# Continuum mechanics problem: Stresses

1. Mar 17, 2010

### MarkusNaslund19

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 $$\sigma_{ij}$$ on the various surfaces of the rod?

(ii) Solve for $$\sigma_{ij}$$.

2. Relevant equations

$$\sigma_{ij}$$ is defined as the stress on the ith surface in the jth direction.

Equilibrium suggests ($$\lambda$$ + 2$$\mu$$)grad(div$$\vec{s}$$) - $$\mu$$ curl(curl$$\vec{s}$$) + $$\rho$$ 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.

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

$$\sigma_{zz} = 0$$ 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).