Theory of Elasticity
弹性力学
Chapter 5
Elastic stress-strain relations
(弹性应力应变关系)
动力学院 闫晓军
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 5
Page
1
Material Behavior- Linear Elastic Solids
5.1 Material Characterization（材料特性）
5.2 Linear Elastic Materials—Hooke’s Law
（广义胡克定律）
5.3 Physical Meaning of Elastic Moduli
（弹性常数的物理意义）
Vocabularies （常用词汇）
Homework（作业)
Chapter 5
Page 2
5.1 Material Characterization
Constitutive Equation (本构方程) :
Relations that characterize the physical properties
of materials are called constitutive equations. (描
述材料物理特性)
Normally defined by constitutive stress- strain relations.
（应力－应变关系）
.
Generally
  f ( ,  , t , T
Scope：linear elastic solid (线弹性体)
Chapter 5
Page
3
)
5.1 Material Characterization
Elastic solid（弹性体）:
recovers its original configuration when the
loadings are removed , does not include rate or
history effects（卸载后回复）
linear elastic solid （线弹性，在本课程中）
linear elasticity predictions have shown good
agreement with experimental data. （采取这样的假设后，
计算结果和试验吻合很好）
Chapter 5
Page 4
5.1 Material Characterization
Testing machines（力学性能测试）
Room Temperature(室温)
Chapter 5
Page 5
High Temperature（高温）
5.1 Material Characterization
Cylindrical or flat stock sample(试件)
Load cell and clip gage（力传感器，应变引伸计）
Definition of stress and stress:
σ=P/A , ε=Δl/l
* Small strain
Chapter
Page 6
5.1 Material Characterization
Typical stress-strain curves（典型应力－应变曲线）
proportional limit（比例极限）
elastic limit（弹性极限）
yield point（屈服点）
ductile materials（韧性
材料）
brittle material （脆性
材料）
Chapter 5
Page 7
5.2 Linear Elastic Materials-Hooke’s Law
Hooke’s Law(虎克定律)
 x  E x
 ij  Cijkl kl
Cijkl elastic moduli (弹性常数)
Units: Stress (force/area).
How many components?
Chapter 5
Page 8
5.2 Linear Elastic Materials-Hooke’s Law
 ij  cijkl kl
Cijkl fourth-order elasticity tensor（四阶弹性张量）
81 Components
symmetry of the stress and strain tensors
36 Components
Chapter 5
Page 9
5.2 Linear Elastic Materials-Hooke’s Law
36 Components of Elastic Moduli
（36个弹性常数）
 x  c11 x  c12 y  c13 z  c14 xy  c15 yz  c16 zx
 y  c21 x  c22 y  c23 z  c24 xy  c25 yz  c26 zx
 z  c31 x  c32 y  c33 z  c34 xy  c35 yz  c36 zx
 xy  c41 x  c42 y  c43 z  c44 xy  c45 yz  c46 zx
 yz  c51 x  c52 y  c53 z  c54 xy  c55 yz  c56 zx
 xz  c61 x  c62 y  c63 z  c64 xy  c65 yz  c66 zx
Chapter
Page
10
5.2 Linear Elastic Materials-Hooke’s Law
isotropic homogenous materials， Need
36 ？各向同性均质材料, 36个?)
 x  c11 x  c12 y  c13 z  c14 xy  c15 yz  c16 zx
 y  c21 x  c22 y  c23 z  c24 xy  c25 yz  c26 zx
 z  c31 x  c32 y  c33 z  c34 xy  c35 yz  c36 zx
 xy  c41 x  c42 y  c43 z  c44 xy  c45 yz  c46 zx
 yz  c51 x  c52 y  c53 z  c54 xy  c55 yz  c56 zx
 xz  c61 x  c62 y  c63 z  c64 xy  c65 yz  c66 zx
For isotropic materials: the influence of εx on σx is the same as
εy on σy; (各向同性材料)
The influence of ε y, εz on σ x is the same.
Chapter 5
Page 11
5.2 Linear Elastic Materials-Hooke’s Law
isotropic homogenous materials(各向同性均质)
only 2 independent elastic constants are needed to describe the
behavior of isotropic materials. (2个弹性常数)
 ij  ij  2 ij
Lame’s constant
拉梅常量
剪切模量
   kk  1   2   3
Chapter 5
shear modulus
Page 12
平均正应变的3倍，体积应变
5.2 Linear Elastic Materials-Hooke’s Law
 ij  ij  2 ij
In individual scalar equations（以单独标量的方式写出）
 x   ( x   y   z )  2  x
 y   ( x   y   z )  2  y
 z   ( x   y   z )  2  z
 xy  2  xy
 yz  2  yz
 zx  2  zx
Chapter
Express the strain in terms of the stress.
（用应力表达应变）
1 

