1
Elasticity
2
3
Chapter 1 Introduction
§ 1-1 The Modeling of the Engineering Mechanics Problem
§ 1-3 The Basic Assumption of the Elasticity Problem
§ 1-4 The Several Basic Concepts of Elasticity
§ 1-5 The Study Method of the Elasticity
§ 1-2 The Basic Contents of the Elasticity
Exercises Lesson
4
第一章 绪 论
§ 1-1 工程力学问题的建模
§ 1-3 弹性力学问题的基本假设
§ 1-4 弹性力学中的几个基本概念
§ 1-5 弹性力学的学习方法
§ 1-2 弹性力学的基本内容
习题课
5
The elasticity is a branch of the solid mechanics,the task of
it is to research the elasticity object’s stress,deformation and
displacement due to external force or change of temperature,
The elasticity is the foundation of studying plasticity,
fracture mechanics and finite element method,
This course shows the mathematics modeling process of
mechanics problems completely,and establishes the basic
equation and boundary condition of the elasticity and
proceeds to beg the solutions of some problem,The
foundation of the elasticity basic equation lays a foundation
for further number method,
6
弹性力学是固体力学的一个分支，研究弹性体由于外
力作用或温度改变等原因而发生的应力、形变和位移。
弹性力学是学习塑性力学、断裂力学、有限元方法的
基础。
本课程较为完整的表现了力学问题的数学建模过程，
建立了弹性力学的基本方程和边值条件，并对一些问题进
行了求解。弹性力学基本方程的建立为进一步的数值方法
奠定了基础。
7
Through the process of
establishing the mechanics model
in the engineering mechanics
problem,generally three parts
should be simplified,
Suffering Force Simplification
Material Simplification
Construction Simplification
1,The Modeling Process of the Engineering Mechanics Problem
§ 1-1 The Modeling of the
Engineering Mechanics Problem
Fig.1-1
8
工程力学问题建立力
学模型的过程中，一般要
对三方面进行简化,
受力简化
材料简化
结构简化
一、工程力学问题的建模过程
§ 1-1 工程力学问题的建模
图 1-1
9
Material is simplified according to these hypothesises of the same
kind,consecution and uniformity in each direction,
（ 3） Material simplification
According to the Saint-Venant’s principle,the complex force
system is simplified to an equivalent force system,
（ 2） Suffering Force Simplification
Such as space problem is simplified to flat surface problem and
symmetry problem in axis,and entity construction is simplified to
plate construction
（ 1） Construction Simplification
10
根据各向同性、连续、均匀等假设进行简化。
（ 3）材料简化
根据圣维南原理，复杂力系简化为等效力系。
（ 2）受力简化
如空间问题向平面问题的简化，向轴对称问题的简化，实
体结构向板、壳结构的简化。
（ 1）结构简化
11
Proceed to handle to the small quantity in high level,
Proceed the linearization that may be linearized,
2,Advertent Problem in Modeling Process
After the model is established,proceed to analyse and
neaten to the result of the computation,and return the
actual problem and proceed the verification,Generally
and mostly it is proceeded through experiment,
（ 2） Experiment Verification
（ 1） Linearization
12
对高阶小量进行处理，能进行线性化的，进行线性化。
二、建模过程中注意的问题
模型建立以后，对计算的结果进行分析整理，返回实际问
题进行验证，一般主要通过实验进行。
（ 2）实验验证
（ 1）线性化
13
The elasticity is a branch of the solid mechanics,the task of it
is to research the elasticity object’s stress,deformation and
displacement due to external force or change of temperature,
§ 1-2 The Basic Contents of the Elasticity
1,Investigative task
The research object of the elasticity is general and
complicated shape structure piece,entity structure,plate shell
etc,
2,Investigative object
14
弹性力学是固体力学的一个分支，研究弹性体由于受
外力作用或由于温度改变等原因而发生的应力、形变和位
移。
§ 1-2 弹性力学的基本内容
一、研究任务
弹性力学的研究对象为一般及复杂形状的构件、实体
结构、板壳等。
二、研究对象
15
Plasticity,plasticity analysis and design of the structure,
3,The relation about the other course,
Material mechanics,research stress and displacement of the
bar structure piece that is pulled,pressed,sheared,bent or
turned,
Theoretical mechanics,Study statics and dynamics of the
rigid body(constraint force,velocity,acceleration),
Structual mechanics,research internal force and displacement
of the bar structure,
Elasticity,stress and displacement analysis of general plane
problem,plate,shell and entity structure etc,
16
塑性力学：结构的塑性分析、设计；
三、与其他学科的关系,
材料力学：研究杆状构件在拉、压、剪、弯、扭状态
下的应力和位移；
理论力学：研究刚体的静、动力学（约束力、速度、
加速度）。
结构力学：研究杆系结构的内力与位移；
弹性力学：一般平面问题、板、壳和实体结构等的
应力和位移分析。
17
§ 1-3 The Basic Assumption of the Elasticity
In elasticity,doing some necessary assumptions under the
premise that can satisfy the practical needing precision and
making the problem solved,
（ 1） Consecution assumption:Some physics measures inside
the object,for example stress,strain and displacement
etc.whose variety regulation may be denoted by continuous
function in coordinate,
（ 2） Ideal elasticity assumption,supposing that the object
is a ideal elastic body,then the elastic body obey the Hooke’s
law---the stress becomes the direct proportation with
homologous deformation.And the elasticity constant doesn’t
change along with the variety of stress and deformation,
（ 3） Even assumption:supposing the object be constituted
by the same material,the elasticity of the object would not
change along with position coordinates change,
The basic assumption of the elasticity,
