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 II: Applications to isotropic media and stress partitioning. Bridging the gap between Biot's and Terzaghi's perspectives

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

Stress partitioning in multiphase porous media is a fundamental problem of solid mechanics, yet not completely understood: no unanimous agreement has been reached on the formulation of a stress partitioning law encompassing all observed experimental evidences in two-phase media, and on the range of applicability of such a law.

A most celebrated stress partitioning law, based on the notion of 'effective stress', is known as 'Terzaghi's principle'. However, while there is agreement on its reliability in describing the behaviour of soils and soft hydrated biological tissues, experimental obser-

vations on certain porous media have been generally interpreted as deviations from such law.

The objective of this study is to perform an analysis on the range of applicability of the notions of effective stress and effective stress principles. This is carried out employing a variational macroscopic theory of porous media (VMTPM) derived in the companion Part I of this study. Such theory predicts that the external stress, the fluid pressure, and the stress tensor workassociated with the macroscopic strain of the solid phase are partitioned according to a relation formally compliant with Terzaghi's law, irrespective of the microstructural and constitutive features of a given medium.

Herein, these laws are applied to the study of stress partitioning in three classes of materials: linear media, media with solid phase having no-tension response, and cohesionless granular media.

It is shown that VMTPM recovers for the dynamics of isotropic media equations having the same structure of Biot's equations. Also, compliance with Terzaghi's principle can be rationally derived as the peculiar behavior of the specialization of VMTPM recovered for cohesionless granular media, in absence of incompressibility constraints. Moreover, it is shown that the experimental observations on saturated sandstones, generally considered as proof of deviations from Terzaghi's law, are predicted by VMTPM. In addition, a rational deduction of the phenomenon of compression-induced liquefaction in cohesionless mixtures is reported: such effect is found to be a natural implication of VMTPM when unilateral contact conditions are considered for the solid above a critical porosity. Finally, a characterization of the phenomenon of crack closure in fractured media is inferred in terms of macroscopic strain and stress paths.

Altogether these results exemplify the capability of VTMPM to describe and predict a large class of linear and nonlinear mechanical behaviors observed in two-phase saturated materials. As a conclusion of this study, a generalized statement of Terzaghi's principle for multiphase problems is proposed.

Introduction

In fluid saturated porous media, the mechanism by which external stresses are partitioned between solid and fluid phases is complex and yet not completely understood [START_REF] Fillunger | Versuche über die zugfestigkeit bei allseitigem wasserdruck, Osterr. Wochenschr[END_REF][START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[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][START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF][START_REF] Jardine | Developments in understanding soil behaviour[END_REF][START_REF] Bowen | Compressible porous media models by use of the theory of mixtures[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 Buhan | On the validity of the effective stress concept for assessing the strength of saturated porous materials: a homogenization approach[END_REF][START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF].

In the first decades of the last century, Terzaghi introduced the concept of 'effective stress', which regulated observable effects in saturated soils, according to a so called 'effective stress principle' [START_REF] Fillunger | Versuche über die zugfestigkeit bei allseitigem wasserdruck, Osterr. Wochenschr[END_REF][START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF]. Ever since its conception, the effective stress has stimulated a large body of theoretical and applied researches in several fields dealing with the problem of stress partitioning in multiphase media [START_REF] De Boer | The development of the concept of effective stresses[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][START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF][START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Rice | Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents[END_REF][START_REF] Pietruszczak | On the mechanical response of saturated cemented materialspart i: theoretical considerations[END_REF][START_REF] Pietruszczak | On the mechanical response of saturated cemented materials -part ii: experimental investigation and numerical simulations[END_REF][START_REF] Hellmich | Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation[END_REF][START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF][START_REF] Oloyede | Complex nature of stress inside loaded articular cartilage[END_REF][START_REF] Mow | Basic orthopaedic biomechanics & mechano-biology[END_REF].

The effective stress expressions so far proposed can mostly be cast in the form of relations equating the external stress applied to the medium to a linear combination of the elastic deformation in the solid skeleton and of the interstitial pressure of the fluid saturating the mixture.

