mmc general

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    P P V

    u

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    u

    grad

    grad

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    grad

    div

    div

    div

    rot

    Dg=

    D

    g

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    t

    S

    Sf Su

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    S Sf Su

    ,

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    t

    {R} = {0, e1, e2, e3} t0 = 0 M X1, X2, X3

    M=

    OM=X1

    e1+ X2e2+ X3

    e3

    M=

    3i=1

    Xiei =Xiei

    x1, x2, x3 m M

    t0 t

    m =3

    i=1xi

    ei =xiei

    xi Xj t

    m = m(M , t)

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    x1= x1(X1, X2, X3, t)x2= x2(X1, X2, X3, t)x3= x3(X1, X2, X3, t)

    t0

    t

    V(m, t) =

    dm

    dt

    Vi = xit (Xk, t) k= 1..3

    (m, t) =

    dV

    dt =

    d2m

    dt2

    i = 2xi

    t2 (Xk, t) k= 1..3

    u =Mm

    V(m, t) =

    u

    t(M , t) =

    du

    dt(M, t)

    {x , t} ou {x1, x2, x3, t}

    G G

    G= G(Xk, t)

    G

    G= g(xk, t)

    G

    G= dG

    dt =

    dG

    dt(Xk, t) =

    G

    t(Xk, t)

    g

    dG

    dt =

    dG

    dt(xk, t) =

    1

    dt

    G

    tdt +

    G

    x1dx1+

    G

    x2dx2+

    G

    x3dx3

    G= dG

    dt =

    G

    t +

    G

    x1

    dx1

    dt +

    G

    x2

    dx2

    dt +

    G

    x3

    dx3

    dt

    Mt

    = 0

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    G Gt

    V =

    dx1dt ,

    dx2dt ,

    dx3dt

    G

    gradG=

    Gx1

    , Gx2 , Gx3

    G=G

    t(xk, t) +

    gradG(xk, t).

    V(xk, t)

    =

    dV

    dt =

    V

    t + grad

    V .

    V

    i = Vi

    t +

    3k=1

    Vk.Vixk

    dM

    dm

    dX

    dx)

    xi = xi(Xk) dxi =

    xiXk

    dXk

    dx1= x1X1

    dX1+ x1X2

    dX2+ x1X3

    dX3dx2=

    x2X1

    dX1+ x2X2

    dX2+ x2X3

    dX3

    dx3= x3X1 dX1+ x3X2 dX2+

    x3X3 dX3

    F

    3 3

    F =gradx =

    x1X1

    x1X2

    x1X3

    x2X1

    x2X2

    x2X3

    x3X1

    x3X2

    x3X3

    {e1,

    e2,e3}

    dx= F .

    dX

    gradx = x =

    x1x2

    x3

    X1

    X2

    X3

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    u =M0M

    x =X+u

    gradx =grad

    X+ gradu

    gradx =F

    grad

    X=I

    F =I+ gradu

    dx

    dX

    dX

    dX

    dX.

    dX

    dX

    dX

    dx= F

    X

    dx =F

    X

    dx.

    dx

    dX.

    dX = [dx]t . [dx] [dX]t . [dX]

    dx.dx dX.dX = ([F] [dX])t . ([F] [dX]) [dX]t [I] [dX]

    dx.

    dx

    dX.

    dX = [dX]t

    [F]t . [F] . [I]

    [dX]

    dx.

    dx

    dX.

    dX = 2 [dX]t [E] [dX]

    [E] = 1

    2 [F]t . [F] . [I]

    E= 1

    2

    I+ gradu

    t.

    I+ gradu

    I

    E=1

    2

    gradu + gradt u + gradt u gradu

    E

    gradX=X

    =

    X1,X1 X1,X2 X1,X3X1,X1 X2,X2 X2,X3

    X1,X1 X3,X2 X3,X3

    =

    1 0 00 1 0

    0 0 1

    =I

    [dx]

    x1x2

    x3

    [dx]t [x1, x2, x3]

    [X]t [I] [X] = [X]t [X] [I]

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    E = 1

    2

    gradu + gradt u

    dx=

    dX+ gradu .

    dX

    dx=

    dX+

    1

    2

    gradu gradt u

    .dX+

    1

    2

    gradu + gradt u

    .dX=

    dX+ .

    dX+ .

    dX

    dx=

    dX+

    dX+ .

    dX

    X

    x

    mm =

    MM +

    dX+ .

    dX

    u (M) =M m

    u (M) =Mm

    u(M) =

    translation

    u(M) +

    rotation

    dX solideindeformable + .

    dX deformation

    =

    0 12

    u1x2

    u2x1

    1

    2 u1x3

    u3x1

    12

    u1x2

    u2x1

    0 12

    u2x3

    u3x2

    1

    2

    u1x3

    u3x1

    1

    2

    u2x3

    u3x2

    0

    = 1

    2 u2x3

    u3x2

    12

    u1x3

    u3x1

    1

    2

    u1x2

    u2x1

    .

    ij = ji

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    =

    11 12 1312 22 2313 23 33

    = x1X1

    12 x1X2 + x2X1 12 x1X3 + x3X1

    12

    x1X2

    + x2X1

    x2

    X212

    x2X3

    + x3X2

    12

    x1X3

    + x3X1

    12

    x2X3

    + x3X2

    x3

    X3

    ij =ji

    = t

    dX

    e

    dX =

    dX.e dx

    X =

    X

    dx2 dX2 = (dx dX) (dx + dX) = 2dX.E

    dX

    dx + dX 2dX e = dxdXdX

    e

    (dx dX) (dx + dX) e.dX.2dX= 2 dX2 e .e

    e =e ..e

    e

    dX

    11=

    u1x1

    est elongation selon x122=

    u2x2

    est elongation selon x233=

    u3x3

    est elongation selon x3

    e=

    e1 e2 e3 11 12 1312 22 23

    13 23 33

    e1e2e2

    103 106 def 106 250def= 250 106 = 2.5 104.

