Editing Linear elasticity (section)

=== Elastodynamics in terms of stresses ===
Elimination of displacements and strains from the governing equations leads to the '''Ignaczak equation of elastodynamics'''<ref>[[Ostoja-Starzewski, M.]], (2018), ''Ignaczak equation of elastodynamics'', Mathematics and Mechanics of Solids. {{doi|10.1177/1081286518757284}}</ref>
<math display="block">\left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - S_{ijkl} \ddot{\sigma}_{kl} + \left( \rho ^{-1} F_{(i}\right) ,_{j)} = 0.</math>

In the case of local isotropy, this reduces to
<math display="block">\left( \rho ^{-1} \sigma _{(ik},_{k}\right) ,_{j)} - \frac{1}{2\mu } \left( 
\ddot{\sigma}_{ij} - \frac{\lambda }{3 \lambda +2\mu }\ddot{\sigma}_{kk}\delta
_{ij}\right) +\left( \rho ^{-1} F_{(i}\right) ,_{j)} = 0. </math>

The principal characteristics of this formulation include: (1) avoids gradients of compliance but introduces gradients of mass density; (2) it is derivable from a variational principle; (3) it is advantageous for handling traction initial-boundary value problems, (4) allows a tensorial classification of elastic waves, (5) offers a range of applications in elastic wave propagation problems; (6) can be extended to dynamics of classical or micropolar solids with interacting fields of diverse types (thermoelastic, fluid-saturated porous, piezoelectro-elastic...) as well as nonlinear media.