However, there is a certain disagreement on the coefficients to be adopted, and several different expressions have been proposed [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[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][START_REF] Jardine | Developments in understanding soil behaviour[END_REF][START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF][START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF][START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF][START_REF] Bishop | The influence of an undrained change in stress on the pore pressure in porous media of low compressibility[END_REF][START_REF] Gray | Analysis of the solid phase stress tensor in multiphase porous media[END_REF][START_REF] Lade | The concept of effective stress for soil, concrete and rock[END_REF]. Comparing different expressions of stress partitioning to one another is difficult since they often derive from different multiphase poroelastic theories which proceed from substantially different physical-mathematical, or engineering, premises to introduce macroscopic stress measures [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][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] Biot | The elastic coefficients of the theory of consolidation[END_REF][START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF][START_REF] Bedford | A variational theory of porous media[END_REF][START_REF] Bishop | The effective stress principle[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] Gray | The solid phase stress tensor in porous media mechanics and the hill-mandel condition[END_REF][START_REF] Lancellotta | Coupling between the evolution of a deformable porous medium and the motion of fluids in the connected porosity[END_REF][START_REF] Schrefler | Mechanics and thermodynamics of saturated/unsaturated porous materials and quantitative solutions[END_REF][START_REF] Wilmanski | On microstructural tests for poroelastic materials and corresponding Gassmann-type relations[END_REF] and from different governing balance equations [START_REF] Bowen | Compressible porous media models by use of the theory of mixtures[END_REF][START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[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] Goodman | A continuum theory for granular materials[END_REF][START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF][START_REF] Wilmanski | A thermodynamic model of compressible porous materials with the balance equation of porosity[END_REF]. Also, there is a certain disagreement on the theoretical accuracy and domain of veracity of the Terzaghi's relation. In fact, as discussed in [START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF][START_REF] Jardine | Developments in understanding soil behaviour[END_REF], no theory of poroelasticity has been able to recover a stress partitioning law for two-phase media in agreement with Terzaghi's postulated one, which can be applied to any biphasic system in absence of specific microstructural or constitutive assumptions and to any experimental condition.

The fulfillment of medium independence for stress partitioning within two-phase continuum poroelasticity theories, framed into the more general problem of the derivation of mediumindependent continum governing equations and boundary conditions, is examined in the companion Part I of this study [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF]. Therein, a rational derivation of medium-independent stress partitioning laws is obtained downstream of a purely-variational and purely-macroscopic deduction of the complete set of momentum balance equations and boundary conditions for the two-phase poroelastic problem in a minimal kinematic setting based on an extrinsic/intrinsic split of volumetric strain measures previously investigated by the authors [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | Formulation of a finite deformation model for the dynamic response of open cell biphasic 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]. The theoretical frame adopted is termed Variational Macroscopic Theory of Porous Media (VMTPM). The stress par-titioning law obtained states that the external stress tensor is always partitioned between solid and fluid phases by a relation formally coincident with the classical tensorial statement of Terzaghi's effective stress principle. In such relation, the role of the effective stress tensor is played by the solid extrinsic stress tensor σ(s) . Notably, this stress law has been derived in absence of any constitutive and/or microstructural hypothesis on the phases, and for this reason it has a general medium-independent validity. However, well-known experimental results deriving from testing of saturated porous media [START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF][START_REF] Jardine | Developments in understanding soil behaviour[END_REF][START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF] are usually interpreted as evidences of deviations from Terzaghi's law for specific classes of two-phase media. The primary objective of this contribution is to demonstrate that VMTPM is capable of predicting such experimental results. The other objectives are: 1) to show that VMTPM can recover results of consolidated use in poroelasticity, such as Terzaghi's stress partitioning principle and Biot's equations; and 2) to show the validity of VMTPM beyond the linear-elastic range.

Accordingly, stress partitioning laws are herein investigated for three classes of isotropic media with volumetric-deviatoric uncoupling subjected to infinitesimal deformations: 1) linear media; 2) media with no-tension response for the solid phase (yet, linear in compression); 3) cohesionless granular media. This is done by analyzing VMTPM predictions for two-phase mixtures in four thought-experiment compression tests, characterized by different loading and drainage conditions.

