# A medium-independent variational macroscopic theory of two-phase porous media – Part I: Derivation of governing equations and stress partitioning laws

Roberto Serpieri, Francesco Travascio

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Roberto Serpieri, Francesco Travascio. A medium-independent variational macroscopic theory of two-phase porous media – Part I: Derivation of governing equations and stress partitioning laws. 2016. [hal-01281182]

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# A medium-independent variational macroscopic theory of two-phase porous media – Part I: Derivation of governing equations and stress partitioning laws

Roberto Serpieri*<fnref n="1" />, Francesco Travascio<fnref n="2" />

**Keywords:** Terzaghi's law, effective stress, VMTPM, least action principle, liquefaction

## Abstract

A macroscopic continuum theory of two-phase saturated porous media is derived by a purely variational deduction based on the least Action principle. The proposed theory proceeds from the consideration of a minimal set of kinematic descriptors and keeps a specific focus on the derivation of most general *medium-independent* governing equations, which have a form independent from the particular constitutive relations and thermodynamic constraints characterizing a specific medium.

The kinematics of the microstructured continuum theory herein presented employs an *intrinsic/extrinsic* split of volumetric strains, and adopts, as an additional descriptor, the *intrinsic* scalar volumetric strain which corresponds to the ratio between solid true densities before and after deformation.

This work completes the Variational Macroscopic Theory of Porous Media (VMTPM) framework, which is currently limited to the derivation of the momentum balances of the solid phase alone: herein, the derivation of a complete set of momentum balances, inclusive of the momentum balance of the fluid phase, is attained on a purely variational basis.

<footnote n="1">¹Dipartimento di Ingegneria, Università degli Studi del Sannio, Piazza Roma, 21 - I. 82100, Benevento, Italy. e-mail: roberto.serpieri@unisannio.it</footnote>

<footnote n="2">²Biomechanics Research Laboratory, University of Miami, Department of Industrial Engineering, 1251 Memorial Drive, MEB 276, Coral Gables, FL, 33146, USA, e-mail: f.travascio@miami.edu</footnote>

<footnote>*Corresponding author</footnote>
More specifically, the present derivation includes: 1) the general three-dimensional equations which must be applied at the macroscopic boundaries of the mixture, comprehensive of the treatment of surfaces where relative solid-fluid motion is either prevented or allowed; and 2) a rational derivation of medium-independent stress partitioning laws.

The demonstration that VMTPM recovers governing equations and results of consolidated use in poroelasticity, such as Terzaghi's stress partitioning principle and Biot's equation, together with established experimental results of poromechanics, is left to a companion paper.
# 1 Introduction

The mechanics of porous media in multiphase physical system has garnered in years a wide range of applications. Traditionally deployed in the field of soil mechanics [1–3], multiphase continuum poroelasticity has become in the last decades an indispensable theoretical tool in biomechanics (see e.g., [4–7]), and, more recently, also in impact engineering [8], due to its importance for the understanding and prediction of several complex physical phenomena occurring in solids interacting with other phases [9–13].

Given the wide range of applications, poroelastic theories face the challenge to deal with a vast array of microstructural features and properties which determine the mechanical behavior of the various types of porous media investigated. This variety of mechanical features and applications has produced in the decades a body of research literature on continuum modelling of multiphase poroelastic problems which is considerably large, to the extent that a comprehensive overview of it may only be gained by the union of several survey works (see for instance [14–18]).

The many theories so far proposed differ by the axiomatic schemes and/or methodological approaches employed to infer governing equations [19–22]. Disagreement is also found in the mathematical and physical significance of some governing equations. In this respect it is worth to recall the problem of the "missing equation", early pointed out by Truesdell and Noll [23] and generally referred to as *closure problem*, as well as the different identifications proposed by several authors for such closure equations (above all, their questioned constitutive and/or thermodynamic nature), [15, 18, 23, 24].

Theories also differ by the structure of the macroscopic governing PDEs [22, 25–29], as well as by the physical-mathematical, or engineering, definition employed to introduce macroscopic stress measures [10, 16, 21, 30–37]. In this respect, it can be observed that, while the existence of a multiplicity of differentiated approaches for studying a given physical problem can be deemed to be physiological in a mature research field, a widely spread opinion in multiphase poroelasticity research recognizes the lack of unanimous convergence over a set of governing equations, or over a hierarchy of governing equations. This disagreement can be stigmatized by the words that De Boer used in 2005 according to whom "*the necessity to attack the problem of developing a consistent general poroelasticity theory is still existent*" [34], and has been remarked even
more recently [29], leaving the impression that multiphase poroelasticity still remains, in some respects, an 'unfinished chapter' of continuum mechanics. Thus, even the simpler two-phase purely-mechanical problem of poroelasticity can be regarded, in some respects, as a still-open problem.

A key point for organizing and establishing interrelations between existing multiphase poroelasticity theories is the assessment of the availability of *medium-independent* equilibrium equations for multiphase problems. By medium-independent equations, we refer to a set of equations regulating the dynamics of multiphase media standing in a form which is completely independent from the subsequent specification of constitutive relations and thermodynamic constraints characterizing a specific medium. The importance of such an issue can be recognized by tracing a parallel with the hierarchy of equations standardly encountered in single-phase continuum theories of solids mechanics: although the class of linear and nonlinear constitutive responses and microstructural features addressed by the many available theories is also very large, the linear momentum balance equation is ordinarily regarded as an unquestioned *universal* equation which holds irrespective of constitutive and microstructural properties and of thermodynamic constraints. Actually, in single-phase theories, either the mathematical structure of the linear momentum balance is unaffected by microstructural and constitutive features (this is the case, for instance, of standard elastoplasticity [38]), or, in case of theories with a greater microstructural content, the necessity to preserve downward compatibility of equilibrium equations with linear momentum balance is tacitly given for granted [39].

For two-phase continuum poromechanic formulations, a similar consensus over a set of equilibrium equations general enough to describe porous media independent of the specific constitutive, microstructural and thermodynamic properties of media (i.e., compressibility of the constituent phases, porosity, etc.) is not found. For instance, several theories identify the closure equation with the Clausius-Duhem inequality [16, 23, 27, 40], while other theories introduce supplemental equations which have a constitutive character [27, 29, 41]. Also, as observed in [42–44], the stress partitioning problem in two-phase saturated media does not admit a medium-independent solution.

The reason why linear momentum balance equations in single-phase continuum mechanics attain medium independence is that they stem as the simplest possible least-Action conditions
when a purely-macroscopic and purely-variational approach is adopted. These equations are
obtained by considering the *minimal* mechanical description in which displacements are the
sole kinematic descriptor field, and a simplest general first-gradient dependence of the strain
energy is assumed [45–47]. Continuum variational approaches, based on Hamilton least-Action
principle [48], are indeed suitable tools to investigate the medium independence problem in
continuum mechanics, for the main reason that in these approaches the least-Action principle
is the sole primitive mechanical concept invoked when deriving momentum balance equations.
The resulting equations stem univocally and unambiguosly from the kinematic descriptors and
from the form of the Action functional adopted. Hence, when natural deformation descriptors
and strain measures are employed for the kinematics, and the form of the Action functional
is *sufficiently general*, governing equations of maximum generality are expected to be derived
[47]. In addition, continuum variational approaches offer several further advantages. Boundary
conditions are simultaneously derived with bulk field equations [12, 49–52] without requiring
further mechanical considerations or ad-hoc hypotheses.

In the light of the discussion above, the objective of this study is the derivation of a *minimal*
medium-independent two-phase poroelastic framework which any more complex theory should
be downward compatible to. Specifically, following the parallel with the variational deriva-
tion of linear momentum balances in single-phase continuum elasticity, the sought poroelastic
framework should have the following features:

* it should have a purely-mechanical, purely-variational and purely-macroscopic character;

* it should proceed from the consideration of a minimum possible number of kinematic descriptors, which should have a clear physical-mechanical meaning, and their experimental characterization should be possible.

Several two-phase and multi-phase continuum poroelasticity theories have been proposed
whose governing equations at a macroscopic level are based, to different extents, on the ap-
plication of classical variational principles, or on some variants of these principles, or even on
the simple application of some variational concepts [9, 15, 20, 21, 29, 53–65]. Reviews specifi-
cally dedicated to this subject are the one by Bedford and Drumheller [15] and the more recent
one in [18]. As remarked in [18], most multiphase poroelastic formulations adopt additional
descriptors such as the volume fractions and the intrinsic strain. However, even remaining in
the purely variational literature, no unanimous consensus over a *minimal* set of governing equa-
tions is found. Specific debated issues are the well-posedness of the variational statement of the
multiphase problem in presence of constraints such as mass balances, and the assessment of the
physical meaning of stress quantities defined with the aid of Lagrange multipliers in relation to
boundary data and to the macroscopic measurement process [15, 19, 29, 66]. The development
of a constraint-free variational statement of the problem, and of a related suitable kinematical
description of open porous systems in which the fluid can freely flow through the porous solid
matrix, are recognized to be relevant problems, in particular, in [66]. Therein a study is pre-
sented on the variational statement of the two-phase poroelastic problem, and on the resulting
boundary conditions, as determined by the replacement, in the kinematic description, of the
fluid macroscopic placement field with another field, defined in the macroscopic solid reference
configuration, which maps solid material points into points of the fluid reference configuration,
which share the same spatial position at the given time instant.

Most recently, a general variational continuum theory with microstructure of two-phase
poroelasticity has been proposed [11, 42, 67–71]. Peculiar feature of this theory, henceforth
abbreviated in Variational Macroscopic Theory of Porous Media (VMTPM), is the resort to an
*extrinsic/intrinsic* split of volumetric strain measures: VMTPM kinematics includes a scalar field
termed *intrinsic* volumetric strain of the solid phase, which essentially corresponds to the ratio
between 'true' densities of solid before and after deformation; such field is independent from
the primary macroscopic volumetric strain measure, which remains instead ordinarily defined
as the determinant of the macroscopic deformation gradient, and accordingly termed *extrinsic*
volumetric strain. Importantly, in the above mentioned references the variational deduction of
VMTPM field equations was only limited to the derivation of the momentum balances of the
solid phase. The purpose of this work is to complete this framework presenting a more general
multiphase variational poroelastic theory by also including the derivation of the fluid linear mo-
mentum balance on a purely variational basis from the least-Action principle. Thus, we aim to
achieve a purely-variational and purely-macroscopic deduction of all momentum balances for
the two-phase poroelastic problem in a *minimal* medium-independent setting, so that the de-
rived equations hold irrespective of the constitutive response and compressibility of the solid
and fluid phases, as well as of thermodynamic constraints. Furthermore, we intend to obtain, on
a purely-variational basis, the following results:

* a rational derivation of the general three-dimensional equations which must be applied at the macroscopic boundaries of the mixture, comprehensive of the treatment of surfaces where relative solid-fluid motion is either prevented or allowed.

* a rational derivation of the medium-independent stress partitioning laws.

To achieve a most general theory, the theoretical derivation hereby proposed will proceed from
finite-deformations to subsequently obtain small-displacement equations as a special case upon
kinematic linearization. Finally, the medium-independent stress-partitioning laws resulting from
this theory will be examined.

Due to page limitations, some of the intermediate developments required for the computa-
tion of the explicit form of the Euler Lagrange equations have been moved in appendix where
notation conventions, useful identities for differential operations and the list of symbols are also
collected.

Also due to page limitations, a specific assessment of the predictive capabilities of VMTPM
theory to isotropic media is reported in a companion paper. Thereby, the field equations herein
obtained are specialized for linear and nonlinear isotropic media subjected to a comprehensive
variety of loading and drainage conditions. The results obtained in the analyses of the com-
panion paper show that VMTPM recovers governing equations and results of consolidated use
in poroelasticity, such as Terzaghi's stress partitioning principle and Biot's equation, and also
predicts established experimental results of poromechanics.

## 2 Variational formulation

The variational formulation is hereby derived following a purely macroscopic approach and
based on the use of the intrinsic strain among the primary kinematic descriptors. In addition, we
employ the hypothesis of complete saturation of space, and proceed from minimal kinematic and
constitutive assumptions substantially analogous to those employed in [11]. The theory herein
presented enhances the one proposed in [11]: this is a fully variational derivation, in that momen-
tum balance equations of both solid and fluid phases are derived on a purely variational basis.
To achieve improved generality and clarity in the derivation, the formulation is first derived in a finite-deformation framework, and then specialized to infinitesimal configuration changes by applying kinematic linearization.

## 2.1 Basic configuration descriptors

We consider a purely-macroscopic description of the change of configuration, under finite deformations, of a two-phase immiscible mixture made of a porous solid with interconnected cells allowing independent relative motion of an interstitial fluid. Complete saturation conditions are considered for the mixture throughout the deformation process.

The reference configuration of the mixture is defined by the macroscopic smooth reference domain of the mixture, $\Omega_0^{(M)}$, and by two scalar fields defining at the macroscopic level the volume fractions of the solid and of the fluid phase, $\Phi_0^{(s)}: X \in \Omega_0^{(M)} \to \Phi_0^{(s)}, \Phi_0^{(f)}: X \in \Omega_0^{(M)} \to \Phi_0^{(f)}$. For the sake of clarity, the description herein considered is based on purely macroscopic quantities whose relation with their microscopic counterparts is pointed out with reference to a microscale Representative Volume Element (RVE). The reference solid volume fraction, $\Phi_0^{(s)}(X)$ in a point X represents, with reference to a RVE $\Omega_0^{RVE}(X)$ centered in the point X, the ratio $V_{0RVE}^{(s)}(X)/V_{0RVE}(X)$ between the volume $V_{0RVE}^{(s)}(X)$ of the subset $\Omega_0^{RVE}(X) \subset \Omega_0^{RVE}(X)$ occupied by the solid phase and the volume $V_{0RVE}^{(f)}(X)$ of $\Omega_0^{RVE}(X)$. The reference fluid volume fraction $\Phi_0^{(f)} = V_{0RVE}^{(f)}(X)/V_{0RVE}(X)$ is analogously defined, with $V_{0RVE}^{(f)}$ being the volume of the subset $\Omega_0^{RVE}(X) \subset \Omega_0^{RVE}(X)$ containing the fluid. Due to the saturation condition holding also at reference configuration, one has for any $X \in \Omega_0^{(M)}$

$$ \Phi_0^{(s)}(X) + \Phi_0^{(f)}(X) = 1 \qquad (1) $$

The deformed configuration of the mixture is defined by two invertible deformation functions, $\bar{\chi}^{(s)}$ and $\bar{\chi}^{(f)}$, termed solid placement and fluid placement, respectively, $\bar{\chi}^{(s)}: X \in \Omega_0^{(M)} \to X \in \Omega^{(M)}$, $\bar{\chi}^{(f)}: X \in \Omega_0^{(M)} \to X \in \Omega^{(M)}$ with common codomain $\Omega^{(M)}$ representing the space occupied by the mixture in the deformed configuration.

The description of the state of the deformed configuration is completed by the spatial fields of current volume fraction, $\phi_x^{(s)}: X \in \Omega^{(M)} \to \phi^{(s)}$, $\phi_x^{(f)}: X \in \Omega^{(M)} \to \phi^{(f)}$. The lowercase subscripts x are added to mark the spatial character of these fields. Field $\phi_x^{(s)}$, while again intro-
duced on the basis of a purely macroscopic description, is such that its value $\phi_x^{(s)}(\mathbf{x})$ in a point $\mathbf{x}$ is related to the small-scale configuration of the microscale RVE, $\Omega(\mathbf{x})$, in the spatial configuration, centered in $\mathbf{x}$ by $\phi^{(s)}(\mathbf{x}) = V_{RVE}^{(s)}(\mathbf{x})/V_{RVE}(\mathbf{x})$, where $V_{RVE}^{(s)}(\mathbf{x})$ is the volume of the solid subset $\Omega^{\text{RVE}}(\mathbf{x}) \subset \Omega(\mathbf{x})$ and $V_{RVE}(\mathbf{x})$ is the volume of $\Omega(\mathbf{x})$. The counterpart relation for the fluid phase is $\phi^{(f)}(\mathbf{x}) = V_{RVE}^{(f)}(\mathbf{x})/V_{RVE}(\mathbf{x})$ with a completely analogous definition. Also, in the current configuration, volume fractions range between 0 and 1 and the relevant saturation condition similar to (1) reads for any $\mathbf{x} \in \Omega^{(M)}$:

$$ \phi^{(s)}(\mathbf{x}) + \phi^{(f)}(\mathbf{x}) = 1 \qquad (2) $$

The description of volume changes achieved in terms of purely macroscopic fields is now examined. The extrinsic volumetric deformation of the solid phase is introduced as the scalar $\bar{J}^{(s)} = \det \partial \bar{\chi}^{(s)} / \partial \mathbf{X}$. This quantity is purely macroscopic and its use is standard in finite deformation poroelasticity, see for instance [14]. Concerning the interpretation of the relation of $\bar{J}^{(s)}(\mathbf{X})$ with the volume changes of the reference and deformed RVEs associated with a point $\mathbf{X} \in \Omega_0^{(M)}$, as shown in [11], this quantity is the ratio $\bar{J}^{(s)}(\mathbf{X}) = V_{RVE}(\mathbf{x})/V_{0RVE}(\mathbf{X})|_{\bar{\chi}^{(s)}}$, where $V_{RVE}(\mathbf{x})$ is the volume of the microscale deformed RVE $\bar{\chi}^{(s)}(\Omega_0^{\text{RVE}}(\mathbf{X}))$ obtained applying the microscopic deformation $\chi^{(s)}$ to the whole domain $\Omega_0^{\text{RVE}}(\mathbf{X})$ upon performing an extrapolation of $\chi^{(s)}$ from $\Omega_0^{\text{RVE}}(\mathbf{X})$ to the whole set $\Omega_0^{\text{RVE}}(\mathbf{X})$. The extrinsic volumetric deformation of the fluid phase is specularly introduced as the scalar $\bar{J}^{(f)} = \det \partial \bar{\chi}^{(f)} / \partial \mathbf{X}$, and the following relation holds: $\bar{J}^{(f)}(\mathbf{X}) = V_{RVE}(\mathbf{x})/V_{0RVE}(\mathbf{X})|_{\bar{\chi}^{(f)}}$. Trivially, for the inverse mappings $(\bar{\chi}^{(s)})^{-1}, (\bar{\chi}^{(f)})^{-1}$, the extrinsic volume ratios are $V_{0RVE}(\mathbf{X})/V_{RVE}(\mathbf{x})|_{(\bar{\chi}^{(s)})^{-1}} = 1/\bar{J}^{(s)}(\mathbf{X})$ and $V_{0RVE}(\mathbf{X})/V_{RVE}(\mathbf{x})|_{(\bar{\chi}^{(f)})^{-1}} = 1/\bar{J}^{(f)}(\mathbf{X})$.

In [11], the necessity to introduce an additional volumetric deformation measure in order to achieve a complete description of the volume changes of the mixture was highlighted. This is in consideration of the fact that $\bar{J}^{(s)}(\mathbf{X})$ can be different from unity even when the solid porous skeleton is undergoing isochoric deformations. Accordingly, consistent with [11], an additional macroscopic field of intrinsic volumetric strain of the solid phase, $\hat{J}^{(s)}$, independent from $\bar{J}^{(s)}$, is introduced to measure the effective volume changes of the solid phase, $\hat{J}^{(s)} : \mathbf{X} \in \Omega_0^{(M)} \to \hat{J}^{(s)} \in \mathbb{R}$. The relation of $\hat{J}^{(s)}$ with the volumes of the microscale RVE in the reference and deformed configurations is $\hat{J}^{(s)}(\mathbf{X}) = V_{RVE}^{(s)}(\mathbf{x})/V_{0RVE}^{(s)}(\mathbf{X})$ with $\mathbf{x} = \bar{\chi}^{(s)}(\mathbf{X})$. The field
of intrinsic volumetric strain of the fluid phase is introduced in completely specular form by replacing the (s) scripts with (f) in the relations above. The intrinsic volumetric strain is related to the RVE volumes by $\hat{J}^{(f)} = V_{RVE}^{(f)}(\mathbf{x})/V_{0RVE}^{(f)}(\mathbf{X})$. According to the macroscopic description for volume changes so far detailed, a homothetic deformation of the solid phase in a point $\mathbf{X}$ is characterized by the condition $\hat{J}^{(s)}(\mathbf{X}) = \bar{J}^{(s)}(\mathbf{X})$. Moreover, current and reference volume fraction fields are related to extrinsic and intrinsic volume deformations by:

$$ \phi^{(s)}(\mathbf{x}) = \frac{V_{RVE}^{(s)}(\mathbf{x})}{V_{RVE}(\mathbf{x})} = \frac{V_{RVE}^{(s)}(\mathbf{x})}{V_{0RVE}^{(s)}(\mathbf{X})} \frac{V_{0RVE}^{(s)}(\mathbf{X})}{V_{0RVE}(\mathbf{X})} \frac{V_{0RVE}(\mathbf{X})}{V_{RVE}(\mathbf{x})} \bigg|_{(\bar{\chi}^{(s)})^{-1}} = \hat{J}^{(s)}(\mathbf{X})\Phi_{0}^{(s)}(\mathbf{X})\frac{1}{\bar{J}^{(s)}(\mathbf{X})} \quad (3) $$

where the correspondence between $\mathbf{x}$ and $\mathbf{X}$ is defined by the solid placement, viz. $\mathbf{x} = \bar{\chi}^{(s)}(\mathbf{X})$. Analogously, for the fluid phase a relation specular to (3) holds:

$$ \phi^{(f)}(\mathbf{x}) = \frac{V_{RVE}^{(f)}(\mathbf{x})}{V_{RVE}(\mathbf{x})} = \frac{V_{RVE}^{(f)}(\mathbf{x})}{V_{0RVE}^{(f)}(\mathbf{X})} \frac{V_{0RVE}^{(f)}(\mathbf{X})}{V_{0RVE}(\mathbf{X})} \frac{V_{0RVE}(\mathbf{X})}{V_{RVE}(\mathbf{x})} \bigg|_{(\bar{\chi}^{(f)})^{-1}} = \hat{J}^{(f)}(\mathbf{X})\Phi_{0}^{(f)}(\mathbf{X})\frac{1}{\bar{J}^{(f)}(\mathbf{X})} \quad (4) $$

with $\mathbf{x}$ and $\mathbf{X}$ being related by $\mathbf{x} = \bar{\chi}^{(f)}(\mathbf{X})$.

From (3) one can also infer a relation for the operative macroscopic measurement of $\hat{J}^{(s)}$ in a point $\mathbf{X}$:

$$ \hat{J}^{(s)}(\mathbf{X}) = \frac{\bar{J}^{(s)}(\mathbf{X})}{\Phi_{0}^{(s)}(\mathbf{X})} \left[ 1 - \phi^{(f)}(\bar{\chi}^{(s)}(\mathbf{X})) \right] \quad (5) $$

Hence, if the field $\bar{\chi}^{(s)}$ is among the known data, so that $\bar{J}^{(s)}$ is known, the measurement of $\hat{J}^{(s)}$ can be related to the measurement of the porosity $\phi^{(f)}$ which, if the void space is completely interconnected, can also be performed by measuring the fluid volume saturating the void space before and after deformation.

