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）
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
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
w0
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
xy
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
xy
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
xy
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
xy
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
xy
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
xy
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  
xy
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