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    N

    M

    n m

    n m NM= 2N .EM

    n m=sinNM

    |n| |m| =

    sinNM(1 + N)(1 + M)

    n m NM

    N

    M

    NM= 2N .E

    M

    def

    12=

    12

    u1x2

    + u2x1

    est la demi distorsion de langle x1,

    x2

    13= 12

    u1x3

    + u3x1

    est la demi distorsion de langle x1,

    x3

    23= 12

    u2x3

    + u3x2

    est la demi distorsion de langle x2,

    x3

    {e1 ,e1 ,

    e1}

    dX1, dX2, d

    X3 dV =

    dX1dX2dX3 dv = dx1dx2dx3

    =v V

    V =

    (1 + 11) dX1. (1 + 22) dX2. (1 + 33) dX3dX1dX2dX3

    = V

    V 11+ 22+ 33=

    u1x1

    +u2x2

    + u3x3

    = tr

    = div u

    M0 {S}

    2 3 3

    tr(A)

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    .n proportionnel a n

    det I= 0

    M0

    = I 0 00 II 00 0 II I {eI ,eII ,eIII }

    KI

    .eK=

    0

    K

    eK = 1

    I II II I

    {eI,eII,

    eII I}

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    det

    I

    = 0

    3 + K12 + K2 + K3= 0

    K1= tr

    K2= 12

    tr2

    tr

    2

    K3= det

    =

    I 0 00 II 00 0 II I

    {eI ,eII ,eIII }

    n = ceI +seII

    t = seI +c

    eII {eI,

    eII} c= cos s= sin

    n eI

    n

    (n ) =n =

    I 0 00 II 00 0 III

    cs0

    = IcIIs

    0

    = IceI+ IIseII (n ) n t

    n

    n ,t

    (n ) =

    = Ic2 + IIs22 = (I II)cs

    (n ) =

    = I+II2

    + III2

    cos(2)

    2 = III

    2 sin(2)

    {eI,eII} M

    (n )

    , 1

    2

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    0

    [II, I] R= 12(I II) ; 0=

    12(I+ II)

    ,

    2

    ,

    n + 2

    +

    +2

    R= LA R= 4LD2

    L

    D

    R+ R= ( + ) 4(L+L)(DD)2

    RR =

    LL

    RR =J

    20m

    2

    2

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    2mm 60mm

    30

    100m

    120 350

    J= 2, 0

    R/R= J.

    Vout= VinR1R3 R2R4

    (R1+ R2) (R3+ R4)

    Vout

    Vin

    R3

    = R

    R

    dR

    Vout= Vin1

    4

    dR

    R

    R3

    R4 R

    3 = meca+therm

    4 = therm

    VoutVin

    =R (R+ dRmeca+ dRtherm) R (R+ dRtherm)

    4R2

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    Vout

    Vin=

    1

    4

    dRmecaR

    def

    0 R

    1

    2

    3

    1

    2

    3

    1

    I 1

    C(o, 0)

    [C, 1]

    2

    [C, 2]

    +2

    [o, 1]

    [C, 3] 2 [o, 1]

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    1 2 3

    o R

    1= o+ R cos(2)2= o+ R cos(2 ( ))3= o+ R cos(2 ( + ))

    1 o= R cos212

    (2+ 3) o= R cos2 cos2

    12

    (2+ 3) o= (1 o)cos2

    o

    si = 4 o=

    2+321cos 22(1cos2)

    si = 4 o=

    12

    (2+ 3)

    = 1

    2arctan

    1

    sin2

    2 31 o

    R=

    1 ocos2

    121= R sin2122= R sin2( )123= R sin2( + )

    =1

    2

    gradu + gradtu

    et =

    1

    2

    gradu gradtu

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    2ij =ui,j+ uj,i 2ij =ui,j uj,i fxk

    =f,k

    2

    gradij

    xk

    2ij,k =ui,jk uj,ik

    uk,ij

    2ij,k = (ui,jk+ uk,ij) (uj,ik+ uk,ij) = (ui,k+ uk,i),j (uj,k+ uk,j),i

    ij,k = ik,j jk,i (a)

    ij,l = il,j jl,i (b)

    xl

    (a) xk (b),

    ij,kl ij,lk = (ik,j jk,i),l (il,j jl,i),k = 0

    ikxjxl

    jkxixl

    ilxjxk

    + jlxixk

    = 0

    211x2

    2

    + 222x2

    1

    2 212

    x1x2= 0

    222x2

    3

    + 233x2

    2

    2 223

    x2x3= 0

    233x2

    1

    + 211x2

    3

    2 231

    x3x1= 0

    211x2x3

    + x1 23x1 31x2 12x3 = 0222x3x1

    + x2

    31x2

    12x3 23x1

    = 0

    233x1x2

    + x3

    12x3

    23x1 31x2

    = 0

    grad

    div

    + gradt

    div

    grad

    grad

    tr

    = 0

    dij =

    ij,k.dxk

    dij =

    (ik,j kj,i) .dxk

    dui =

    ui,j .dxj =

    (ij ij) .dxj

    Su

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    S Sf Su

    {S}

    {S}

    {Sf}

    fs(P)

    {Su}

    {Su}

    u =u0 sur {Su}

    ui = u0i sur {Su} a vec i [1, 2, 3]

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    1

    1/137

    106

    1038

    {FextS}

    DS/R0

    {FextS}G =

    DS/R0G

    F (ext S) =

    S

    dv= MGMG(ext S) =

    S

    GM dv =

    G

    1040

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    dv

    M

    G G

    {FextS}G = {0}

    F (ext S) =

    0

    MG(ext S) =

    0

    S1

    S2 S2

    S1 {Fext S}

    (S) (S1) (S2)

    (S)

    {Fext S} =

    DS/R0

    (S1) (S2) (S1) {S2 S1}

    {Fext S1} + {S2 S1} = DS1/R0

    (S2) (S1) (S2) {S1 S2}

    {Fext S2} + {S1 S2} =

    DS2/R0

    [{Fext S1} + {Fext S2}] +{S2 S1} + {S1 S2} =

    DS1/R0

    +

    DS2/R0

    {Fext S}+{S2 S1}+{S1 S2} =

    DS/R0

    {S2 S1} = {S1 S2}

    (S2) (S1)

    (S1) (S2)

    S1 S2 S2 S1

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    (S)

    (S)

    (S2) (S1)

    +

    (S)

    (S1)

    (S2) (S1) ()

    (S)

    (S2) (S1)

    M

    (S1) n

    M n (S1)

    (S2) (S1)

    M n

    F.L2

    1P a = 1N/m2

    1M P a= 106P a

    T(M, n )

    : (M,

    n )

    T(M,

    n )

    M

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    T(M, n )

    T(M, n ) =n

    n + nt

    n n

    nt n

    t =

    T(M, n ) n

    n

    t

    n n

    (S)