18
§ 1-3 弹性力学的基本假设
在弹性力学中，在满足实用所需精度的前提下做一些
必要的假设，使问题得以求解。
（ 1）连续性假设：这样物体内的一些物理量，例如
应力、应变和位移等可用坐标的连续函数表示它们的变
化规律。
（ 2）完全弹性假设：假定物体为完全弹性体，则
服从虎克定律 ---应力和相应的形变成正比，弹性常数
不随应力或形变的大小而变化。
（ 3）均匀性假设：假定物体由同一材料组成，这
样物体的弹性不随位置坐标而变化。
弹性力学的基本假设为,
19
（ 4） Isotropy assumption,The elastic properties of one
point in object are the same in every direction,
（ 5） Assumption of small deformation,supposing
displacement and deformation is very small.Then using the
dimension before deformation instead of the one after deformation,
The small quantity in high level may be ignored when
investigating strain and displacement of the object.Which is very
important to the linearization of the equation,
The assumptions above are suitable for many problems in
engineering,but they exist errors much differently for some
problems,then it is necessary to use another brief method.But it
is still the same for the basic theories of many concepts.The
elasticity is the foundation of the subjects of learning
plasticity,fracture mechanics and finite element method and etc,
20
（ 4）各向同性假设：物体内一点的弹性性质在所有
各个方向都相同。
（ 5）小变形假设：假定位移和形变是微小的。这样，
可以用变形前的尺寸代替变形后的尺寸，在考察物体的
应变和位移时，可以略去高阶小量，这对于方程的线性
化十分重要。
以上的假设对于工程中不少问题是适用的，但对于
一些问题的误差太大，就必须用另外的简化方案，但许
多概念基本理论仍然是共同的，弹性力学是学习塑性力
学、断裂力学、有限元方法等学科的基础。
21
§ 1-4 The Several Basic Concepts of the Elasticity
It can be divided into the stress of volume and plane according
to the different distribution of the external function,which are
called volumetric force and surface force respectively,
（ 2） Property,volumetric force is
different from the different location
in a point;the volumetric force is
continuous in distribution,
1.External stress
1.Volumetric force （ 1） Definition,It is the stress
distributed in volume of the object that
is called volumetric force,for
example,gravity and inertia force.It is
shown in Fig.1-2,Q?
z
x
y
△ V
O
P X
Y
Q?F
Fig.1-2
Z
22
§ 1-4 弹性力学中的几个基本概念
按照外力作用的不同分布方式，可分为体积力和表面力，
分别简称体力和面力。
（ 2）性质：体力随点的位置不同
而不同；体力是连续分布的。
（一）外力
1.体力
（ 1）定义：所谓体力是分布在物
体体积内的力，如重力和惯性力。
如图 1－ 2所示 。 Q?
图 1-2
z
x
y
△ V
O
P X
Y
Q?FZ
23
（ 3） Gather degree,
（ 4） The component of volumetric force,
The force of F is resolved along with the three
coordinates,which will get the three components of
straight intersection,kZjYiXF ??? ???
X,Y,Z are called the components of volumetric force at
point P.Plus sign and negative sign are separately
determined by the direction of components,and then
is[Force][Length]-3。
The average gather degree of volumetric force,
V
Q
?
?
The gather degree of volumetric force at point P,
V
QF
V ?
??
??lim 0
Q?The direction of F is the limited one of
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（ 3）集度,
（ 4）体力分量,
将 F沿三个坐标轴分解，可得到三个正交的分力,
kZjYiXF ??? ???
X,Y,Z称为物体在 P点的体力分量，正负号视分
力指向而定，因次是 [力 ][长度 ]-3。
体力的平均集度为,
V
Q
?
?
P点所受体力的集度为,
V
QF
V ?
??
??lim 0
F Q?的方向就是 的极限方向。
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2,Surface force
S?
SQ??
(3) Gather degree of surface force:the average gather degree
of the surface force above,
The gather degree of the surface
force at P,
S
QF
S ?
??
?? 0l i m
(4) The components of surface force,
The components of surface force
are,,,and then are
[Force][Length]-2
X Y Z
x
y
z
P
△ S
X
Y
Z
F
Q?
Fig.1-3
(2) Property:In general,surface force is the function of located
coordinates at point in the surface of the object,
(1) Definition:surface force is distributed one in the surface of the
object.For instance,liquid stress and contact stress,
It is shown in Fig.1-3,
Q?
26
2,面力
S? 上面力的平均集度为,
S
Q
?
?（ 3）面力集度,
P点所受面力的集度为,
S
QF
S ?
??
?? 0
l i m
（ 4）面力分量,
P点的面力分量为,,，
因次是 [力 ][长度 ]-2。
X Y Z
x
y
z
P
△ S
X
Y
Z
F
Q?
图 1-3
（ 2）性质：面力一般是物体表面点的位置坐标的函数。
（ 1）定义：分布在物体表面上的力。如流体压力和接触力
。如图 1－ 3所示 。 Q?
27
2.stress
3.Stress gather degree,
The average gather degree of internal force above,
AQ??
A?
The stress at point P,
A
Qs
A ?
??
?? 0lim
??
---Positive stress ---Shearing strength
And then are[Force][Length]-2
The stress component at point P is,,??
2.Property,The same point in the object,whose stress of different cross sections is
different,
1.Definition,The object bears the external force function.Additional internal force is
produced among every cross sections of the object interior.For displaying these internal
forces,we use a cross section to cut the object open,and then take out a part among
them.The function of a part to another part among them that expresses for internal force,
which are resultant force of distributed forces that distribute on the cross section.When the
area of cross section incline to the zero,the distributed force on the cross section is shown as
Fig.1-4s,
x
y
z
A
B
P
o
△ A
?
? s Q?
n
m
Fig.1-4
28
（二）应力
3.应力集度,
上的内力的平均集度为,AQ??A?
P点的应力为,
A
Qs
A ?
??
?? 0l i m
?? ---正应力 ---切应力
因次是 [力 ][长度 ]-2。
P点的应力分量为, ??
