連続体力学の基礎 - nagaokaut.ac.jp › gigaku-press › wp-content ›...
200
Transcript of 連続体力学の基礎 - nagaokaut.ac.jp › gigaku-press › wp-content ›...
連続体力学の基礎れんぞくたいりきがくのきそ
古口日出男・永澤 茂
共 著
Fundamentals of Continuum Mechanics
1.
, .
,
. , ,
.
.
2.
, , , ,
. , ,
.
, .
3. (keywords)
Internet ,
(tensile) (compressive) (shear)
(strain),
(elasticity) (plasticity),
(viscosity) (velocity),
, , ,
(vector) (tensor),
(basic vector),
(bound vector) (free vector),
－ I －
本書の目的と使い方PrefaceContents
(differentiation of functions with several variables)
(integration),
( coordinate transformation),
(material coordinate)
(material derivative) = (substantial derivative),
(index), (summation convention),
(space) (time),
(deformation) , (field) (equilibrium),
(free body),
(divergence) (rotation),
(constitutive equation),
Leibniz (Leibniz integral rule)
Gau (Gauss ’divergence theorem),
(total differential) (increment),
(partial derivative) (ordinary differential),
(chain rule for partial derivative),
(multiplication of a matrix by a vector ),
(multiplication of matrices)
4.
,
.
.
,
.
. , , ,
. ,
.
－ II －
連続体力学の基礎
本書の目的と使い方 （How to use this book and purposes） ............................................................. i
1. 本書の目的 ......................................................................................................................................... i
2. 達成したい目標 ................................................................................................................................. i
3. 核となる用語（keywords） ............................................................................................................... i
4. 本書の内容および本書を使った学習の方法 ............................................................................... ii
１.連続体力学の考え方 （Defi nition of the continuum mechanics） ...............................................1
1.1 連続体力学の歴史 （History of the continuum mechanics） ...........................................................1
1.2 連続体の概念を理解するための取り組み方（Approach to philosophy） ..................................2
1.3 力学とは ? （What is mechanics?） ...................................................................................................2
1.4 連続体の性質（properties of a continuum body） ..........................................................................4
1.5 空間内の物質の連続分布 : 密度と平滑化 （Distribution and smoothness） .................................4
1.6 微小要素の大きさについて （Size of infi nitesimal elements） ......................................................7
1.7 連続量（物理量） ............................................................................................................................7
1,8 場の量 （時空間で定義される連続量）と表記法 .......................................................................9
1.9 連続体の分類 .................................................................................................................................10
1.10 連続体力学の学習で必要な考え方 ...........................................................................................12
1.11 例題 多質点系の平衡方程式 ......................................................................................................14
練習問題 ...............................................................................................................................................16
復習 基礎知識 .....................................................................................................................................17
練習問題 ...............................................................................................................................................21
２.ベクトルとテンソル （Vectors and tensors） ............................................................................23
2.1 ベクトルの演算（積）と座標変換則
（Vector product and the transformation rule of coordinates） ..................................................25
2.1.1 位置ベクトルと束縛ベクトル ..............................................................................................25
2.1.2 ベクトルの等価性 （Equality of vectors） ..............................................................................26
2.1.3 ベクトル場 （Vector fi eld） ......................................................................................................27
目 次
－ III －
目 次PrefaceContents
2.1.4 ベクトルの演算 （Operation rules in vector space） ...............................................................29
（1） スカラー積，内積 （inner product） .....................................................................................29
例題 2.1 ...................................................................................................................................29
例題 2.2 ...................................................................................................................................29
例題 2.3 ...................................................................................................................................30
（2） ベクトル積 , 外積（outer product） .....................................................................................31
例題 2.4 ...................................................................................................................................31
（3） 基底ベクトルの外積演算 ....................................................................................................32
（4） 座標変換則 （Rules of translation and rotation of coordinates） ...........................................33
例題 2.5 ...................................................................................................................................34
（5） 演算規則に慣れよう ............................................................................................................35
2.1.5 まとめ ......................................................................................................................................36
練習問題 ...............................................................................................................................................37
2.2 テンソルの演算（積）と座標変換則
（Tensor product and the transformation rule of coordinates） ......................................................39
2.2.1 ベクトルの演算（積）再考 ..................................................................................................39
2.2.2 テンソル積 （dyadic, tensor product） と基底 （diad） ............................................................40
2.2.3 テンソル積によって表現される物理量 ..............................................................................43
2.2.4 テンソル積の座標変換則 ......................................................................................................45
2.2.5 関数としてのベクトル量 [ 復習 ] ........................................................................................47
練習問題 ...............................................................................................................................................48
３.応力 （Stress） ..................................................................................................................................51
3.1 応力ベクトル （stress vector） ........................................................................................................51
3.2 応力テンソル（stress tensor） .......................................................................................................52
3.3 法線応力とせん断応力（normal and shear stress） .....................................................................55
3.4 せん断応力の対称性 （symmetry of shear stress components ） ...................................................55
3.5 表面力の釣り合い .........................................................................................................................56
例題 3.1 .............................................................................................................................................59
例題 3.2 .............................................................................................................................................63
例題 3.3 .............................................................................................................................................65
3.6 Mohr の応力円（Mohr's stress circle） ..........................................................................................66
－ IV －
連続体力学の基礎
例題 3.4 .............................................................................................................................................68
例題 3.5 .............................................................................................................................................69
練習問題 ...............................................................................................................................................70
４.ひずみ（Strain） .............................................................................................................................73
4.1 1 次元の伸長変形 .........................................................................................................................73
4.2 運動記述のための座標系の選択 ................................................................................................74
4.3 3 次元変形への導入 .....................................................................................................................76
4.4 変形勾配 ........................................................................................................................................76
4.5 変形計量とひずみ ........................................................................................................................78
4.6 変位場を用いたひずみの表現 .....................................................................................................79
例題 4.1 .............................................................................................................................................80
例題 4.2 ............................................................................................................................................ 81
4.7 体積ひずみ（dilatation） ...............................................................................................................82
4.8 相対変位の分解（並進と剛体回転）（Linear motion and rigid rotation） .................................82
練習問題 ...............................................................................................................................................84
５.構成方程式 , 固体弾・塑性の基礎 （Plasticity and elasticity on metal） ...........................87
5.1 固体の機械的性質 ........................................................................................................................87
5.2 １次元材料引張試験 ................................................................................................................... 87
5.3 金属の結晶と塑性変形 （結晶のすべりによる変形） ............................................................... 90
5.4 真応力と対数ひずみの構成近似モデル .................................................................................... 93
5.5 弾性変形における実用弾性率 .....................................................................................................95
練習問題 ...............................................................................................................................................96
６.固体・流体の構成方程式 （Constitutive equations on solid and fl uid） .................................99
6.1 流体および固体の力学的特性の記述 .........................................................................................99
6.2 完全流体 （perfect fl uid） ...............................................................................................................99
6.2.1 理想気体 （ideal gas） ..............................................................................................................99
6.2.2 非圧縮性流体 （Incompressible fl uid） ..................................................................................100
6.3 Newton 流体（Newtonian fl uid） ................................................................................................100
6.4 Stokes 流体（Stokes fl uid） .........................................................................................................100
－ V －
目 次PrefaceContents
6.5 現実挙動の流体の状態方程式 ...................................................................................................101
6.6 Hooke 弾性固体（Hookean elastic solid） ..................................................................................102
6.6.1 等方性弾性体 （Isotropic elasticity） .....................................................................................103
6.6.2 体積弾性係数K .....................................................................................................................104
例題 6.1 ......................................................................................................................................105
6.6.3 ポアソン比 ............................................................................................................................106
6.6.4 横弾性係数 ............................................................................................................................107
6.6.5 温度効果（Effect of temperature） .......................................................................................108
6.7 線形粘弾性体（Linear viscoelastic bodies） ...............................................................................108
6.7.1 Maxwell 模型 ........................................................................................................................109
6.7.2 Voigt 模型 ..............................................................................................................................110
6.7.3 Maxwell 模型のクリープ関数（creep function） .............................................................. 111
6.7.4 Maxwell 模型の緩和関数 （relaxation function） .................................................................111
練習問題 ............................................................................................................................................ 112
７.保存則と場の方程式 （Field equation and laws of conservation）
7.1 各種保存法則の復習 .................................................................................................................. 112
7.1.1 エネルギー保存則 ................................................................................................................113
（a） 現実世界の事象と変形過程の重要な性質 ......................................................................113
（b） 熱の性質 ..............................................................................................................................114
7.1.2 物体の運動の性質 ................................................................................................................114
7.1.3 線形（並進） 運動量保存則 .................................................................................................115
7.1.4 角運動量保存則 ................................................................................................................... 116
7.2 連続体力学から見たエネルギー保存則 （Balance of energy） .................................................117
7.3 Gauss の定理 ................................................................................................................................119
7.3.1 関数の定積分の公式 ............................................................................................................119
7.3.2 経路に沿った線積分による誘導 ........................................................................................120
7.3.3 Gauss 定理の応用 ................................................................................................................ 123
（a） 物理量 A が速度（uj） の場合 ............................................................................................123
（b） 物理量 A がポテンシャル関数Ψの場合 ........................................................................ 124
（c） 物理量 A が速度（uj） の場合（再び） ............................................................................. 124
（d） 物理量 A が 2 階テンソル Cij である場合 ...................................................................... 125
－ VI －
連続体力学の基礎
7.4 物質導関数 （Material derivative） .............................................................................................. 125
7.5 体積積分の物質導関数 （Material derivative of volume integral） ............................................129
7.6 補足 , Leibniz の積分公式 （Essential rules of Leibniz integration） ..........................................131
7.7 連続と運動の方程式 （Equations of continuity and motion ） ....................................................132
7.7.1 質量保存則と連続の方程式 ................................................................................................133
7.7.2 運動量保存則と運動方程式 ................................................................................................133
7.8 エネルギー保存則の展開 （Description of energy balance） ......................................................135
練習問題 .............................................................................................................................................139
８.固体の弾性理論と応力関数 （Theory of elasticity and stress function） .............................141
8.1 変位場解法の序 ..........................................................................................................................141
8.2 平面応力状態 （Plain stress state） ..............................................................................................144
8.3 平面ひずみ状態 （Plain strain state） ..........................................................................................145
8.4 Airy の応力関数 （Airy stress function） .....................................................................................146
例題 8.1 調和関数φと重調和関数Φの関係 .............................................................................150
例題 8.2 応力関数Φから変位場の解析 .....................................................................................151
例題 8.3 応力場と平衡方程式の関係 .........................................................................................154
例題 8.4 3 次元丸棒の変形 ..........................................................................................................157
例題 8.5 応力場から適合条件の照合 .........................................................................................160
8.5 サンブナンの原理 （Principle of Saint-Venant） ........................................................................161
練習問題 .............................................................................................................................................163
９.流体の場の方程式と境界条件 （Field equation of fl uid and boundary condition）
9.1 Navier-Stokes 方程式 ..................................................................................................................165
9.2 境界条件について ......................................................................................................................166
9.3 Reynolds 数と相似則 .................................................................................................................169
練習問題 （1）,（2） ...........................................................................................................................170
9.4 水平な水路あるいは管内の流れ ...............................................................................................171
例題 9.1 ...........................................................................................................................................172
例題 9.2 ...........................................................................................................................................177
例題 9.3 ...........................................................................................................................................178
練習問題 （3）, （4） ............................................................................................................................179
－ VII －
目 次PrefaceContents
9.5 完全流体 .......................................................................................................................................179
9.6 渦度と循環 ...................................................................................................................................181
9.7 渦無し流れ ...................................................................................................................................183
例題 9.4 ...........................................................................................................................................184
練習問題（5）,（6）,（7）.......................................................................................................................186
－ VIII －
連続体力学の基礎
1.
1. 1
. ( , ), ( ,
), ( , ) ,
. ,
( ) .
, , 2
. , . ,
, .
, (velocity) .
, ,
(displacement) .
, 17 19 .
, . (17 , ), (18 ,
, , , ; 19 , , ),
(19 , ), (19
, ).
.
. (1653 , ), (1687, ),
( , ),
(1826, ), (1847, ).
1
1. 2
2 .
(1) ,
.
(2) ,
, .
(1) , , .
, (2) ,
,
. (1) (2) ,
,
, (2) . (
) , . ,
, .
. ,
,
.
1. 3
, .
2
, .
, .
, , , . , (
) (material body, object, body)
. , , .
.
.
, ( ) ,
,
. ,
. ,
, .
, ,
.
,
.
, .
. , (
) , ,
. ,
.
, .
, ,
, (rigid body)
.
. ,
, .
, .
(material) (atom)(material body, body)
3
, .
, .
, ,
.
1. 4
a, b a <b [a, b]
( [a, b])
(continuum body)
, [1, 2] 1. 5
. ,
, .
1. 5 1. 5 .
[1, 2] [1. 5, 1. 512345]
[1. 500000000000001, 1. 512000000000000009] . . . . . .
3
, ( ) , 3 (x, y, z) .
, (
) . ( )
[ ] .
.
.
1. 5
,
4
P (x, y, z) dV= dxdydz , dx, dy, dz
. ( ) , 3 dxdy,
dydz, dzdx 1 dV , ,
.
(mass density)
j=0 4
j ,
Vj Mj
1.
V0V1
V2V4dV4
V3
x
y P
5
j 0, 1, 2, 3, 4
2 2 dV
dV . j=0, 1 ,
,
= (M0+M1)/(V0+V1) (1.1)
. j=0 ,
= M0/V0 (1.2)
. j=0, 1 ,
.
, ( )
. ,
.
j P (x, y, z) j (x, y, z) , (1.
3) dMj dVj dMj/dVj . 1
j=4 . (x, y, z) , Vj P
dMj dVj , dVj
( ). ,
.
P :
j
j
0d dd
limP,,j V
Mzyx
V(1. 3)
j j . dVj
(index)
,
6
, (1. 3)
. ( ) Q , j
.
