Constitutive relations for wave propagation in a double porosity solids

• 3 Longitudinal and 1 transverse waves in a double porosity solid.
• Constitutive relations corrected to ensure interaction-free reflection.
• Reflection/refraction studied at fluid-porous (im)pervious boundary.

This study considers the propagation of harmonic plane waves in a double porosity solid saturated with a non-viscous fluid. Existence of three longitudinal waves and a transverse wave is explained through the Christoffel equations, which define the phase velocities and the polarizations of constituent particles. Reflection of plane waves is studied at the stress-free plane surface of the composite medium. A numerical example is solved to calculate the partition of incident energy among the reflected waves. Conservation of the incident energy could be achieved only through the share of interaction energy. The presence of interaction energy is something unexpected, when the medium behaves non-dissipative to the propagation of elastic waves. The reason lies in the constitutive relations being used for double porosity medium, which are not symmetric in elastic coupling, as required by the Betti’s reciprocal theorem. Corrections to the constitutive relations are proposed and the correct relations are used to study the reflection and refraction phenomena at the boundaries of double porosity solids. Dispersion in velocity and attenuation is studied for the four attenuated waves in double porosity solid saturated with viscous fluid.