1. 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

2. Theory of Elasticity Content （内容） Introduction（概述）
Chapter
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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）
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） 5 1

3. Material Behavior- Linear Elastic Solids
Theory of Elasticity
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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) 5 2

4. 5.1 Material Characterization
Theory of Elasticity
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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
Scope：linear elastic solid (线弹性体) 5 3

5. 5.1 Material Characterization
Theory of Elasticity
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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） 4 5

6. 5.1 Material Characterization
Theory of Elasticity
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5.1 Material Characterization
Testing machines（力学性能测试）
Room Temperature(室温)
High Temperature（高温） 5 5

7. 5.1 Material Characterization
Theory of Elasticity
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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 6

8. 5.1 Material Characterization
Theory of Elasticity
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5.1 Material Characterization
Typical stress-strain curves（典型应力－应变曲线）
proportional limit（比例极限）
ductile materials（韧性材料）
elastic limit（弹性极限）
yield point（屈服点）
brittle material （脆性材料） 5 7

9. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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5.2 Linear Elastic Materials-Hooke’s Law
Hooke’s Law(虎克定律)
Cijkl elastic moduli (弹性常数)
Units: Stress (force/area).
How many components? 5 8

10. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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5.2 Linear Elastic Materials-Hooke’s Law
Cijkl fourth-order elasticity tensor（四阶弹性张量）
81 Components
symmetry of the stress and strain tensors
36 Components 5 9

11. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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5.2 Linear Elastic Materials-Hooke’s Law
36 Components of Elastic Moduli
（36个弹性常数） 10

12. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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5.2 Linear Elastic Materials-Hooke’s Law
isotropic homogenous materials， Need 36 ？各向同性均质材料, 36个?)
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. 5 11

13. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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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)
Lame’s constant
拉梅常量
shear modulus
剪切模量
平均正应变的3倍，体积应变 5 12

14. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
Chapter
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5.2 Linear Elastic Materials-Hooke’s Law
In individual scalar equations（以单独标量的方式写出）
Express the strain in terms of the stress.
（用应力表达应变） 13

15. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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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 5 14

16. 5.2 Linear Elastic Materials-Hooke’s Law
Theory of Elasticity
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5.2 Linear Elastic Materials-Hooke’s Law
Hooke’s Law(虎克定律) 5 15

17. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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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) 5 16

18. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli
G or μ
G or μ is called the shear modulus.(切变模量) 5 17

19. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli
K and σm and θ
σm is called average stress. (平均应力)
θis called the bulk strain, related to the change in volume of material element.(体应变) 5 18

20. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli
K and σm and θ
K is called the bulk modulus of elasticity (体积模量.)
课本，P52，公式错 19

21. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli
Pure Shear（纯剪切）
this modulus μor G is simply the
slope of the shear stress-shear strain curve
a thin-walled cylinder
torsional loading 5 20

22. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli
Hydrostatic Compression (or Tension)(静水压力/拉)
the bulk modulus of elasticity（体积弹性模量） 5 21

23. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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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） 5 22

24. 4.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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4.3 Physical Meaning of Elastic Moduli

25. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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5.3 Physical Meaning of Elastic Moduli 5 24

26. 5.3 Physical Meaning of Elastic Moduli
Theory of Elasticity
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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。) 5 25

27. Theory of Elasticity Vocabulary （常用词汇） constitutive equation 本构方程
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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 5 26

28. Theory of Elasticity
Chapter
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Homework
思考题：4-1
习 题： 4-3 27 5