Relations (3) and (4) can be straightforwardly linearized and it can be easily recognized that they correspond to standard kinematic relations also derived in [21] and [67] for infinitesimal deformations. Specifically, recalling from identity (164) reported in the Appendix 4.2 that the relation between an infinitesimal increment of the volumetric extrinsic strain measure $d\bar{J}^{(s)}$ and infinitesimal displacement field $d\bar{\chi}^{(s)}$ is

$$ d\bar{J}^{(s)} = \bar{J}^{(s)} \frac{\partial d\bar{\chi}_{i}^{(s)}}{\partial x_{i}}, \quad (6) $$
the extrinsic spatial infinitesimal volumetric strain measures $\bar{e}^{(s)}$ and the intrinsic spatial infinitesimal volumetric strain measures $\hat{e}^{(s)}$ are defined as:

$$ \bar{e}^{(s)} = \frac{d\bar{J}^{(s)}}{\bar{J}^{(s)}} = \frac{\partial d\bar{\chi}_{i}^{(s)}}{\partial x_{i}}, \qquad \hat{e}^{(s)} = \frac{d\hat{J}^{(s)}}{\hat{J}^{(s)}} \qquad (7) $$

According to these definitions, linearization of the porosity-strain relation (3) is computed as follows:

$$ \begin{aligned} d\phi^{(s)} &= \partial_{(\bar{\chi}^{(s)}, \hat{J}^{(s)})} \phi^{(s)} [d\bar{\chi}^{(s)}, d\hat{J}^{(s)}] \\ &= \Phi_{0}^{(s)} \left( \frac{1}{\bar{J}^{(s)}} \partial_{\hat{J}^{(s)}} \hat{J}^{(s)} [d\hat{J}^{(s)}] + \hat{J}^{(s)} \partial_{\bar{\chi}^{(s)}} \left( \frac{1}{\bar{J}^{(s)}} \right) [d\bar{\chi}^{(s)}] \right) \\ &= \Phi_{0}^{(s)} \left( \frac{1}{\bar{J}^{(s)}} \partial_{\hat{J}^{(s)}} \hat{J}^{(s)} [d\hat{J}^{(s)}] - \hat{J}^{(s)} \frac{1}{(\bar{J}^{(s)})^2} \partial_{\bar{\chi}^{(s)}} \bar{J}^{(s)} [d\bar{\chi}^{(s)}] \right) \\ &= \Phi_{0}^{(s)} \left( \frac{1}{\bar{J}^{(s)}} \hat{e}^{(s)} - \frac{\hat{J}^{(s)}}{\bar{J}^{(s)}} \bar{e}^{(s)} \right) \end{aligned} \qquad (8) $$

When the deformation is infinitesimal, so that $\bar{J}^{(s)} \simeq 1$ and $\hat{J}^{(s)} \simeq 1$, the previous relation yields:

$$ d\phi^{(s)} = \Phi_{0}^{(s)} (\hat{e}^{(s)} - \bar{e}^{(s)}) \qquad (9) $$

which corresponds to Equation (61) derived in [67].

A completely analogous computation of the linearized form of (4) yields for the fluid phase:

$$ d\phi^{(f)} = \Phi_{0}^{(f)} (\hat{e}^{(f)} - \bar{e}^{(f)}) \qquad (10) $$

with definitions for $\bar{e}^{(s)}$ and $\hat{e}^{(s)}$ analogous to those in (7):

$$ \bar{e}^{(f)} = \frac{\partial d\bar{\chi}_{i}^{(f)}}{\partial x_{i}}, \qquad \hat{e}^{(f)} = \frac{d\hat{J}^{(f)}}{\hat{J}^{(f)}} \qquad (11) $$

Equations (3) and (4) are written in terms of relation between fields as follows:

$$ \phi_{\mathbf{x}}^{(s)} \circ \bar{\chi}^{(s)} = \frac{\hat{J}^{(s)}}{\bar{J}^{(s)}} \Phi_{0}^{(s)}, \qquad \phi_{\mathbf{x}}^{(f)} \circ \bar{\chi}^{(f)} = \frac{\hat{J}^{(f)}}{\bar{J}^{(f)}} \Phi_{0}^{(f)} \qquad (12) $$

where an x subscript is used to denote spatial fields, all remaining fields with no subscripts are defined in the reference configuration, and symbol 'o' indicates function composition. From
(12) one infers:

$$ \phi_{\mathbf{x}}^{(s)} = \frac{\hat{J}^{(s)}}{\bar{J}^{(s)}} \Phi_{0}^{(s)} \circ (\bar{\chi}^{(s)})^{-1}, \quad \phi_{\mathbf{x}}^{(f)} = \frac{\hat{J}^{(f)}}{\bar{J}^{(f)}} \Phi_{0}^{(f)} \circ (\bar{\chi}^{(f)})^{-1} \quad (13) $$

whereby the saturation condition (2) is expressed in terms of field relations in the following form termed Finite deformation saturation constraint:

$$ \frac{\hat{J}^{(s)}}{\bar{J}^{(s)}} \Phi_{0}^{(s)} \circ (\bar{\chi}^{(s)})^{-1} + \frac{\hat{J}^{(f)}}{\bar{J}^{(f)}} \Phi_{0}^{(f)} \circ (\bar{\chi}^{(f)})^{-1} = 1 \quad (14) $$

The same saturation condition expressed in (14) can be also written in terms of field values for a space point x in the following way:

$$ \frac{\hat{J}^{(s)}(\mathbf{X}^{(s)})}{\bar{J}^{(s)}(\mathbf{X}^{(s)})} \Phi_{0}^{(s)}(\mathbf{X}^{(s)}) + \frac{\hat{J}^{(f)}(\mathbf{X}^{(f)})}{\bar{J}^{(f)}(\mathbf{X}^{(f)})} \Phi_{0}^{(f)}(\mathbf{X}^{(f)}) = 1 \quad (15) $$

specifying that x is the common image of the reference points $\mathbf{X}^{(s)}$ and $\mathbf{X}^{(f)}$, respectively via $\bar{\chi}^{(s)}$ and $\bar{\chi}^{(f)}$, viz.:

$$ \mathbf{x} = \bar{\chi}^{(s)}(\mathbf{X}^{(s)}) = \bar{\chi}^{(f)}(\mathbf{X}^{(f)}) \quad (16) $$

Relation (14) represents the saturation condition in relation to the volumetric deformation measures. Such relation implies that, under the saturation hypothesis and for a given reference configuration defined by $\Omega_{0}^{(M)}$, $\Phi_{0}^{(s)}$ and $\Phi_{0}^{(f)}$, the quantities $\hat{J}^{(s)}$, $\bar{J}^{(s)}$, $\hat{J}^{(f)}$, and $\bar{J}^{(f)}$ are not independent.

## 2.2 Variational formulation

### 2.2.1 Selection of unconstrained kinematic descriptors

In this subsection, a variational formulation is developed based on the configuration description detailed in Section 2.1. To this end, attention is herein taken in properly selecting a set of independent kinematic descriptor fields which are not constrained to respect further equations of saturation constraints and mass balances. This choice represents a precise element of distinction of the theory presently developed from the variational formulation proposed by Bedford and Drumheller in [21]. In particular, the absence of constraints for the primary descriptors in VMTPM does not require Lagrange multipliers for defining stress quantities. Accordingly,
fields $\bar{\chi}^{(s)}$, $\hat{J}^{(s)}$, $\bar{\chi}^{(f)}$, and $\hat{J}^{(f)}$ cannot be taken altogether as primary kinematic descriptors since quantities $\hat{J}^{(s)}$, $\bar{J}^{(s)}$, $\hat{J}^{(f)}$, and $\bar{J}^{(f)}$ are mutually related by the saturation constraint (14). Several options are available for selecting a suitable subset of three kinematically independent descriptors of the change of configuration of the mixture. Hereby, we choose $\bar{\chi}^{(s)}$, $\bar{\chi}^{(f)}$, and $\hat{J}^{(s)}$. Such a choice retains both descriptors of the solid phase deformation while only the macroscopic displacements are included for the fluid phase. Denoting by $[t_0, t_f]$ the time interval of interest, the domain of these fields is the set $\Omega_0^{(M)} \times [t_0, t_f]$. Once these three fields are specified, the intrinsic strain in a point $x = \bar{\chi}^{(s)}(\mathbf{X}^{(s)}) = \bar{\chi}^{(f)}(\mathbf{X}^{(f)})$, which is the common image of two reference points $\mathbf{X}^{(s)}$ and $\mathbf{X}^{(f)}$ (respectively via $\bar{\chi}^{(s)}$ and $\bar{\chi}^{(f)})$ can be computed on account of (15):

$$ \hat{J}^{(f)}(\mathbf{X}^{(f)}) = \frac{\bar{J}^{(f)}(\mathbf{X}^{(f)})}{\Phi_0^{(f)}(\mathbf{X}^{(f)})} \left( 1 - \Phi_0^{(s)}(\mathbf{X}^{(s)}) \frac{\hat{J}^{(s)}(\mathbf{X}^{(s)})}{\bar{J}^{(s)}(\mathbf{X}^{(s)})} \right) \quad (17) $$

In this way, $\hat{J}^{(f)}$ is treated as a field indirectly related to $\bar{\chi}^{(s)}$, $\bar{\chi}^{(f)}$, and $\hat{J}^{(s)}$ via the saturation constraint. This choice is well posed, and the kinematic independence of $\bar{\chi}^{(s)}$, $\bar{\chi}^{(f)}$, and $\hat{J}^{(s)}$ is preserved only if $\Phi_0^{(f)} \neq 0$. When porosity is null, the solid volume fraction has a unit value. Hence, relation (3) yields coincidence of $\bar{J}^{(s)}$ and $\hat{J}^{(s)}$, so that independence is lost. This condition, which needs special treatment (see Section 3 in [68]) is herein excluded assuming $\Phi_0^{(f)} \neq 0$ over $\Omega_0^{(M)}$. Conversely, the existence of entirely fluid regions of the mixture with $\Phi_0^{(f)} = 1$ is not ruled out, see Figure 1.

Accordingly, the argument functions of the considered Lagrange function $L_0^{(M)}$ are the descriptor space-time fields $\bar{\chi}^{(s)}(\mathbf{X}, t)$, $\bar{\chi}^{(f)}(\mathbf{X}, t)$, and $\hat{J}^{(s)}(\mathbf{X}, t)$ of the mixture, viz.:

$$ L_0^{(M)} = L_0^{(M)}(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}) \quad (18) $$

The use of primary kinematic fields which are deliberately all based in the reference configuration follows a statement of the least-action problem also considered by Leech [72] and Bedford and Drumheller [21], and expresses the Action functional in relation to the set of possible evolution histories of a fixed mass of mixture. This particular choice, besides being compliant with the original application of this principle to Lagrangian systems (which have fixed mass), yields the desirable effects of excluding the necessity to uncouple mass balances from the equilibrium equations to be fulfilled. Actually, the densities in the deformed configuration, $\bar{\rho}^{(s)}$, $\bar{\rho}^{(f)}$, are
secondary variables which are related to the reference densities, $\bar{\rho}_0^{(s)}$, $\bar{\rho}_0^{(f)}$, by:

$$ \bar{\rho}^{(s)} = \frac{\bar{\rho}_0^{(s)}}{\bar{J}^{(s)}}, \qquad \bar{\rho}^{(f)} = \frac{\bar{\rho}_0^{(f)}}{\bar{J}^{(f)}} \qquad (19) $$

and that can, as such, be post-computed once the least-action boundary value problem is solved.

The condition expressed by (17) can be rewritten in terms of a relation of functional dependence between fields. To this end we introduce the definition:

$$ \phi_{\bar{\chi}^{(\beta)}}^{(\alpha)} = \phi_{\bar{x}}^{(\alpha)} \circ \bar{\chi}^{(\beta)} \qquad (20) $$

which represents the volume fraction of phase $\alpha$ in the deformed configuration at the space point $\bar{x}$ corresponding to point $\bar{X}^{(\beta)}$ as a result of the deformation $\bar{\chi}^{(\beta)}$. According to this definition, we have:

$$ \phi_{\bar{\chi}^{(s)}} := \phi_{\bar{x}}^{(s)} \circ \bar{\chi}^{(s)} = \frac{\hat{J}^{(s)}}{\hat{J}^{(f)}} \Phi_{0}^{(s)} \qquad (21) $$

and

$$ \phi_{\bar{\chi}^{(f)}}^{(s)} = \phi_{\bar{x}}^{(s)} \circ \bar{\chi}^{(f)} = \phi_{\bar{\chi}^{(s)}}^{(s)} \circ (\bar{\chi}^{(s)})^{-1} \circ \bar{\chi}^{(f)} \qquad (22) $$

whereby (17) is expressed as the following relation between fields:

$$ \hat{J}_{\text{sat}\bar{\chi}^{(f)}}^{(f)} := \frac{\bar{J}^{(f)}}{\Phi_{0}^{(f)}} \left(1 - \phi_{\bar{\chi}^{(f)}}^{(s)}\right) \qquad (23) $$

### 2.2.2 Mechanical framework and form of the Action functional

The mechanical framework is hereby derived proceeding with a purely variational deduction of the governing equations by following standard methodologies for the application of Hamilton's least Action principle in continuum mechanics [21, 46]. A purely mechanical statement of the problem is first considered, excluding solid-fluid interaction phenomena which require a thermodynamic treatment. Darcy forces describing solid-fluid interactions will be subsequently added to the resulting purely mechanical Euler-Lagrange equations (see Subsection 2.5) once these will have been derived for the purely mechanical problem.

#### Constitutive response - Medium independence

From the constitutive point of view, in order to derive governing equations of broadest gen-
erality, a completely general functional dependence of the Action functional upon the primary descriptors is considered. This choice allows to retrieve the most general set of equations which apply irrespective of the underlying constitutive responses of the solid phase and of the fluid phase (and hence also irrespective of the degree of anisotropy of the solid phase and from the degree of nonlinearity of the solid and fluid constitutive responses). This feature of the formula- tion is referred to as medium independence.

The constitutive relations are defined in terms of functions of strain energy densities per unit reference space $\bar{\psi}_0^{(s)}$ and $\bar{\psi}_0^{(f)}$. These are related to the true, or effective, reference densities of strain energy, $\hat{\psi}_0^{(s)}$ and $\hat{\psi}_0^{(f)}$, by:

$$ \bar{\psi}_0^{(s)} = \Phi_0^{(s)} \hat{\psi}_0^{(s)}, \quad \bar{\psi}_0^{(f)} = \Phi_0^{(f)} \hat{\psi}_0^{(f)} \qquad (24) $$

For both solid and fluid phases, standard local constitutive responses of first-gradient type (i.e. depending upon the first gradient of the macroscopic placement) are considered. Accord- ingly, for the solid phase, a generic dependence of type:

$$ \bar{\psi}_0^{(s)}(\mathbf{X}) = \bar{\psi}_0^{(s)}\left(\mathbf{X}, \left.\frac{\partial \bar{\mathcal{X}}^{(s)}}{\partial \mathbf{X}}\right|_{\mathbf{X}}, \hat{J}^{(s)}(\mathbf{X})\right) \qquad (25) $$

is considered, where dependence upon $\mathbf{X}$ is introduced to address a constitutive response which can be a macroscopically nonhomogeneous function of space. Nonhomogeneity of the consti- tutive response of the solid phase is specifically accounted for, in order to address the possible space nonhomogeneity of $\Phi_0^{(s)}$. In particular, function $\bar{\psi}_0^{(s)}$ must vanish in the points $\mathbf{X} \in \Omega_0^{(M)}$ such that $\Phi_0^{(s)} = 0$, i.e., where the solid is absent and space is completely saturated by the fluid alone.

For the fluid phase, a generic inviscid behavior is considered and it is assumed that in the reference configuration the fluid is in a homogeneous state, so that the following strain energy density can be adopted:

$$ \hat{\psi}_0^{(f)}(\mathbf{X}) = \hat{\psi}_0^{(f)}(\hat{J}^{(f)}(\mathbf{X})) \qquad (26) $$

### Kinetic energy

For the reference field of density of kinetic energy of the solid phase, $\bar{\kappa}_0^{(s)}$, in addition to the quadratic term of translational kinetic energy associated with the solid velocity, $\bar{\nu}^{(s)} =$
$\partial \bar{\chi}^{(s)}(\mathbf{X}, t)/\partial t$, a further microinertia term $\bar{\kappa}_{0 add}^{(s)}$ is considered:

$$ \bar{\kappa}_{0}^{(s)}(\mathbf{X}) = \frac{1}{2} \bar{\rho}_{0}^{(s)} ||\bar{\mathbf{v}}^{(s)}(\mathbf{X})||^{2} + \bar{\kappa}_{0 add}^{(s)} \quad (27) $$

Microinertia terms are essentially introduced to retrieve kinetic additional terms in the governing equations which are comparable to those considered in [20,21,57]. However, it should be remarked that the addition of $\bar{\kappa}_{0 add}^{(s)}$ related terms entails some form of constitutive or microscale assumption. For this reason, their introduction lies outside the sought medium-independent treatment (see the considerations in appendix C of [15], or the deduction of these terms as reported in [11]). For this reason, while the computation of microinertia related terms is included for completeness, the resulting terms will be subsequently removed from a final summary of the purely medium-independent part of the theory. In particular, herein the microinertia function is defined according to the expression employed in [11] so as to vanish in presence of a rate of deformation having in $\mathbf{X}$ the shape of an homothety (i.e., when $\dot{J}^{(s)} = \bar{J}^{(s)}$). To this end, it is set $\bar{\kappa}_{0 add}^{(s)} = \bar{\kappa}_{0 add}^{(s)}(\bar{J}^{(s)}(\mathbf{X}))$ with $\bar{J}^{(s)}(\mathbf{X}) = \bar{J}^{(s)}(\mathbf{X}) - \dot{J}^{(s)}(\mathbf{X})$. For simplicity, also for $\bar{\kappa}_{0 add}^{(s)}$ a quadratic expression is chosen:

$$ \bar{\kappa}_{0 add}^{(s)} = \frac{1}{2} \bar{\rho}_{add.0}^{(s)} (\dot{J}^{(s)})^2 = \frac{1}{2} \bar{\rho}_{add.0}^{(s)} (\bar{J}^{(s)} - \dot{J}^{(s)})^2 \quad (28) $$

where $\bar{\rho}_{add.0}^{(s)}$ is a microinertia density parameter assumed henceforth, for simplicity, a constant reference field.

For the fluid phase, expressions analogous to (27) and (28) are considered:

$$ \bar{\kappa}_{0 add}^{(f)} = \frac{1}{2} \bar{\rho}_{add.0}^{(f)} (\dot{J}^{(f)})^2 = \frac{1}{2} \bar{\rho}_{add.0}^{(f)} (\bar{J}^{(f)} - \dot{J}^{(f)})^2 \quad (29) $$

$$ \bar{\kappa}_{0}^{(f)}(\mathbf{X}) = \frac{1}{2} \bar{\rho}_{0}^{(f)} ||\bar{\mathbf{v}}^{(f)}(\mathbf{X})||^{2} + \bar{\kappa}_{0 add}^{(f)} \quad (30) $$

The remaining choices for the definition of the Lagrange function are the standard ones. Accordingly, denoting by $T_{0}^{(M)}$ the kinetic energy of the mixture and the potential energy by $U_{0}^{(M)}$, we write:

$$ L_{0}^{(M)} = T_{0}^{(M)} - U_{0}^{(M)} \quad (31) $$

The potential energy of the mixture $U_{0}^{(M)}$ is divided into three contributions: the solid phase
strain energy $U_0^{(s)}$, the fluid phase strain energy $U_0^{(f)}$, and the potential energy due to external actions $U_0^{ext}$:

$$U_0^{(M)} = U_0^{(s)} + U_0^{(f)} + U_0^{ext} \qquad (32)$$

According to (25), the functional dependence of the total potential energy of the solid phase turns out to be:

$$U_0^{(s)}(\bar{\chi}^{(s)}, \hat{J}^{(s)}) = \int_{\Omega_0^{(M)}} \bar{\psi}_0^{(s)}(\bar{\chi}^{(s)}, \hat{J}^{(s)}) dV_0 \qquad (33)$$

The functional dependence for the strain energy of the fluid phase $U_0^{(f)}$ is more complex since $\hat{J}^{(f)}$ is not among the primary descriptor fields and this term depends upon all three primary kinematic fields via (23). By combining (23) and (26), $U_0^{(f)}$ is provided by the following integral:

$$U_0^{(f)}(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}) = \int_{\Omega_0^{(M)}} \Phi_0^{(f)} \cdot [\hat{\psi}_0^{(f)} \circ \hat{J}_{sat\bar{\chi}^{(f)}}(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)})] dV_0 \qquad (34)$$

In (34), $\hat{\psi}_0^{(f)}$ is a predetermined constitutive function which does not depend from the primary fields (although the value $\hat{\psi}_0^{(f)}(\mathbf{X})$ attained by $\hat{\psi}_0^{(f)}$ in a point $\mathbf{X}$ depends from the kinetic descriptors $\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}$, via the function combination $\hat{\psi}_0^{(f)} \circ \hat{J}_{sat\bar{\chi}^{(f)}}$). Similarly, $\Phi_0^{(f)}$, as a reference configuration field, does not depend from the kinematic descriptors.

Potential $U_0^{ext}$ is decomposed into the sum of a potential energy term $U_{\Omega_0}^{ext}$, due to external actions associated with the state of the interior points of $\Omega_0^{(M)}$, plus a second term $U_{\partial\Omega_0}^{ext}$ associated with the external actions across the boundary $\partial\Omega_0^{(M)}$, viz.:

$$U_0^{ext} = U_{\Omega_0}^{ext} + U_{\partial\Omega_0}^{ext} \qquad (35)$$

where

$$U_{\Omega_0}^{ext} = \int_{\Omega_0^{(M)}} \bar{\psi}_{0ext}^{(s)}(\bar{\chi}^{(s)}) dV_0 + \int_{\Omega_0^{(M)}} \bar{\psi}_{0ext}^{(f)}(\bar{\chi}^{(f)}) dV_0 \qquad (36)$$

and

$$U_{\partial\Omega_0}^{ext} = \int_{\partial\Omega_0^{(M)}} \bar{\psi}_{0ext}^{(\partial)} dA_0 \qquad (37)$$

The kinetic energy of the mixture is split into solid ($T_0^{(s)}$) and fluid ($T_0^{(f)}$) contributions, viz.:

$$T_0^{(M)} = T_0^{(s)} + T_0^{(f)} \qquad (38)$$
with

$$T_0^{(s)}(\bar{\chi}^{(s)}, \hat{J}^{(s)}) = \int_{\Omega_0^{(M)}} \bar{\kappa}_0^{(s)}(\bar{\chi}^{(s)}, \hat{J}^{(s)}) dV_0 \quad (39)$$

$$T_0^{(f)}(\bar{\chi}^{(f)}) = \int_{\Omega_0^{(M)}} \bar{\kappa}_0^{(f)}(\bar{\chi}^{(f)}, \hat{J}^{(f)}) dV_0 \quad (40)$$

where $\bar{\kappa}_0^{(s)}$ and $\bar{\kappa}_0^{(f)}$ are specified by (27) and (30).

