Theory of Elasticity
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Transcript Theory of Elasticity
Theory of Elasticity
弹性力学
Chapter 7
Two-Dimensional Formulation
平面问题基本理论
Content(内容)
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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)
Material Behavior- Linear Elastic Solids(弹性应力应变关系)
Formulation and Solution Strategies(弹性力学问题求解)
Two-Dimensional Formulation (平面问题基本理论)
Two-Dimensional Solution (平面问题的直角坐标求解)
Two-Dimensional Solution (平面问题的极坐标求解)
Three-Dimensional Problems(三维空间问题)
Bending of Thin Plates (薄板弯曲)
Plastic deformation – Introduction(塑性力学基础)
Introduction to Finite Element Mechod(有限元方法介绍)
Chapter 1
Page
1
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)
Chapter 7
Page 2
7.1 Plane Stress (平面应力)
Example: thin elastic plate(弹性薄板)
h, is small in comparison to other dimensions
z =±h, are stress free
z z h 0
zx z h 0
zy z h
0
Not only on the surface,
but also throughout
the entire domain.
(整个实体)
z xz yz 0
x x ( x, y), y y ( x, y), xy xy ( x, y)
Chapter 7
Page 3
7.1 Plane Stress (平面应力)
Field equations(基本方程)
z xz yz 0
x x ( x, y), y y ( x, y), xy xy ( x, y)
Hooke’s law
strain-displacement equations
1
x y
E
1
y y x
E
x
z
x y
xy
E
1
1
xy
xy , xz yz 0
E
x
y
yz
xz
The equilibrium equations
x xy
(平衡方程)
x
xy
x
Chapter 7
u
u
u
, y
, z
x
y
z
1 u
(
)
2 y x
1 w
(
)0
2 z y
1 u w
(
)0
2 z x
x
y
y
y
Page 4
Fx 0
Fy 0
7.1 Plane Strain (平面应变)
Example: long cylindrical body (长圆柱体)
(1) A prismatic body whose length is much larger
than any in-plane dimension, L Rmax .
(2) In-plane loads are independent of the out-ofplane coordinate z.
(3) Absence of normal strain z 0, in a direction
perpendicular to the plane.
u u( x, y), v v( x, y), w 0
all cross-sections have identical displacements(横截面位移相同)
u
u
1 u
x , y , xy ( )
3-D
Chapter 7
2-D
x
y
z xz yz 0
Page 5
2 y
x
7.1 Plane Strain (平面应变)
Plain Strain Examples
Chapter 7
Page 6
7.1 Plane Strain (平面应变)
Field equations(基本方程)
u u( x, y), v v( x, y), w 0
strain-displacement equations
u
1 u
, y
, xy ( )
x
y
2 y x
z xz yz 0
x
the equilibrium equations
Chapter 7
Page 7
Hooke’s law
x ( x y ) 2 x
y ( x y ) 2 y
z ( x y ) ( x y )
xy 2 xy , xz yz 0
x xy
Fx 0
x
y
xy y
Fy 0
x
y
7.1 Plane Stress and Plane Strain
Difference
z xz yz 0
w0
x y xy
x y xy
Plane “Stress”
6 component , 3 are zero
Chapter 7
Page 8
Plane “Strain”
6 component , 3 are zero
7.1 Plane Stress and Plane Strain
Problems:
Plain Stress
平面应力问题
Plain Strain
平面应变问题
非平面问题
Not Plain
Problem
Chapter 7
Page 9
7.2 Displacement Formulation (位移法)
Displacements Formulation(Navier equations
for plane stress)
2u
E
u v
Fx 0
2(1 v) x x y
2v
E
u v
Fy 0
2(1 v) y x y
+ u ub ( x, y), v vb ( x, y)
Chapter 7
Page 10
(B.C.)
7.2 Displacement Formulation (位移法)
Displacements Formulation( Navier equations
for plane strain)
2u ( )
u v
Fx 0
x x y
u v
v ( ) Fy 0
y x y
2
+
u ub ( x, y), v vb ( x, y)
Chapter 7
Page 11
(B.C.)
7.3 Stress Formulation (应力法)
Stress Formulation(for plane stress)
x xy
Fx 0
x
y
xy y
Fy 0
x
y
+
+
T T
( x, y ) n y n
Chapter 7
Page 12
(b )
y
(b )
y
2 xy
or
F Fy
2 ( x y ) (1 v) x
y
x
(b )
Txn Tx(b ) ( x, y ) x(b ) nx xy
ny
n
y
2 y
x
2 2
2
y
x
xy
2
(b )
xy x
(B.C.)
7.3 Stress Formulation (应力法)
Stress Formulation( for plane strain)
2
2
2
xy
x
y
2
2
2
F 0
y
x
xy
x
y
xy
x
x
xy
x
y
y
Fy 0
+
or
1 Fx Fy
( x y )
(1 v) x y
2
+
(b )
Txn Tx(b ) ( x, y ) x(b ) nx xy
ny
T T
n
y
Chapter 7
(b )
y
( x, y ) n y n
(b )
y
Page 13
(b )
xy x
(B.C.)
7.3 Stress Formulation (应力法)
Difference in solution
the equilibrium equations
(平衡方程)
Compatibility Equations
(相容方程)
Plain Stress
xy
x
Fx 0
x
y
xy
y
Fy 0
x
y
2
2 xy
2 x y
2 2
2
y
x
xy
Plain Strain
F Fy
2 ( x y ) (1 v) x
x
y
2 ( x y )
1 Fx Fy
(1 v) x
y
Which factor causes the difference?