P (x, y, z) , , (P) ,
.
.
1. 6
,
. dV
, P(x. y, z) .
dV , ,
. , ,
.
.
, 1
= 5×10-6 cm
, .
1. 7 ( )
.
, (linear momentum) , (angular
momentum), (energy) .
.
P(x, y, z) . dV=dxdydz ,
7
v=(vx, vy, vz) 3*,
dp =(dpx, dpy, dpz) O dh =(dhx, dhy, dhz) (1. 4), (1. 5) 4*. OP=r= (x, y, z) O
P (position vector) . (1. 5)
, (outer product) . ,
.
Vdd vp (1. 4)
Vdd vrh (1. 5)
(1. 4) , P v (1. 6)
.
dVd
VVMomentumzyx
VV
pv0d0d
limd
dinlim,, (1. 6)
(1. 4) dV , v .
. ,
, .
,
dV , dp , dV
. dV +0 ,
.
dV
. ,
.
v 3 3(bold) v
v dV
8
3
4
1. 8
, t (1 ) 3 r=(x, y, z) (3
) r
x 4
(r, t) .
, .
u(r, t), v(r, t), (r, t), p(r, t), T(r, t)
, (vector quantity) (bold) ,
, , (scalar quantity) .
, u ( )
.
(index)5* ,
.
: r = (x, y, z) = (x1, x2, x3) = (xk) = xk , ( k=1, 2, 3)
: u = (u1, u2, u3) = (uk) = uk, ( k=1, 2, 3) (1.7)
: v = (v1, v2, v3) = (vk) = vk, ( k=1, 2, 3)
k , (free index)
, u
. k, j .
: uk (xj, t) ( k=1, 2, 3, j=1, 2, 3) (1.8)
, .
, .
: u = [ u1 (x1, x2, x3, t), u2 (x1, x2, x3, t),
u3 (x1, x2, x3, t) ] (1.9)
0,1,2,3,…
9
5
, .
, ,
. ,
, . 6*.
1. 9
1.1 ,
. , .
, (solid) (fluid) .
, (displacement) .
(strain) .
, (velocity) . ,
(rate of strain) . ,
.
, , . ,
.
1 1 , 0
.
: , : =f( ) (1.10)
: .
: =E (E: ) (1.11)
.
( , ductile material), (plastics), (clay soils)
.
.
(component)(base vector)
10
6
( , ) ( )
.
. d /dt ( =
d /dt , : ) .
(glacier), , (glue, adhesives),
(Mayonnaise) . ( ) (
)
(Rheology) 7*.
( )
(normal strain) , ( )
(shear strain) 2 ( ,
)
( )
N/m2 = Pa N/mm2 = MPa .
(normal stress) ,
(shear stress) .
(force, structural load) (1 )
(elasticity)
(viscosity)
3
11
7
1. 10
,
. ,
.
(dynamics of multiple material points):
. ( ) ,
,
( ) , ,
.
(free body): ( ) .
(free body diagram) .
.
.
. 1.2(a) , m, ,
, ( )
1.2(b) ,
W=mg F, P
, . 3 W, F, P 0
1.2
mW=mg
FP
(a) (b)
F
PW
12
,
(constitutive equation):
. , .
: Hooke’s law ( =C C ) .
, d dt t
.
(equilibrium equation): .
,
. . ,
, , ,
, . , (
) .
.
( ) 0 D’Alembert (Lagrange-d’Alembert principle):
( ) –dp/dt
.
, –dp/dt .
. p: , t:
(moment of force): . (torque)
. , × ,
. ,
r F r F .
: Newton . ,
.
13
: Newton . ,
.
( ) .
: Newton 3 . ,
.
,
. , ,
.
1.11
( )
K
0 I
, ( ) )(eIF , J I ( )
IJF (1.12)
KIK
JIJ
eI ,,2,10
1
)( FF (1.12)
I 0
(1.12) I=1,2,3,…, K (1.13), (1.14)
1.3 I, J
P(x,y,z)
QI
rI
rJQJ
rJ rIFIJFJI
,
,
,
14
01 11
)(K
I
K
JIJ
K
I
eI FF (1.13)
01
)(K
I
eIF (1.14)
JIIJ FF(1.14)
3 , P
I QI IPQ Ir (1.12)
Ir ( ) 0 ( ) Ir
Ir ( )
(1.15)
0
,,2,10
1
)(
1
)(
K
JIJI
eII
K
JIJ
eII KI
FrFr
FFr
(1.15)
I =1,2,3…K (1.15) (1.16) .
01 11
)(K
I
K
JIJI
K
I
eII FrFr (1.16)
2 JIIJ FF I=J
0FIJ
JI rr IJF ( )
2
0Frr IJJI (1.17)
I, J =1,2,3...K
01 11 1
K
I
K
JIJJ
K
I
K
JIJI FrFr
15
JIIJ FF
01 1
K
I
K
JIJI Fr (1.18)
(1.18) (1.16) (1.19)
01
)(K
I
eII Fr (1.19)
I Ir
, F D
2 , (Couple)
FD
(1) 1.4 O
. , F =(-5N, 10N, 2N) P
0.6 cos 45 m,0.6sin 45 m, 0.75m
(2)
(3)
1.4 1.5 F
x
y
z
0.1m
0.4m
PF1m
4
O LL-aa
Fx
F
R1 R2
S
-S-M
M
,
,
16
(4) ( )
(5) 1.5 a F L
x
, S
M ,
x S
M F a, L
R1, R2
(6) , 1.5
1) f(x, t) .
,
. ,
t x
(x0, t0)
: ft
x0 , t0
limt 0
f x0 , t0 t f x0 , t0
t (1.18)
ft
x,t limt 0
f x,t t f x,tt
(1.19)
,
,
17
n ft n x,t lim
t 0
1t
n 1 f x,t tt n 1
n 1 f x,tt n 1 (1.20)
2 ft x
x,t limt 0
1t
f x, t tx
f x,tx
limx 0
1x
f x x,tt
f x, tt
2 fx t
(1.21)
2)
3 .
1, 2, 3
)3,2,1(,,,, 321 kxxxxxzyx kkxr (1.22)
A 1, 2, 3
)3,2,1(,,,, 321 kAAAAAAAA kkzyxA (1.23)
A, B (inner product, scalar product) A, B
iii
iik
kk BABABABBB
AAA3
1
3
13
2
1
321BA
(1.24)
, 1 2
, . (1.24)
i, k (dummy index) , i, k
. ,
18
(summation convention) . .
3)
, alpha , beta , gamma
, delta , epsilon , eta , phi ,
psi , xi zeta lambda mu
nu pi theta , rho sigma
omega , a, b, g, d, e, h, f, y, x, z, l, m, n, p, q, r, s, w
. , .
,
A, B, G, D, E, H, F, Y, X, Z, L, M, N, P, Q, R, S, W .
,
.
http://e-words. jp/p/r-greek. html
, d, D, . x,t y=f(x,t)
df Df f (differential), (total differential)
. (increment) dx
Dx x x . ,
.
x,t f ,
.
x f x [ x f
] , df/dx . f
x1, x2, x3, … , xk (k=1, 2, 3, …)
, df/dxk (k=1, 2, 3, …) , kxf / .
. xk
df .
19
, , ,
. , df, Df, f f .
( ) .
4)
, x1, x2, x3, … , x
. xk (k=1, 2, 3…)
. k , (free
index) .
1 10 k ,
, (dummy index) . ,
. (1.24) k, i
102110
1.... xxxx
k k (1.25)
,
. , (stress)
. .
, ,
.
, ( ) . x 2 k, j
,
xkj (1.26)
. k j ,
, .
k , j X (1.26)
20
(7) , .
(8) x,y,z P(0,0,0) 10kg F (1N, -2N,
2N)
(9)
yxxeye
xyx
yx cossin)ii(
)sin()i(
2
: (i) x+y=z . (ii) y x
(10) f(x,y) df P (x,y)
f x, y yfxf /,/ Q(x+dx, y+dy)
df (11) z= f(x,y) x,y u,v
)()(
)()(
yz
xz
uz
)()(
)()(
yz
xz
vz
(12) z =f(x,y) z=f(y/x)
yzy
xzx
(13) z=f(x,y) x= r cos( ), y= r sin( )
/,/ zrz
22
yz
xz
(14) A, B, C 332211 eeeA aaa , 332211 eeeB bbb ,
21
332211 eeeC ccc BAC
(15) A i k a ik
(index, subscript)
15.1 3 (matrix) B b12, b31, b33
982153764
B
15.2 b11+b22+b33 3
1i
i
ikikb .
15.3 3
12
kkb
(16) A, B , .
zyx
ihgfedcba
BA :3,:3
16.1 A B AB 16.2
amn : A m n , bh : B h
3
1kkikba
16.3 A, B BT, AT BTAT
(17) A A-1
213301421
A
22
2.
÷
3
2 2
1 (vector) 0 (scalar)
0,1,2 , 3,4,5
. n
. , 2
4
23
÷ 3
,
24
2.1
(magnitude) (direction), (sense)
) , ( ) , , ,
(magnitude)
) ,
2.1.1
O P(x1,x2 ,x3) .
(2.1) ek (k=1,2,3)
, 1**
jjj
j jj
xx
xxxx
er
eeeerOP 3
1332211 (2.1)
2 ,
1** ( ) 1
2.1 OP P a(P)
O
P (x1, x2, x3)
x1 (x)x2 (y)
x3 (z)
r
x1
x2
x3
OP (x1, x2, x3)
x1 (x)x2 (y)
x3 (z)
r
x1
x2
x3
RQ
a(P)
a(R)a(Q)
,
,
,
,
,
25
1
O . , O (bound vector) .
, .
P , . P . Q, R
, ( ) . (vector
field) .
jjj aa ea . (free
vector) . , .
, P a
.
2.1.2
2 a, b , , , a=b
. a, b a=b
. 2 a, b
, . ,
, 2, .
2.2
A
B
FFM
L
L/2
G
,
,
,
,
,
26
,
,
2.2 2L A F , G
M = F L/2 (2.2)
G F , .
a Fa , Fa
b , b
Fa Fb . ,
Fa Fb . , .
, .
2.1.3
. P
.
: P , ( ) .
, , P
. , .
, ,
. , .
,
,
,
,
,
,
,
,
27
. ,
. ,
.
( ),
, .
, , . ,
. , . ,
2.3
2.4
,
,
,
,
28
r O
.
2.1.4
(1) (inner product) 2 a, b (2.3) .
cos|||| baba (2.3)
a b b a ( )
.
2.1
a=1e1 + 0e2= e1 , a b = b1e1 + b2e2
(2.4) .
, 2.5 ,
cos e1
b e1 x1
e1 b = b1 = |b |cos (2.4)
, , .
2.2
a b = a1b1+ a2b2+a3b3 (2.5)
.
a= a i e i , b= bk ek .
i ,k . ,
2.5 a b
ab
,
29
,
,
2** ,
, (2.6) .
a b = a i e i bk ek = a i bk e i ek (2.6)
, i=k
e i ek= 12 cos0 = 1 (2.7)
. i k
e i ek= 12 cos90 = 0 (2.8)
. (2.7), (2.8) ,
ik , (2.9)
e i ek= ik (2.9)
a b = a i bk i k = ak bk
a b = a1 b1+ a2 b2+ a3 b3 .
2.3
. .
F ( ) dx
WAB: xA xB , (2.10)
.
2**
c
30
2
2
c
c
xB
xA
xB
xAAB dxFdxFdxFdW 332211xF (2.10)
(2) (outer product)
2 a, b ,
(2.11) c . a, b
.
|c | = |a | |b | sin (2.11)
|c| ,
. ,
(2.12)
.
122131331223321
21
213
31
312
32
321
321
321
321
bababababababbaa
bbaa
bbaa
bbbaaa
eee
eee
eeeba
(2.12)
2.4 .
m r,
v (i) ( , ) M:
2.6 a, b
b
a
c=a b
|a| sina
bS = |a| |b| sin
c
31
,
,
,
M = r × F ( ) (2.13)
(ii) L: L= r × (mv) ( ) (2.14)
( , )
,
(iii) ( ) : v = × r ( ) (2.15)
(3)
O-x1x2x3 x1 , x2 , x3
ek (k=1,2,3) , ( ) . , 3 .
e i × e j (= ek ) (2.16)
i,j 1,2,3
i j , |e i | = |e j | =1, sin =sin90°=1 |ek |=1 . 3
, i j 3
, k . , i, j , k 33** ,
. i=j , sin = sin0 = 0 e i × e j= 0 (
0 ). , , e i × e j = e j × e i
.
1(permulation symbol) i jk
( ) , 3 (i,j,k) ,
3** 3 1, 2, 3 , (1, 2, 3)
.
32
3
2 5
x3 (counter-clockwise)
2.7 , 321O xxx x3
321O xxx . x3 (2.23),
(2.24)
3333 , eexx (2.23)
p13=p23=0, p33=1 (2.24)
),( 21 ee ),( 21 ee
.
e1 ),( 21 ee,
.
211 eee
e1 1e
cos
cos01121111 eeeeee
e1 2e cos(90°+ )
sin)2/cos()2/cos(
)2/cos(10222121 eeeeee
.
2.7
x2
e1
e2
x1O
34
211 )sin(cos eee (2.25)
p11= cos , p21= sin , p31=0 (2.26)
e2 .
, .
212 cossin eee (2.27)
p12= sin , p22= cos , p32=0 (2.28)
, (2.22) p i j , (2.29)
1000cossin0sincos
321321 eeeeee (2.29)
(5)
(2.29) , P
.
,
.
, kjkj pee ,
k .
. (2 .30) (2.29)
3
2
1
3
2
1
1000cossin0sincos
eee
eee
(2.30)
(6)
v (entity)
,
,
35
,
,
ek vk
(2.31)
kkjj vv eev (2.31)
(2.21): kjkj pee ,
.
kkkjkj vpv eev kkjj vpv (2.32)
(2.33)
3
2
1
3
2
1
1000cossin0sincos
vvv
vvv
(2.33)
(2.33) , P , . ,
. P , P
tP
. .
