theory of elasticity 弹性力学
DESCRIPTION
Theory of Elasticity 弹性力学. Chapter 7 Two-Dimensional Formulation 平面问题基本理论. Theory of Elasticity. Chapter. Page. Content （内容）. Introduction （概述） Mathematical Preliminaries （数学基础） Stress and Equilibrium （应力与平衡） Displacements and Strains （位移与应变） - PowerPoint PPT PresentationTRANSCRIPT
Theory of Elasticity弹性力学Chapter 7
Two-Dimensional Formulation
平面问题基本理论
Chapter Page
Content （内容）
1 1
1. Introduction （概述）2. Mathematical Preliminaries （数学基础）3. Stress and Equilibrium （应力与平衡）4. Displacements and Strains （位移与应变）5. Material Behavior- Linear Elastic Solids （弹性应力应变关
系 )6. Formulation and Solution Strategies （弹性力学问题求
解）7. Two-Dimensional Formulation （平面问题基本理论）8. Two-Dimensional Solution （平面问题的直角坐标求解）9. Two-Dimensional Solution （平面问题的极坐标求解）10. Three-Dimensional Problems （三维空间问题）11. Bending of Thin Plates （薄板弯曲）12. Plastic deformation – Introduction （塑性力学基础）13. Introduction to Finite Element Mechod （有限元方法介
绍）
Chapter Page
Two-Dimensional Formulation
• 7.1 Plane Stress and Plane Strain ( 平面应力和平面应变 )• 7.2 Displacement Formulation ( 位移求解 )• 7.3 Stress Formulation and Airy Stress
Function ( 应力求解与应力函数 )• 7.4 Photoelastic stress measurement ( 光弹应力测试 )
7 2
Chapter Page
7.1 Plane Stress （平面应力）
z =±h, are stress free
h, is small in comparison to other dimensions
Not only on the surface, but also throughout the entire domain. ( 整个实体 )
0 hzz
0 hzzx
0 hzzy
7 3
Example: thin elastic plate( 弹性薄板 )
),(),,(),,(
0
yxyxyx xyxyyyxx
yzxzz
Chapter Page
7.1 Plane Stress （平面应力）
0,1
1
1
1
yzxzxyxy
yxyxz
xyy
yxx
E
E
E
E
0)(2
1
0)(2
1
)(2
1
,,
x
w
z
u
y
w
z
xy
u
z
u
y
u
x
u
xz
yz
xy
zyx
0 yzxzz Hooke’s law strain-displacement equations
The equilibrium equations（平衡方程）
7 4
Field equations(基本方程 )),(),,(),,( yxyxyx xyxyyyxx
0
0
yyxy
xxyx
Fyx
Fyx
7.1 Plane Strain （平面应变）
0
)(2
1,,
yzxzz
xyyx xy
u
y
u
x
u
Chapter Page
all cross-sections have identical displacements （横截面位移相同）
3-D 2-D
(1) A prismatic body whose length is much larger than any in-plane dimension, .
(2) In-plane loads are independent of the out-of-plane coordinate z.
(3) Absence of normal strain , in a direction perpendicular to the plane.
maxRL
0z
7 ５
Example: long cylindrical body ( 长圆柱体 )
0),,(),,( wyxvvyxuu
Chapter Page
7.1 Plane Strain （平面应变）
Plain Strain Examples Plain Strain Examples
7 ６
7.1 Plane Strain （平面应变）
0
)(2
1,,
yzxzz
xyyx xy
u
yx
u
0,2
)()(
2)(
2)(
yzxzxyxy
yxyxz
yyxy
xyxx
0
0
yyxy
xxyx
Fyx
Fyx
Chapter Page
Hooke’s lawstrain-displacement equations
the equilibrium equations
7 7
Field equations( 基本方程 )0),,(),,( wyxvvyxuu
7.1 Plane Stress and Plane Strain
0w
Chapter Page
x y xy x y xy
Plane “Stress”6 component , 3 are zero
Plane “Strain”6 component , 3 are zero
Difference
7 8
0 yzxzz
7.1 Plane Stress and Plane Strain
Chapter Page
Plain Stress
平面应力问题
Plain Strain平面应变问题
非平面问题
Not Plain Problem
7 9
Problems ：
Chapter Page
7.2 Displacement Formulation ( 位移法 )
0)1(2
0)1(2
2
2
y
x
Fy
v
x
u
yv
Ev
Fy
v
x
u
xv
Eu
Displacements Formulation （ Navier equations for plane stress ）
+
7 10
),(),,( yxvvyxuu bb (B.C.)
Chapter Page
+
7 11
Displacements Formulation （ Navier equations
for plane strain ）
7.2 Displacement Formulation ( 位移法 )
),(),,( yxvvyxuu bb (B.C.)
0)(
0)(
2
2
y
x
Fy
v
x
u
yv
Fy
v
x
u
xu
Chapter Page
7.3 Stress Formulation ( 应力法 )
yxxyxyyx
2
2
2
2
2
2
y
F
x
Fv yx
yx )1()(2
0
0
yyxy
xxyx
Fyx
Fyx
Stress Formulation （ for plane stress ）
+ or
+
7 12
xbxyy
by
by
ny
ybxyx
bx
bx
nx
nnyxTT
nnyxTT)()()(
)()()(
),(
),(
(B.C.)