The paper is organized as follows. Governing equations and boundary conditions defining the two-phase medium-independent boundary-value poroelastic problem in linearized kinematics, as framed in VMTPM, are recalled from Part I in Section 2, together with the relevant medium-independent stress partitioning laws. In Section 3 this boundary-value description is specialized to isotropic media with volumetric-deviatoric uncoupling, presenting the relevant elastic moduli (Section 3.1), the corresponding governing PDEs (showing the recovery of Biot's equations) (Section 3.2) and also reporting useful Composite Spheres Assemblage (CSA) homogenization estimates ( Section 3.4). The isotropic boundary-value description is deployed in Section 4 to analyze the stress partitioning response in terms of strain and stress paths in the planes of normalized strain and stress volumetric coordinates in four types of ideal compression tests: a Jacketed Drained test (JD), an Unjacketed test (U), a Jacketed Undrained test (JU), and a (so-called) Creep test with constant confining stress and Controlled Fluid Pressure (CCFP), for three classes of isotropic media assuming volumetric-deviatoric uncoupling and infinitesimal deformations: 1) linear media; 2) media with solid phase having no-tension response; 3) cohesionless granular media. Next, in Section 5, the elemental responses analyzed in Section 4 are employed to identify the specific set-up boundary conditions used in experimental compression tests on porous water-saturated Weber sandstone [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], and analyze stress partitioning in these experiments. In Section 6 the results of Sections 4 and 5 are reconsidered in a comprehensive discussion on the range of validity of Terzaghi's effective stress principle and on the meaningfulness of the concept of effective stress. Conclusive remarks are finally reported in a final dedicated section. Details on the notation and the list of symbols used in the present paper, with related descriptions are reported in Appendix, Section 8.

Two-phase medium-independent boundary value VMTPM problem

The statement of the boundary value problem as derived in Part I [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF] is hereby recalled. The framework adopted is medium-independent: it does not require any specific constitutive or microstructural hypotheses on the media, and neglects microinertia terms. The kinematically-linear equations are considered to describe the problem for infinitesimal deformations. It is important to remark that kinematic linearization is the only simplificative restriction applying to the equations recalled in this subsection. Conversely, no restrictions are applied with respect to constitutive nonlinearity: the equations remain ordinarily applicable to describe any nonlinear constitutive response of constituent phases. Accordingly, the deformation is described by the infinitesimal displacement fields of the solid phase ū(s) , of the fluid phase ū(f) , and by the infinitesimal intrinsic strain field ê(s) . The scalar field ê(s) is a primary kinematic descriptor which corresponds to the specialization to infinitesimal kinematics of the finite macroscopic field of intrinsic volumetric strain Ĵ(s) . Field Ĵ(s) is a finite-deformation primary descriptor in VMTPM corresponding to the ratio ρ(s) /ρ (s) 0 between true densities of solid before and after deformation. Such field can be also operatively defined by its experimental characterization in terms of changes of solid volume fractions before (Φ (s) 0 ) and after deformation (φ (s) ), viz.:

Ĵ(s) = ρ(s) ρ(s) 0 = J(s) φ (s) Φ (s) 0 (1)
with J(s) = det ∂ χ(s) /∂X denoting the Jacobian of the macroscopic placement field of the solid phase. In linearized kinematics, relation (1) specializes to:

ê(s) = dφ (s) φ (s) + ∂ ū(s) i ∂x i (2)
and links the characterization of ê(s) (see Part I of this study) to the measurement of infinitesimal porosity changes dφ (s) .

The kinematic descriptor fields ū(s) , ū(f) and ê(s) are defined over the domain of the mixture Ω (M ) which, due to the infinitesimal kinematics, represents both undeformed and deformed configurations. The undeformed configuration is also defined by the fields of solid and fluid volume fractions (i.e., φ (s) and φ (f ) ). The domain Ω (M ) is partitioned in two subsets defined as follows: Ω (f ) containing only fluid (φ (f ) = 1), and the complementary subset

Ω (s) ⊂ Ω (M )
where φ (s) = 0.