2.性质,在物体内的同一点，不同截面上的应力是不同的。
1.定义,物体承受外力作用，物体内部各截面之间产生附加内
力，为了显示出这些内力，我们用一截面截开物体，并取出其
中一部分，其中一部分对另一部分的作用，表现为内力，它们
是分布在截面上分布力的合力。当截面面积趋于零时截面上的
分布力。如图 1－ 4所示 。 s
x
y
z
A
B
P
o
△ A
?
?
s
Q?
n
m
图 1-4
29
4.The component of the stress
Stress is relevant with not only the position of point but also
the direction of the cross section,It is not a general vector but is
two rank tensor,
The component of the stress on
the plane is equal in size but
contrary in direction at the meaning
of omitting the small quantity in
high level,
(1) For analyzing the state of one
point,one small positive parallel
hexahedron is taken out from the
point.The component of the stress of
each section along with coordinates
axis that is called the component of
the stress,
x
y
z
o
Fig.1-5
A
B
C
P
30
4.应力分量
应力不仅和点的位置有关，和截面的方位也有关，不是
一般的矢量，而是二阶张量。
相对平面上的应力分量在略
去高阶小量的意义上大小相等，
方向相反。
（ 1）为了分析一点的应力状
态，在这一点从物体内取出一个
微小的正平行六面体，各面上的
应力沿坐标轴的分量称为应力分
量。
x
y
z
o
图 1-5
A
B
C
P
31
The drawing shows that the normal of the
surface of the unit is y,it is called surface
y.The stress that the stress component
plumbs the surface of the unit is called the
positive stress,
The positive stress is recorded σy,the
positive direction along y axis is
positive,whose suffix means the direction
along coordinates axis,
（ 2） Symbol provision,
σ y
x y
z
o
yx?
Fig.1-6
yz?
The stress paralleling the surface of the unit is
called the slicing stress,which is showed by,
and whose the first suffix y means the flat
surface of the place and the second suffix x,z
mean respectively along the direction of the
coordinates axis.,is showed in Fig.1-6,
yx?
yx?
yz?
yz?
32
图示单元体面的法线为 y,称
为 y面，应力分量垂直于单元体
面的应力称为正应力。
正应力记为 σ y,沿 y轴的正
向为正,其下标表示所沿坐标轴
的方向。
σ y
x y
z
o
yx?
图 1-6
（ 2）符号规定,
yz?
yz?
平行于单元体面的应力称为
切应力，用, 表示，其第
一下标 y表示所在的平面，第二下
标 x,z分别表示沿坐标轴的方向。
如图 1－ 6所示的, 。
yx?
yx?
yz?
yz?
33
The components of the
stress on other x,z positive
surface is shown in Fig.1-
7,
The stress on positive
surface is positive along the
positive direction of
coordinates,and is negative
athwart the positive direction
of coordinates,
Fig.1-7
xy?
34
其它 x,z正面上的
应力分量的表示如图 1－
7所示。
凡正面上的应力沿坐
标正向为正，逆坐标正向
为负。
图 1-7
35
The stress that paralleling the
surface of the unit is shown
likeτyx,τyz in Fig.1-8,which
is positive along the negative
direction of x axis and z axis,Fig.1-8
Fig.1-8 shows that the normal of
the surface of the unit is the
negative direction of y,the
positive stress is
recorded,which is positive
along the negative direction of y
axis,
y?
36
平行于单元体面
的应力如图示的 τ yx、
τ yz，沿 x轴,z轴的
负向为正。 图 1-8
图 1－ 8所示单元体
面的法线为 y的负向，正
应力记为,沿 y轴负向
为正。
y?
37
Elasticity
（ 3） Noticing the slicing
stress sign of elasticity is
distinguishing with material
mechanics.Generally the slicing
stress is positive in elasticity,
such as the Fig.1-
9shows.However,the sign
closing both sides is different in
material mechanics,
Material
Mechanics
We should draw stress circle
according to the sign provision
of material mechanics,
Fig.1-9
38
弹性力学
材料力学
（ 3）注意弹性力学
切应力符号和材料力学
是有区别的，图 1－ 9中，
弹性力学里，切应力都
为正，而材料力学中相
邻两面的的符号是不同
的。
在画应力圆时，应
按材料力学的符号规定。
图 1-9
39
2.Shearing stress,In Fig.1-5,
the orthogonal changes of the
line segment PA,PB,PC are
means with the radian,which is
called shearing stress.Shearing
stress is shown respectively
by,,,
yz? zx? xy?
3.Strain
The strain is the changes of the shape.The strain of the object
may come down to the changes of the length and the angle,
x? y?
1.Positive strain,In Fig.1-5 the flex of the line segment
PA,PB,PC per-unit length,namely unit flexible or opposite flex,
which is called positive strain,positive strain is shown
respectively by,,,
z?
P
Fig.1-5
A
B
C
P
40
2.切应变：图 1-5中线段
PA,PB,PC之间的直角的改
变，用弧度表示，称为切应变。
分别用,, 表示。
yz? zx? xy?
（三）形变（应变）
形变 就是形状的改变。物体的形变可以归结为长度的改
变和角度的改变。
x? y?
1.正应变：图 1-5中线段 PA,PB,PC每单位长度的伸
缩，即单位伸缩或相对伸缩，称为正应变。分别用,,
表示。 z?