### External volume forces

In the reference configuration, the external volume forces, $\bar{b}_0^{(s,ext)}$ and $\bar{b}_0^{(f,ext)}$, are related to the external potential energy by:

$$\bar{b}_0^{(s,ext)} := -\frac{\partial \bar{\psi}_{0ext}^{(s)}}{\partial \bar{\chi}^{(s)}}, \quad \bar{b}_0^{(f,ext)} := -\frac{\partial \bar{\psi}_{0ext}^{(f)}}{\partial \bar{\chi}^{(f)}} \quad (41)$$

Spatial fields of external volume forces $\bar{b}_0^{(s,ext)}$ and $\bar{b}_0^{(f,ext)}$ are provided by the push-forward of the vector fields $\bar{b}_0^{(s,ext)}$ and $\bar{b}_0^{(f,ext)}$:

$$\bar{b}^{(s,ext)} = \frac{1}{\bar{J}^{(s)}} \bar{b}_{0}^{(s,ext)}, \quad \bar{b}^{(f,ext)} = \frac{1}{\bar{J}^{(f)}} \bar{b}_{0}^{(f,ext)} \quad (42)$$

### Definition of internal stress measures

The definitions of stress measures are introduced by an extension to the present two-phase framework of the standard mathematical definitions employed in the context of finite-deformation elasticity for single-continuum problems.

Since the theory herein described is purely mechanical, finite stress measures are defined in the usual form in terms of work-association as Lie derivatives [73, 74] of the density of strain energy of the solid and the fluid phase with respect to the associated strain measures. These definitions are hereby given. In the next subsection, their relation with the Cauchy stress tensor of the solid phase and the interstitial fluid pressure $p$ are recalled.

For the solid phase, denoting by $\bar{F}^{(s)} = \frac{\partial \bar{\chi}^{(s)}}{\partial \bar{X}}$ the deformation gradient, the work-conjugate stress measures are:

$$\check{P}_{iK}^{(s)} = \frac{\partial \bar{\psi}_{0}^{(s)}}{\partial \bar{F}_{iK}^{(s)}}, \quad \hat{\Pi}^{(s)} = -\frac{\partial \bar{\psi}_{0}^{(s)}}{\partial \hat{J}^{(s)}} \quad (43)$$

The tensor $\check{P}^{(s)}$ is a two-point stress tensor termed *extrinsic first Piola-Kirchhoff stress tensor*. The scalar quantity $\hat{\Pi}^{(s)}$ is the material (i.e., reference) stress measure work-associated with
$\hat{J}^{(s)}$, denominated (reference) intrinsic solid pressure. The negative sign in the definition of $\hat{\Pi}^{(s)}$ is purposefully introduced to treat the stress measure associated with intrinsic strains as a pressure quantity in a way similar to the treatment employed in [68], limited therein to the context of infinitesimal deformations.

The spatial counterparts of (43) are the result of push-forward operations of the quantities in (43) along the deformed solid configuration defined by the solid placement $\bar{\chi}^{(s)}$:

$$ \tilde{\sigma}_{ij}^{(s)} = \frac{1}{\bar{J}^{(s)}} \tilde{P}_{iK}^{(s)} \frac{\partial \bar{\chi}_{j}^{(s)}}{\partial X_{K}} = \frac{1}{\bar{J}^{(s)}} \frac{\partial \bar{\psi}_{0}^{(s)}}{\partial \bar{F}_{iK}^{(s)}} \frac{\partial \bar{\chi}_{j}^{(s)}}{\partial X_{K}} \quad (44) $$

$$ \hat{p}^{(s)} = \frac{1}{\bar{J}^{(s)}} \hat{\Pi}^{(s)} = - \frac{1}{\bar{J}^{(s)}} \frac{\partial \bar{\psi}_{0}^{(s)}}{\partial \hat{J}^{(s)}} \quad (45) $$

and are the spatial forms of the extrinsic stress tensor and of the intrinsic solid pressure, respectively.

The scalar stress measure for the fluid phase are similarly defined as the Lie derivatives of the strain energy of the fluid phase with respect to the primary strain measure of the fluid phase $\hat{J}^{(f)}$. Two alternate stress measures of the fluid phase (in pressure form) associated with the reference configuration are the following:

$$ \hat{\Pi}^{(f)} = - \frac{\partial \hat{\psi}_{0}^{(f)}}{\partial \hat{J}^{(f)}}, \qquad \hat{\Pi}^{(f)} = - \frac{\partial \bar{\psi}_{0}^{(f)}}{\partial \hat{J}^{(f)}} \quad (46) $$

which, owing to (24), are related by $\hat{\Pi}^{(f)} = \Phi_{0}^{(f)} \hat{\Pi}^{(f)}$. The spatial counterparts of (46) are again provided by a push forward operation:

$$ \hat{p}^{(f)} = \frac{1}{\hat{J}^{(f)}} \hat{\Pi}^{(f)}, \qquad \hat{p}^{(f)} = \frac{1}{\hat{J}^{(f)}} \hat{\Pi}^{(f)} \quad (47) $$

## Mechanical interpretation of stress measures

Within a purely mechanical formulation, the use of definitions (44), (45) and (47), based on work association with primary strain measures and on the concept of Lie derivative, represent a standard in solid continuum mechanics for defining stress measures. Such definitions, which are self-consistent in that they require no additional mechanical arguments, ensure that all stress measures above introduced are physically founded and mathematically well-posed [73].
It is however convenient to report some additional side-considerations in order to gain physi-cal and engineering insight on the mechanical meaning of the stress quantities above introduced to elucidate their relation with the standard notions of (macroscopic) Cauchy stress tensor for the solid phase $\sigma^{(s)}$, and of interstitial fluid pressure $p$.

As a general consideration, insights on $\hat{\Pi}^{(f)}$, $\hat{\Pi}^{(s)}$, and $\check{P}$ are obtained by examining the role of these quantities in the momentum balance equations and boundary conditions derived next (see Section 2.4), and their relation with boundary data (78). In particular, in the companion paper, a thorough mechanical interpretation of internal stress quantities is gained by applying the governing equations and boundary conditions in the analysis of experimental tests including a comprehensive set of loading and drainage conditions at the boundary. Such scenarios provide explicit relations with the primary measured physical stress and loading quantities during the experiments (i.e., the macroscopic strain, the fluid pressure, and the stress at the load cell).

It is however possible to anticipate that in a point $x$ the pressure quantity $\hat{p}^{(f)}$ corresponds to the interstitial fluid pressure $p$ under the reasonable hypothesis that the state of the interstitial fluid is microscopically uniform at such point. The term 'microscopically uniform' in a point $x$ refers to the condition that the state of the fluid has negligible fluctuations in the interstitial space of the deformed RVE $\Omega^{\text{RVE}(f)}(x)$ centered in $x$. In this condition $\psi^{(f)}$ is uniform in $\Omega^{\text{RVE}(f)}(x)$, and the microscale Jacobian $J^{(f)}$ and microscale strain energy density $\psi_0^{(f)}$ are uniform inside $\Omega_0^{\text{RVE}(f)}(\mathbf{X})$. As a consequence of such microscale uniformity, macroscopic and microscale quantities coincide inside $\Omega_0^{\text{RVE}(f)}(\mathbf{X})$:

$$ \hat{J}^{(f)} = J^{(f)}, \quad \hat{\psi}_{0}^{(f)} = \psi_{0}^{(f)} \qquad (48) $$

where the left hand sides are macroscopic quantities, and the right hand sides are microscale quantities. The direct consequence of (48) is that the macroscopic fluid pressure $\hat{p}^{(f)}$, defined as:

$$ \hat{p}^{(f)} = -\frac{1}{\hat{J}^{(f)}} \frac{\partial \hat{\psi}_{0}^{(f)}}{\partial \hat{J}^{(f)}} = \frac{1}{\hat{J}^{(f)}} \hat{\Pi}^{(f)} \qquad (49) $$

and the interstitial fluid pressure in the interior points of $\Omega_0^{\text{RVE}(f)}(\mathbf{X})$, defined instead as:

$$ p = -\frac{1}{J^{(f)}} \frac{\partial \psi_{0}^{(f)}}{\partial J^{(f)}} \qquad (50) $$
become coincident, viz.:

$$ \hat{p}^{(f)} \equiv p. \qquad (51) $$

On account of the consequences of microscopic uniformity, quantities $\hat{p}^{(f)}$ and $p$ are both simply referred to as 'fluid pressure'.

Also, the relation of the extrinsic stress tensor $\tilde{\sigma}^{(s)}$ with the ordinary Cauchy stress tensor $\sigma^{(s)}$ can be elucidated based on considerations analogous to those reported in [42] for the special case of infinitesimal kinematics: within VMTPM, $\tilde{\sigma}^{(s)}$ is mechanically defined by (44) based on a more general kinematic definition where the deformation state in a point is defined by the couple $(\bar{\mathbf{F}}^{(s)}, \hat{J}^{(s)})$. Accordingly, a computation of the partial derivative $\frac{\partial \bar{\psi}_{0}^{(s)}}{\partial \bar{\mathbf{F}}^{(s)}}$ in (44) is carried out, as such, by keeping fixed the other degree of freedom $\hat{J}^{(s)}$. The extrinsic stress tensor $\tilde{\sigma}^{(s)}$ is thus recognized to be work-associated with an infinitesimal volumetric deformation mode where no intrinsic volume variation of the solid phase takes place, i.e. with an isochoric strain mode. For this reason, the alternative denomination of isochoric stress tensor for identifying $\tilde{\sigma}^{(s)}$ is used in [42].

As observed in [42], in absence of interstitial fluid (i.e., when the fluid pressure is null $p = 0$), the extrinsic stress tensor $\tilde{\sigma}^{(s)}$ corresponds to the standard notion of macroscopic Cauchy stress tensor $\sigma^{(s)}$ for porous media with voids. This is recognized by conveniently anticipating equations (120), (122), and (123) (rationally deduced in the sequel) which, for $p = 0$, specialize to:

$$ p = 0 \qquad \frac{\partial \tilde{\sigma}_{ij}^{(s)}}{\partial x_j} + \bar{b}_i^{(s,ext)} = 0, \qquad \hat{p}^{(s)} = 0, \qquad \tilde{\sigma}^{(s)} \mathbf{n} = \mathbf{t}^{(ext)} \qquad (52) $$

Relation (52) states that, when $p = 0$, tensor $\tilde{\sigma}^{(s)}$ has to formally satisfy the same interior and boundary equations pertinent to the standard notion of Cauchy stress, and makes these two tensors coincident, viz., $\tilde{\sigma}^{(s)} \equiv \sigma^{(s)}$, in conditions of null fluid pressure.

### Boundary conditions

The whole boundary of the mixture $\partial\Omega_{0}^{(M)}$ is assumed impermeable consistently with the above stated choice of circumscribing a fixed mass of mixture in the statement of the least Action principle, and bilateral boundary conditions are considered. Accordingly, the solid and fluid macroscopic placements are constrained to be coincident over $\partial\Omega_{0}^{(M)}$, viz.:

$$ \bar{\chi}^{(s)} = \bar{\chi}^{(f)}, \qquad \text{over } \partial\Omega_{0}^{(M)} \qquad (53) $$
Denoting by $\bar{\chi}^{(u)}$ the common value of the placement at the boundary ($\bar{\chi}^{(u)} = \bar{\chi}^{(s)} = \bar{\chi}^{(f)}$), the vector field of external tractions over $\partial\Omega_0^{(M)}$ in the reference configuration is defined as:

$$ t_0^{(ext)} := - \frac{\partial \bar{\psi}_{0ext}^{(\partial)}}{\partial \bar{\chi}^{(u)}} \qquad (54) $$

where the push forward of $t_0^{(ext)}$ is the spatial traction

$$ t^{(ext)} = \left( \frac{dA_0}{dA} \right) \bar{t}_0^{(s)(ext)} \qquad (55) $$

To address a variety of relevant boundary conditions, a specific partition, $\Omega_0^{(M)} = \Omega_0^{(s)} \cup \Omega_0^{(f)}$, is considered in the reference configuration of $\Omega_0^{(M)}$ into two subsets: a 'fluid-saturated solid porous' (reference) subset $\Omega_0^{(s)}$ characterized by $\Phi_0^{(s)} \neq 0$, and a (reference) fluid subset $\Omega_0^{(f)}$ with $\Phi_0^{(s)} = 0$ (i.e., where the solid phase is absent and space is completely saturated by the fluid phase alone). A sketch of this partition is shown in Figure 1 with the corresponding partition in the deformed configuration $\Omega^{(M)}$. We assume that both $\Omega_0^{(s)}$ and $\Omega_0^{(f)}$ are simply connected regular domains with piecewise smooth boundaries $\partial\Omega_0^{(s)}$ and $\partial\Omega_0^{(f)}$. Also, in order to address a variety of boundary conditions and interface conditions, we assume non-null the intersection $\partial\Omega_0^{(MU)} = \partial\Omega_0^{(M)} \cap \partial\Omega_0^{(s)}$, as shown in Figure 1. The remaining part of the solid boundary $\partial\Omega_0^{(s)}$ surface, which does not belong to $\partial\Omega_0^{(M)}$, $S_0^{(sf)} = \partial\Omega_0^{(s)} \setminus \partial\Omega_0^{(MU)}$, is termed free solid-fluid macroscopic interface (see dashed line in Figure 1), with its deformed counterpart $S^{(sf)}$. The term 'interface' is purposefully adopted for $S_0^{(sf)}$ considering that, since this surface does not belong to $\partial\Omega_0^{(s)}$, it is not a boundary of the physical system of the mixture.

It is important to remark that, since relation (53) is a constraint, on the boundary $\partial\Omega_0^{(M)}$ where it applies, the virtual variations of placements $\delta\bar{\chi}^{(s)}$ and $\delta\bar{\chi}^{(f)}$ cannot be independently assigned but must satisfy the relation:

$$ \delta\bar{\chi}^{(s)} = \delta\bar{\chi}^{(f)}, \quad \text{over } \partial\Omega_0^{(M)} \qquad (56) $$

On this basis, boundary conditions of displacement-type and of stress-type are standardly applied. In particular, when displacements are prescribed throughout $\partial\Omega_0^{(M)}$ in all cartesian
<figure><img src="image_2.png" /><figcaption>Figure 1: Sketch of the partition of the macroscopic mixture domains into subsets. $\Omega_0^{(M)}$: reference domain of the mixture; $\Omega^{(M)}$: deformed domain of the mixture; $\Omega_0^{(s)}$: macroscopic reference domain of the solid subset ($\Phi_0^{(s)} \neq 0$); $\Omega_0^{(f)}$: macroscopic domain entirely occupied by the fluid at reference configuration ($\Phi_0^{(s)} = 0$); $\partial\Omega_0^{(M)}$: mixture boundary at reference configuration; $\partial\Omega_0^{(MU)}$: part of the boundary $\partial\Omega_0^{(M)}$ which also belongs to the boundary $\partial\Omega_0^{(s)}$; $S_0^{(sf)}$: free solid-fluid macroscopic interface. Omitted '0' subscripts denote the corresponding current configurations.</figcaption></figure>

directions, these conditions read:

$$ \bar{\chi}^{(s)} = \bar{\chi}^{(f)} = \bar{\chi}^{(u)}, \quad \delta\bar{\chi}^{(s)} = \delta\bar{\chi}^{(f)} = \mathbf{o} \qquad (57) $$

and, when boundary conditions of traction type are prescribed in all cartesian directions, these read:

$$ \bar{\chi}^{(s)} = \bar{\chi}^{(f)}, \quad \delta\bar{\chi}^{(s)} = \delta\bar{\chi}^{(f)} = \delta\bar{\chi}^{(u)}, \quad \mathbf{t}_0^{(ext)} = \bar{\mathbf{t}}_0^{(ext)} \qquad (58) $$

where $\bar{\mathbf{t}}_0^{(ext)}$ is the prescribed traction vector field. A more articulated specification of boundary conditions along each cartesian direction can be also introduced in the customary way by considering three partitions $\partial\Omega_0^{(M)} = {}^d\partial\Omega_{0i}^{(M)} \cup {}^s\partial\Omega_{0i}^{(M)}$ per each i-th Cartesian direction ($i = 1, 2, 3$), where ${}^d\partial\Omega_{0i}^{(M)}$ is the subset where boundary conditions of displacement type are
applied along the i-th Cartesian direction:

$$ \bar{\chi}_i^{(s)} = \bar{\chi}_i^{(f)} = \bar{\chi}_i^{(u)}, \quad \delta\bar{\chi}_i^{(s)} = \delta\bar{\chi}_i^{(f)} = 0 \quad \text{over } d\partial\Omega_{0i}^{(M)} \quad (59) $$

and $s\partial\Omega_{0i}^{(M)}$ is the subset of stress-type boundary conditions for the i-th Cartesian direction:

$$ \bar{\chi}_i^{(s)} = \bar{\chi}_i^{(f)}, \quad \delta\bar{\chi}_i^{(s)} = \delta\bar{\chi}_i^{(f)} = \delta\bar{\chi}_i^{(u)}, \quad t_{0i}^{\text{ext}} = \bar{t}_{0i}^{\text{ext}} \text{ over } s\partial\Omega_{0i}^{(M)} \quad (60) $$

where $\bar{\chi}_i^{(u)}$ and $\bar{t}_{0i}^{\text{ext}}$ are the scalar fields of boundary data, defined over $d\partial\Omega_{0i}^{(M)}$ and $s\partial\Omega_{0i}^{(M)}$, respectively.

## 2.3 Integral equations

*General expression of the least Action condition*

Proceeding from the continuum description detailed in Section 2.2.1, the general condition of least Action, defined as:

$$ \delta \int_{t_0}^{t_f} L_0^{(M)}(t) \, dt = 0 \quad (61) $$

is more explicitly stated on account of (173) (see Appendix 4.3) in the following form:

$$ \begin{aligned} & \frac{d}{dt} \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] - \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] \\ & + \frac{d}{dt} \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] - \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] \\ & + \frac{d}{dt} \partial_{\hat{j}^{(s)}} L_0^{(M)} [\delta \hat{j}^{(s)}] - \partial_{\hat{j}^{(s)}} L_0^{(M)} [\delta \hat{j}^{(s)}] \\ & + \partial_{\bar{\chi}^{(u)}} U_{\partial\Omega_0}^{\text{ext}} [\delta \bar{\chi}^{(u)}] = 0 \end{aligned} \quad (62) $$

For simplicity, in the following, only conditions of complete stress-type (58) are considered over the whole boundary, having in particular for displacements and virtual displacements:

$$ \bar{\chi}^{(s)} = \bar{\chi}^{(f)}, \quad \delta\bar{\chi}^{(s)} = \delta\bar{\chi}^{(f)} = \delta\bar{\chi}^{(u)}, \quad \text{over } \partial\Omega_0^{(M)} \quad (63) $$
Based on the definitions of Section 2.2.1, the terms in (62) are split into:

$$
\begin{align}
-\partial_{\bar{\chi}^{(s)}} L_0^{(M)} &= \partial_{\bar{\chi}^{(s)}} U_0^{(M)} = \partial_{\bar{\chi}^{(s)}} U_0^{(s)} + \partial_{\bar{\chi}^{(s)}} U_0^{(f)} + \partial_{\bar{\chi}^{(s)}} U_0^{\text{ext}} \nonumber \\
-\partial_{\bar{\chi}^{(f)}} L_0^{(M)} &= \partial_{\bar{\chi}^{(f)}} U_0^{(M)} = \partial_{\bar{\chi}^{(f)}} U_0^{(s)} + \partial_{\bar{\chi}^{(f)}} U_0^{(f)} + \partial_{\bar{\chi}^{(f)}} U_0^{\text{ext}} \tag{64} \\
-\partial_{j^{(s)}} L_0^{(M)} &= \partial_{j^{(s)}} U_0^{(M)} = \partial_{j^{(s)}} U_0^{(s)} + \partial_{j^{(s)}} U_0^{(f)} + \partial_{j^{(s)}} U_0^{\text{ext}} \nonumber
\end{align}
$$

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}_i} L_0^{(M)} [\delta \bar{\chi}_i^{(s)}] &= \frac{d}{dt} \partial_{\bar{\chi}_i} T_0^{(s)} [\delta \bar{\chi}_i^{(s)}] \nonumber \\
\frac{d}{dt} \partial_{\bar{\chi}_i} L_0^{(M)} [\delta \bar{\chi}_i^{(f)}] &= \frac{d}{dt} \partial_{\bar{\chi}_i} T_0^{(f)} [\delta \bar{\chi}_i^{(f)}] \tag{65} \\
\frac{d}{dt} \partial_{\hat{j}^{(s)}} L_0^{(M)} [\delta \hat{J}^{(s)}] &= \frac{d}{dt} \partial_{\hat{j}^{(s)}} T_0^{(s)} [\delta \hat{J}^{(s)}] + \frac{d}{dt} \partial_{\hat{j}^{(s)}} T_0^{(f)} [\delta \hat{J}^{(f)}] \nonumber
\end{align}
$$

The computation of each of the terms on the right hand sides of (64) and (65) is reported
in Appendix 4.4. In particular, a key development for obtaining the explicit form of the Euler-
Lagrange equations is the computation of the variation $\delta U_0^{(f)}$ and of the associated variation of
intrinsic volumetric fluid strain $\delta \hat{J}_{\text{sat}\bar{\chi}^{(f)}}$ (see (34)). The final computed expressions of these
two terms are hereby recalled from Appendix 4.4:

$$
\begin{align}
\delta \hat{J}^{(f)} &= \partial_{(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, j^{(s)})} \hat{J}_{\text{sat}\bar{\chi}^{(f)}}^{(f)} [\delta \bar{\chi}^{(s)}, \delta \bar{\chi}^{(f)}, \delta \hat{J}^{(s)}] \nonumber \\
&= -\frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \left[ -\phi_{\bar{\chi}^{(s)}}^{(s)} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \circ (\bar{\chi}^{(s)})^{-1} \circ \bar{\chi}^{(f)} \right] \nonumber \\
&\quad -\frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \left[ -\frac{\partial \phi_{\bar{\chi}^{(s)}}^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \delta \bar{\chi}_i^{(s)} \circ \bar{\chi}^{(f)} \right] \nonumber \\
&\quad +\frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \left[ \phi_{\bar{\chi}^{(f)}}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \frac{\partial \delta \bar{\chi}_i^{(f)}}{\partial X_J} \right] \nonumber \\
&\quad +\frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \left[ \frac{\partial \phi_{\bar{\chi}^{(f)}}^{(f)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \delta \bar{\chi}_i^{(f)} \right] \nonumber \\
&\quad -\frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \delta \hat{J}^{(s)} \tag{66}
\end{align}
$$

It should be noted that the expression (66) provides a relation written in terms of fields placed
in the reference configuration $\Omega_0^{(M)}$. The same relation expressed by (66) is more concisely
written in terms of spatial variables associated with the space point x, viz.:

$$ \delta \hat{J}^{(f)} = \frac{\bar{J}^{(f)}}{\Phi_{0}^{(f)}} \left( \phi^{(s)} \frac{\partial \delta \bar{\chi}_{i}^{(s)}}{\partial x_{i}} + \frac{\partial \phi^{(s)}}{\partial x_{i}} \delta \bar{\chi}_{i}^{(s)} + \phi^{(f)} \frac{\partial \delta \bar{\chi}_{i}^{(f)}}{\partial x_{i}} + \frac{\partial \phi^{(f)}}{\partial x_{i}} \delta \bar{\chi}_{i}^{(f)} - \frac{\Phi_{0}^{(s)}}{\bar{J}^{(s)}} \delta \hat{J}^{(s)} \right) \quad (67) $$

Accounting in (67) for the product derivatives rule, one obtains the *Linearized Saturation Constraint* for virtual deformations associated with a finite deformation, see [68] and [42]. This constraint expresses the relation that virtual infinitesimal variations must fulfill in order to preserve the condition of space saturation (14):

$$ \Phi_{0}^{(s)} \frac{\delta \hat{J}^{(s)}}{\hat{J}^{(s)}} + \Phi_{0}^{(f)} \frac{\delta \hat{J}^{(f)}}{\hat{J}^{(f)}} = \frac{\partial \phi^{(s)} \delta \bar{\chi}_{i}^{(s)}}{\partial x_{i}} + \frac{\partial \phi^{(f)} \delta \bar{\chi}_{i}^{(f)}}{\partial x_{i}} \quad (68) $$

When linearization is performed in the reference configuration, one can refer to the relation (7), written for virtual deformations, and apply $\Phi_{0}^{(s)} \simeq \phi^{(s)}$ and $\Phi_{0}^{(f)} \simeq \phi^{(f)}$, obtaining:

$$ \phi^{(s)} \delta \hat{e}^{(s)} + \phi^{(f)} \delta \hat{e}^{(f)} = \frac{\partial \phi^{(s)} \delta \bar{\chi}_{i}^{(s)}}{\partial x_{i}} + \frac{\partial \phi^{(f)} \delta \bar{\chi}_{i}^{(f)}}{\partial x_{i}} \quad (69) $$

In particular, when the fields $\phi^{(s)}$ are $\phi^{(f)}$ are uniform in space, the relation (69) specializes to:

$$ \phi^{(s)} \delta \hat{e}^{(s)} + \phi^{(f)} \delta \hat{e}^{(f)} = \phi^{(s)} \frac{\partial \delta \bar{\chi}_{i}^{(s)}}{\partial x_{i}} + \phi^{(f)} \frac{\partial \delta \bar{\chi}_{i}^{(f)}}{\partial x_{i}} \quad (70) $$

Relation (70) corresponds to the form of the saturation constraint obtained in [42, 68] for infinitesimal deformations, and has been previously reported in a form combined with mass balances also by Bedford and Drumheller [21]. It is worth remarking that the derivation herein reported for this condition elucidates its pure kinematic significance independent of mass balances. In this respect, it is also worth observing that equation (70) is also derived by taking the sum of (9) and (10).