    M

    e1 ,e2 ,

    e3

    1

    dS1

    e1

    1 =T(M, e1) +

    e1

    1 1

    2

    3

    k k= 1..3

    k = k1

    e1+ k2e2+ k3

    e3 kj

    k

    j

    k, j = 1..3

    e1 dS1= 12

    dx2dx3 1= (11

    e1+ 12e2+ 13

    e3)

    e2 dS2= 12

    dx3dx1 2= (21

    e1+ 22e2+ 23

    e3)

    e3 dS3= 12

    dx1dx2 1= (31

    e1+ 32e2+ 33

    e3)

    n =n1e1+ n2

    e2+ n3e3 dS

    T(M, n ) =T1

    e1+ T2e2+ T3

    e3

    dV g

    dV

    dS1= dS.n1 dS2= dS.n2 dS3= dS.n3 dV =

    16

    dx1dx2dx3

    T dS

    1dS1

    2dS2

    3dS3+

    g dV = dV

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    T

    1n1

    2n2

    3n3

    dS+ g dV = dV

    T

    1n1

    2n2

    3n3

    0

    T1T2T3

    = 11 12 1321 22 2331 32 33

    n1n2n3

    :(M, n )

    T(M, n )

    T(M, n ) =(M).n

    (M)

    M

    (S) (S)

    D (S)

    (S) {S}

    {D} {S} (D)

    P

    n {D}

    n M

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    R0

    D

    fv(M)dv+D

    T(P,n ).dS=

    D

    (M)dv

    D

    fv(M)dv+

    D

    (P).dS=

    D

    (M)dv

    D

    fv(M).dv+

    D

    div ().dv=

    D

    (M).dv

    {D} {S}

    div +

    fv =

    div + g =

    0

    R0

    O

    D

    OM

    fv(M)dv+

    D

    OP

    T(P, n ).dS=

    D

    OM (M)dv

    D

    OM

    fv(M)dv+

    D

    OP

    dS=

    D

    OM

    div +

    fv

    dv

    D

    OP

    dS=

    D

    OM

    div dv

    f.dS

    div f.dv

    dS

    divdv

    divf =

    f =fk,k

    div =

    div = ik,k.

    ei k

    f dv = 0 f 0.

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    e1

    D

    OP

    n

    .e1dS=

    D

    OM

    div

    .e1dv

    D

    (x23j x32j) .njdS=

    D

    (x23j,j x32j,j) dv

    D

    xj(x23j x32j) dv =

    D

    x2

    3jxj

    x32jxj

    dv D

    2j3j + x23jxj

    3j2j x32jxj

    =x23jxj

    x32jxj

    2j3j 3j2j = 0

    ajFbj =Fab

    23 32= 0

    ij =ji

    = t

    ei ej ij

    e i e j ji

    e j e i

    23 32

    .n = 11n1+ 12n2+ 13n3 = 1ini

    2ini3ini

    OP .n =

    x23ini x32inix31ini x13ini

    x123ini x21ini

    f.nidS

    fxi

    dv

    ij

    ij = 0si i =j

    ij = 1si i= j

    xixj

    =ij

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    det

    I

    = 0

    III II I

    e I,e I I,

    e I II

    III III

    e J ( JI).e J =

    0 (J =

    I , I I , I I I )

    =

    I 0 00 II 00 0 III

    {eI ,eII ,eIII }

    det I= 3 + L12 + L2 + L3

    L1= tr

    L2= 12

    tr2

    tr

    2

    L3= det

    e 1 , e 2 , e 3

    L1= kk =1+ 2+ 3L2=

    12

    (iijj ijij)L3= det

    e I,e I I,

    e I II

    L1= KK=I+ II+ II IL2= III+ IIII I+ II IIL3= IIIIII

    = II I = p p

    = det

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    S

    D

    = S+ D tr D= 0 tr s= tr = 13 kk S= 13 kkI D = 13tr I

    =

    13(211 22 33) 12 1312 13(222 33 11) 2313 23

    13

    (233 11 22)

    D

    +1

    3(11+ 22+ 33)

    1 0 00 1 00 0 1

    S

    J1= tr

    D

    = 0 pardefinition

    J2= 12tr

    2D

    J3= det D

    D =

    sI 0 00 sII 00 0 sIII

    {eI ,eII ,eIII }

    J1= 0

    J2= sIsII+ sIIsII I+ sII IsIJ3= sIsIIsII I=

    1

    6 (I II)2 + (II II I)2 + (II I I)2

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    M

    (S)

    n n1, n2, n3 {e I,

    e I I,e I II}

    M

    T(M, n ) =(M).n

    123

    = I 0 00 II 0

    0 0 II I

    n1n2n3

    = In1IIn2

    II In3

    avec I> II> II I

    T =

    T(M, n ) =n

    n + nt

    n

    t

    n =

    T .n =In21+ IIn

    22+ IIIn

    23

    n =T nn = T2 2n

    2n+

    2n =

    2In

    21+

    2IIn

    21+

    2II In

    21

    n = n21+ IIn22+ II In

    23

    n21+ n22+ n

    23= 1

    I> II> II I

    n21= 2n+(nII )(nIII)

    (III )(IIII )

    n22= 2n+(nIII )(nI)(IIIII )(III)

    n23= 2n+(nI)(nII )(IIII)(IIIII )

    et I> II> III

    n2j 0 2n+ (n II) (n II I) 02n+ (n III) (n I) 02n+ (n I) (n II) 0

    T

    {, } (n, n)

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    (dS)

    (dSII I) {e I,

    e I I}

    II I

    (dS)

    (dSIII) e I II

    n e I II n3 =0

    n {e I,

    e I I}

    n3 = 0

    2n+ (n I) (n II) = 0

    n

    I+ II

    2

    2+ 2n =

    I II

    2

    2

    0=

    12

    (I+ II) ; 0

    R = 12(I II)

    =

    I 0 00 II 00 0 0

    {eI ,

    eII ,eIII }

    =

    I 0

    0 II

    {eI ,eII}

    eIII

    eII I n =n1

    eI+ n2eII

    III = 0

    3

    II III

    det I = 0

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    n1, n2 n e I,

    e I I n = cos eI+

    sin eII n

    eI = eI,

    n

    c= cos s= sin

    T1= IcIIs {e I, e I I}

    n =

    cs

    t =

    s

    c

    +

    2 n eII I

    T1= 1

    n + 1t avec 1= Ic

    2 + IIs2 et 1= cs (I II)