P
图 1-5
A
B
C
P
41
（ 2） Among all points of the object have the opposite displacement,so the
object produce distortion.In the elasticity,studying primarily the displacement
that the distortion of the object causes,
（ 1） The displacement that the movement of the whole object like a rigid
body proceeding causes,generally which includes the translation and the
rotation,In this way the displacement do not make the opposite distance of the
shape and particle of the object change.(Object only contain outside effect but
have no inside effect)
1,When the positions of all point of the object change,thinking generally
it is constituted by the displacement of two kinds of property,
4.Displacement
Displacement,When the object transform,the amount of changes of the all
point position calls the displacement,
2.The expressing method of the displacement
The displacement of random point in object,whose projection of u,v,w in the
axis of x,y,z are used to mean it.Positive direction along coordinates axis is
positive,negative direction along coordinates axis is negative.These three
projections are called the component of the displacement of the point,
42
（ 2）物体的各点间有相对位移，因而物体产生了变形。
弹性力学中主要研究物体由变形而引起的位移。
（ 1）整个物体象一个刚体一样进行的运动所引起的位移，
一般包括平移和转动。这样位移并不使物体的形状、质点间
的相对距离发生变化。（物体只有外效应而无内效应）。
1.当物体各点发生位置改变时，一般认为是由
两种性质的位移组成,
（四）位移
位移,物体变形时，各点位置的改变量称为位移。
2.位移的表示方法
x y z
u v w
物体内任意一点的位移，用它在,, 轴上的投
影,, 来表示，以沿坐标轴正向为正，沿坐标轴负向为
负。这三个投影称为该点的位移分量。
43
dis
place
men
t
def
orm
ati
on
stres
s
Vo
lum
etri
c
stres
s
Plan
e s
tres
s
Geometrical
equation
Physical equation Equation of
equilibrium
Boundary condition
Fig.1-10
5.The relation of physics quantity
44
位
移
形
变
应
力
体
力
面
力
几何方程 物理方程 平衡方程
边界条件
图 1-10
（五）各物理量之间的关系
45
The deducing of the formula of the elasticity is more complicated.The meaning
of the formula is not clear and definite,and the formula is not easy to memorize,so
the beginner will feel difficult,
Don’t stand on points without measure and fix attention on the main process of
the deduce.The deduce and the memory of the formula had better to pass the
matrix form,
§ 1-5 The study method of the elasticity
Because the basic equation is a system of partial differential equation and
contact is less,the beginner’s comprehension is difficult.The direct solution of the
system of partial differential equation is very difficult,only when the boundary
condition is simple,the solution may be solved.Most problems need to be solved
through the numerical method,so the meaning of the basic equation is in order to
lay the foundation for the study of the future,
In deduce process,Be good at utilizing the small deformation to omit the small
quantity in high level,In boundary condition,distinguishing the primary boundary
and the subordinate boundary.In the subordinate boundary,using the condition of
the equivalent force system to instead according to the Saint-Venant principle,
At the rear of each chapter,attaching some exercises.And these exercises may
deepen the comprehension of the concept and the method,
46
弹性力学的公式推导比较繁复，公式的意义不明确，不
便记忆，因此初学者会感到困难。
在学习中，不要过分拘泥于细节，应着眼于推导的主要过
程，公式的推导和记忆，最好通过矩阵形式和张量。
§ 1-5 弹性力学的学习方法
由于基本方程是偏微分方程组，接触较少，理解有困难。
偏微分方程组的直接求解是十分困难的，只有在边界条件比
较简单时，可以解出，大多需要通过数值方法求解，因此基
本方程的意义很大程度上是为将来的学习打基础。
在推导过程中，善于利用小变形略去高阶小量，在边界条
件中，要分清主要边界和次要边界，在次要边界上根据圣维
南原理，用等效力系的条件进行替代。
在每章的最后，附有一些习题，通过练习，加深对概念和
方法的理解。
47
,Introduction,exercise lesson
[exercise 1]What is the research object and content of the elasticity?What are
the different and the similar comparing with material mechanics?
Answer:The elasticity research the stress,the strain and the displacement of the
object in the stage of elasticity under the outside factor influence,whose research
object is the component,the entity structure,the plate shell etc,of general and
complicated shape.But material mechanics study the stress and the displacement
of the bar under the state of pull,press,shear,bend,twist,
[exercise 2]What is the basic assumption in elasticity?
Answer,In order to simplify computation,as follows basic assumptions are
adopted in elasticity,
（ 1） Consecution assumption（ 2） Ideal elasticity assumption（ 3） Even
assumption（ 4） Isotropy assumption（ 5） Assumption of small deformation
[exercise 3]What is assumption of small deformation?What simplifications do the
assumption of small deformation bring?
