Constitutive equations and reciprocity

Abstract

The differential equations that must be satisfied for most fields (physical variables) in geophysics are determined primarily by conservation equations which relate the divergence of the flux of the field, the field's time rate of change, and its sources and sinks. These conservation (or equilibrium) equations do not provide sufficient constraints to determine the fluxes and fields even when boundary conditions (both in space and time) are specified. To constrain the fields completely, it is necessary to introduce the properties of the media; that is, the constitutive equations. Because the conservation equations can be determined without considering the properties of the media, these equations are valid for the most general media; that is, heterogeneous, anisotropic, time-varying, non-linear, etc. media in which many field variables can interact or be coupled. When the fields can be described by ‘self-adjoint’ differential equations in space–time, these media exhibit reciprocity; that is, upon interchange of ‘sources’ and ‘detectors’, the same result is obtained. We show that viscoelastic, elastodynamic problems relating to generalized Kelvin–Voigt and generalized Maxwell media satisfy the conditions for reciprocity. In addition, we show that the introduction of tensor ‘densities’ (which relate the inertial-force density to the particle-acceleration, particle-velocity and particle-displacement vectors in the inertial force's constitutive equation) do not invalidate the reciprocity conditions. The two constitutive equations (the stress/strain and the inertial-force ones) lead to dispersion and attenuation in the propagation of the fields even though none of the material constants in the constitutive or conservation equations is complex (i.e. with real and imaginary parts). Complex material properties cannot exist in nature for actual materials or media; nor can the material constants or properties be functions of frequency. However, ‘apparent’ or ‘equivalent’ properties may be complex and functions of frequency if time-harmonic fields are assumed to exist.