# Priniciple of Minimum Energy vs. Maxwell equations

1. Jun 21, 2012

### Hassan2

Dear all,

I'm studying a paper on coupled magneto-mechanical problems.

Suppose we have a "deformable" ferromagnetic bar placed in an initially uniform magnetic field. Both ends of the bar are clamped. The bar has magnetostriction property, so it may expand or contract depending on the the magnitude of the flux density.

In the paper, in order to find the magnetic field numerically, it takes the approach of energy minimization and here the energy is the sum of magnetic and mechanical energies. Both magnetic field and deformation is obtained this way.

I'm trying to solve the problem using Maxwell equation ( ∇×(B/μ)=0) rather than energy minimization, but it seems the Maxwell equation does not take into account the change in mechanical energy due to the induced magnetization even though the permeability may depend on the mechanical stress.

I wonder if the Maxwell equation incomplete when it comes to deformable media? This would be disappointing to me.

I would really appreciate if you helped me overcome the problem.

Thanks

2. Jul 3, 2012

### Hassan2

I have derived the following equation for magneto static case and I call it Amper's law for stressed bodies:

$\nabla \times H = J_{f}-\sigma_{ij}\frac{\partial \epsilon_{ij}}{\partial A}$

$\sigma$ and $\epsilon$ are the elastic stress and strain respectively and Einstein's summation notation has been used. $A$ is magnetic vector potential.

My professor became angry at me for saying this because he can't believe that Ampere's low is invalid for stressed deformable bodies. He can't disprove it and he doesn't want to see my proof either.

I am not 100% sure of the equation but the derivation is straight forward and It makes sense too. I wonder if any of you has seen such an equation ?

Thanks