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Continuum mechanics problem: Stresses

  1. Mar 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Mechanics of Deformable Media (Bhatia and Singh), 5.2:

    Consider a long rod of elastically isotropic material of L standing vertically in a vacuum in equilibrium under the gravitational field of the earth, then:

    (i) What are the boundary conditions for [tex]\sigma_{ij}[/tex] on the various surfaces of the rod?

    (ii) Solve for [tex]\sigma_{ij}[/tex].

    2. Relevant equations

    [tex]\sigma_{ij}[/tex] is defined as the stress on the ith surface in the jth direction.

    Equilibrium suggests ([tex]\lambda[/tex] + 2[tex]\mu[/tex])grad(div[tex]\vec{s}[/tex]) - [tex]\mu[/tex] curl(curl[tex]\vec{s}[/tex]) + [tex]\rho[/tex] F = 0

    where lambda and mu are constants, s is the displacement field, and F is the body force and rho the density of the material.

    3. The attempt at a solution

    The force IN the rod is the reactant force from it's weight exerted by the surface it rests on. It is purely in the z-axis, in cylindrical polar coordinates.

    [tex]\sigma_{zz} = -\frac{mg}{A}[/tex] at z=0, the end of the rod touching the surface

    [tex]\sigma_{zz} = 0[/tex] at z=L, the top of the rod since we're in a vacuum

    All other stresses at the boundaries are 0.

    I am not sure if my boundaries above are correct and it is not clear to me how to find the stress tensor for part (ii).
  2. jcsd
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