Two-point tensor

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Two-point tensors, or double vectors, are tensor-like quantities which transform as vectors with respect to each of their indices and are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates. Examples include 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.

[edit] 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,

Q{_p}{_q}(e{_p}\otimes e{_q}),

transforms a vector u to a vector v such that

v = Qu

where v and u are measured in the same coordinate system (denoted by the "e"). In contrast, a two-point tensor, G will be written as

G_{pq}(e_p\otimes E_q)

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

v = GU.

[edit] 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

v'p = Qpqvq.

For tensors suppose we then have

T_{pq}(e_p \otimes e_q).

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

T'_{pq}(e'_p \otimes e'_q).

We can say

T'ij = QipQjrTpr.

Then

T' = QTQT

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

F_{pq}(e'_p \otimes e_q)

which transforms as

F' = QF.

[edit] 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

v'p = Qpquq.

Now, writing out in full,

u = uqeq

and also

v = v'pep.

This then requires Q to be of the form

Q_{pq}(e'_p \otimes e_q).

By definition of tensor product,

(e'_p\otimes e_q)e_q=(e_q.e_q) e'_p = e'_p\qquad(1)

So we can write

u_p e_p = (Q_{pq}(e'_p \otimes e_q))(v_q e_q)

Thus

u_p e_p = Q_{pq} v_q(e'_p \otimes e_q) e_q

Incorporating (1), we have

upep = Qpqvqep.
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