Relationship Between Stress and Strain

Within the elastic region (see the graph below), there exist a relationship between the force applied and the change in the physical state of the object. Experiments have shown that the elastic behavior can be described by the following relations with the condition that the change in the physical state is small in compared to the original. It can be shown that the stress is directly proportional to the strain within the straight line region. This relationship is expressed by Hooke's Law in two ways.

1. The equation F = - kx

2. The equations of stress and strain.

In the region where the stress is directly proportional to the strain, relationship are obtained. The stress is directly proportional to the strain.

In tensile/ compressive we have

In the Bulk stress/strain we have:

Since P = F/A then the equation simplifies to P = -B(ΔV/V0)

In the Shear stress/strain we have:

Example 1

A tow truck is pulling a car out of a ditch by means of a steel cable that is 9.1 m long. and has radius of 0.50 cm. When the car just begins to move, the tension in the cable is 890 N. How much has the cable stretched?

Example 2

The pressure increases by 1.0x104 N/m2 for every meter of depth beneath the surface of the ocean. At what depth does the volume of a Pyrex glass tube , 1.0 x 10-2 m on an edge at the ocean's surface decreases by 1.0 x 10-10m3?

Example 3

A copper cube, 0.3m on a side is subjected to two shearing forces each of magnitude F = 6.0 x106 N. Find the angle in degrees which is one measure of how the shape of the block has been altered by shear deformation.