In the kinematically linearized theory, the macroscopic strain of the solid is defined by ê(s)

and by the infinitesimal extrinsic strain tensor:

ε(s) = sym ū(s) ⊗ ∇ (3)
while infinitesimal volumetric strain measures are the extrinsic volumetric strains of solid and fluid:

ē(s) = ∂ ū(s) i ∂x i = trε (s) , ē(f) = ∂ ū(f) i ∂x i (4)
plus the intrinsic volumetric strains ê(s) and ê(f) .

The complete saturation hypothesis reads in the infinitesimal case:

φ (s) + φ (f ) = 1, dφ (s) + dφ (f ) = 0 (5)
and, as shown in Part I, it implies the following dimensionless saturation relation between volu-metric infinitesimal strains:

φ (s) ê(s) + φ (f ) ê(f) = ∂φ (s) ū(s) i ∂x i + ∂φ (f ) ū(f) i ∂x i (6)
whereby ê(f) can be treated as a derived field depending from ū(s) , ū(f) and ê(s) .

The primary stress measures are the fluid pressure p, the extrinsic stress tensor of the solid phase σ(s) , and the intrinsic pressure of the solid phase p(s) . These quantities are defined by work association:

σ(s) ij = ∂ ψ(s) ∂ ε(s) ij , p(s) = - ∂ ψ(s) ∂ê (s) , p = - ∂ ψ(f) ∂ê (s) (7)
where ψ(s) and ψ(f) are the macroscopic strain energy densities, and ψ(f) is defined by the

relation ψ(f) = φ (f ) ψ(f) .
Momentum balances are obtained by applying a kinematic linearization to the stationarity conditions stemming from the least-Action principle. Hereby we consider the most general medium-independent equations obtained ruling out microinertia terms:

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 (8)
Linear momentum balance of the fluid phase:

-φ (f ) ∂p ∂x i + b(fs) i + b(f,ext) i = ρ(f) ü(f) i (9)
Intrinsic momentum balance: Equations ( 8), ( 9) and ( 10) express stationarity of the Action in relation to the displacement fields of the solid phase, of the fluid phase, and to the intrinsic volumetric strain, respectively.

p(s) -φ (s) p = 0 (
When inertia terms are negligible Equations ( 8), ( 9) and ( 10) specialize as follows:

∂ σ(s) ij ∂x j -φ (s) ∂p ∂x i + b(sf) i + b(s,ext) i = 0 (11) -φ (f ) ∂p ∂x i + b(fs) i + b(f,ext) i = 0 (12) p(s) -φ (s) p = 0 (13)

Boundary conditions with bilateral contact

The achievement of bilateral contact conditions at the boundaries can be determined by gluing the external surfaces of the specimen with the internal surfaces of the environment, or by applying a compressive prestress across the boundaries. Stress-type bilateral boundary conditions over ∂Ω (M ) are also obtained from a variational deduction [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF], and turn out to be:

σ(s) ij -pδ ij n j = t ext i over ∂Ω (M ) (14)
where n denotes as usual the unit outward normal to the boundary. Boundary conditions of displacement-type are:

ū(s) = ū(f) = u (ext) over ∂Ω (M ) (15)
Conditions over free solid-fluid macroscopic interfaces

In several mixture problems, such as in unjacketed tests, it is necessary to consider the condition which characterizes those interior macroscopic surfaces which, although not belonging to the true boundary ∂Ω (M ) , are part of the boundary ∂Ω (s) of the macroscopic physical subdomain Ω (s) where φ (s) = 0. Such surfaces have been termed free solid-fluid macroscopic interfaces, see [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF]. Their mathematical definition is S (sf ) = ∂Ω (s) \ ∂Ω (M ) . In points interior to Ω (s) a mixture of solid and fluid is present (being both φ (s) = 0 and φ (f ) = 0), while, in the external points belonging to Ω (f ) , space is entirely occupied by the fluid alone, φ (f ) = 1.

The following condition holds over S (sf ) [37]:

σ(s) ij n j = 0 over S (sf ) (16)
It is worth remarking that the condition σ(s) n = o over S (sf ) does not entail absence of mechanical interaction between the solid phase interior to Ω (s) and the fluid external to Ω (s) in the points of S (sf ) , since, in these points, coupling between the solid and the surrounding fluid regions of Ω (f ) still remains mediated by the intrinsic stress entering [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF]. Hence condition [START_REF] Hellmich | Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation[END_REF], although formally similar, has not to be confused with the condition which in single-phase elasticity involves the Cauchy stress tensor σ (s) holding in a point of the boundary surface of a solid domain (i.e., σ (s) n = o). This last condition states instead that there is no mechanical interaction between the interior solid and the external environment at that point.