The variation of the fluid potential energy, expressed as:

$$ \delta U_{0}^{(f)} = \partial_{\hat{J}^{(f)}} U_{0}^{(f)} \left[ \delta \hat{J}^{(f)} \right] = \partial_{\left(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}\right)} U_{0}^{(f)} \left[ \delta \bar{\chi}^{(s)}, \delta \bar{\chi}^{(f)}, \delta \hat{J}^{(s)} \right] \quad (71) $$

is computed differentiating relation (34) via the rule for the variation of function composition,
see (154), (177) and (178). Its explicit expression is:

$$
\begin{align}
\delta U_0^{(f)} &= \int_{\Omega_0^{(M)}} \overbrace{\frac{\partial \hat{\psi}_0^{(f)}}{\partial \hat{J}^{(f)}}}^{-\hat{\Pi}^{(f)}} \partial(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}) \hat{J}^{(f)}_{sat\bar{\chi}^{(f)}} [\delta\bar{\chi}^{(s)}, \delta\bar{\chi}^{(f)}, \delta\hat{J}^{(s)}] dV_0 \nonumber \\
&= -\int_{\Omega_0^{(M)}} \hat{\Pi}^{(f)} \bar{J}^{(f)} \phi_{\bar{\chi}^{(s)}} \frac{\partial \delta\bar{\chi}_i^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} dV_0 \nonumber \\
&\quad -\int_{\Omega_0^{(M)}} \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(s)}}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \delta\bar{\chi}_i^{(s)} dV_0 \tag{72} \\
&\quad -\int_{\Omega_0^{(M)}} \hat{\Pi}^{(f)} \bar{J}^{(f)} \phi_{\bar{\chi}^{(f)}} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \frac{\partial \delta\bar{\chi}_i^{(f)}}{\partial X_J} dV_0 \nonumber \\
&\quad -\int_{\Omega_0^{(M)}} \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(f)}}}{\partial X_J} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \delta\bar{\chi}_i^{(f)} dV_0 \nonumber \\
&\quad + \int_{\Omega_0^{(M)}} \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \delta\hat{J}^{(s)} dV_0 \nonumber
\end{align}
$$

Collecting (72) with the required expressions reported in Appendix 4.4 for $\delta U_0^{(s)}$ and $\delta U_0^{ext}$, and for the remaining terms entering (65), the following explicit expression of equation (62) is obtained:

$$
\begin{align}
& \int_{\Omega_0^{(M)}} \left[ \check{P}_{iJ} - \bar{J}^{(f)} \phi_{\bar{\chi}^{(s)}}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} + \bar{\rho}_{\text{add.0}}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{s}} - \hat{J}^{\ddot{s}}) \right] \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left( -\bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(s)}}^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} - \bar{b}_{0i}^{(s,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{\ddot{(s)}} \right) \delta \bar{\chi}_i^{(s)} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{J}^{(f)} \phi_{\bar{\chi}^{(f)}}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} + \bar{\rho}_{\text{add.0}}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{f}} - \hat{J}^{\ddot{f}}) \right] \frac{\partial \delta \bar{\chi}_i^{(f)}}{\partial X_J} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(f)}}^{(f)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} - \bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(f)} \bar{\chi}_i^{\ddot{(f)}} \right] \delta \bar{\chi}_i^{(f)} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\hat{\Pi}^{(s)} + \frac{\bar{J}^{(f)}}{\bar{J}^{(s)}} \Phi_0^{(s)} \hat{\Pi}^{(f)} - \bar{\rho}_{\text{add.0}}^{(s)} (\bar{J}^{\ddot{s}} - \hat{J}^{\ddot{s}}) + \frac{\bar{J}^{(f)} \Phi_0^{(s)}}{\Phi_0^{(f)} \bar{J}^{(s)}} \bar{\rho}_{\text{add.0}}^{(f)} (\bar{J}^{\ddot{f}} - \hat{J}^{\ddot{f}}) \right] \delta \hat{J}^{(s)} dV_0 \nonumber \\
& - \int_{\partial \Omega_0^{(M)}} t_{0i}^{\text{ext}} \delta \bar{\chi}_i^{(u)} dA_0 = 0 \tag{73}
\end{align}
$$

Application of the divergence theorem to the terms in (73) containing $\frac{\partial \delta \bar{\chi}^{(s)}}{\partial \bar{\chi}}$ and $\frac{\partial \delta \bar{\chi}^{(f)}}{\partial \bar{\chi}}$
yields (see Appendix 4.4 for details):

$$
\begin{align}
& \int_{\Omega_0^{(M)}} \left[ -\frac{\partial \check{P}_{iJ}}{\partial X_J} + \phi^{(s)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) \right] \delta \bar{\chi}_i^{(s)} \, dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{b}_{0i}^{(s,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} - \frac{\partial}{\partial X_J^{(s)}} \left( \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right) \right] \delta \bar{\chi}_i^{(s)} \, dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ \phi^{(f)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) - \frac{\partial}{\partial X_J} \left( \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right) \right] \delta \bar{\chi}_i^{(f)} \, dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(f)} \bar{\chi}_i^{(f)} \right] \delta \bar{\chi}_i^{(f)} \, dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\hat{\Pi}^{(s)} + \frac{\bar{J}^{(f)}}{\bar{J}^{(s)}} \Phi_0^{(s)} \hat{\Pi}^{(f)} \delta \hat{J}^{(s)} - \bar{\rho}_{add.0}^{(s)} (\bar{J}^{(s)} - \hat{J}^{(s)}) + \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \bar{\rho}_{add.0}^{(f)} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] \delta \hat{J}^{(s)} \, dV_0 \nonumber \\
& + \int_{\partial \Omega_0^{(M)}} \left[ \check{P}_{iJ} - \bar{J}^{(f)} \phi^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right] N_J \delta \bar{\chi}_i^{(s)} \, dV_0 \nonumber \\
& + \int_{\partial \Omega_0^{(M)}} \left[ -\phi^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} + \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] \delta \bar{\chi}_i^{(f)} N_J \, dV_0 \nonumber \\
& - \int_{\partial \Omega_0^{(M)}} t_{0i}^{ext} \delta \bar{\chi}_i^{(u)} \, dA_0 = 0 \tag{74}
\end{align}
$$

where the boundary integrals have been collected in the last three rows of (74).

## 2.4 Strong form equations

Strong form equations are inferred from (74) via application of the fundamental lemma of calculus of variations. Field equations are obtained considering virtual displacements vanishing at the boundary, such that $\delta \bar{\chi}^{(s)} = \mathbf{0}$, $\delta \bar{\chi}^{(f)} = \mathbf{0}$, and $\delta \hat{J}^{(s)} = 0$ over $\partial \Omega_0^{(M)}$. In particular, by selecting virtual variations of type $\delta \bar{\chi}^{(s)} \neq \mathbf{0}$, $\delta \bar{\chi}^{(f)} = \mathbf{0}$, and $\delta \hat{J}^{(s)} = 0$ over $\Omega_0^{(M)}$ one obtains the linear momentum balance of the solid phase:

$$
\begin{align}
& \frac{\partial \check{P}_{iJ}}{\partial X_J^{(s)}} - \phi^{(s)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) + \bar{b}_{0i}^{(s,ext)} = \tag{75} \\
& = \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} - \frac{\partial}{\partial X_J^{(s)}} \left( \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right) \nonumber
\end{align}
$$
Similarly, by selecting virtual variations vanishing at the boundary, such that $\delta\bar{\chi}^{(s)} = \mathbf{0}$, $\delta\bar{\chi}^{(f)} \neq \mathbf{0}$, and $\delta\hat{J}^{(s)} = 0$ over $\Omega_0^{(M)}$, one obtains the linear momentum balance of the fluid phase:

$$ 
\begin{aligned}
& -\phi^{(f)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) + \bar{b}_{0i}^{(f,ext)} = \\
& = \bar{\rho}_0^{(f)} \bar{\chi}_i^{(f)} - \frac{\partial}{\partial X_J} \left[ \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right]
\end{aligned}
\quad (76) $$

An analogous selection with $\delta\bar{\chi}^{(s)} = \mathbf{0}$, $\delta\bar{\chi}^{(f)} = \mathbf{0}$ and $\delta\hat{J}^{(s)} \neq 0$ yields the intrinsic momentum balance:

$$ \hat{\Pi}^{(s)} - \frac{\bar{J}^{(f)}}{\bar{J}^{(s)}} \Phi_0^{(s)} \hat{\Pi}^{(f)} = -\bar{\rho}_{add.0}^{(s)} (\bar{J}^{(s)} - \hat{J}^{(s)}) + \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \bar{\rho}_{add.0}^{(f)} (\bar{J}^{(f)} - \hat{J}^{(f)}) \quad (77) $$

For the boundary integrals in (74) considering variation fields at the boundary $\partial\Omega_0^{(M)}$ such that $\delta\bar{\chi}^{(s)} = \delta\bar{\chi}^{(f)} = \delta\bar{\chi}^{(u)}$ one obtains:

$$ 
\left[ \begin{aligned}
& \check{P}_{iJ} - \bar{J}^{(f)} \phi^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} - \phi^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} + \\
& \qquad + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) + \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \end{aligned} \right] N_J = t_{0i}^{ext} \quad (78) $$

### Strong-form equations associated with virtual isochoric deformations

Equations (75)-(78) provide a complete statement of the boundary value problem of the dynamic evolution of the mixture as dictated by the least Action principle. These equations are complete since they allow to determine, upon integrating in space-time the boundary value problem, the updated configuration of the mixture in any space-time point. In this respect, no additional equations need to be supplemented to (75)-(78) besides optional switch to the desired displacement-type or stress-type boundary constraints with the aid of (59) and (60). Actually, in the statement of the evolution problem given by (75)-(78), mass balances are not required among the governing equations because updated densities can be post-computed from the macroscopic displacement fields by (19).

It is however convenient to derive from the general stationarity condition (62) supplementary strong form conditions that the extrinsic Piola stress tensor of the solid phase $\check{P}$ must comply with over the free surface $S_0^{(sf)}$ (see Figure 1). It is worth noting that for such conditions the
denomination 'boundary conditions' should be avoided since $S_0^{(sf)}$ is not properly the boundary of the space domain of the mixture $\Omega_0^{(M)}$. Rather, $S_0^{(sf)}$ is the macroscopic surface geometrically delimiting the physical subsystem of the solid phase.

In order to derive a convenient strong-form relation at the free solid-fluid macroscopic interface $S_0^{(sf)}$ for $\tilde{P}$, we now compute the strong form equations which are inferred from (62) when the subclass of virtual isochoric deformations (i.e., deformations which are characterized by null intrinsic virtual deformations $\delta \hat{J}^{(s)} = 0$ and $\delta \hat{J}^{(f)} = 0$) are selected.

Let us observe from (68) that, in order to preserve the saturation condition (68), the macroscopic displacement fields in a virtual isochoric deformation must fulfill the condition:

$$ \frac{\partial \phi^{(s)} \delta \bar{\chi}_i^{(s)}}{\partial x_i} + \frac{\partial \phi^{(f)} \delta \bar{\chi}_i^{(f)}}{\partial x_i} = 0 \qquad (79) $$

A virtual isochoric deformation compatible with the condition of impermeable external boundary expressed by (56) can be constructed by choosing an arbitrary vector field $\delta \tilde{u}^{(s)}$ such that $\delta \tilde{u}^{(s)} = \mathbf{0}$ at the boundary $\partial \Omega_0^{(M)}$, and by setting the variations of the primary fields to:

$$ \delta \bar{\chi}^{(s)} = \delta \tilde{u}^{(s)}, \quad \delta \bar{\chi}^{(f)} = \delta \tilde{u}^{(f)} = - \frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}^{(s)}, \quad \delta \hat{J}^{(s)} = 0 \qquad (80) $$

When virtual deformations in (62) are set to the form specified in (80), the corresponding virtual intrinsic strain of the fluid is null (i.e., $\delta \hat{J}^{(f)} = 0$). Then (66) vanishes with the sum of the terms on its right hand side being zero. It stems from (71) and (72) that also $\delta U_0^{(f)}$ is null, and the terms contained in the right hand side of (72) cancel each other.

The effects determined by setting virtual deformations contained in the explicit integral equations (73) and (74) to the form specified in (80) are listed below:

* since $\delta \hat{J}^{(f)} = 0$, all terms provided by (72) cancel each other, since their sum is null;

* since $\delta \hat{J}^{(s)} = 0$, all terms multiplying $\delta \hat{J}^{(s)}$ are cancelled;

* considering the partition $\Omega_0^{(M)} = \Omega_0^{(s)} \cup \Omega_0^{(f)}$ sketched by Figure 1 (with $\Phi_0^{(s)} = 0$ over $\Omega_0^{(f)}$), and the associated additive split of the integrals

$$ \int_{\Omega_0^{(M)}} (\cdot) dV_0 = \int_{\Omega_0^{(s)}} (\cdot) dV_0 + \int_{\Omega_0^{(f)}} (\cdot) dV_0, \qquad (81) $$
it is recognized that all the integrals over $\Omega_0^{(M)}$ in(73) and (74) containing $\delta\bar{\chi}^{(f)}$ reduce to the only integral over the subset $\Omega_0^{(s)}$ since, as one can infer from (80), $\delta\bar{\chi}^{(f)} = \mathbf{0}$ over $\Omega_0^{(f)}$;

* the boundary integrals in (74) all vanish due to (80) and to the property that $\delta\tilde{\mathbf{u}}^{(s)} = \mathbf{0}$ over $\partial\Omega_0^{(M)}$;

In addition to the simplifications in the bullet list, it is also always true that energy densities for the solid phase must be null functions in the points where $\Phi_0^{(s)} = 0$. Hence, the domain integrals of the solid phase vanish throughout $\Omega_0^{(f)}$, and reduce to only integrals over $\Omega_0^{(s)}$.

Accounting for all of the above listed simplifications, when virtual isochoric deformations are considered into the integral equation (74), this specializes to:

$$ 
\begin{aligned}
& \int_{\Omega_0^{(s)}} \left[-\frac{\partial \tilde{P}_{iJ}}{\partial X_J}\right] \delta \tilde{u}_i^{(s)} dV_0 \\
& + \int_{\Omega_0^{(s)}} \left[-\bar{b}_{0i}^{(s,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} - \frac{\partial}{\partial X_J^{(s)}} \left(\bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)})\right)\right] \delta \tilde{u}_i^{(s)} dV_0 \\
& + \int_{\Omega_0^{(s)}} \left[-\frac{\partial}{\partial X_J} \left(\bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)})\right)\right] \left(-\frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}_i^{(s)}\right) dV_0 \\
& + \int_{\Omega_0^{(s)}} \left[-\bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(f)} \bar{\chi}_i^{(f)}\right] \left(-\frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}_i^{(s)}\right) dV_0 \\
& + \int_{\partial \Omega_0^{(s)}} \left[\tilde{P}_{iJ} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)})\right] N_J \delta \tilde{u}_i^{(s)} dV_0 \\
& + \int_{\partial \Omega_0^{(s)}} \left[\bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)})\right] \left(-\frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}_i^{(s)}\right) N_J dV_0 = 0
\end{aligned} 
\quad (82) $$

The previous relation must hold for any $\delta\tilde{\mathbf{u}}^{(s)}$ and can be used to infer strong form equations, again on account of the fundamental lemma of calculus of variations, by a standard localization technique. In particular, a domain strong form equation is first obtained by considering in (82) the subclass of virtual isochoric deformations vanishing at the boundary of the solid macroscopic domain, i.e., such that $\delta\tilde{\mathbf{u}}^{(s)} = \mathbf{0}$ over $\partial\Omega_0^{(s)}$. By this choice, the last two rows in (82) are cancelled and localization by the fundamental lemma yields the following equation holding in
the interior points of $\Omega_0^{(s)}$:

$$ 
\begin{aligned}
& -\frac{\partial \check{P}_{iJ}}{\partial X_J} - \bar{b}_{0i}^{(s,ext)} + \frac{\phi^{(s)}}{\phi^{(f)}} \bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} - \frac{\phi^{(s)}}{\phi^{(f)}} \bar{\rho}_0^{(f)} \bar{\chi}_i^{(f)} \\
& - \frac{\partial}{\partial X_J} \left( \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right) \\
& + \frac{\phi^{(s)}}{\phi^{(f)}} \frac{\partial}{\partial X_J} \left( \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right) = 0
\end{aligned} 
\quad (83) $$

As a second step, substitution of (83) into (82) yields:

$$ 
\begin{aligned}
& \int_{\partial \Omega_0^{(s)}} \left[ \check{P}_{iJ} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right] N_J \delta \tilde{u}_i^{(s)} dV_0 \\
& + \int_{\partial \Omega_0^{(s)}} \left[ \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] \left( -\frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}_i^{(s)} \right) N_J dV_0 = 0
\end{aligned} 
\quad (84) $$

The previous relation must hold true for any field $\delta \tilde{u}^{(s)}$ simultaneously fulfilling (63) and (80). Since the simultaneous fulfillment of these two equations necessarily implies $\delta \tilde{u}^{(s)} = 0$ over $\partial \Omega_0^{(MU)}$, it is inferred from (82) the following second integral equation which differs from the previous one since integration is carried out only on the subset $S_0^{(sf)} = \partial \Omega_0^{(s)} \setminus \partial \Omega_0^{(MU)}$, which is termed free solid-fluid macroscopic interface:

$$ 
\begin{aligned}
& \int_{S_0^{(sf)}} \left[ \check{P}_{iJ} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right] N_J \delta \tilde{u}_i^{(s)} dV_0 \\
& + \int_{S_0^{(sf)}} \left[ \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] \left( -\frac{\phi^{(s)}}{\phi^{(f)}} \delta \tilde{u}_i^{(s)} \right) N_J dV_0 = 0
\end{aligned} 
\quad (85) $$

for any arbitrary field $\delta \tilde{u}^{(s)}$ on $S_0^{(sf)}$. This last integral condition can be straightforwardly localized into the following strong form equation holding on $S_0^{(sf)}$:

$$ 
\left[ \check{P}_{iJ} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) - \frac{\phi^{(s)}}{\phi^{(f)}} \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] N_J = 0 
\quad (86) $$

Equation (86) is of significant practical relevance when the physical domain of interest is $\Omega_0^{(s)}$ and the objective of the analysis is the direct determination of the stress state inside the
solid domain equation. In this case equation (86) represents the boundary condition for the extrinsic stress $\tilde{P}$ to be applied over free solid-fluid interfaces $S_0^{(sf)}$.

Surface conditions to be fulfilled over $\partial\Omega_0^{(s)}$ can be specialized for the static case, when inertia terms are negligible. In this condition, relations (78) and (86) provide:

$$ \left( \tilde{P}_{iJ} - \bar{J}^{(f)} \phi^{(s)} \hat{\Pi}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} - \bar{J}^{(f)} \hat{\Pi}^{(f)} \phi^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \right) N_J = 0 \quad \text{over } \partial\Omega_0^{(M)} \\ \tilde{P}_{iJ} N_J = 0 \quad \text{over } S_0^{(sf)} \qquad (87) $$

It is worth observing that the condition $\tilde{P}N = 0$ over $S_0^{(sf)}$ and the formally similar boundary relation for single-phase continuum elasticity $PN = 0$ (where $P$ is the classical first Piola tensor) have very different physical meanings. The condition $PN = 0$ in a point of the boundary surface of the solid domain states that there is no mechanical interaction between the interior solid and the external environment at that point. Conversely, within the present two-phase continuum description, condition $\tilde{P}N = 0$ over $S_0^{(sf)}$ does not imply that mechanical interaction between the interior solid and the fluid external to $\Omega_0^{(s)}$ is absent in points of $S_0^{(sf)}$. As a matter of fact, $S_0^{(sf)}$ is not strictly a boundary of the physical system and, over this surface, coupling between solid and fluid is also mediated by the intrinsic solid stress $\hat{\Pi}^{(s)}$ via the intrinsic momentum balance (77). Recalling that $\tilde{P}$ is work-associated with isochoric strains, it is recognized that vanishing of $\tilde{P}$ in a point of $S_0^{(sf)}$ indicates that virtual isochoric strains applied to the solid in that point produce no strain energy exchange between the solid and the fluid. However coupling between the solid and the surrounding fluid regions still remains mediated by the intrinsic stress entering (77).

