Roberto Serpieri
email: roberto.serpieri@unisannio.it

Francesco Travascio
email: f.travascio@miami.edu

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

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

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 momentum balances, inclusive of the momentum balance of the fluid phase, is attained on a purely variational basis.

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 [START_REF] Fillunger | Erdbaumechanik?[END_REF][START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF][START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF], multiphase continuum poroelasticity has become in the last decades an indispensable theoretical tool in biomechanics (see e.g., [START_REF] Andreaus | A 2-d continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time[END_REF][START_REF] Ateshian | Multigenerational interstitial growth of biological tissues[END_REF][START_REF] Ehlers | Porous media: theory, experiments and numerical applications[END_REF][START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF]), and, more recently, also in impact engineering [START_REF] Markert | A biphasic continuum approach for viscoelastic high-porosity foams: comprehensive theory, numerics, and application[END_REF], due to its importance for the understanding and prediction of several complex physical phenomena occurring in solids interacting with other phases [START_REF] Lopatnikov | Macroscopic lagrangian formulation of poroelasticity with porosity dynamics[END_REF][START_REF] Gray | The solid phase stress tensor in porous media mechanics and the hill-mandel condition[END_REF][START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF][START_REF] Madeo | A continuum model for deformable, second gradient porous media partially saturated with compressible fluids[END_REF][START_REF] De Buhan | On the validity of the effective stress concept for assessing the strength of saturated porous materials: a homogenization approach[END_REF].

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 [START_REF] Boer | Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory[END_REF][START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF][START_REF] Schrefler | Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions[END_REF][START_REF] Gray | Averaging theory for description of environmental problems: What have we learned?[END_REF][START_REF] Serpieri | Variational theories of two-phase continuum poroelastic mixtures: a short survey[END_REF]).

The many theories so far proposed differ by the axiomatic schemes and/or methodological approaches employed to infer governing equations [START_REF] Truesdell | The classical field theories[END_REF][START_REF] Cowin | A variational principle for granular materials[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF]. 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 [START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF] 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), [START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF][START_REF] Serpieri | Variational theories of two-phase continuum poroelastic mixtures: a short survey[END_REF][START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF][START_REF] Gajo | A general approach to isothermal hyperelastic modelling of saturated porous media at finite strains with compressible solid constituents[END_REF].

Theories also differ by the structure of the macroscopic governing PDEs [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF][START_REF] Albers | Influence of coupling through porosity changes on the propagation of acoustic waves in linear poroelastic materials[END_REF][START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF][START_REF] Bowen | Compressible porous media models by use of the theory of mixtures[END_REF][START_REF] Goodman | A continuum theory for granular materials[END_REF][START_REF] Lopatnikov | Poroelasticity-i: governing equations of the mechanics of fluidsaturated porous materials[END_REF], as well as by the physical-mathematical, or engineering, definition employed to introduce macroscopic stress measures [START_REF] Gray | The solid phase stress tensor in porous media mechanics and the hill-mandel condition[END_REF][START_REF] Schrefler | Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Biot | The theory of consolidation[END_REF][START_REF] Bishop | The effective stress principle[END_REF][START_REF] Coussy | From mixture theory to Biot's approach for porous media[END_REF][START_REF] De Boer | The development of the concept of effective stresses[END_REF][START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF][START_REF] Gray | Unsaturated flow theory including interfacial phenomena[END_REF][START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF][START_REF] Skempton | Terzaghi's discovery of effective stress, From Theory to Practice in Soil Mechanics: Selections from the Writings of Karl Terzaghi[END_REF]. 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" [START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF], and has been remarked even more recently [START_REF] Lopatnikov | Poroelasticity-i: governing equations of the mechanics of fluidsaturated porous materials[END_REF], 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 [START_REF] Simo | Computational inelasticity[END_REF]), 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 [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF].

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 [START_REF] Schrefler | Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions[END_REF][START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF][START_REF] Bowen | Compressible porous media models by use of the theory of mixtures[END_REF][START_REF] Hassanizadeh | Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries[END_REF], while other theories introduce supplemental equations which have a constitutive character [START_REF] Bowen | Compressible porous media models by use of the theory of mixtures[END_REF][START_REF] Lopatnikov | Poroelasticity-i: governing equations of the mechanics of fluidsaturated porous materials[END_REF][START_REF] Drumheller | The theoretical treatment of a porous solid using a mixture theory[END_REF]. Also, as observed in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Jardine | Developments in understanding soil behaviour[END_REF][START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF], 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 [START_REF] Lanczos | The variational principles of mechanics[END_REF][START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF][START_REF] Auffray | Analytical continuum mechanics á la hamilton-piola least action principle for second gradient continua and capillary fluids[END_REF]. 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 [START_REF] Auffray | Analytical continuum mechanics á la hamilton-piola least action principle for second gradient continua and capillary fluids[END_REF]. In addition, continuum variational approaches offer several further advantages. Boundary conditions are simultaneously derived with bulk field equations [START_REF] Madeo | A continuum model for deformable, second gradient porous media partially saturated with compressible fluids[END_REF][START_REF] Dell'isola | Variational principles are a powerful tool also for formulating field theories[END_REF][START_REF] Dell'isola | Edge contact forces and quasi-balanced power[END_REF][START_REF] Eremeyev | The nonlinear theory of elastic shells with phase transitions[END_REF][START_REF] Pietraszkiewicz | Extended non-linear relations of elastic shells undergoing phase transitions[END_REF] 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 derivation 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 application of classical variational principles, or on some variants of these principles, or even on the simple application of some variational concepts [START_REF] Lopatnikov | Macroscopic lagrangian formulation of poroelasticity with porosity dynamics[END_REF][START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF][START_REF] Cowin | A variational principle for granular materials[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Lopatnikov | Poroelasticity-i: governing equations of the mechanics of fluidsaturated porous materials[END_REF][START_REF] Biot | Theory of finite deformations of porous solids[END_REF][START_REF] Biot | Variational lagrangian-thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion[END_REF][START_REF] Kenyon | Thermostatics of solid-fluid mixtures[END_REF][START_REF] Aizicovici | A variational theorem in the linear theory of mixtures of two elastic solids. the quasi-static case[END_REF][START_REF] Nunziato | On ideal multiphase mixtures with chemical reactions and diffusion[END_REF][START_REF] Passman | Mixtures of granular materials[END_REF][START_REF] Bedford | A variational theory of immiscible mixtures[END_REF][START_REF] Gouin | Hamilton's principle and rankine-hugoniot conditions for general motions of mixtures[END_REF][START_REF] Gouin | Hamiltonian principle in binary mixtures of euler fluids with applications to the second sound phenomena[END_REF][START_REF] Lopatnikov | Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body[END_REF][START_REF] Lopatnikov | Poroelasticity-iii: conditions on the interfaces[END_REF][START_REF] Sciarra | Dilatancy and compaction around a cylindrical cavern leached-out in a fluid saturated salt rock[END_REF][START_REF] Dell'isola | Boundary conditions at fluid-permeable interfaces in porous media: A variational approach[END_REF]. Reviews specifically dedicated to this subject are the one by Bedford and Drumheller [START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF] and the more recent one in [START_REF] Serpieri | Variational theories of two-phase continuum poroelastic mixtures: a short survey[END_REF]. As remarked in [START_REF] Serpieri | Variational theories of two-phase continuum poroelastic mixtures: a short survey[END_REF], 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 equations 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 [START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF][START_REF] Truesdell | The classical field theories[END_REF][START_REF] Lopatnikov | Poroelasticity-i: governing equations of the mechanics of fluidsaturated porous materials[END_REF][START_REF] Dell'isola | Boundary conditions at fluid-permeable interfaces in porous media: A variational approach[END_REF]. 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 [START_REF] Dell'isola | Boundary conditions at fluid-permeable interfaces in porous media: A variational approach[END_REF]. Therein a study is presented 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 [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF][START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF][START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF][START_REF] Travascio | Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach[END_REF][START_REF] Serpieri | Fundamental solutions for a coupled formulation of porous biphasic media with compressible solid and fluid phases[END_REF][START_REF] Travascio | Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach[END_REF]. 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 momentum 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 derived 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 computation 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 companion 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.

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 [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF]. The theory herein presented enhances the one proposed in [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF]: this is a fully variational derivation, in that momentum 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.

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, Ω (M ) 0

, and by two scalar fields defining at the macroscopic level the volume fractions of the solid and of the fluid phase,

Φ (s) 0 : X ∈ Ω (M ) 0 → Φ (s) 0 , Φ (f ) 0 : X ∈ Ω (M ) 0 → Φ (f ) 0 .
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,

Φ (s) 0 (X) in a point X represents, with reference to a RVE RVE Ω 0 (X) centered in the point X, the ratio V (s) 0 RV E (X)/V 0 RV E (X) between the volume V (s) 0 RV E (X) of the subset RVE Ω (s) 0 (X) ⊂ RVE Ω 0 (X)
(f ) 0 = V (f ) 0 RV E (X)/V 0 RV E (X) is analogously defined, with V (f ) 0 RV E being the volume of the subset RVE Ω (f ) 0 (X) ⊂ RVE Ω 0 (X)
Φ (s) 0 (X) + Φ (f ) 0 (X) = 1 (1)
The deformed configuration of the mixture is defined by two invertible deformation functions, χ(s) and χ(f) , termed solid placement and fluid placement, respectively, χ(s) : ) with common codomain Ω (M ) representing the space occupied by the mixture in the deformed configuration.

X ∈ Ω (M ) 0 → x ∈ Ω (M ) , χ(f) X ∈ Ω (M ) 0 → x ∈ Ω (M
The description of the state of the deformed configuration is completed by the spatial fields of current volume fraction, φ

x :

x ∈ Ω (M ) → φ (s) , φ (f )
x : x ∈ Ω (M ) → φ (f ) . The lowercase subscripts x are added to mark the spatial character of these fields. Field φ (s)

x , while again intro-duced on the basis of a purely macroscopic description, is such that its value φ (s)

x (x) in a point

x is related to the small-scale configuration of the microscale RVE, RVE Ω (x), in the spatial configuration, centered in x by φ

(s) (x) = V (s) RV E (x)/V RV E (x), where V (s) RV E (x) is the volume of the solid subset RVE Ω (s) (x) ⊂ RVE Ω (x) and V RV E (x) is the volume of RVE Ω (x). The counterpart relation for the fluid phase is φ (f ) (x) = V (f ) RV E (x)/V RV E (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 x ∈ Ω (M ) :

φ (s) (x) + φ (f ) (x) = 1 (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

J(s) = det ∂ χ(s) /∂X.
This quantity is purely macroscopic and its use is standard in finite deformation poroelasticity, see for instance [START_REF] Boer | Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory[END_REF]. Concerning the interpretation of the relation of J(s) (X) with the volume changes of the reference and deformed RVEs associated with a point

X ∈ Ω (M ) 0
, as shown in [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF], this quantity is the ratio J

(s) (X) = V RV E (x)/V 0 RV E (X)| χ(s) ,
where

V RV E (x)
(f) (X) = V RV E (x)/V 0 RV E (X)| χ(f) . Trivially, for the inverse mappings χ(s) -1 , χ(f) -1 , the extrinsic volume ratios are V 0 RV E (X)/V RV E (x)| (χ (s) ) -1 = 1/ J(s) (X) and V 0 RV E (X)/V RV E (x)| (χ (f ) ) -1 = 1/ J(f) (X).
In [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF], 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 J(s) (X) can be different from unity even when the solid porous skeleton is undergoing isochoric deformations. Accordingly, consistent with [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF], an additional macroscopic field of intrinsic volumetric strain of the solid phase, Ĵ(s) , independent from J(s) , is introduced to measure the effective volume changes of the solid phase, Ĵ(s) :

X ∈ Ω (M ) 0 → Ĵ(s) ∈ R.
The relation of Ĵ(s) with the volumes of the microscale RVE in the reference and

deformed configurations is Ĵ(s) (X) = V (s) RV E (x)/V (s)
0 RV E (X) with x = χ(s) (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

Ĵ(f) = V (f ) RV E (x)/V (f )
0 RV E (X). According to the macroscopic description for volume changes so far detailed, a homothetic deformation of the solid phase in a point X is characterized by the condition Ĵ(s) (X) = J(s) (X). Moreover, current and reference volume fraction fields are related to extrinsic and intrinsic volume deformations by:

φ (s) (x) = V (s) RV E (x) V RV E (x) = V (s) RV E (x) V (s) 0 RV E (X) V (s) 0 RV E (X) V 0 RV E (X) V 0 RV E (X) V RV E (x) (χ (s) ) -1 = Ĵ(s) (X)Φ (s) 0 (X) 1 J(s) (X) (3)
where the correspondence between x and X is defined by the solid placement, viz. x = χ(s) (X).