Boundary conditions with unilateral contact

The typical condition for several tests on different classes of fluid saturated materials is more properly described by unilateral contact since, in most experimental setups, specimens are ordinarily not bilaterally constrained to the walls of the confining chamber [START_REF]ASTM standard d 2435-96. standard test method for one-dimensional consolidation[END_REF][START_REF] Armstrong | An analysis of the unconfined compression of articular cartilage[END_REF].

Unilateral contact in a point x ∈ ∂Ω (M ) is addressed by combining bilateral (closed contact) boundary conditions, corresponding to full adhesion between the solid macroscopic external surface and the container wall boundaries expressed, with open contact conditions, corresponding to the solid phase boundary moving off the wall boundaries. Closed contact conditions are expressed by equations ( 14) and ( 15) (with u (ext) representing the displacement of the containerwall boundary). In open contact conditions, the solid macroscopic external surface upon moving off the wall boundaries after deformation, are converted into a free solid-fluid macroscopic interface, of type S (sf ) subjected to [START_REF] Hellmich | Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation[END_REF], with the wall boundaries remaining in contact only with the fluid phase. The combination of these two conditions is achieved by extending the standard description of contact in single-continuum problems by employing a set-valued law and a gap function [46,[START_REF] Studer | Numerics of unilateral contacts and friction: modeling and numerical time integration in non-smooth dynamics[END_REF]. Accordingly, the gap function, g, is defined over the boundary surface as:

g = u (ext) -ū(s) • n. (17)
In presence of closed contact conditions in x ∈ ∂Ω (M ) , which correspond to g = 0 in a boundary point x, the behavior of the boundary is stated by [START_REF] Pietruszczak | On the mechanical response of saturated cemented materialspart i: theoretical considerations[END_REF]. Conversely, open contact in a point x ∈ ∂Ω (M ) corresponds to the attainment of condition g > 0 in such a point, with separation of surfaces S (sf ) (where (16) applies) and ∂Ω (M ) (where (14) applies). In infinitesimal displacements, the undeformed and deformed configurations are superimposed. Hence, although S (sf ) and ∂Ω (M ) are distinct surfaces, they are superposed in a neighborhood of x. Thus, when open contact is attained in a point x ∈ ∂Ω (M ) , both conditions ( 16) and ( 14) apply in such a point. Consequently, in open contact, one has simultaneously that -pn = t (ext) and σ(s) n = o.

The first of these two relations implies that the external tractions are all transfered to the fluid phase interposing between the solid and the wall. The second relation indicates that the solid behaves as a free solid-fluid macroscopic interface. Note that this second condition is formally similar to the condition of open contact for standard unilateral contact in Cauchy single-phase continua: σ (s) n = o, where σ (s) is the Cauchy stress tensor [46], although it is different since it involves the extrinsic stress tensor σ(s) .

Summary of unilateral boundary conditions for stresses

closed contact: σ(s) n -pn = t (ext) if g = u (ext) -ū(s) • n = 0 open contact: -pn = t (ext) σ(s) n = o if g = u (ext) -ū(s) • n > 0 (18)
Medium-independent general stress partitioning law Two general stress partitioning laws of medium-independent character have been derived in Part I [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF].

The first one applies to regions Ω (h) where the stress state is macroscopically uniform and with null relative solid-fluid motion at the boundary Ω (h) . By denoting σ(s) h and p h the constant values of fields σ(s) and p inside Ω (h) , the external traction field t (ext) (x, n) over ∂Ω (h) , can be represented by a single constant tensor σ (ext) associated with domain Ω (h) :

t (ext) (x, n) = σ (ext) n, x ∈ ∂Ω (h) (19)
This external stress tensor σ (ext) is always partitioned in compliance with the following general law, irrespective of the particular constitutive and microstructural features of the medium considered [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF]:

σ (ext) = σ(s) h -p h I (20)
A second general stress partitioning law, formally similar to [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF], applies to regions undergoing conditions of undrained flow. In such regions, hereby denoted as Ω (u) , the macroscopic relative solid-fluid motion is prevented across any surface. In this case, the traction in any point

x ∈ Ω (u) over a surface of unit normal n turns out to be expressed as:

t (ext) (x, n) = σ (ext) n, ∀x ∈ Ω (u) , ∀n (21)
where the tensor field σ (ext) , defined over Ω (u) , has the expression:

σ (ext) = σ(s) -pI (22)
As previously observed [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF][START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[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], relations [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF] and ( 22) coincide, from a formal point of view, with the classical tensorial statement of Terzaghi's principle upon identifying σ(s) with the effective stress tensor.

3 Linear Isotropic formulation with volumetric-deviatoric uncoupling This section examines the specialization of the linear formulation of the previous section under hypotheses of isotropy and volumetric-deviatoric uncoupling. In Section 3.1, as a consequence of these assumptions, suitable elastic moduli are introduced and general representations are derived of the stess-strain law of the solid phase. The corresponding linear PDE governing the response of isotropic media for negligible inertia forces are derived in Section 3.2. Finally, in Section 3.4, bounds for the elastic moduli are estimated by deploying a simple Composite Spheres Assemblage (CSA) homogenization approach [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF][START_REF] Hashin | Analysis of composite materials -a survey[END_REF].

Linear elastic isotropic laws

As a consequence of the assumption of volumetric-deviatoric uncoupling for the solid phase and linear elastic response, the strain energy density ψ(s) achieves the following quadratic form:

ψ(s) ε(s) dev , ē(s) , ê(s) = 1 2 K(s) dev ε(s) dev : ε(s) dev + 1 2 ē(s) ê(s) K (s) iso    ē(s) ê(s)    (23)
where the volumetric-deviatoric split is introduced for strains and energy in the usual way:

ε(s) = ε(s) dev + ε(s) sph , ε(s) sph = 1 3 trε (s) I = 1 3 ē(s) I , ε(s) dev = ε(s) - 1 3 ē(s) I (24) ψ(s) ε(s) , ê(s) = ψ(s) dev ε(s) dev + ψ(s) sph ē(s) , ê(s) (25)
Standard variationally-consistent definitions for elastic coefficientsis are considered. These are introduced as the second derivatives [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]:

K (s) iso =    K ē(s) ē(s) K ê(s) ē(s) K ê(s) ē(s) K ê(s) ê(s)    =      ∂ 2 ψ(s) ∂ē (s) ∂ē (s) ∂ 2 ψ(s) ∂ê (s) ∂ē (s) ∂ 2 ψ(s) ∂ê (s) ∂ē (s) ∂ 2 ψ(s) ∂ê (s) ∂ê (s)      , K(s) dev = ∂ 2 ψ(s) ∂ ε(s) dev ∂ ε(s) dev . (26)
Owing to these definitions, the stress-strain relations for the solid phase are written as follows:

σ(s) dev = K(s) dev ε(s) dev (27)    p(s) p(s)    = - K (s) iso    ē(s) ê(s)    (28)
where the primary volumetric and deviatoric stresses (introduced in the standard work-associationcompliant form) are:

σ(s) dev = ∂ ψ(s) ∂ε (s) dev , σ(s) sph = ∂ ψ(s) ∂ε (s) sph . ( 29
)
The relations involved in the volumetric-deviatoric split for stresses are the usual ones:

σ(s) = σ(s) dev + σ(s) sph (30) σ(s) sph = -p (s) I σ(s) dev = σ(s) + p(s) I (31)
In particular, the auxiliary extrinsic pressure-like scalar stress p(s) is the stress quantity work associated with -ē (s) , and, owing to [START_REF] Lade | The concept of effective stress for soil, concrete and rock[END_REF], it is one third of the trace of the extrinsic stress tensor:

p(s) = - ∂ ψ(s) ∂ē (s) = - 1 3 tr σ(s) (32)

