z [73]. . - , , - . , (, ). , , - () h . , constr = ( r -

    15

    ), ( - ). ,

    0= 0=r , 2/ r . ij

    kl , :

    klijklij C = (1.1) .. [36],

    (1.1) , - , - ( ). , , - -, - ijklC - .

    - , , - - .

    x , y , z Oz . Oz . - constr = [] h . h . -, -, , - , .. Oz . , -, r - h (.1.1).

    , , 1.1:

  • 16

    krtg = (1.2) hk /2= - .

    2r

    hh

    r

    z

    .1.1. h r .

    -

    , - h .

    .1.2 - : constr = (), constz = (-, Oz ), const= (, - Oz ), zr eee ,, . - , - : eeer ,, ( zr eee ,, ) re ( 2/ ), - . -, , :

    )(r - , Oz ; )( - h ; )( - , -

    ; 17

    er,er

    ee

    eze

    z

    y

    =const

    r=const

    x

    0

    z=const

    .1.2. :

    re

    - ;

    e

    - ;

    e

    - .

    ( zr eee ,, ) -

    ( eeer ,, ) -

    jiji eCe = (1.) },,{ ri , },,{ zrj

  • 18

    =

    sincos0cossin0

    001

    ijC (1.4)

    1.2.2

    -

    pqjpipij CC = , (1.5) :

    +==

    +==

    +==

    +=

    ++=

    =

    )cos(sincossin)(

    ;sincos;cossin

    ;sinsin2cos

    ;cossin2sin

    22

    22

    22

    zz

    rzrrr

    rzrzr

    zz

    zz

    rr

    (1.6)

    (1.6) -

    -: .,,,,, rrzzzr (1.6) - : .,,,,, rrr - - - , , -- .

    19

    , - .

    . 1.3 - . - - .

    ,,r - .

    r

    r

    r

    r

    . 1.3. - ,

    - :

  • 20

    +==

    ==

    ==

    ++=

    +=

    =

    )cos(sincossin)(

    ;sincos;cossin

    ;sincossin2cos

    ;coscossin2sin

    ;

    22

    22

    22

    zz

    rrzrrz

    rrrr

    z (1.7)

    -

    ,

    ij , -

    }3,2,1{, =lxl , :

    +

    =i

    j

    j

    iij x

    uxu

    21 , (1.8)

    { }3,2,1, =kuk - .

    :

    zzrz

    zr

    rzrr

    21

    21

    21

    21

    21

    21

    (1.9)

    :

    21

    21

    21

    21

    21

    21

    21

    r

    r

    rrr

    (1.10)

    (1.5) -

    - :

    +==

    ==

    ==

    ++=

    +=

    =

    )cos(sincossin)(2

    ;sincos;cossin

    ;sincossincos

    ;coscossinsin

    ;

    22

    22

    22

    zz

    rzrrr

    rzrrr

    zz

    zz

    (1.11)

    +==

    ==

    ==

    ++=

    +=

    =

    )cos(sincossin)(2

    ;sincos;cossin

    ;sincossincos

    ;coscossinsin

    ;

    22

    22

    22

    zz

    rrzrrz

    rrr

    z

    rr

    (1.12)

    [36] -

    :

  • 22

    +

    =

    +

    =

    +

    =

    =

    +

    =

    =

    ru

    ruu

    zu

    ru

    urz

    uz

    ur

    uur

    ru

    rz

    rzrz

    z

    zz

    r

    rr

    21

    ;

    ;1

    ;

    ;1

    ;

    (1.13)

    - . eeer ,, :

    eueueueueueuu rrzzrr ++=++= (1.14) (1.3) (1.14) :

    )sincos()cossin( zzrr eeueeueuu ++++= (1.15) (1.14) (1.15)

    =

    ==

    sincos;cossin

    ;

    uuuuuu

    uu

    z

    rr

    (1.16)

    :

    23

    =

    ==

    sincos;cossin

    ;

    z

    z

    rr

    uuuuuu

    uu (1.17)

    (1.16) (1.13) (1.11) (1.2), - :

    +

    +

    +

    =

    +

    +

    +

    =

    ++

    +

    +

    =

    +

    +

    +

    =

    +

    +

    +

    =

    =

    ru

    ru

    ru

    zu

    zu

    rku

    ru

    ru

    zu

    ru

    rku

    ru

    ru

    zu

    ru

    ru

    zu

    ru

    ru

    zu

    rur

    u

    rrr

    rrr

    r

    r

    rr

    2sinsin

    coscossin

    ;sin1coscoscos

    sincos

    ;sin1coscossin

    cossin

    ;cossin

    coscos

    ;sin

    cossin

    ,

    2

    2

    2

    2

    2

    (1.18)

  • 24

    (1.18) (1.17) - , :

    ( )

    +

    +

    +

    =

    +

    +

    +

    +

    =

    +

    +

    +

    +

    =

    +

    +

    +

    =

    +

    +

    +

    +

    +

    =

    =

    22

    2

    2

    2

    2

    cossin1

    cossin12

    ;sin

    cos1

    ;cos

    sin1

    ;sincossin1

    cos1

    ;cossin1

    sin1

    ,

    z

    rz

    rz

    rr

    rz

    rr

    zz

    r

    zz

    r

    rr

    urz

    ur

    uurz

    uz

    ur

    ur

    ur

    uur

    zu

    ru

    ru

    ruu

    r

    zuu

    rzu

    ruu

    r

    zuu

    rzu

    ruu

    r

    ru

    (1.19)

    - , , : - .

    25

    , , - , -.

    1.2.3

    (41) - c :

    =+

    ++

    +

    =+

    ++

    +

    =+

    +

    +

    +

    01

    ;021

    ;01

    zzrzrzz

    zrr

    rzrrr

    Xrrrz

    Xzrrr

    Xzrrr

    (1.20)

    zr XXX ,, - . (1.7) (1.2) -

    (2.20), - :

    ++

    +

    +

    +

    =+

    +

    +

    +

    +

    +++

    3

    22

    22

    coscossincos2

    sin2coscossin2sin1

    ;0sin

    coscossin1

    )coscossin2sin(1

    krr

    r

    Xz

    zr

    rr

    rrr

    r

    rr

    rr

    rrr

    rr

  • 26

    ( )

    ( )

    ) 0sincossincos

    cossincossin1

    coscossinsin

    cossincossin2cos

    ;0cossincossin

    cossincossin

    22

    32

    22

    22

    2

    =++++

    +

    +

    +

    ++

    +

    +

    +

    +

    +

    =+

    +

    +

    ++

    XX

    r

    kkr

    rzzz

    XXz

    zzk

    rr

    rrr

    r

    r

    (1.21)

    1.2.4

    - , - , - .,, eeer

    - , (1.1) - - . :

    ==

    =+=

    ==

    .1;1

    ;1;1

    ;1;1

    rr

    rrr

    r

    rr

    rrr

    r

    rrr

    rr

    GEEE

    GEEE

    GEEE

    (1.22)

    27

    (1.22)

    : EEEr ,, -

    ;,, r

    rr GGG ,, -:

    ij , { } ,,, rji = - , - i , -

    j . , ,

    , r - -, - . - :

    ===

    ===

    ====

    21

    12

    21

    ;;;

    ;;

    GGGGGEE

    EEE

    rr

    rr