連続体力学の基礎 -...
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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
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1.
, .
,
. , ,
.
.
2.
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. , ,
.
, .
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
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(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
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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
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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
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連続体力学の基礎
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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
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目 次PrefaceContents
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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
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連続体力学の基礎
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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
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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
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連続体力学の基礎
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, .
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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
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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)
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Mzyx
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, (1. 3)
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v=(vx, vy, vz) 3*,
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pv0d0d
limd
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, .
,
dV , dp , dV
. dV +0 ,
.
dV
. ,
.
v 3 3(bold) v
v dV
8
3
4
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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
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, .
, ,
. ,
, . 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
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( , ) ( )
.
. 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
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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
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,
(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
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: 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
![Page 25: 連続体力学の基礎 - nagaokaut.ac.jplib.nagaokaut.ac.jp/gigaku-press/wp-content/uploads/2015/...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](https://reader033.vdocuments.site/reader033/viewer/2022052718/5f0622b47e708231d41677f9/html5/thumbnails/25.jpg)
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
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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
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(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
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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
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(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
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, , ,
. , 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
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(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
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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
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2.
÷
3
2 2
1 (vector) 0 (scalar)
0,1,2 , 3,4,5
. n
. , 2
4
23
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÷ 3
,
24
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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
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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
,
![Page 37: 連続体力学の基礎 - nagaokaut.ac.jplib.nagaokaut.ac.jp/gigaku-press/wp-content/uploads/2015/...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](https://reader033.vdocuments.site/reader033/viewer/2022052718/5f0622b47e708231d41677f9/html5/thumbnails/37.jpg)
,
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
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. ,
. ,
.
( ),
, .
, , . ,
. , . ,
2.3
2.4
,
,
,
,
28
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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
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,
,
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
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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
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,
,
,
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
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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
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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
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,
,
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
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(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
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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
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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
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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
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(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
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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
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(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
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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
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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
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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
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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
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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
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(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
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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
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| 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
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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
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(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
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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
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) ( ) ,
, 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
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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
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. 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
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, , (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
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, 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
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)
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
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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
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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
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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
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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
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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
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( 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
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, 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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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: 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
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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
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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
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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
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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
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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
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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
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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
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5.
5.1
4.3 ()
, ( )
2
(a) (elasticity):
(Hooke's
law) .
(b) (plasticity):
( )
5.2
87
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,
, ,
, ,
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
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(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
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, ,
, ,
, ,
,
,
: H’ ( )
( ): B
( ): eB
( )
( )
(engineering strain, nominal strain) Green
(logarithmic strain) Euler
5.3
( )1
1 (dislocation) , (lattice defacts) .
5.3
90
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45°
,
( )5.4 5.5
5.4
1 ( 5.5
5.4
5.5
b : =
91
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5.6
( )
5.7 , , 5.8 ,
5.6
5.7
92
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(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
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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
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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
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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
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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
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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
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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
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, ,
,
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
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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
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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
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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
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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
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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
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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
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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
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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
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, (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
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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
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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
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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
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7.
7.1
1.10
7.1.1
(a)
,
,
113
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(b)
7.1.2
NEWTON
114
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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
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, 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
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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
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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
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, 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
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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
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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
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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
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(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
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.
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
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(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
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(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
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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
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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
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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
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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
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(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
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(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
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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
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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 .
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(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
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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)
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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) .
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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
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(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
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Gauss Cauchy ,
( )
(2
)
140
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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
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, (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
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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
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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
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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
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.
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
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, ,
(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
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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
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(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
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(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
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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
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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
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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
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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
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(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
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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
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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
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(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
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(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
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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
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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
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(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
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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 .
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)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
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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
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*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
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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)
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[ ( )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
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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
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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)
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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)
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(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)
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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)
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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)
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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
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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
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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)
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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
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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
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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)
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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
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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)
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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
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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
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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
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(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
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=
(a) c
(b) y
(c) Arn sin n
(d) sin
r
(7) (5)
(a) 0
(b)
187
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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
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ISBN:978-4-907996-3