r, Ń
" A, B, n, v..
"
(x, y, z) �! (x1, x2, x3) �! (xi)
"
" i, j, k �! ei
" a � b = aibi = akbk
" �ij = ei � ej
"
ńł
1 i, j, k 1, 2, 3
�ł
= -1 i, j, k 1, 2, 3
ijk
ół
0
" c = a � b ci = ajbk
ijk
" (a � b) � c = aibjck
ijk
" A,i = "A/"xi
" gradĆ = Ć,i divA = Ai,i rotA = Ak,j ei
ijk
"Ć = Ć,ii
"
A � n dS = divAdV
S V
" dI/dt I = M(x, t)dV
V (t)
dI "M "M
= dV + M (v � n) dS = + grad M � v + Mdivv dV
dt "t "t
V S V
d
M(x, t)dV = @ + M divv dV
dt
V
V (t)
 t = t0 B0 t B
P0 P
u = x - X ui = xi - Xi
x3
B
0
Q0
P0 dX
u+du
B
Q
u
d x
X
P
x
x2
x1
x = �(X, t) xi = �i(X1, X2, X3, t)
.P.
P'
P
..
P'
P
.
.P'
"� "xi
J = | | = | | > 0
"X "Xj
x = AX + c
A, c detA > 0
�
konfiguracja
odniesienia
u
X
konfiguracja
x
aktualna
ds1 ds2
=
dS1 dS2
konfiguracja
odniesienia
dS
1
�
dS2
d s1
d s2
konfiguracja
aktualna
F dX
dx
"�(X)
dx = �(X + dX) - �(X) H" dX a" F dX
"X
"�
F =
"X
x1 = X1 + X2(et - 1)
x2 = X1(e-t - 1) + X2
x3 = X3
-x1 + x2(et - 1)
X1 =
1 - et - e-t
x1(et - 1) - x2
X2 =
1 - et - e-t
X3 = x3
u1 = X2(et - 1)
u2 = X1(e-t - 1
u3 = 0
-x1 + x2(et - 1)
u1 = x1 -
1 - et - e-t
x1(et - 1) - x2
u2 = x2 -
1 - et - e-t
u3 = 0
x1 = X1
x2 = X2 + AX3
x3 = X3 + AX2
A = const detF = 1 - A2 = 0 A = ą1

X1 = x1
x2 - Ax3
X2 =
1 - A2
x3 - Ax2
X3 =
1 - A2
u1 = 0
u2 = AX3
u3 = AX2
u1 = 0
x3 - Ax2
u2 = A
1 - A2
x2 - Ax3
u3 = A
1 - A2
P0, Q0 P, Q
2 2 2
|P0Q0|2 = dS2 = dX1 + dX2 + dX3 = dXjdXj = �jk dXjdXk = dX � dX
|P Q|2 = ds2 = dx2 + dx2 + dx2 = dxidxi = dx � dx
1 2 3
"xi
dxi = dXj = FdX
"Xj
"xi "xi
ds2 - dS2 = dXj dXk - �jk dXjdXk = 2 Ejk dXjdXk = dX (2 E) dX
"Xj "Xk
1 "xi "xi 1
Ejk = - �jk = FT F - 1
2 "Xj "Xk 2
E
"ui "xi
= - �ik
"Xj "Xj
1 "uj "uk "ui "ui
Ejk = + +
2 "Xk "Xj "Xj "Xk
1
T T
E = u + u + u � u
2
1 "uj "uk "ui "ui
"
Ejk = + -
2 "xk "xj "xj "xk
ńł
x1 = X1 + aX2
�ł
x2 = X2
ół
x3 = X3
ńł
X1 = x1 - ax2
�ł
X2 = x2
ół
X3 = x3
1 a 0
J = 0 1 1
0 0 1
�ł �ł
0 a/2 0
1 "xi "xi
�ł łł
Ejk = - �jk = a/2 a2/2 0
2 "Xj "Xk
0 0 0
�ł �ł
1 "Xi "Xi �ł 0 a/2 0 łł
"
Ejk = �jk - = a/2 -a2/2 0
2 "xj "xk
0 0 0
ńł ńł
u1 = aX2 u1 = ax2
�ł �ł
u2 = 0 u2 = 0
ół ół
u3 = 0 u3 = 0
1 "uj "uk "ui "ui
Ejk = + +
2 "Xk "Xj "Xj "Xk
 a2/2
"
Ejk Ejk
�ł �ł
0 a/2 0
�ł łł
�jk = a/2 0 0
0 0 0
"u(X , t)
v =
"t
a = t2
ńł
v1 = 2tX2
�ł
v2 = 0
ół
v3 = 0
"u(x , t) "u
v = + v
"t "x
a = t2
ńł
v1 = 2tx2 + t2v2
�ł
v2 = 0
ół
v3 = 0
ńł
v1 = 2tx2
�ł
v2 = 0
ół
v3 = 0
v2 = 0
E
P Q
|PQ| - |P0Q0| ds - dS
= a" P Q
|P0Q0| dS
P Q > 0 P Q < 0 �i = dXi/dS
(ds - dS)(ds + dS) (ds - dS)(ds - dS + 2 dS) dXi dXk