2.1.5
, O
P .
, (a)
, (b)
. ,
(b) ,
36
(a)
,
. P .
P 1
,
. 1 ,
.
. ,
.
, (Bold)
, .
, .
,
. , ,
,
.
(1) (E.2.1) .
,
.
62
ijkijk
illjkijk
jlimjmillmkijk
(E.2.1)
(2) a=(1, 2, 3), b=(0, -1, 1) , .
(i) a×b (ii) a b
,
37
E2.1
,
(iii) 3a 2b (3) .
30° , 9.
8N 3m .
(4) 980 N .
. (5) P=[p i j ] tP P
(6) e1, e2 2 O-x1x2
30° 21 xxO
21 xxO 21 , ee e1, e2
(7) 2 321O xxx 321O xxx a, kkjj aa eea (j,k:1,2,3)
, ka ja , (E.2.2) 4**.
kjjkjjk apaa ee (E.2.2)
(8) 1,1,0a , 1,,2 xxr , a rx
(9) e1, e2, e3 . u
= e1 2e2+3e3 u
.
(10) E2.1 O-x1x2
O 60
O-x '1x '2
BAA , B OB=(2,1), OA=(0.5, 2) [
4** P pk j , k j .
O
B
A
x1
x2
x’2
x’1
=60E.2.1
38
4
4
2.2
2.2.1
2 a, b a b
. 2 1 . 2 a=akek ,
b= b je j (k,j:1,2,3 ) ab
a b =(akek) (b je j)= akb j eke j (2.34)
= a1b1 e1e1+ a1b2 e1e2+a1b3 e1e3
+a2b1 e2e1+ a2b2 e2e2+a2b3 e2e3
+a3b1 e3e1+ a3b2 e3e2+a3b3 e3e3
(a1e1, a1e2…)
332313
322212
312111
321
3
2
1
bababababababababa
bbbaaa
ab (2.35)
2 eke j (k,j: 1,2,3 )
, (k j )
.
( ) eke j ,
k,j
k=j , eke j = 0.
k j, k,j : eke j = e jek
39
2 , (2.16)
(2.34)
a b = a1b2 e1e2+a1b3 e1e3+a2b1 e2e1+ a2b3 e2e3
+a3b1 e3e1+ a3b2 e3e2
= (a1b2 a2b1)e1e2+(a2b3 a3b2)e2e3+(a3b1 a1b3)e3e1 (2.36)
(2.36) ,
e1e2 e3, e2e3 e1, e3e1 e2
. (2.16) ( )
“ “ ,
e1 e2 = e3, e2 e3 = e1, e3 e1 = e2
.
k,j
k=j , eke j = 1. ek e j = 1
k j , eke j = . ek e j = 0
ab = a1b1 +a2b2 + a3b3 (2.37)
(2.34) (2.35)
2.2.2 (dyadic, tensor product) (diad)
2 a=akek ,b=b je j (k,j:1,2,3 )
40
(dyadic) a b b a 5**.
332313
322212
312111
)(bababababababababa
ba jkba
(2.38)
332313
322212
312111
)(ababababababababab
ab jkab (2.39)
a b b a ,
(2.38),(2.39) ,
,
333323231313
323222221212
313121211111
33221133
3322112233221111
332211332211
332211332211
)()()(
)()(,
eeeeeeeeeeee
eeeeeeeeee
eeeeeeeeeeeeeeba
eeebeeea
babababababa
babababbba
bbbabbbabbbaaa
bbbaaa
(2.40)
.
( ) ,
mkmkmmkk baba eeeeba )()( (2.41)
k,m = 1,2,3
ak bm
(diad) ek em 5** , a, b ab
41
5
5
6**
a=akek (k=1,2,3 : ) ,
2 ( ) ek em
(diadic) (diad)
(diad) ( )
( )
(i) a ei e j (i,j=1,2,3: )
aaa jijiji eeeeee (2.42)
(ii) a, b, c,
d, f (i,j,k=1,2,3: )
, .
jkijii
jkiijkijii
fcbdafdcbaeeeeee
eeeeeeeeee)()(
(2.43)
(iii) (diad) z
(i,j,k=1,2,3: )
kjjikjji
kjjikjji
babababa
eezeezeeeezzeezeezeeee
)()()(
)()()( (2.44)1 ,2
kjjikjji
kjjikjji
babababa
eezeezeeeezzeezeezeeee
)()()(
)()()(
(2.45) 1 ,2
(2.44),(2.45) 6** 2
( ) , 2 (x,y) (ax+bx , y)=a (x,y)+b (x,y), (x,ay+by)=a (x,y)+b (x,y)
42
6
6
(dyadic) (diad)
2.2.3
(dyadic) a b (diad) e i e j , 2
1
a b e i e j
. 2 a,b
2 (tensor of the second rank)
2 (the
second rank) 2 a,b
, n (=0,1,2,3,4…)
n n=1
, 1
(diadic) a b , 2 ( ).
(diad) e i e j , 2 e i , e j .
(diadic) a b z
(dyadic) a b z (dyadic)
a b = a ib j e i e j (2 )
T = T i j e i e j … e i e j : T i j (2.46)
T : 2 , T i j : 2
2.2.3
(diadic) a b
( )
43
2.8
(
) 3
3
F
F
, (F1, F2,
F3)
3 A3
1 1 2 2
3 3 3
3
333
3
232
3
131 ,,
AF
AF
AF
(2.47)
(2.47) (2.48)
3
333
3
232
3
131 ,,
AF
AF
AF
(2.48)
k Ak j
F j : k j
: kk Aa /)( b j=F j .
k
jj
kjk A
FF
Aba )(
(2.49)
(2.49) ( )
2.8 3
x3
x1
x2
A3
A1
A2
x1: 1
x3: 3
x2: 2
F
F1
F3
F2
44
2.8
2.82.8
. (2.49) , ak b j
(dyadic) 7**
,
,
(dyadic) ,
. 2
2
3 4
(2.44)
(2.45)
,
2.2.4
2.1 2
T
2 321 xxxO 321 xxxO
),,( 321 eee , ),,( 321 eee 2 T
kjkj TT , (k,j=1,2,3: )
jkkjjkkj TT eeeeT (k,j=1,2,3: ) (2.50)
7** akb j= kj AF / b jak= kj AF /)(
45
7
7
kjkj TT , , (2.44)
jkkjjkkj TT eTeeTe , (2.51)
2 :
jkjk p ee
(2.52)
):,;:,(
)()(
jkmwwmjmkw
mwjmkw
mjmwkwjkkj
Tpppp
ppTeTe
eTeeTe
(2.52)
2
(2.52) 2
2 P 1
, tP 1 , 2
(2.52) , (2.53)
332313
322212
312111
333231
232221
131211
333231
232221
131211
333231
232221
131211
ppppppppp
TTTTTTTTT
ppppppppp
TTTTTTTTT
(2.53)
(1 ) 1 (magnitude) 1
(direction) (sense) , 2 , 1
2
2
n n
46
n P
2.2.5
A t
A
A (t)= Ak(t) ek = ( Ak(t) ) (2.54)
t
t Ak(t) t
dAk/dt Ak
dA (t)/dt = dAk(t)/dt ek = ( dAk(t)/dt ) (2.55)
A t
dttAdtt kk )()( eA (2.56)
a(t) , b(t) t da/dt , db/dt
(2.57)
: a+b , a b ,
c a : ca
: a b
: a b
(2.57)
47
dtdbab
dtda
dtbad
dtd
dtd
dtd
dtdbab
dtda
dtbad
dtd
dtd
dtd
dtdaca
dtdc
dtcad
dtdc
dtdc
dtcd
dtdb
dtda
dtbad
dtd
dtd
dtd
kjijkk
jijk
kjijk
kkk
kkk
kk
k
kkkk
)()(
)()(
)()(
)()(
bababa
bababa
aaa
baba
(2.57)1 ,2 ,3 ,4
(11) (2.44) (2.51)
(12)
ek(t) v vk
dv/dt
(13) i jk
2 jk 2jk =
(14) =(1,0,0), = (0, 1,0) , =(( 1, 0,2),
(1,2,3), (0, 2, 1)) t .
120321201
120321201
.
(15) ek (k=1,2,3) a= e1,
b= -e2 2 T= e1 e1 +2e1 e3 +e2 e1 +2e2 e2
+3e2 e3 2e3 e2 +e3 e3 T a b T .
48
(16) 2,1,1,,,1,1,2, 22 cba yyxx
(i) ba (ii) bac cba
(17) .
1~3 .
jm
k
k
jjjk
k
ii z
tx
tx
xadXXxdx (c)(b)(a) 3
(18)
a)
3332231133
3322221122
3312211111
,,
nnntnnntnnnt
b) 333332323131232322222121131312121111 bababababababababa
c) x1y1+ x2y2+ x3y3
d)
3
1
3
1
3
1i j kkjiijk xxxc
e)
2
1kkk xau
f) 2221212
2121111
xaxayxaxay
(19) (mechanical work) , F
( ) dr
rF d . F , ek (k=1,2,3: )
. F= 3312
22121 )()( eee xxxxx
49
E.2
3. (Stress)
1 1.2
(1) , 2 2
,
[ ] 2
(1) ( )
. , ( , )
(2) ( , )
. : = ÷
(2) , ,( ) .
, 1.11
( ) ,
3.1 (stress vector)
3.11*** P(x1 ,x2 ,x3)
S ( : ) f
,
1*** (bold)
E.2
51
1
1
| S|= S S
+0
P(x1,x2 ,x3)
kk
S
t
dSd
S
e
t
ff0
lim
(3.1)
(3.1) t , f = f j e j
(e j) S = S f S
2 (2.48)
, S
3 3
.
3.2
,
P
, 3.2
dA3 ( x3 3=(0,0,1) , dA3=
dx1dx2 ) d f 3 3dA3 3= 3k ek
(k= 1,2, 3: )
3332313
3
3.1 S f
S= S
Sf
P
3.2 dA3
d f 3= 3dA3
3331 32
dx1
dx2
32 dx1dx2
33 dx1dx2
df3
31dx1dx2
T
T
52
tT
T
tT
T
t
321332213212131
21333232131
213333
eee
eee
f
dxdxdxdxdxdx
dxdx
dxdxdAd
(3.2)
3 dA1, dA2, dA3 3
d f 1, d f 2, d f 3
i = i j e j (i: , j: ) (3.3)
i j i j 3
i (i=1,2,3) T
333231
232221
131211
333231
232221
131211
3
2
1
T (3.4)
, (dyadic)
(3.5) 2***
T = k k= e k kj e j = kj e k e j (k,j: ) (3.5)
P
3.3
k=1,2,3 k
k = e k k
(dyadic) , k j kj
2*** 3
ek e j
53
(3.5) . T P
T ( )
2 : (dyadic)
2 T . T
(3.4), (3.5) 2
: (2.52)
, , 2
3.5
,
3.3 ( )
Stress component
x1 x2 x3x1 11 12 13
x2 21 22 23
x3 31 32 33x1
x2x3 Symbol of
each component
• Stress [ ij] , 2nd rank tensor• 11, 22, 33 ( ii): normal stress• 13 ( ij): shear stress
11
22
33
31
32
21
23
12
13
EX. 1-plane, 2-component 12
)( ji
54
1
, (
) (3 2 = 6 )
3 1
. ( ) ,
,
,
. ( )
, 3.3
3.3
3.3
(normal stress)
(shear stress) . 3.1 S t
3.4
,
3.4 ,
ABCD
( 3
3.4
dx2 dx1
dx1 21
dx1 21
dx2 12 dx2 12
B C
A D
55
) ( ) ,
, 2
.
ABCD
AB, CD : dx2 12 , .
AD, CB : dx1 21 , .
AB-CD 2 , dx1 : dx1 (dx2 12) . , AD-CB 2 , dx2 : dx2 (dx1 21)
. , 2
0 .
dx2 (dx1 21) dx1 (dx2 12) = 0 (3.6)
12= 21 1 , 2
, 13= 31, 32= 23 , (3.7)
i j= j i (i,j=1,2,3: ) (3.7)
3.5
3.5
( ) P( )
. ,
, ,
. 3.5
SP
SftS
S
S
56
0 S
(1.13)
( ) = 0 (3.8)
2 3***3
3.6 ( ) 2 OAB
( ) OAB ( ) AB
( )AO ( )BO
, ( :1).
AB
= ( 1, 2) = (cos , sin ) (3.9)
( : , : )
AB: t = (t1, t2) (3.10)
3*** 3 33, 31, 32 0
(plane stress)
3.6 OAB
x1
x2
12
A
BO
t
11
1 2221
2
t1
t2
57
. AO+BO:
1= (- 11, - 12), 2= (- 1, - 22) (3.11)
.
AO, BO . , + 1, 2
( )
x1 :
AB: t1dS, AO: - 11dS1, BO: - 21dS2 … 0
12
211
111221111 tdSdS
dSdSdStdSdS (3.12)
x2 : AB: t2dS, AO: - 12dS1, BO: - 22dS2 … 0 .
22
221
122222112 tdSdS
dSdSdStdSdS (3.13)
(3.14)
AB: dS , AO: dS1=dS cos , BO: dS2=dS sin
.
dSdSdSdS
dtt
2
1
2
1
2
1
2212
2111
sincos
,sincos
S (3.14)
, k j AB t j
, (3.9) ,
(3.14) , (3.15), (3.16) .
58
, , (3.17) .
Cauchy
2
1
2212
2111
2
1
tt
(3.15)
(3.15) , (3.16) .
2221
12112121 tt (3.16)
kkjjt (k=1,2: , j=1,2: ) (3.17)
, (3.18) .
