unimportant: we can consider the body to always be in static equilibrium,
i.e. we can assume that at all times the body satisfies
@f{eqnarray*}
- - \textrm{div}\ ( C \varepsilon(\mathbf{u})) &=& \mathbf{f}
+ - \textrm{div}\ ( C \varepsilon(\mathbf{u})) &=& \mathbf{f}(\mathbf{x},t)
+ \qquad
\textrm{in}\ \Omega,
\\
\mathbf{u}(\mathbf{x},t) &=& \mathbf{d}(\mathbf{x},t)
introduce a tensorial stress variable $\sigma$, and write the differential
equations in terms of the stress:
@f{eqnarray*}
- - \textrm{div}\ \sigma &=& \mathbf{f}
+ - \textrm{div}\ \sigma &=& \mathbf{f}(\mathbf{x},t)
+ \qquad
\textrm{in}\ \Omega(t),
\\
\mathbf{u}(\mathbf{x},t) &=& \mathbf{d}(\mathbf{x},t)
have to solve the following system:
@f{eqnarray*}
- \textrm{div}\ C \varepsilon(\Delta\mathbf{u}^n) &=& \mathbf{f} + \textrm{div}\ \sigma^{n-1}
+ \qquad
\textrm{in}\ \Omega(t_{n-1}),
\\
\Delta \mathbf{u}^n(\mathbf{x},t) &=& \mathbf{d}(\mathbf{x},t_n) - \mathbf{d}(\mathbf{x},t_{n-1})