= = 2Eik
dS.dS dS.dS dS dS
P Q(P Q + 2) = 2 Eik�i�k
dS OX1 �j = �1j (�1 = 1, �2 = �3 = 0)
11(11 + 2) = 2 E11 �! 11 = 1 + 2E11 - 1
dS = dX1 E11
1
P0Q0 X1, P Q = 11; �1 = (1, 0, 0), �i = �1i
2
P0R0 X2, P R = 22; �2 = (0, 1, 0), �i = �2i
X
2
R
R
ds2
dS2
Q
ds1
dS1 Q X1 P
P
dXi dxi
"
�i = , �i =
dS ds
"ui
xi = ui + Xi �! dxi = + �ik dXk
"Xk
"ui
+ �ik
"Xk "ui dXk dS
"
�i = dXk = + �ik
ds "Xk dS ds
"ui
+ �ik
"Xk "ui �k
"
�i = dXk = + �ik
ds "Xk 1 + 
cos �12 = 0
2E12
" "
�(1) � �(2) = cos �" =
12
(1 + 2E11)(1 + 2E22)
Eij Eij = 0 P Q = 0 �" = 0
�!
du d
v(X, t) = = �(X, t)
dt dt
d
v(x, t) = �[�-1(x, t), t]
dt
x x + dx
"v
dv = dx
"x
L = v = F-1
L
D = 0.5 (L + LT ) �!
� = 0.5 (L - LT ) �!
F = 1 + u
1
T
� H" ( u + u)
2
1 d
T
D H" ( v + v) = �
2 dt
gradu
1
T
� = ( u - u)
2
1 "uj "uk "ui "ui 1
Ejk = + + = ejk + (eij + �ij)(eik - �ik)
2 "xk "xj "xj "xk 2
Ejk H" �jk �jk �jk
Y,y
(x,y)
�
(X,Y)
R
�
ż
X,x
X = R cos Ń0 x = R cos (Ń + Ń0)
Y = R sin Ń0 y = R sin (Ń + Ń0)
u(X, Y ) = x - X = R cos(Ń + Ń0) - R cos Ń0 = X (cos Ń - 1) - Y sin Ń
v(X, Y ) = y - Y = X sin Ń + Y (cos Ń - 1)
�xx = cos Ń - 1
�yy = cos Ń - 1
�xy = 0
�xx = �yy = 0

2 2
"u "u "v
Exx = + (1/2) + = (cos Ń - 1) + (1/2) (cos Ń - 1)2 + sin2 Ń = 0
"X "X "X
u v
� +
=
x y
u v
x y
�xy = v + u
2
y
x
u
v
x
y
u
v
2�xy x -
=
y
v
u
-
x y
� �
(�ij - � �ij) nj = 0
�3 - I��2 + II�� - III� = 0
I� = �kk
II� = 1/2( �ii�jj - �ij�ji)
III� = det �ij = (1/6) �im �jn �kp
ijk mnp
"V - "V0
= �kk = divu
"V0
�ł �ł �ł �ł
3 -1 0 4 0 0
�ł-1 3 0łł .10-3 �! �ł0 2 0łł .10-3
� =
0 0 1 0 0 1
�ij u
�ki,hj = 0
phi mjk
ijk
�ij,kl + �kl,ij - �il,jk - �jk,il = 0
"2�11 "2�22 "2�12
+ = 2
"x2 "x2 "x1 "x2
2 1
"2�22 "2�33 "2�23
+ = 2
"x2 "x2 "x2 "x3
3 2
"2�33 "2�11 "2�31
+ = 2
"x2 "x2 "x3 "x1
1 3
" "�23 "�31 "�12 "2�11
- + + =
"x1 "x1 "x2 "x3 "x2"x3
" "�23 "�31 "�12 "2�22
- + =
"x2 "x1 "x2 "x3 "x3"x1
" "�23 "�31 "�12 "2�33
+ - =
"x3 "x1 "x2 "x3 "x1"x2
" [kg/m3]
dm
=
dV
Ł
F
s
1
F
n
V
F
2
M
M = dm = dV
V
= const M = V
"
s
d m
d V
d P
V
dm dV dP
dP
b =
dm
dP
b" =
dV
b" = b
dP = dm g g
g = 9, 81m/s2 b = g b" = g
"
(n)
T
n
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Mxxxxx "P
"s xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxx
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xxxxx
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n
"S "P
T(n)
"P
T(n) = lim
"S0
"S
"
T(n) n
T(n) = �n + �
n
�n �
n
�n = T(n) � n = Ti(n)ni
�n = T(n) � T(n) - (�n)2
�n
(n)
T
n
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Mxxxxx