## 2.5 Additional solid-fluid interaction

The momentum balance equations derived in Subsection 2.4 have been obtained based on the mechanical hypotheses of Subsection 2.2.2 where only the individual strain energies $U_0^{(s)}$ and $U_0^{(f)}$ have been specified, and which still do not contemplate a specific solid-fluid interaction apart from the interaction of geometric/volumetric type determined by the saturation constraint (14) (see the coupling term in Equation (75) containing the gradient of $\hat{\Pi}^{(f)}$).
In multiphase porous media the interaction between phases is not limited to this volumetric coupling due to saturation. An important role is also played by forces which solid and fluid mutually exchange, which are experimentally found to be dependent on the macroscopic relative solid-fluid flow. This additional interaction, which includes the well-known phenomenological Darcy-Forchheimer laws [75], is herein referred to as *drag interaction*, and is represented by an additional term, $L_0^{(sf)}$, in Lagrange function. Such interaction is examined aiming at its simplest and most general description to preserve the medium independence of the present theoretical approach. Accordingly, in order to achieve a kernel of equations of maximum generality conveniently embracing the widest possible class of two-phase media, minimal assumptions are introduced for $L_0^{(sf)}$. Specifically, for the drag interaction, the only two features that are herein considered are: 1) the *internal force character*; and 2) the *local short-range character*.

Specifically, by *internal force character* we refer to the property of a given term of the Lagrange function, characteristic of internal forces in isolated systems, to fulfill homogeneity of space. Space homogeneity of a Lagrange function $L$ is the invariance of $L$ to parallel translations $\mathbf{v}$ applied to the physical system [48]. This property, which implies the conservation of linear momentum, is fulfilled by the internal strain energies $U_0^{(s)}$ and $U_0^{(f)}$. Actually, definitions (34) and (33) and the first-gradient nature of $\bar{\psi}_0^{(s)}$ and $\hat{\psi}_0^{(f)}$ imply that the following property holds for any finite uniform translation field $\mathbf{v}$:

$$ U_0^{(s)}(\bar{\chi}^{(s)}, \hat{J}^{(s)}) = U_0^{(s)}(\bar{\chi}^{(s)} + \mathbf{v}, \hat{J}^{(s)}) \quad (88) $$

$$ U_0^{(f)}(\bar{\chi}^{(s)}, \bar{\chi}^{(f)}, \hat{J}^{(s)}) = U_0^{(f)}(\bar{\chi}^{(s)} + \mathbf{v}, \bar{\chi}^{(f)} + \mathbf{v}, \hat{J}^{(s)}) \quad (89) $$

This property is also stated in terms of variations as follows:

$$ \partial_{\bar{\chi}^{(s)}} U_0^{(s)}[\delta \mathbf{v}] = 0, \quad \partial_{\bar{\chi}^{(s)}} U_0^{(f)}[\delta \mathbf{v}] + \partial_{\bar{\chi}^{(f)}} U_0^{(f)}[\delta \mathbf{v}] = 0 \quad (90) $$

where $\delta \mathbf{v}$ is a uniform infinitesimal virtual variation field. In particular, accounting for the null gradient of $\delta \mathbf{v}$, the first of (90) is directly inferred from (174) (see Appendix), while the second
of (90) is deduced from (67) observing that:

$$ \partial_{\bar{\chi}^{(s)}} \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)}[\delta\mathbf{v}] + \partial_{\bar{\chi}^{(f)}} \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)}[\delta\mathbf{v}] = \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\partial(\phi^{(s)} + \phi^{(f)})}{\partial x_i} \delta v_i = 0 \quad (91) $$

since $\phi^{(s)} + \phi^{(f)} = 1$ by (2).

The requirement that also $L_0^{(sf)}$ must have an internal force character is stated in terms analogous to (90):

$$ \partial_{\bar{\chi}^{(s)}} L_0^{(sf)}[\delta\mathbf{v}] + \partial_{\bar{\chi}^{(f)}} L_0^{(sf)}[\delta\mathbf{v}] = 0 \quad (92) $$

The second hypothesis of short-range character implies that, similar to $U_0^{(s)}$ and $U_0^{(f)}$ which are expressed in terms of local strain energy density, also $L_0^{(sf)}$ can be expressed as the space integral of a macroscopic local energy density function $\bar{\psi}_{\mathbf{x}}^{(sf)}$, viz.:

$$ L_0^{(sf)} = \int_{\Omega^{(M)}} \bar{\psi}_{\mathbf{x}}^{(sf)} dV \quad (93) $$

The integral form provided by (92) allows one to write:

$$ \int_{\Omega^{(M)}} \left( \frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}_{i}^{(s)}} \delta v_i + \frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}_{i}^{(f)}} \delta v_i \right) dV = 0 \quad (94) $$

The direct consequence of hypotheses 1) and 2) is that the drag interaction can be described by a field of local drag volume forces $\bar{b}^{(sf)}$ and $\bar{b}^{(fs)}$, defined as:

$$ \bar{b}^{(sf)}(\mathbf{x}) = -\frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}^{(s)}}, \quad \bar{b}^{(fs)}(\mathbf{x}) = -\frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}^{(f)}} \quad (95) $$

Such forces must have equal magnitude and opposite direction. Actually, since Equation (94) is a spatial relation holding for any subdomain $\Omega \subseteq \Omega^{(M)}$, and since it must also hold for any vector $\delta\mathbf{v}$, one infers that:

$$ \frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}^{(s)}} + \frac{\partial \bar{\psi}^{(sf)}}{\partial \bar{\chi}^{(f)}} = 0, \quad \text{over } \Omega^{(M)} \quad (96) $$

which reads (in field format):

$$ \bar{b}_{\mathbf{x}}^{(fs)} = -\bar{b}_{\mathbf{x}}^{(sf)} \quad (97) $$

Inclusion of $L_0^{(sf)}$ into the model requires the addition of the terms $\partial_{\bar{\chi}^{(s)}} L_0^{(sf)}[\bar{\chi}^{(s)}]$ and
$\partial_{\bar{\chi}^{(f)}} L_0^{(sf)} [\delta \bar{\chi}^{(f)}]$ into the general expression of the variation (62). These terms, when expressed as integrals over the reference domain of the mixture $\bar{\chi}^{(s)}$ read:

$$ \partial_{\bar{\chi}^{(s)}} L_0^{(sf)} [\bar{\chi}^{(s)}] = - \int_{\Omega_0^{(M)}} \bar{J}^{(s)} \bar{b}_{\bar{\chi}^{(s)}}^{(sf)} \delta \bar{\chi}^{(s)} dV_0, \quad \partial_{\bar{\chi}^{(f)}} L_0^{(sf)} [\bar{\chi}^{(f)}] = - \int_{\Omega_0^{(M)}} \bar{J}^{(f)} \bar{b}_{\bar{\chi}^{(f)}}^{(fs)} \delta \bar{\chi}^{(f)} dV_0 \quad (98) $$

where, consistent with (99), we have:

$$ \bar{b}_{\bar{\chi}^{(s)}}^{(sf)} \circ (\bar{\chi}^{(s)})^{-1} + \bar{b}_{\bar{\chi}^{(f)}}^{(fs)} \circ (\bar{\chi}^{(f)})^{-1} = 0 \quad (99) $$

The strong form equations resulting from the inclusion of this medium-independent solid fluid interaction are summarized below. To place emphasis on medium independence, microinertia terms, which have in some respects a constitutive medium-dependent character, are purposefully not included.

Linear momentum balance of the solid phase:

$$ \frac{\partial \tilde{P}_{iJ}}{\partial X_J^{(s)}} - \phi^{(s)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) + \bar{b}_{0i}^{(s,ext)} + \bar{J}^{(s)} \bar{b}_i^{(sf)} = \bar{\rho}_0^{(s)} \bar{\chi}_i^{\ddot{(s)}} \quad (100) $$

Linear momentum balance of the fluid phase:

$$ -\phi^{(f)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) + \bar{b}_{0i}^{(f,ext)} + \bar{J}^{(f)} \bar{b}_i^{(fs)} = \bar{\rho}_0^{(f)} \bar{\chi}_i^{\ddot{(f)}} \quad (101) $$

Intrinsic momentum balance:

$$ \bar{J}^{(s)} \hat{\Pi}^{(s)} - \bar{J}^{(f)} \Phi_0^{(s)} \hat{\Pi}^{(f)} = 0 \quad (102) $$

The domain equations above are completed by boundary conditions (87).

## 2.6 The kinematically-linear medium-independent problem

The strong form equations of Section 2.4 are now specialized for the kinematically linear static boundary value problem, suitable for the description of problems with infinitesimal deformations. This specialization is carried out in a standard way as a first-order Taylor series truncation
of deformation and strain measures, and has the primary effect of making the reference and current configurations coincide:

$$ \Omega_0^{(M)} \simeq \Omega^{(M)}, \quad \Omega_0^{(s)} \simeq \Omega^{(s)}, \quad \Omega_0^{(f)} \simeq \Omega^{(f)} \qquad (103) $$

The coincidence of reference and current configurations is reflected in the notation henceforth used by dropping the lowercase/uppercase distinction previously applied to subscripts. Also, the prefix d is used to denote infinitesimal quantities. To achieve a less dense notation, such prefix is omitted for primary descriptor fields, and for symbols ordinarily employed to denote infinitesimal strain measures: $\bar{\varepsilon}^{(s)}$, $\bar{e}^{(s)}$, $\bar{e}^{(f)}$, $\hat{e}^{(s)}$, and $\hat{e}^{(f)}$ (where the prefix is redundant).

Primary descriptors of the linearized formulation are the solid and fluid infinitesimal displacements $\bar{\mathbf{u}}^{(s)} = d\bar{\chi}^{(s)}$ and $\bar{\mathbf{u}}^{(f)} = d\bar{\chi}^{(f)}$ and the infinitesimal intrinsic volumetric strain $\hat{e}^{(s)}$.

Finite volumetric strain measures are replaced by linearized volumetric strain measures $\bar{e}^{(s)}$, $\bar{e}^{(f)}$, $\hat{e}^{(s)}$, and $\hat{e}^{(f)}$, previously defined in (7) and (11). Their relation is:

$$ \bar{J}^{(s)} \simeq 1 + \bar{e}^{(s)} = 1 + \frac{\partial d\bar{\chi}^{(s)}}{\partial x_i}, \quad \bar{J}^{(f)} \simeq 1 + \bar{e}^{(f)} = 1 + \frac{\partial d\bar{\chi}^{(f)}}{\partial x_i}, \quad \hat{J}^{(f)} \simeq 1 + \hat{e}^{(f)}, \quad \hat{J}^{(s)} \simeq 1 + \hat{e}^{(s)} \qquad (104) $$

In the linearized theory, the extrinsic strain of the solid is defined by the infinitesimal extrinsic strain tensor

$$ \bar{\varepsilon}^{(s)} = \mathrm{sym} \left( \bar{\mathbf{u}}^{(s)} \otimes \nabla \right) \qquad (105) $$

and by $\hat{e}^{(s)}$.

For stress measures, the simplifications stemming from the coincidence of reference and current configurations:

$$ \frac{\partial \bar{\chi}_i^{(s)}}{\partial X_J} \simeq \delta_{ij}, \quad \frac{\partial \bar{\chi}_i^{(f)}}{\partial X_J} \simeq \delta_{ij}, \quad \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} = \delta_{ji}, \quad \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} = \delta_{ji}, \quad \bar{J}^{(s)} \simeq 1, \quad \bar{J}^{(f)} \simeq 1, \quad \hat{J}^{(s)} \simeq 1, \quad \hat{J}^{(f)} \simeq 1 \qquad (106) $$

are applied to Equations (44), (45), and (48)-(51), obtaining the following identifications:

$$ \tilde{P}_{ij}^{(s)} \simeq \check{\sigma}_{ij}^{(s)}, \quad \hat{\Pi}^{(s)} \simeq \hat{p}^{(s)}, \quad \hat{\Pi}^{(f)} \simeq p \qquad (107) $$
so that the fields of spatial extrinsic solid stress tensor, spatial intrinsic solid pressure and fluid pressure provide a complete description of the stress state in the mixture.

## Quadratic forms of strain energy

In linearized kinematics, the choice of a linear constitutive theory determines strain energy densities which are quadratic forms in the relevant strain measures [68]:

$$ \bar{\psi}^{(s)} = \frac{1}{2} A_{ijkl} \bar{\varepsilon}_{ij}^{(s)} \bar{\varepsilon}_{kl}^{(s)} + B_{ij} \bar{\varepsilon}_{ij}^{(s)} \hat{e}^{(s)} + \frac{1}{2} C (\hat{e}^{(s)})^2 \quad (108) $$

$$ \bar{\psi}^{(f)} = \phi^{(f)} \frac{1}{2} \hat{k}_f (\hat{e}^{(f)})^2 \quad (109) $$

where the elastic coefficients of the solid and the fluid phases, $A_{ijkl}$, $B_{ij}$, $C$ and $\hat{k}_f$, correspond to the second-order derivatives of the corresponding strain energy densities, and with respect to the relevant primary infinitesimal strain measures. In particular, $\hat{k}_f$ is the fluid intrinsic stiffness. Also, the elastic coefficients in the energy of solid must respect the so-called major and minor symmetries (i.e., $A_{ijkl} = A_{klij} = A_{ijklk}$, $B_{ij} = B_{ji}$) which stem from Schwarz's theorem and from the requirement of objectivity. Stress-strain relations for the solid are:

$$ \tilde{\sigma}_{ij}^{(s)} = \frac{\partial \bar{\psi}^{(s)}}{\partial \bar{\varepsilon}_{ij}^{(s)}} = A_{ijkl} \bar{\varepsilon}_{ij}^{(s)} + B_{ij} \hat{e}^{(s)}, \quad \hat{p}^{(s)} = - \frac{\partial \bar{\psi}^{(s)}}{\partial \hat{e}^{(s)}} = -B_{ij} \bar{\varepsilon}_{ij}^{(s)} - C \hat{e}^{(s)} \quad (110) $$

and the relation between fluid pressure and intrinsic strain is:

$$ \hat{k}_f = \frac{\partial^2 \bar{\psi}^{(f)}}{\partial \hat{e}^{(f)} \partial \hat{e}^{(f)}}, \quad p = -\hat{k}_f \hat{e}^{(f)} \quad (111) $$

## Linearized saturation condition

The saturation condition for the kinematic linear problem can be obtained applying the linearization provided by (69) to real deformations, i.e., replacing $\delta\bar{\chi}^{(s)}$ and $\delta\bar{\chi}^{(f)}$ with $\bar{u}^{(s)}$ and $\bar{u}^{(f)}$ in (69). Accordingly, the following linearized saturation constraint is obtained:

$$ \phi^{(s)} \hat{e}^{(s)} + \phi^{(f)} \hat{e}^{(f)} = \frac{\partial \phi^{(s)} \bar{u}_i^{(s)}}{\partial x_i} + \frac{\partial \phi^{(f)} \bar{u}_i^{(f)}}{\partial x_i} \quad (112) $$

Equation (112) is reported also by Bedford and Drumheller ( [21] (see Eq. (42) therein) in a form combined with the mass balances, see also [42]. It is interesting to observe that, herein,
this condition is derived independently from any consideration on mass balances. It is also worth noting that, under the hypothesis that the fields $\phi^{(s)}$ and $\phi^{(f)}$ are uniform, the same relation can be directly obtained by combining (9) and (10) with the linearization of (2):

$$d\phi^{(s)} + d\phi^{(f)} = 0. \qquad (113)$$

## Momentum balances with inertia terms

The momentum balances for the kinematic linear problem are obtained specializing (75), (76) and (77), by applying simplifications (106). This yields the following kinematically linear momentum balances:

### linear momentum balance of the solid phase

$$\frac{\partial \check{\sigma}_{ij}^{(s)}}{\partial x_j} - \phi^{(s)} \frac{\partial p}{\partial x_i} + \bar{b}_i^{(sf)} + \bar{b}_i^{(s,ext)} = \bar{\rho}^{(s)} \bar{u}_i^{(\ddot{s})} + \frac{\partial}{\partial x_j} \left[ \bar{\rho}_{add.}^{(s)} \left( \frac{\partial \bar{u}_i^{(\ddot{s})}}{\partial x_i} - \hat{e}^{(\ddot{s})} \right) \right] \quad (114)$$

### linear momentum balance of the fluid phase:

$$-\phi^{(f)} \frac{\partial p}{\partial x_i} + \bar{b}_i^{(fs)} + \bar{b}_i^{(f,ext)} = \bar{\rho}^{(f)} \bar{u}_i^{(\ddot{f})} - \frac{\partial}{\partial x_j} \left[ \bar{\rho}_{add.}^{(f)} \left( \frac{\partial \bar{u}_i^{(\ddot{f})}}{\partial x_i} - \hat{e}^{(\ddot{f})} \right) \right] \quad (115)$$

### intrinsic momentum balance:

$$\frac{\hat{p}^{(s)}}{\phi^{(s)}} - p = - \frac{1}{\phi^{(s)}} \bar{\rho}_{add.}^{(s)} \left( \frac{\partial \bar{u}_i^{(\ddot{s})}}{\partial x_i} - \hat{e}^{(\ddot{s})} \right) + \frac{1}{\phi^{(f)}} \bar{\rho}_{add.}^{(f)} \left( \frac{\partial \bar{u}_i^{(\ddot{f})}}{\partial x_i} - \hat{e}^{(\ddot{f})} \right) \quad (116)$$

It is interesting to observe that the intrinsic momentum balance (116) can be written in a notation symmetric with respect to indexes s and f. Actually, if one introduces the counterpart of $\hat{p}^{(f)}$ for the solid phase by the symmetric definition $\hat{p}^{(s)} = -\frac{\partial \hat{\psi}_0^{(s)}}{\partial \hat{e}^{(s)}} = \frac{\hat{p}^{(s)}}{\phi^{(s)}},$ then, recalling also (51), Equation (116) achieves the symmetric notation:

$$\hat{p}^{(s)} - \hat{p}^{(f)} = - \frac{1}{\phi^{(s)}} \bar{\rho}_{add.}^{(s)} \left( \frac{\partial \bar{u}_i^{(\ddot{s})}}{\partial x_i} - \hat{e}^{(\ddot{s})} \right) + \frac{1}{\phi^{(f)}} \bar{\rho}_{add.}^{(f)} \left( \frac{\partial \bar{u}_i^{(\ddot{f})}}{\partial x_i} - \hat{e}^{(\ddot{f})} \right) \quad (117)$$

We prefer, however, to maintain the unsymmetric notation with stress quantities $\hat{p}^{(s)}$ and $p$ in
consideration of their more direct physical identification, and to avoid proliferation of stress notations.

### Boundary conditions

The boundary conditions of the kinematic linear problem are obtained from the specialization of (78) provided by (106). Recalling that $\phi^{(s)} + \phi^{(f)} = 1$, these turn out to be:

$$ \left[ \tilde{\sigma}_{ij}^{(s)} - p\delta_{ij} + \bar{\rho}_{add.}^{(s)} \left( \frac{\partial \bar{u}_i^{(s)}}{\partial x_i} - \hat{e}^{(s)} \right) \delta_{ij} + \bar{\rho}_{add.}^{(f)} \left( \frac{\partial \bar{u}_i^{(f)}}{\partial x_i} - \hat{e}^{(f)} \right) \delta_{ij} \right] n_j = t_i^{ext} \quad \text{over } \partial\Omega^{(M)} \quad (118) $$

A derivation of (118) based on simplified arguments in a 1D setting has been also previously reported [68].

Over free solid-fluid interfaces $S^{(sf)}$ (Figure 1), the relation inferred from (86) and (106) is:

$$ \left[ \frac{1}{\phi^{(s)}} \tilde{\sigma}_{ij}^{(s)} + \frac{1}{\phi^{(s)}} \bar{\rho}_{add.}^{(s)} \left( \frac{\partial \bar{u}_i^{(s)}}{\partial x_i} - \hat{e}^{(s)} \right) \delta_{ij} - \frac{1}{\phi^{(f)}} \bar{\rho}_{add.}^{(f)} \left( \frac{\partial \bar{u}_i^{(f)}}{\partial x_i} - \hat{e}^{(f)} \right) \delta_{ij} \right] n_j = 0 \quad \text{over } S^{(sf)} \quad (119) $$

## 2.7 Equations for static and quasi-static problems

The kinematically linear equations are hereby specialized to their form suitable for problems in which inertia terms can be neglected. Accordingly, upon neglecting inertia terms in (114), (115) and (117), one obtains:

### Linear momentum balance of the solid phase (static and quasi-static problems)

$$ \frac{\partial \tilde{\sigma}_{ij}^{(s)}}{\partial x_j} - \phi^{(s)} \frac{\partial p}{\partial x_i} + \bar{b}_i^{(sf)} + \bar{b}_i^{(s,ext)} = 0 \quad (120) $$

### Linear momentum balance of the fluid phase (static and quasi-static problems):

$$ -\phi^{(f)} \frac{\partial p}{\partial x_i} + \bar{b}_i^{(fs)} + \bar{b}_i^{(f,ext)} = 0 \quad (121) $$

### Intrinsic momentum balance (Static and quasi-static problems):

$$ \hat{p}^{(s)} - \phi^{(s)} p = 0 \quad (122) $$
Observe that Equations (120)-(122) have a medium-independent character since they can be obtained by a direct linearization of (100)-(102).

The boundary and surface conditions for static and quasi-static problems are obtained from (118). For stresses, one obtains a relation which, based on simplified arguments in a 1D setting, has been previously reported [68]:

$$ (\tilde{\sigma}_{ij}^{(s)} - p\delta_{ij}) n_j = t_i^{ext} \quad \text{over } \partial\Omega^{(M)} \qquad (123) $$

while the boundary condition for displacements is

$$ \bar{u}^{(s)} = \bar{u}^{(f)} = u^{(ext)} \qquad (124) $$

In particular, over $\partial\Omega^{(M)} \setminus \partial\Omega^{(MU)}$, where $\phi^{(s)} = 0$ and, consequently, $\tilde{\sigma}^{(s)} = 0$, equation (123) specializes to:

$$ -pn_i = t_i^{ext} \quad \text{over } \partial\Omega^{(M)} \setminus \partial\Omega^{(MU)} \qquad (125) $$

Over free solid-fluid interfaces $S^{(sf)}$ (Figure 1), the relation inferred from (119) is:

$$ \tilde{\sigma}_{ij}^{(s)} n_j = 0 \quad \text{over } S^{(sf)} \qquad (126) $$

It is worth to recall that the considerations previously reported on the physical meaning of a vanishing normal extrinsic stress traction (see Section 2.4) apply also to (126), which is a special case of (87).

### 2.7.1 Medium-independent stress partitioning law

In a domain $\Omega^{(h)}$ (with null relative solid-fluid motion at the boundaries) for macroscopically uniform stress states, i.e., states such that fields $\tilde{\sigma}^{(s)}$ and $p$ have constant values $\tilde{\sigma}_h^{(s)}$, $p_h$, Equation (123) specializes to

$$ t^{(ext)}(\mathbf{x}, \mathbf{n}) = \tilde{\sigma}_h^{(s)}\mathbf{n} - p_h\mathbf{n}, \quad \mathbf{x} \in \partial\Omega^{(h)} \qquad (127) $$
so that the external traction field $t^{(ext)}(\mathbf{x}, \mathbf{n})$ in (127) can be represented by a single constant tensor $\sigma^{(ext)}$:

$$t^{(ext)}(\mathbf{x}, \mathbf{n}) = \sigma^{(ext)}\mathbf{n}, \quad \mathbf{x} \in \partial\Omega^{(h)} \qquad (128)$$

where $\sigma^{(ext)}$ is a tensor associated with $\Omega^{(h)}$ and defined as

$$\sigma^{(ext)} = \tilde{\sigma}_{h}^{(s)} - p_{h}\mathbf{I} \qquad (129)$$

Hence, when a traction is applied over the (impermeable) boundary $\partial\Omega^{(h)}$ so as to produce a uniform stress state $\sigma^{(ext)}$ inside $\Omega^{(h)}$, it is then partitioned in compliance with relation (129). Such partitioning is independent from the particular constitutive and microstructural features of the medium considered.