Tt (3.18)
T 2 (3.5) ,
k j= jk (3.15),(3.16)
3.1
AB t AB ( x 1 )
( x 2 ) ,
3.2
,
3.7
t O-x1x2
2
1
12
11
cossinsincos
tt
x1
x2
11
2221
12A
BO
t
t1
t211
x1
x2
12
59
, AB 2 ,
x 1 AB 11, 12 . 3.7 t , 2 O-x1x2 O- 21xx
2121 , eeee ,
12
1121
2
121 eeeet
tt
(3.19)
(2.29) 2
.
kjkj pee
eeee ,cossinsincos
2121
t (3.19)
jkjk tptt
12
1
12
11 ,cossinsincos
(3.20)
Cauchy (3.15)
sincos
cossinsincos
2212
2111
12
11 (3.21)
x 2 21, 22
x 2 Cauchy
+ /2 2
O-x1x2 O- 21xx 2121 , eeee
60
)
2sin(
)2
cos(
cossinsincos
2212
2111
22
21 (3.22)
cos)2
sin(,sin)2
cos(
cossinsincos
cossinsincos
2212
2111
2212
2111 (3.23)
O-x1x2 O- 21xx
P 2
(2.53) .
2
2
, 2,
.
cossinsincos
cossinsincos
2221
1211
2221
1211 (3.24)
[ ]
2
2 (3.23) ,
.
(3.23) , ,
( ) 2.
61
2 ( )
Cauchy
3.7 ABO AB t
tk
, 11, 12
, t tk 2 1k
2 ,
3
, 3.8
( ) PABC
O-x1x2xO x1x2
O x1x2x1
cossinsincos
cossinsincos
2212
2111
2212
2111
sincos
cossinsincos
2212
2111
12
11
x2
O-x1x2x
Cauchy
3.8
Ax1
x2
x3
C
BP
t1
3
2
62
t
t
t
. 2 dS: ABC
dS1: PBC, dS2:
PCA, dS3: PAB , PABC
, , .
t dS = 1dS1 + 2dS2 + 3dS3 (3.25)
dSk dS .
dSk = kdS (k=1,2,3: ) (3.26)
k , dS:ABC
(3.4) T (3.27),(3.28)
.
t = 1 1 + 2 2 + 3 3 = T3
2
1
321 (3.27)
,
kkjkjkjt (j=1,2,3: , k=1,2,3: ) (3.28)
, (3.17) Cauchy , 3
3.2
O O-x1x2x3 x3 O- 321 xxx 2 (3.24)
O- 321 xxx O-x1x2x3
cos c, sin s
63
3 , x3= x 3, e3= e 3 . (3.24) 3
3 (3.29)
: 12= 21, 13= 31, 23= 32
3332313231
2322212221
1312111211
333231
232221
131211
333231
232221
131211
10000
10000
10000
cssccssccssc
cssc
cssc
cssc
(3.29)
,
3333
23323132
13323131
231323
122
222
112221121122
122221121121
231313
221211222221121112
122
222
112221121111
2
2
cs
sc
cs
sccscsccss
sccscs
sc
sccscsscsc
scscscsscc
(3.30)1-9
(3.30) 33= 31= 32=0 . 0313233 .
(3.24)
, (3.31)
. (3.31) , Mohr
64
2cos2sin2
,2sin2cos22
2sin2
2cos12
2cos1
,2sin2cos22
2sin2
2cos12
2cos1
122211
2112
1222112211
12221122
1222112211
12221111
(3.31)1 ,2 , 3
3.3
3.9 x1 W , OB
W A x1 =
W/A 0 O-x1x2
(3.32)
22122111 000000
eeeeeeeeT (3.32)
OB
(3.24) 11, 12 , (3.33)
3.9 x1
A WWO
AB
x1x2
65
cossincos
sincos
000
cossinsincos 2
(3.33)
= sin cos = ( /2)sin2 OB
3.9 =45°(= /4 rad.)
/2
3.6 (Mohr's stress circle)
, 3.2 3.3
, 2
2 , 3
, 3.2 3
, 2 (3.31)
(3.24) (3.31)
3.7
OAB OA ( ) (= 11)
( ) - (=- 12) ( )
OA
AB ( )=( 11, 12)
2
(3.31) cos2 , sin2
( 11, 12) ( 11, 12)
66
( 22, 21) ( 22, 21)
(3.31)
2cos2sin2
,2sin2cos22
,2sin2cos22
122211
2112
1222112211
22
1222112211
11
(3.34)1,2 ,3
(3.34)1 ,3 2 , (3.35)1 (3.34)2 , 3
2 , (3.35)2
212
222112
12
22211
22
212
222112
12
22211
11
22
,22
(3.35)1,2
( ) , 2
P 1211 , Q 2122 , (3.35)
R, ( ave ,0) 4*** AB , (3.35)
R (3.36)
2
,2
2211
212
22211
ave
R (3.36)1,2
4*** P, Q
2 ( 2 =180°)
( ) P 1211 , ,Q 2122 ,
67
, O-x1x2 11, 22, 12 . 3.10 (3.35)
O-x1x2 A* O- 21xx
P Mohr
, 11> 22 . 11< 22
, A* G
3.4
3.10 , 2
(3.34) P,Q A*,G (i) =0 22221111 , 1212 . (ii) =90° , cos(2×90°) = 1, sin(2×90°) = 0 ,
11222211 , . 1212 . P=G
(iii) =45° cos(2×45°) = , sin(2×45°) = 1 , :
12221211 , aveave : 2/221112
3.10 Mohr
Q
F
E
B*
D*
ave
C*
max = R
ave R
( 11 ave )2 + 122 = R2
O-x1x2 :P( 11, 12 )
P
A*
2
O-x1x2:A*: ( 11 , 12 )
G
G:( 22 , + 12 )
3.11
68
3.11
3.11
=0,45,90° ,
3.11(a) 2
3.5 3.6 1 2 1p, 2p p1 p2
. max s1
( )
0
3.11(b) =0 B* 1 ( )
D* 2
( )
A*C*B* = 2 p1
A*C*D*=2 p2=2 p1+90°
1p= ave + R
2p= ave R
(a) =0,45,90° (b)
3.11 Mohr
F
E
B*
D*
C*
P
A*
2
O x1x2
A*: ( 11 , - 12 )
ave
12
P( 11, 12 )
11
22
12
12 ( 22- 11)/2
= 0
= 90= 45
F
E
B*
D*
GQ
ave
12
12
A*
C*
P
2
1p
2p
max = R
A*C* = R
A*C*B* = 2 p1
p1
A*C*D* = 2 p2 = 2 p1 +
2 s1
A*C*E = 2 s1
= ave+R
ave R11
22
69
R (3.36)1 . , max = R
A*C*E= 2 s1
A*C*F=2 s1+90°
(1) 3.12
( : j k jk ).
(2) ( ) P (x1,x2)
T
MPa PP (x1,x2)
( ) : x1+ 3x2 = 1 ( 0 )
3.13 ( ) ( )
Cauchy ( )
(3) 3.14
3.12
( )
3.13
x1
x2
0 1
31
a= (1,3)x1+ 3x2=1
2110
T
70
P
PP
7 AB, BC, CD, DE, EF, FG, GA
AB BC, CD, DE,
EF, FG, GA tAB, tBC, tCD, tDE, tEF, tFG, tGA
3.14 2
3
MPa cm
(4) 3.15
.
2
1
2221
1211
00
AB
.
(5) (3.34)
.
1222
1211 2,2 (3.37)
(6) (3.34) 012
3.14 ABCDEFG
DE
EF
A B
CD
E
FG
3.03.8
1.5
1.5 2.8
3.6
AB
FG
GACD
BC
x1
x2
22012001911910
333231
232221
131211
3.15 2
O
A
B
x1x2
71
2211
1222tan (3.38)
, (3.38) 0
(principal direction) .
(7) (3.34)3 , 12 ,
.
12
2211
22tan (3.39)
(8) (3.39) , (3.38)
45 °
(9) O-x1x2 ,
11 = 80 MPa, 22 = 50 MPa, 12 = 25 MPa
30° O-x1’x2’
4.1
72
4.1
4.1
4. (Strain)
4.1 1
( )
( ) 2 2
( )
4.1 ( )
X
x
u = x- X
u ( ) ,
(displacement) ( ) u= 0,
( ) u= u
(4.1) F
.
XxEF x1 (4.1)
Xu
XXxEx x
ux
Xxx (4.2)1,2
4.1 1 ( )
X
x
FxFx
A A0
A0
A
u= x X
73
(4.2)1 Ex X Green
, x
x Euler u
x X Ex x 2
x Fx ( ) A0, A
(true stress) x (nominal stress) x0
2
00
0 AF
AAF
AF x
xxx
x (4.3)
4.2
, ( )
(configuration)
(motion)
.
t0 : X
t > t0 X : x
x X t , ( )
.
x=x(X,t) (4.4)1
xk= xk (X j, t) (k,j=1,2,3: ) (4.4)2
, ,
. x , 1 1 ( )
74
X=X(x,t) (4.5)1
Xa = Xa(xb, t) (a,b=1,2,3: ) (4.5)2
X , , (material
coordinate) . , x
(spatial coordinate)
4.1
Xk
(Lagrangian description)
(material description)
, X t v X
t x
XX
XxXvt
tt ,),( (4.6)
,
(4.5) x v(x,t)
(Eulerian description)
(spatial description)
( )
(material time derivative)
v(X,t)
75
v(x,t)
7
4.3 3
3 3
(stretching)
(simple shear) (twisting)
(bending)
,
, 0 ( ) ,
, 3 ,
3 2 9 (
6 ) 4.1 1
3
3 2 dX, dx
3
2
2
4.4
3 P Q
76
P X = ( Xk ) , P' x = ( xk )
u = ( uk )
X + u = x (4.7)
Q 4.2
2 P, Q dX= dXmem dx =dxmem u= x X , X x :
x = x(X) dX dx (4.9)
j
kkj X
xFF (4.8)
(deformation gradient) .
XFx ddXXxx
dXdXdX
Xx
Xx
Xx
Xx
Xx
Xx
Xx
Xx
Xx
dxdxdx
jj
kk ,dd
,
3
2
1
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
3
2
1
(4.9)
4.2 P,Q PP', QQ'
x1 x2
x3
PQ
P’Q’u
u+du
uk = xk Xk: P P’
O
OP:X = Xm em dX
OQ:X+dX
x= xmemdx
x+dx
dX = dXmem
77
: x X ( X = (x) ) F -1
xFX ddxxXX j
j
kk
1,dd (4.10)
4.5
2 PQ
( ) 2 ds2-dS2 PQ: dX=dXkek 2 : dS2 P'Q': dx=dXkek
2 : ds2 , (4.11)1,2
jiijmm
jkkjkk
dxdxdxdxddds
dXdXdXdXdddS
xx
XX2
2
(4.11)1,2
F F -1
hmh
k
m
k
hh
jm
m
kkjjkkj
hmh
k
m
k
hh
jm
m
kkjjkkj
dXdXXx
Xx
dXXx
dXXxdxdxds
dxdxxX
xX
dxxX
dxxXdXdXdS
2
2
(4.12)1,2
2 ds2 dS2 , (4.12) (4.13),(4.14)
2
dX, dx dXkdX j (dyadic) dX dX
2
78
kj
j
s
k
ttskj
jkkj
Xx
XxEwhere
dXdXEdSds
21
,222
(4.13)1,2
j
s
k
ttskjkj
jkkj
xX
xXewhere
dxdxedSds
21
,222
(4.14)1,2
(dyadic) dXkdX j Ekj
2 ekj 21
Ekj Green's
(Lagrangian) strain tensor ekj
Almansi's (Eulerian) strain tensor
F 2
kjj
s
k
s
j
s
k
tis
kjj
s
k
s
j
s
k
iis
cxX
xX
xX
xX
CXx
Xx
Xx
Xx
(4.15)1,2
,
4.6
u= x-X
j
iij
j
i
Xu
Xx,
)(
XuIF
IFIXxXx
XXu
(4.16)
1
79
1
j
k
k
j
k
i
j
ijk
ikj
i
k
iij
k
i
j
iikij
k
iik
j
iij
k
i
j
i
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xx
Xx
j
m
k
m
j
k
k
jkj
j
m
k
m
j
k
k
jkj
xu
xu
xu
xu
e
Xu
Xu
Xu
Xu
E
21
,21
(4.17)1,2
1k
m
Xu 2
j
m
k
m
j
m
k
m
xu
xu
Xu
Xu ,
kjkj
jk
j
jk
jj
jk
j
k xxxXX
xXuX
xXx
X)(
j
k
k
j
j
k
k
jkjkj x
uxu
Xu
Xu
eE21
21
(4.18)
, Green Ekj Euler
ekj Green Euler
2
4.1
80
4.3 x1 du1
(du1 dX1)
1
111
1111
2111
21
111
2111
22
222
22
xue
dxedxdxe
dSdsdxedudSds
dxedxdxedSds jkkj
(4.19)1,2,3
E11 = e11 = u1/ X1 1
(4.2) Ex = u/ X
4.2
4.4
2112
2
1
1
21212 2
xu
xue (4.20)
2 : 2e12 x1-O-x2
4.3
dX1x1
x2
u1 u1+du1
dx2=dx3=0
ds=dx1=dX1+du1dS = dX1
4.4
x1
x2
21212
1 tanxu
12121
2 tanxu
O
81
4.7 dilatation
e
e= e11+e22+e33 = ekk (k=1,2,3: ) (4.21)
e i j=0 (i j) 1
V , (4.22)
V = (1+e11)(1+e22)(1+e33) (4.22)
, V=V 1 e= e11+e22+e33 .
4.8 ( )
4.2 , u+du , X+dX u
. u(X+dX) = u(X) +du (4.23)
du , dX ,
j
j
kkkk dX
Xuududu
ddd
XXX
XXuXuXXuu
(4.24)1,2
(4.8) F , (4.24) jk Xu // XuL (2 )
,
82
jk
j
j
kj
k
j
j
kjkjk dX
Xu
XudX
Xu
XudXLdu
21
21 (4.25)
XWEXLu
XE
XW
ddd
ddXedXXu
Xu
ddXwdXXu
Xu
jkjjk
j
j
k
jkjjk
j
j
k
,21
,21
(4.26)1,2 ,3
(4.25),(4.26) kj Xu / +/
, du (4.24)2
(4.25) 1 ekj , (4.18)
, 1 ekj k,j (symmetric part)
, 2 wk j
, k,j
(antisymmetric part)
XW d (4.27) .
