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Principal stress

Zhigang Suo's picture

These notes are written to supplement ES 240 Solid Mechanics.

Also see notes on linear algebra.

 

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good lecture notes

it is difficult for me to understand the third rank tensor.

The send rank tensor is stress, so what is the third rank tensor.

Zhigang Suo's picture

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

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