Answer:supposing that the displacement of all points of the whole object are far
smaller than the
48
,绪论, 习题课
[练习 1]弹性力学的研究对象、内容是什么？与材料力学比较
有何异同？
答,弹性力学研究物体在外界因素影响下处于弹性阶段的应力、
应变和位移，其研究对象为一般及复杂形状的构件、实体结构、
板壳等。而材料力学是研究杆件在拉、压、剪、弯、扭状态下
的应力和位移。
[练习 2]弹性力学中基本假设是什么？
答,为了简化计算，弹性力学中采用如下基本假设,
（ 1）连续性假设，（ 2）完全弹性假设，（ 3）均匀性假设，
（ 4）各向同性假设，（ 5）小变形假设。
[练习 3]什么是小变形假设？小变形假设带来那些简化？
答,假定物体受力以后，整个物体所有各点的位移都远远小
49
Original dimension of the object,this is the assumption of small
deformation.When establishing the equilibrium equation after the
distortion of the object,using the dimension before deformation
instead of the one after deformation.All the quadratic power or the
product of the corner and the displacement may be omitted when
investigating deformation and displacement of the object,In this
way the algebraic equation and the differential equation of the
elasticity are simplified to the linear equation,
50
于物体原来的尺寸，就是小变形假设。小变形假设，在建立
物体变形以后的平衡方程时，可以用变形以前的尺寸来代替
变形以后的尺寸，并且，在考察物体的形变及位移时，转角
和位移的二次幂或乘积都可以略去不计。这样可使弹性力学
中的代数方程和微分方程简化为线性方程。
51
52
Elasticity
2
3
Chapter 1 Introduction
§ 1-1 The Modeling of the Engineering Mechanics Problem
§ 1-3 The Basic Assumption of the Elasticity Problem
§ 1-4 The Several Basic Concepts of Elasticity
§ 1-5 The Study Method of the Elasticity
§ 1-2 The Basic Contents of the Elasticity
Exercises Lesson
4
第一章 绪 论
§ 1-1 工程力学问题的建模
§ 1-3 弹性力学问题的基本假设
§ 1-4 弹性力学中的几个基本概念
§ 1-5 弹性力学的学习方法
§ 1-2 弹性力学的基本内容
习题课
5
The elasticity is a branch of the solid mechanics,the task of
it is to research the elasticity object’s stress,deformation and
displacement due to external force or change of temperature,
The elasticity is the foundation of studying plasticity,
fracture mechanics and finite element method,
This course shows the mathematics modeling process of
mechanics problems completely,and establishes the basic
equation and boundary condition of the elasticity and
proceeds to beg the solutions of some problem,The
foundation of the elasticity basic equation lays a foundation
for further number method,
6
弹性力学是固体力学的一个分支，研究弹性体由于外
力作用或温度改变等原因而发生的应力、形变和位移。
弹性力学是学习塑性力学、断裂力学、有限元方法的
基础。
本课程较为完整的表现了力学问题的数学建模过程，
建立了弹性力学的基本方程和边值条件，并对一些问题进
行了求解。弹性力学基本方程的建立为进一步的数值方法
奠定了基础。
7
Through the process of
establishing the mechanics model
in the engineering mechanics
problem,generally three parts
should be simplified,
Suffering Force Simplification
Material Simplification
Construction Simplification
1,The Modeling Process of the Engineering Mechanics Problem
§ 1-1 The Modeling of the
Engineering Mechanics Problem
Fig.1-1
8
工程力学问题建立力
学模型的过程中，一般要
对三方面进行简化,
受力简化
材料简化
结构简化
一、工程力学问题的建模过程
§ 1-1 工程力学问题的建模
图 1-1
9
Material is simplified according to these hypothesises of the same
kind,consecution and uniformity in each direction,
（ 3） Material simplification
According to the Saint-Venant’s principle,the complex force
system is simplified to an equivalent force system,
（ 2） Suffering Force Simplification
Such as space problem is simplified to flat surface problem and
symmetry problem in axis,and entity construction is simplified to
plate construction
（ 1） Construction Simplification
10
根据各向同性、连续、均匀等假设进行简化。
（ 3）材料简化
根据圣维南原理，复杂力系简化为等效力系。
（ 2）受力简化
如空间问题向平面问题的简化，向轴对称问题的简化，实
体结构向板、壳结构的简化。
（ 1）结构简化
11
Proceed to handle to the small quantity in high level,
Proceed the linearization that may be linearized,
2,Advertent Problem in Modeling Process
After the model is established,proceed to analyse and
neaten to the result of the computation,and return the
actual problem and proceed the verification,Generally
and mostly it is proceeded through experiment,
（ 2） Experiment Verification
（ 1） Linearization
12
对高阶小量进行处理，能进行线性化的，进行线性化。
二、建模过程中注意的问题
模型建立以后，对计算的结果进行分析整理，返回实际问
题进行验证，一般主要通过实验进行。
（ 2）实验验证
（ 1）线性化
13
The elasticity is a branch of the solid mechanics,the task of it
is to research the elasticity object’s stress,deformation and
displacement due to external force or change of temperature,
§ 1-2 The Basic Contents of the Elasticity
1,Investigative task
The research object of the elasticity is general and
complicated shape structure piece,entity structure,plate shell
etc,
2,Investigative object
14
弹性力学是固体力学的一个分支，研究弹性体由于受
外力作用或由于温度改变等原因而发生的应力、形变和位
移。
§ 1-2 弹性力学的基本内容
一、研究任务
弹性力学的研究对象为一般及复杂形状的构件、实体
结构、板壳等。
二、研究对象
15
Plasticity,plasticity analysis and design of the structure,
3,The relation about the other course,
Material mechanics,research stress and displacement of the
bar structure piece that is pulled,pressed,sheared,bent or
turned,
Theoretical mechanics,Study statics and dynamics of the
rigid body(constraint force,velocity,acceleration),
Structual mechanics,research internal force and displacement
of the bar structure,
Elasticity,stress and displacement analysis of general plane
problem,plate,shell and entity structure etc,
16
塑性力学：结构的塑性分析、设计；
三、与其他学科的关系,
材料力学：研究杆状构件在拉、压、剪、弯、扭状态
下的应力和位移；
理论力学：研究刚体的静、动力学（约束力、速度、
加速度）。
结构力学：研究杆系结构的内力与位移；
弹性力学：一般平面问题、板、壳和实体结构等的
应力和位移分析。
17
§ 1-3 The Basic Assumption of the Elasticity
In elasticity,doing some necessary assumptions under the
premise that can satisfy the practical needing precision and
making the problem solved,
（ 1） Consecution assumption:Some physics measures inside
the object,for example stress,strain and displacement
