Chapter 1. Introduction
1.1
1.2
1.3
1.4
Content of Theory of Elasticity
Important Concept in Theory of Elasticity
Basic Assumptions
Problems
1.1 Contents of Theory of Elasticity
Theory of Elasticity  is the branch of Solid Mechanics
which deals with the stress and displacements in elastic
solids produced by external forces or changes in temperature.

The purpose of study  is to check the sufficiency of
the strength, stiffness and stability of structural and machine
elements.

Solid Mechanics I ---- bar
(Mechanics of Materials)

Solid Mechanics II ---- bar system
(Structure Mechanics)
bars
plates
Solid Mechanics Solid Mechanics III ---- blocks
(Theory of Elasticity)
dams
shells



Solid Mechanics VIII
(Theory of Plasticity)



beam
mech. of mater.
beam
theory of elasticity
plate
mech of mater.
plate
theory of elast.
For example
For example
Joint application of the above three branches of solid
mechanics
------------- Finite Element Method (FEM)
1.2 Some important concepts in theory of elasticity

External forces

Stresses (internal force)

deformations --- strains and displacement
There are two kinds of external forces that act on the
bodies
(1) Body forces
gravitational force
inertia forces (in motion)
definition of body force:
Q
F = V 0 V
lim
(vector quantity)
Component of F --- X, Y, Z, the projections of F
on x, y, z axis
Dimension is [force][length] 3 , e.x., N/m 3 .
Fig. 1.2.1
pressure (in water, atmosphere)
(2) Surface force
contact force
definition:
Q
S  0 S
F  lim
[force] [length] 2
Components of F along x, y, z axes denoted by
Fig. 1.2.2
X,Y , Z
(3) The internal forces produced by external forces
Q
lim
Stress at a point: definition S = A0
A
-- normal stress (normal component)
 -- shear stress (shear component )
(4) The stress state at a point
Definition of the stress component and its sign
( Note : differences with the definition in solid mechanics II)
Relations between shear stresses
 yz   zy,
 zx   xz,
 xy   yx
We will show that the stress state on any section through the
point can be calculated if we know the 6 stress components,
i.e., the 6 stress components completely define the stress
state at a point.
(5) Deformation:
shape of a body
By deformation we mean the change of
 x ,  y ,  z ,  xy ,  yz ,  xz
6
strain
components
completely define the deformation condition (or strain
condition) at that point
(6) Displacement: By displacement (unit: length) we
mean the change of position, the displacement components in
the x, y, z axes are denoted by u, v, w respectively.
 ij, ui at a point vary with the
All the above  ij ,
position of the point considered, so they are functions of
coordinates in space.
1.3 Basic assumptions in theory of elasticity
 ij, ui can be
(1) The body is continuous, so  ij ,
expressed by continuous functions in space
(2) The body is perfectly elastic---- wholly obeys
Hook's law of elasticity ---- linear relations between
stress components and strain components.
(3) The body is homogeneous , i.e., the elastic
properties are the same throughout the
body-elastic constants will be independent
of
the
location in the body.
(4) The body is isotropic so that the elastic properties
are the
same in all directions, thus the elastic
constants will be
independent of the orientation
of coordinate axes.
example:

polycrystalline ceramics and steels

wood and fiber reinforced composite
(5) The displacements and strains are small, i.e., the
displacements components of all points of the body
during deformation are very small compared with
its original dimensions.
Problems (Exercise):
1.1.4, 1.1.2, new:
Chapter 2 Theory of Plane Problems
2.1 Plane Stress and Plane Strain

spatial problems

plane problem ---- plane stress and plane strain problems
(1) plane stress problem
and plane stress condition
 x  f1 ( x, y )
 y  f2 ( x, y )
(2) Plane strain problem
and plane strain condition
 x  f1 ( x, y )
 y  f2 ( x, y )
 xy  f3 ( x, y )
 xy  f3 ( x, y )
z  0
 xz   yz  0
z  0
 xz   yz  0
Example: thin plate
Example: dam
2.2 Equation of Equilibrium in Plane Problems