, (4.28) (nabla)
(4.29): u , (4.30)
3
2
1
2
3
3
2
1
3
3
1
2
3
3
2
1
2
2
1
1
3
3
1
1
2
2
1
021
21
210
21
21
210
dXdXdX
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
Xu
(4.27)
kk X
e (4.28)
83
2
1
1
23
1
3
3
12
3
2
2
31 X
uXu
Xu
Xu
Xu
Xu
Xu
Xuu
X iij
kijki
j
kkjkk
jj
eee
eeeeeeu
(4.29)
3
2
1
12
13
23
312212311312332
021
21
210
21
21
210
21
21
21
21
21
21
dXdXdX
dXdXdXdXdXdX
dXdXd ikjijkkkjj
eee
eeeXu
(4.30)
(4.30) , (4.27) u
( ) ,
XXW dd2
, 2/
(1) P X x(X) xk X j
33122211 ,2,23 XxXXxXXx
(E.4.1)
F ( )
, j
kkj X
xFF
F
84
F
F
F
F
F
(2) xk X j
33222
11 ,,3
XxXxXXx (E.4.2)
F , 1
(3) ( ) P (x1, x2) (u1, u2)
eij (i,j=1,2: ) (u1, u2)
(u1, u2) 2 (4) u u i) (i=1,2,3)
(a) 3211321 5,2,3,, axaxaxaxuuu (E.4.3)
(b) 212
22
22
1321 cossin,,,, 333 xxexexxeuuu xxx (E.4.4)
(5) , (E.4.5)
(4.18)
,
,,
2
31
1
23
3
12
321
332
1
23
3
12
2
31
213
222
3
12
2
31
1
23
132
112
xe
xe
xe
xxxe
xe
xe
xe
xxxe
xe
xe
xe
xxxe
X1
X2
O 1
1
x1
x2
O
85
5.
5.1
4.3 ()
, ( )
2
(a) (elasticity):
(Hooke's
law) .
(b) (plasticity):
( )
5.2
87
,
, ,
, ,
5.1(a) 5.1(b) ,
, (
) 2
(i) ( )
(ii) .
.
, .
(normal stress):
AF ( )
(normal strain):
0Lue ( ),
L
LdLu
0
e
L0 e=u/L0
(a) (b)
5.1
u
F
88
(nominal strain, engineering strain)
L dL/L
(logarithmic
strain) ,
N
L
L
L
LN
LL
LdL
LL
LdLe
1lnln
1)(
0
00
0
0
(5.1),(5.2)
(5.1), (5.2) , e
5.2 ,
: E, ( )
( ): E
( ): Y
0.2% : 0.2 ( Y )
5.2
e
B
0.2%
H’
E
Y
89
, ,
, ,
, ,
,
,
: H’ ( )
( ): B
( ): eB
( )
( )
(engineering strain, nominal strain) Green
(logarithmic strain) Euler
5.3
( )1
1 (dislocation) , (lattice defacts) .
5.3
90
45°
,
( )5.4 5.5
5.4
1 ( 5.5
5.4
5.5
b : =
91
5.6
( )
5.7 , , 5.8 ,
5.6
5.7
92
(work hardening)
5.4
- ( )
(5.3)
n Ludwik [1909]: =C n (0 n<1) (5.4)
n: , : = n
)('
)(
00
0
EH
EE
YY
Y
5.8
( )( )
93
n : = 0 + K n (5.5)
n Swift[1952]: =C ( 0+ )n (5.6)
Voce [1948]: = C(1 m -ne ) (5.7)
C(1-m)= Y0 ,
Ramberg and Osgood[1946]: = ( /E) {1+ ( /B)n} (5.8)
/
Chakrabarty [1987]: ( Ludwik )
(5.9)
)/()/(
)/( 0
01
0 EE
EEE
Y
Yn
Y
5.9 (a)~(e)
n=0
n=1
n
n
E
H’
Y0
:H’=0, 0<E<+:H’ >0 0<E<+
E Y0
E
n (Ludwik,Ramberg Osgood)
Y0
(Ludwik, Swift )
94
5.9
5.9
5.9
Hart[1967]: (5.10)
Backofen[1964]: (5.11)
(Newton ): (5.12)
( ) .
dtd
m
5.6
(1) ( ) e Hooke's law
5.10(a)
uL
AF
eE ( 0
0
LLdLuL
L) (5.13)
( Young's modulus) .
(compliance with stretch)… J= E 1
.
5.5
(1) ( )
5.10(a) ,
e Hooke’s law
E (Young’s modulus) .
,
5.10
L 0
L
A 0=
1b 0
F
u
(a)
Au F
h
A=1
b (b)
FKC
m
mn
95
L
L
LLdLuuL
AF
eE
0
0 (5.13)
(2) ( )
5.10(b) F
(shear) 2et
Hooke's law
AF
AF
uh
AF
eG
t tan1
2 (5.14)
(rigidity), (shear modulus)
, x2 , x1
= 21, e t= e21=h
uxu
221
2
1 (5.15)1,2
(3)
5.10(a) , e
ec
eec (5.16)
(Poisson's ratio) , m
= 1/
(1) 5.3 2
1 , 2
1 =45
5.3
96
5.3
5.3
( ) 2 (E.5.1) =45°
,
cossinsincos
000
cossinsincos 1
2221
1211 (E.5.1)
(2)
(3) 5.10(b) e12 (5.15)2
(4) 5.10(a) b, b0, F, L, L0
L>L0, b<b0
(5) 5.10(a) , F
E (E.5.2)
22
22 EeEE (E.5.2)
( ) U = AL E = L
LFdL
0
5.3
O
A
B
x1x2
97
00
LLdLuL
L du= dL
(6) O-x1x2x3 P (x1 , x2 ,0)
0000000
QP
kj (E.5.3)
O-x1x2x3 x3
O-x'1x'2 x'3
[ ' kj] ( ) cos( )= c, sin( )= s 3 3.2
(3.24), ' kj= ' jk
10000
0000000
10000
'''''''''
333231
232221
131211
cssc
QP
cssc
(E.5.4)
(7) ( ) ( )
3
(8)
98
6.
(Constitutive equation)
5
6 ,
3
1
6.2
,
( ) (
) ij
ijij p ij (6.1)
pp
p
ij
000000
p
6.2.1 (ideal gas)
p, T,
,
99
, ,
,
p RT ( ) (6.2)
=1/v: (v: ) T : R :
6.2.2 (Incompressible fluid)
p T
= constant ( (6.3)
6.3 Newton
(6.4)
klijklijij VDp (6.4)
ij : Vkl12
uk
xl
ul
xk
Dijkl uk :
Newton , Dijkl ,
3 4 34= 81
2
jkiljlikklijijklD
(6.5)
100
ijkkijij
kljkiljlikklijij
klijklijij
VVpVp
VDp
2
(6.6)
, (6.7)
kkkk Vp 233 (6.7)
(Contraction) ,
6.4 Stokes
Stokes Vkk
3 2 0
23
(6.8)
(6.9)
ijkkijijij VVp322 (6.9)
Stokes3kk p
6.2.2
Vkk=0 ijijij Vp 2 (6.10)
0 , , (6.1)
6.5
,
ijijkkij ee 2
eij Lame ,
101
2 ,
,
, ,
, ,
, ,
, ,
,
,
(6.2) ,
f (p, , T) = 0 (6.11)
2
(6.2) van der
Waals (6.12)
, V 2 1
p
V 2 V RT (6.12)
(6.3)
,
( ) ,
,
( )
2 .
Newton
.
Newton
.
6.6 Hooke Hookean elastic solid)
3 ,
1 , Y.C.Fung: A first course in continuum mechanics, third edition (1994), page 182 (Prentice Hall) .
102
klijklij eC (6.13)
ij: , ekl: , Cijkl
(4 ) , 34=81 .
(4.18) ,
jiij Cijkl C jikl (6.14)
ekl elk Cijkl C jikl Cijlk
, , 6
Cijkl , 62=36
Isotropic elastic materials)
6.6.1
2 .
Cijkl ij kl ik jl il jk ik jl il jk (6.15)
ik jl il jk =0
Cijkl ij kl ik jl il jk (6.16)
ijkkij
jiijkkij
kljkiljlikklij
klijklij
eeeee
eeC
2
(6.17)
(6.17) , x1, x2, x3 ( x,y,z ) ( G ), (6.18), (6.19)
2 Lame .
103
31313333221133
23232233221122
12121133221111
2,22,2
2,2
eeeeeeeeee
eeeee (6.18)1~6
zxzxzzzzyyxxzz
yzyzyyzzyyxxyy
xyxyxxzzyyxxxx
eeeeeeeeeeeeeee
2,2
2,22,2
(6.19)1~6
i = j
kkiikkii eee 2323 (6.20)
ii= 11+ 22+ 33, eii=e11+ e 22+ e 33, ekk= e 11+ e 22+ e 33
23kk
kke (6.18) eij
kkijijije2322
1 (6.21)
Lame EE
1 1 2E 2 1
3 3 2 1
ijkkij
kkijijij
EE
e
112
1
(6.22)
6.6.2 K
3 (10) .
104
dx1, dx2, dx3
dx1(1+e11), dx2(1+e22), dx3(1+e33)
V
V= dx1(1+e11) dx2(1+e22) dx3(1+e33) - dx1 dx2 dx3
= dx1dx2dx3(1+e11+ e22+ e33+ e11 e22+ e22 e33+ e11 e33 +e11 e22 e33) - dx1dx2 dx3
dx1dx2dx3(e11+ e22+ e33) ….. ( )
,
V/V = (e11+ e22+ e33) = ekk (6.23)
(6.21) ,
pkk 3332211 (6.24)
i=j)
233
2323
21
233233
21
23313
21
323
3321
23221
pp
pp
ppe kkiiiiii
(6.25)
ii 11 22 33 3 , (6.26)
pekk
23
K (6.26)
6.1
ekk
e e=e11+e22+e33
105
V0 dx1dx2dx3
V 1 e11 dx1 1 e22 dx2 1 e33 dx3
, (6.27) .
332211113333222211332211
332211113333222211332211
321
321332211
321
321333222111
0
0
11
1111
111
eeeeeeeeeeeeeeeeeeeeeeee
dxdxdxdxdxdxeee
dxdxdxdxdxdxdxedxedxe
VVVe
(6.27)
, 2
, 1
ekk = e11+e22+e33
6.6.3
S
x3 33 (6.18)
, 03231122211
313333221133
2322332211
1211332211
20,220,2020,20
eeeeeeeeeeeeeee
2 e11 e22 2 e33 0 ,
e11 e22 e33 (6.28)
3 33=…
106
33
333333
23
221
e
ee
3333 23e (6.29)
(6.28) 11332211 20 eeee
e33 1 2 e11 0 e33 2e11e11
e33 2 (6.30)1
(6.28) 22332211 20 eeee ,
e22
e33 2 (6.30)2
, ,
e11
e33
e22
e33 2 (6.31)
( ) E 33 e33
, (6.29) (6.32) .
23
33
33
eE
(6.32)
, E, Lame
.
6.6.4
2
107
12 0
.
0, 312333221112 (6.33)
(6.18)
3133332211
2322332211
1211332211
20,2020,202,20
eeeeeeeeeeeeeee
e12 2 2e12
(6.34)
G G (Modulus of Rigidity)
6.6.5
Hooke Duhamel-NeumannCijkl ekl
T0 , T(6.35)
0TTeC ijklijklij (6.35)
ijijijkkij TTee 02 (6.36)
, ,
= (3 +2 ) = 21
E (6.37)
.
6.7
108
, (linear spring)
(dashpot) , 2 ( : Maxwell model, Voigt model) 3 , 4 ,
6.7.1 Maxwell
6.1 Maxwell .
s d
, ,
= s + d (6.38)
. ,
s E s (6.39)
dd d
dt (6.40)
E ,
t s d (6.38)
, (6.41) .
ddt
d s
dtd d
dt (6.41)
(6.39)d s
dt1E
d s
dt(6.40)
d d
dtd (6.41)
(6.42) .
6.1 Maxwell model (2 )
109
ddt
1E
d s
dtd (6.42)
s d ( ).
E
ddt
ddt
(6.43)
E (6.44)
6.7.2 Voigt
, 6.2
Voigt
: S
: d
2
: = S + d (6.45)
, (6.39), (6.40) 2
= S = d
E ddt
(6.46)
Maxwell Voigt
3 (Maxwell )
(6.47)
dtdE
dtd
dRe (6.47)
6.2 Voigt model (2 )
110
e, d 2 , ER
6.7.3 Maxwell
(t) u(t) (t)
(creep function) c(t)
. Creep ,
, Maxwell (6.42)4
ssss
sssss
1 (6.48)
(t)=u(t) 1/s , (6.48) s ,
sEssssss 11111111
222 (6.49)
tctu
Ett 11
(t>0) (6.50)
(creep) c(t)= (t/ +1/E)u(t)
6.7.4 Maxwell
Maxwell (t)=u(t) (t)
(relaxation function) k(t)
(6.48) (s) 1/s
(s)
4 t f(t) Laplace f (s) s
, 0
)()( dtetfsf st
111
111
11
11 sE
ssssss
sss (6.51)
/tEet (t>0) (6.52)
( ) k t Eet
(1) Voigt c(t), k(t)
. (t) u(t) ,
)()()(
)(11)( /
tEuttk
tueE
tc t
(E.6.1)1,2
(2)
(3) Newton
(4) Newton
(5)
(6) ( )
(7) zz=0( ) (O-xy)
(8) Maxwell Voigt .