A second important consequence of (123) is inferred for regions undergoing undrained flow conditions, which are those regions where macroscopic relative solid-fluid motion is null everywhere so that any surface is impermeable. In the domain of one of these regions, say $\Omega^{(u)}$, relation (123) applies to the boundary $\partial\bar{\Omega}^{(u)}$ of any subset domain $\bar{\Omega}^{(u)} \subseteq \Omega^{(u)}$ since $\bar{\Omega}^{(u)}$ is impermeable. Due to the arbitrariness of $\bar{\Omega}^{(u)}$, the condition of homogeneity can be recovered, as a limit, considering arbitrarily small domains $\bar{\Omega}_{\mathbf{x}}^{(u)} \subseteq \Omega^{(u)}$ centered in a point $\mathbf{x}$. Consequently, it is recognized that, in undrained regions, relations (128), (129) are recovered in a stronger form associated with any point $\mathbf{x} \in \Omega^{(u)}$

$$t^{(ext)}(\mathbf{x}, \mathbf{n}) = \sigma^{(ext)}\mathbf{n}, \quad \forall \mathbf{x} \in \Omega^{(u)}, \forall \mathbf{n} \qquad (130)$$

where $\sigma^{(ext)}$ is a tensor now associated with any point $\mathbf{x} \in \Omega^{(u)}$, defined as

$$\sigma^{(ext)} = \tilde{\sigma}^{(s)} - p\mathbf{I} \qquad (131)$$

As observed also in these previous works, relations (129) and (131) coincide, from a formal point of view, with the classical statement of Terzaghi's principle if one identifies $\tilde{\sigma}^{(s)}$ with the effective stress tensor. However, it is important to remark that, herein, this condition has been derived in absence of any constitutive hypothesis on the phases, thus representing a stress partitioning law of general validity for homogeneous stress states, not limited to soils.
### 3 Discussion and conclusions

A variational theory of two-phase saturated porous media has been derived proceeding from the adoption of the VMTPM kinematic framework [11, 42, 67, 68], and applying standard concepts of variational continuum mechanics [45, 46, 48] and of continuum theories with microstructure [21, 39].

VMTPM kinematics is based on an extrinsic/intrinsic split of volumetric strain measures: the additional descriptor field is the intrinsic scalar volumetric strain measure $\hat{J}^{(s)}$, and corresponds to the ratio $\hat{\rho}_{0}^{(s)}/\hat{\rho}^{(s)}$ between true densities of solid before and after deformation. This field is independent from $\bar{J}^{(s)}$, which remains customarily defined in VMTPM as the determinant of the macroscopic deformation gradient. The experimental characterization of $\hat{J}^{(s)}$ is possible by measuring the changes of the porosity field via the relation linking the intrinsic volumetric strain to the porosities before ($\Phi_{0}^{(f)}$) and after deformation ($\phi^{(f)}$):

$$ \hat{J}^{(s)} = \bar{J}^{(s)} \left(1 - \phi^{(f)}\right) / \left(1 - \Phi_{0}^{(f)}\right) \qquad (132) $$

The choice of kinematic descriptors in VMTPM is minimal since it consists of the least possible set of fields ensuring the fulfillment of the saturation condition, without adding artificial incompressibility constraints. Previous works have adopted porosity as the additional descriptor [9, 20, 21]. In this work, the use of $\hat{J}^{(s)}$ is preferred alongside with $\bar{J}^{(s)}$ since these fields share analogous properties. In particular, both $\bar{J}^{(s)}$ and $\hat{J}^{(s)}$ are naturally defined in the reference configuration of the solid phase; also they jointly achieve a unit value in presence of rigid deformations, and allow to characterize a homotetic deformation by the condition $\bar{J}^{(s)} = \hat{J}^{(s)}$. These common properties are found to be convenient when introducing the stress measures and stiffness quantities work-associated with $\bar{J}^{(s)}$ and $\hat{J}^{(s)}$, allowing their easier correlation and physical interpretation (see also the companion Part II).

In previous VMTPM derivations [11, 42, 67–71], variational deductions of the governing equations were limited to the derivation of the momentum balances of the solid phase. In the present contribution, also the fluid momentum balance has been derived on variational basis. Accordingly, the present theory is purely-variational in that no arguments other than least Action conditions are used to derive all momentum balance equations. Also, similar to background works, the derivation herein reported has a purely-macroscopic character, in that no consider-
ations on the microstructural features of the medium have been introduced. All the equations have been obtained avoiding any assumption on the microscale geometric or constitutive features of the medium. To this end, an Action functional of maximum generality has been employed with kinematic descriptors having a universal character (i.e., the displacements and the intrinsic strain can be ordinarily measured for any deforming medium). Accordingly this theory meets the sought requirement of *medium independence* and, as such, can be applied irrespective of the degree of anisotropy of the medium and of its constitutive linear or nonlinear features, in a way similar to the linear momentum balance for single-phase continua.

This variational formulation has been developed based on the configuration description adopt-
ing a set of independent kinematic descriptor fields which are not constrained to respect further
equations expressing saturation or mass balances. This choice represents a precise element of
distinction of this theory from the variational formulation of Bedford and Drumheller [21], and
ensures the well-posedness of the variational statement of the problem. In particular, the absence
in VMTPM of constraints for the primary descriptors allows to ordinarily define stress measures
based on explicit work-association, as standardly done in the single-continuum elasticity theory,
making no recourse to Lagrange multipliers for defining stress quantities.

Importantly, the framework hereby derived is downward compatible with the single contin-
uum Cauchy linear momentum balance equations. More specifically, recalling Equation (52), it
is observed that when the fluid pressure is null (i.e., $\tilde{\sigma}^{(s)} \equiv \sigma^{(s)}$), the solid intrinsic pressure
is null, and the extrinsic stress tensor becomes coincident with the standard notion of Cauchy
stress tensor. Moreover, as reported in relation (52), the extrinsic momentum balance becomes
formally coincident with Cauchy linear momentum balances.

The present study also shows that the "*missing equations*" for the *closure* of the two-phase
poroelastic boundary value problem are naturally identified when the problem is approached in
purely variational terms. The closure equations are represented by the saturation constraint and
by the intrinsic momentum balance. Notably, neither of these equations has a constitutive nature
or a thermodynamic nature.

In a similar way, a derivation of the general three-dimensional conditions which must be
applied at the macroscopic surfaces of the mixture is achieved in purely-variational terms. The
reported derivation is general in that it is comprehensive of impermeable boundaries as well as
of free solid-fluid macroscopic surfaces. The derivation proves the variational consistency of the boundary equations obtained, which generalize those reported in [68]. In this respect, the derivation of general surface conditions for free solid-fluid macroscopic interfaces, stemming as the strong-form equations obtained when virtual isochoric deformations are considered, also represents an original result which may be denominated *principle of virtual isochoric deformations*, see Section 2.4.

Result relevant to stress partitioning have been also derived. It was shown that, whenever external tractions applied over the (impermeable) boundary induce a deformation in the medium which is (macroscopically) uniform in space, an external stress tensor can be defined such that its partition between the two phases in compliance with relation (129), see Section 2.7. From a formal point of view, such result coincides with the classical statement of Terzaghi's principle, and the extrinsic stress tensor $\tilde{\sigma}^{(s)}$ is identified with the largely employed notion of *effective stress* tensor. However, it is important to remark that, in this work, such relation has been derived in absence of any constitutive hypothesis on the phases, thus representing a stress partitioning law of general validity for homogeneous stress states in saturated two-phase media not limited to saturated soils. A related result is the proof that in regions of the medium undergoing undrained flow conditions (i.e., with macroscopic relative solid-fluid motion prevented), an *external stress* field can be meaningfully considered with the external stress tensor being partitioned between the two phases in a relation formally coincident with Terzaghi's law, see Equations (130)-(131). A less general proof of this result has been previously reported in [42].

The results obtained on stress partitioning deserve a final consideration. Some experimental results in poromechanics have been interpreted as evidences of deviations from Terzaghi's law for specific classes of two-phase media [36]. Since the general character of the partitioning law herein derived excludes the possibility of such deviations, a dedicated study needs to be conducted to assess the capability of VMTPM to predict such experimental results. Such investigation is reported in the companion paper which is Part II of this study, together with a specific assessment of the predictive capabilities of VMTPM theory in relation to the description of tests on biphasic specimens subjected to a comprehensive variety of loading and drainage conditions. The companion study shows that governing equations and results of consolidated use in poroelasticity, such as Terzaghi's stress partitioning principle and Biot's equations, are ordi-
narily recovered by VMTPM, and presents an application of the variational governing equations
combined with the employment of a nonlinear constitutive response for the solid phase.

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# 4 Appendix A - Notation and identities for differential operations

Notation conventions, definitions and identities employed for the operations of calculus of variations used in this study are hereby synoptically collected. These are first summarized in Section 4.1 for single scalar real-valued functions and for the related functionals. Section 4.2 reports conventions and identities for multiple variable functions and for the related functionals and functional operators. All mathematical hypotheses necessary for the definitions and operations herein reported to make sense are given for granted.

## 4.1 Single variable functions

Symbol $\varphi$ is used to refer to a functional operating from a space of real valued functions $\mathcal{F}$ into $\mathbb{R}$, such that $\alpha = \varphi(f)$ with $\alpha \in \mathbb{R}$ and $f \in \mathcal{F}$, being $f$ a real valued function such that $y = f(x)$ with $x, y \in \mathbb{R}$.

An engineering notation is used to denote variations, employing the prefixes $\delta$ or $d$ for differentials. In particular, in the body of the paper prefix $d$ is used to denote real variations in linearized theories, while $\delta$ is used to denote virtual variations when expressing stationarity conditions; henceforth in this appendix only prefix $\delta$ is used. An infinitesimal variation $\delta f$ of function $f$ is the function $x \to \delta y$ with $\delta y \in \mathbb{R}$ being the variation in the codomain variable $y$.

An engineering notation is also adopted to indicate the result of the variation, $\partial_f \varphi$, of a functional $\varphi$ with respect to its argument function $f$. This variation is written in the form of a linear relation among the infinitesimal variation of the argument function $\delta f$, reported under square brackets, and the increment $\delta \alpha$ of $\alpha$ with the following notation:

$$ \delta \alpha = \partial_f \varphi [\delta f] \qquad (133) $$

and $\partial_f \varphi$ is such that, for any $\delta f$

$$ \partial_f \varphi [\delta f] = \lim_{\lambda \to 0} \frac{\varphi(f + \lambda \delta f) - \varphi(f)}{\lambda} \qquad (134) $$

### Variation of function composition

Let $a: (f, x) \to y$ be a functional application relating a real function $f \in \mathcal{F}$ and $x \in \mathbb{R}$ to a scalar $y \in \mathbb{R}$ and let $b: (f, y) \to z$ be a second functional application relating a real function
$f \in \mathcal{F}$ and $y \in \mathbb{R}$ to a scalar $z \in \mathbb{R}$. Consider the following composition of these two functional applications $c = b \circ a$ with :

$$c(f, x) = b(f, a(f, x)) \qquad (135)$$

Computation of the total variation with respect to $f$ by application of the chain rule yields

$$\partial_{fc}[\delta f] = \partial_{fb}[\delta f] \circ a + \frac{\partial b}{\partial y} \partial_{fa}[\delta f] \qquad (136)$$

In particular, when $b$ is independent from $f$ so that $\partial_{fb}[\delta f] = 0$, one obtains as a special case of (136)

$$\partial_{fc}[\delta f] = \frac{\partial b}{\partial y} \partial_{fa}[\delta f] \qquad (137)$$

Variation of the application relating functions to their inverses

By choosing in (136), for $a$, the application such that $a(f, x) = f(x)$, and, for $b$, the application such that $b(f, y) = f^{-1}(y)$, with $f \in \mathcal{F}$ being $\mathcal{F}$ a set of invertible functions, then the resulting composed application $c$ turns out to be

$$c(f, x) = f^{-1}(f(x)) = x \qquad (138)$$

This particular choice makes $c$ an application no longer dependent on $f$, with trivially

$$\partial_{fc}(f, x)[\delta f] = 0 \quad \forall f \in \mathcal{F}, x \in \mathbb{R}, \delta f \in \delta \mathcal{F} \qquad (139)$$

Furthermore, the chosen application $a$ is already linear in the argument function $f$, so that its linearization in $f$ coincides with $a$ itself and:

$$\partial_{fa}(f, x)[\delta f] = \partial_{ff}(x)[\delta f] = \delta f(x) \qquad (140)$$

According to the previous identity, and using the following notation for the application of $\partial_{fb}$ to $\delta f$

$$\partial_{fb}(f, y)[\delta f] = \partial_{ff^{-1}}(y)[\delta f] \qquad (141)$$
the application of (136) can be written as follows

$$ \partial_f f^{-1}(y) [\delta f] \circ a + \frac{\partial f^{-1}}{\partial y} \delta f(x) = 0 \quad (142) $$

Since $\frac{\partial f^{-1}}{\partial y}|_y = \left(\frac{\partial f}{\partial x}\right)_x^{-1}$ with $y = f(x)$, and since $\partial_f f^{-1}(y) [\delta f] \circ a = \partial_f f^{-1}(f, x) [\delta f]$ one computes the following formula for the variation of the inverse function:

$$ \partial_f f^{-1}(f, x) [\delta f] = - \left( \left. \frac{\partial f}{\partial x} \right|_x \right)^{-1} \delta f(x) \quad (143) $$

## 4.2 Multi-variable functions

Consider a functional $\varphi$ operating from a space of three-dimensional invertible mappings $\mathcal{M}$ into $\mathbb{R}$, such that $\alpha = \varphi(\chi)$ with $\alpha \in \mathbb{R}$ and $\chi \in \mathcal{M}$, being $\chi$ a three-dimensional mapping. Specifically $\chi$ defines the correspondence $\mathbf{x} = \chi(\mathbf{X})$ with $\mathbf{X}, \mathbf{x} \in \mathbb{R}^3$ being points of the reference and current configurations, respectively.

An infinitesimal variation $\delta\chi$ of mapping $\chi$ is the function $\mathbf{X} \to \delta\mathbf{x}$ where $\delta\mathbf{x} \in \mathbb{R}^3$ is the variation of the codomain point $\mathbf{x}$. The variation of functional $\varphi$ with respect to the argument mapping $\chi$ is indicated by $\partial_{\chi}\varphi$. Its components $\partial_{\chi_i}\varphi$ are the linear functionals such that, for a given variation function $\delta\chi_i$ of the $i$-th component of mapping $\delta\chi$, the limit condition holds

$$ \partial_{\chi_i} \varphi [\delta \chi_i] = \lim_{\lambda \to 0} \frac{\varphi(\chi + \lambda \delta \chi_i e^{(i)}) - \varphi(\chi)}{\lambda} \quad (144) $$

being $e^{(i)}$ is the unit vector of the $i$-th Cartesian director.

The relation between the scalar variation $\delta\alpha \in \mathbb{R}$ and $\delta\chi = \sum_{i=1}^3 \delta\chi_i e^{(i)}$ is notated as usual by placing $\delta\chi$ under square brackets:

$$ \delta\alpha = \partial_{\chi} \varphi [\delta\chi] = \sum_{i=1}^{3} \partial_{\chi_i} \varphi [\delta\chi_i] \quad (145) $$

### Variation of composition of functional applications

Let $\mathbf{A}^{(a)} : (\chi, \mathbf{X}) \to \mathbf{x}$ be a functional application relating a real function $\chi \in \mathcal{M}$ and a
point $\mathbf{X} \in \mathbb{R}^3$ to points $\mathbf{Y} \in \mathbb{R}^3$, viz.:

$$ \mathbf{Y} = \mathbf{A}^{(a)}(\chi, \mathbf{X}) \qquad (146) $$

and let $\mathbf{A}^{(a)} : (\chi, \mathbf{Y}) \to \mathbf{Z}$ be a second functional application relating a mapping $\chi \in \mathcal{M}$ and a point $\mathbf{Y} \in \mathbb{R}^3$ to points $\mathbf{Z} \in \mathbb{R}^3$:

$$ \mathbf{Z} = \mathbf{A}^{(b)}(\chi, \mathbf{Y}) \qquad (147) $$

Consider the composition of such two functional applications $\mathbf{A}^{(c)} = \mathbf{A}^{(b)} \circ \mathbf{A}^{(a)}$:

$$ \mathbf{A}^{(c)}(\chi, \mathbf{X}) = \mathbf{A}^{(b)}(\chi, \mathbf{A}^{(a)}(\chi, \mathbf{X})) \qquad (148) $$

Computation of the total variation of $\mathbf{A}^{(c)}$ with respect to $\chi$, $\delta \mathbf{Z} = \partial_{\chi} \mathbf{A}^{(c)}(\chi, \mathbf{X}) [\delta \chi]$, by application of the chain rule yields

$$
\begin{align}
\delta Z_i &= \partial_{\chi} A_i^{(c)} (\chi, \mathbf{X}) [\delta \chi] \nonumber \\
&= \sum_{j=1}^{3} \partial_{\chi_j} A_i^{(c)} (\chi, \mathbf{X}) [\delta \chi_j] \nonumber \\
&= \sum_{j=1}^{3} \partial_{\chi_j} A_i^{(b)} (\chi, \mathbf{X}) [\delta \chi_j] \circ \mathbf{A}^{(a)} (\chi, \mathbf{X}) \nonumber \\
&\quad + \sum_{j=1}^{3} \sum_{k=1}^{3} \frac{\partial A_i^{(b)}}{\partial Y_k} \partial_{\chi_j} A_k^{(a)} (\chi, \mathbf{X}) [\delta \chi_j] \tag{149}
\end{align}
$$

Special cases and variants of (149) are now examined. A first similar case of (151) is obtained by replacing $\mathbf{A}^{(b)}$ with an application having a scalar codomain with images $\mathbf{Z}$ in place of $\mathbf{Z}$:

$$
\begin{align}
\delta Z &= \sum_{j=1}^{3} \partial_{\chi_j} A^{(b)}(\chi, \mathbf{X}) [\delta \chi_j] \circ \mathbf{A}^{(a)}(\chi, \mathbf{X}) \nonumber \\
&\quad + \sum_{j=1}^{3} \sum_{k=1}^{3} \frac{\partial A^{(b)}}{\partial Y_k} \partial_{\chi_j} A_k^{(a)}(\chi, \mathbf{X}) [\delta \chi_j] \tag{150}
\end{align}
$$

A second simple special case of (149) is retrieved when $\mathbf{A}^{(b)}$ is independent from $\chi$, so that
$\partial_{\chi} \mathbf{A}^{(b)} [\delta\chi] = \mathbf{o}$. We have:

$$ \delta Z_i = \sum_{j=1}^{3} \sum_{k=1}^{3} \frac{\partial A_i^{(b)}}{\partial Y_k} \partial_{\chi_j} A_k^{(a)}(\chi, \mathbf{X}) [\delta \chi_j] \quad (151) $$

and, in case the external function has scalar codomain, the further specialization holds:

$$ \delta Z = \sum_{j=1}^{3} \sum_{k=1}^{3} \frac{\partial A^{(b)}}{\partial Y_k} \partial_{\chi_j} A_k^{(a)}(\chi, \mathbf{X}) [\delta \chi_j] \quad (152) $$

A variant of (151) is obtained when $\mathbf{Z}$ is replaced with a scalar function with images $\mathbf{Z}$ and when $\mathbf{A}^{(a)}$ is function of several independent mappings $\chi^{(1)}, \chi^{(2)}, \dots, \chi^{(N_m)}$, and of several additional scalar functions $\varphi^{(1)}, \varphi^{(2)}, \dots, \varphi^{(N_s)}$:

$$ \mathbf{A}^{(a)} = \mathbf{A}^{(a)}(\chi^{(1)}, \chi^{(2)}, \dots, \chi^{(N_m)}, \varphi^{(1)}, \varphi^{(2)}, \dots, \varphi^{(N_s)}) \quad (153) $$

In this case one has:

$$ \delta Z = \sum_{j=1}^{3} \sum_{k=1}^{3} \frac{\partial A^{(b)}}{\partial Y_k} \left( \sum_{h=1}^{N_m} \partial_{\chi_j^{(h)}} A_k^{(a)}(\chi, \mathbf{X}) [\delta \chi_j^{(h)}] + \sum_{h=1}^{N_s} \partial_{\varphi^{(h)}} A_k^{(a)}(\chi, \mathbf{X}) [\delta \varphi^{(h)}] \right) \quad (154) $$

For the composition of a scalar function $A^{(b)}(\mathbf{T})$ with codomain in $\mathbb{R}$ defined in a domain of second order tensors $\mathbf{T}$, with components $T_{ij}$, and a functional application over three-dimensional mappings $\chi, \mathbf{A}^{(a)}(\chi)$, with codomain in the space of second order tensors, developments analogous to those leading to (151) yield for the variation of the composed functional application $A^{(c)}(\chi) = A^{(b)}(\mathbf{A}^{(a)}(\chi)):

$$ \delta Z = \partial_{\chi} A^{(c)}(\chi) [\delta\chi] = \sum_{j=1}^{3} \sum_{h=1}^{3} \sum_{k=1}^{3} \frac{\partial A^{(b)}}{\partial T_{hk}} \partial_{\chi_j} A_{hk}^{(a)}(\chi) [\delta\chi_j] \quad (155) $$

For the composition $A^{(c)}(t) = A^{(b)}(\chi(t))$ of a functional $A^{(b)}(\chi)$, with codomain in $\mathbb{R}$ defined in a domain of 3D mappings $\chi \in \mathcal{M}$, with a 3D motion, defined as a correspondence between $t \in \mathbb{R} \to \chi$, the total time derivative $\dot{A}^{(c)}$ is computed by the chain rule written as it follows:

$$ \dot{A}^{(c)} = \sum_{j=1}^{3} \partial_{\chi_j} A^{(b)}(\chi) [\dot{\chi}_j] \quad (156) $$
Variation of the inverse of a 3D mapping

By choosing in (149), for $\mathbf{A}^{(a)}$, the application such that $\mathbf{A}^{(a)}(\chi, \mathbf{X}) = \chi(\mathbf{X})$, and, for $\mathbf{A}^{(b)}$, the application such that $\mathbf{A}^{(b)}(\chi, \mathbf{x}) = \chi^{-1}(\mathbf{x})$, with $\chi \in \mathcal{M}$, being $\mathcal{M}$ a set of invertible mappings, then the resulting composed application $\mathbf{A}^{(c)}$ turns out to be:

$$ \mathbf{A}^{(c)}(\chi, \mathbf{X}) = \chi^{-1}(\chi(\mathbf{X})) = \mathbf{X} \qquad (157) $$

Since the images of $\mathbf{A}^{(c)}$ are no longer dependent on $\chi$, one has:

$$ \delta\mathbf{Z} = \partial_{\chi}\mathbf{A}^{(c)}(\chi, \mathbf{X})[\delta\chi] = \mathbf{0} \quad \forall \chi \in \mathcal{M}, \mathbf{X} \in \mathbb{R}^3, \delta\chi \in \delta\mathcal{M} \qquad (158) $$

where $\delta\mathcal{M}$ is the tangent space to set $\mathcal{M}$. Moreover, considering that the selected application $\mathbf{A}^{(a)}$ is already linear in the argument $\chi$, it turns out to be:

$$ \partial_{\chi}\mathbf{A}^{(a)}(\chi, \mathbf{X})[\delta\chi] = \partial_{\chi}\chi(\mathbf{X})[\delta\chi] = \delta\chi(\mathbf{X}) \qquad (159) $$

which in components reads:

$$ \sum_{j=1}^{3} \partial_{\chi_j} A_{i}^{(a)}(\chi, \mathbf{X})[\delta\chi_j] = \delta_{ij}\delta\chi_j = \delta\chi_i \qquad (160) $$

According to the previous identity, and denoting the variation of $\mathbf{A}^{(b)}$ with respect to $\chi$ applied to $\delta\chi$ as:

$$ \partial_{\chi}\mathbf{A}^{(b)}(\chi, \mathbf{x})[\delta\chi] = \partial_{\chi}\chi^{-1}(\mathbf{x})[\delta\chi] \qquad (161) $$

application of (149) reads:

$$ \partial_{\chi_j}\chi_I^{-1}(\mathbf{x})[\delta\chi_j] \circ \mathbf{A}^{(a)}(\chi, \mathbf{X}) + \sum_{j=1}^{3} \frac{\partial\chi_I^{-1}}{\partial x_j}(\mathbf{X})\delta\chi_j = 0 \qquad (162) $$

Recalling that $\frac{\partial\chi_I^{-1}}{\partial x_j}\big|_{\mathbf{x}} = \left(\frac{\partial\chi}{\partial\mathbf{X}}\bigg|_{\mathbf{x}}\right)_{Ij}^{-1}$ with $\mathbf{x} = \chi(\mathbf{X})$, the formula of the variation of the inverse mapping is computed:

$$ \partial_{\chi_j}\chi_I^{-1}(\chi, \mathbf{X})[\delta\chi_j] = -\sum_{j=1}^{3} \frac{\partial\chi_I^{-1}}{\partial x_j}\bigg|_{\mathbf{x}} \delta\chi_j(\mathbf{X}) = -\sum_{j=1}^{3} \left(\frac{\partial\chi}{\partial\mathbf{X}}\bigg|_{\mathbf{x}}\right)_{Ij}^{-1} \delta\chi_j(\mathbf{X}) \qquad (163) $$
Henceforth, and throughout the manuscript, in order to achieve an abbreviated notation, sum-
mation symbols are omitted and the summation convention over repeated indices is applied.