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xxxxx
"s xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
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xxxxx
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"
M n
Tn M
T(i) (i = 1, 2, 3)
ei
T(n) = T(i) ni
x
3
(3)
T
�33
(2)
�31 �32 T
(1)
T
�13 �23
�21 �22
x
�11 �12
2
x
1
n
 �
T(n) = � n �! Ti(n) = �ijnj
M
�ł �ł
7 0 2
�ł0 5 0łł (MP a)
2 0 4
T(n) n
n = (2/3, -2/3, -1/3)
�I = 8, �II = 5, �III = 3 (MP a) T(n) = (4, -10/3, 0)(MP a)
�n = 44/9MP a = 5, 2MP a cos � H" 0.94; � H" 200
"
n M
T(n) = �n
� n - �n = (� - �1)n = 0 �! (�ij - � �ij) nj = 0
n n � n = nini = n1 + n2 + n2 = 1
2 2 3
(�ij - � �ij) = 0
�3 - I� �2 + II� � - III� = 0
I�, II�, III�
I� = �ii
1
�11 �12 �22 �23 �33 �31
II� = (�ii �jj - �ij �ji) = + +
�21 �22 �32 �33 �13 �11
2
1
III� = �ip �jq �kr = (�ij)
ijk pqr
6
I�, II�, III�
�k
n(k)
I� = �1 + �2 + �3
II� = �1�2 + �2�3 + �3�1
III� = �1�2�3
�ł �ł
3 1 1
�ł1 0 2łł (MP a)
1 2 0
�3 - 3�2 - 6� + 8 = �3 + 23 - 3�(� + 2) = (� + 2)(� - 4)(� - 1) = 0 �I = 4, �II =
1, �III = -2 (MP a)
" "
nI = (0, 1/ 2, -1/ 2)
" " "
nII = (1/ 3, -1/ 3, -1/ 3)
" " "
nIII = (-2/ 6, -1/ 6, -1/ 6)
"
1
p = (�11 + �22 + �33)
3
p�ij
�ł �ł
p 0 0
�ł0 p 0łł
0 0 p
sij = �ij - p �ij
�ł �ł �ł �ł
s11 s12 s13 �11 - p �12 �13
�łs21 s22 s23łł = �ł
sij = �21 �22 - p �23 łł
s31 s32 s33 �31 �32 �33 - p
(sij - s �ij) nj = 0
s3 - IIs s - IIIs = 0
Is = sii
1
IIs = sij sji
2
1
IIIs = sij sjk ski = (sij)
3
Is = s1 + s2 + s3 = 0
IIs = -(s1s2 + s2s3 + s3s1)
IIIs = s1s2s3
Is = 0
1 1
2
IIs = (I� - 3 II�) = (�1 - �2)2 + (�2 - �3)2 + (�3 - �1)2
3 6
1
3
IIIs = (2 I� - 9 I� II� + 27 III�)
27
b0 dA
0
b da N
dA n da
konfiguracja
odniesienia
N
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dA
dP
n
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da
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konfiguracja
aktualna
J = F
b0 = J b
= J
0
n da = J N F-1 dA
dP
n � = lim
da0 da
�! �
dP
N S = lim
dA0 dA
�! S = J F-1�
F-1dP
N Ą = lim
dA0 dA
�! Ą = J F-1�F-T
" �
" S
"
 D F E
ą = � � D = S �  = Ą � 

F = 1 + u F H" 1 + u
1 1
� H" (1 + u) S = (1 + u) Ą (1 + u)T
1 + u 1 + u
� H" S H" Ą
dI
= fdV + gdS
dt
V S
I = M(x, t)dV M, f V g S
V
"
M = dV f = g = 0
V
d
dV = 0
dt
V
Ł + divv = 0
"
M = v v
f = F g = T(n)
d
vdV = bdV + T(n)dS
dt
V V S
d( v)
+ ( v) v dV = bdV + (�n)dS
dt
V V S
Ł
Łv + v + ( v) (v) = b + div �
Ł
( Ł + v ) v + v = b + div �
Ł
div� + b = v
 b" = b
dv "v