    2

    1= I+II

    2 + III

    2 cos21= III2 sin2

    1=

    I+ II

    2

    o

    +

    I II

    2

    R

    cos(2)

    1=

    I II

    2

    R

    sin(2)

    {, } 1

    T1

    0(o, 0)

    R o= 12(I+ II) R=

    12(I II)

    e I [o] 2 O

    =

    11 12

    12 22

    {e1,e2}

    T1

    n 1=e 1

    T2

    n 2=e 2

    T1=

    11 12

    12 11

    10

    =

    11

    12

    T1= 11

    e1+ 12e2 =11

    n1+ 12t1

    =

    cos2= cos2 sin2 = c2 s2 sin2 = 2cos sin

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    T2=

    11 12

    12 11

    01

    =

    1222

    T2= 12e1+ 22

    e2 =22

    n2 12

    t2 car

    t2 = e1

    n1,n2

    2 1

    2

    centre o=

    12

    (11+ 22)

    rayon R=

    (11 22)

    2+ 4212

    I=o+ R II=o R

    tan(2) = 121o =12

    arctan 121o

    n 1 =e 1

    eI

    {S}

    {Sf}

    T(P) =(P).

    n(P) =

    fs(P) P {Sf}

    (P) = 0

    n(P)

    T(P) =(P).

    n(P)

    T(P) =

    fs(P)

    x= f(u, v)y = g (u, v)z= h (u, v)

    (x , y , z) = 0. Oz R x= R cos , y = R sin , z = h h x2 +y2 =R2

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    S Sf Su

    I II= N

    Ce

    N

    C

    e

    [C] =

    L2.F1

    1Brewster =

    1012 P a1

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  • 7/22/2019 Mmc General

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    = E.

    e

    < 0

    d = E.d

    p

    E= d/d

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    t0

    J(t0, t) (t) =J(t0, t).0

    th = T

    11

    22

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    F L 11

    = F/A

    A

    {11, 11} = {, }

    e

    360 M P a

    1600 M P a

    /= E 200 000M P a= 200 GP a

    22/11 22= 11

    0.3

    11= E11 22= 11

    / p

    s > e

    dd =E

    s

    1m3

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    11cm h22cm 16cm h32cm

    = 11

    22 F

    A = 11 = F/A

    11=E11 E 40 GP a

    22 = 11

    0.2

    ,

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    G

    G E

    2 (1 + )

    thermiqueij =.T direction {i} si i= jthermiqueij = 0 si i =j

    acier =

    beton = 10.106K1

    = S

    0

    + T

    T0

    0

    S

    =(1 + )

    E

    Etr

    I+ Tth

    I

    E [E] =

    F.L2

    P a

    MP a

    GP a

    [0 ; 0.5]

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    [] =

    T1

    K1

    T

    T0

    = 2 + tr

    I TI

    12 12 = 212

    = G

    [] =

    F.L2

    P a

    [] =

    F.L2

    P a

    thermique = p = T

    [] =

    F.L2.T1

    Pa/K

    = C . = S.

    C

    S

    F.L2

    =

    11 12 1312 22 2313 23 33

    = 112233

    23= 2313= 1312= 12

    =

    11 12 1312 22 2313 23 33

    =

    112233

    23= 2.23

    13= 2.1312= 2.12

    T0

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    |11 1| |22 2| |33 3| |23 ou 32 4| |13 ou 31 5| |12 ou 21 6|

    2

    = S

    1122332313

    12

    =

    1E

    E

    E 0 0 0

    E

    1E

    E 0 0 0

    E

    E

    1E 0 0 0

    0 0 0 2(1+)E 0 0

    0 0 0 0 2(1+)E 0

    0 0 0 0 0 2(1+)E

    .

    1122332313

    12

    S 6 6 S

    = C

    1122332313

    12

    =

    2 + 0 0 0 2 + 0 0 0 2 + 0 0 00 0 0 0 00 0 0 0 0

    0 0 0 0 0

    .

    1122332313

    12

    = E(1+)(12)= G = E2(1+)

    = E12

    2 + = E 1(1+)(12)

    3 + 2= E12

    E= 3+2+=

    2(+)

    = 3+2

    11+ =

    2(+)3+2

    1+

    = 3+2

    = E1 +

    + 1 2

    tr I E1 2

    T I

    2ij ij

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    = 1

    2

    3 + 2tr

    I

    +

    3 + 2T I

    PPV

    div +

    fv =

    u (S)

    S

    div .u dv+

    S

    fv

    u dv=

    S

    .u dv

    div .u =ij,j .u

    i = (ij .u

    i ),j ij .u

    i,j

    ij .u

    i,j = ji .u

    j,i

    ij .ui,j = ij .u

    j,i ij .u

    i,j = ij .

    12

    uj,i+ u

    i,j

    ij .u

    i,j = ij .

    ij

    = 1

    2

    gradu + gradt u

    S

    div .u : dv+ S

    fvu dv=S

    .u dv

    S

    .u

    .dS+

    S

    :

    dv+

    S

    fv

    u dv=

    S

    .u dv

    S

    S= Su Sf

    Su .

    n .u dS+ Sf .n .u dS+ S :

    dv+

    S

    fv

    u dv= S

    .u dv

    Pe Su

    fu.

    u dS

    Pel

    +

    S

    fs .

    u dS

    Pes

    +

    S

    fv

    u dv

    Pev

    +

    Pi S

    :

    dv=

    Pa S

    .u dv

    Pic u

    ijij

    :

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    Pe + Pi =P

    a

    Pel =

    Su

    fu.

    u dS

    fu

    Pes=Sf

    fs .

    u dS

    Pev =S

    fv

    u dv

    Pi =S

    :

    dv

    Pa =

    S

    .u dv

    {u }

    Ru.u = 0

    Pel = 0

    u

    {u } {u } Pa =S

    .u dv =S

    dudt .

    u dv

    Pa =

    S

    12

    ddt

    u2

    dv = ddtS

    12

    u2

    dv

    Ec

    Pe+ Pi = dEc

    dt

    Pes=Sf

    fs .

