Longitudinal and Transverse Waves in Finite Elastic Strain. Hadamard and Green Materials

This work is concerned with the propagation of purely longitudinaland purely transverse waves in homogeneously deformed isotropicelastic materials. Two types of compressible material are alsodiscussed. A Hadamard material, so called by John in the hyperelasticcase, is one in which longitudinal waves may propagate in everydirection when the material is homogeneously deformed. A secondmaterial, called a "Green material" is introduced. In it twotransverse waves can propagate in every direction when the materialis homogeneously deformed. It is seen that a Mooney materialis the only isotropic incompressible elastic material in whichtwo transverse waves can propagate in every direction when itis homogeneously deformed, while the pressure stays constantthroughout the material. The propagation of finite amplitudewaves in these materials is discussed. Finally, it is shownthat the only motions which can be maintained in all homogeneouscompressible elastic Hadamard materials under the action ofsurface forces alone, are necessarily homogeneous and accelerationless.