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

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Abstract
This work is concerned with the propagation of purely longitudinal and purely transverse waves in homogeneously deformed isotropic elastic materials. Two types of compressible material are also discussed. A Hadamard material, so called by John in the hyperelastic case, is one in which longitudinal waves may propagate in every direction when the material is homogeneously deformed. A second material, called a “Green material” is introduced. In it two transverse waves can propagate in every direction when the material is homogeneously deformed. It is seen that a Mooney material is the only isotropic incompressible elastic material in which two transverse waves can propagate in every direction when it is homogeneously deformed, while the pressure stays constant throughout the material. The propagation of finite amplitude waves in these materials is discussed. Finally, it is shown that the only motions which can be maintained in all homogeneous compressible elastic Hadamard materials under the action of surface forces alone, are necessarily homogeneous and accelerationless.

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