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Proving equations for a plane stress condition

  1. Jul 29, 2010 #1
    1. The problem statement, all variables and given/known data
    For a plane stress condition (stress z-axis = 0 ), prove the following relations if strain x-axis and strain y-axis are determined by experiments.

    σ_x=(∈_x+v∈_y/1-v^2)E

    &

    σ_y=(∈_y+v∈_x/1-v^2)E

    where:
    σ_x = stress in x-axis
    σ_y = stress in y-axis
    ∈_x = strain in x-axis
    ∈_x = strain in y-axis
    E = modulus of elasticity


    2. Relevant equations
    ∈_z=-(∈_x+∈_y)(v/1-v)

    Poission's ratio
    ∈_x=σ_x/E

    ∈_y=∈_z=-vσ_x/E

    v=-lateral strain/axial strain

    v=-∈_y/∈_x=-∈_z/∈_x


    3. The attempt at a solution
    I'm not sure which equations I should be using to solve the problem. Can I use Hooke's Law for multi-axial loading since it is a plane stress condition?

    I have tried rearranging Poisson's ratio equation with no luck.
     
  2. jcsd
  3. Jul 29, 2010 #2

    Mapes

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    Hooke's Law only applies to uniaxial loading of a rod. You might find generalized Hooke's Law useful:

    [tex]\epsilon_x=\frac{\sigma_x}{E}-\frac{\nu\sigma_y}{E}-\frac{\nu\sigma_z}{E}[/tex]

    It applies in 3-D and in any isotropic material. More http://john.maloney.org/Papers/Generalized%20Hooke%27s%20Law%20%283-12-07%29.pdf" [Broken].
     
    Last edited by a moderator: May 4, 2017
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