Stress and Strain
Up to now we have been studying
the dynamics of rigid bodies, that is, idealized objects that have
a definite size and shape, but one in which the particles making up the
object are constrained so that the relative positions of the particles
never changes. In other words, the rigid body does not ever stretch, squeeze
or twist. However, we know that in reality this does occur, and we need
to find a way to describe it. This is done by the concepts of stress,
strain and elastic modulus. Stress is a measurement of the
strength of a material, strain is a measure of the change in the shape
of the object that is undergoing stress and elastic modulus is a measurement
of the amount of stress needed to change the shape of the object.
There are three main types
of stress. If we stretch or compress an object, we are subjecting it to
a tensile stress. If an object is subjected to a force along an
entire surface, changing its volume, then it is said to be experiencing
a bulk stress. Finally, if the force is acting tangentially to the
surface, causing it to twist, then we are subjecting it to a shear stress.
Tensile Stress
Consider a bar of cross sectional
area A being subjected to equal and opposite forces F pulling
at the ends. If this were a rope, we would say that it is experiencing
a tension force. Taking this concept over, we say that the bar is under
tension, and is experiencing a stress that we define to be the ratio of
the force to the cross sectional area
This stress is called the tensile stress
because every part of the object is subjected to a tension. The SI
unit of stress is the Newton per square meter, which is called the Pascal
1 Pascal = 1 Pa = 1 N/m^{2}
Example:
A 250 kg bob is attached
to a steel cable with a diameter of 0.05 m. If we take the cable to be
essentially massless, what is the tensile stress experienced by the cable?
The stress is just the force
divided by the area
If the bar is being pressed instead of pulled,
then we say that it is undergoing compressive stress instead of
tensile stress.
Tensile Strain
The fractional amount that an
object stretches when it is subjected to a tensile stress is called the
tensile strain. Mathematically, we write this as


(63) 
where l_{0} is the original unstressed
length of the bar.
Elastic Modulus
Robert Hooke found that, when
the forces are not too large, the amount of strain experience by an object
was directly proportional to the stress. This is another example of Hooke's
law. Define the elastic modulus to be


(64) 
Using the definitions of stress and strain, this
can be rearranged to yield
For tensile stress, the elastic modulus is called
the Young's modulus and is denoted by Y.
When a material is stressed,
the dimensions perpendicular to the direction of the stress become smaller
by an amount proportional to the fractional change in length. This can
be written as


(65) 
where r
is a dimensionless constant called Poisson's ratio. Like Young's
modulus, it is a property of the material and can be used to characterize
it.
Example:
A 10000 kg box hangs by a
20 m long cable which has a cross sectional area of 0.15 m^{2}.
When an additional 250 kg is added to the box, the cable is seen to stretch
0.001 mm. What is the stress, strain and Young's modulus for the cable?
What is the material used in the cable?
Comparing this with a standard chart of material
characteristics, we see that the cable was probably made of tungsten.
Shear Stress and Strain
Now consider a force that is
applied tangentially to an object
The ratio of the shearing force to the area A
is called the shear stress
If the object is twisted through an angle q,
then the strain is
Shear Strain = tanq
Finally, we can define the shear modulus, M_{S},
as
The shear modulus is also known as the torsion
modulus.