    u dS

    Pev =S

    fv

    u dv

    Pi =S

    : dv

    Ec =S

    12

    u 2dv

    pi = :

    dt

    dw = pi.dt= : dt= : d

    = S

    dw= : Sd= 1

    2d

    : S

    dw= 12

    d

    :

    t0= 0 t

    we =1

    2 : =

    1

    2: S =

    1

    2: C

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    We=

    S

    we dv=

    S

    1

    2 : dv=

    S

    1

    2: Cdv

    we =1

    2 =

    1

    2S=

    1

    2C

    we =1

    2

    2ijij +

    2

    we= 1

    2

    1 +

    E ijij

    E2

    (S)

    t0= 0 t

    Tf =We+ Ec

    I

    te > 0

    II I

    c

    e < 0

    II

    II I ce et I

    te

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    max e =

    12

    e

    max= 1

    2 max {|I II| , |II III| , |II I I|}

    1

    2e I II I e

    T =I II I

    T e

    s

    J1= 0J2= sIsII+ sIIsII I+ sII IsIJ3= sIsIIsII I

    = 1

    6

    (I II)

    2+ (II III)

    2+ (II I I)

    2

    J3 J2

    II=III= 0 J2= 13

    2Iet I e

    J2 1

    3

    2e (I II)2 + (II II I)

    2 + (II I I)2 22e

    VM=

    1

    2

    (I II)

    2 + (II II I)2 + (III I)

    2

    VM e

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    {S}

    fv(M)

    (S)

    (Su)

    (Sf)

    fs(P)

    (M)

    (M)

    u(M)

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    ij =ji

    div +

    fv =

    0

    T(P) =(P).

    n(P) =

    fs(P) P {Sf} et

    n {S}sortant

    = t ij =ji

    graddiv

    + gradtdiv

    gradgrad

    tr

    = 0

    =1

    2

    gradu + gradt u

    ui = u0i sur {Su}

    = (1 + )

    E

    Etr

    I+ T

    thI

    = E

    1 +

    +

    1 2tr

    I

    E

    1 2T I

    = 2 + tr

    I TI

    =

    1

    2 3 + 2tr I+ 3 + 2 T I

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    11 12 13 22 23 33 u1 u2 u3 11 12 13 22 23 33

    u(M)

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    div +

    fv =

    0

    div

    2 + tr

    I

    +fv =

    0

    2div

    12

    gradu + gradt u

    +

    div

    divu .I

    +

    fv =

    0

    div

    w.I

    =

    grad (w)

    div

    gradt u

    =

    graddivu

    div

    gradu

    = u

    u + ( + )

    graddiv u +

    fv =

    0

    u =

    graddivu

    rot

    rotu

    ( + 2)graddivu

    rot

    rotu +fv =

    0

    u + ( + )

    graddivu +

    fv =

    0

    ( + 2)graddivu

    rot

    rotu +fv =

    0

    ui = u0i sur {Su}

    div

    w.I

    =

    w 0 00 w 0

    0 0 w

    .

    x1

    x2

    x3

    =

    wx1wx2wx3

    = gradw

    div

    gradtu

    =

    u1x1

    u2x1

    u3x1

    u1x2

    u2x2

    u3x2

    u1x3

    u2x3

    u3x3

    .

    x1

    x2

    x3

    =

    u1,11+ u2,21+ u3,31u1,12+ u2,22+ u3,32

    u1,13+ u2,23+ u3,33

    =

    (u1,1+ u2,2+ u3,3),1(u1,1+ u2,2+ u3,3),2

    (u1,1+ u2,2+ u3,3)3

    =

    graddivu

    div

    gradu

    =

    u1x1

    u1x2

    u1x3

    u2x1

    u2x2

    u2x3

    u3x1

    u3x2

    u3x3

    x1

    x2

    x3

    =

    u1,11+ u1,22+ u1,33u2,11+ u2,22+ u2,33

    u3,11+ u3,22+ u3,33

    =

    u1u2

    u3

    = u

    u v w

    =

    u .w

    v

    u .v

    w

    u

    =.u

    2u

    T u + (+)

    graddivu +

    fv

    (3+ 2) gradT =

    0

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    {u }

    ui =u0i sur {Su}

    NOTE

    grad div + grad

    t

    div grad

    grad tr = 0

    = (1 + )

    E

    Etr

    I tr

    = 1 2

    E tr

    et =(1 + )

    E

    E tr

    I

    =tr

    graddiv

    (1 + ) I

    + gradt

    div

    (1 + ) I

    grad

    grad ((1 2) )

    (1 + ) I

    = 0

    (1 + )grad

    div + gradt

    div

    2gradgrad

    grad

    grad

    + gradt

    grad

    (1 2)grad

    grad

    + I= 0

    graddiv + gradt

    div

    1

    (1 + ) grad grad I= 0

    gradgrad

    2

    xixj=

    2

    xjxi

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    divu + ( + )

    div

    grad div u + div

    fv = 0

    div

    grad f = f

    div

    u = div

    u

    div

    u =tr = = 12E

    (2 + )1 2

    E + div

    fv =

    0

    2 +

    2 + 3 + div

    fv =

    0

    1

    1 +

    + divfv =

    0

    + 1

    1 + grad

    grad +grad

    fv+ gradt

    fv+

    1 div

    fvI= 0

    + 1

    1 + grad

    grad = 0

    + 1

    1 + grad

    grad = 0

    div fv = 0

    2= 0

    = 0

    T

    + 1

    1 +grad

    grad + grad

    fv+ gradt

    fv+

    1 div

    fvI+

    E

    1 + grad

    gradT+ T I= 0

    gradfv+ gradt

    fv+ +

    1

    1 + grad

    grad

    1 +I= 0

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    div + fv = 0

    T(P) =(P).

    n(P) =

    fs(P) P {Sf} et

    n {S}

    NOTE

    {F1} (D) 1, 1,

    u 1

    {F2} (D)

    2, 2,u 2

    {F1+ F2}

    1+ 2, 1+ 2,u 1+

    u 2

    (D) {F} 1, 1, u 1 2, 2, u 2

    1 2, 1 2,

    u 1 u 2

    (D)

    We =

    S

    1

    2

    1 2

    : S

    1 2

    dv= 0

    We 1 2= 0

    u

    {u }

    u ,

    We

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    V

    V(u) =Ve(u) + Vf(u

    )

    {u}

    Ve

    Ve(u) =

    S

    1

    2

    :C

    .dv=

    S

    1

    2 :C .dv

    Vf

    Vf(u) =

    S

    fv .

    u.dv

    forces volumiques

    +

    S

    fs .

    u.dS

    forces surfaciques

    {u} {u}

    {u} = {u} + {u} avec{u}CA.