div� + b" = = + v v
x
dt "t
X x
"v
divS + b" = 0
0
"t
X
"v
div(Ą FT) + b" = 0
0
"t
X
" � = �T
-
-
M = r � v r = OM f =
r � b g = r � T(n) O
d
r � vdV = r � b dV + r � T(n)dS
dt
V V S
d
r � vdV = r � bdV + r � (� n)dS
dt
V V S
(v � v) + r � ( v). + (r � v) div v = r � b + (� - �T) + r � �
r � ( v). + (r � v) div v = r � b + (� - �T) + r � �
Ł
r � Łv + r � v + (r � v) div v = r � b + (� - �T) + r � �
Ł
r � Łv + (r � v) div v = (� - �T ) + r � � + r � b - r � v
T
� = �
(n)
�ijkxjb"dV + �ijkxjTk dS = 0
k
V S
�ijkxjb"dV + �ijkxj(�lknl)dS = 0
k
V S
�ijkxjb"dV + �ijkxj,l�lkdV + �ijkxj�lk,ldV = 0
k
V V V
�ijk [xj(�lk,l + b") + �jl�lk] dV = 0
k
V
�ijk�jk = 0
�jk = �kj
� = �T
(FS) = (FS)T
Ą = ĄT
"
u M = (u+1/2v�v)
f = b � v g = g1 + g2
g1 = T(n) � v g2 = -q � n
q
[q] = [J/m2 s]
d 1
u + v � v dV = b � vdV + T(n) � vdS- q � n dS
dt 2
V V S S
dE dC d W d Q
+ +
dt dt dt dt
d'W
dt
dE dC
,
dt dt
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d'Q
dt
1 1
u + Łu + v � v + Ł v � v + ( u + v � v) v = b � v + div� � v + � � v - q
Ł Ł
2 2
u = � � D - div q
Ł
D
1
T
D = ( v + v)
2
�ij,j + bi = vŁi
vi
�ij,j vi + bi vi = vŁi vi
(�ij vi),j - �ij vi,j + bivi = vŁi vi
�ij vjdV - (�ij Dij)dV + bividV = vŁi vidV
V V V V
Ł
ą a" Tn v dS + b � vdV = �D dV + v � v dV
S V V V
d 1
ą = (�L)dV + v � v dV
dt 2
V V
d 1
W = (S)dV + v � v dV
0
dt 2
V0 V0
d 1
W = (Ą
)dV + v � v dV
0
dt 2
V0 V0
"
T (T > 0)
S s
S = sdV
V
dS = dS(e) + dS(i)
dQ
ds(e) =
T
dS(i) e" 0 dS(i) = 0
dS(i) = dS - dS(e) e" 0
dS 1 dQ
- e" 0
dt T dt
 T1 T2
dS1 = -dQ/T1
dS2 = dQ/T2
dS = dQ(1/T2 - 1/T1) e" 0
T1 > T2
d q � n q � n
s dV + dS = a dV + dS e" 0
dt T T
V S V S
q
a dV + dV e" 0
T
V V
1 1
a + q - q � T e" 0
2
T T
1 1
a + qi , i - qi T, i e" 0
2
T T
qi, i = � � D - u
Ł
1
� � D - (u - T a) - qi T, i e" 0
Ł
T
D = �ij Dij - (u - T a) � = u - T s D �
Ł
p
e
�ij Dij = �ij Dij + �ij Dij
� = �(�e, T, �)
e
�
= Dij �
ij
� = u - T a - s j
Ł Ł
u - T a = � + s j
Ł Ł
"� "� "�
� = �
+ j + �
Ł Ł
"�e ij "T "�
ij
"� "�
s = - �ij = -
"T "�ij
"�
p
D = �ij Dij + �
Ł
"�
q = -k T
 k [k] = [J/(m K s)] k > 0
cv [c] = [J/(kg K)]
- q = cv j
ui = vi ui vi
Ł
�ij = 1/2(ui, j + uj, i) �ij
Dij = 1/2(vi, j + vj, i) Dij
Ł + vi, i = 0
�ij, j + bi = vŁi �ij
u = �ij Dij - qi, i u, qi
Ł
1 1
a + qi , i - qi T, i e" 0 s, T
2
T T
qi = -k T, i
-qi, i = cv j
�ij = fij(�kl, T )
�ij = fij (�kl)
� E
X
X
E
T
u(E, s) s
u(E, s) s(E, T )
� = u - T s u � Ą E
" E E + dE s s + ds
"u "u
du = dE + ds
"E s=const "s
E=const
 du = dW + dQ.
