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Understanding the strain energy function invariant term

  1. May 24, 2012 #1
    Hi, Dear all,

    Facing problem to understand strain energy function invariant terms
    A typical strain energy function consist of strain invariant can be defined as followed
    W(I1,I4)=C0+C1(I1-3)(I4-1)+C2(I1-3)^2+C3(I1-4)^2+C4(I1-3)+C5(I4-1),
    I1 and I4 are so called invariants of Green's strain tensor. (large deformation)
    I1=trC=λ1^2+λ2^2+λ3^2
    I4=N1*λ1^2*N1+N2*λ2^2*N2+N3*λ3^2*N3.

    here is the complete link taken http://www.engin.umich.edu/class/bme456/ch6fitelasticmodelconstant/bme456fitmodel.htm [Broken]

    1. I read from article that N is a unit vector along the stretch direction, so can i conclude that
    I4 consist of unit vector multiply with principal stretch?

    2. the lamda in the formula is stretch ratio or principal stretch?
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. May 27, 2012 #2
    After giving it a quick read,

    2)[itex]\lambda_1 ,\lambda_2 ,\lambda_3[/itex] are the "normal stretch", where the link calls them that. but I think they are just the stretch coefficients for force in the same direction as the normal plane vector.

    1) Yes, It seems so then, assuming you have "principle"[itex]\iff[/itex]"normal"

    That's my take on it.
     
  4. May 30, 2012 #3
    Hi, jfy4,

    I actually try to read more, but cannot find resources.
    1. so all the N1, N2, N3,should always equal to 1? or under any condition they will change?
    2. or can I call them as right stretch tensor?sorry, as i cannot differentiate left and right stretch tensor, so cannot evaluate more for you.
     
  5. May 30, 2012 #4
    well, [itex]N_i[/itex] is a unit vector, a vector whose magnitude is 1. That is [itex]N_i N_i=N_{1}^{2}+N_{2}^{2}+N_{3}^{2}=1[/itex]. But, it is a vector, not a constant, so it's direction can vary. It says:
     
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