    {u}

    = champ dedeformation reel

    + champdesperturbationsde deformation

    Ve(u) =

    S

    1

    2 : C + 2 : C

    +: C dv

    Vf(u) =

    S

    fv .

    u.dv+

    S

    fs .

    u.dS

    S

    fv .

    u .dv+

    S

    fs .

    u .dS

    S

    t0

    f(t).

    du

    dv

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    V(u) =

    S

    1

    2: C.dv

    S

    fv .

    u .dv+

    S

    fs .

    u .dS

    V(u)

    Pi

    S

    : .dv +

    Pe S

    fv .

    u.dv+

    S

    fs .

    u.dS

    Pi+Pe=0

    +

    S

    1

    2 : C.dv

    forme bilineaire symetrique0

    V(u)

    V(u) V(u) u CA.

    k

    F

    u

    We= 12ku

    2

    TF = 12F u

    We = TF u=

    Fk

    Ve=

    12

    ku2

    VF = F u

    V= 12ku2 F u

    dVdu u=ureel = 0 ku F = 0 u= Fk

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    u (M) =ni=1

    Zi.i(x,y,z) aveci CA.

    Zi i

    u

    V(u )

    V(Zi) V

    V

    VZj

    = 0 j

    K11 K12 K12 K22

    Kn,n1Kn,n1 Kn,n

    .Z1Z2

    Zn

    = F1F2

    Fn

    [K] [Z] = [F]

    [K]

    [Z]

    Zi

    [F]

    [Zi] [Z] = [K]1 . [F]

    u (M) =n

    i=1 Zi.i(M)

    = 12

    gradu + gradt u

    = 2

    + I

    u , ,

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    z

    uz

    zz = zx = zy = 0

    xx, yy , xy

    zz = +.(xx+ yy)

    z

    z

    Z=.z

    xx,yy,xy

    zz zz = .(xx+ yy)

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    (M) = 11(x1, x2) 12(x1, x2) 012(x1, x2) 22(x1, x2) 00 0 0

    (M) = 11(x1, x2) 12(x1, x2) 012(x1, x2) 22(x1, x2) 00 0 0

    (M) =

    11(x1, x2) 12(x1, x2)12(x1, x2) 22(x1, x2)

    (M) =

    11(x1, x2) 12(x1, x2)12(x1, x2) 22(x1, x2)

    (M) =

    11(x1, x2) 12(x1, x2) 012(x1, x2) 22(x1, x2) 00 0 33(x1, x2)

    (M) =

    11(x1, x2) 12(x1, x2) 012(x1, x2) 22(x1, x2) 00 0 33(x1, x2)

    e3

    33= (11+ 22)

    e3

    33= +(11+ 22)

    (M) =

    11(x1, x2) 12(x1, x2)12(x1, x2) 22(x1, x2)

    (M) =

    11(x1, x2) 12(x1, x2)12(x1, x2) 22(x1, x2)

    211x2

    2

    + 222x2

    1

    2 212

    x1x2= 0

    222x2

    3

    + 233x2

    2

    2 223

    x2x3= 0

    233x2

    1

    + 211x2

    3

    2 231

    x3x1= 0

    211x2x3

    + x1 23x1

    31x2 12x3 = 0222

    x3x1+ x2

    31x2

    12x3 23x1

    = 0233x1x2

    + x3

    12x3

    23x1 31x2

    = 0

    31= 32= 0

    x3 0,

    211x2

    2

    + 222x2

    1

    2 212

    x1x2= 0

    233x2

    2

    = 0233x2

    1

    = 0233x1x2

    = 0

    33

    211x22

    +222

    x21 2

    212x1x2

    = 0

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    33(x1, x2)

    33(x1, x2) = ax1+ bx2+ c.

    tr

    I

    = 2.

    =11+ 22=

    = (1 ) (11+ 22) = (1 )

    = (1 + ) (11+ 22) = (1 + )

    = 11+ 22=

    11x1

    + 12x2 + fv1 = 0

    12x1

    + 22x2 + fv2 = 0

    fv3 = 0

    div

    +

    fv =

    0

    11x1

    + 12x2 + fv1 = 0

    12x1

    + 22x2 + fv2 = 0

    fv3 = 0

    div

    +

    fv =

    0

    (S)

    T(P, n ) =(P).n =

    11 12 012 22 00 0 0

    . n1n2

    0

    T1T20

    = fs1fs2

    fs3

    e3

    (S)

    T(P, n ) =(P).n =

    11 12 012 22 00 0 33

    . n1n2

    0

    T1T20

    = fs1fs2

    fs3

    = 11+ 22

    11= 1122

    E22=

    2211E

    12= (1+)E 12

    = (1 + ) (11+ 22)

    11= 1 +

    E 11

    E(1 + ) (11+ 22)

    11=1 2

    E

    11

    1 22

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    22=

    1 2

    E

    22

    1 11

    12= 1 +

    E 12=

    1 2

    E

    1

    1

    12

    12=

    1 2

    E

    1 +

    1

    12

    E=E 1 + 2

    (1 + )2 et =

    1 +

    E = E

    1 2 et =

    1

    11= 11

    22E

    22= 22

    11E

    12=(1+)E 12

    = E

    1 +

    +

    1 I

    = 1 +

    E

    EI

    = 1 +

    E

    I

    = 1 +

    E

    EI

    112212

    = 1E

    1 0 1 00 0 2 (1 + )

    112212

    1122

    12 = E

    1 2

    1 0 1 0

    0 0 12(1 ) 1122

    12 1122

    12 = E

    1 + 112

    12 0

    12

    112 0

    0 0 12 1122

    12

    we =1

    2(11.11+ 22.22+ 12.12)

    E E = E/12 = /1

    I=

    1 00 1

    tr

    I

    = 2

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    1E2

    x22

    [11 22] + 1E2

    x21

    [22 11]

    21+E212x1x2

    = 0

    (11+ 22)

    (1 + )211x2

    1

    + 222x2

    2

    + 2 212

    x1x2

    = 0

    div+

    fv =

    0

    11x1 + 12x2 + fv1 = 0 x112x1

    + 22x2 + fv2 = 0

    x2

    211x21

    +222

    x22+ 2

    212x1x2

    + divfv = 0

    + (1 + )divfv = 0

    + (1 + )divfv = 0

    + 1

    1 div

    fv = 0

    div+ fv = 0

    fv =

    gradV fv1 = V/x1 et f

    v2 = V/x2

    11x1

    + 12x2 + Vx1

    = 012x1

    + 22x2 + Vx2

    = 0

    (11+V)

    x1= 12x2

    (22+V)x2

    = 12x1

    e1 fv =g

    e1 V =g x1

    2hx1x2

    = 2h

    x2x1 h

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    11+ V =

    x2

    22+ V = x1

    12= x1

    12= x2

    x1 =

    x2

    =

    x2et =

    x1

    11+ V = x2

    = 2x2

    2

    22+ V = x1

    = 2x2

    1

    12= 2x1x2

    + (1 + )divfv = 0

    + (1 + )divfv = 0

    =11+ 22= 2V

    ( 2V) + (1 + )V = 0

    2 + (1 )V = 0

    2 + (1 )V = 0

    2 +1 2

    1 V = 0

    V = 0.

