Continuum mechanics problem: Stresses

  • #1

Homework Statement



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].


Homework 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.

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).
 

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