How to Simplify a Double Dot Product of Tensors?

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Discussion Overview

The discussion revolves around the simplification of a double dot product involving tensors, specifically a second-order tensor and a fourth-order tensor. Participants explore the properties of tensor operations, particularly in the context of linear algebra and continuum mechanics.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to understand how to expand a double dot product of basis vectors involving a second-order tensor and a fourth-order tensor.
  • Another participant questions whether the original poster is missing definitions or clarity in their problem, suggesting a lack of familiarity with the double dot product in this context.
  • A participant mentions that the double dot product is rarely used in modern differential geometry, advocating for the use of tensor products and contractions instead.
  • One participant asserts that the double dot product is indeed used in continuum mechanics, providing an example of its application in representing the orientation of rigid fibers and the velocity gradient of a flow field.
  • The original poster questions whether their interpretation of a fourth-order tensor as a dyad of two second-order tensors is correct, asking about the requirements for such a representation, particularly regarding symmetry.
  • Another participant provides a mathematical expression to illustrate the contraction of a fourth-order tensor to a second-order tensor through the double dot product.

Areas of Agreement / Disagreement

Participants express differing views on the relevance and application of the double dot product in modern contexts, with some asserting its utility in specific fields like continuum mechanics while others suggest it is less common in differential geometry. The discussion remains unresolved regarding the clarity of definitions and the correctness of the original poster's interpretations.

Contextual Notes

There are indications of missing definitions and assumptions regarding the properties of tensors and the double dot product, which may affect the clarity of the discussion. The scope of the discussion appears to be limited to specific applications in linear algebra and continuum mechanics.

nikphd
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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?

(e_ie_je_ke_l):(e_me_n)=?

and

(e_ie_j):(e_ke_le_me_n)=?

Thanks a lot!
 
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Are you missing the definitions or what exactly is the problem?
I haven't seen this being used before.
 
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!
 
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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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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!
 
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
 

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