Analogously, for the fluid phase a relation specular to (3) holds:

φ (f ) (x) = V (f ) RV E (x) V RV E (x) = V (f ) RV E (x) V (f ) 0 RV E (X) V (f ) 0 RV E (X) V 0 RV E (X) V 0 RV E (X) V RV E (x) (χ (f ) ) -1 = Ĵ(f) (X)Φ (f ) 0 (X) 1 J(f) (X) (4)
with x and X being related by x = χ(f) (X).

From (3) one can also infer a relation for the operative macroscopic measurement of Ĵ(s) in a point X:

Ĵ(s) (X) = J(s) (X) Φ (s) 0 (X) 1 -φ (f ) ( χ(s) (X)) (5)
Hence, if the field χ(s) is among the known data, so that J(s) is known, the measurement of Ĵ(s) can be related to the measurement of the porosity φ (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 [START_REF] Bedford | A variational theory of porous media[END_REF] and [START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF] 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 J(s) and

infinitesimal displacement field d χ(s) is d J(s) = J(s) ∂d χ(s) i ∂x i , (6)
the extrinsic spatial infinitesimal volumetric strain measures ē(s) and the intrinsic spatial infinitesimal volumetric strain measures ê(s) are defined as:

ē(s) = d J(s) J(s) = ∂d χ(s) i ∂x i , ê(s) = d Ĵ(s) Ĵ(s) (7)
According to these definitions, linearization of the porosity-strain relation (3) is computed as follows:

dφ (s) = ∂ ( χ(s) , Ĵ(s) ) φ (s) d χ(s) , d Ĵ(s) = Φ (s) 0 1 J(s) ∂ Ĵ(s) Ĵ(s) d Ĵ(s) + Ĵ(s) ∂ χ(s) 1 J(s) d χ(s) = Φ (s) 0 1 J(s) ∂ Ĵ(s) Ĵ(s) d Ĵ(s) -Ĵ(s) 1 J(s) 2 ∂ χ(s) J(s) d χ(s) = Φ (s) 0 1 J(s) ê(s) - Ĵ(s) J(s) ē(s) (8)
When the deformation is infinitesimal, so that J(s) 1 and Ĵ(s) 1, the previous relation yields:

dφ (s) = Φ (s) 0 ê(s) -ē(s) (9)
which corresponds to Equation (61) derived in [START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF].

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

dφ (f ) = Φ (f ) 0 ê(f) -ē(f) (10)
with definitions for ē(s) and ê(s) analogous to those in [START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF]:

ē(f) = ∂d χ(f) i ∂x i , ê(f) = d Ĵ(f) Ĵ(f) (11)
Equations ( 3) and (4) are written in terms of relation between fields as follows:

φ (s) x • χ(s) = Ĵ(s) J(s) Φ (s) 0 , φ (f ) x • χ(f) = Ĵ(f) J(f) Φ (f ) 0 (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 '•' indicates function composition. From [START_REF] Madeo | A continuum model for deformable, second gradient porous media partially saturated with compressible fluids[END_REF] one infers:

φ (s) x = Ĵ(s) J(s) Φ (s) 0 • χ(s) -1 , φ (f ) x = Ĵ(f) J(f) Φ (f ) 0 • χ(f) -1 (13)
whereby the saturation condition (2) is expressed in terms of field relations in the following form termed Finite deformation saturation constraint:

Ĵ(s) J(s) Φ (s) 0 • χ(s) -1 + Ĵ(f) J(f) Φ (f ) 0 • χ(f) -1 = 1 (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:

Ĵ(s) X (s) J(s) X (s) Φ (s) 0 X (s) + Ĵ(f) X (f ) J(f) X (f ) Φ (f ) 0 X (f ) = 1 (15)
specifying that x is the common image of the reference points X (s) and X (f ) , respectively via χ(s) and χ(f) , viz.:

x = χ(s) X (s) = χ(f) X (f ) (16)
Relation [START_REF] Boer | Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory[END_REF] 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

Ω (M ) 0 , Φ (s)
0 and Φ (f ) 0 , the quantities Ĵ(s) , J(s) , Ĵ(f) , and J(f) are not independent.

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 [START_REF] Bedford | A variational theory of porous media[END_REF]. In particular, the absence of constraints for the primary descriptors in VMTPM does not require Lagrange multipliers for defining stress quantities. Accordingly, fields χ(s) , Ĵ(s) , χ(f) , and Ĵ(f) cannot be taken altogether as primary kinematic descriptors since quantities Ĵ(s) , J(s) , Ĵ(f) , and J(f) are mutually related by the saturation constraint [START_REF] Boer | Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory[END_REF].

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 χ(s) , χ(f) , and Ĵ(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

Ω (M ) 0 ×[t 0 , t f ].
Once these three fields are specified, the intrinsic strain in a point x = χ(s) X (s) = χ(f) X (f ) , which is the common image of two reference points X (s) and X (f ) (respectively via χ(s) and χ(f) ) can be computed on account of (15):

Ĵ(f) X (f ) = J(f) X (f ) Φ (f ) 0 X (f ) 1 -Φ (s) 0 X (s) Ĵ(s) X (s) J(s) X (s) (17)
In this way, Ĵ(f) is treated as a field indirectly related to χ(s) , χ(f) , and Ĵ(s) via the saturation constraint. This choice is well posed, and the kinematic independence of χ(s) , χ(f) , and Ĵ(s)

is preserved only if Φ (f ) 0 = 0.
When porosity is null, the solid volume fraction has a unit value. Hence, relation (3) yields coincidence of J(s) and Ĵ(s) , so that independence is lost. This condition, which needs special treatment (see Section 3 in [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF]) is herein excluded assuming

Φ (f ) 0 = 0 over Ω (M ) 0
. Conversely, the existence of entirely fluid regions of the mixture with Φ (f ) 0 = 1 is not ruled out, see Figure 1.

Accordingly, the argument functions of the considered Lagrange function L (M ) 0 are the descriptor space-time fields χ(s) (X, t), χ(f) (X, t), and Ĵ(s) (X, t) of the mixture, viz.:

L (M ) 0 = L (M ) 0 χ(s) , χ(f) , Ĵ(s) (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 [START_REF] Leech | Hamilton's principle applied to fluid mechanics[END_REF] and Bedford and Drumheller [START_REF] Bedford | A variational theory of porous media[END_REF], and expresses the Action functional in relation to the set of possibile evolution histories of a fixed mass of mixture. This particular choice, besides being compliant with the originary application of this principle to Langrangian 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, ρ(s) , ρ(f) , are secondary variables which are related to the reference densities, ρ(s) 0 , ρ(f) 0 , by:

ρ(s) = ρ(s) 0 J(s) , ρ(f) = ρ(f) 0 J(f) (19)
and that can, as such, be post-computed once the least-action boundary value problem is solved.

The condition expressed by [START_REF] Gray | Averaging theory for description of environmental problems: What have we learned?[END_REF] can be rewritten in terms of a relation of functional dependence between fields. To this end we introduce the definition:

φ (α) χ(β) = φ (α) x • χ(β) (20)
which represents the volume fraction of phase α in the deformed configuration at the space point

x corresponding to point X (β) as a result of the deformation χ(β) . According to this definition, we have:

φ (s) χ(s) := φ (s) x • χ(s) = Ĵ(s) J(s) Φ (s) 0 (21) and φ (s)
χ(f) = φ (s) x • χ(f) = φ (s) χ(s) • χ(s) -1 • χ(f) (22)
whereby [START_REF] Gray | Averaging theory for description of environmental problems: What have we learned?[END_REF] is expressed as the followiing relation between fields:

Ĵ(f) sat χ(f) := J(f) Φ (f ) 0 1 -φ (s) χ(f) (23)

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 [START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF]. 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 formulation is referred to as medium independence.

The constitutive relations are defined in terms of functions of strain energy densities per unit reference space ψ(s) 0 and ψ(f) 0 . These are related to the true, or effective, reference densities of strain energy, ψ(s) 0 and ψ(f) 0 , by:

ψ(s) 0 = Φ (s) 0 ψ(s) 0 , ψ(f) 0 = Φ (f ) 0 ψ(f) 0 (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. Accordingly, for the solid phase, a generic dependence of type:

ψ(s) 0 (X) = ψ(s) 0 X, ∂ χ(s) ∂X X , Ĵ(s) (X) (25)
is consdiered, where dependence upon X is introduced to address a constitutive response which can be a macroscopically nonhomogeneous function of space. Nonhomogeneity of the constitutive response of the solid phase is specifically accounted for, in order to address the possible space nonhomogeneity of Φ (s) 0 . In particular, function

ψ(s) 0 must vanish in the points X ∈ Ω (M ) 0 such that Φ (s) 0 = 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 an homogeneous state, so that the following strain energy density can be adopted:

ψ(f) 0 (X) = ψ(f) 0 Ĵ(f) (X) (26)

Kinetic energy

For the reference field of density of kinetic energy of the solid phase, κ(s) 0 , in addition to the quadratic term of translational kinetic energy associated with the solid velocity, v(s) = ∂ χ(s) (X, t)/∂t, a further microinertia term κ(s) 0 add is considered:

κ(s) 0 (X) = 1 2 ρ(s) 0 ||v (s) (X)|| 2 + κ(s) 0 add (27)
Microinertia terms are essentially introduced to retrieve kinetic additional terms in the governing equations which are comparable to those considered in [START_REF] Cowin | A variational principle for granular materials[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Nunziato | On ideal multiphase mixtures with chemical reactions and diffusion[END_REF]. However, it should be remarked that the addition of κ(s) 0 add 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 [START_REF] Bedford | Theories of immiscible and structured mixtures[END_REF], or the deduction of these terms as reported in [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF]). 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 [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF] so as to vanish in presence of a rate of deformation having in X the shape of an homothety (i.e., when

J (s) = J (s) ). To this end, it is set κ(s) 0 add = κ(s) 0 add J (s) (X) with J(s) (X) = J(s) (X) -Ĵ(s) (X).
For simplicity, also for κ(s) 0 add a quadratic expression is chosen:

κ(s) 0 add = 1 2 ρ(s) add. 0 J (s) 2 = 1 2 ρ(s) add. 0 J (s) -J (s) 2 (28)
where ρ(s) add. 0 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:

κ(f) 0 add = 1 2 ρ(f) add. 0 J (f ) 2 = 1 2 ρ(f) add. 0 J (f ) -J (f ) 2 (29)
κ(f) 0 (X) = 1 2 ρ(f) 0 ||v (f ) (X)|| 2 + κ(f) 0 add (30)
The remaining choices for the definition of the Lagrange function are the standard ones.