 ij 
 ij   ij ii
E
E
Page 13
5.2 Linear Elastic Materials-Hooke’s Law
E: the modulus of elasticity or Young’s modulus
杨氏模量
νPoisson’s ratio. 泊松比
 (3  2  )
E



2(   )
Chapter 5
Page 14
5.2 Linear Elastic Materials-Hooke’s Law
Hooke’s Law(虎克定律)
 ij  ij  2 ij
1 

 ij 
 ij   ij ii
E
E
Chapter 5
Page 15
1
 x  [ x   ( y   z )]
E
1
 y  [ y   ( x   z )]
E
1
 z  [ z   ( y   x )]
E
1
 xy 
 xy
2
1
 yz 
 yz
2
1
 zx 
 zx
2
5.3 Physical Meaning of Elastic Moduli
E and ν
E is called the modulus of elasticity or Yang’s
modulus, and ν is referred to as Poisson’s ratio. (弹
性模量(杨氏模量)与泊松比)
 (3  2  )
E



2(   )
Chapter 5
Page 16
5.3 Physical Meaning of Elastic Moduli
G or μ
G or μ is called the shear modulus.(切变模量)
  G

E
G
2(1   )
Chapter 5
Page 17

5.3 Physical Meaning of Elastic Moduli
K and σm and θ
σm is called average stress. (平均应力)
1
 m  ( x   y   z )
3
θis called the bulk strain, related to the change in
volume of material element.(体应变)
V
1  2

 3 m   x   y   z 
( x   y   z )
V0
E
Chapter 5
Page 18
5.3 Physical Meaning of Elastic Moduli
K and σm and θ
m
K 

K is called the bulk modulus of elasticity (体
积模量.)
课本，P52，公式错
Chapter
Page
19
5.3 Physical Meaning of Elastic Moduli
Pure Shear（纯剪切）
0  0 
 ij   0 0 
 0 0 0 
 0

 ij    2

 0

2
0
0
   / 2 xy   /  xy
a thin-walled cylinder
torsional loading
Chapter 5
this modulus μor G is simply the
slope of the shear stress-shear
strain curve
Page 20
0


0

0

5.3 Physical Meaning of Elastic Moduli
Hydrostatic Compression (or Tension)(静
水压力/拉)
  m
 ij   0
 0
m
m
m
K 
0
 m
0
0   1  2
 E  m

0 
 ij  
0


 m 



0
0

1  2
m
E
0




0


1  2

m 

E
m

the bulk modulus of elasticity（体积弹性模量）
Chapter 5
Page 21
0
5.3 Physical Meaning of Elastic Moduli
• for isotropic materials，led to the definition
of five constants.（5个常数）
• Only 2 of these are needed to characterize
the material. （2个独立常数）
• if any two are given, the remaining three can
be determined by using simple formulae.
（给定2个能求出余下的3个）
Chapter 5
Page 22
4.3 Physical Meaning of Elastic Moduli
Chapter
Page
5.3 Physical Meaning of Elastic Moduli
m
K

Chapter 5
Page
24
5.3 Physical Meaning of Elastic Moduli
The forms of Hooke’s law in curvilinear coordinates?
For isotropic materials, the elasticity tensor Cijkl is the
same in all coordinate frames, and thus the structure
of Hooke’s law remains the same in any orthogonal
curvilinear system.
(正交坐标系下的虎克定律，和直角坐标下相同。)
Chapter 5
Page 25
Vocabulary （常用词汇）
constitutive equation
本构方程
Independent
独立的
isotropic
各向同性
Hydrostatic
静水的
Pure shear
纯剪切
Bulk modulus
体积 模量
ductile materials
韧性材料
brittle material
脆性材料
Bulk strain
体应变
Chapter 5
Page 26
Homework
思考题：4-1
习 题： 4-3
Chapter 5
Page 27