0
dW = Ą � dE dQ
dQ = T ds
0
"u "u
Ą(E, s) = 0 T =
"E s=const "s
E=const
" q = 0
E E + dE T T + dT
"� "�
d� = dE + dT
"E T =const "T
E=const
d� = du - T ds - sdT
1 1
d� = dW + T ds - T ds - sdT = Ą � dE - sdT
0 0
"� "�
Ą(E, T ) = 0 s = -
"E T =const "t
E=const
"u "�
Ą(E, s) = 0 Ą(E, T ) = 0
"E s=const "E T =const
" E
"Ąij "Ąij
Ąij(E, s) = Ekl + ds
"Ekl s=const "s
E=const
"Ąij "Ąij
Ąij(E, T ) = Ekl + dT
"Ekl T =const "T
E=const
"Ąij "2u
a
Cijkl = = 0
"Ekl s=const "Eij"Ekl s=const
"Ąij "2�
i
Cijkl = =
"Ekl T =const 0 "Eij"Ekl T =const
"Ąij "2u
bij = =
"s "s2 s=const
s=const
"Ąij "2�
aij = =
2
"T "T
T =const T =const
" u �
"
Cijkl = Cjikl Ąij = Ąji
"2 "2
Cijkl = Cklij =
"Eij"Ekl "Ekl"Eij
Cijlk = Cjikl
Cijl 21
�ł �ł �ł �ł�ł �ł
�11 C1111 C1122 C1133 C1112 C1123 C1131 �11
�ł�22�ł �łC1122 C2222 C2233 C2212 C2223 C2231�ł�ł �22 �ł
�ł �ł �ł �ł�ł �ł
�ł�33�ł �łC1133 C2233 C3333 C3312 C3323 C3331�ł�ł �33 �ł
�ł �ł �ł �ł�ł �ł
=
�ł�12�ł �łC1112 C2212 C3312 C1212 C1223 C1231�ł�ł2�12�ł
�ł �ł �ł �ł�ł �ł
�ł�23łł �łC1123 C2223 C3323 C1223 C2323 C2331łł�ł2�23łł
�31 C1131 C2231 C3331 C1231 C2331 C3131 2�31
Ox2 x3
x1 = -x1, x2 = x2, x3 = x3
u1 = -u1, u2 = u2, u3 = u3
�21 = -�21 �31 = -�31
�33 = C1133�11 + C2233�22 + C3333�33 + 2C3312�12 + 2C3323�23 + C3331�31
�33 = C1133�11 + C2233�22 + C3333�33 - 2C3312�12 + 2C3323�23 - 2C3331�31
�33 = C1133�11 + C2233�22 + C3333�33 + 2C3312�12 + 2C3323�23 + C3331�31
�33 = �33 C3312 = C3331 = 0
C1112 = C1113 = C2212 = C2213 = C3312 = C3313 = 0
13
�ł �ł �ł �ł�ł �ł
�11 C1111 C1122 C1133 0 C1123 0 �11
�ł�22�ł �ł �ł�ł
C2222 C2233 0 C2223 0 �22 �ł
�ł �ł �ł �ł�ł �ł
�ł�33�ł �ł �ł�ł
C3333 0 C3323 0 �33 �ł
�ł �ł �ł �ł�ł �ł
=
�ł�12�ł �ł
C1212 0 C1231�ł�ł2�12�ł
�ł �ł �ł �ł�ł �ł
�ł�23łł �ł łł�ł2�23łł
(symetria) C2323 0
�31 C3131 2�31
9
�ł �ł �ł �ł�ł �ł
�11 C1111 C1122 C1133 0 0 0 �11
�ł�22�ł �ł �ł�ł
C2222 C2233 0 0 0 �22 �ł
�ł �ł �ł �ł�ł �ł
�ł�33�ł �ł �ł�ł
C3333 0 0 0 �33 �ł
�ł �ł �ł �ł�ł �ł
=
�ł�12�ł �ł �ł�ł2�12�ł
C1212 0 0
�ł �ł �ł �ł�ł �ł
�ł�23łł �ł łł�ł2�23łł
(symetria) C2323 0
�31 C3131 2�31
"
Ąij H" �ij Ekl H" �kl T = T0 + "T s = s0 + "s
i
�ij = Cijkl + bij"T
kl
a
�ij = Cijkl + aij"s
kl
i a
Cijkl Cijkl
-1
�ij = Cijkl�kl + ąij"T
-1
�ij = Cijkl�kl + �ij"s
-1
Cijkl Cijkl ąij
"
Ś
"Ś
�ij =
"�ij
1 1
Ś = �ij �ij = Cijkl �ij �kl
2 2
" Cijkl
Cijkl = �ij�kl + �(�ik�jl + �il�jk)
�ł �ł �ł �ł�ł �ł
�11  + 2�   0 0 0 �11
�ł�22�ł �ł
  + 2�  0 0 0�ł�ł �22 �ł
�ł �ł �ł �ł�ł �ł
�ł�33�ł �ł
   + 2� 0 0 0�ł�ł �33 �ł
�ł �ł �ł �ł�ł �ł
=
�ł�12�ł �ł
0 0 0 � 0 0�ł�ł2�12�ł
�ł �ł �ł �ł�ł �ł
�ł�23łł �ł
0 0 0 0 � 0łł�ł2�23łł
