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Tensor Strains

  1. Feb 24, 2010 #1
    Ok I wan't to start by saying I'm in a ridiculous solid state physics class where the stuff we are learning is either poorly explained by our text book or even non-existent in the text. My teacher asked me the following question ... A single cube-shaped crystal of a simple cubic metal with face normals [100][010][001] has a value of the elastic stiffness constant c11 = cxx = 293 GPa. Write down the generalized form of Hooke's Law, in the tensor form and in the reduced index (matrix) form.

    So I just want to say, outside of this class I've never seen a tensor in my life except in classical dynamics which made sense for rotations. We went over tensors for like 20 minutes, where the basic idea of a tensor was described as being an object that calls a particular value a number of times. I have a somewhat ok idea of what a tensor is, though never in my life have I ever had to use them. I have no idea what my teacher is talking about and I can't find good information on the web or in my horrid little Intro to solids Charles Kittel book.

    Apparently hookes law is a fourth rank tensor and I have no freakin clue why that is.

    Thanks for the replies
     
  2. jcsd
  3. Feb 25, 2010 #2

    Mapes

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    Stiffness is represented by a fourth-rank tensor because it couples (relates) two second-rank tensors, stress and strain. And these are tensors because they each couple two vectors (aka first-rank tensors): the stress tensor relates a force vector to a vector representing the direction normal to an area, and the strain tensor relates an initial undeformed vector to the deformed version. Does this help?

    http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Constitutive_relations" [Broken] might be helpful.
     
    Last edited by a moderator: May 4, 2017
  4. Feb 25, 2010 #3

    turin

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    Hi, Mapes. Thanks for the link. I was wondering if you could confirm for me a factor-of-two error in that article. Here is the equation sequence that I believe has the error:

    [tex]\sigma_{ij}=\frac{\partial{w}}{\partial{\epsilon_{ij}}}=C_{ijkl}\ \epsilon_{kl}[/tex]

    [tex]w=C_{ijkl}\ \epsilon_{ij}\ \epsilon_{kl}[/tex]

    where C is the stiffness tensor, ε is the strain tensor, σ is the stress tensor, and w is the strain energy density. I would replace the last equation with:

    [tex]w=\frac{1}{2}\ C_{mnkl}\ \epsilon_{mn}\ \epsilon_{kl}[/tex]

    which leads to:

    [tex]\frac{\partial{w}}{\partial\epsilon_{ij}}
    =\frac{1}{2}\ C_{mnkl}\ \frac{\partial\left(\epsilon_{mn}\ \epsilon_{kl}\right)}{\partial\epsilon_{ij}}
    =\frac{1}{2}\ C_{mnkl}\ \frac{\partial\epsilon_{mn}}{\partial\epsilon_{ij}}\ \epsilon_{kl}\ +\ \frac{1}{2}\ C_{mnkl}\ \epsilon_{mn}\ \frac{\partial\epsilon_{kl}}{\partial\epsilon_{ij}}
    =\frac{1}{2}\ C_{ijkl}\ \epsilon_{kl}\ +\ \frac{1}{2}\ C_{mnij}\ \epsilon_{mn}[/tex]

    and then using the symmetry of C and renaming the dummy indices mn → kl:

    [tex]\frac{\partial{w}}{\partial\epsilon_{ij}}=C_{ijkl}\ \epsilon_{kl}[/tex]

    Maybe I don't understand their notation.
     
  5. Feb 25, 2010 #4

    Mapes

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    Yep, looks like a missing 1/2 to me too.
     
  6. Feb 25, 2010 #5
    turin,

    Nothing is wrong with the final expression since [tex] C_{ijkl} = C_{klij} [/tex] due to symmetry. After summing on the indices from 1 to 3 the two terms end up being equal. Intuitively, it makes sense because the first derivative of energy wrt object-position gives the object-force or in this case the stress on a unit cell. Which is what you end up with.

    modey3
     
    Last edited: Feb 25, 2010
  7. Feb 25, 2010 #6
    sol66,

    The best place to learn tensor applications in mechanics is by reading a continuum mechanics book. I recommend getting the Schaums Outline for continuum mechanics. That book has served me well in my graduate career.

    modey3
     
  8. Feb 25, 2010 #7

    Mapes

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    Modey3, just to be clear, turin suggests that the "Wikiversity" equation [itex]\frac{1}{2}\bold{\epsilon C\epsilon}= C_{ijkl}\epsilon_{ij}\epsilon_{kl}[/itex] should be [itex]\frac{1}{2}\bold{\epsilon C\epsilon}= \frac{1}{2}C_{ijkl}\epsilon_{ij}\epsilon_{kl}[/itex]; is this what you're disagreeing with?
     
  9. Feb 25, 2010 #8
    I'm not disagreeing with turin. The guy who wrote the Wiki forgot to add the 1/2.

    modey3
     
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