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A state of stress is a linear map

Zhigang Suo's picture

A state of stress in a body.  Components of stress.  Subject to a load, a body develops internal forces.  The distribution of the internal forces in the body is usually inhomogeneous. For example, when a rod is bent, part of the rod is in tension, and the other part of the rod is in compression.

If the internal forces are uniformly distributed in the body, the body is said to be in a homogeneous state of stress.  For brevity, we often say a state of stress, or just stress.

It has been a common practice to define a state of stress by defining its components.  Here is the procedure.  Draw a free-body diagram using a rectangular part of the body.  Represent the internal forces in the body by a force on each face of the rectangular block.  The force is a vector of three components, one being normal to the face, and the other two being tangential to the face.  A component of the force divided by the area of the face defines a component of stress.

One state of stress, many sets of components. A state of stress in the body is a fact.  A choice of block in our mind is an artifact.  For a body in a homogeneous state of stress, the components of stress are independent of the location, size and shape of the rectangular block.  But the components of stress do depend on the orientation of the block.  The state of stress is, of course, independent of the choice of block. Here are the key points:

  • One state of stress in a body
  • Many blocks in different orientations
  • Many sets of components of stress on the faces of various blocks

We resolve this issue by relating the components of stress on the faces of a block in one orientation to the components of stress on the faces of a block in another orientation.    

But why should we even bother with any block at all? We now define a state of stress without using any block.  To do so requires us to invoke the fundamental ideas in linear algebra: vector space, and linear map between vector spaces.

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