The constraint of incompressibility has led to the discovery of several
exact solutions in isotropic finite elasticity, most notably the controllable or
universal solutions of Rivlin and others. Ericksen has examined the problem of
finding all such solutions. He has also proved that there are no controllable
finite deformations in isotropic compressible elasticity, except for homogeneous
In this talk some related questions are examined for three special classes of compressible isotropic elastic materials, one of which is the class of harmonic materials. Several closed form solutions, similar to the Rivlin solutions, are obtained. In particular, some classes of controllable deformations are obtained, e.g. some deformations possible in all harmonic materials. The question of finding all such controllable deformations is addressed, with rather surprising results. For instance, it is shown that every harmonic scalar function generates a deformation that is controllable for harmonic materials. Finally, analysis of the controllability conditions suggests that the three classes of strain energy functions considered may be the only ones for which results of this type can be obtained.
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