Two-point Tensor
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Two-point Tensor

Two-point tensors, or double vectors, are tensor-like quantities which transform as euclidean vectors with respect to each of their indices and are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates.[1] Examples include the deformation gradient and the first Piola-Kirchhoff stress tensor.

As with many applications of tensors, Einstein summation notation is frequently used. To clarify this notation, capital indices are often used to indicate reference coordinates and lowercase for present coordinates. Thus, a two-point tensor will have one capital and one lower-case index; for example, AjM.

Continuum mechanics

A conventional tensor can be viewed as a transformation of vectors in one coordinate system to other vectors in the same coordinate system. In contrast, a two-point tensor transforms vectors from one coordinate system to another. That is, a conventional tensor,

,

actively transforms a vector u to a vector v such that

where v and u are measured in the same space and their coordinates representation is with respect to the same basis (denoted by the "e").

In contrast, a two-point tensor, G will be written as

and will transform a vector, U, in E system to a vector, v, in the e system as

.

The transformation law for two-point tensor

Suppose we have two coordinate systems one primed and another unprimed and a vectors' components transform between them as

.

For tensors suppose we then have

.

A tensor in the system . In another system, let the same tensor be given by

.

We can say

.

Then

is the routine tensor transformation. But a two-point tensor between these systems is just

which transforms as

.

The most mundane example of a two-point tensor

The most mundane example of a two-point tensor is the transformation tensor, the Q in the above discussion. Note that

.

Now, writing out in full,

and also

.

This then requires Q to be of the form

.

By definition of tensor product,

 

 

 

 

So we can write

Thus

Incorporating (1), we have

.

In the equation following (1) there are four q's !?

See also

References

  1. ^ Humphrey, Jay D. Cardiovascular solid mechanics: cells, tissues, and organs. Springer Verlag, 2002.

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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