Why Do Some Tensor Functions Appear Isotropic in Component Notation?

In summary, the conversation discusses the definition of an isotropic tensor function and how it applies to Cartesian tensors. It is mentioned that any tensor function can be considered isotropic in component notation, but this is proven to be false. An example of an anisotropic tensor function is given to demonstrate this.
  • #1
Andrea
Hello,

I consider only Cartesian tensors in the following. The definition of
isotropic tensor function I know is

1) T = F ( G )

such that, for any rotation ( ' = transpose),

2) O F( G ) O' = F( O G O' )

But, if I change to component notation, it seem to me that any tensor
function is isotropic, which cannot obviously be. Denoting the
components in the new basis with ^*, I have

3a) T_ij^* = O_ir T_rs (O_sj)'

3b) G_ij^* = O_ir G_rs (O_sj)'

since T and G are tensors. Then, by 1),

4a) T_rs = F_rs ( G_mn )

4b) T_rs^* = F_rs^* ( G_mn^* )

Then, substituting 3a) and 3b) into 4b), I get

5) O_ir T_rs (O_sj)' = F_rs^* ( O_mk G_kl (O_ln)' )

Finally, substituting 4a) into 5), I have

6) O_ir F_rs ( G_mn ) (O_sj)' = F_rs^* ( O_mk G_kl (O_ln)' )

that is,

O F( G ) O' = F (O G O' )

So any tensor function would be isotropic. Clearly that's false, but I
don't see where the error is. Can you help me find it? Thanks,

Andrea
 
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  • #2
Here's an example of a tensor valued, anisotropic function

[tex]H = g(T) = (T\cdot{n\otimes{n}})n\otimes{n}[/tex]
where n is a fixed unit vector.

Then
[tex]
g(QTQ^T) = (QTQ^T\cdot{n\otimes{n}})n\otimes{n} = (T \cdot (Q^Tn \otimes Q^Tn)) n\otimes{n}
[/tex]
which is not the same as
[tex]
Qg(T)Q^T = (T\cdot{n\otimes{n}})Qn\otimes{Qn}
[/tex]
for general orthogonal Q, only if n is a proper vector of Q.
Im basically extrapolating from Jaunzemis' Continuum Mechanics book, pg 287.
 
  • #3


Hello Andrea,

Thank you for your question. The key point here is that not all tensor functions are isotropic. In order for a tensor function to be isotropic, it must satisfy the condition in equation 2) of your definition: O F( G ) O' = F( O G O' ). This means that the function must remain unchanged when the coordinates are rotated.

In your example, you are assuming that both T and G are isotropic tensors, which is not necessarily true. In fact, most tensor functions are not isotropic. For example, the stress tensor in a solid material is not isotropic, as it changes with the orientation of the material.

In order to find the error in your reasoning, you should consider a specific example of a tensor function that is not isotropic, and see if it still satisfies equation 2). This will help you understand the limitations of isotropic tensor functions and where your error lies.

I hope this helps clarify the concept of isotropic tensor functions. Let me know if you have any further questions.


 

Related to Why Do Some Tensor Functions Appear Isotropic in Component Notation?

1. What is an isotropic tensor function?

An isotropic tensor function is a mathematical function that has the same value at any point in space, regardless of the direction or orientation of the coordinate system. It is also known as a scalar function.

2. How is an isotropic tensor function different from a regular tensor function?

A regular tensor function has different values at different points in space, depending on the direction or orientation of the coordinate system. An isotropic tensor function, on the other hand, has the same value at any point in space.

3. What are some examples of isotropic tensor functions?

Examples of isotropic tensor functions include distance, mass, and temperature. These quantities have the same value at any point in space, regardless of the coordinate system used to measure them.

4. How are isotropic tensor functions used in science?

Isotropic tensor functions are used in various fields of science, such as physics, engineering, and mathematics. They are particularly useful in modeling physical systems and solving equations involving symmetry and invariance, such as in fluid mechanics and electromagnetism.

5. Can an isotropic tensor function have more than one variable?

Yes, an isotropic tensor function can have more than one variable. For example, temperature may vary with both time and position, but it is still considered an isotropic tensor function because it has the same value at any point in space and time.

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