Chapter 7
Page 14
7.3 Stress Formulation (应力法)
The difference in Physical Equation
between Plain Stress and Plain Strain
Plain Stress
Plain Strain
1
x ( x v y )
E
1
y ( y v x )
E
(1 v)
xy
xy
E
1 v2
v
x
( x
y)
E
1 v
1 v2
v
y
( y
x)
E
1 v
(1 v)
xy
xy
E
Chapter 7
Page 15
7.3 Stress Formulation (应力法)
Plain Stress
Plain Strain
Plain Strain
E
E
1 2
E
1
Chapter 7
Page 16
Plain Stress
E (1 2 )
(1 ) 2
1
7.3 Airy Stress Function (应力函数)
Solution of plain problems(平面问题的应力求解)
x xy
Fx 0
x
y
xy y
Fy 0
x
y
2
2 xy
2 x y
2 2
2
y
x
xy
l ( x ) s m( xy ) s X
m( y ) s l ( xy ) s Y
Single Connected (单连通域)
3 unknowns
Solution is not easy
Chapter 7
Plain Strain
1 Fx Fy
( x y )
1 v x
y
2
Plain Stress
Fx Fy
( x y ) (1 v)
y
x
2
2
2
x 2 y 2
( x y ) 0
employs the Airy stress function
Single unknown
Page 17
7.3 Airy Stress Function (按应力求解)
方程的解
x xy
Fx 0
x
y
xy y
Fy 0
x
y
齐次方程通解
xy
x
0
x
y
xy
y
0
x
y
非齐次方程的特解
全解 = 齐次方程通解+
x FX x, y Fy y, xy 0;
+非齐次方程的特解。
x 0, y 0, xy FX y FY x
Chapter 7
Page 18
7.3 Airy Stress Function (应力函数)
xy
x
0
x
y
xy
y
0
x
y
也必存在一函数 B(x,y),使得
y
xy
x
( xy )
x
y
y
由微分方程理论,必存在一函
数 A(x,y),使得
A( x, y )
x
y
A( x, y)
xy
x
y
yx
( yx )
y
x
x
Chapter 7
B( x, y)
x
xy
B( x, y)
y
A( x, y ) B( x, y )
x
y
由微分方程理论,必存在一函
数 φ(x,y),使得
A( x, y )
( x, y )
( x, y)
B( x, y)
y
x
2
2
2
y 2 , x 2 , xy
x
y
xy
齐次方程的通解
Page 19
7.3 Airy Stress Function (应力函数)
xy
x
0
x
y
xy
y
0
x
y
通解
2
2
2
y 2 , x 2 , xy
x
y
xy
特解
x FX x, y Fy y, xy 0;
2
2
2 2 ( x y ) 0
x y
x 2 FX x
y
2
y 2 FY y
x
2
xy
xy
2
满足相容方程
4
4
4
2 2 2 4 0
4
x
x y
y
biharmonic equation
+边界条件+单值条件
Chapter 7
Page 20
7.3 Airy Stress Function (应力函数)
3D
15 unknowns including 3 displacements, 6 strains, and 6 stresses.
2
2
2
y 2
x
( x y ) 0
2 2 2 4 0
4
x
x y
y
4
4
1 unknowns
Chapter 7
Page 21
4
2D
7.4 Photoelastic stress Measurement (光弹应力测试)
Solution of plain problems(平面问题的应力求解)
x xy
x
xy
x
y
y
y
Fx 0
Fy 0
2
2
2 2 ( x y ) 0
x y
Stress distribution doesn’t depend
on material constants
Photoelastic stress measurement
(光弹应力测试)
l ( x ) s m( xy ) s X
m( y ) s l ( xy ) s Y
Single Connected (单连通域)
Chapter 7
Page 22
7.4 Photoelastic stress Measurement (光弹应力测试)
Photoelastic experiment(光弹性实验)
Ch(1 2 )
光程差 模型厚度
Chapter 7
主应力差值
Page 23
7.4 Photoelastic stress Measurement (光弹应力测试)
Example:
Chapter 7
Page 24
7.4 Photoelastic stress Measurement (光弹应力测试)
Chapter 7
Page 25
7.4 Photoelastic stress Measurement (光弹应力测试)
Example:
indirect tension test
(ASTM D-4123 1987)
bituminous and other brittle materials such as concrete,
asphalt, rock, and ceramics.
Chapter 7
Page 26
7.4 Photoelastic stress Measurement (光弹应力测试)
Example:
Chapter 7
Page 27
7.4 Photoelastic stress Measurement (光弹性测试)
Example: FEM
Chapter 7
Page 28
7.4 Photoelastic stress Measurement (光弹性测试)
Example: granular(颗粒状) materials
Chapter 7
Page 29
7.4 Photoelastic stress Measurement (光弹性测试)
Example:
Photoelastic studies of the stress distribution around the tip of a crack
Chapter 7
Page 30
Vocabulary(词汇)
Plane stress
Plane strain
Photoelastic stress measurement
Airy Stress Function
biharmonic equation
Chapter 6
Page
31
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
Homework
思考题:
6-1
6-5
Chapter 7
Page 32