Principal stress

Most tensors are formed from other tensors.  

A scalar is a tensor of order 0.  Examples include mass, energy, and entropy.  These tensors are difficult to make.  We find them in nature.

A vector is a tensor of order 1.  Examples include displacement, force, electric field.

Start with tensors you know, you can make new tensors by recipes like these: 

• A linear map that maps tensor 1 to tensor 2 is a new tensor.
• A bilinear map that maps tensor 1 and tensor 2 to tensor 3 is a new tensor. 

You can invent your own recipes, so long as you start with known tensors and use linear maps.

Stress.  By force balance, stress is a linear map that maps an area vector to the force acting on the area.  See my notes on finite elasticity.  The area vector is a tensor of order 1, and the force is a tensor of order 1.  Consequently, the stress is a tensor of order 2.

Piezoelectricity.  When a stress is applied on a piezoelectric material, an electric field arises.  This phenomenon is modeled by a linear map that maps the stress to the electric field.  The stress is a tensor of order 2, and the electric field is a tensor of order 1.  Thus, the map is a tensor of order 3.

Please take a look at the recent thread on linear algebra and tensors.