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Two-point tensor

From Wikipedia, the free encyclopedia

Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They 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.

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Transcription

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

.

Simple example

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,

 

 

 

 

(1)

So we can write

Thus

Incorporating (1), we have

.

See also

References

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

External links

This page was last edited on 17 March 2023, at 17:37
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.