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# Principal stress

Wed, 2013-09-18 09:55 - Zhigang Suo

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## Comments

## good lecture notes

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.

## Recipe to make new tensors

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:

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.