4th order tensor double product

In summary, the conversation is about the use of double dot product in tensor algebra and differential geometry. The main question is whether a 4th order tensor can be represented as a dyad of two 2nd order tensors and what are the requirements for this representation. The conversation also discusses the use of double dot product in continuum mechanics and the suggestion to move the question to the linear algebra section.
  • #1
nikphd
6
0
Hello,

Been a long time lurker, but first time poster. I hope I can be very thorough and descriptive. So, I have been battling with a double inner product of a 2nd order tensor with a 4th order one. So my question is:

How do we expand (using tensor properties) a double dot product of the basis vectors to a simpler one?

[tex](e_ie_je_ke_l):(e_me_n)=?[/tex]

and

[tex](e_ie_j):(e_ke_le_me_n)=?[/tex]

Thanks a lot!
 
Last edited:
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  • #2
Are you missing the definitions or what exactly is the problem?
I haven't seen this being used before.
 
  • #3
What exactly haven't you seen been used before? The double dot product of a tensor of n=4 with one of n=2? You mean you have only seen it being used for tensors of equal order?

The properties I am referring to, is actually expanding the double dot product to two single dot products!
 
Last edited:
  • #4
The "double dot product". What I am saying is that you will rarely find this being used in modern differential geometry, for the plain reason that we have tensor products, contractions, etc. . I believe that this is the reason why you are not getting any responses. Moreover, this is a linear algebra question, not a geometry one.

Either way, if you provide the definitions in terms of said concepts and show how far you got, I am sure people will be able to help you.
 
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  • #5
Double dot is used extensively in continuum mechanics, even in 2014! For example, 4th order tensors represent orientation of rigid fibers in a 3D space, and 2nd order tensor is the velocity gradient of a flow field.
So should I move my question to the linear algebra section?
My basic question is this actually: "Is the following statement correct? D:uuuu=(D:uu)uu, meaning can i represent a 4th order tensor as a dyad of two 2nd order tensors?And if yes which are the requirements? Symmetry?" I thought it was a pretty straightforward question!
 
  • #6
The product contracts the order of the 4th order tensor to a 2nd order tensor. i.e.
Aijkl ei ej ek el : Bmn em en = Aijkl Bmn ei ej dkm dln = Aijkl Bkl ei ej
 

1. What is a 4th order tensor double product?

A 4th order tensor double product is a mathematical operation that involves multiplying two 4th order tensors to produce a 4th order tensor as the result. It is similar to a matrix multiplication but involves higher dimensions.

2. How is a 4th order tensor double product calculated?

To calculate a 4th order tensor double product, you first need to expand the tensors into their component form. Then, you can use the index notation and Einstein summation convention to perform the multiplication and summation of the components to obtain the resulting 4th order tensor.

3. What are some applications of 4th order tensor double product?

4th order tensor double product has various applications in physics and engineering, such as in solid mechanics, fluid dynamics, and electromagnetism. It is used to represent stress and strain tensors, fluid stress tensors, and electromagnetic stress tensors, among others.

4. Can a 4th order tensor double product be simplified?

Yes, a 4th order tensor double product can be simplified by using symmetry properties of the tensors involved. For example, a symmetric 4th order tensor double product can be reduced to a 2nd order tensor double product, which is easier to work with.

5. Are there any limitations or challenges in using 4th order tensor double product?

One challenge in using 4th order tensor double product is the complexity of the calculations involved, which can become even more challenging for higher order tensors. Additionally, the interpretation of the resulting tensor can also be difficult, as it represents a higher order of mathematical abstraction.

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