Editing Virtual work (section)

===Proof of equivalence between the principle of virtual work and the equilibrium equation===

We start by looking at the total work done by surface traction on the body going through the specified deformation:
<math display="block"> \int_{S} \mathbf u \cdot \mathbf T dS = \int_{S} \mathbf u \cdot \boldsymbol \sigma \cdot \mathbf n dS </math>

Applying divergence theorem to the right hand side yields:
<math display="block"> \int_S \mathbf{u \cdot \boldsymbol \sigma \cdot n} dS = \int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV </math>

Now switch to indicial notation for the ease of derivation.
<math display="block">\begin{align}
\int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV 
  &= \int_V \frac{\partial}{\partial x_j} \left( u_i \sigma_{ij} \right) dV \\
  &= \int_V \left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV
\end{align}</math>

To continue our derivation, we substitute in the equilibrium equation <math> \frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0 </math>. Then
<math display="block">\int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV
       = \int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV</math>

The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows:
<math display="block">\begin{align}
\int_V\left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV
       &= \int_V\left( \frac12 \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) 
          + \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i \right) dV \\
       &= \int_V \left( \left[ \epsilon_{ij} 
          + \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i\right)  dV \\
       &= \int_V\left( \epsilon_{ij} \sigma_{ij} - u_i f_i \right) dV\\
       &= \int_V \left( \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f \right)  dV
\end{align}</math>
where <math> \boldsymbol\epsilon </math> is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.

Now recap. We have shown through the above derivation that
<math display="block"> \int_{S} \mathbf{u \cdot T} dS = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV - \int_V \mathbf u \cdot \mathbf f dV </math>

Move the 2nd term on the right hand side of the equation to the left:
<math display="block"> \int_{S} \mathbf{u \cdot T} dS + \int_V \mathbf u \cdot \mathbf f dV = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV </math>

The physical interpretation of the above equation is, ''the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains''.

For practical applications:

* In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.
* In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.

These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.