�31 0 0 0 0 0 � 2�31
�ij =  �kk�ij + 2� �ij
"
�ij = -�m�ij �m/(- ) a" K
kk
K =  + 2/3 �
K
�11 = -fc �ij = 0 �11/�11 = E
�(3 + 2�)
E =
� + 
-�22/�11 = �

� =
2( + �)
 �12 = �21 = 0

�12/2�12 = G
G = �
"
Ś = 0.5 �ij > 0 = 0

ij ij
Ś
1
2
Ś = I� - 2(1 + �)II�
2E
1
2 2 2
Ś = (1 + �)(�1 + �2 + �3) - �(�1 + �2 + �3)2
2E
1
Ś = (1 + �)[(�1 - �2)2 + (�2 - �3)2 + (�3 - �1)2)] + (1 - 2�)(�1 + �2 + �3)2]
2E
�1 = -1; �2 = �3 = 0; Ś > 0 �! E > 0
�1 = -(�2 + �3) = 1; Ś > 0 �! 1 + � > 0
�1 = �2 = �3 = 1; Ś > 0 �! 1 - 2� > 0
E > 0, 1 - 2� > 0, (1 + �) > 0 Ś > 0
oddialywania
przemieszczenie u
zewnetrzne i
predkosc vi
F Ł
i
i
rownowaga warunki zgodnosci
(ruch) odksztalcen
odksztalcenie
naprezenie
�
� ij
ij
zwiazki
konstytutywne
Ł
S
�
ij S�
u
i
�
V
ij
F
Su
" V S = Su + S�
Cijkl b u �
� V u(X, t0) =
Ł
u(0)(X); u(X, t0) = v(0)(X) V u(X, t) = u"(X, t) Su
�n = Ł S�
�ij = �(�) + �(T )
ij ij
�(�)
ij
�(T )
ij
1 
�(�) = �ij - �ij �kk
ij
2� 3 + 2�
�(T ) = ą(T - T0 )�ij
ij
E�
 =
(1 + �)(1 - 2�)
E
� =
2(1 + �)
E � ą
1 
�ij = �ij - �ij �kk + ą(T - T0 )�ij
2� 3 + 2�
�ij =  �ij �kk + 2� �ij - (3 + 2�) ą �ij (T - T0)
1
�ij = [ (1 + �) �ij - � �kk �ij ]
E
E �
�ij = �ij + �kk �ij
1 + � 1 - 2�
dQ
ds = ds(e) =
T
dQ
T a = = qi, i
dt
"s "s
T �Ł + j = qi, i
"�ij ij "T
"s "2�
cv = T = -T
2
"T "T
"s "2� 1 "�ij
= - = -
"�ij "�ij "T "T
-qi, i = k "T qi = -k T, i
"�ij
= -(3 + 2�) ą �ij
"T
k "T = cv j + (3 + 2�) ą T �kk
Ł
�ij = 1/2(ui, j + uj, i)
Ł + ui, i = 0
Ł
�ij, j + bi = �i
k "T = cv j + (3 + 2�) ą T �kk
Ł
�ij =  �ij �kk + 2� �ij - (3 + 2�) ą �ij (T - T0)
(1); ui(3); �ij(6); �ij(6), T (1)
1 1 1 1
W = �ij �ij = sij + �kk �ij eij + �mm �ij
2 2 3 3
1 1
W = sijeij + �kk�mm
2 6
1 1 sij IIs 1 + �
(1)
W = sijeij = sij = = IIs
2 2 2G 2G E
2
1 1 1 I� I� 1 - 2�
(2) 2
W = �kk�mm = I� I� = I� = = I�
6 6 6 3K 18K 6E
�T = ą "T
ą (� 10-6/K)
19, 1 � 22, 2
18 � 21
5 � 11
7 � 14
10 � 18
�
d
ą(T - T ) dx
0
1
T1
h/2
Tsred dx
h/2
T2
ą(T - T ) dx
2
0
T1 h
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
xxxxxxxxx
T2
x dx
T > T1
2
� = E� = E ą "T E = 280MP a "T = 100K
280 � 10 � 10-6 � 100 = 280kP a
"l = ą "T l
T1 + T2
Tśred =
2
T1 + T2
"l = ą (Tśred - T0) l = ą ( ) l
2
h d� = ą (T2 - T0) dx - ą (T1 - T0) dx
d� (T2 - T1)
= ą
dx h
d2w M (T2 - T1)
= + ą
dx2 EJ h
d2w ą T0 T0ą
T2 - T1 = T0 x = x w = x3 + C1x + C2 w = 0 x =
dx2 h 6h
0 x = l
�ij = ��ij + 2��ij � = �kk = ui,i
1
�ij = (ui,j + uj,i)
2
xj
�ij,j = �,j�ij + �ui,jj + �uj,ji
�ij,j + bi = 0
( + �)�,i + �"ui + bi = 0
( + �) u + �"u + b = 0
xj
�("ui),j + ( + �)�,ij + ( bi),j = 0
i j; j i
