1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finite strain

  1. Apr 6, 2013 #1
    For the deformation field given by
    $$
    x_1 = X_1 + \alpha X_2,\quad x_2 = X_2 - \alpha X_1,\quad x_3 = X_3
    $$
    where ##\alpha## is a constant, determine the matrix form of the tensors ##\mathbf{E}## and ##\mathbf{e}##, and show that the circle of particles ##X_1^2+ X_2^2 = 1## deforms into the circle ##x_1^2 + x_2^2 = 1 + \alpha^2##.

    How do I find ##\mathbf{e}## and the solution is
    $$
    \frac{1}{2(1+\alpha^2)}\begin{bmatrix}
    -\alpha^2 & 0 & 0\\
    0 & -\alpha^2 & 0\\
    0 & 0 & 0
    \end{bmatrix}
    $$
    First, we will find ##\mathbf{E}##.
    $$
    \mathbf{F} =
    \begin{bmatrix}
    1 & \alpha & 0\\
    -\alpha & 1 & 0\\
    0 & 0 & 1
    \end{bmatrix}
    $$
    Then ##\mathbf{C} = \mathbf{F}^T\mathbf{F}##. So we have that
    $$
    \mathbf{C} =
    \begin{bmatrix}
    1 + \alpha^2 & 0 & 0\\
    0 & 1 + \alpha^2 & 0\\
    0 & 0 & 1
    \end{bmatrix}
    $$
    which leads to
    $$
    \mathbf{E} = \frac{1}{2}
    \begin{bmatrix}
    \alpha^2 & 0 & 0\\
    0 & \alpha^2 & 0\\
    0 & 0 & 0
    \end{bmatrix}.
    $$
     
  2. jcsd
  3. Apr 7, 2013 #2
    I found ##\mathbf{e} = \mathbf{F}^{-T}\mathbf{E}\mathbf{F}^{-1}## but this returns the negative of the book answer for ##\mathbf{e}##.
    Assuming the book has a typo, how do I show ##X_1^2+X_2^2 = 1## deforms into ##x_1^2 + x_2^2 = 1 +\alpha^2##?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Finite strain
  1. Angular strains (Replies: 2)

  2. Stress / strain (Replies: 1)

  3. Torsional Strain (Replies: 12)

  4. Stresses and Strains (Replies: 1)

  5. Strain equation (Replies: 3)

Loading...