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4th order tensor double product

  1. Jul 15, 2014 #1
    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: Jul 15, 2014
  2. jcsd
  3. Jul 15, 2014 #2
    Are you missing the definitions or what exactly is the problem?
    I haven't seen this being used before.
     
  4. Jul 15, 2014 #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 refering to, is actually expanding the double dot product to two single dot products!
     
    Last edited: Jul 15, 2014
  5. Jul 15, 2014 #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.
     
  6. Jul 16, 2014 #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!
     
  7. Mar 3, 2015 #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
     
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