�("uj),i + ( + �)�,ji + ( bj),i = 0
2�"�ij + 2( + �)�ij + [( bi),j + ( bj),i] = 0
Ś = �kk
2( + �) 
"�ij + Ś,ij - �ij "Ś + [( bi),j + ( bj),i] = 0
3 + 2� 3 + 2�
l
"�zz/"z + g = 0 z = l
�xz = �yz = �zz = 0
�zz
�zz = - g(l - z)
�
�xx = �yy = g(l - z)
E
g(l - z)
�zz = -
E
�xy = �yz = �zx = 0
�
ux = g(l - z) x
E
�
uy = g(l - z) y
E
g
uz = [z2 - 2lz + �(x2 + y2)]
2E
�13 = �23 = �33 = 0; b3 = 0
�33 = 0
�
�33 = - (�11 + �22)
E
y
�yy
�xx
x
�xy
X3
z
X
2
X
1
ńł
1
�ł
�ł
�11 = (�11 - � �22)
�ł
�ł
E
�ł
�ł
�ł
�ł
�ł
1
�22 = (�22 - � �11)
�ł
E
�ł
�ł
�ł
�ł
�ł
�ł
1 + �
�ł
ół
�12 = �12
E
Ox1x2
u1 = u1(x1, x2); u2 = u2(x1, x2); u3 = 0
�ij = 0.5(ui,j + uj,i)
"u1 "u2 1 "u1 "u2
�11 = ; �22 = ; �12 = ( + )
"x1 "x2 2 "x2 "x1
�11, �22, �12 x3 �23 = �31, �33 = 0
"u1 "u2
�1 = +
"x1 "x2
y
�yy y
y
�xx 2�xy
x x
x
y
�yy
2
�xy
�xx
x
z
ńł
1 + � �
�ł
�ł
�11 = �11 - (�11 + �22 + �33)
�ł
�ł
E E
�ł
�ł
�ł
�ł
�ł
�ł
1 + � �
�ł
�ł
�ł �22 = �22 - (�11 + �22 + �33)
�ł
�ł
E E
�ł
�ł
�ł
�ł
�ł
�ł
1 + � �
�ł
�ł
�33 = �33 - (�11 + �22 + �33)
�ł
�ł
E E
�ł
�ł
1 + �
�ł
�ł
�12 = �12
�ł
�ł
E
�ł
�ł
�ł
�ł
�ł
�ł
�ł 1 + �
�ł
�ł �13 = �13
�ł
�ł
E
�ł
�ł
�ł
�ł
�ł
�ł
1 + �
�ł
ół
�23 = �23
E
�33 = �(�11 + �22); �13 = �23 = 0
( )3
ńł
1
�ł
�ł
�11 = (�11 - �1 �22)
�ł
�ł
�ł E1
�ł
�ł
�ł
�ł
�ł
1
�22 = (�22 - �1 �11)
�ł E1
�ł
�ł
�ł
�ł
�ł
�ł
1 + �1
�ł
�ł
ół �12 = �12
E1
E �
E1 = , �1 =
1 - �2 1 - �
ńł
"�11 "�12
�ł
�ł
+ + b1 = 0
�ł
�ł
"x1 "x2
�ł
�ł
�ł
�ł
"�21 "�22
+ + b2 = 0
�ł
�ł
"x1 "x2
�ł
�ł
�ł
�ł
�ł
ół
b3 = 0
x3
"2�11 "2�22 "2�12
+ = 2
"x2 "x2 "x1"x2
2 1
"(�11 + �22) = 0
"2 "2
" = +
"x2 "x2
1 2
ńł
"�11 "�12
�ł
�ł
+ = 0
�ł
�ł
"x1 "x2
�ł
�ł
�ł
�ł
"�21 "�22
+ = 0
�ł
�ł
"x1 "x2
�ł
�ł
�ł
�ł
�ł
ół
"(�11 + �22) = 0
Ś
"2Ś
�11 =
"x2
2
"2Ś
�22 =
"x2
1
"2Ś
�12 = -
"x1"x2
Ś
"4Ś "4Ś "4Ś
""Ś a" + 2 + = 0
"x4 "x2"x2 "x4
1 1 2 2
�13 = �23 = �33 = 0 �13 = �23 = 0
�11, �12, �22 �11, �12, �22, �33
�13 = �23 = 0 �13 = �23 = �33 = 0
�11, �12, �22, �33 �11, �12, �22
r, Ń
x = r cos Ń; y = r sin Ń
"�rr 1 "�rŃ �rr - �ŃŃ
+ + = 0
"r r "Ń r
"�rŃ 1 "�ŃŃ 2�rŃ
+ + = 0
"r r "Ń r
y
���
�rr
�r�
r
�
x
1
�rr = (�rr - ��ŃŃ)
E
1
��� = (�ŃŃ - ��rr)
E
2(1 + �)
�r Ń = �rŃ
E
Ś = Ś(r, Ń)
1 "Ś 1 "2Ś
�rr = +
r "r r2 "Ń2
"2Ś
�ŃŃ =
"r2
" 1 "Ś
�rŃ = -
"r r "Ń
""Ś = 0
1 " 1 " "2 1 "Ś 1 "Ś "2Ś
+ + + + = 0
r "r r2 "Ń2 "r2 r "r r2 "Ń2 "r2
�
,
. ,
sprzezystosc
idealna , ,
wzmocnienie
plastycznosc
D
C
B
oslabienie
1
ET
A
x
zniszczenie
E
1
�
H
K
O
p e
� �
� � �
E1
1
E E E
1 1 1
� � �
f(�ij) = F (�ij) - k2 = 0, (k = const)
"f
df = d�ij = 0
"�ij
"f
df = d�ij < 0
"�ij
f(�ij) = F (�ij) - k2 = 0
f
d�ij
�ij
(� )
f =k
ij
.
