Understanding the governing equations of one-dimensional finite strain consolidation of saturated soils

Various forms of governing equations are available for one-dimensional finite strain consolidation of saturated soils. They are expressed using either Eulerian or Lagrangian description, each with different dependent variables, including porosity, void ratio, strain, consolidation ratio, and excess pore pressure. To better understand the applicability of these equations, the coordinate transformation relationship between the two descriptions is established in terms of porosity, void ratio and displacement, considering the compressibility of solid. The transformation relationship between the time derivatives of dependent variable in the two descriptions is also established. The applicability of the two descriptions to solving consolidation problems is analyzed. Considering the compressibility and inertia of solid and liquid, a governing equation system for one-dimensional finite strain consolidation in Eulerian description is derived, including continuity equation, momentum balance equation, and Darcy's law. After neglecting the compressibility and inertia of solid and liquid, the system of equations is simplified into a differential equation with a single dependent variable. Through coordinate transformation and time derivative transformation, the consolidation differential equation with Lagrangian description is also obtained. The differential equations are degenerated into various existing forms of finite strain consolidation governing equations, and the applicability of these governing equations is clarified through the basic assumptions involved in the degenerating process.