( )E
(9) (6.17) ,
ijjiijkljkiljlik eeee 2 (E.6.2)
(10) (6.31), (6.32) , .
)1(2E ,
)1)(21(E (E.6.3)1,2
112
7.
7.1
1.10
7.1.1
(a)
,
,
113
(b)
7.1.2
NEWTON
114
1. ( ).
2.
= × ( ) 3. (
).
7.1.3 ( )
N j p j , (7.1) .
mj : j , v j : j ( ) .
v j ,
.
p j mj v j (7.1)
p (7.2) .
N
jj
N
jj m
11vpp (7.2)
N ( )
,
, ( 3 )
N
3 , v3 , v3
, d p = m3 v 3
, 3
115
, d p
1 ,
7.1.4
3 ( ) ,
1 1.11 I F I , F I t
. F IJ p IJ=F IJ t
t , I J .
, .
1.11 , F IJ p IJ=F IJ t
01
tIJ
N
I
N
JI Fr (7.3)
(torque)
(moment of inertia)
( )
116
7.1
V
S
T
,
7.2
7.1 V, S Ss
T F V ,
Sh h
dV ,
, dV
V S
3
:
K
G
E
V
Vjj dVvvK
21
(j: ) (7.4)
117
vj dV
V
G x dVV
(7.5)
xx gz
V
E dVV
(7.6)
V (7.4),(7.5),(7.6) K+G+E
K+G+E
(7.7)
Q, W (.)
D( )/Dt 4.2
WQEGKDtD
(7.7)
QW
(7.7)
, 2 vjvj
( )
( ) 2
dV
118
, V dV
(7.7) , dV
V (7.7)
Gauss
7.3 Gauss
7.3.1
Gauss
( )
(7.8)
Gauss
,
.
y= y(xk) y/ xk
y= y(x) y(xk) (k=1,2,3) (7.9)
y x=(x1, x2, x3)
y/ xk L
dy= ( y/ xk)dxk
LL
dxxydx
xydx
xydyyy 3
32
21
112 x (7.10)
(nabla)
y = ek ( y/ xk )
y , (7.11)
119
y = f(x) (7.11)
dy/dx = f ' (x)
2
1
)('12
x
x
dxxfyy (7.12)
1
y ( ) y x [ x1 ,
x2] y x1, x2
, y2 y1
,
(7.10) y(x) , F= y
, ( y2 y1
)
7.3.2
,
Gauss
A(xk) (k=1,2,3) F= A/ xk ,
dA
33
22
11
dxxAdx
xAdx
xAdx
xAdA k
k
(7.13)
, x2, x3 x1 dA1 , L: [a,b]
, (7.14) .
b
aab
A
Adx
xAAAdAb
a1
1111
1
1
(7.14)
120
7.2 V , L
dx2dx3
(7.14) dx2dx3
3211
3211 dxdxdxxAdxdxAA
b
aab (7.15)
x2x3 V
V
b
aVab dxdxdx
xAdxdxAA 321
13211 (7.16)
L V
, ( A/ x1) dx1dx2dx3 V
L (x1=a, b)
A dx2dx3 V
A 1b A1a 1 L
x 1 x1
7.2 V L a,b
x1
x2
x3x1=x1
dx2dx3
x1=bx1=a
L
V x2x3
() V
bdSb
adSa
nb
na
P(x1,x2,x3)
: V: S
e1
121
A L , 1
(7.16) (7.17)
VVab dxdxdx
xAdxdxAA 321
132 (7.17)
7.2 , L a, b
, dSa, dSb n a, n b
, ( ) a,b ,
d S e 1
a,b n1= dx2dx3/dS ,
d S e 1 = dx2dx3 = n e 1dS = n1 dS (7.18)
a,b n x 1 1 x1=b n1>0 x1=a n1<0
a,b (7.17)
(Ab Aa) dx2dx3 = Abn1b dSb + Aan1a dSa (7.19)
a,b V :
(7.17) V S d S
, A d S e 1 =An1 dS S
(7.17) (7.20)
VS
dxdxdxxAdSAn 321
11 (7.20)
V A
1 , e1 n n1 , n1
. V S
122
(7.20) , x1
V S .
, k=1 k=2,3
(7.20) , , k=1,2,3 (7.21)
dV= dx1dx2dx3
V kSk dV
xAdSAn (k=1,2,3) (7.20)
Vk
S k dVxAdSAn (k=1,2,3) (7.21)
(7.20), (7.21) Gauss 2 . Gauss
A F F = A/ xk ,
J =V
FdV (7.22)
J
, (7.4)~(7.6)
7.3.3 Gauss
(a) A (uj)
A , u= (uj) ,
(7.21) A= uj (j=1,2,3) .
Vk
j
S kj dVxu
dSnu (7.23)
j=k , (7.24) , Gauss
2 Green , Stokes , Gauss
(7.21)
123
.
dVxu
xu
xudS
dVxudSnu
VS
Vk
kS kk
3
3
2
2
1
1nu (7.24)
(7.24) V S
S V
,
0
uk/ xk (= div u) . V
div u >0
(b) A
A= , ek (7.25) .
Vk
kS kk dVx
dSn ee (7.25)
(gradient) , grad (= )
(7.10) 3
(c) A (uj) ( )
(7.23) , j, k , ei ijk uk/ xj
(2.20) ei ijk uk/ xj = u (= rot u) rot u
, u
VSS
Vj
kijkiS jkiijk
dVdSdS
dVxudSnu
unuun
ee (7.26)
124
(7.26)
rot u
( )
u , 2 = rot u
.
,
(d) A 2 Cij
Vk
ij
S kij dVxC
dSnC (7.27)
i=k
7.4
O-X1X2X3
t=t0 P
X1=a1, X2=a2. X3=a3
(a1,a2,a3)
P
(7.28) P’
xi xi a1,a2,a3,t i 1,2,3 (7.28)
7.3
X3, x3
X2, x2
X1, x1
t=0
t=t
P(a1, a2, a3)
P’(x1, x2, x3)
D
D’
xk=xk (a1, a2, a3, t), k=1,2,3
Xk=ak
125
(7.28) D (a1,a2,a3) D' (x1,x2,x3)
1 1 . D
D'
xi (a1,a2,a3,t) 1 . ,
, dt=t t0 , 1
(
) , D
(Jacobian determinant) (7.29) 0
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
ax
ax
ax
ax
ax
ax
ax
ax
ax
ax
(7.29)
(7.28) (material description) Lagrange
a1,a2,a3 vi
i (7.30), (7.31) .
vi a1,a2 ,a3,txi
t a1 ,a2 a3
(7.30)
i a1,a2 ,a3,tvi
t a1 ,a2a3
2xi
t 2a1 ,a2 a3
(7.31)
x=(x1,x2,x3) t (spatial
description) [Euler ]
t x
vi (x1,x2,x3,t)
(7.32) .
txvvt
tvt
dtdvt
k
ik
iii ,,,, xxxx (7.32)
126
t x=(xi) t+dt xi+vidt
. x=(xk) , vk dxk=vkdt
Taylor , t+dt ( )
, .
txvdtvx
txvdtt
txvtxv
txvdttdtvxvdtdt
tdvtdv
kikk
kikiki
kikkii
i
,,,,
,,,, xx (7.33)
(7.33) (7.32) . (7.32)
( ) ( )
. (7.32)
f(x1,x2,x3,t)
t f(x,t)
(Material derivative) , Df/Dt (7.32)
(7.33) d( )/dt , D( )/Dt
f(x,t) (7.34)
Df x1, x2 , x3, tDt
ft x fixed
fx1
x1
tfx2
x2
tfx3
x3
t
ft x fixed
fx1
v1fx2
v2fx3
v3
ft x fixed
fxk
vk (7.34)
,
. ,
(7.28) xk , t : vk =
dxk/dt = dak/dt , t0 ak
127
Euler f , xk t
, f xk t . f f ,
.
f = ( f/ t) t + ( f/ xk) xk (7.35)
xk xk xk+ xk
t xk/ t ,
f/ t ( t +0)
f/ t , xk/ t xk
vk .
f = ( f/ t) t + ( f/ xk) vk t (7.36)
t, xk t t+ t , vk t
xk/ t = vk
, t ,
xk+vk t t t
, Euler
,
xk/ t = vk Lagrange
Euler , xk/ t = vk
f/ t ,
. ,
. Lagrange (
) D( )/Dt
.
Df/Dt = ( f/ t) + vk ( f/ xk) (7.37)1
Df/Dt = ( f/ t) +v f (7.37)2
? ,
,
128
dV , A
V :
VAdVtJ )(
, , .
( ) ,
,
, (FEM)
(FDM)
7.5
A J(t)
J(t)
V V
dVtAdxdxdxtxxxAtJ ),(),,,()( 321321 x (7.38)
),,,( 321 txxxA xk , t (k=1,2,3)
t dt t+dt J(t)
t , V , t+dt V+ V
dt
(7.39)
VV Vdt
dVtAdVdttAdtd
DtDJ ),(),(lim
0xx (7.39)
V 7.4
V V (7.40) (7.40)
V A
(7.40) 7.4
V .
129
VV Vdt
V V Vdt
dVdttAdtddVtAdVdttA
dtd
dVtAdVdttAdVdttAdtd
DtDJ
),(),(),(lim
),(),(),(lim
0
0
xxx
xxx
(7.40)
(7.40)
dV=vjnjdSdt (i=1,2,3) (7.41)
. dxi=vj dt dS
nj dS vj
. (7.40) (7.42)
V Sjj
V Vjjdt
dSnvtAdVtA
dSdtnvtAdtddV
tA
DtDJ
),(
),(lim0
x
x(7.42)1,2
Sjj
Sjj dSdtnvdSdxnV
7.4 vj V
130
(7.42) Leibniz
(7.42) . Avj Gauss
(7.43)
DDt
AdVV
At
dVV xi
VAvi dV (7.43)
(7.44), (7.45)
V j
jj
jV
dVxv
AvxA
tAAdV
DtD
DtDJ
(7.44)
VV j
j
V
dVADtDAdV
xv
ADtDAAdV
DtD vdiv
(7.45)
(7.45) A J
v
7.6 , Leibniz
Gauss , 7.3.1
Leibniz ,
. (7.42) (3 )
1
tattaF
tbttbFdx
tF
dxtxFxt
xdxtFdxtxF
ttb
ta
tb
ta
tb
ta
tb
ta
)),(()),((
),(),(
)(
)(
)(
)(
)(
)(
)(
)(
(7.46)
3 http://en.wikipedia.org/wiki/Leibniz_integral_rule .
131
(7.46) x t F(x,t) x=[a,b]
t Leibniz . a, b
t (7.46) (7.42)
FS
jdSAn (7.47)
a, b
F vj V (7.47) V 7.4
dt +0
, (7.46) (7.42) ,
.
(7.46)
, x = [a(t),b(t)] , t
.
tFF
tx
tFF
xtx
tFF
tF
t
(7.48)
(7.48) Leibniz (
) (7.42)
(7.42)
,
7.7
132
7.7.1
t V m , V
(7.49) .
m dVV
(7.49)
x,t t x
m 0
DmDt
0 (7.50)
(7.42)
tdV
VvinidS
S0 (7.51)
(7.43) .
0dVxv
tV i
i (7.52)
V 0
.
0i
i
xv
t (7.53)
(7.51) (7.53) , (equations of continuity)
133
7.7.2
Newton
Pi Fi (7.54)
DPi
DtFi (i=1,2,3) (7.54)
, dVvPV
ii (7.55)
, V
iS
ii dVXdSTF (7.56)
vi Xi
Ti dS
nj , ji Cauchy
jjii nT (j=1,2,3: , i=1,2,3: ) (7.57)
(7.54) (7.55),(7.56)
Vi
Si
Vi dVXdSTdVv
DtD
i=1,2,3) (7.58)
vi (7.45) .
(7.57) (7.27)
dVXx
dVvvxt
v
Vi
j
ji
Vji
j
i (7.59)
(7.60) (7.60)
0 0 4 (3.17) 3 .
134
(7.61) (j=1,2,3: , i=1,2,3: ) .
0dVXx
vvxt
v
Vi
j
jiji
j
i (7.60)
ij
jiji
j
i Xx
vvxt
v(7.61)
(7.61)
ij
ji
j
ij
i
j
ji X
xxvv
tv
xv
tv (7.62)
(7.62) , (7.53) 0 . ,
Euler (Eulerian equation of motion) (7.63) .
ij
jii
j
ij
i XxDt
Dvxvv
tv
(7.63)
O-x1,x2,x3 (7.64)
33
33
2
23
1
133
23
32
2
22
1
122
13
31
2
21
1
111
XxxxDt
Dv
XxxxDt
Dv
XxxxDt
Dv
(7.64)
(7.64) 0 . (7.59)~(7.64) , : ij= ji i, j
(7.64)
.
7.8
135
7.2 (7.7) ,
7.1 h=(hi)
ni dS nihidS
(7.65)
V i
i
Sii dV
xhdSnhQ (7.65)
7.1 ,
vi , (7.66) .
dVvx
dVvF
dSvndVvFdSvTdVvFW
Viji
jVii
Sijji
Vii
Sii
Vii
(7.66)
Fi ( ), iT ni
dS , (.) (7.4)~(7.6) (7.66) (7.7) ,
dVvx
dVvFdVxhdVdVdVvv
DtD
Viji
jVii
V i
i
V VVii2
1
(7.67)
dVvx
vFxhdVvv
DtD
Viji
jii
i
i
Vii2
1
(7.67)
136
K, G, E
dVDt
Dxv
DtD
dVxv
DtD
DtD
DtDE
dVDtD
xv
DtD
dVxv
DtD
DtD
DtDG
dVDt
vvDvvxv
DtD
dVxvvv
DtvvDvv
DtD
DtDK
V k
k
V k
k
V k
k
V k
k
V
iiii
k
k
V k
kii
iiii
21
21
21
21
21
(7.53) , 0 (7.67)
dVvx
vFxhdV
DtD
DtDvv
DtD
Viji
jii
i
i
Vii2
1
(7.68)
j
iji
j
jiiii
i
iii x
vx
vvFxh
DtD
DtDvv
DtD
21
(7.69)
(7.63)
j
ijiii
ii
i
iii x
vFXDtDvv
xh
DtD
DtDvv
DtD
21
(7.70)
Xi
(7.71) (7.72) .