.
obciazenie
�
ij
.
odciazenie
�
O
.
sprezystosc
(� )
f <0
ij
d�ij = d�e + d�p
ij ij
d�ij = d�e
ij
1
d�e = Dijkl d�kl = [ (1 + �) d�ij - � d�kk �ij ]
ij
E
g(�ij)
"g
d�p = d
ij
"�ij
g = f
"f
d�p = d
ij
"�ij
d�p f(�ij) = 0
ij
f = f(�ij, �p , k)
ij
�p k
ij
f = 0 f < 0
k
�e
�e = �e( ) � = �(�)
e e
�e = 3 IIs
"
p
dW = �e d p
"
p p
d = 2/3 d
p
ij ij
f(�ij, �p , k) = F (�ij, �p ) - k2( ) = 0
p
ij ij
g(�ij, �p , k)
ij
"g
d�p = d
ij
"�ij
f(�ij, �p , k) = 0
ij
f = 0 f + df = 0
"f "f "f
df = d�ij + d�p + dk = 0
"�ij "�p ij "k
ij
k k = k(�p )
ij
d�ij = d�e + d�p
ij ij
e
d�ij = Cijkl d�e
kl
"g
e e
d�ij = Cijkl (d�kl - d�e ) = Cijkl d�kl - d
kl
"�kl
"f "g "f "g "f "k "g
e
Cijkl d - + d + d = 0
kl
"�ij "�kl "�p "�ij "k "�p "�ij
ij ij
e
("f/"�ij) Cijkl d�kl
d =
e
h + ("f/"�mn) Cmnpq("g/"�pq)
h
"f "g "f "k "g
h = - -
"�p "�ij "k "�p "�ij
ij ij
d�ij
e
"g ("f/"�rs) Crskl d�kl "g
d�p = d =
ij
e
"�ij h + ("f/"�mn) Cmnpq("g/"�pq) "�ij
p
e
d�ij = (Cijkl + Cijkl) d�kl
e e
Cijtu ("f/"�rs) ("g/"�tu) Crskl
p
Cijkl = -
e
h + ("f/"�mn) Cmnpq ("g/"�pq)
p p
Cijkl = Cklij f = g

f = g
e e
Cijtu ("f/"�rs) ("f/"�tu) Crskl
p
Cijkl = - .
e
h + ("f/"�mn) Cmnpq ("f/"�pq)
� Q � u
� = f(�)
� = E� E
S0
� = 0 � < S0 � > 0 � = S0
� = ��
Ł
 � - �
s e
relaksacja
pelzanie
czas czas
E
�
�
�
� �
Ł
� = +
Ł
E �
� = �0 = const � = 0
Ł
� �
Ł
0 = +
E �
E
� = C exp - t
�
C
t = 0- t = �
1 1
�(�) - �(0-) = [�(�) - �(0-)] + �śred.�
E �
�(0-) = �(0-) = 0 �(�) = �0 �(�) = �0 C = �0 = E�0
E
� = E�0 exp - t
�
�0
E
R(t) = E exp - t
�
 tR = �/E t " � = 0
� = �0 = const
1 1
�(t) = t + �0
E �
�0
1 1
P (t) = t +
E �
s
e
czas
czas
E
�
�
�
�(t) = E�(t) + ��(t)
Ł
� = const 0- �
� � �
�dt = E�dt + ��dt
Ł
0- 0- 0-
�
��dt = �[�(�) - �(0-)] = ��0
Ł
0-
�
E�dt = 0
0-
 �(t)
�
�(t)dt = 1
0-
�(t) = ��0�(t)
�(t) = E�0 + ��0�(t)
R(t) = E + � �(t)
� = �0
�0 E
= �(t) + �(t)
Ł
� �
�0 E
�(t) = + C exp - t
E �
t = 0 � = �0 C = -�0/E
�0 E
�(t) = 1 - exp - t
E �
1 E
P (t) = 1 - exp - t
E �