    2 = 0

    divgradh= h

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    (x1, x2)

    2 = 0

    11=

    2

    x2

    2

    V

    22= 2x2

    1

    V

    12= 2x1x2

    T(P, n ) =

    fs sur Sf

    u = u 0sur Su

    =

    ix

    n1xm2

    z

    r, z

    r

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    g

    xi=g,i ou

    g

    xi=ig

    aibi =3

    i=1 aibi

    RR R c= ab = a.b = a b

    R R3 R3 v =u =.u

    R3 R3 R3 w = u v

    R3 R3 R a= u v = u .v

    R3 R3 R3 R a= (u , v , w ) = (u v) .w

    R3 R3 R3 R3 (u v) w = (u .w ) .v (v .w ) .u

    R3 R3 R32 A= u v

    R23 R23 R23 C=AB = A.B

    R23 R3 R3 v =Au =A.u

    R23 R23 R

    a= A : B

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    M =x1

    e1+ x2e2+ x3

    e3

    M=rer + z

    ez

    dM=dx1

    e1+ dx2e2+ dx3

    e3

    dM=drer + rd

    e + dzez

    dv= dx1dx2dx3

    dv= rdrddz

    ejxi

    =0 i, j

    err =

    0

    er =

    0

    ezr =

    0

    er =

    ee =

    erez =

    0

    erz =

    0

    ez =

    0

    ezz =

    0

    {e1 ,e2 ,

    e3}

    g= g (x1, x2, x3)

    u = u (x1, x2, x3)

    A= A (x1, x2, x3)

    grad

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    grad

    dg =

    grad g.

    dM

    grad g= gx1

    e1+ gx2

    e2+ gx3

    e3

    grad g= g

    r

    er + 1

    r

    g

    e + g

    z

    ez

    grad g=

    .g

    =

    x1

    e1+

    x2

    e2+

    x3

    e3 =

    123

    {e1,e2,e3}

    grad

    du= gradu .

    dM

    A

    B =

    A1A2A3

    B1B2B3

    = B1 B2 B3

    A1 A1B1 A1B2 A1B3A2 A2B1 A2B2 A2B3A3 A3B1 A3B2 A3B3

    = A1B1 A1B2 A1B3A2B1 A2B2 A2B3

    A3B1 A3B2 A3B3

    gradu = u

    gradu = u1u2u3

    x1x2x3

    = u1

    x1

    u1

    x2

    u1

    x3u2x1

    u2x2

    u2x3

    u3x1

    u3x2

    u3x3

    gradu =

    urr 1r ur u urzur

    1r

    u

    + ur uz

    uzr

    1ruz

    uzz

    div

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    div

    div u =

    .u

    div u = uixi =ui,i = u1x1

    + u2x2 + u3x3

    div u = 1r

    r (r.ur) +

    1ru +

    uzz

    div

    div =

    =

    T11 T12 T13T21 T22 T23T31 T32 T33

    x1x2x3

    div =

    T11x1

    + T12x2 + T13x3

    T21x1

    + T22x2 + T23x3

    T31x1

    + T32x2 + T33x3

    div =

    Trrr +

    1rTr +

    TrrTr +

    Trzz

    Trr +

    1rT +

    Tr+Trr +

    Tzz

    Tzrr + 1r T

    z

    + Tzz

    r + Tzz

    z

    rot

    rotu = u

    rotu =

    x1x2x3

    u1u2u3

    =

    u3x2

    u2x3u1x3

    u3x1u2x1

    u1x2

    {e1,e2,e3}

    rotu = 1r uz uzur

    z uzr

    1rr (ru)

    1rur

    {er,e,ez}

    g= g= 2g g= div grad g

    g =

    2gx2

    k

    =g,kk = 2gx2

    1

    + 2gx2

    2

    + 2gx2

    3

    g=

    2gr2 +

    1rgr +

    1r22g2 +

    2gz2 =

    1rr

    r gr

    + 1r2

    2g2 +

    2gz2

    u =

    2u =

    u

    u =

    divgradu

    u =

    2

    xii

    u =

    u1u2u3

    u =

    ur 1r2 ur+ 2u u 1r2 u 2ur uz

    = 1r r r urr + 1r2 2ur2 +

    2urz2

    2r2u

    urr2

    1rr

    r ur

    + 1r2

    2u2 +

    2uz2 +

    2r2ur

    ur2

    1rr

    r uzr

    + 1r2

    2uz2 +

    2uzz2

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    =2T

    T = T11 T12 T13T21 T22 T23

    T31 T32 T33

    rotgrad g=

    0

    div

    rotu =0

    rot

    rotu

    =

    graddivu

    u

    g= divgrad g

    grad (f g) =f

    grad g+ g

    grad f

    div (fu ) =grad f.u + f.div u

    div (u v) =

    rotu .v

    rotv .u

    rot (fu ) =f.

    rotu +grad f u

    Dg =

    D

    g

    D

    D

    dV

    dS n

    D dS

    dS= dS.n

    ds

    D t

    D ds

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    D

    u .dS=

    D

    div u .dV

    D

    .dS=

    D

    .n dS=

    D

    div .dV

    D

    f.dS=D

    grad f.dV

    D

    f.dSi =

    D

    f.ni.dS=

    D

    f

    xi.dV

    D

    u

    dS=D

    rot

    u .dV

    D

    f.

    grad g g.

    grad f

    dS=

    D

    (f. g g. f) .dV

    D

    f.grad f.

