弹性力学
Theory of Elasticity
闫晓军，Xiaojun Yan, Ph.D.
Associate Professor
School of Jet Propulsion
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
About the course
The lectures will be given in Chinese, but the
lecture will be written in English and Chinese ，
and Chinese textbook is selected.（中文讲述，中
英文课件，中文教材）
The homework will take 20% of the overall score,
the mid-term exam another 20%, while the final
exam will take the rest 60%. ( 平时20% ，期中
20% ，期末60% )
Chapter 1
Page 2
About the course
References:
1 杨桂通，《弹性力学》，高等教育出版社，2002 1
2 徐芝纶， 《弹性力学简明教程》，高等教育出版社，
1983
3 S. P. Timoshenko, J.N. Goodier, “Theory of
Elasticity”,
McGraw-Hill Book Company, 1970
4 Martin H. Sadd, “Elasticity”, Elsevier Butterworth–
heinemann, 2005
Chapter 1
Page 3
Introduction
•
•
•
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What is Elasticity （弹性力学？）
History of Elasticity（历史）
Applications of Elasticity （应用）
Basic Assumptions（基本假设）
Chapter 1
Page 4
What is Elasticity
• Solid Mechanics: deals with the
deformations set up in a solid as a result of
a system of forces.
固体力学：研究固体在外力载荷作用下
物体内部所产生的变形等。
Chapter
Page 5
What is Elasticity
固体：“A material is called solid
rather than fluid if it can also
support a substantial shearing
force over the time scale of
some natural process or
technological application of
interest.”
J. R. Rice
Difference between solids and
fluids
Mechanics of Solids, The New
Encyclopedia of Britannica, 15th
edition, Vol. 23, pp. 734-747, 2002,
Chapter
Page 6
What is Elasticity
Elasticity is an elegant and fascinating
subject that deals with determination of
the stress,strain, and displacement
distribution in an elastic solid under the
influence of external forces.
研究弹性体在外力载荷或者温度改变
（外界因素）时，物体内部所产生的位移、变
形和应力分布等。
Chapter 1
Page 7
What is Elasticity
F
F
Elastic
Chapter
D
Page 8
Perfectly Elastic
D
History of Elasticity
• Newton
• Galileo
• Hooke
• Bernoullis
• Navier
• Poisson
Chapter
Cauchy
Saint-Venant
Kirchhoff
Love
Timoshenko
Theodore Von Karman
G. I. Taylor
Griffith , Irwin , Rice
Courant ,Clough
Ziekiewitz, Feng Kang
Page 9
History(1600-1700)
• Leonardo da Vinci
• System for equalizing the release of a spring
the possible length dependence of
the wire strength due to statistical
defects distribution
Chapter
Page
10
History (1600-1700)
• Galileo
Rods under tension: Independent of length , proportional
to the cross section.
Chapter 1
Page 11
History (1600-1700)
•Galileo’ test
•A deformable body ,First time
•Geometric dependence : beam length,the
bending resistance of cross-section.
•stress distribution was wrong
d
L
W
Chapter 1
Page
12
History(1600-1700)
Isaac Newton
Chapter 1
Page 13
History (1600-1700)
• Robert Hooke (1635-1703)
Willen Church
The only building in
existence that Robert
Hooke designed and that
is in original condition.
Hooke memorial window, St. Helen's,
Bishopsgate, City of London.
Chapter 1
Page 14
History (1600-1700)
• Hooke Law
• 郑玄（公元127－200）
tension is proportional to the stretch
Hooke’s law established the notion of
(linear) elasticity but not yet in a way that
was expressible in terms of stress and
strain.
每加物一石，则张一尺。
Chapter 1
Page
15
History (1700-1880)
• Bernoullis
Jacob Bernoulli
1654-1705
  E
Chapter 1
Johann Bernoulli
1667-1748
Daniel Bernoulli
1700-1782
Leohard Euler, Johann Bernoulli’s student, 1727
Page
16
History (1700-1880)
• Clande-Louis-MarieHenri Navier (1821)
“Equilibrium and motion
of elastic solids”


C  2 ui  2u k ,ki  f i  0
u: displacement vector
C a measure of elastic modulus
f the body force vector.
Chapter 1
Page
17
History (1700-1880)
• Simon Denis Poisson
Poisson’s ratio （1829）
Foam structures with
a negative Poisson's
ratio, Science, 235
1038-1040 (1987).
Longitudinal and
transverse waves
Chapter 1
Page
18
History (1700-1880)
• Augustin-Louis Cauchy
formalized the concept of
stress in the context of a
generalized threedimensional theory (提出
了三维空间应力的概念)
Chapter 1
Page
19
History (1700-1880)
• Augustin-Louis Cauchy
 Concepts of princprincipal stresses and principal
strains（主应力和主应变）
 Generalized Hooke’s law（广义虎克定律）
 Equations of motion in terms of components of
stress with their boundary conditions（以应力表
示的运动方程）
 Cauchy’s relation of elasticity tensor （弹性张量
的关系式）
Chapter 1
Page
20
History (1700-1880)
• Adhemer Jean Claude Barre
de Saint-Venant (a student of Navier )
 Saint-Venant Principle （圣维南原理）
Semi-inverse solution(1853) Beam
Bending
Bending and torsion of a noncircular prismatic rod
Chapter 1
Page
21
History (1700-1880)
• Gustav Robert Kirchhoff
Founding master of electromagnetism
Kirchhoff plate theory
Chapter 1
Page
22
History (1880-1950)
• Love
 the point source theory and
Love wave
 A treatise on the
mathematical theory of
elasticity(1892-1893)
Chapter 1
Page
23
History (1880-1950)
• S.P.Timoshenko (work
with Prandtl, the father of
aerodynamics. )
 Beams on elastic foundation
 Timoshenko beam theory
 Mechanics of plates and
shells
 Elastic vibration
a scientist and an engineer
Chapter 1
Page
24
History (1880-1950)
• Theodore Von
Karman
H. S. Tsien and
W. Z. Chien
Werner Heisenberg
• Large deflection
and buckling
Chapter 1
Page
25
History (1950-)
• Theodore Von Karman
and G. I. Taylor
Establishing IUTAM
Chapter 1
Page
26
History (1950-)
• Griffith (1921), Irwin (1957),
Rice (1968)
• fracture mechanics (G, K, J)
Chapter 1
Page
27
History (1950-)
• Fracture mechanics
Chapter 1
Page
28
History (1950-)
• Courant (1943)
• Clough, Ziekiewitz, Feng Kang, Finite Element
Method
Chapter 1
Page
29
Applications of Elasticity
• Construction
Chapter 1
Page
30
Applications of Elasticity
• Aeronautic Engineering
Chapter 1
Page
31
Applications of Elasticity
• Astronautic Engineering
Chapter 1
Page
32
Applications of Elasticity
Integrated Circuit
Elasticity plays a central role in estimating the
integrity of ICs.
Chapter 1
Page
33
Applications of Elasticity
Nanotechnology
Chapter 1
Page
34
Applications of Elasticity
Computational Microscope
Chapter 1
Page
35
Applications of Elasticity
Biology and Biomechanics
The wing of a dragonfly and topological
optimization
Dragonfly wings - note the more reinforced
structure in the leading third of each wing.
The front wing beats approximately 1/4 of a
cycle ahead of the rear wing.
Chapter 1
Page
36
Applications of Elasticity
• Sports
Archery
Chapter 1
Page
37
Basic Assumptions
•
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Continuous(连续性)
Homogeneous（均匀）
Isotropic（各向同性）
perfectly elastic（理想弹性）
Small deformation（小变形）
Chapter
Page
38