The traditional theory of the solid state rests on two false assumptions. One is the principle of a constant relaxed (or standard) state. The other is the principle of relaxability-in-the-large, first formulated mathematically by de Saint-Venant. His equations are essentially identical with Riemann's equations expressing the condition that a geometry be Euclidean-in-the-large. It is shown that a principle of relaxability-in-the-small is sufficient for the geometry of strain—which then becomes a three-dimensional Riemannian geometry. The kinematics of strain is next developed without introducting the principle of a constant relaxed state.

The ground is thus cleared for the construction of a classical theory of anelasticity. It is first shown that the variability of the relaxed state makes the density of the substance independent of the strains. Consequently, the internal energy can depend on the specific volume as well as on the strains and the entropy. The expression for the rate of increase of entropy is then derived. As usual, this suggests, but does not uniquely determine the form of the dissipative laws. In addition to the usual equations for the viscous stresses, one is led to another set of equations involving parameters that quantitatively describe the anelastic properties of the substance.

These results are used to derive the equations for the waves of distortion and dilation in an ideal isotropic anelastic medium. When the two coefficients of viscosity and two coefficients of anelasticity all vanish, these reduce to the usual equations for such waves. In addition to the four dissipative coefficients, the isotropic anelastic substance is characterized by four elastic moduli, rather than by the two that characterize an isotropic elastic substance. Thus, there are eight parameters whose values can be adjusted to describe the particular substance. When the four dissipative parameters do not vanish, the propagation of the waves can be described by five relaxation times. The ideal isotropic anelastic substance thus has a relaxation spectrum of about the same complexity as those of actual substances.

The kinematic independence of the density and the strains causes a hydrostatic pressure to have different dynamic effects than a uniaxial pressure. This is in accord with experiment—the latter being much more effective in producing anelastic deformation than the former.

• Received 4 November 1947

DOI:https://doi.org/10.1103/PhysRev.73.373