Statics and dynamics in elasticity : how to add time in elasticity In elasticity, there is are static relations of the form $R(\sigma,\epsilon)=0$. In fluid dynamics, there is a dynamics relation with the conservation of momentum leading to Navier-Stokes equation.
My question is : how to describe the dynamics of the deformation in elasticity, since all the relations are static ?
I know why way to do it is to introduce a viscosity, and to make the material viscoelastic. But it seems to me a bit ad hoc. Are there more rigorous ways to introduce time ? 
For example, defining an energy and writing not far from equilibrium $\frac{\partial u}{\partial t}=-\frac{\delta F}{\delta u}$, with $u$ the displacement ? Are there models for that ?
 A: In fluid dynamics, we substitute the relationship between the stress tensor and the rate of deformation tensor (expressed in terms of velocity gradients) for a Newtonian fluid into the conservation of momentum equation to obtain the time-dependent Navier-Stokes equations.  For solid dynamics, we substitute the relationship between the stress tensor and the strain tensor (expressed in terms of displacement gradients) for a Hookean solid into the conservation of momentum equation to obtain the time-dependent equations for a linearly elastic solid.
A: The dynamic equations for the components of the stress tensor $\sigma_{ij}(x_1,x_2,x_3,t)$ are not static, they are of the form:
$$\sum_{j} \frac{\partial \sigma_{ij}}{\partial x_j} + b_i = \rho \frac{\partial v_i}{\partial t}$$
where $v_i(x_1,x_2,x_3,t)$ is the field of velocities. So, the stress components are functions of time and the strain relations can be computed algebraically from:
$$R(\sigma_{ij}(\boldsymbol{x},t),\varepsilon_{ij}(\boldsymbol{x},t)) = 0$$