Accordingly, denoting by T (M ) 0

the kinetic energy of the mixture and the potential energy by

U (M ) 0
, we write:

L (M ) 0 = T (M ) 0 -U (M ) 0 (31)
The potential energy of the mixture U (M ) 0 is divided into three contributions: the solid phase strain energy U (s) 0 , the fluid phase strain energy U (f ) 0 , and the potential energy due to external actions U ext 0 :

U (M ) 0 = U (s) 0 + U (f ) 0 + U ext 0 (32)
According to [START_REF] Albers | Influence of coupling through porosity changes on the propagation of acoustic waves in linear poroelastic materials[END_REF], the functional dependence of the total potential energy of the solid phase turns out to be:

U (s) 0 χ(s) , Ĵ(s) = Ω (M ) 0 ψ(s) 0 χ(s) , Ĵ(s) dV 0 (33)
The functional dependence for the strain energy of the fluid phase U (f ) 0 is more complex since Ĵ(f) is not among the primary descriptor fields and this term depends upon all three primary kinematic fields via [START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF]. By combining ( 23) and ( 26), U (f ) 0 is provided by the following integral:

U (f ) 0 χ(s) , χ(f) , Ĵ(s) = Ω (M ) 0 Φ (f ) 0 • ψ(f) 0 • Ĵ(f) sat χ(f) χ(s) , χ(f) , Ĵ(s) dV 0 (34)
In [START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF], ψ(f) 0 is a predetermined constitutive function which does not depend from the primary fields (although the value , viz.:

ψ(f) 0 (X) attained by ψ(f) 0 in a point X depends from the kinetic de- scriptors χ(s) , χ(f) , Ĵ(s) , via the function combination ψ(f) 0 • Ĵ(f) sat χ(f) ). Similarly, Φ (f )
U ext 0 = U ext Ω0 + U ext ∂Ω0 (35)
where

U ext Ω0 = Ω (M ) 0 ψ(s) 0ext ( χ(s) ) dV 0 + Ω (M ) 0 ψ(f) 0ext ( χ(f) ) dV 0 (36)
and

U ext ∂Ω0 = ∂Ω (M ) 0 ψ(∂) 0ext dA 0 (37)
The kinetic energy of the mixture is split into solid (T

T (M ) 0 = T (s) 0 + T (f ) 0 (38)
with

T (s) 0 χ (s) , J (s) = Ω (M ) 0 κ(s) 0 χ (s) , J (s) dV 0 (39)
T (f ) 0 χ (f ) = Ω (M ) 0 κ(f) 0 χ (f ) , J (f ) dV 0 (40)
where κ(s) 0 and κ(f) 0 are specified by ( 27) and [START_REF] Biot | The theory of consolidation[END_REF].

External volume forces

In b(s,ext)

0 := - ∂ ψ(s) 0ext ∂ χ(s) , b(f,ext) 0 := - ∂ ψ(f) 0ext ∂ χ(f) (41)
0 : b(s,ext) = 1 J(s) b(s,ext) 0 , b(f,ext) = 1 J(f) b(f,ext) 0 (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 [START_REF] Marsden | Mathematical foundations of elasticity[END_REF][START_REF] Guo | Time derivatives of tensor fields in nonlinear continuum mechanics[END_REF] 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 F(s) = ∂ χ(s) ∂X the deformation gradient, the work-conjugate stress measures are:

P (s) iK = ∂ ψ(s) 0 ∂ F (s) iK , Π(s) = - ∂ ψ(s) 0 ∂ Ĵ(s) (43)
The tensor P(s) is a two-point stress tensor termed extrinsic first Piola-Kirchhoff stress tensor.

The scalar quantity Π(s) is the material (i.e., reference) stress measure work-associated with Ĵ(s) , denominated (reference) intrinsic solid pressure. The negative sign in the definition of Π(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 [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF], limited therein to the context of infinitesimal deformations.

The spatial counterparts of ( 43) are the result of push-forward operations of the quantities in [START_REF] Jardine | Developments in understanding soil behaviour[END_REF] along the deformed solid configuration defined by the solid placement χ(s) :

σ(s) ij = 1 J(s) P (s) iK ∂ χ(s) j ∂X K = 1 J(s) ∂ ψ(s) 0 ∂ F (s) iK ∂ χ(s) j ∂X K (44)
p(s) = 1 J(s) Π(s) = - 1 J(s) ∂ ψ(s) 0 ∂ Ĵ(s) (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 Ĵ(f) . Two alternate stress measures of the fluid phase (in pressure form) associated with the reference configuration are the following:

Π(f) = - ∂ ψ(f) 0 ∂ Ĵ(f) , Π(f) = - ∂ ψ(f) 0 ∂ Ĵ(f) (46)
which, owing to [START_REF] Gajo | A general approach to isothermal hyperelastic modelling of saturated porous media at finite strains with compressible solid constituents[END_REF], are related by

Π(f) = Φ (f ) 0 Π(f)
. The spatial counterparts of ( 46) are again provided by a push forward operation:

p(f) = 1 Ĵ(f) Π(f) , p(f) = 1 Ĵ(f) Π(f) (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 [START_REF] Marsden | Mathematical foundations of elasticity[END_REF].

It is however convenient to report some additional side-considerations in order to gain physical 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 σ (s) , and of interstitial fluid pressure p.

As a general consideration, insights on Π(f) , Π(s) , and 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 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 RVE

Ω (f ) (x) centered in x. In this condition ψ (f ) is uniform in RVE Ω (f ) (x),
and the microscale Jacobian J (f ) and microscale strain energy density ψ

(f ) 0 are uniform inside RVE Ω (f ) 0 (X).
As a consequence of such microscale uniformity, macroscopic and microscale quan-

tities coincide inside RVE Ω (f ) 0 (X): Ĵ(f) = J (f ) , ψ(f) 0 = ψ (f ) 0 (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 p(f) , defined as:

p(f) = - 1 Ĵ(f) ∂ ψ(f) 0 ∂ Ĵ(f) = 1 Ĵ(f) Π(f) (49)
and the interstitial fluid pressure in the interior points of RVE Ω (f ) 0 (X), defined instead as:

p = - 1 J (f ) ∂ψ (f ) 0 ∂J (f ) (50)
become coincident, viz.:

p(f) ≡ p. (51)
On account of the consequences of microscopic uniformity, quantities p(f) and p are both simply referred to as 'fluid pressure'.

Also, the relation of the extrinsic stress tensor σ(s) with the ordinary Cauchy stress tensor σ (s) can be elucidated based on considerations analogous to those reported in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF] for the special case of infinitesimal kinematics: within VMTPM, σ(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 ( F(s) , Ĵ(s) ). Accordingly, a computation of the partial derivative 44) is carried out, as such, by keeping fixed the other degree of freedom Ĵ(s) . The extrinsic stress tensor σ(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.

∂ ψ(s) 0 ∂ F(s) in (
For this reason, the alternative denomination of isochoric stress tensor for identifying σ(s) is used in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF].

As observed in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF], in absence of interstitial fluid (i.e., when the fluid pressure is null p = 0), the extrinsic stress tensor σ(s) corresponds to the standard notion of macroscopic Cauchy stress tensor σ (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 ∂ σ(s) ij ∂x j + b(s,ext) i = 0, p(s) = 0, σ(s) n = t (ext) (52)
Relation [START_REF] Pietraszkiewicz | Extended non-linear relations of elastic shells undergoing phase transitions[END_REF] states that, when p = 0, tensor σ(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., σ(s) ≡ σ (s) , in conditions of null fluid pressure.

Boundary conditions

The whole boundary of the mixture ∂Ω , viz.: in the reference configuration is defined as:

χ(s) = χ(f) , over ∂Ω (M ) 0 (53
t (ext) 0 := - ∂ ψ(∂) 0ext ∂ χ(u) (54)
where the push forward of t (ext) 0

is the spatial traction

t (ext) = dA 0 dA t(s)(ext) 0 (55)
To address a variety of relevant boundary conditions, a specific partition, Ω

(M ) 0 = Ω (s) 0 ∪ Ω (f ) 0 , is
Ω (f ) 0 with Φ (s) 0 = 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 Ω (M ) . We assume that both Ω , 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 (sf ) 0

considering that, since this surface does not belong to ∂Ω (s) 0 , 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 ∂Ω

(M ) 0
where it applies, the virtual variations of placements δ χ(s) and δ χ(f) cannot be independently assigned but must satisfy the relation:

δ χ(s) = δ χ(f) , over ∂Ω (M ) 0 (56)
On this basis, boundary conditions of displacement-type and of stress-type are standardly applied. In particular, when displacements are prescribed throughout ∂Ω : free solid-fluid macroscopic interface. Omitted '0' subscripts denote the corresponding current configurations. directions, these conditions read:

χ(s) = χ(f) = χ(u) , δ χ(s) = δ χ(f) = o (57)
and, when boundary conditions of traction type are prescribed in all cartesian directions, these read:

χ(s) = χ(f) , δ χ(s) = δ χ(f) = δ χ(u) , t (ext) 0 = text 0 ( 58
)
where text 0 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 ∂Ω

(M ) 0 = d ∂Ω (M ) 0i ∪ s ∂Ω (M ) 0i
per each i-th Cartesian direction

(i = 1, 2, 3), where d ∂Ω (M ) 0i
is the subset where boundary conditions of displacement type are applied along the i-th Cartesian direction:

χ(s) i = χ(f) i = χ(u) i , δ χ(s) i = δ χ(f) i = 0 over d ∂Ω (M ) 0i (59)
and s ∂Ω (M ) 0i

is the subset of stress-type boundary conditions for the i-th Cartesian direction:

χ(s) i = χ(f) i , δ χ(s) i = δ χ(f) i = δ χ(u) i , t ext 0i = text 0i over s ∂Ω (M ) 0i (60)
where χ(u)

i and text 0i are the scalar fields of boundary data, defined over d ∂Ω (M ) 0i

and s ∂Ω (M ) 0i , respectively.

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:

δ t f t 0 L (M ) 0 (t) dt = 0 (61)
is more explicitly stated on account of (173) (see Appendix 4.3) in the following form:

d dt ∂ χ (s) L (M ) 0 δ χ(s) -∂ χ(s) L (M ) 0 δ χ(s) + d dt ∂ χ(f) L (M ) 0 δ χ(f) -∂ χ(f) L (M ) 0 δ χ(f) + d dt ∂ J(s) L (M ) 0 δ Ĵ(s) -∂ Ĵ(s) L (M ) 0 δ Ĵ(s) + ∂ χ(u) U ext ∂Ω0 δ χ(u) = 0 (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:

χ(s) = χ(f) , δ χ(s) = δ χ(f) = δ χ(u) , over ∂Ω (M ) 0 (63)
Based on the definitions of Section 2.2.1, the terms in ( 62) are split into:

-∂ χ(s) L (M ) 0 = ∂ χ(s) U (M ) 0 = ∂ χ(s) U (s) 0 + ∂ χ(s) U (f ) 0 + ∂ χ(s) U ext 0 -∂ χ(f) L (M ) 0 = ∂ χ(f) U (M ) 0 = ∂ χ(f) U (s) 0 + ∂ χ(f) U (f ) 0 + ∂ χ(f) U ext 0 -∂ Ĵ(s) L (M ) 0 = ∂ Ĵ(s) U (M ) 0 = ∂ Ĵ(s) U (s) 0 + ∂ Ĵ(s) U (f ) 0 + ∂ Ĵ(s) U ext 0 ( 64
)
d dt ∂ χ(s) i L (M ) 0 δ χ(s) i = d dt ∂ χ(s) i T (s) 0 δ χ(s) i d dt ∂ χ(f) i L (M ) 0 δ χ(f) i = d dt ∂ χ(f) i T (f ) 0 δ χ(f) i d dt ∂ J(s) L (M ) 0 δ Ĵ(s) = d dt ∂ J(s) T (s) 0 δ Ĵ(s) + d dt ∂ J(s) T (f ) 0 δ Ĵ(s) (65)
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 δU (f ) 0 and of the associated variation of intrinsic volumetric fluid strain δ Ĵ(f) sat χ(f) (see [START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF]). The final computed expressions of these two terms are hereby recalled from Appendix 4.4:

δ Ĵ(f) = ∂ (χ (s) , χ(f) , Ĵ(s) ) Ĵ(f) sat χ(f) δ χ(s) , δ χ(f) , δ Ĵ(s) = - J(f) Φ (f ) 0 -φ (s) χ(s) ∂δ χ(s) i ∂X J ∂ χ(s) -1 J ∂x i • χ(s) -1 • χ(f) - J(f) Φ (f ) 0   - ∂φ (s) χ(s) ∂X J ∂ χ(s) -1 J ∂x i δ χ(s) i • χ(f)   + J(f) Φ (f ) 0 φ (f ) χ(f) ∂ χ(f) -1 J ∂x i ∂δ χ(f) i ∂X J + J(f) Φ (f ) 0   ∂φ (f ) χ(f) ∂X J ∂ χ(f) -1 J ∂x i δ χ(f) i   - J(f) Φ (f ) 0 Φ (s) 0 J(s) δ Ĵ(s) (66)
It should be noted that the expression (66) provides a relation written in terms of fields placed in the reference configuration Ω (M ) 0

. The same relation expressed by ( 66) is more concisely written in terms of spatial variables associated with the space point x, viz.:

δ Ĵ(f) = J(f) Φ (f ) 0 φ (s) ∂δ χ(s) i ∂x i + ∂φ (s) ∂x i δ χ(s) i + φ (f ) ∂δ χ(f) i ∂x i + ∂φ (f ) ∂x i δ χ(f) i - Φ (s) 0 J(s) δ Ĵ(s) (67)
Accounting in [START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF] for the product derivatives rule, one obtains the Linearized Saturation Constraint for virtual deformations associated with a finite deformation, see [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF] and [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF]. This constraint expresses the relation that virtual infinitesimal variations must fulfill in order to preserve the condition of space saturation ( 14):

Φ (s) 0 δ Ĵ(s) Ĵ(s) + Φ (f ) 0 δ Ĵ(f) Ĵ(f) = ∂φ (s) δ χ(s) i ∂x i + ∂φ (f ) δ χ(f) i ∂x i ( 68
)
When linearization is performed in the reference configuration, one can refer to the relation [START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF], written for virtual deformations, and apply Φ (s) 0 φ (s) and Φ (f ) 0 φ (f ) , obtaining:

φ (s) δê (s) + φ (f ) δê (f ) = ∂φ (s) δ χ(s) i ∂x i + ∂φ (f ) δ χ(f) i ∂x i (69)
In particular, when the fields φ (s) are φ (f ) are uniform in space, the relation (69) specializes to:

φ (s) δê (s) + φ (f ) δê (f ) = φ (s) ∂δ χ(s) i ∂x i + φ (f ) ∂δ χ(f) i ∂x i (70)
Relation [START_REF] Serpieri | Fundamental solutions for a coupled formulation of porous biphasic media with compressible solid and fluid phases[END_REF] corresponds to the form of the saturation constraint obtained in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF] for infinitesimal deformations, and has been previously reported in a form combined with mass balances also by Bedford and Drumheller [START_REF] Bedford | A variational theory of porous media[END_REF]. 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 [START_REF] Gray | The solid phase stress tensor in porous media mechanics and the hill-mandel condition[END_REF].

The variation of the fluid potential energy, expressed as:

δU (f ) 0 = ∂ Ĵ(f) U (f ) 0 δ Ĵ(f) = ∂ (χ (s) , χ(f) , Ĵ(s) ) U (f ) 0 δ χ(s) , δ χ(f) , δ Ĵ(s) (71)
is computed differentiating relation [START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF] via the rule for the variation of function composition, see (154), ( 177) and (178). Its explicit expression is:

δU (f ) 0 = Ω (M ) 0 Φ (f ) 0 - Π(f) ∂ ψ(f) 0 ∂ Ĵ(f) ∂ (χ (s) , χ(f) , Ĵ(s) ) Ĵ(f) sat χ(f) δ χ(s) , δ χ(f) , δ Ĵ(s) dV 0 = - Ω (M ) 0 Π(f) J(f) φ (s) χ(s) ∂δ χ(s) i ∂X J ∂ χ(s) -1 J ∂x i dV 0 - Ω (M ) 0 Π(f) J(f) ∂φ (s) χ(s) ∂X J ∂ χ(s) -1 J ∂x i δ χ(s) i dV 0 - Ω (M ) 0 Π(f) J(f) φ (f ) χ(f) ∂ χ(f) -1 J ∂x i ∂δ χ(f) i ∂X J dV 0 - Ω (M ) 0 Π(f) J(f) ∂φ (f ) χ(f) ∂X J ∂ χ(f) -1 J ∂x i δ χ(f) i dV 0 + Ω (M ) 0 Π(f) J(f) Φ (s) 0 J(s) δ Ĵ(s) dV 0 (72)
Collecting [START_REF] Leech | Hamilton's principle applied to fluid mechanics[END_REF] with the required expressions reported in Appendix 4.4 for δU (s) 0 and δU ext 0 , and for the remaining terms entering (65), the following explicit expression of equation ( 62) is obtained:

Ω (M ) 0 PiJ -J(f) φ (s) χ(s) ∂ χ(s) -1 J ∂x i Π(f) + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) ∂δ χ(s) i ∂X J dV 0 + Ω (M ) 0   -J(f) ∂φ (s) χ(s) ∂X J ∂ χ(s) -1 J ∂x i Π(f) - b(s,ext) 0i + ρ(s) 0 χ (s) i   δ χ(s) i dV 0 + Ω (M ) 0 -J(f) φ (f ) χ(f) ∂ χ(f) -1 J ∂x i Π(f) + ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) ∂δ χ(f) i ∂X J dV 0 + Ω (M ) 0   -J(f) ∂φ (f ) χ(f) ∂X J ∂ χ(s) -1 J ∂x i Π(f) - b(f,ext) 0i + ρ(f) 0 χ (f ) i   δ χ(f) i dV 0 + Ω (M ) 0 -Π(s) + J(f) J(s) Φ (s) 0 Π(f) - ρ(s) add. 0 J (s) -J (s) + J(f) Φ (f ) 0 Φ (s) 0 J(s) ρ(f) add. 0 J (f ) -J (f ) δ Ĵ(s) dV 0 - ∂Ω (M ) 0 t ext 0i δ χ(u) i dA 0 = 0 (73)
Application of the divergence theorem to the terms in (73) containing ∂δ χ(s) ∂X and ∂δ χ(f) ∂X yields (see Appendix 4.4 for details):

Ω (M ) 0 - ∂ PiJ ∂X J + φ (s) χ(s) ∂ ∂X J J(f) ∂ χ(s) -1 J ∂x i Π(f) δ χ(s) i dV 0 + Ω (M ) 0 - b(s,ext) 0i + ρ(s) 0 χ (s) i - ∂ ∂X (s) J ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) δ χ(s) i dV 0 + Ω (M ) 0 φ (f ) χ(f) ∂ ∂X J J(f) ∂ χ(s) -1 J ∂x i Π(f) - ∂ ∂X J ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) δ χ(f) i dV 0 + Ω (M ) 0 - b(f,ext) 0i + ρ(f) 0 χ (f ) i δ χ(f) i dV 0 + Ω (M ) 0 -Π(s) + J(f) J(s) Φ (s) 0 Π(f) δ Ĵ(s) - ρ(s) add. 0 J (s) -J (s) + J(f) Φ (f ) 0 Φ (s) 0 J(s) ρ(f) add. 0 J (f ) -J (f ) δ Ĵ(s) dV 0 + ∂Ω (M ) 0 PiJ -J(f) φ (s) χ(s) ∂ χ(s) -1 J ∂x i Π(f) + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) N J δ χ(s) i dV 0 + ∂Ω (M ) 0 -φ (f ) χ(f) J(f) ∂ χ(f) -1 J ∂x i Π(f) + ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) δ χ(f) i N J dV 0 - ∂Ω (M ) 0 t ext 0i δ χ(u) i dA 0 = 0 (74)
where the boundary integrals have been collected in the last three rows of (74).

Strong form equations

Strong form equations are inferred from [START_REF] Guo | Time derivatives of tensor fields in nonlinear continuum mechanics[END_REF]

∂ PiJ ∂X (s) J -φ (s) ∂ ∂X J J(f) ∂ χ(s) -1 J ∂x i Π(f) + b(s,ext) 0i = = ρ(s) 0 χ (s) i - ∂ ∂X (s) J ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) (75)
Similarly, by selecting virtual variations vanishing at the boundary, such that δ χ(s) = o, δ χ(f) = o, and δ Ĵ(s) = 0 over Ω (M ) 0

, one obtains the linear momentum balance of the fluid phase:

-φ (f ) ∂ ∂X J J(f) ∂ χ(f) -1 J ∂x i Π(f) + b(f,ext) 0i = = ρ(f) 0 χ (f ) i - ∂ ∂X J ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) (76)
An analogous selection with δ χ(s) = o, δ χ(f) = o and δ Ĵ(s) = 0 yields the intrinsic momentum balance:

Π(s) - J(f) J(s) Φ (s) 0 Π(f) = -ρ (s) add. 0 J (s) -J (s) + J(f) Φ (f ) 0 Φ (s) 0 J(s) ρ(f) add. 0 J (f ) -J (f ) (77)
For the boundary integrals in [START_REF] Guo | Time derivatives of tensor fields in nonlinear continuum mechanics[END_REF] considering variation fields at the boundary ∂Ω

(M ) 0 such that δ χ(s) = δ χ(f) = δ χ(u) one obtains: PiJ -J(f) φ (s) ∂ χ(s) -1 J ∂x i Π(f) -φ (f ) J(f) ∂ χ(f) -1 J ∂x i Π(f) + +ρ (s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) + ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) N J = t ext 0i (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 [START_REF] Gouin | Hamilton's principle and rankine-hugoniot conditions for general motions of mixtures[END_REF]. 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 [START_REF] Truesdell | The classical field theories[END_REF].

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 P must comply with over the free surface S (sf ) 0

(see Figure 1). It is worth noting that for such conditions the denomination 'boundary conditions' should be avoided since S (sf ) 0

is not properly the boundary of the space domain of the mixture Ω (M ) 0

. Rather, S

(sf ) 0

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 (sf ) 0

for 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 δ Ĵ(s) = 0 and δ Ĵ(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:

∂φ (s) δ χ(s) i ∂x i + ∂φ (f ) δ χ(f) i ∂x i = 0 (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 δ ũ(s) such that δ ũ(s) = o at the boundary ∂Ω (M ) 0

, and by setting the variations of the primary fields to:

δ χ(s) = δ ũ(s) , δ χ(f) = δ ũ(f) = - φ (s) φ (f ) δ ũ(s) , δ Ĵ(s) = 0 (80)
When virtual deformations in [START_REF] Lopatnikov | Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body[END_REF] are set to the form specified in (80), the corresponding virtual intrinsic strain of the fluid is null (i.e., δ Ĵ(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 δU (f ) 0 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 δ Ĵ(f) = 0, all terms provided by (72) cancel each other, since their sum is null;

• since δ Ĵ(s) = 0, all terms multiplying δ Ĵ(s) are cancelled;

• considering the partition Ω

Ω (M ) 0 (•) dV 0 = Ω (s) 0 (•) dV 0 + Ω (f ) 0 (•) dV 0 , (81)
it is recognized that all the integrals over Ω (M ) 0 in( 73) and ( 74) containing δ χ(f) reduce to the only integral over the subset Ω (s) 0 since, as one can infer from (80), δ χ(f) = o over Ω (f ) 0 ;

• the boundary integrals in [START_REF] Guo | Time derivatives of tensor fields in nonlinear continuum mechanics[END_REF] all vanish due to (80) and to the property that δ ũ(s) = o over ∂Ω (M ) 0

;

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

Ω (s) 0 - ∂ PiJ ∂X J δ ũ(s) i dV 0 + Ω (s) 0 - b(s,ext) 0i + ρ(s) 0 χ (s) i - ∂ ∂X (s) J ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) δ ũ(s) i dV 0 + Ω (s) 0 - ∂ ∂X J ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) - φ (s) φ (f ) δ ũ(s) i dV 0 + Ω (s) 0 - b(f,ext) 0i + ρ(f) 0 χ (f ) i - φ (s) φ (f ) δ ũ(s) i dV 0 + ∂Ω (s) 0 PiJ + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) N J δ ũ(s) i dV 0 + ∂Ω (s) 0 ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) - φ (s) φ (f ) δ ũ(s) i N J dV 0 = 0 (
∂ PiJ ∂X J - b(s,ext) 0i + φ (s) φ (f ) b(f,ext) 0i + ρ(s) 0 χ (s) i - φ (s) φ (f ) ρ(f) 0 χ (f ) i - ∂ ∂X (s) J ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) + φ (s) φ (f ) ∂ ∂X J ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) = 0 (83)
As a second step, substitution of (83) into (82) yields:

∂Ω (s) 0 PiJ + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) N J δ ũ(s) i dV 0 + ∂Ω (s) 0 ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) - φ (s) φ (f ) δ ũ(s) i N J dV 0 = 0 (84)
The previous relation must hold true for any field δ ũ(s) simultaneously fulfilling ( 63) and (80).

