Chapter 1. Introduction
1.1 Content of Theory of Elasticity
1.2 Important Concept in Theory of Elasticity
1.3 Basic Assumptions
1.4 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 MechanicsSolid Mechanics III ---- blocks
(Theory of Elasticity) dams
shells
Solid Mechanics VIII
(Theory of Plasticity)
beam mech. of mater.
For example
beam theory of elasticity
plate mech of mater.
For example
plate theory of elast.
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
gravitational force
(1) Body forces inertia forces (in motion)
definition of body force:
F = (vector quantity)
Component of F --- X, Y, Z, the projections of F on x, y, z axis
Dimension is [force][length], e.x., N/m.
Fig. 1.2.1
pressure (in water, atmosphere)
(2) Surface force
contact force
definition: [force] [length]
Components of F along x, y, z axes denoted by
Fig. 1.2.2
(3) The internal forces produced by external forces
Stress at a point: definition S =
-- 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
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: By deformation we mean the change of shape of a body
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.
All the above at a point vary with the position of the point considered, so they are functions of coordinates in space.
1.3 Basic assumptions in theory of elasticity
(1) The body is continuous, so can be
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 (2) Plane strain problem
and plane stress condition and plane strain condition
Example: thin plate Example: dam
2.2 Equation of Equilibrium in Plane Problems