Variation of the Jacobian of a mapping

With the previously introduced notation and identities at hand, the variation of the determi-
nant of an invertible mapping χ is computed from (155)

$$
\partial_{\chi_i} \bar{J}[\delta\chi_i] = \partial \frac{\partial \chi_h}{\partial X_J} \det \left( \frac{\partial \chi}{\partial \mathbf{X}} \right) \partial_{\chi_i} \left( \frac{\partial \chi_h}{\partial X_J} \right) [\delta\chi_i] = \bar{J} \frac{\partial (\chi)_J^{-1}}{\partial x_i} \frac{\partial \delta\chi_i}{\partial X_J} = \bar{J} \frac{\partial \delta\chi_i}{\partial x_i} \quad (164)
$$

Time rate of the Jacobian of a mapping

Application of (156) and account of (164) provide

$$
\dot{\bar{J}} = \partial_{\chi_j} \bar{J}(\chi) [\dot{\chi}_j] = \bar{J} \frac{\partial (\chi)_J^{-1}}{\partial x_i} \frac{\partial^2 \chi_i}{\partial t \partial X_J} = \bar{J} \frac{\partial \dot{\chi}_i}{\partial x_i} \quad (165)
$$

According to (165) the variation with respect to the deformation rate χ˙ of the time rate of the
Jacobian of a mapping simply turns out to be

$$
\partial_{\dot{\chi}_i} \dot{\bar{J}}[\delta\chi_i] = \bar{J} \frac{\partial (\chi)_J^{-1}}{\partial x_i} \frac{\partial \delta\chi_i}{\partial X_J} \quad (166)
$$

4.3 Euler-Lagrange equations

With the notation of Section 4.2 at hand, the developments yielding the strong form Euler-Lagrange equations from the generic least-action condition are hereby recalled for a continuum system whose state is defined by one scalar continuum field $\varphi : \Omega_0^{(M)} \times [t_0, t_f] \to \varphi \in \mathbb{R}$.

All mathematical hypotheses necessary for the developments below reported to make sense are given for granted.

Proceeding from the generic statement of the least-Action condition for a Lagrange function

$$
L_0^{(M)} \qquad \delta \int_{t_0}^{t_f} L_0^{(M)} (\varphi(t), \dot{\varphi}(t)) \, dt = 0 \tag{167}
$$

with $\varphi(t_0)$ and $\varphi(t_f)$ being given initial and final states, respectively. Account of the dependence
$L_0^{(M)} = L_0^{(M)}(\varphi, \dot{\varphi})$ up to first-order time derivatives of $\varphi$ and application of (145) with $\chi_1 = \varphi$
and $\chi_2 = \dot{\varphi}$ yields

$$
\int_{t_0}^{t_f} \left\{ \partial_{\varphi} L_0^{(M)}[\delta\varphi] + \partial_{\dot{\varphi}} L_0^{(M)}[\delta\dot{\varphi}] \right\} dt = 0 \quad (168)
$$
Considering the property $\delta\dot{\varphi} = \frac{d\delta\varphi}{dt}$ and that for a linear operator $A^{(l)}$ over a variable $\delta\varphi$ the identity holds $\frac{d}{dt}(A^{(l)}[\delta\varphi]) = (\frac{d}{dt}A^{(l)})[\delta\varphi] + A^{(l)}[\frac{d\delta\varphi}{dt}]$, relation (168) can be equated to:

$$ \int_{t_0}^{t_f} \left\{ \partial_{\varphi} L_0^{(M)}[\delta\varphi] + \frac{d}{dt} \left( \partial_{\dot{\varphi}} L_0^{(M)}[\delta\varphi] \right) - \left( \frac{d}{dt} \partial_{\dot{\varphi}} L_0^{(M)} \right) [\delta\varphi] \right\} dt = 0 \quad (169) $$

and hence to

$$ \int_{t_0}^{t_f} \left\{ \partial_{\varphi} L_0^{(M)}[\delta\varphi] - \left( \frac{d}{dt} \partial_{\dot{\varphi}} L_0^{(M)} \right) [\delta\varphi] \right\} dt + \partial_{\dot{\varphi}} L_0^{(M)}(t_f) [\delta\varphi(t_f)] - \partial_{\dot{\varphi}} L_0^{(M)}(t_0) [\delta\varphi(t_0)] = 0 \quad (170) $$

Taking into account that $\varphi(t_0)$ and $\varphi(t_f)$ are given, so that their variations $\delta\varphi(t_f)$ and $\delta\varphi(t_0)$ are null, the following generic format of the Euler-Lagrange equation associated with field $\varphi$ is inferred from (169):

$$ \left( \frac{d}{dt} \partial_{\dot{\varphi}} L_0^{(M)} - \partial_{\varphi} L_0^{(M)} \right) [\delta\varphi] = 0 \quad (171) $$

Since the previous relation must hold for any function $\delta\varphi$ one infers the strong-form equation:

$$ \frac{d}{dt} \partial_{\dot{\varphi}} L_0^{(M)} - \partial_{\varphi} L_0^{(M)} = 0 \quad (172) $$

which is complemented by constraints for field $\varphi$ and its variations $\delta\varphi$ over the space-time boundary $\partial\Omega_0^{(M)} \times [t_0, t_f]$.

For mechanical systems depending on a finite number of continuum fields $\varphi_1, \varphi_2, \dots, \varphi_N$, the weak statement provided by (171) generalizes to:

$$ \sum_{k=1}^{N} \left( \frac{d}{dt} \partial_{\dot{\varphi}_k} L_0^{(M)} - \partial_{\varphi_k} L_0^{(M)} \right) [\delta\varphi_k] = 0 \quad (173) $$

and the derivation of the system of strong-form PDE is subordinated to the specification of the space-time boundary constraints for fields $\varphi_1, \varphi_2, \dots, \varphi_N$ (and associated constraints for their variations $\delta\varphi_1, \delta\varphi_2, \dots, \delta\varphi_N$) over $\partial\Omega_0^{(M)} \times [t_0, t_f]$.

## 4.4 Variation of individual terms in Lagrange function

Hereby the computation of the variations of the individual terms $\delta U_0^{(s)}$, $\delta U_0^{(f)}$, $\delta T_0^{(s)}$, $\delta T_0^{(f)}$ required to obtain the explicit form of (64) and (65) is reported. In the developments below, it
is accounted for the property that the domain $\Omega_0^{(M)}$ is independent from kinematic descriptors $\bar{\chi}^{(s)}$, $\bar{\chi}^{(f)}$, $\hat{J}^{(s)}$, and that, consequently, all variations can be directly transferred to the integrand functions inside the integrals $\int_{\Omega_0^{(M)}} (\cdot) dV_0$.

Computation of $\delta U_0^{(s)}$

Application of (145) to (33) accounting for (25) and (43) yields:

$$ 
\begin{aligned}
\delta U_0^{(s)} &= \partial_{\bar{\chi}_i^{(s)}} U_0^{(s)} [\delta \bar{\chi}_i^{(s)}] + \partial_{\hat{J}^{(s)}} U_0^{(s)} [\delta \hat{J}^{(s)}] = \\
&= \int_{\Omega_0^{(M)}} \partial_{\bar{\chi}_i^{(s)}} \bar{\psi}_0^{(s)} [\delta \bar{\chi}_i^{(s)}] dV_0 + \int_{\Omega_0^{(M)}} \partial_{\hat{J}^{(s)}} \bar{\psi}_0^{(s)} [\delta \hat{J}^{(s)}] dV_0 \quad (174) \\
&= \int_{\Omega_0^{(M)}} \check{P}_{kJ}^{(s)} \frac{\partial \delta \bar{\chi}_k^{(s)}}{\partial X_J} dV_0 - \int_{\Omega_0^{(M)}} \hat{\Pi}^{(s)} \delta \hat{J}^{(s)} dV_0
\end{aligned}
$$

where the following trivial identities have been taken into account:

$$ \partial_{\bar{\chi}_i^{(s)}} \bar{\psi}_0^{(s)} [\delta \bar{\chi}_i^{(s)}] = \frac{\partial \bar{\psi}_0^{(s)}}{\partial \bar{F}_{kJ}^{(s)}} \partial_{\bar{\chi}_i^{(s)}} \bar{F}_{kJ}^{(s)} [\delta \bar{\chi}_i^{(s)}] \quad (175) $$

and

$$ \partial_{\bar{\chi}_i^{(s)}} \bar{F}_{kJ}^{(s)} [\delta \bar{\chi}_i^{(s)}] = \frac{\partial}{\partial X_J} \partial_{\bar{\chi}_i^{(s)}} \bar{\chi}_k^{(s)} [\delta \bar{\chi}_i^{(s)}] = \frac{\partial}{\partial X_J} \delta_{ik} [\delta \bar{\chi}_i^{(s)}] = \frac{\partial \delta \bar{\chi}_k^{(s)}}{\partial X_J} \quad (176) $$

Computation of $\delta U_0^{(f)}$

Due to the independence of domain $\Omega_0^{(M)}$ and field $\Phi_0^{(f)}$ from the kinematic descriptors, the variation of (34) turns out to be

$$ \delta U_0^{(f)} = \int_{\Omega_0^{(M)}} \Phi_0^{(f)} \delta \left( \hat{\psi}_0^{(f)} \circ \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} \right) dV_0 \quad (177) $$

Applying the chain rule in the form provided by (154) to the variation term in the integrand of (177), accounting for (23) and recalling also (46), one obtains

$$ \delta \left( \hat{\psi}_0^{(f)} \circ \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} \right) = \frac{\partial \hat{\psi}_0^{(f)}}{\partial \hat{J}^{(f)}} \delta \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} = - \hat{\Pi}^{(f)} \delta \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} \quad (178) $$

with

$$ \delta \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} = \partial_{\bar{\chi}_i^{(s)}} \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} [\delta \bar{\chi}_i^{(s)}] + \partial_{\bar{\chi}_i^{(f)}} \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} [\delta \bar{\chi}_i^{(f)}] + \partial_{\hat{J}^{(s)}} \hat{J}_{sat\bar{\chi}^{(f)}}^{(f)} [\delta \hat{J}^{(s)}] \quad (179) $$
The addends in the RHS of (179) are now separately computed based on the definition of $\hat{J}^{(f)}_{sat\bar{\chi}^{(f)}}$ provided by (22) and (23). For the first term it is computed

$$ \partial_{\bar{\chi}_i} \hat{J}^{(f)}_{sat\bar{\chi}^{(f)}} = - \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \partial_{\bar{\chi}_i} \phi^{(s)}_{\bar{\chi}^{(f)}} \quad (180) $$

The variation $\partial_{\bar{\chi}^{(s)}} \phi^{(s)}_{\bar{\chi}^{(f)}}$ in (180) has a complex kinematic meaning since it represents the variation of solid volume fraction experienced in a point which follows the macroscopic motion of the fluid phase, as it is induced by an infinitesimal isochoric variation of the solid phase deformation. Such term is however straightforwardly computed based on formulas (150) and (163). Actually, application of (150) to (22) yields

$$ \begin{aligned} \partial_{\bar{\chi}_i} \phi^{(s)}_{\bar{\chi}^{(f)}} [\delta \bar{\chi}_i^{(s)}] &= \partial_{\bar{\chi}_i} \left( \phi^{(s)}_{\bar{\chi}^{(s)}} \circ (\bar{\chi}^{(s)})^{-1} \circ \bar{\chi}^{(f)} \right) [\delta \bar{\chi}_i^{(s)}] \\ &= \left( \partial_{\bar{\chi}_i} \phi^{(s)}_{\bar{\chi}^{(s)}} [\delta \bar{\chi}_i^{(s)}] \right) \circ (\bar{\chi}^{(s)})^{-1} \circ \bar{\chi}^{(f)} \quad (181) \\ &\quad + \frac{\partial \phi^{(s)}_{\bar{\chi}^{(s)}}}{\partial X_J} \left\{ \partial_{\bar{\chi}_i} (\bar{\chi}^{(s)})^{-1}_J [\delta \bar{\chi}_i^{(s)}] \right\} \circ \bar{\chi}^{(f)} \end{aligned} $$

Recalling (21) and (164), the variation in the second row of (181) is recognized to be equal to

$$ \begin{aligned} \partial_{\bar{\chi}_i} \phi^{(s)}_{\bar{\chi}^{(s)}} [\delta \bar{\chi}_i^{(s)}] &= -\Phi_0^{(s)} \frac{\hat{J}^{(s)}}{(\bar{J}^{(s)})^2} \partial_{\bar{\chi}_i} \bar{J}^{(s)} [\delta \bar{\chi}_i^{(s)}] \\ &= -\Phi_0^{(s)} \frac{\hat{J}^{(s)}}{(\bar{J}^{(s)})^2} \bar{J}^{(s)} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial x_i} [\delta \bar{\chi}_i^{(s)}] \quad (182) \\ &= -\phi^{(s)}_{\bar{\chi}^{(s)}} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial x_i} [\delta \bar{\chi}_i^{(s)}] \end{aligned} $$

The variation in the third row of (181) is computed invoking (163)

$$ \partial_{\bar{\chi}_i} (\bar{\chi}^{(s)})^{-1} [\delta \bar{\chi}_i^{(s)}] = - \frac{\partial (\bar{\chi}^{(s)})^{-1}_J}{\partial x_i} \delta \bar{\chi}_i^{(s)} \quad (183) $$

Accordingly, one obtains for (181)

$$ \partial_{\bar{\chi}_i} \phi^{(s)}_{\bar{\chi}^{(f)}} [\delta \bar{\chi}_i^{(s)}] = -\phi^{(s)}_{\bar{\chi}^{(s)}} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial x_i} \circ (\bar{\chi}^{(s)})^{-1} \circ \bar{\chi}^{(f)} - \frac{\partial \phi^{(s)}_{\bar{\chi}^{(s)}}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})^{-1}_J}{\partial x_i} \delta \bar{\chi}_i^{(s)} \circ \bar{\chi}^{(f)} \quad (184) $$

As a side comment, the minus sign appearing in the first addend in (184) is explained consid-
ering that a macroscopic isochoric volumetric dilatation induces a decrease in the solid volume fraction. The second minus sign is justified by the consideration that a variation of solid dis- placement determines in a fixed space point a convected variation of porosity. This convected porosity variation is negative when directed along the porosity gradient.

For the second addend in the RHS of (179) one computes from (23)

$$
\partial_{\bar{\chi}_i} \hat{J}^{(f)}_{sat\bar{\chi}^{(f)}} [\delta \bar{\chi}_i^{(f)}] = \frac{\phi^{(f)}_{\bar{\chi}^{(f)}}}{\Phi_0^{(f)}} \partial_{\bar{\chi}_i^{(f)}} \bar{J}^{(f)} [\delta \bar{\chi}_i^{(f)}] - \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \partial_{\bar{\chi}_i^{(f)}} \phi^{(s)}_{\bar{\chi}^{(f)}} [\delta \bar{\chi}_i^{(f)}] \quad (185)
$$

The variation of the first term in the RHS of (185) is provided by (164)

$$
\partial_{\bar{\chi}_i} \bar{J} [\delta \bar{\chi}_i^{(f)}] = \bar{J}^{(f)} \frac{\partial \delta \bar{\chi}_i^{(f)}}{\partial x_i} \quad (186)
$$

The variation in the second term in the RHS of (185) contains the variation of solid volume
fraction measured in a point which follows the macroscopic motion of the fluid phase, as the
effect of the application of an infinitesimal variation to the fluid placement. It is computed by
application of (152) to the composition of functions $A^{(b)} = \phi^{(s)}_{\bar{\chi}^{(s)}} \circ (\bar{\chi}^{(s)})^{-1}$ and $A^{(a)} = \bar{\chi}^{(f)}$
appearing in (22), and turns out to be:

$$
\partial_{\bar{\chi}_i} \phi_{\bar{\chi}^{(f)}}^{(s)} [\delta \bar{\chi}_i^{(f)}] = \frac{\partial \phi_{\bar{\chi}^{(f)}}^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})^{-1}_J}{\partial x_i} \delta \bar{\chi}_i^{(f)} \quad (187)
$$

The variation of the third term in the RHS of (179) is provided again by the chain rule

$$
\partial_{\hat{J}(s)} \hat{J}^{(f)}_{sat\hat{\chi}^{(f)}} [\delta \hat{J}^{(s)}] = \frac{\partial \hat{J}^{(f)}_{sat\hat{\chi}^{(f)}}}{\partial \phi^{(s)}_{\hat{\chi}^{(f)}}} \frac{\partial \phi^{(s)}_{\hat{\chi}^{(f)}}}{\partial \hat{J}^{(s)}} [\delta \hat{J}^{(s)}] = - \frac{\bar{J}^{(f)} \Phi_0^{(s)}}{\Phi_0^{(f)} \bar{J}^{(s)}} [\delta \hat{J}^{(s)}] \quad (188)
$$

Addition of all the above computed variation terms appearing in the RHS of (179) provides the explicit expression for $\delta U_0^{(f)}$ shown by Equation (66) in the body of the paper. Moreover, composition of (66) with (177) and (178) provides the sought explicit expression for $\delta U_0^{(f)}$ shown by Equation (72) in the body of the paper.

Variation of $\delta U_{\Omega 0}^{ext}$
The variation $\delta U_{\Omega 0}^{ext}$ is computed from (35), (36) and (41), and turns out to be

$$ \begin{aligned} \delta U_{\Omega 0}^{ext} &= \partial_{\bar{\chi}_i^{(s)}} U_{\Omega 0}^{ext} [\delta \bar{\chi}_i^{(s)}] + \partial_{\bar{\chi}_i^{(f)}} U_{\Omega 0}^{ext} [\delta \bar{\chi}_i^{(f)}] \\ &= -\int_{\Omega_0^{(M)}} \bar{b}_{0i}^{(s,ext)} \delta \bar{\chi}_i^{(s)} dV_0 - \int_{\Omega_0^{(M)}} \bar{b}_{0i}^{(f,ext)} \delta \bar{\chi}_i^{(f)} dV_0 \end{aligned} \quad (189) $$

Variation terms associated with kinetic energy

The variation terms associated with $T_0^{(s)}$ and $T_0^{(f)}$, appearing in (65), are also computed applying the chain rules of Section 4.2 to Equations (28)-(30) and (39)-(40). Specifically, one computes for the terms associated with $T_0^{(s)}$

$$ \begin{aligned} \frac{d}{dt} \partial_{\bar{\chi}_i^{(s)}} T_0^{(s)} [\delta \bar{\chi}_i^{(s)}] &= \int_{\Omega_0^{(M)}} \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} \ddot{\delta \bar{\chi}_i^{(s)}} dV_0 \\ &\quad + \int_{\Omega_0^{(M)}} \frac{d}{dt} \partial_{\bar{\chi}_i^{(s)}} \bar{\kappa}_{0add}^{(s)} [\delta \bar{\chi}_i^{(s)}] dV_0 \end{aligned} \quad (190) $$

Computation of the microinertia term with $\bar{\kappa}_{0add}^{(s)}$ is carried out accounting for (28) with the aid of (165) and (166), according to which it results:

$$ \begin{aligned} \frac{d}{dt} \partial_{\bar{\chi}_i^{(s)}} \bar{\kappa}_{0add}^{(s)} [\delta \bar{\chi}_i^{(s)}] &= \bar{\rho}_{add.0}^{(s)} \frac{d}{dt} \left( \dot{\bar{J}}^{(s)} \partial_{\bar{\chi}_i^{(s)}} \dot{\bar{J}}^{(s)} [\delta \bar{\chi}_i^{(s)}] \right) \\ &= \bar{\rho}_{add.0}^{(s)} \frac{d}{dt} \left( \dot{\bar{J}}^{(s)} \frac{\partial \dot{\bar{J}}^{(s)}}{\partial \bar{J}^{(s)}} \partial_{\bar{\chi}_i^{(s)}} \bar{J}^{(s)} [\delta \bar{\chi}_i^{(s)}] \right) \quad (191) \\ &= \bar{\rho}_{add.0}^{(s)} \frac{d}{dt} \left( \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})^{-1}_J}{\partial x_i} \dot{\bar{J}}^{(s)} \right) \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J} \end{aligned} $$

Change of index $s$ to $f$ in (191) provides its counterpart associated with $T_0^{(f)}$.

