Incompressibility

A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. (Rubber and much living tissue (which is composed mostly of water) are examples of incompressible materials.) Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they admit some tricky a priori bounds and they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton's Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton's Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs, which have a nonstandard structure, admit, under favorable assumptions, periodic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.