Since the simultaneous fulfillment of these two equations necessarily implies δ ũ(s) = o over

∂Ω (M U ) 0
, it is inferred from (82) the following second integal equation which differs from the previous one since integration is carried out only on the subset S

(sf ) 0 = ∂Ω (s) 0 \ ∂Ω (M U ) 0
, which is termed free solid-fluid macroscopic interface:

S (sf ) 0 PiJ + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) N J δ ũ(s) i dV 0 + S (sf ) 0 ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) - φ (s) φ (f ) δ ũ(s) i N J dV 0 = 0 (85)
for any arbitrary field δ ũ(s) on S (sf ) 0

. This last integral condition can be straightforwardly localized into the following strong form equation holding on S (sf ) 0

:

PiJ + ρ(s) add. 0 J(s) ∂ χ(s) -1 J ∂x i J (s) -J (s) - φ (s) φ (f ) ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) N J = 0 (86)
Equation ( 86) is of significant practical relevance when the physical domain of interest is

Ω (s)
0 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 P to be applied over free solid-fluid interfaces S (sf ) 0

.

Surface conditions to be fulfilled over ∂Ω (s) 0 can be specialized for the static case, when inertia terms are negligible. In this condition, relations (78) and (86) provide: 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 Π(s) via the intrinsic momentum balance (77). Recalling that P is work-associated with isochoric strains, it is recognized that vanishing of P in a point of S (sf ) 0

PiJ -J(f) φ (s) Π(f) ∂ χ(s) -1 J ∂x i -J(f) Π(f) φ (f ) ∂ χ(f) -1 J ∂x i N J = 0 over ∂Ω (M ) 0 PiJ N J = 0 over S (sf ) 0 (
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).

Additional solid-fluid interaction

The momentum balance equations derived in Subsection 2. 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 [START_REF] Markert | A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua[END_REF], is herein referred to as drag interaction, and is represented by an additional term, L (sf ) 0 , 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 (sf ) 0 . 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 v applied to the physical system [48]. This property, which implies the conservation of linear momentum, is fulfilled by the internal strain energies U (s) 0 and U (f ) 0 . Actually, definitions [START_REF] De Boer | Theoretical poroelasticity -a new approach[END_REF] and [START_REF] De Boer | The development of the concept of effective stresses[END_REF] and the first-gradient nature of ψ(s) 0 and ψ(f) 0 imply that the following property holds for any finite uniform translation field v:

U (s) 0 χ(s) , Ĵ(s) = U (s) 0 χ(s) + v, Ĵ(s) (88)
U (f ) 0 χ(s) , χ(f) , Ĵ(s) = U (f ) 0 χ(s) + v, χ(f) + v, Ĵ(s) (89)
This property is also stated in terms of variations as follows:

∂ χ(s) U (s) 0 [δv] = 0, ∂ χ(s) U (f ) 0 [δv] + ∂ χ(f) U (f ) 0 [δv] = 0 ( 90
)
where δv is a uniform infinitesimal virtual variation field. In particular, accounting for the null gradient of δv, the first of (90) is directly inferred from (174) (see Appendix), while the second

∂ χ(s) Ĵ(f) sat χ(f) [δv] + ∂ χ(f) Ĵ(f) sat χ(f) [δv] = J(f) Φ (f ) 0 ∂ φ (s) + φ (f ) ∂x i δv i = 0 (91) since φ (s) + φ (f ) = 1 by (2).
The requirement that also L (sf ) 0

must have an internal force character is stated in terms analogous to (90):

∂ χ(s) L (sf ) 0 [δv] + ∂ χ(f) L (sf ) 0 [δv] = 0 (92)
The second hypothesis of short-range character implies that, similar to U

L (sf ) 0 = Ω (M ) ψ(sf) x dV (93)
The integral form provided by (92) allows one to write:

Ω (M ) ∂ ψ(sf) ∂ χ(s) i δv i + ∂ ψ(sf) ∂ χ(f) i δv i dV = 0 (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 b(sf) and b(fs) , defined as:

b(sf) (x) = - ∂ ψ(sf) ∂ χ(s) , b(fs) (x) = - ∂ ψ(sf) ∂ χ(f) (95)
Such forces must have equal magnitude and opposite direction. Actually, since Equation ( 94) is a spatial relation holding for any subdomain Ω ⊆ Ω (M ) , and since it must also hold for any vector δv, one infers that: δ χ(f) into the general expression of the variation [START_REF] Lopatnikov | Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body[END_REF]. These terms, when expressed as integrals over the reference domain of the mixture χ(s) read:

∂ ψ(sf) ∂ χ(s) + ∂ ψ(sf) ∂ χ(f) = 0, over Ω (M ) (96
∂ χ(s) L (sf ) 0 χ(s) = - Ω (M ) 0 J(s) b(sf) χ(s) δ χ(s) dV 0 , ∂ χ(f) L (sf ) 0 χ(f) = - Ω (M ) 0 J(f) b(fs) χ(f) δ χ(f) dV 0 (98)
where, consistent with (99), we have:

b(sf) χ(s) • χ(s) -1 + b(fs) χ(f) • χ(f) -1 = 0 (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:

∂ PiJ ∂X (s) J -φ (s) ∂ ∂X J J(f) ∂ χ(s) -1 J ∂x i Π(f) + b(s,ext) 0i + J(s) b(sf) i = ρ(s) 0 χ (s) i ( 100
)
Linear momentum balance of the fluid phase:

-φ (f ) ∂ ∂X J J(f) ∂ χ(f) -1 J ∂x i Π(f) + b(f,ext) 0i + J(f) b(fs) i = ρ(f) 0 χ (f ) i (101)
Intrinsic momentum balance:

J(s) Π(s) -J(f) Φ (s) 0 Π(f) = 0 (102)
The domain equations above are completed by boundary conditions (87).

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:

Ω (M ) 0 Ω (M ) , Ω (s) 0 Ω (s) , Ω (f ) 0 Ω (f ) (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: ε(s) , ē(s) , ē(f) , ê(s) , and ê(f) (where the prefix is redundant).

Primary descriptors of the linearized formulation are the solid and fluid infinitesimal displacements ū(s) = d χ(s) and ū(f) = d χ(f) and the infinitesimal intrinsic volumetric strain ê(s) .

Finite volumetric strain measures are replaced by linearized volumetric strain measures ē(s) , ē(f) , ê(s) , and ê(f) , previously defined in ( 7) and [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF]. Their relation is:

J(s) 1+ē (s) = 1+ ∂d χ(s) i ∂x i , J(f) 1+ē (f ) = 1+ ∂d χ(f) i ∂x i , Ĵ(f) 1+ê (f ) , Ĵ(s) 1+ê (s) (104)
In the linearized theory, the extrinsic strain of the solid is defined by the infinitesimal extrinsic strain tensor

ε(s) = sym ū(s) ⊗ ∇ (105)
and by ê(s) .

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

∂ χ(s) i ∂X J δ ij , ∂ χ(f) i ∂X J δ ij , ∂ χ(s) -1 J ∂x i = δ ji , ∂ χ(f) -1 J ∂x i = δ ji , J(s) 1, J(f) 1, Ĵ(s) 1, Ĵ(f) 1 (106)
are applied to Equations ( 44), [START_REF] Lanczos | The variational principles of mechanics[END_REF], and ( 48)-( 51), obtaining the following identifications:

P (s) iJ σ(s) ij , Π(s) p(s) , Π(f) p (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 [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF]:

ψ(s) = 1 2 A ijkl ε(s) ij ε(s) kl + B ij ε(s) ij ê(s) + 1 2 C ê(s) 2 (108) ψ(f) = φ (f ) 1 2 kf ê(f) 2 (109)
where the elastic coefficients of the solid and the fluid phases, A ijkl , B ij , C and kf , correspond to the second-order derivatives of the corresponding strain energy densities, and with respect to the relevant primary infinitesimal strain measures. In particular, kf 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 ijlk , B ij = B ji ) which stem from Schwarz's theorem and from the requirement of objectivity. Stress-strain relations for the solid are:

σ(s) ij = ∂ ψ(s) ∂ ε(s) ij = A ijkl ε(s) ij + B ij ê(s) , p(s) = - ∂ ψ(s) ∂ê (s) = -B ij ε(s) ij -C ê(s) (110)
and the relation between fluid pressure and intrinsic strain is:

kf = ∂ 2 ψ(f) ∂ê (f ) ∂ê (f ) , p = -kf ê(f) (111)

Linearized saturation condition

The saturation condition for the kinematic linear problem can be obtained applying the linearization provided by [START_REF] Travascio | Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach[END_REF] to real deformations, i.e., replacing δ χ(s) and δ χ(f) with ū(s) and ū(f) in ( 69). Accordingly, the following linearized saturation constraint is obtained:

φ (s) ê(s) + φ (f ) ê(f) = ∂φ (s) ū(s) i ∂x i + ∂φ (f ) ū(f) i ∂x i (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 [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF]. 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 φ (s) and φ (f ) are uniform, the same relation can be directly obtained by combining ( 9) and ( 10) with the linearization of (2):

dφ (s) + dφ (f ) = 0. ( 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

∂ σ(s) ij ∂x j -φ (s) ∂p ∂x i + b(sf) i + b(s,ext) i = ρ(s) ü (s) i + ∂ ∂x j   ρ(s) add.   ∂ ü (s) i ∂x i -ë (s)     (114)
linear momentum balance of the fluid phase:

-φ (f ) ∂p ∂x i + b(fs) i + b(f,ext) i = ρ(f) ü (f ) i - ∂ ∂x j   ρ(f) add.   ∂ ü (f ) i ∂x i -ë (f )     ( 115
)
intrinsic momentum balance:

p(s) φ (s) -p = - 1 φ (s) ρ(s) add.   ∂ ü (s) i ∂x i -ë (s)   + 1 φ (f ) ρ(f) add.   ∂ ü (f ) i ∂x i -ë (f )   (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 p(f)

for the solid phase by the symmetric definition s) , then, recalling also [START_REF] Eremeyev | The nonlinear theory of elastic shells with phase transitions[END_REF], Equation (116) achieves the symmetric notation:

p(s) = - ∂ ψ(s) 0 ∂ê (s) = p(s) φ (
p(s) - p(f) = - 1 φ (s) ρ(s) add.   ∂ ü (s) i ∂x i -ë (s)   + 1 φ (f ) ρ(f) add.   ∂ ü (f ) i ∂x i -ë (f )   (117)
We prefer, however, to maintain the unsymmetric notation with stress quantities 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 φ (s) + φ (f ) = 1, these turn out to be:

  σ(s) ij -pδ ij + ρ(s) add.   ∂ ü (s) i ∂x i -ë (s)   δ ij + ρ(f) add.   ∂ ü (f ) i ∂x i -ë (f )   δ ij   n j = t ext i over ∂Ω (M ) (118)
A derivation of (118) based on simplified arguments in a 1D setting has been also previously reported [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF].

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

  1 φ (s) σ(s) ij + 1 φ (s) ρ(s) add.   ∂ ü (s) i ∂x i -ë (s)   δ ij - 1 φ (f ) ρ(f) add.   ∂ ü (f ) i ∂x i -ë (f )   δ ij   n j = 0 over S (sf ) (119)

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)

∂ σ(s) ij ∂x j -φ (s) ∂p ∂x i + b(sf) i + b(s,ext) i = 0 (120)
Linear momentum balance of the fluid phase (static and quasi-static problems):

-φ (f ) ∂p ∂x i + b(fs) i + b(f,ext) i = 0 (121)
Intrinsic momentum balance (Static and quasi-static problems):

p(s) -φ (s) p = 0 (122)
Observe that Equations (120)-( 122) have a medium-inidependent 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 [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF]:

σ(s) ij -pδ ij n j = t ext i over ∂Ω (M ) (123)
while the boundary condition for displacements is

ū(s) = ū(f) = u (ext) (124)
In particular, over ∂Ω (M ) \ ∂Ω (M U ) , where φ (s) = 0 and, consequently,

σ(s) = o, equation (123) specializes to:
-

pn i = t ext i over ∂Ω (M ) \ ∂Ω (M U ) (125)
Over free solid-fluid interfaces S (sf ) (Figure 1), the relation inferred from (119) is:

σ(s) ij n j = 0 over S (sf ) (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).

