ON THE DERIVATION OF THE ELASTIC EQUATION OF MOTION

Abstract

Several key points in elastodynamics are reexamined. The variational principle is the most fundamental axiom of classical mechanics; for a point particle originally at r, the virtual displacement r→r+δu leads to Newton's second law, f=ma. On the other hand, elastodynamics considers a continuum that has already been subjected to a displacement u before the virtual displacement r+u→r+u+δu. The present paper shows that because of this difference in the nature of the virtual displacement, the variational principle for continua does not in general lead to Newton's second law. The present paper also shows that the generalized form of Hooke's Law with 36 independent elastic constants always leads to the existence of a strain energy function, because, as a consequence of elementary results in linear algebra, the strain energy due to the 15 antisymmetric components of the elastic tensor is identically zero.