    dS=

    D

    f. g+

    grad 2f

    .dV

    D

    u .ds=

    D

    rotu .dS

    Du .dS=

    Ddivu .dV

    D D D

    D n D

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    =

    11 12 1312 22 2313 23 33

    {e1,e2,e3}

    =

    rr r rzr zrz z zz

    {er,e,ez}

    T(M,n ) = (M) .

    n

    div +

    fv =

    0

    11x1

    + 12x2 + 13x3

    + fv1 = 0

    12x1 + 22

    x2 + 23

    x3 + fv2 = 013x1

    + 23x2 + 33x3

    + fv3 = 0

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    rrr +

    1rr +

    rrr +

    rzz + f

    vr = 0

    rr +

    1r +

    2rr +

    zz + f

    v = 0

    rzr +

    1rz +

    rzr +

    zzz + f

    vz = 0

    = t

    dans tout systemedeprojection

    T(P, n ) =(P).n = fs sur (Sf)

    (Sf)

    n

    T

    fs

    u = u 0 sur (Su)

    =

    1

    2 gradu + gradt u

    =

    11=

    u1x1

    12= 12

    u1x2

    + u2x1

    13=

    12

    u1x3

    + u3x1

    12 22=

    u2x2

    23= 12

    u2x3

    + u3x2

    13 23 33=

    u3x3

    {e1,e2,e3}

    =

    rr = urr r = 12 1r ur + ur ur rz = 12 uzr + urz r = 1r ur+ u z = 12 uz + 1r uz rz z zz =

    uzz

    {er,e,ez}

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    grad div + gradt div grad grad tr = 0

    211x2

    2

    + 222x2

    1

    2 212

    x1x2= 0

    222x2

    3

    + 233x2

    2

    2 223

    x2x3= 0

    233x2

    1

    + 211x2

    3

    2 231

    x3x1= 0

    211x2x3

    + x1

    23x1

    31x2 12x3

    = 0

    222x3x1

    + x2

    31x2

    12x3 23x1

    = 0

    233x1x2

    + x3

    12x3

    23x1 31x2

    = 0

    max |I, II, III| e

    max |I, II, II I| e

    1

    6

    (I II)

    2 + (II II I)2 + (III I)

    2

    1

    32e

    max |I II, II II I, II I I| e

    = (1 + )

    E

    Etr

    I

    = 2 + tr

    I

    = E

    (1+)(12)

    = G = E2(1+)

    E= 3+2+

    = 2(+)

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    u + ( + )

    graddivu +

    fv =

    0

    ui+ ( + )

    xi

    tr

    + fvi = 0

    g= 2gx2

    1

    + 2gx2

    3

    + 2gx2

    3

    tr

    = u1x1 + u2x2

    + u3x3

    ur

    1r2

    ur+ 2

    u

    + ( + ) r tr

    + fvr = 0

    u

    1r2

    u 2

    ur

    + ( + ) 1r

    tr

    + fv = 0

    uz+ ( + )

    z

    tr + fvz = 0 g= 1r r r gr + 1r2 2g2 + 2gz2tr

    = div u = 1rr (rur) +

    1r2u +

    uzz

    + 1

    1 + grad

    grad + grad

    fv + gradt

    fv +

    1 div

    fvI= 0

    ij + 1

    1 +

    2

    xixj+

    1

    fvkxk

    +fvi

    xj+

    fvjxi

    = 0

    rr 2r2

    (rr ) + 11+

    2r2

    + 1

    divfv + 2

    fvrr

    = 0

    2r2 (rr ) +

    11+

    1rr +

    1

    divfv + 2

    fvrr = 0

    zz + 11+

    2z2 +

    1

    divfv + 2

    fvzr = 0

    r 4r2 r

    1rfv +

    fvr = 0

    rz 1r2 rz +

    11+

    2zr +

    fvzr +

    fvrz = 0

    z 1

    r2 z +

    fv

    z = 0 g= 1r

    r

    r gr

    + 1r2

    2g2 +

    2gz2

    = tr

    = rr + + zz

    divfv =

    1rr (rf

    vr) +

    1r2fv +

    fvzz

    V

    fv =

    gradV

    V = 0

    fvx =

    Vx

    fvy = Vy

    sur {x , y }

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    fvr =

    Vr

    fv = 1rV

    sur {er ,e}

    =

    xx(x, y) xy(x, y)xy(x, y) yy(x, y)

    {x ,y }

    zz =

    +(xx+ yy) deformations planes0 contraintes planes

    =

    rr(r, ) r(r, )r(r, ) (r, )

    {er,e}

    zz =

    +(rr + ) deformations planes0 contraintes planes

    (xx+V)

    x +

    xyy

    = 0xyx +

    (yy+V)y = 0

    r

    (rr + V) + 1rr

    + rrr

    = 0rr +

    1r ( + V) +

    2rr = 0

    u (x, y) =u(x, y).x + v(x, y).y

    u (r, ) =u(r, ).er + v(r, ).e

    =

    xx(x, y) xy(x, y)xy(x, y) yy(x, y)

    {x ,y }

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    xx = ux

    yy = vy

    xy = 12

    uy +

    vx

    zz = 0 deformations planes(xx+ yy) contraintes planes

    =

    rr(r, ) r(r, )r(r, ) (r, )

    {er,e}

    rr = ur

    = ur +

    1rv

    r = 12

    1rv +

    vr

    vr

    zz = 0 deformations planes(rr + ) contraintes planes

    xx =

    E(12)

    (xx+ yy)

    yy = E(12)(yy+ xx)

    xy = E1+xy

    rr =

    E(12)

    (rr + )

    = E(12)

    ( + rr)

    r = E1+

    r

    xx = 1+E ((1 ) xx yy)

    yy = 1+E ((1 ) yy xx)

    xy = 1+E xy

    rr= 1+

    E ((1 ) rr )

    = 1+E ((1 ) rr)

    r = 1+E r

    (x, y)

    xx+ V =

    2y2

    yy + V = 2

    x2

    xy = 2xy

    (r, )

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    rr + V = 1r

    r +

    1r22

    + V =

    2r2

    r = r

    1r

    V

    2 =

    2

    x2+

    2

    y2

    2

    x2 +

    2

    y2

    =

    4

    x4 + 2

    4

    x2y2+

    4

    y4 = 0

    2 = 2r

    +1r

    r

    + 1r2

    22

    2r

    +1r

    r

    + 1r2

    22

    = 0