137
iii x
FX (7.71)
j
ijii
i
i
iii x
vvDtDv
xh
DtD
tvv
DtD
21
(7.72)
12
Dvi2
DtDvi
Dtvi t
0
j
iji
i
i
xv
xh
DtD
(7.73)
ji i,j ,
Vij , (7.75)
i
j
j
ijiij x
vxvVV
21
(7.74)
jijii
i Vxh
DtD
(7.75)
(7.75) ,
(7.7)
( )
138
(1) .
(2) ( )
.
(3) V
dVtxAJ ),(
DtDJ
.
3.1 Leibniz , A t
V t .
3.2 Gauss ,
DJ/Dt .
(4) , 3.1, 3.2 .
(5) F = (2x1)e1+(3x1x2 + x3x22) e3 , V S
J
V : P(0,0,0) Q(1,1,1) 1
S: V 6
SS
dxdxFdxdxFdxdxFdJ 213132321SF (E.7.1)
J .
(6) (5) , Gauss J .
(7) V S .
S
kkS
dSnxdJ Sx (E.7.2)
x=(x1, x2, x3) nk dS
(8) (7.52)
0k
k
xv
DtD
(E.7.3)
(9) (7.63) 0 ,
139
Gauss Cauchy ,
( )
(2
)
140
8.
8.1
,
.
.
, uj
vj (j=1,2,3)
, .
Euler Hooke .
.
.
Almansi (Euler ) , . , (4.17)2
eij12
u j
xi
uix j
ukxi
ukx j
(8.1)
2
eij12
u j
xi
uix j
(8.2)
. vi ui ,
vi=Dui t, x j
Dtuit
x j
tuix j
uit
v juix j
(8.3)
vj ui/ xj , 2
141
, (8.4) .
viuit
(8.4)
i vi . ,
i=Dvi t, x j
Dtvit
x j
tvix j
vit
v jvix j
(8.5)
vj vi/ xj , 2
, (8.6) .
ivit
(8.6)
Euler
ij
iji X
x (8.7)
Hooke
i
j
j
iij
k
kijijkkij x
uxu
Gxu
Gee 2 (8.8)
ijj
i
ik
k
iji
j
jj
iij
jk
k
ii
j
j
iij
k
k
ji
Xxx
uG
xxu
G
Xxx
uxx
uGxx
u
Xxu
xu
Gxu
x
22
222
(8.9)
142
e = uk/ xk
ijj
i
i
i Xxx
uG
xeG
tu 2
2
2
(8.10)
. Navier . (8.10)
(nabula) (8.11) .
G 2ui1
1 2exi
Xi
2uit2 (i=1,2,3) (8.11)
, (E.6.3) ( +G) .
(8.11) tui ,x 2 .
,
ui x = (x1,x2,x3) t
2 .
* : . ( )
( =0) , .
* : . ( )
, , .
2 .
(8.11) .
.
1. ( , ).
2. , .
3. ( 3 = 0 ) , ( 3
= 0) . 2 1 ,
.
4. ( ).
143
8.2
0323133 .
.
x3 x1-x2 .
Hooke 3 Hooke
.
313122113333
232311332222
121233221111
21,1
21,1
21,1
Ge
Ee
Ge
Ee
Ge
Ee
(8.12)1~6
0323133 , (8.13) .
0,
0,121,1
31221133
23112222
1212221111
eE
e
eE
eG
eE
e
(8.13)1~6
(8.13) , .
1212
1122222
2211211
1
1
1
eE
eeE
eeE
(8.14)1~3
2 Euler (8.15) .
(8.16)
144
22
2
22
22
1
12
21
2
12
21
1
11
tuX
xx
tuX
xx (8.15)1,2
2
1
1
212
2
222
1
111 2
1,,xu
xue
xue
xue (8.16)
u1,u2 Navier (8.17) [
] .
22
2
22
2
1
1
222
22
21
22
21
2
12
2
1
1
122
12
21
12
11
11
tuX
xu
xu
xG
xu
xuG
tuX
xu
xu
xG
xu
xuG
(8.17)
8.3
, 3
0,0 33
2
3
1 uxu
xu
(8.18)
u1,u2 x1,x2 .
0323133 eee (8.19)
. 3 3
. 3
3 (x1x2 ) .
Hooke . (8.12)
, (8.19) e33 =0
221133 (8.20)
145
.
112222112
112222
221122112
221111
111
111
EEe
EEe
(8.21)
(8.21)
221122
221111
1121
1121
eeE
eeE
(8.22)
. 2 Euler (8.15) (8.22) ,
(8.16) u1,u2 Navier (8.23)
.
22
2
22
2
1
1
222
22
21
22
21
2
12
2
1
1
122
12
21
12
211
211
tuX
xu
xu
xG
xu
xuG
tuX
xu
xu
xG
xu
xuG
(8.23)
(8.23) (8.17) , 2
,
8.4 Airy
2 (8.17), (8.23)
( ) ,
. , .
146
, ,
(8.14),(8.21) .
.
(stress
function)
,
, 2 ( ,
) Airy
, (8.15)
0 ,
, (8.24)
0j
ji
x (8.24)
. , 2
x1=x, x2=y .
, u1=u, u2=v
0
0
2
22
1
12
2
21
1
11
xx
xx (8.25)
0
0
yx
yx
yyxy
yxxx
(8.25)’
(x,y) , (8.26)
. x y .
147
yxxy xyyyxx
2
2
2
2
2
,, (8.26)
(8.25)’
x
2
y2 y
2
x y
3
x y2
3
x y2 0 (8.27)1
x
2
x y y
2
x2
3
x2 y
3
y x2 0 (8.27)2
(8.26)
x,y
(8.26) ,
(8.13), (8.21)
, 1*
. 3
2 ,
:
. (8.16) u, v , (8.28)
u,v
(8.29) .
yu
xve
yve
xue xyyyxx 2
1,, (8.28)
2exy
x y12
2
x yuy
2
x yvx
12
2
y2
ux
2
x2
vy
12
2exx
y2
2eyy
x2
(8.29)
1* (E.4.5) 3 .
148
(8.29) Hooke (8.13) .
ExEyGyxxxyyyyxxxy
2
2
2
22
22
2G=E/(1+ ) ,
2
2
2
2
2
2
2
22
2
2
2
22
12
12
xxyyyx
xyyx
xxyyyyxxxy
xxyyyyxxxy
(8.30)
. , (8.25)’
000
222
2
2
2
22
2
2
2
2
yxxxyy
yxxyyxxy
xyxxxyyy
xyxxyyxyyyxx
(8.31)
(8.32) .
022
2
2
2
2
yxxyxyyyxx (8.32)
(8.26) ,
4
y4
4
x4 24
x2 y2 0 (8.33)
, (8.33) (8.34)
.
02
,
2
x2
2
y2 (8.34)
149
(8.34) (bi-harmonic function)
. (8.26)
, Airy .
u, v
,
(harmonic function)
1 (8.35)
0 (8.35)
8.1
8.1
(x,y)
.
x y
x , (8.36) .
2
x2
2
y2 x
2 xx2
2 xy2 x
xx y
xy
xx
xx
y y
2x
x2
x2 x2
y2
(8.36)
150
0004
24
222
2222
22
2
22
2
2
2
2
2
2
2
2
2
2
2
2
2
4
4
22
4
4
4
2
2
2
2
2
2
2
2
4
4
22
4
4
4
2
2
2
2
2
3
3
3
4
4
22
4
4
4
2
2
2
2
4
4
22
4
22
4
2
3
2
3
4
4
3
3
2
2
2
2
3
3
2
3
2
3
2
2
3
3
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
yxx
yxyxx
x
yxyxx
yxx
yxxyxyxx
yxx
yxxyxyxx
xyx
yx
xyx
yxx
yxyxxx
xxyx
yx
yxyx
y
yxx
yxxx
xxxyx
yx
xx
xyyx
xx
xx
yx
xx
xyx
(8.37)
, , 0
, x
y
8.2
a22
x2 b2xy c22
y2
(8.38)
Airy . ,
. 0 .
151
22
222
222
222
222
2
2
222
222
2
2
2
2222
22
ca
ycxybxay
ycxybxax
ycxybxayx
, 2 =0
. , .
(8.38) (8.26) ,
222 ,, bac xyyyxx (8.39)
.
,
2211 acEEx
ue yyxxxx (8.40)
2211 caEEy
ve xxyyyy (8.41)
22 caEE
e yyxxzz (8.42)
G
bGx
vyue xy
xy 2221 2 (8.43)
u , (8.40) x
u x, y xE
c2 a2 c1 y (8.44)
v , (8.41) y
v x, y yE
a2 c2 c3 x (8.45)
c1(y), c3(x) , (8.44), (8.45)
(8.43) , (8.46) .
152
12
uy
vx
12
dc1dy
dc3dx
b22G
(8.46)
(8.46) 1 , c1(y) y 1 c3(x) x
1 . .
c1 c4y c5c3 c6x c7
(8.47)
c4~c7 (8.46) .
c4 c6b2G
(8.48)
( ) , 0 .
c5 c7 0 (8.49)
. (8.48) ,
u x, yxE
c2 a2 c4y
v x, yyE
a2 c2 c4b2G
x (8.50)
2 0 .
, (8.51) .
z12
vx
uy
(8.51)
(8.51) (8.50)
vx
uy
c4b2
Gc4 0 (8.52)
c4 .
153
c4b2
2G (8.53)
, u, v (8.54) .
u x, yxE
c2 a2b22G
y
v x, yyE
a2 c2b22G
x (8.54)
8.3
, 2
. , x=y=0 u=v=0
.
AxyAyAxb
AxyyAyyxAa
xyyyxx
xyyyxx
2,,)(
,3
,32)(
22
2332
(8.55)1,2
.
(8.25)’ .
(a)
03
0
022320
223
2
232
AyAyyy
Axyx
Ayx
AxyAxyxyy
Ayyxx
Ayx
yyxy
yxxx
(b)
02220
42220
2
2
AyAyyy
Axyx
Ayx
AxAxAxxyy
Axx
Ayx
yyxy
yxxx
154
(a) (b)
.
(a) . 2 Hooke
(a) ,
2
323
332
122
1
32
311
31
321
AxyEGx
vyue
yyxyEA
Eyve
yyyxEA
Exue
xyxy
xxyyyy
yyxxxx
(8.56)1~3
. u (8.56)1 x .
ycxyxyyxEA
dxyyyxEAdx
xuu
1333
332
332
31
332
(8.57)
v (8.56)2 y
xcyyxyEA
dyyyxyEAdy
yvv
24
224
323
122
2121
32
31
(8.58)
(8.57) (8.58) c1 c2 .
(8.56)3 0 0 .
(8.56)3 (8.57), (8.58) ,
xcdxdyyxy
xEA
ycdydxyxyyx
yEA
xv
yu
24
224
1333
122
2121
332
31
155
2
2123
12
1231
AxyE
xcdxdyc
dydxyx
EA
xv
yu
3
3
21x
EAxc
dxdyc
dyd (8.59)
, (8.60)
xv
yu
(8.60)
0 xv
yu , .
xcdxdxy
EAyc
dydxyxyx
EA
22
1223 2
31
2321 2
31 xyx
EAxc
dxdyc
dyd (8.61)
(8.59), (8.61) , c1, c2
.
231 3
1 xyxEAyc
dyd (8.62)
(8.63) . c3
333
1 3cxyyx
EAc (8.63)
156
c2
22 xy
EAxc
dxd (8.64)
x , (8.65) . c4
422
2 2cyx
EAc (8.65)
u, v (8.57), (8.58) c1, c2
33
333333
13
32
3
cxyEA
cxyyxEAxyxyyx
EAu
(8.66)
422
4
4224224
16
212
262
61
2
cyxyEA
cyxEAyyxy
EAv
(8.67)
c3, c4 ,
: x=y=0 u=v=0 , c3=c4=0
. u, v .
8.4 3
L a z=0~L .
(1) (u,v,w) . c
u= czy, v= czx, w=0 (8.68)
(2) .
(3) .
157
(4) z=L ( ) .
1) (8.68) , .
20
21
21,
20
21
21
,021
21,00
,0,0
cycyzu
xwecxcx
yw
zve
czczxv
yue
zzwe
czxyy
veczyxx
ue
zxyz
xyzz
yyxx
Hooke (8.12) 0 .
yz, zx .
GcyGcxGe zxxzyzyzzy
yxxyzzyyxx
,2
,0,0
2) (7.64)
00
,000
,000
zGcx
yGcy
xzyx
Gcxzyxzyx
Gcyzyxzyx
zzyzxz
zyyyxy
zxyxxx
,
3) : z=L n (0,0,1) , z=0
(0.0,-1)
Cauchy
z=L : 00
0000
100 GcxGcyGcxGcy
GcxGcy
, z=0 z=L
, r=(x, y) 90°
158
(r=a) n ,
(8.69) .
0,sin,cos0,,ay
axn (8.69)
a , x , x=a
cos , y=a sin Cauchy
0cossincossinsincos
,000
0000
0sincos
GcaxyGct
tGaxGcy
GcxGcy
z
z
(8.70)
, 0
4) z=L .
t = (-Gcy, Gcx, 0) dS=dxdy
0
0
,00
22
22
22
22
xa
xa
a
aAA
zy
xa
xa
a
aAA
zx
AAzz
dyxdxGcdxdyGcxdxdy
ydydxGcdxdyGcydxdy
dxdydxdy
(8.71)1~3
(x, y, L) , dS dM ,
r = (x, y, 0)
32222
3
321
00 eeeee
trM
dSyxGcdSGcyGcxdSGcxGcy
yx
dSd
(8.72)
159
dS dM , M =(0, 0, Mz)
442
00
3
22
22
4aGcaGcddrrGc
dxdyyxGcdMM
aAA
zz
(8.73)
8.5
,
0,, 22zzxyyzxzyyxx xy
0000
,000
,000
2
2
zyxzyx
zx
yxzyx
zyy
xzyx
zzyzxz
zyyyxy
zxyxxx
, ,
(8.74) .