etc.whose variety regulation may be denoted by continuous
function in coordinate,
（ 2） Ideal elasticity assumption,supposing that the object
is a ideal elastic body,then the elastic body obey the Hooke’s
law---the stress becomes the direct proportation with
homologous deformation.And the elasticity constant doesn’t
change along with the variety of stress and deformation,
（ 3） Even assumption:supposing the object be constituted
by the same material,the elasticity of the object would not
change along with position coordinates change,
The basic assumption of the elasticity,
18
§ 1-3 弹性力学的基本假设
在弹性力学中，在满足实用所需精度的前提下做一些
必要的假设，使问题得以求解。
（ 1）连续性假设：这样物体内的一些物理量，例如
应力、应变和位移等可用坐标的连续函数表示它们的变
化规律。
（ 2）完全弹性假设：假定物体为完全弹性体，则
服从虎克定律 ---应力和相应的形变成正比，弹性常数
不随应力或形变的大小而变化。
（ 3）均匀性假设：假定物体由同一材料组成，这
样物体的弹性不随位置坐标而变化。
弹性力学的基本假设为,
19
（ 4） Isotropy assumption,The elastic properties of one
point in object are the same in every direction,
（ 5） Assumption of small deformation,supposing
displacement and deformation is very small.Then using the
dimension before deformation instead of the one after deformation,
The small quantity in high level may be ignored when
investigating strain and displacement of the object.Which is very
important to the linearization of the equation,
The assumptions above are suitable for many problems in
engineering,but they exist errors much differently for some
problems,then it is necessary to use another brief method.But it
is still the same for the basic theories of many concepts.The
elasticity is the foundation of the subjects of learning
plasticity,fracture mechanics and finite element method and etc,
20
（ 4）各向同性假设：物体内一点的弹性性质在所有
各个方向都相同。
（ 5）小变形假设：假定位移和形变是微小的。这样，
可以用变形前的尺寸代替变形后的尺寸，在考察物体的
应变和位移时，可以略去高阶小量，这对于方程的线性
化十分重要。
以上的假设对于工程中不少问题是适用的，但对于
一些问题的误差太大，就必须用另外的简化方案，但许
多概念基本理论仍然是共同的，弹性力学是学习塑性力
学、断裂力学、有限元方法等学科的基础。
21
§ 1-4 The Several Basic Concepts of the Elasticity
It can be divided into the stress of volume and plane according
to the different distribution of the external function,which are
called volumetric force and surface force respectively,
（ 2） Property,volumetric force is
different from the different location
in a point;the volumetric force is
continuous in distribution,
1.External stress
1.Volumetric force （ 1） Definition,It is the stress
distributed in volume of the object that
is called volumetric force,for
example,gravity and inertia force.It is
shown in Fig.1-2,Q?
z
x
y
△ V
O
P X
Y
Q?F
Fig.1-2
Z
22
§ 1-4 弹性力学中的几个基本概念
按照外力作用的不同分布方式，可分为体积力和表面力，
分别简称体力和面力。
（ 2）性质：体力随点的位置不同
而不同；体力是连续分布的。
（一）外力
1.体力
（ 1）定义：所谓体力是分布在物
体体积内的力，如重力和惯性力。
如图 1－ 2所示 。 Q?
图 1-2
z
x
y
△ V
O
P X
Y
Q?FZ
23
（ 3） Gather degree,
（ 4） The component of volumetric force,
The force of F is resolved along with the three
coordinates,which will get the three components of
straight intersection,kZjYiXF ??? ???
X,Y,Z are called the components of volumetric force at
point P.Plus sign and negative sign are separately
determined by the direction of components,and then
is[Force][Length]-3。
The average gather degree of volumetric force,
V
Q
?
?
The gather degree of volumetric force at point P,
V
QF
V ?
??
??lim 0
Q?The direction of F is the limited one of
24
（ 3）集度,
（ 4）体力分量,
将 F沿三个坐标轴分解，可得到三个正交的分力,
kZjYiXF ??? ???
X,Y,Z称为物体在 P点的体力分量，正负号视分
力指向而定，因次是 [力 ][长度 ]-3。
体力的平均集度为,
V
Q
?
?
P点所受体力的集度为,
V
QF
V ?
??
??lim 0
F Q?的方向就是 的极限方向。
25
2,Surface force
S?
SQ??
(3) Gather degree of surface force:the average gather degree
of the surface force above,
The gather degree of the surface
force at P,
S
QF
S ?
??
?? 0l i m
(4) The components of surface force,
The components of surface force
are,,,and then are
[Force][Length]-2
X Y Z
x
y
z
P
△ S
X
Y
Z
F
Q?
Fig.1-3
(2) Property:In general,surface force is the function of located
coordinates at point in the surface of the object,
(1) Definition:surface force is distributed one in the surface of the
object.For instance,liquid stress and contact stress,
It is shown in Fig.1-3,
Q?
26
2,面力
S? 上面力的平均集度为,
S
Q
?
?（ 3）面力集度,
P点所受面力的集度为,
S
QF
S ?
??
?? 0
l i m
（ 4）面力分量,
P点的面力分量为,,，
因次是 [力 ][长度 ]-2。
X Y Z
x
y
z
P
△ S
X
Y
Z
F
Q?
图 1-3
（ 2）性质：面力一般是物体表面点的位置坐标的函数。
（ 1）定义：分布在物体表面上的力。如流体压力和接触力
。如图 1－ 3所示 。 Q?
27
2.stress
3.Stress gather degree,
The average gather degree of internal force above,
AQ??
A?
The stress at point P,
A
Qs
A ?
??
?? 0lim
??
---Positive stress ---Shearing strength
And then are[Force][Length]-2
The stress component at point P is,,??
2.Property,The same point in the object,whose stress of different cross sections is
different,
1.Definition,The object bears the external force function.Additional internal force is
produced among every cross sections of the object interior.For displaying these internal
forces,we use a cross section to cut the object open,and then take out a part among
them.The function of a part to another part among them that expresses for internal force,
which are resultant force of distributed forces that distribute on the cross section.When the
area of cross section incline to the zero,the distributed force on the cross section is shown as
Fig.1-4s,
x
y
z
A
B
P
o
△ A
?
? s Q?
n
m
Fig.1-4
28
（二）应力
3.应力集度,
上的内力的平均集度为,AQ??A?
P点的应力为,
A
Qs
A ?
??
?? 0l i m
?? ---正应力 ---切应力
因次是 [力 ][长度 ]-2。
P点的应力分量为, ??
