# Poroelasticity-II: On the Equilibrium State of the Fluid-Filled Penetrable Poroelastic Body

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## Abstract

Part-II of the paper titled “Poroelasticity-II” is devoted to defining the joint equilibrium state of a fluid and elastic penetrable material, encapsulated in a rigid volume, Ω. One can ask different questions in respect to such an equilibrium state, such as: (1) In what physical state will the fluid be, if the porous volume of the poroelastic material is filled with a given mass of fluid? (2) What is the capacity of a volume filled with a penetrable sponge in respect to the fluid being in specific state? etc. The point is that, in the frame of the most popular theory of poroelastic materials, Biot’s theory, such a question cannot be addressed because this theory is formulated as a linear theory in the neighborhood of some equilibrium state. In this article, we continue to describe our theory that can be applied to solve such problems. We consider a simplified case of a linear-elastic matrix and a fluid having a Van der Waals equation of state and address the problem of the equilibrium state.

## Keywords

Poroelastic materials Fluid Van der Waals Equilibrium## List of Symbols

- \({\varsigma}\)
Biot’s “fluid content” parameter

**u**Vector of matrix displacement in Biot’s nomenclature

**U**Vector of fluid displacement in Biot’s nomenclature

**ξ**_{S}Vector of matrix displacement

**ξ**_{f}Vector of the fluid displacement

**V**_{f}Fluid particle velocity

- \({\varphi}\)
Porosity as the argument of the equation of state

- \({\varphi_0}\)
Equilibrium porosity of the poroelastic material body filled with a fluid

- \({\varphi_{00}}\)
Porosity of the “empty” poroelastic material body (non-filled with a fluid)

- Ω
Volume of the container

*M*_{f}Mass of fluid

*ρ*_{f}Density of the fluid as the argument of equation of state

*ρ*_{f0}Density of the fluid in the equilibrium state

*ρ*_{S00}Density of matrix material in the “empy” state (not filled with a fluid)

*a*and*b*Van der Waals constants

*p*_{f}Pressure of the fluid

*T*Temperature

- \({\bar{{\varepsilon}}_{ij}}\)
Tensor of the internal strain (See Part I)

- \({\bar{{\varepsilon}}}\)
Bulk part of the tensor of the internal strain

- \({\bar{{\varepsilon}}_0}\)
Bulk part of the tensor of the internal strain in an equilibrium state of fluid-filled poroelastic material material

- \({I_2 = \bar{{\varepsilon}}_{ij}\bar{{\varepsilon }}_{ij}}\)
Second invariant of the strain tensor

*e*_{S}Internal energy of the matrix, which we consider (in neglection of heat effects) a function of \({\bar{{\varepsilon}}_{ij}}\) and \({\varphi }\)

*δ*_{ij}Kronecker delta

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