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How to find the Piola-Kirchhoff stress tensor

  1. Nov 1, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello,
    I am supposed to show that the quantity

    TR=JTF-t

    satisfies

    TR=∂W/∂F

    for some scalar function W(X, F, θ) in my continuum mechanics homework. The task is to identify this scalar function W(X, F, θ).


    2. Relevant equations
    This is part b) of a question. In part a), we get the following results which may or may not be relevant to the solution of the problem:

    Constitutive equations are:
    T=T(F,θ)
    η=η(F,θ)
    A=A(F,θ)
    q=q(F,θ,gradθ)

    where
    η = -∂A/∂θ
    TF-T = ρ*∂A/∂F

    3. The attempt at a solution
    By substitution of the results in 2., I got

    TR=JTF-t
    =J*ρ*∂A/∂F
    R(X)*∂A(F,θ)/∂F(X)

    Is it as easy as saying that

    W(X, F, θ) = ρR(X)*A(F,θ) ?

    That's the solution I got but it seems too easy/I'm unsure... :D

    I would greatly appreciate any advice.

    Thank you so so much!!!
     
    Last edited: Nov 1, 2016
  2. jcsd
  3. Nov 5, 2016 #2
    I assume that [itex]T_R[/itex] is the First Piola-Kirchhoff Stress (PK1), T is the Cauchy Stress, F is the Deformation Gradient, J is the determinant of F, and W is the Strain Energy Density...

    [itex]T_R=\frac{\partial W}{\partial F}[/itex] is another way of saying that the PK1 Stress and the Deformation Gradient are "work conjugate." Indeed they are -- i.e. they behave in the same way under a rigid body rotation. Similarly, the Cauchy Stress is work-conjugate with the Almansi Strain and the PK2 Stress is work conjugate with the Green Strain, etc... This is important, because a FEA software may choose to calculate the PK1 Stress for a given element, and it would presumably be able to do so if the Deformation Gradient of that element is known. If the Green Strain is known, on the other hand, then that would be insufficient by itself to obtain the PK1 Stress (although you could presumably obtain the PK2 stress in this case).

    I believe you're being asked to show that the PK1 stress ([itex]\mathbf{T}_R[/itex]) and F are work-conjugate.
    You say that you're being asked to actually find W, but that doesn't make much sense to me. W is different for metals, rubber, etc... [itex]T_R=\frac{\partial W}{\partial F}[/itex] is true regardless of the material...

    [itex]W(\mathbf{X},\mathbf{F},\theta)=\rho_R\mathbf{X}*A(\mathbf{F},\theta)[/itex] is totally meaningless to me. Can't help there.
    I might be able to help more if you define your variables: A, [itex]\theta[/itex], and [itex]\rho[/itex]...
     
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