2.性质,在物体内的同一点，不同截面上的应力是不同的。
1.定义,物体承受外力作用，物体内部各截面之间产生附加内
力，为了显示出这些内力，我们用一截面截开物体，并取出其
中一部分，其中一部分对另一部分的作用，表现为内力，它们
是分布在截面上分布力的合力。当截面面积趋于零时截面上的
分布力。如图 1－ 4所示 。 s
x
y
z
A
B
P
o
△ A
?
?
s
Q?
n
m
图 1-4
29
4.The component of the stress
Stress is relevant with not only the position of point but also
the direction of the cross section,It is not a general vector but is
two rank tensor,
The component of the stress on
the plane is equal in size but
contrary in direction at the meaning
of omitting the small quantity in
high level,
(1) For analyzing the state of one
point,one small positive parallel
hexahedron is taken out from the
point.The component of the stress of
each section along with coordinates
axis that is called the component of
the stress,
x
y
z
o
Fig.1-5
A
B
C
P
30
4.应力分量
应力不仅和点的位置有关，和截面的方位也有关，不是
一般的矢量，而是二阶张量。
相对平面上的应力分量在略
去高阶小量的意义上大小相等，
方向相反。
（ 1）为了分析一点的应力状
态，在这一点从物体内取出一个
微小的正平行六面体，各面上的
应力沿坐标轴的分量称为应力分
量。
x
y
z
o
图 1-5
A
B
C
P
31
The drawing shows that the normal of the
surface of the unit is y,it is called surface
y.The stress that the stress component
plumbs the surface of the unit is called the
positive stress,
The positive stress is recorded σy,the
positive direction along y axis is
positive,whose suffix means the direction
along coordinates axis,
（ 2） Symbol provision,
σ y
x y
z
o
yx?
Fig.1-6
yz?
The stress paralleling the surface of the unit is
called the slicing stress,which is showed by,
and whose the first suffix y means the flat
surface of the place and the second suffix x,z
mean respectively along the direction of the
coordinates axis.,is showed in Fig.1-6,
yx?
yx?
yz?
yz?
32
图示单元体面的法线为 y,称
为 y面，应力分量垂直于单元体
面的应力称为正应力。
正应力记为 σ y,沿 y轴的正
向为正,其下标表示所沿坐标轴
的方向。
σ y
x y
z
o
yx?
图 1-6
（ 2）符号规定,
yz?
yz?
平行于单元体面的应力称为
切应力，用, 表示，其第
一下标 y表示所在的平面，第二下
标 x,z分别表示沿坐标轴的方向。
如图 1－ 6所示的, 。
yx?
yx?
yz?
yz?
33
The components of the
stress on other x,z positive
surface is shown in Fig.1-
7,
The stress on positive
surface is positive along the
positive direction of
coordinates,and is negative
athwart the positive direction
of coordinates,
Fig.1-7
xy?
34
其它 x,z正面上的
应力分量的表示如图 1－
7所示。
凡正面上的应力沿坐
标正向为正，逆坐标正向
为负。
图 1-7
35
The stress that paralleling the
surface of the unit is shown
likeτyx,τyz in Fig.1-8,which
is positive along the negative
direction of x axis and z axis,Fig.1-8
Fig.1-8 shows that the normal of
the surface of the unit is the
negative direction of y,the
positive stress is
recorded,which is positive
along the negative direction of y
axis,
y?
36
平行于单元体面
的应力如图示的 τ yx、
τ yz，沿 x轴,z轴的
负向为正。 图 1-8
图 1－ 8所示单元体
面的法线为 y的负向，正
应力记为,沿 y轴负向
为正。
y?
37
Elasticity
（ 3） Noticing the slicing
stress sign of elasticity is
distinguishing with material
mechanics.Generally the slicing
stress is positive in elasticity,
such as the Fig.1-
9shows.However,the sign
closing both sides is different in
material mechanics,
Material
Mechanics
We should draw stress circle
according to the sign provision
of material mechanics,
Fig.1-9
38
弹性力学
材料力学
（ 3）注意弹性力学
切应力符号和材料力学
是有区别的，图 1－ 9中，
弹性力学里，切应力都
为正，而材料力学中相
邻两面的的符号是不同
的。
在画应力圆时，应
按材料力学的符号规定。
图 1-9
39
2.Shearing stress,In Fig.1-5,
the orthogonal changes of the
line segment PA,PB,PC are
means with the radian,which is
called shearing stress.Shearing
stress is shown respectively
by,,,
yz? zx? xy?
3.Strain
The strain is the changes of the shape.The strain of the object
may come down to the changes of the length and the angle,
x? y?
1.Positive strain,In Fig.1-5 the flex of the line segment
PA,PB,PC per-unit length,namely unit flexible or opposite flex,
which is called positive strain,positive strain is shown
respectively by,,,
z?
P
Fig.1-5
A
B
C
P
40
2.切应变：图 1-5中线段
PA,PB,PC之间的直角的改
变，用弧度表示，称为切应变。
分别用,, 表示。
yz? zx? xy?
（三）形变（应变）
形变 就是形状的改变。物体的形变可以归结为长度的改
变和角度的改变。
x? y?
1.正应变：图 1-5中线段 PA,PB,PC每单位长度的伸
缩，即单位伸缩或相对伸缩，称为正应变。分别用,,
表示。 z?