A similar computation of the variation associated with the rate of the intrinsic solid strain yields

$$ \begin{aligned} \frac{d}{dt} \partial_{\bar{J}^{(s)}} L_0^{(M)} [\delta \bar{J}^{(s)}] &= \frac{d}{dt} \int_{\Omega_0^{(M)}} \left( \frac{\partial \bar{\kappa}_{0add}^{(s)}}{\partial \bar{J}^{(s)}} \frac{\partial \dot{\bar{J}}^{(s)}}{\partial \dot{\bar{J}}^{(s)}} \partial_{\bar{J}^{(s)}} \dot{\bar{J}}^{(s)} [\delta \bar{J}^{(s)}] + \frac{\partial \bar{\kappa}_{0add}^{(f)}}{\partial \bar{J}^{(f)}} \frac{\partial \dot{\bar{J}}^{(f)}}{\partial \dot{\bar{J}}^{(f)}} \partial_{\bar{J}^{(f)}} \dot{\bar{J}}^{(f)}_{sat\bar{\chi}^{(f)}} [\delta \bar{J}^{(s)}] \right) dV_0 \\ &= - \int_{\Omega_0^{(M)}} \left[ \bar{\rho}_{add.0}^{(s)} \frac{d}{dt} (\bar{J}^{(s)} - \dot{\bar{J}}^{(s)}) - \frac{d}{dt} \left( \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \bar{\rho}_{add.0}^{(f)} (\bar{J}^{(f)} - \dot{\bar{J}}^{(f)}) \right) \right] \delta \bar{J}^{(s)} dV_0 \end{aligned} \quad (192) $$

where, in the computation of (192), account has been taken of the property $\partial_g \dot{f} = \partial_g f$.

For simplicity, in the above computed variations associated with microinertia terms $\bar{\kappa}_{0add}^{(s)}$ and $\bar{\kappa}_{0add}^{(f)}$, the terms containing second time rates are considered to be dominant with respect
to terms containing only first time derivatives so that the latter are ruled out. Accordingly, the following inertia terms are finally computed for (65):

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}_i}^{(s)} L_0^{(M)} [\delta \bar{\chi}_i^{(s)}] &= \int_{\Omega_0^{(M)}} \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} \ddot{\delta \bar{\chi}_i^{(s)}} dV_0 + \int_{\Omega_0^{(M)}} \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J} (\bar{J}^{(s)} - \hat{J}^{(s)}) dV_0 \nonumber \\
\frac{d}{dt} \partial_{\bar{\chi}_i^{(f)}} L_0^{(M)} [\delta \bar{\chi}_i^{(f)}] &= \int_{\Omega_0^{(M)}} \bar{\rho}_0^{(s)} \bar{\chi}_i^{(f)} \ddot{\delta \bar{\chi}_i^{(f)}} dV_0 + \int_{\Omega_0^{(M)}} \bar{\rho}_{add.0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J} (\bar{J}^{(s)} - \hat{J}^{(s)}) dV_0 \nonumber \\
\frac{d}{dt} \partial_{\hat{J}^{(s)}} L_0^{(M)} [\delta \hat{J}^{(s)}] &= - \int_{\Omega_0^{(M)}} \left[ \bar{\rho}_{add.0}^{(s)} (\bar{J}^{(s)} - \hat{J}^{(s)}) - \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \bar{\rho}_{add.0}^{(f)} (\bar{J}^{(f)} - \hat{J}^{(f)}) \right] \delta \hat{J}^{(s)} dV_0 \tag{193}
\end{align}
$$

To facilitate the assembly of all the above computed variations of the individual terms of $\delta U_0^{(s)}$, $\delta U_0^{(f)}$, $\delta T_0^{(s)}$, $\delta T_0^{(f)}$ in (62), as an intermediate step towards the computation of (73) and (74), these individual terms are grouped in the order appearing in the first three rows of (62). We also report for each group the application of integration by parts and of the divergence theorem, required to transform (73) in (74).

Accordingly, the first row of (62) turns out to be equal to

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] - \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] &= \nonumber \\
&= \int_{\Omega_0^{(M)}} \left[ \check{P}_{iJ} - \bar{J}^{(f)} \phi_{\bar{\chi}^{(s)}}^{(s)} \hat{\Pi}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} + \bar{\rho}_{add.0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{(s)} - \hat{J}^{(s)}) \right] \frac{\partial \delta \bar{\chi}_i^{(s)}}{\partial X_J^{(s)}} dV_0 \nonumber \\
&\quad + \int_{\Omega_0^{(M)}} \left( -\hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(s)}}^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} - \bar{b}_{0i}^{(s,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{(s)} \ddot{\delta} \right) \delta \bar{\chi}_i^{(s)} dV_0 \tag{194}
\end{align}
$$

Applying integration by parts and the divergence theorem and accounting for the identity

$$
\bar{J}^{(f)} \phi_{\bar{\chi}^{(s)}}^{(s)} \hat{\Pi}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} - \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(s)}}^{(s)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} = \phi_{\bar{\chi}^{(s)}}^{(s)} \frac{\partial}{\partial X_J} \left( \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \right) \quad (195)
$$
one finally computes:

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] - \partial_{\bar{\chi}^{(s)}} L_0^{(M)} [\delta \bar{\chi}^{(s)}] = & \nonumber \\
& - \int_{\partial \Omega_0^{(M)}} \left[ \hat{P}_{iJ} - \bar{J}^{(f)} \phi_{\bar{\chi}^{(s)}}^{(s)} \hat{\Pi}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} + \bar{\rho}_{\text{add.}0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{(s)}} - \hat{J}^{\ddot{(s)}}) \right] N_J \delta \bar{\chi}_i^{(s)} dV_0 + \nonumber \\
& - \int_{\Omega_0^{(M)}} \left[ \frac{\partial \hat{P}_{iJ}}{\partial X_J^{(s)}} - \phi_{\bar{\chi}^{(s)}}^{(s)} \frac{\partial}{\partial X_J} \left( \hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} \right) \right] \delta \bar{\chi}_i^{(s)} dV_0 + \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{b}_{0i}^{(s,ext)} + \bar{\rho}_0^{(s)} \bar{\chi}_i^{\ddot{(s)}} - \frac{\partial}{\partial X_J^{(s)}} \left( \bar{\rho}_{\text{add.}0}^{(s)} \bar{J}^{(s)} \frac{\partial (\bar{\chi}^{(s)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{(s)}} - \hat{J}^{\ddot{(s)}}) \right) \right] \delta \bar{\chi}_i^{(s)} dV_0 \tag{196}
\end{align}
$$

In a similar way, collection of terms appearing in the second row of (62) yields:

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] - \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] = & \nonumber \\
& - \int_{\Omega_0^{(M)}} \left[ -\hat{\Pi}^{(f)} \phi_{\bar{\chi}^{(f)}}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} + \bar{\rho}_{\text{add.}0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{(f)}} - \hat{J}^{\ddot{(f)}}) \right] \frac{\partial \delta \bar{\chi}_i^{(f)}}{\partial X_J} dV_0 + \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\hat{\Pi}^{(f)} \bar{J}^{(f)} \frac{\partial \phi_{\bar{\chi}^{(f)}}^{(f)}}{\partial X_J} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} - \bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(f)} \bar{\chi}_i^{\ddot{(f)}} \right] \delta \bar{\chi}_i^{(f)} dV_0 \tag{197}
\end{align}
$$

Applying derivation by parts and the divergence theorem one obtains:

$$
\begin{align}
\frac{d}{dt} \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] - \partial_{\bar{\chi}^{(f)}} L_0^{(M)} [\delta \bar{\chi}^{(f)}] = & \nonumber \\
& - \int_{\partial \Omega_0^{(M)}} \left[ -\hat{\Pi}^{(f)} \phi_{\bar{\chi}^{(f)}}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} + \bar{\rho}_{\text{add.}0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{(f)}} - \hat{J}^{\ddot{(f)}}) \right] N_J \delta \bar{\chi}_i^{(f)} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ \phi_{\bar{\chi}^{(f)}}^{(f)} \frac{\partial}{\partial X_J} \left( \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} \hat{\Pi}^{(f)} \right) - \bar{\rho}_{\text{add.}0}^{(f)} \bar{J}^{(f)} \frac{\partial (\bar{\chi}^{(f)})_J^{-1}}{\partial x_i} (\bar{J}^{\ddot{(f)}} - \hat{J}^{\ddot{(f)}}) \right] \delta \bar{\chi}_i^{(f)} dV_0 \nonumber \\
& + \int_{\Omega_0^{(M)}} \left[ -\bar{b}_{0i}^{(f,ext)} + \bar{\rho}_0^{(f)} \bar{\chi}_i^{\ddot{(f)}} \right] \delta \bar{\chi}_i^{(f)} dV_0 \tag{198}
\end{align}
$$

Finally, the third row of of (62) turns out to be:

$$
\begin{align}
\frac{d}{dt} \partial_{\hat{J}^{(s)}} L_0^{(M)} [\delta \hat{J}^{(s)}] - \partial_{\hat{J}^{(s)}} L_0^{(M)} [\delta \hat{J}^{(s)}] = & \nonumber \\
& - \int_{\Omega_0^{(M)}} \left[ -\hat{\Pi}^{(s)} + \frac{\bar{J}^{(f)}}{\bar{J}^{(s)}} \Phi_0^{(s)} \hat{\Pi}^{(f)} \delta \hat{J}^{(s)} - \bar{\rho}_{\text{add.}0}^{(s)} (\bar{J}^{\ddot{(s)}} - \hat{J}^{\ddot{(s)}}) + \frac{\bar{J}^{(f)}}{\Phi_0^{(f)}} \frac{\Phi_0^{(s)}}{\bar{J}^{(s)}} \bar{\rho}_{\text{add.}0}^{(f)} (\bar{J}^{\ddot{(f)}} - \hat{J}^{\ddot{(f)}}) \right] \delta \hat{J}^{(s)} dV_0 \tag{199}
\end{align}
$$
# 5 List of Symbols

Vector and tensor quantities are notated in bold. Null vector quantities and null tensor quantities are indicated by **o** and **O**, respectively. Gibbs nabla vector ∇ is used, when necessary, for implicit shorthand representation of differential operations. For all microscale quantities, of kinematic, stress and stiffness type, no accent is used. In general, an overbar is used to denote macroscopic kinematic quantities, except for kinematic quantities of *intrinsic* type which are denoted by a hat accent. Macroscopic stress quantities are all notated with accents different from the overbar, with the only exceptions of macroscopic Cauchy stresses and of the fluid pressure which have no accent. Intrinsic macroscopic stresses are denoted by a hat accent while extrinsic macroscopic stresses are denoted by a check accent (a reversed hat). In general, subscripts indicate the Cartesian indexes of scalar components. The phase to which a given quantity belongs is denoted by bracketed superscripts, *s* or *f*, if not otherwise stated. Specific items are listed in the following tables.
<table><caption>Table 1: List of symbols</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>Ω<sub>0</sub><sup>(M)</sup></td><td>macroscopic reference domain of the mixture</td></tr><tr><td>Ω<sup>(M)</sup></td><td>macroscopic current domain of the mixture</td></tr><tr><td>Ω<sub>0</sub><sup>(s)</sup></td><td>macroscopic reference subset domain of the mixture with nonvanishing solid phase</td></tr><tr><td>Ω<sup>(s)</sup></td><td>macroscopic current subset domain of the mixture with nonvanishing solid phase</td></tr><tr><td>Ω<sub>0</sub><sup>(f)</sup></td><td>macroscopic reference subset domain of the mixture entirely occupied by the fluid</td></tr><tr><td>Ω<sup>(f)</sup></td><td>macroscopic current subset domain of the mixture entirely occupied by the fluid</td></tr><tr><td>∂Ω<sub>0</sub><sup>(M)</sup></td><td>boundary of macroscopic reference domain of the mixture</td></tr><tr><td>∂Ω<sup>(M)</sup></td><td>boundary of macroscopic current domain of the mixture</td></tr><tr><td>∂Ω<sub>0</sub><sup>(MU)</sup></td><td>surface common to the boundaries of the reference domain of the mixture and of its solid subset</td></tr><tr><td>S<sub>0</sub><sup>(sf)</sup></td><td>free solid-fluid macroscopic interface in the reference configuration</td></tr><tr><td>S<sup>(sf)</sup></td><td>free solid-fluid macroscopic interface in the current configuration</td></tr><tr><td>X</td><td>generic point of the macroscopic reference domain of the mixture</td></tr><tr><td>x</td><td>generic point of the macroscopic current domain of the mixture</td></tr><tr><td>RVE<br>Ω<sub>0</sub><sup>(X)</sup></td><td>microscale RVE of the mixture in the reference configuration, centered in the point X</td></tr><tr><td>RVE<sup>(s)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>solid subset of the X-centered microscale reference RVE of the mixture</td></tr><tr><td>RVE<sup>(f)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>fluid subset of the X-centered microscale reference RVE of the mixture</td></tr><tr><td>RVE<br>Ω<sup>(X)</sup></td><td>microscale RVE of the mixture in the current configuration, centered in the point x</td></tr><tr><td>RVE<sup>(s)</sup><br>Ω<sup>(X)</sup></td><td>solid subset of the x-centered microscale RVE of the mixture in the current configuration</td></tr><tr><td>RVE<sup>(f)</sup><br>Ω<sup>(X)</sup></td><td>fluid subset of the x-centered microscale RVE of the mixture in the current configuration</td></tr><tr><td>χ<sup>(s)</sup></td><td>macroscopic placement vector of the solid phase</td></tr><tr><td>χ<sup>(f)</sup></td><td>macroscopic placement vector of the fluid phase</td></tr><tr><td>V<sub>0RVE</sub><sup>(X)</sup></td><td>volume of the X-centered microscale reference RVE of the mixture</td></tr><tr><td>V<sub>0RVE</sub><sup>(s)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>volume of the solid subset of the X-centered microscale reference RVE of the mixture</td></tr><tr><td>V<sub>0RVE</sub><sup>(f)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>volume of the fluid subset of the X-centered microscale reference RVE of the mixture</td></tr><tr><td>V<sub>RVE</sub><sup>(X)</sup></td><td>volume of the x-centered microscale deformed RVE</td></tr><tr><td>V<sub>RVE</sub><sup>(s)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>volume of the solid subset of x-centered microscale deformed RVE</td></tr><tr><td>V<sub>RVE</sub><sup>(f)</sup><br>Ω<sub>0</sub><sup>(X)</sup></td><td>volume of the fluid subset of x-centered microscale deformed RVE</td></tr><tr><td>Φ<sub>0</sub><sup>(s)</sup></td><td>reference solid volume fraction</td></tr><tr><td>Φ<sub>0</sub><sup>(f)</sup></td><td>reference fluid volume fraction</td></tr><tr><td>φ<sup>(s)</sup></td><td>current solid volume fraction</td></tr><tr><td>φ<sup>(f)</sup></td><td>current fluid volume fraction</td></tr><tr><td>ŷ<sup>(s)</sup></td><td>finite macroscopic intrinsic volumetric strain of the solid phase</td></tr><tr><td>ŷ<sup>(f)</sup></td><td>finite macroscopic intrinsic volumetric strain of the fluid phase</td></tr><tr><td>ȳ<sup>(s)</sup></td><td>finite macroscopic extrinsic volumetric strain of the solid phase</td></tr><tr><td>ȳ<sup>(f)</sup></td><td>finite macroscopic extrinsic volumetric strain of the fluid phase</td></tr><tr><td>J<sup>(s)</sup></td><td>finite microscale volumetric strain of the fluid phase</td></tr><tr><td>ρ̂<sub>0</sub><sup>(s)</sup></td><td>macroscopic true density of the solid phase in the reference configuration</td></tr><tr><td>ρ̂<sup>(s)</sup></td><td>macroscopic true density of the solid phase in the current configuration</td></tr><tr><td>ρ̄<sub>0</sub><sup>(s)</sup></td><td>apparent reference solid density</td></tr><tr><td>ρ̄<sub>0</sub><sup>(f)</sup></td><td>apparent reference fluid density<sub>68</sub></td></tr><tr><td>ρ̄<sup>(s)</sup></td><td>apparent current solid density</td></tr><tr><td>ρ̄<sup>(f)</sup></td><td>apparent current fluid density</td></tr></tbody></table>
<table><caption>Table 2: List of symbols</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>φ<sub>x</sub><sup>(s)</sup></td><td>spatial field of solid volume fraction</td></tr><tr><td>φ<sub>x</sub><sup>(s)</sup></td><td>spatial field of fluid volume fraction</td></tr><tr><td>J&#770;<sup>(f)</sup><sub>satχ<sup>(f)</sup></sub></td><td>saturation-based field relating J&#770;<sup>(f)</sup> to primary kinematic descriptors χ&#770;<sup>(s)</sup>, χ&#770;<sup>(f)</sup>, J&#770;<sup>(s)</sup></td></tr><tr><td>F&#770;<sup>(s)</sup></td><td>macroscopic deformation gradient of the solid phase</td></tr><tr><td>ū<sup>(s)</sup></td><td>infinitesimal macroscopic displacement vector of the solid phase</td></tr><tr><td>ū<sup>(f)</sup></td><td>infinitesimal macroscopic displacement vector of the fluid phase</td></tr><tr><td>ê<sup>(s)</sup></td><td>infinitesimal macroscopic intrinsic volumetric strain of the solid phase</td></tr><tr><td>ê<sup>(f)</sup></td><td>infinitesimal macroscopic intrinsic volumetric strain of the fluid phase</td></tr><tr><td>ē<sup>(s)</sup></td><td>infinitesimal macroscopic extrinsic volumetric strain of the solid phase</td></tr><tr><td>ē<sup>(f)</sup></td><td>infinitesimal macroscopic extrinsic volumetric strain of the fluid phase</td></tr><tr><td>ε&#770;<sup>(s)</sup></td><td>infinitesimal extrinsic strain tensor</td></tr><tr><td>L<sub>0</sub><sup>(M)</sup></td><td>Lagrange function of the mixture</td></tr><tr><td>L<sub>0</sub><sup>(sf)</sup></td><td>term of Lagrange function describing drag solid-fluid interaction</td></tr><tr><td>T<sub>0</sub><sup>(M)</sup></td><td>kinetic energy of the mixture</td></tr><tr><td>T<sub>0</sub><sup>(s)</sup></td><td>kinetic energy of the solid phase</td></tr><tr><td>T<sub>0</sub><sup>(f)</sup></td><td>kinetic energy of the fluid phase</td></tr><tr><td>U<sub>0</sub><sup>(M)</sup></td><td>potential energy of the mixture</td></tr><tr><td>U<sub>0</sub><sup>(s)</sup></td><td>strain energy of the solid phase</td></tr><tr><td>U<sub>0</sub><sup>(f)</sup></td><td>strain energy of the fluid phase</td></tr><tr><td>U<sub>0</sub><sup>ext</sup></td><td>potential energy of the mixture due to external actions</td></tr><tr><td>U<sub>0</sub><sup>ext</sup></td><td>part of the potential energy due to external actions, associated with volume forces</td></tr><tr><td>U<sub>0</sub><sup>ext</sup></td><td>part of the potential energy due to external actions, associated with boundary forces</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(s)</sup></td><td>apparent solid strain energy density per unit reference space</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(f)</sup></td><td>apparent fluid strain energy density per unit reference space</td></tr><tr><td>ψ<sub>0</sub><sup>(f)</sup></td><td>microscale fluid strain energy density per unit reference space</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(s)</sup></td><td>true solid strain energy density per unit reference space</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(f)</sup></td><td>true fluid strain energy density per unit reference space</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(s)</sup></td><td>reference potential energy density of solid associated with external volume forces</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(f)</sup></td><td>reference potential energy density of fluid associated with external volume forces</td></tr><tr><td>ψ&#770;<sub>0</sub><sup>(∂)</sup></td><td>apparent reference surface density of potential energy associated with boundary forces</td></tr><tr><td>κ&#770;<sub>0</sub><sup>(s)</sup></td><td>apparent density of kinetic energy of the solid phase</td></tr><tr><td>κ&#770;<sub>0</sub><sup>(f)</sup></td><td>apparent density of kinetic energy of the fluid phase</td></tr><tr><td>κ&#770;<sub>0</sub><sup>(s)</sup></td><td>microinertia-associated part of the apparent density of kinetic energy of the solid phase</td></tr><tr><td>κ&#770;<sub>0</sub><sup>(f)</sup></td><td>microinertia-associated part of the apparent density of kinetic energy of the fluid phase</td></tr><tr><td>ρ&#770;<sub>add. 0</sub><sup>(s)</sup></td><td>microinertia-associated density parameter of the solid phase</td></tr><tr><td>ρ&#770;<sub>add. 0</sub><sup>(f)</sup></td><td>microinertia-associated density parameter of the fluid phase</td></tr><tr><td>b&#770;<sub>0</sub><sup>(s,ext)</sup></td><td>solid external volume forces in the reference configuration</td></tr><tr><td>b&#770;<sub>0</sub><sup>(f,ext)</sup></td><td>fluid external volume forces in the reference configuration</td></tr><tr><td>b&#770;<sub>0</sub><sup>(s,ext)</sup></td><td>solid external volume forces in the current configuration</td></tr><tr><td>b&#770;<sub>0</sub><sup>(f,ext)</sup></td><td>fluid external volume forces in the current configuration</td></tr></tbody></table>
<table><caption>Table 3: List of symbols</caption><thead><tr><th>Symbol</th><th>Description</th></tr></thead><tbody><tr><td>b<sup>(fs)</sup>, b<sup>(sf)</sup></td><td>drag volume forces associated with solid-fluid interaction</td></tr><tr><td>b<sub>x</sub><sup>(fs)</sup>, b<sub>x</sub><sup>(sf)</sup></td><td>spatial vector fields of drag volume forces associated with solid-fluid interaction</td></tr><tr><td>P<sup>(s)</sup></td><td>extrinsic first Piola-Kirchhoff stress tensor of the solid phase</td></tr><tr><td>Π<sup>(s)</sup></td><td>intrinsic solid pressure</td></tr><tr><td>σ<sup>(s)</sup></td><td>spatial form of the extrinsic solid stress tensor</td></tr><tr><td>p</td><td>spatial form of the intrinsic solid pressure</td></tr><tr><td>σ<sup>(f)</sup></td><td>reference form of the true intrinsic fluid pressure</td></tr><tr><td>p<sup>(f)</sup></td><td>reference form of the intrinsic fluid pressure</td></tr><tr><td>p<sup>(ext)</sup></td><td>spatial form of the true intrinsic fluid pressure</td></tr><tr><td>p<sup>(ext)</sup></td><td>spatial form of the intrinsic fluid pressure</td></tr><tr><td>φ<sub>x</sub><sup>(s)</sup>, φ<sub>x</sub><sup>(f)</sup>, φ<sub>y</sub><sup>(s)</sup>, φ<sub>y</sub><sup>(f)</sup></td><td>reference maps of volume fraction fields along phase placements</td></tr><tr><td>A<sub>ijkl</sub>, B<sub>ij</sub>, C</td><td>generic elastic moduli</td></tr><tr><td>k<sub>f</sub></td><td>fluid intrinsic stiffness</td></tr><tr><td>σ<sup>(ext)</sup></td><td>external stress tensor</td></tr><tr><td>σ<sub>h</sub><sup>(s)</sup></td><td>value attained by the extrinsic stress tensor in a region with uniform stress state</td></tr><tr><td>p<sub>h</sub></td><td>value attained by the fluid pressure in a region with uniform stress state</td></tr><tr><td>Ω<sup>(h)</sup></td><td>macroscopic space region with uniform stress state</td></tr><tr><td>Ω<sup>(u)</sup></td><td>macroscopic space region undergoing undrained flow conditions</td></tr></tbody></table>