7.3 Stress Formulation ( 应力法 )
Chapter Page
Stress Formulation （ for plane strain ）
+ or
+
7 13
yxxyxyyx
2
2
2
2
2
2
0
0
yyxy
xxyx
Fyx
Fyx
y
F
x
F
vyx
yx )1(
1)(2
xbxyy
by
by
ny
ybxyx
bx
bx
nx
nnyxTT
nnyxTT)()()(
)()()(
),(
),(
(B.C.)
7.3 Stress Formulation ( 应力法 )
y
F
x
Fv yx
yx )1()(2
yxxyxyyx
2
2
2
2
2
2
Chapter Page
Difference in solution
the equilibrium equations ( 平衡方程 )
Compatibility Equations （相容方程）
Which factor causes the difference?
7 14
y
F
x
F
vyx
yx )1(
1)(2
Plain StrainPlain Stress
0
0
yyxy
xxyx
Fyx
Fyx
7.3 Stress Formulation ( 应力法 )
Chapter Page
The difference in Physical Equation between Plain Stress and Plain Strain
7 15
xyxy
xyy
yxx
E
v
vE
vE
)1(
)(1
)(1
Plain Stress
xyxy
xyy
yxx
E
vv
v
E
v
v
v
E
v
)1(
)1
(1
)1
(1
2
2
Plain Strain
7.3 Stress Formulation ( 应力法 )
Chapter Page
1
21 E
E
Plain Stress Plain Strain
1
2)1(
)21(
E
E
Plain Strain Plain Stress
7 16
7.3 Airy Stress Function （应力函数）
Ylm
Xml
sxysy
sxysx
)()(
)()(
0
0
yyxy
xxyx
Fyx
Fyx
yxxyxyyx
2
2
2
2
2
2
Chapter Page
Solution of plain problems( 平面问题的应力求解 )
Single Connected ( 单连通域 )
7 17
y
F
x
F
vyx
yx 1
1)(2
Plain Strain
y
F
x
Fv yx
yx )1()(2
Plain Stress
0)(2
2
2
2
yxyx
3 unknownsSolution is not easy employs the Airy stress function
Single unknown
7.3 Airy Stress Function （按应力求解）
;0,, xyyyXx yFxF
xFyF YXxyyx ,0,0
Chapter Page
方程的解
非齐次方程的特解
齐次方程通解
0
0
yx
yx
yxy
xyx
7 18
0
0
yyxy
xxyx
Fyx
Fyx
全解 = 齐次方程通解＋
+ 非齐次方程的特解。
xyxB
y ),(
yyxA
x ),(
Chapter Page
7.3 Airy Stress Function （应力函数）
)( xyxyx
yyx
)( yxyxy
xxy
0
0
yx
yx
yxy
xyx
由微分方程理论，必存在一函数 A(x,y) ，使得
xyxA
xy ),(
也必存在一函数 B(x,y) ，使得
yyxB
xy ),(
yyxB
xyxA
),(),(
由微分方程理论，必存在一函数 φ(x,y) ，使得
xyx
yxB
),(),(
yyx
yxA
),(),(
yxxy
2
,2
2
yx
,
2
2
xy
齐次方程的通解
7 19
7.3 Airy Stress Function （应力函数）
Chapter Page
0
0
yx
yx
yxy
xyx
yxxy
2
,2
2
yx
,
2
2
xy
通解
;0,, xyyyXx yFxF 特解
yFx Yy
2
2
xFy Xx
2
2
yxxy
2
满足相容方程
0)(2
2
2
2
yxyx 02
4
4
22
4
4
4
yyxx
+ 边界条件＋单值条件biharmonic equation
7 20
7.3 Airy Stress Function （应力函数）
0)(2
2
2
2
yxyx
15 unknowns including 3 displacements, 6 strains, and 6 stresses.3 D
2 D02
4
4
22
4
4
4
yyxx
1 unknowns
Chapter Page 7 21
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
0)(2
2
2
2
yxyx
Chapter Page
Ylm
Xml
sxysy
sxysx
)()(
)()(
Solution of plain problems （平面问题的应力求解）
Single Connected ( 单连通域 )
Stress distribution doesn’t depend on material constants
Photoelastic stress measurement ( 光弹应力测试 )
7 22
0
0
yyxy
xxyx
Fyx
Fyx
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
Chapter Page
Photoelastic experiment （光弹性实验）
7 23
)( 21 Ch
光程差 模型厚度 主应力差值
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
Chapter Page 7 24
Example:
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
Chapter Page 7 25
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
Chapter Page 7 26
Example:
indirect tension test
(ASTM D-4123 1987)
bituminous and other brittle materials such as concrete,asphalt, rock, and ceramics.
7.4 Photoelastic stress Measurement ( 光弹应力测试 )
Chapter Page 7 27
Example:
7.4 Photoelastic stress Measurement ( 光弹性测试 )
Chapter Page 7 28
Example: FEM
7.4 Photoelastic stress Measurement ( 光弹性测试 )
Chapter Page 7 29
Example: granular （颗粒状） materials
7.4 Photoelastic stress Measurement ( 光弹性测试 )
Chapter Page 7 30
Example:
Photoelastic studies of the stress distribution around the tip of a crack
Vocabulary( 词汇 )
Chapter Page 6 31
Plane stressPlane strain Photoelastic stress measurementAiry Stress Functionbiharmonic equation
平面应力平面应变 光弹应力测试艾里应力函数双调和方程
