So we can write s = E e .  This equation is known as Hooke’s Law.

 

At the top of the proportional region is the proportional limit.

 

Most materials also exhibit an elastic region.  When a material is loaded with a stress within the elastic region and then unloaded, the material will return to the original size and shape, i.e., there is no permanent deformation.

 

The elastic region is generally slightly larger than the proportional region, but for our purposes can be considered to be the same.  The top of the elastic region, called the elastic limit, is then basically the same as the proportional limit.

 

We also define a yield point.  This is the value of stress at which we start to see large increases in strain for no increase in stress.  This point is generally very near the elastic limit and proportional limit.  Mild steel exhibits a well defined yield point, s_y .  Not all materials do.  For our purposes, the yield point, elastic limit and proportional limit will be taken to be the same.

 

If we stress a material beyond the yield point and unload, there will be some permanent deformation.  The unloading path is nearly linear and parallel to the proportional region.  Therefore, there is an elastic strain which is recovered when the material is unloaded, e_E = s / E.  Subtracting the elastic strain from the total strain gives the permanent or plastic strain, e_P .

 

If we reload the material, the behavior will follow the unloading path back to the original curve.  So we have essentially increased the size of the proportional region and raised the yield point.  This phenomenon is called work or strain hardening.

 

 

 

 

 

 

One way we classify materials is as either being ductile or brittle.  A ductile material generally has a large plastic region, i.e., it can experience considerable strain beyond the yield point.  A brittle material generally has a small plastic region.  In other words, a ductile material gives a significant warning that failure is impending, but a brittle material will give very little warning of failure.

 

When a material is strain hardened, its yield strength is increased, but the material becomes more brittle since the plastic region is reduced.

Lecture #5, page 2                   Back                        Lecture #5, page 3