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
 
Stress = F/A (62)
 
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/m2

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?

drawing

    The stress is just the force divided by the area

formula

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
 
formula (63)
where l0 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
 
formula (64)
Using the definitions of stress and strain, this can be rearranged to yield
formula

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
 

formula (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 m2. 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?

drawing
formula
formula
formula

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
drawing

The ratio of the shearing force to the area A is called the shear stress

formula

If the object is twisted through an angle q, then the strain is

Shear Strain = tanq

Finally, we can define the shear modulus, MS, as

formula

The shear modulus is also known as the torsion modulus.