021,1
021,11
021,11
313122
232322
121222
Gexy
EEe
Geyx
EEe
Gexy
EEe
yyxxzzzz
xxzzyyyy
zzyyxxxx
(8.74)1~6
(8.74) , (E.4.5) .
,
160
00,0
00,0
00,0
yye
xe
ze
zyxe
yxe
ze
ye
yxze
xze
ye
xe
xzye
zxxyzxyzz
yzxyzxyy
xyzxyzxx
EEze
xe
xze
EEye
ze
zye
EEExe
ye
yxe
xxzzzx
zzyyyz
yyxxxy
202,02
220,02
422,02
2
2
2
2
2
2
2
2
2
2
2
2
. , .
, .
,
.
8.5 (Principle of Saint-Venant)
A
A’ A A’
,
2
.
.
161
(a)
(b)
8.1
A
A
p0
F=p0A
p0
p0F
p0
A
A
M=FdL
F
F
dLdLF
FMM h
h
8.1 (a) (b)
(a) 2
( ) ( )
(b) 2
162
2
(1) Airy (x,y) xx, yy, xy
1. = A y2 + B x2
2. = ex sin(y)
3. = x ex cos(y)
(2) =Ay3+Cx3 Airy
xx, yy, xy .
exx, eyy, exy .
(3) Airy 4
.
(4) (Saint-Venant)
(5) (E.8.1) ,
.
Vijij dVeE
21
(E.8.1)
(6.18), (6.19) .
(6) u
a, b, c , E
u u1,u2,u3
ax by,by cz,cz ax (E.8.2)
(7) Airy .
163
)cosh()sin( kykx (E.8.3)
, 2
)cosh(kyky eeky , k
(8) O (0,0,0) x3
RR
xA
Rxx
AR
xxAuuu 43,,,, 3
23
332
331
321 , (E.8.4)
A , 23
22
21 xxxR
: ,)3,2,1,:(, 21 kn
Rnx
xRnRR
xRx
xR
nk
k
nn
k
k
k
52
21
32
5121
323
21
1
33 RxxRxRxxxRxRxx
x
(9) Airy ( 11, 22, 12) (u1,u2)
21
222
1
22xpxp
(E.8.5)
(x1, x2)= (0,0) (u1,u2)= (0,0) , x1=x,
x2=y
164
9.
9. 1
6
Newton Newton1*
ij p ij Vkk ij 2 Vij p ijvk
xkij
vi
x j
vj
xi
(9.1)
Euler (7.63)
Dvi
DtXi
pxi xi
vk
xk xk
vk
xi xk
vi
xk
(9.2)
Xi
( : ) (7.53)
t
vk
xk
0 vk
xk
0 (9.3)
(9.2)
Dvi
DtXi
pxi
2vi
xk xk
(9.4)
3 /
/ 3 Laplace
ij (6.6)
165
1
*2 vi (i=1,2,3), p 4
(9.3) (9.4) 4
9. 2
vn s
vn fvt f
vt s0
0
it̂
ijiji tt ˆ (9.5)
Newton (9.1) (9.5)
Laplace2
x12
2
x22
2
x32
2
xk xk
166
jiji
ji
j
j
ijij
k
kjiji
npn
nxv
xvn
xvnpt
'
ˆ (9.6)
ij'
0 ii pnt̂
9.1(a) 9.1(b)
9.1(c) 9.1(d)
167
[ ( )2 ] = [ ]
p1 p2
9.1(a) dx dy
9.1(b)
x-z y-z
x-z 9.1(c) dy
9.1(d) dyd
R1 dx R1d d dx/R1
dyd dydx/R1 y-z
dydx/R2 dxdy
dydx/R1+ dydx/R2 dxdy
1R1
1R2
p1 p2 (9.7)
Laplace
R1 R2 p1- p 2>0
R1= R2= p1= p 2
nk ik2
ik1 0
Laplace
168
nk ik2
ik1 1
R1
1R2
ni (9.8)
p1 p2 ni ik2 ' ik
1 ' nk1R1
1R2
ni (9.9)
ij ' 0 (9.7)
9. 3 Reynolds
V L L V
(9.4)
LVttVppVvuLxx iiii /',/',/',/' 2 (9.10)
''
'''
'' 2
kk
i
i
i
xxu
VLxp
DtDu
(9.11)
2
Reynolds Re
Re VL VL
vi
169
ui 'xi '
0 (9.12)
(9.11) (9.12)
Reynolds
Reynolds
(9.11)
Reynolds
Reynolds
u2
u y V2
V/L
V 2
V LVL
(9.13)
Reynolds
(1) 10 2mm/sec
Reynolds m 1.4
(1.4 10 2 g / cm sec )
(2)
Reynolds 50km/h 6m
Reynolds 20 1.808 10 4
g/cm sec) = 0.150st (cm2/sec)
170
9. 4
Navier-Stokes Reynolds
Navier-Stokes
pxi
2vi
xk xk
0 (9.14)
2h
9.2
x 0
v1=v(x2), v2=v3=0, v1(h)=0, v1(-h)=0
Navier-Stokes
px1
2v1
x22 0 (9.15)
px2
0 (9.16)
px3
0 (9.17)
171
(9.16) (9.17) p x1 (9.15)
x12 p x1
2 0 p x1
(9.15)
d2v1
dx22
1 dpdx1
const (9.18)
x2
v1 A Bx2x2
2
2dpdx1
(9.19)
A B A h2 2 dp dx1 , B 0
v11
2dpdx1
h2 x22 (9.20)
Navier-Stokes
Dvi
DtXi
ij
x j
(E9.1)
r, ,z x,y,z
x r cosy r sinz z
rx
xr
cos ,ry
yr
sin
xyr2
sinr
,y
xr2
cosr
(E9.2)
172
x, y r,
xrx r x
cosr
sinr
yry r y
sinr
cos(E9.3)
vx vr cos v sinvy vr sin v cosvz vz
(E9.4)
rr xx cos2yy sin2
xy sin 2
xx sin2yy cos2
xy sin 2
r yy xx cos sin xy cos 2 (E9.5)
zr zx cos zy sin
z zx sin zy cos
zz zz
xxvx
x, yy
vy
y, zz
vz
z,
xy12
vx
yvy
x, yz
12
vy
zvz
y, zx
12
vz
xvx
z
(E9.6)
(E9.3) (E9.4) (E9.6)
173
xx cosr
sinr
vr cos v sin
cos2 vr
rsin2 vr
r1r
vcos sin
vr
1r
vr vr
,
yy sin2 ur
rcos2 vr
r1r
vcos sin
vr
1r
vr vr
,
xysin 2
2vr
r1r
v vr
rcos 2
rvr
1r
vr vr
(E9.7)
zz, zy, zx
rrvr
r,
vr
r1r
v, zz
vz
z
r12
1r
vr vr
vr
, zr12
vr
zvz
r,
z12
1r
vz vz
(E9.8)
Dvi
Dtai x
axvx
tvx
vx
xvy
vx
yvz
vx
z(E9.9)
(E9.4) (E9.5) (E9.9)
174
ax tvr cos v sin
vr cos v sin cosr
sinr
vr cos v sin
vr sin v cos sinr
cosr
vr cos v sin
vz zvr cos v sin
cosvr
tvr
vr
rvr
vr v2
rvz
vr
z
sinvt
vrvr
vr
v vrvr
vzvz
(E9.10)
ax ar cos a sin (E9.11)
arvr
tvr
vr
rvr
vr v2
rvz
vr
z
arvt
vrvr
vr
v vrvr
vzvz
(E9.12)
azvz
tvr
vz
rvr
vz vzvz
z(E9.13)
ij
x j
(E9.3) (E9.5)
xx
xxy
yxz
z
175
rr
r1r
r rr
rrz
zcos
1r
r
r2 r
rz
zsin
(E9.14)
vr
tvr
vr
rvr
vr v2
rvz
vr
zXr
rr
r1r
r rr
rrz
zvt
vrvr
vr
v vrvr
vzvz
X1r
r
r2 r
rz
z
vz
tvr
vz
rvr
vz vzvz
zXz
zr
r1r
z zz
zrz
r
(E9.15)
rr p 2 err p 2vr
r
p 2 e p 2vr
r1r
v
zz p 2 ezz p 2vz
z
r 2 er1r
vr vr
vr
z 2 e z1r
vz vz
zr 2 ezrvr
zvz
r
(E9.16)
(E9.15) Navier-Stokes
176
vr
tvr
vr
rvr
vr v2
rvz
vr
zXr
pr
2vrvr
r2
2r2
v
vt
vrvr
vr
v vrvr
vzvz
X1r
p 2v2r2
vr vr2
vz
tvr
vz
rvr
vz vzvz
zXz
pz
2vz
(E9.17)
22
r2
1r r
1r2
2
2
2
z2 (E9.18)
R
U
vz = vr , vr r = vr z
2vr
z2
pr
,pz
0 (E9.21)
1r
rvr
rvz
z0 (E9.22)
z 0 vr vz 0,z h vr 0, vz U,r R p p0
(E9.23)
177
h p0 (1)
vr1
2pr
z z h (E9.24)
(2) z
U1r
ddr
rvr dz0
h h3
12 rddr
ddpdr
(E9.25)
p p03 Uh3 R2 r2 (E9.26)
F3 UR4
2h3 (E9.27)
h
x-y x z
9.3
178
vx=v(z) Navier-Stokes
d 2vdz2 g sin 0,
dpdz
g cos 0 (E9.31)
zx= dv/dz=0 zz=-p=-p0 p0
z=0 v=0
p p0 g h z cos , vgsin2
z 2h z (E9.32)
y
Q vdz0
h gh2 sin3
(E9.33)
(3) Couette
9.4
rad/sec
T
R1
R2
(4) Navier-Stokes
a b
9. 5
9.4 Couette
179
ij p ij (9.21)
Dvi
DtXi
pxi
(9.22)
p vi Xi
v1
x1
v2
x2
v3
x3
0vi
xi
0 (9.23)
(9.23)
v3=0 v1 v2 x1 x2
(9.23)
u1
x1
v2
x2
0 (9.24)
x1, x2 v1,v2
v1 x2
, v2 x1
(9.25)
(9.24) x1, x2
9.25) (9.22)
180
2
t x2 x2
2
x1 x2 x1
2
x22 X1
1 px1
2
t x1 x2
2
x12 x1
2
x1 x2
X21 p
x2
(9.26)
0 p
t2
x2
2
x1 x1
2
x2
0 (9.27)
22
x12
2
x22 (9.28)
dx1
v1
dx2
v2
(9.29)
v2dx1 v1dx2 0 (9.25)
x1
dx1 x2
dx2 d 0 (9.30)
x1, x2
9. 6
181
I C v dlC
viCdxi (9.31)
C v dl
Stokes C
I C v n dSS
curlv i ni dSS (9.32)
S C ni
curlv i eijkv j ,k curlv
C
DDt
v dlc
(9.33)
C
DDt
vi dxiC
DDtC
vidxiDvi
Dtdxi vi
Ddxi
DtC(9.34)
Ddxi Dt dxi dvi
(9.22) Dvi/Dt (9.34)
DDt
vi dxiC
1 pxi
Xi dxi vidviC
dpC
Xi dxiC
12
dv2
C
(9.35)
182
3 v2
0 2 0
DDt
vi dxiC
dpC
(9.36)
(barotropic) (9.36) 0
DDt
vi dxiC0 (9.37)
Helmholtz
0 0
0
(9.32) 0
9. 7
0
v curlv 0 (9.38)
eijkvj ,k 0 (9.39)
2
3
183
v1
x2
v2
x1
0 (9.40)
(9.25) (9.40)
2
x12
2
x22 0 (9.41)
Laplace
3
0,0,03
1
1
3
2
3
3
2
1
2
2
1
xv
xv
xv
xv
xv
xv
(9.42)
v grad
(9.42)
2
x12
2
x22
2
x32 0 (9.43)
Laplace
Euler
184
0
9.4 O
xy x,y,z 2
ax by cz Ax2 By2 Cz2 Dxy Eyz Fzx
=0
z=0 vz z 0 x y z 0 x y 0a = b = c = 0, C = - A - B, E = F = 0
Dxy x y
Ax2 By2 A B z2
z A=B
A x2 y2 2z2
vx 2Ax, vy 2Ay, vz 4Az
dxvx
dyvy
dzvz
dx2Ax
dy2Ay
dz4Az
x2z=c1 y2z=c2 3
y B=0
A x2 y2
xz=const
185
(5) z
(a) 1
4log x2 y2 1
2log r r2 x2 y2
(b) x
(c) Arn cos n tan 1 yx
(d) cos
r
ur r, u
1r
(6) 2
v1 y, v2 x
, v3 0
9.4
186
=
(a) c
(b) y
(c) Arn sin n
(d) sin
r
(7) (5)
(a) 0
(b)
187
本書は 2011 年 4 月 1日、長岡技術科学大学より UD Project の一冊として刊行
されたものの復刊です。
連続体力学の基礎
れんぞくたいりきがくのきそ
Fundamentals of Continuum Mechanics
2015 年 6 月 19 日 初版発行©
著 者 古 口 日出男
永 澤 茂
発行者 福 田 雅 夫
発行所 GIGAKU Press
(長岡技術科学大学出版会)
〒940-2188 新潟県長岡市上富岡町 1603-1
長岡技術科学大学内
電話 0258-47-9266
Printted in Japan
ISBN978-4-907996-09-3
ISBN:978-4-907996-3
