On the strain-energy function for isotropic bodies

Abstract

The strain-energy function φ is assumed to be made up of a sum of homogeneous polynomials of the first, second, third, etc. degrees in the components of the strain tensor eαβ. The expansion is carried only so far as to include the third degree group of terms and then this expression for φ is studied from the point of view of the isotropy of space. Isotropy of space having been defined by means of rectangular trihedrons (or orthogonal ennuples), the effect on the "structure-tensor" cαβγδρσ (which forms with eαβ the third degree terms) of the assumption of isotropy is obtained. It is found that the third degree group introduces, for a completely isotropic state, only two coefficients of elasticity, which when taken together with the corresponding Lame's coefficients λ, μ of the second degree group gives only four elastic moduli in the complete expression for φ. The expression for φ thus developed is introduced into a set of equations obtained by F. D. Murnaghan connecting the stress and strain components by means of φ; and so we obtain relations between stress and strain which are applicable to deformations of higher order than those allowed by Hooke's law. These latter relations are then applied to the case of normal and uniform compression and these further applied to P. W. Bridgman's experiments on the change of volume of sodium under pressure at 30°C. In this application the four elastic moduli are reduced to two.

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