Medium-independent stress partitioning law

In a domain Ω (h) (with null relative solid-fluid motion at the boundaries) for macroscopically uniform stress states, i.e., states such that fields σ(s) and p have constant values σ(s) h , p h , Equation (123) specializes to

t (ext) (x, n) = σ(s) h n -p h n, x ∈ ∂Ω (h) (127)
so that the external traction field t (ext) (x, n) in (127) can be represented by a single constant tensor σ (ext) :

t (ext) (x, n) = σ (ext) n, x ∈ ∂Ω (h) (128)
where σ (ext) is a tensor associated with Ω (h) and defined as

σ (ext) = σ(s) h -p h I (129)
Hence, when a traction is applied over the (impermeable) boundary ∂Ω (h) so as to produce a uniform stress state σ (ext) inside Ω (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 Ω (u) , relation (123) applies to the boundary ∂ Ω(u) of any subset domain Ω(u) ⊆ Ω (u) since Ω(u) is impermeable. Due to the arbitrariness of Ω(u) , the condition of homogeneity can be recovered, as a limit, considering arbitrarily small domains Ω(u)

x ⊆ Ω (u) centered in a point x. Consequently, it is recognized that, in undrained regions, relations (128), (129) are recovered in a stronger form associated with any point x ∈ Ω (u)

t (ext) (x, n) = σ (ext) n, ∀x ∈ Ω (u) , ∀n (130)
where σ (ext) is a tensor now associated with any point x ∈ Ω (u) , defined as

σ (ext) = σ(s) -pI (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 σ(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.

A variational theory of two-phase saturated porous media has been derived proceeding from the adoption of the VMTPM kinematic framework [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF][START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF][START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF], and applying standard concepts of variational continuum mechanics [START_REF] Lanczos | The variational principles of mechanics[END_REF][START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF]48] and of continuum theories with microstructure [START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Mindlin | Micro-structure in linear elasticity[END_REF].

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

Ĵ(s) = J(s) 1 -φ (f ) / 1 -Φ (f ) 0 (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 [START_REF] Lopatnikov | Macroscopic lagrangian formulation of poroelasticity with porosity dynamics[END_REF][START_REF] Cowin | A variational principle for granular materials[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF]. In this work, the use of Ĵ(s) is preferred alongside with J(s) since these fields share analogous properties. In particular, both J(s) and Ĵ(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 J(s) = Ĵ(s) . These common properties are found to be convenient when introducing the stress measures and stiffness quantities work-associated with J(s) and Ĵ(s) , allowing their easier correlation and physical interpretation (see also the companion Part II).

In previous VMTPM derivations [START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic media[END_REF][START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | A rational procedure for the experimental evaluation of the elastic coefficients in a linearized formulation of biphasic media with compressible constituents[END_REF][START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF][START_REF] Travascio | Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach[END_REF][START_REF] Serpieri | Fundamental solutions for a coupled formulation of porous biphasic media with compressible solid and fluid phases[END_REF][START_REF] Travascio | Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach[END_REF], 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 adopting 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 [START_REF] Bedford | A variational theory of porous media[END_REF], 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 continuum Cauchy linear momentum balance equations. More specifically, recalling Equation (52), it is observed that when the fluid pressure is null (i.e., σ(s) ≡ σ (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 [START_REF] Pietraszkiewicz | Extended non-linear relations of elastic shells undergoing phase transitions[END_REF], 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 [START_REF] Serpieri | General quantitative analysis of stress partitioning and boundary conditions in undrained biphasic porous media via a purely macroscopic and purely variational approach[END_REF]. 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. A less general proof of this result has been previously reported in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF].

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 [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]. 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 conventions and identities for 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.

Single variable functions

Symbol ϕ is used to refer to a functional operating from a space of real valued functions F into R, such that α = ϕ (f ) with α ∈ R and f ∈ F, being f a real valued function such that

y = f (x) with x, y ∈ R.
An engineering notation is used to denote variations, employing the prefixes δ or d for differentials. In particular, in the body of the paper prefix d is used to denote real variations in linearized theories, while δ is used to denote virtual variations when expressing stationarity conditions; henceforth in this appendix only prefix δ is used. An infinitesimal variation δf of function f is the function x → δy with δy ∈ R being the variation in the codomain variable y.

An engineering notation is also adopted to indicate the result of the variation, ∂ f ϕ, of a functional ϕ 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 δf , reported under square brackets, and the increment δα of α with the following notation:

δα = ∂ f ϕ [δf ] (133)
and ∂ f ϕ is such that, for any δf

∂ f ϕ [δf ] = lim λ→0 ϕ(f + λδf ) -ϕ(f ) λ (
c(f, x) = b(f, a(f, x)) (135)
Computation of the total variation with respect to f by application of the chain rule yields

∂ f c [δf ] = ∂ f b [δf ] • a + ∂b ∂y ∂ f a [δf ] (136)
In particular, when b is independent from f so that ∂ f b [δf ] = 0, one obtains as a special case of (136)

∂ f c [δf ] = ∂b ∂y ∂ f a [δf ] (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 ∈ F being F a set of invertible functions, then the resulting composed application c turns out to be

c(f, x) = f -1 (f (x)) = x (138)
This particular choice makes c an application no longer dependent on f , with trivially

∂ f c (f, x) [δf ] = 0 ∀f ∈ F, x ∈ R, δf ∈ δF (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:

∂ f a (f, x) [δf ] = ∂ f f (x) [δf ] = δf (x) (140)
According to the previous identity, and using the following notation for the application of

∂ f b to δf ∂ f b (f, y) [δf ] = ∂ f f -1 (y) [δf ] (141)
the application of (136) can be written as follows

∂ f f -1 (y) [δf ] • a + ∂f -1 ∂y δf (x) = 0 (142) Since ∂f -1 ∂y y = ∂f ∂x x -1
with y = f (x), and since

∂ f f -1 (y) [δf ] • a = ∂ f f -1 (f, x) [δf ]
one computes the following formula for the variation of the inverse function:

∂ f f -1 (f, x) [δf ] = - ∂f ∂x x -1 δf (x) (143)

Multi-variable functions

Consider a functional ϕ operating from a space of three-dimensional invertible mappings M into R, such that α = ϕ (χ) with α ∈ R and χ ∈ M, being χ a three-dimensional mapping.

Specifically χ defines the correspondence x = χ (X) with X , x ∈ R 3 being points of the reference and current configurations, respectively.

An infinitesimal variation δχ of mapping χ is the function X → δx where δx ∈ R 3 is the variation of the codomain point x. The variation of functional ϕ with respect to the argument mapping χ is indicated by ∂ χ ϕ. Its components ∂ χ i ϕ are the linear functionals such that, for a given variation function δχ i of the i-th component of mapping δχ, the limit condition holds

∂ χ i ϕ [δχ i ] = lim λ→0 ϕ(χ + λδχ i e (i) ) -ϕ (χ) λ (144)
being e (i) is the unit vector of the i-th Cartesian director.

The relation between the scalar variation δα ∈ R and δχ = 3 i=1 δχ i e (i) is notated as usual by placing δχ under square brackets:

δα = ∂ χ ϕ [δχ] = 3 i=1 ∂ χ i ϕ [δχ i ] (145)

Variation of composition of functional applications

Let A (a) : (χ, X) → x be a functional application relating a real function χ ∈ M and a point X ∈ R 3 to points Y ∈ R 3 , viz.:

Y = A (a) (χ, X) (146)
and let A (a) : (χ, Y) → Z be a second functional application relating a mapping χ ∈ M and a point Y ∈ R 3 to points Z ∈ R 3 :

Z = A (b) (χ, Y) (147)
Consider the composition of such two functional applications

A (c) = A (b) • A (a)
:

A (c) (χ, X) = A (b) χ, A (a) (χ, X) (148)
Computation of the total variation of A (c) with respect to χ, δZ = ∂ χ A (c) (χ, X) [δχ], by application of the chain rule yields

δZ i = ∂ χ A (c) i (χ, X) [δχ] = 3 j=1 ∂ χ j A (c) i (χ, X) [δχ j ] = 3 j=1 ∂ χ j A (b) i (χ, X) [δχ j ] • A (a) (χ, X) + 3 j=1 3 k=1 ∂A (b) i ∂Y k ∂ χ j A (a) k (χ, X) [δχ j ] (149)
Special cases and variants of (149) are now examined. A first similar case of (151) is obtained by replacing A (b) with an application having a scalar codomain with images Z in place of Z:

δZ = 3 j=1 ∂ χ j A (b) (χ, X) [δχ j ] • A (a) (χ, X) + 3 j=1 3 k=1 ∂A (b) ∂Y k ∂ χ j A (a) k (χ, X) [δχ j ] (150)
A second simple special case of (149) is retrieved when A (b) is independent from χ, so that b) [δχ] = o. We have:

∂ χ A (
δZ i = 3 j=1 3 k=1 ∂A (b) i ∂Y k ∂ χ j A (a) k (χ, X) [δχ j ] (151)
and, in case the external function has scalar codomain, the further specialization holds:

δZ = 3 j=1 3 k=1 ∂A (b) ∂Y k ∂ χ j A (a) k (χ, X) [δχ j ] (152)
A variant of ( 151) is obtained when Z is replaced with a scalar function with images Z and when A (a) is function of several independent mappings χ (1) , χ (2) , . . . , χ (Nm) , and of several additional scalar functions ϕ (1) , ϕ (2) , . . . , ϕ (Ns) :

A (a) = A (a) χ (1) , χ (2) , . . . , χ (Nm) , ϕ (1) , ϕ (2) , . . . , ϕ (Ns) (153)

In this case one has:

δZ = 3 j=1 3 k=1 ∂A (b) ∂Y k Nm h=1 ∂ χ (h) j A (a) k (χ, X) δχ (h) j + Ns h=1 ∂ ϕ (h) A (a) k (χ, X) δϕ (h) (154)
For the composition of a scalar function A (b) (T) with codomain in R defined in a domain of second order tensors T, with components T ij , and a functional application over threedimensional mappings χ, A (a) (χ), 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) (χ) = A (b) A (a) (χ) :

δZ = ∂ χ A (c) (χ) [δχ] = 3 j=1 3 h=1 3 k=1 ∂A (b) ∂T hk ∂ χ j A (a) hk (χ) [δχ j ] (155)
For the composition A (c) (t) = A (b) (χ (t)) of a functional A (b) (χ), with codomain in R defined in a domain of 3D mappings χ ∈ M, with a 3D motion, defined as a correspondence between t ∈ R → χ, the total time derivative Ȧ(c) is computed by the chain rule written as it follows:

Ȧ(c) = 3 j=1 ∂ χ j A (b) (χ) [ χj ] (156)
Variation of the inverse of a 3D mapping By choosing in (149), for A (a) , the application such that A (a) (χ, X) = χ(X), and, for A (b) , the application such that A (b) (χ, x) = χ -1 (x), with χ ∈ M, being M a set of invertible mappings, then the resulting composed application A (c) turns out to be:

A (c) (χ, X) = χ -1 (χ(X)) = X (157)
Since the images of A (c) are no longer dependent on χ, one has:

δZ = ∂ χ A (c) (χ, X) [δχ] = o ∀χ ∈ M, X ∈ R 3 , δχ ∈ δM ( 158
)
where δM is the tangent space to set M. Moreover, considering that the selected application A (a) is already linear in in the argument χ, it turns out to be:

∂ χ A (a) (χ, X) [δχ] = ∂ χ χ(X) [δχ] = δχ (X) (159)
which in components reads:

3 j=1 ∂ χ j A (a) i (χ, X) [δχ j ] = δ ij δχ j = δχ i (160)
According to the previous identity, and denoting the variation of A (b) with respect to χ applied to δχ as:

∂ χ A (b) (χ, x) [δχ] = ∂ χ χ -1 (x) [δχ] (161)
application of (149) reads:

∂ χ j χ -1 I (x) [δχ j ] • A (a) (χ, X) + 3 j=1 ∂χ -1 I ∂x j (X) δχ j = 0 (162)
Recalling that ∂χ -1

I ∂x j x = ∂χ ∂X X - 1
Ij with x = χ(X), the formula of the variation of the inverse mapping is computed:

∂ χ j χ -1 I (χ, X) [δχ j ] = - 3 j=1 ∂χ -1 I ∂x j x δχ j (X) = - 3 j=1 ∂χ ∂X X -1 Ij δχ j (X) (163)
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 determinant of an invertible mapping χ is computed from (155)

∂ χ i J [δχ i ] = ∂ ∂χ h ∂X J det ∂χ ∂X ∂ χ i ∂χ h ∂X J [δχ i ] = J ∂ (χ) -1 J ∂x i ∂δχ i ∂X J = J ∂δχ i ∂x i ( 164
)
Time rate of the Jacobian of a mapping Application of (156) and account of (164) provide

J = ∂ χ j J (χ) [ χj ] = J ∂ (χ) -1 J ∂x i ∂ 2 χ i ∂t∂X J = J ∂ χi ∂x i (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

∂ χi J [δχ i ] = J ∂ (χ) -1 J ∂x i ∂δχ i ∂X J (166)

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 ϕ :

Ω (M ) 0 × [t 0 , t f ] → ϕ ∈ 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 (M ) 0 δ t f t 0 L (M ) 0 (ϕ(t), φ(t)) dt = 0 (167)
with ϕ(t 0 ) and ϕ(t f ) being given initial and final states, respectively. Account of the dependence

L (M ) 0 = L (M ) 0
(ϕ, φ) up to first-order time derivatives of ϕ and application of (145) with χ 1 = ϕ and χ 2 = φ yields

t f t 0 ∂ ϕ L (M ) 0 [δϕ] + ∂ φL (M ) 0 [δ φ] dt = 0 (168)
is accounted for the property that the domain Ω (M ) 0 is independent from kinematic descriptors χ(s) , χ(f) , Ĵ(s) , and that, consequently, all variations can be directly transferred to the integrand functions inside the integrals

Ω (M ) 0 (•) dV 0 .
Computation of δU (s) 0

Application of (145) to [START_REF] De Boer | The development of the concept of effective stresses[END_REF] accounting for ( 25) and ( 43) yields:

δU (s) 0 = ∂ χ(s) i U (s) 0 δ χ(s) i + ∂ Ĵ(s) U (s) 0 δ Ĵ(s) = = Ω (M ) 0 ∂ χ(s) i ψ(s) 0 δ χ(s) i dV 0 + Ω (M ) 0 ∂ Ĵ(s) ψ(s) 0 δ Ĵ(s) dV 0 = Ω (M ) 0 P (s) kJ ∂δ χ(s) k ∂X J dV 0 - Ω (M ) 0 Π(s) δ Ĵ(s) dV 0 (174)
where the following trivial identities have been taken into account:

∂ χ(s) i ψ(s) 0 δ χ(s) i = ∂ ψ(s) 0 ∂ F (s) kJ ∂ χ(s) i F (s) kJ δ χ(s) i (175)
and

∂ χ(s) i F (s) kJ δ χ(s) i = ∂ ∂X J ∂ χ(s) i χ(s) k δ χ(s) i = ∂ ∂X J δ ik δ χ(s) i = ∂δ χ(s) k ∂X J (176)
Computation of δU and field Φ (f ) 0 from the kinematic descriptors, the variation of (34) turns out to be δU

(f ) 0 = Ω (M ) 0 Φ (f ) 0 δ ψ(f) 0 • Ĵ(f) sat χ(f) dV 0 (177)
Applying the chain rule in the form provided by (154) to the variation term in the integrand of (177), accounting for [START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF] and recalling also [START_REF] Berdichevsky | Variational principles of continuum mechanics[END_REF], one obtains

δ ψ(f) 0 • Ĵ(f) sat χ(f) = ∂ ψ(f) 0 ∂ Ĵ(f) δ Ĵ(f) sat χ(f) = - Π(f) δ Ĵ(f) sat χ(f) (178) with δ Ĵ(f) sat χ(f) = ∂ χ(s) i Ĵ(f) sat χ(f) δ χ(s) i + ∂ χ(f) i Ĵ(f) sat χ(f) δ χ(f) i + ∂ Ĵ(s) Ĵ(f) sat χ(f) δ Ĵ(s) (179) 61
The addends in the RHS of (179) are now separately computed based on the definition of Ĵ(f) sat χ(f) provided by ( 22) and [START_REF] Truesdell | noll the non-linear field theories of mechanics[END_REF]. For the first term it is computed

∂ χ(s) i Ĵ(f) sat χ(f) = - J(f) Φ (f ) 0 ∂ χ(s) i φ (s) χ(f) (180)
The variation ∂ χ(s) φ (s) χ(f) in (180) has a complex kinematic meaning since it represents the variation of solid volume fraction experimented 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 [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF] yields

∂ χ(s) i φ (s) χ(f) δ χ(s) i = ∂ χ(s) i φ (s) χ(s) • χ(s) -1 • χ(f) δ χ(s) i = ∂ χ(s) i φ (s) χ(s) δ χ(s) i • χ(s) -1 • χ(f) + ∂φ (s) χ(s) ∂X J ∂ χ(s) i χ(s) -1 J δ χ(s) i • χ(f) (181)
Recalling ( 21) and ( 164), the variation in the second row of ( 181) is recognized to be equal to

∂ χ(s) i φ (s) χ(s) δ χ(s) i = -Φ (s) 0 Ĵ(s) J(s) 2 ∂ χ(s) i J(s) δ χ(s) i = -Φ (s) 0 Ĵ(s) J(s) 2 J(s) ∂δ χ(s) i ∂x i δ χ(s) i = -φ (s) χ(s) ∂δ χ(s) i ∂x i δ χ(s) i (182)
The variation in the third row of (181) is computed invoking (163)

∂ χ(s) i χ(s) -1 δ χ(s) i = - ∂ χ(s) -1 J ∂x i δ χ(s) i (183)
Accordingly, one obtains for (181)

∂ χ(s) i φ (s) χ(f) δ χ(s) i = -φ (s) χ(s) ∂δ χ(s) i ∂x i • χ(s) -1 • χ(f) - ∂φ (s) χ(s) ∂X J ∂ χ(s) -1 J ∂x i δ χ(s) i • χ(f) (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 displacement 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)

∂ χ(f) i Ĵ(f) sat χ(f) δ χ(f) i = φ (f ) χ(f) Φ (f ) 0 ∂ χ(f) i J(f) δ χ(f) i - J(f) Φ (f ) 0 ∂ χ(f) i φ (s) χ(f) δ χ(f) i (185)
The variation of the first term in the RHS of ( 185) is provided by (164)

∂ χ(f) i J δ χ(f) i = J(f) ∂δ χ(f) i ∂x i (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 [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF], and turns out to be:

A (b) = φ (s) χ(s) • χ(s) -1 and A (a) = χ(f) appearing in
∂ χ(f) i φ (s) χ(f) δ χ(f) i = ∂φ (s) χ(f) ∂X J ∂ χ(s) -1 J ∂x i δ χ(f) i (187)
The variation of the third term in the RHS of (179) is provided again by the chain rule

∂ Ĵ(s) Ĵ(f) sat χ(f) δ Ĵ(s) = ∂ Ĵ(f) sat χ(f) ∂φ (s) χ(f) ∂φ (s) χ(f) ∂ Ĵ(s) δ Ĵ(s) = - J(f) Φ (f ) 0 Φ (s) 0 J(s) δ Ĵ(s) (188)
Addition of all the above computed variation terms appearing in the RHS of (179) provides the explicit expression for δU (f ) 0

shown by Equation [START_REF] Dell'isola | Boundary conditions at fluid-permeable interfaces in porous media: A variational approach[END_REF] in the body of the paper. Moreover, composition of (66) with (177) and (178) provides the sought explicit expression for δU (f ) 0

shown by Equation [START_REF] Leech | Hamilton's principle applied to fluid mechanics[END_REF] in the body of the paper.

Variation of δU

ext Ω0 63
The variation δU ext Ω0 is computed from ( 35), ( 36) and ( 41), and turns out to be

δU ext Ω0 = ∂ χ(s) i U ext Ω0 δ χ(s) i + ∂ χ(f) i U ext Ω0 δ χ(f) i = = -Ω (M ) 0 b(s,ext) 0i δ χ(s) i dV 0 -Ω (M ) 0 b(f,ext) 0i δ χ(f) i dV 0 (189)
Variation terms associated with kinetic energy

The variation terms associated with T (s) 0

and T (f ) 0 , appearing in [START_REF] Dell'isola | Boundary conditions at fluid-permeable interfaces in porous media: A variational approach[END_REF], 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 (s) 0 d dt ∂ χ(s) i T (s) 0 δ χ(s) i = Ω (M ) 0 ρ(s) 0 χ (s) i δ χ(s) i dV 0 + Ω (M ) 0 d dt ∂ χ(s) i κ(s) 0 add δ χ(s) i dV 0 (190)
Computation of the microinertia term with κ(s) 0 add is carried out accounting for [START_REF] Goodman | A continuum theory for granular materials[END_REF] with the aid of ( 165) and (166), according to which it results:

d dt ∂ χ(s) i κ(s) 0 add δ χ(s) i = ρ(s) add. 0 d dt J (s) ∂ χ(s) i J (s) δ χ(s) i = ρ(s) add. 0 d dt J (s) ∂ J (s) ∂ J (s) ∂ χ(s) i J (s) δ χ(s) i = ρ(s) add. 0 d dt J(s) ∂ χ(s) -1 J ∂x i J (s) ∂δ χ(s) i ∂X J (191)
Change of index s to f in (191) provides its counterpart associated with T

A similar computation of the variation associated with the rate of the intrinsic solid strain yields 192) where, in the computation of (192), account has been taken of the property ∂ ġ ḟ = ∂ g f . For simplicity, in the above computed variations associated with microinertia terms κ(s) in [START_REF] Lopatnikov | Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body[END_REF], 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 [START_REF] Lopatnikov | Poroelasticity-ii: on the equilibrium state of the fluid-filled penetrable poroelastic body[END_REF]. We also report for each group the application of integration by parts and of the divergence theorem, required to transform [START_REF] Marsden | Mathematical foundations of elasticity[END_REF] in [START_REF] Guo | Time derivatives of tensor fields in nonlinear continuum mechanics[END_REF].

d dt ∂ J(s) L (M ) 0 δ Ĵ(s) = d dt Ω (M ) 0 ∂κ (s) 0 add ∂ J (s) ∂ J (s) ∂ J (s) ∂ J(s) J (s) δ Ĵ(s) + ∂κ (f ) 0 add ∂ J (f ) ∂ J (f ) ∂ J (f ) ∂ J(s) J(f) sat χ(f) δ Ĵ(s) dV 0 = - Ω (M ) 0 ρ(s) add. 0 d dt J (s) -J (s) - d dt J(f) Φ (f ) 0 Φ (s) 0 J(s) ρ(f) add. 0 J (f ) -J (f ) δ Ĵ(s) dV 0 (
Accordingly, the first row of ( 62) turns out to be equal to

