Supplementary material from "Variational principles of liquid–porous material interactions with applications to approximate solutions and finite-element modelling"

For liquid–porous material interactions (LPMI), three variational principles are developed in Biot's theory of porous materials in association with Darcy's Law. First one takes solid acceleration, and fluid velocity, acceleration, pressure with time derivative as five variables, while second and third, respectively, have three and two variables when Darcy's Law used as variational constrain reducing variables. These principles provide a means to derive approximate solutions and finite-element (FE) models for LPMI. Each LPMI element includes fluid and solid together with no longer necessarily to distinguish fluid domain from solid. Examples illustrate applications to establish FE model for numerical solutions of LPMI. Solution of a one-dimensional problem reveals that LPMI behaves a coupled-frequency separation measured by a defined-coupling factor. Dynamic response explores coupled-dynamic absorber mechanism in LPMI system, from which it is found that the external force at fluid does not contribute to fluid response but fully absorbed by solid local resonance if the force frequency equals the solid natural frequency; while in the reverse case when the force frequency equals the fluid one, the force at solid does not produce the solid response but fully absorbed by fluid local resonance. New findings with defined LPMI coupling factor provide guidelines designing LPMI system with expected dynamic behaviour meeting engineering requirements.