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Deformation (continuum mechanics)

  1. Aug 23, 2009 #1
    1. The problem statement, all variables and given/known data

    A body which in the reference configuration is a unit cube with its edges parallel to the coordinate axes undergoes the following deformation:

    x1=a1(X1+sX2), x2=a2X2, and x3=a3X3
    (where a1,a2,a3,s are constants).

    determine the lengths of its edges after the deformation

    2. Relevant equations



    3. The attempt at a solution

    I think i need to know the x1, x2, and x3 before the deformation, as in when it was a unit cube. so if x1=1 then the new length (length after deformation) will be:
    sqrt[(1-a1(X1+sX2))^2]

    but this is not working for me since the answer is:
    lengths: a1, sqrt[(s^2)(a1^2)+a2^2], a3
     
  2. jcsd
  3. Aug 23, 2009 #2

    tiny-tim

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    Hi sara_87! :smile:

    (try using the X2 tag just above the Reply box :wink:)
    You're getting very confused :confused:

    X1 and X2 will not appear in the final result …

    and the ends of your edges are {X1 = 1, X2 = X3 = 0} etc :wink:
     
  4. Aug 23, 2009 #3
    how did you know that X1=1 and X2=X3=0 ?
    and are these for the reference configuration or after the deformation?
    I am confused :(
     
  5. Aug 23, 2009 #4

    tiny-tim

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    Because it's "a unit cube with its edges parallel to the coordinate axes" …

    so the ends of three edges (before the deformation) are {1 0 0} {0 1 0} and {0 0 1} :smile:
     
  6. Aug 23, 2009 #5
    thanks.
    so say for one of the edges, it used to be (1,0,0) then after deformation, it's
    (a1, s*a1, 0)
    so the length is: sqrt[(a1-1)^2+(sa1)^2]

    i think im making a big mistake but i dont know what it is.
     
  7. Aug 24, 2009 #6

    tiny-tim

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    No, you're not applying the formula …

    for example, the new X2 should be a2(old X2), = a20 :smile:
     
  8. Aug 24, 2009 #7
    oh ok, thank you.
    also, how do i find the angles between these edges (after deformation)?
     
  9. Aug 24, 2009 #8

    tiny-tim

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    dot product divided by moduli = … ? :smile:
     
  10. Aug 24, 2009 #9
    =cos(theta), thank you very much :)
     
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