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

J(f) φ (s) χ(s) Π(f) ∂ χ(s) -1 J ∂x i - Π(f) J(f) ∂φ (s) χ(s) ∂X J ∂ χ(s) -1 J ∂x i = φ (s) χ(s) ∂ ∂X J Π(f) J(f) ∂ χ(s) -1 J ∂x i (
In a similar way, collection of terms appearing in the second row of (62) yields:

d dt ∂ χ(f) L (M ) 0 δ χ(f) -∂ χ(f) L (M ) 0 δ χ(f) = = Ω (M ) 0 - Π(f) φ (f ) χ(f) J(f) ∂ χ(f) -1 J ∂x i + ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) ∂δ χ(f) i ∂X J dV 0 + + Ω (M ) 0   - Π(f) J(f) ∂φ (f ) χ(f) ∂X J ∂ χ(f) -1 J ∂x i - b(f,ext) 0i + ρ(f) 0 χ (f ) i   δ χ(f) i dV 0 (197)
Applying derivation by parts and the divergence theorem one obtains:

d dt ∂ χ(f) L (M ) 0 δ χ(f) -∂ χ(f) L (M ) 0 δ χ(f) = = ∂Ω (M ) 0 - Π(f) φ (f ) χ(f) J(f) ∂ χ(f) -1 J ∂x i + ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) N J δ χ(f) i dV 0 + Ω (M ) 0 φ (f ) χ(f) ∂ ∂X J J(f) ∂ χ(f) -1 J ∂x i Π(f) - ρ(f) add. 0 J(f) ∂ χ(f) -1 J ∂x i J (f ) -J (f ) δ χ(f) i dV 0 + Ω (M ) 0 - b(f,ext) 0i + ρ(f) 0 χ (f ) i δ χ(f) i dV 0 (198)
Finally, the third row of of ( 62) turns out to be: macroscopic reference domain of the mixture Ω (M ) macroscopic current domain of the mixture Ω (s) 0 macroscopic reference subset domain of the mixture with nonvanishing solid phase Ω (s) macroscopic current subset domain of the mixture with nonvanishing solid phase Ω current solid volume fraction φ (f ) current fluid volume fraction Ĵ(s) finite macroscopic intrinsic volumetric strain of the solid phase Ĵ(f) finite macroscopic intrinsic volumetric strain of the fluid phase J(s) finite macroscopic extrinsic volumetric strain of the solid phase J(f) finite macroscopic extrinsic volumetric strain of the fluid phase J (f ) finite microscale volumetric strain of the fluid phase ρ(s)

  occupied by the solid phase and the volume V 0 RV E (X) of RVE Ω 0 (X). The reference fluid volume fraction Φ

  containing the fluid. Due to the saturation condition holding also at reference configuration, one has for any X ∈ Ω (M ) 0

  is the volume of the microscale deformed RVE χ(s) ( RVE Ω 0 (X)) obtained applying the microscopic deformation χ (s) to the whole domain RVE Ω 0 (X) upon performing an extrapolation of χ (s) from RVE Ω (s) 0 (X) to the whole set RVE Ω 0 (X). The extrinsic volumetric deformation of the fluid phase is specularly introduced as the scalar J(f) = det ∂ χ(f) /∂X, and the following relation holds: J

0

  , as a reference configuration field, does not depend from the kinematic descriptors. Potential U ext 0 is decomposed into the sum of a potential energy term U ext Ω0 , due to external actions associated with the state of the interior points of Ω (M ) 0 , plus a second term U ext ∂Ω0 associated with the external actions across the boundary ∂Ω (M ) 0

  the reference configuration, the external volume forces, b(s,ext)

  Spatial fields of external volume forces b(s,ext) and b(f,ext) are provided by the push-forward of the vector fields b(s,ext) 0 and b(f,ext)

  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 ∂Ω (M ) 0

) 21 Denoting

 21 by χ(u) the common value of the placement at the boundary ( χ(u) = χ(s) = χ(f) ), the vector field of external tractions over ∂Ω (M ) 0

  considered in the reference configuration of Ω (M ) 0 into two subsets: a 'fluid-saturated solid porous' (reference) subset Ω (s) 0 characterized by Φ (s) 0 = 0, and a (reference) fluid subset

  connected regular domains with piecewise smooth boundaries ∂Ω (s) 0 and ∂Ω (f ) 0 . Also, in order to address a variety of boundary conditions and interface conditions, we assume non-null the intersection ∂Ω shown in Figure 1. The remaining part of the solid boundary ∂Ω (s) 0 surface, which does not belong to ∂Ω

Figure 1 :

 1 Figure 1: Sketch of the partition of the macroscopic mixture domains into subsets. Ω (M ) 0 : reference domain of the mixture; Ω (M ) : deformed domain of the mixture; Ω (s) 0 : macroscopic reference domain of the solid subset (Φ (s) 0 = 0); Ω (f ) 0 : macroscopic domain entirely occupied by the fluid at reference configuration (Φ (s) 0 = 0); ∂Ω (M ) 0 : mixture boundary at reference configuration; ∂Ω (M U ) 0 : part of the boundary ∂Ω (M ) 0 which also belongs to the boundary ∂Ω (s) 0 ; S (sf ) 0

  via application of the fundamental lemma of calculus of variations. Field equations are obtained considering virtual displacements vanishing at the boundary, such that δ χ(s) = o, δ χ(f) = o, and δ Ĵ(s) = 0 over ∂Ω (M ) 0 . In particular, by selecting virtual variations of type δ χ(s) = o, δ χ(f) = o, and δ Ĵ(s) = 0 over Ω (M ) 0 one obtains the linear momentum balance of the solid phase:

  and the associated additive split of the integrals

  82)The previous relation must hold for any δ ũ(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 δ ũ(s) = o over ∂Ω (s) 0 . By this choice, the last two rows in (82) are cancelled and localization by the fundamental lemma yields the following equation hodling in the interior points of Ω

  87) It is worth observing that the condition PN = o over S (sf ) 0 and the formally similar boundary relation for single-phase continuum elasticity PN = o (where P is the classical first Piola tensor) have very different physical meanings. The condition PN = o 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 PN = o over S (sf ) 0does not imply that mechanical interaction between the interior solid and the fluid external to Ω

  [START_REF] Andreaus | A 2-d continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time[END_REF] have been obtained based on the mechanical hypotheses of Subsection 2.2.2 where only the individual strain energies U , and which still do not contemplate a specific solid-fluid interaction apart from the interaction of geometric/volumetric type determined by the saturation constraint[START_REF] Boer | Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory[END_REF] ( see the coupling term in Equation (75) containing the gradient of Π(f) ).

  expressed in terms of local strain energy density, also L (sf ) 0 can be expressed as the space integral of a macroscopic local energy density function ψ(sf)x , viz.:

  the model requires the addition of the terms ∂ χ(s) L (sf ) 0 χ(s) and ∂ χ(f) L (sf ) 0

  [START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF]. From a formal point of view, such result coincides with the classical statement of Terzaghi's principle, and the extrinsic stress tensor σ(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).

  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-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

134 )

 134 Variation of function compositionLet a : (f, x) → y be a functional application relating a real function f ∈ F and x ∈ R to a scalar y ∈ R and let b : (f, y) → z be a second functional application relating a real function f ∈ F and y ∈ R to a scalar z ∈ R. Consider the following composition of these two functional applications c = b • a with :

0J

  add and κ(s) 0 add , the terms containing second time rates are considered to be dominant with respect 64 to terms containing only first time derivatives so that the latter are ruled out. Accordingly, the following inertia terms are finally computed for (65):(f ) -J (f ) δ Ĵ(s) dV 0(193)To facilitate the assembly of all the above computed variations of the individual terms of δU

J

  (f ) -J (f ) δ Ĵ(s) dV 0

  domain of the mixture entirely occupied by the fluid Ω (f ) macroscopic current subset domain of the mixture entirely occupied by the fluid ∂Ω (M ) 0 boundary of macroscopic reference domain of the mixture ∂Ω (M ) boundary of macroscopic current domain of the mixture ∂Ω (M U ) 0 surface common to the boundaries of the reference domain of the mixture and of its solid subset S (sf ) 0 free solid-fluid macroscopic interface in the reference configuration S (sf ) free solid-fluid macroscopic interface in the current configuration X generic point of the macroscopic reference domain of the mixture x generic point of the macroscopic current domain of the mixture RVE Ω 0 (X) microscale RVE of the mixture in the reference configuration, centered in the point X RVE Ω (s) 0 (X) solid subset of the X-centered microscale reference RVE of the mixture RVE Ω (f ) 0 (X) fluid subset of the X-centered microscale reference RVE of the mixture RVE Ω (x) microscale RVE of the mixture in the current configuration, centered in the point x solid subset of the x-centered microscale RVE of the mixture in the current configuration RVE Ω fluid subset of the x-centered microscale RVE of the mixture in the current configuration χ(s) macroscopic placement vector of the solid phase χ(f) macroscopic placement vector of the fluid phase V 0 RV E (X) volume of the X-centered microscale reference RVE of the mixture V (s) 0 RV E (X) volume of the solid subset of the X-centered microscale reference RVE of the mixture V (f ) 0 RV E (X) volume of the fluid subset of the X-centered microscale reference RVE of the mixture V RV E (x) volume of the x-centered microscale deformed RVE V (s) RV E (x) volume of the solid subset of x-centered microscale deformed RVE V (f ) RV E (x) volume of the fluid subset of x-centered microscale deformed RVE Φ

0

  macroscopic true density of the solid phase in the reference configuration ρ(s) macroscopic true density of the solid phase in the current configuration ρ(s) 0 apparent reference solid density ρ(f) fluid volume fraction Ĵ(f) sat χ(f) saturation-based field relating Ĵ(f) to primary kinematic descriptors χ(s) , χ(f) , Ĵ(s) F(s) macroscopic deformation gradient of the solid phase ū(s) infinitesimal macroscopic displacement vector of the solid phase ū(f) infinitesimal macroscopic displacement vector of the fluid phase ê(s) infinitesimal macroscopic intrinsic volumetric strain of the solid phase ê(f) infinitesimal macroscopic intrinsic volumetric strain of the fluid phase ē(s) infinitesimal macroscopic extrinsic volumetric strain of the solid phase ē(f) infinitesimal macroscopic extrinsic volumetric strain of the fluid phase ε(s) infinitesimal extrinsic strain tensor L (M ) 0 Lagrange the mixture L (sf ) 0 term of Lagrange function describing drag solid-fluid interaction T (M ) 0 kinetic energy of the mixture T (s) 0 kinetic energy of the solid phase T the mixture due to external actions U ext Ω0 part of the potential energy due to external actions, associated with volume forces U ext ∂Ω0 part of the potential energy due to external actions, associated with boundary forces ψ(s) 0 apparent solid strain energy density per unit reference space ψ(f) 0 apparent fluid strain energy density per unit reference space ψ (f ) 0 microscale fluid strain energy density per unit reference space ψ(s) 0 true solid strain energy density per unit reference space ψ(f) 0 true fluid strain energy density per unit reference space ψ(s) 0ext reference potential energy density of solid associated with external volume forces ψ(f) 0ext reference potential energy density of fluid associated with external volume forces ψ(∂) 0ext apparent reference surface density of potential energy associated with boundary forces κ(s) 0 apparent density of kinetic energy of the solid phase κ(f) 0 apparent density of kinetic energy of the fluid phase κ(s) 0 add microinertia-associated part of the apparent density of kinetic energy of the solid phase κ(f) 0 add microinertia-associated part of the apparent density of kinetic energy of the fluid phase ρ(s) add. 0 microinertia-associated density parameter of the solid phase ρ(f) add. 0 microinertia-associated density parameter of the fluid phase b(s,ext) 0 solid external volume forces in the reference configuration b(f,ext) 0 fluid external volume forces in the reference configuration b(s,ext) solid external volume forces in the current configuration b(f,ext) fluid external volume forces in the current configuration

Table 2 :

 2 List of symbols

Considering the property δ φ = dδϕ dt and that for a linear operator A (l) over a variable δϕ the identity holds d dt A (l) [δϕ] = d dt A (l) [δϕ] + A (l) dδϕ dt , relation (168) can be equated to:

and hence to

Taking into account that ϕ(t 0 ) and ϕ(t f ) are given, so that their variations δϕ (t f ) and δϕ (t 0 ) are null, the following generic format of the Euler-Lagrange equation associated with field ϕ is inferred from (169):

Since the previous relation must hold for any function δϕ one infers the strong-form equation:

which is complemented by constraints for field ϕ and its variations δϕ over the space-time boundary ∂Ω

For mechanical systems depending on a finite number of continuum fields ϕ 1 , ϕ 2 , . . . , ϕ N , the weak statement provided by (171) generalizes to:

and the derivation of the system of strong-form PDE is subordinated to the specification of the space-time boundary constraints for fields ϕ 1 , ϕ 2 , . . . , ϕ N (and associated contraints for their variations δϕ 1 , δϕ 2 , . . . , δϕ N ) over ∂Ω

Variation of individual terms in Lagrange function

Hereby the computation of the variations of the individual terms δU

required to obtain the explicit form of ( 64) and ( 65) is reported. In the developments below, it

reference maps of volume fraction fields along phase placements A ijkl , B ij , C generic elastic moduli kf fluid intrinsic stiffness σ (ext) external stress tensor σ(s) h value attained by the extrinsic stress tensor in a region with uniform stress state p h value attained by the fluid pressure in a region with uniform stress state Ω (h) macroscopic space region with uniform stress state Ω (u) macroscopic space region undergoing undrained flow conditions