P
图 1-5
A
B
C
P
41
（ 2） Among all points of the object have the opposite displacement,so the
object produce distortion.In the elasticity,studying primarily the displacement
that the distortion of the object causes,
（ 1） The displacement that the movement of the whole object like a rigid
body proceeding causes,generally which includes the translation and the
rotation,In this way the displacement do not make the opposite distance of the
shape and particle of the object change.(Object only contain outside effect but
have no inside effect)
1,When the positions of all point of the object change,thinking generally
it is constituted by the displacement of two kinds of property,
4.Displacement
Displacement,When the object transform,the amount of changes of the all
point position calls the displacement,
2.The expressing method of the displacement
The displacement of random point in object,whose projection of u,v,w in the
axis of x,y,z are used to mean it.Positive direction along coordinates axis is
positive,negative direction along coordinates axis is negative.These three
projections are called the component of the displacement of the point,
42
（ 2）物体的各点间有相对位移，因而物体产生了变形。
弹性力学中主要研究物体由变形而引起的位移。
（ 1）整个物体象一个刚体一样进行的运动所引起的位移，
一般包括平移和转动。这样位移并不使物体的形状、质点间
的相对距离发生变化。（物体只有外效应而无内效应）。
1.当物体各点发生位置改变时，一般认为是由
两种性质的位移组成,
（四）位移
位移,物体变形时，各点位置的改变量称为位移。
2.位移的表示方法
x y z
u v w
物体内任意一点的位移，用它在,, 轴上的投
影,, 来表示，以沿坐标轴正向为正，沿坐标轴负向为
负。这三个投影称为该点的位移分量。
43
dis
place
men
t
def
orm
ati
on
stres
s
Vo
lum
etri
c
stres
s
Plan
e s
tres
s
Geometrical
equation
Physical equation Equation of
equilibrium
Boundary condition
Fig.1-10
5.The relation of physics quantity
44
位
移
形
变
应
力
体
力
面
力
几何方程 物理方程 平衡方程
边界条件
图 1-10
（五）各物理量之间的关系
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The deducing of the formula of the elasticity is more complicated.The meaning
of the formula is not clear and definite,and the formula is not easy to memorize,so
the beginner will feel difficult,
Don’t stand on points without measure and fix attention on the main process of
the deduce.The deduce and the memory of the formula had better to pass the
matrix form,
§ 1-5 The study method of the elasticity
Because the basic equation is a system of partial differential equation and
contact is less,the beginner’s comprehension is difficult.The direct solution of the
system of partial differential equation is very difficult,only when the boundary
condition is simple,the solution may be solved.Most problems need to be solved
through the numerical method,so the meaning of the basic equation is in order to
lay the foundation for the study of the future,
In deduce process,Be good at utilizing the small deformation to omit the small
quantity in high level,In boundary condition,distinguishing the primary boundary
and the subordinate boundary.In the subordinate boundary,using the condition of
the equivalent force system to instead according to the Saint-Venant principle,
At the rear of each chapter,attaching some exercises.And these exercises may
deepen the comprehension of the concept and the method,
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弹性力学的公式推导比较繁复，公式的意义不明确，不
便记忆，因此初学者会感到困难。
在学习中，不要过分拘泥于细节，应着眼于推导的主要过
程，公式的推导和记忆，最好通过矩阵形式和张量。
§ 1-5 弹性力学的学习方法
由于基本方程是偏微分方程组，接触较少，理解有困难。
偏微分方程组的直接求解是十分困难的，只有在边界条件比
较简单时，可以解出，大多需要通过数值方法求解，因此基
本方程的意义很大程度上是为将来的学习打基础。
在推导过程中，善于利用小变形略去高阶小量，在边界条
件中，要分清主要边界和次要边界，在次要边界上根据圣维
南原理，用等效力系的条件进行替代。
在每章的最后，附有一些习题，通过练习，加深对概念和
方法的理解。
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,Introduction,exercise lesson
[exercise 1]What is the research object and content of the elasticity?What are
the different and the similar comparing with material mechanics?
Answer:The elasticity research the stress,the strain and the displacement of the
object in the stage of elasticity under the outside factor influence,whose research
object is the component,the entity structure,the plate shell etc,of general and
complicated shape.But material mechanics study the stress and the displacement
of the bar under the state of pull,press,shear,bend,twist,
[exercise 2]What is the basic assumption in elasticity?
Answer,In order to simplify computation,as follows basic assumptions are
adopted in elasticity,
（ 1） Consecution assumption（ 2） Ideal elasticity assumption（ 3） Even
assumption（ 4） Isotropy assumption（ 5） Assumption of small deformation
[exercise 3]What is assumption of small deformation?What simplifications do the
assumption of small deformation bring?
Answer:supposing that the displacement of all points of the whole object are far
smaller than the
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,绪论, 习题课
[练习 1]弹性力学的研究对象、内容是什么？与材料力学比较
有何异同？
答,弹性力学研究物体在外界因素影响下处于弹性阶段的应力、
应变和位移，其研究对象为一般及复杂形状的构件、实体结构、
板壳等。而材料力学是研究杆件在拉、压、剪、弯、扭状态下
的应力和位移。
[练习 2]弹性力学中基本假设是什么？
答,为了简化计算，弹性力学中采用如下基本假设,
（ 1）连续性假设，（ 2）完全弹性假设，（ 3）均匀性假设，
（ 4）各向同性假设，（ 5）小变形假设。
[练习 3]什么是小变形假设？小变形假设带来那些简化？
答,假定物体受力以后，整个物体所有各点的位移都远远小
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Original dimension of the object,this is the assumption of small
deformation.When establishing the equilibrium equation after the
distortion of the object,using the dimension before deformation
instead of the one after deformation.All the quadratic power or the
product of the corner and the displacement may be omitted when
investigating deformation and displacement of the object,In this
way the algebraic equation and the differential equation of the
elasticity are simplified to the linear equation,
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于物体原来的尺寸，就是小变形假设。小变形假设，在建立
物体变形以后的平衡方程时，可以用变形以前的尺寸来代替
变形以后的尺寸，并且，在考察物体的形变及位移时，转角
和位移的二次幂或乘积都可以略去不计。这样可使弹性力学
中的代数方程和微分方程简化为线性方程。
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