Download - 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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= 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