Elastic moduli

As shown 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], a convenient representation of the solid linear elastic law can be achieved in terms of standard Lamè and bulk moduli of the dry porous medium (i.e., in absence of the fluid phase) when inertia forces are negligible:

kV = K ē(s) ē(s) - ( K ê(s) ē(s) ) 2 K ê(s) ê(s) , μ = K(s) dev 2 , λ = kV - 2 3 μ (33)
Upon introducing the three auxiliary moduli kr , kr , and ks :

kr = K ê(s) ē(s) K ê(s) ê(s) , kr = φ (s) kr = φ (s) K ê(s) ē(s) K ê(s) ê(s) , ks = K ê(s) ê(s) φ (s) (34)
the stiffness matrix K (s) iso can be expressed as:

K (s) iso =      kV + kr 2 ks φ (s) kr ks kr ks ks φ (s)      (35)
The resulting representation for the stress-strain law is the following:

σ(s) = 2με (s) + λē (s) I - kr φ (s) p(s) I (36)
p(s) = -ks kr ē(s) + φ (s) ê(s) (37)
In particular, in view of relation [START_REF] Rice | Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents[END_REF], equations ( 36) and (37) achieve a convenient expression in terms of fluid pressure [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) = 2με (s) + λē (s) I -kr pI (38) φ (s) ks p = -kr ē(s) -φ (s) ê(s) (39)
The inverse of [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF] provides the compliance matrix C (s) iso :

C (s) iso = K (s) iso -1 =      1 kV -kr φ (s) kV -kr φ (s) kV 1 φ (s)
1 ks

+ ( kr) 2 φ (s) kV      (40)
whereby the spherical stress-strain relation reads:

   ē(s) ê(s)    = - C (s) iso    p(s) p(s)    (41)
Using relations ( 24), ( 27), ( 31), ( 33), ( 40) and ( 41), the inverse strain-stress law can be reconstructed:

ε(s) = ε(s) dev + 1 3 ē(s) I = 1 2μ σ(s) dev + 1 3 ē(s) I = 1 2μ σ(s) + p(s) I + 1 3 ē(s) I (42)
ε(s) = 1 + ν Ē σ(s) - ν Ē tr σ(s) I + kr 3φ (s)k V p(s) I (43)
ê(s) = kr 3φ (s)k V tr σ(s) - 1 φ (s) 1 ks + kr 2 φ (s)k V p(s) (44)
where

ν = 3 kV -2μ 2 3 kV + μ , Ē = 9 kV μ 3 kV + μ (45)
Relation [START_REF] Travascio | Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach[END_REF] recovers the Lamé inverse elastic laws when p(s) = 0. Also, for static problems, use of (10) allows expressing the relations [START_REF] Travascio | Articular cartilage biomechanics modeled via an intrinsically compressible biphasic model: Implications and deviations from an incompressible biphasic approach[END_REF] and [START_REF]ASTM standard d 2435-96. standard test method for one-dimensional consolidation[END_REF] as functions of p:

ε(s) = 1 + ν Ē σ(s) -ν Ē tr σ(s) I + kr 3 kV pI (46) ê(s) = kr 3φ (s)k V tr σ(s) - 1 ks + kr 2 φ (s)k V p (47)
For the fluid phase, the quadratic strain energy is written as:

ψ(f) = φ (f ) 1 2 kf ê(f) 2 ( 48
)
where kf is the fluid intrinsic bulk modulus, whose definition is recalled below together with the fluid pressure-intrinsic strain relation:

kf = ∂ 2 ψ(f) ∂ê (f ) ∂ê (f ) , p = -kf ê(f) . ( 49
)

Governing PDE for the isotropic linear problem

Governing equations ( 8)-( 10) are hereby specialized on account of the isotropic constitutive laws with volumetric-deviatoric uncoupling obtained in Section 2. For simplicity, henceforth, space uniformity of porosities, densities and of elastic and inertial coefficients is assumed, and external volume forces are excluded (i.e., b(f,ext) = o and b(s,ext) = o).

ū(s) -ū (f ) PDE with inertial terms

The domain equations ( 5)-( 10) are combined with the isotropic stress-strain laws (49), ( 27), [START_REF] Gray | Unsaturated flow theory including interfacial phenomena[END_REF] and [START_REF] Hassanizadeh | General conservation equations for multi-phase systems: 1. averaging procedure[END_REF], and are solved to obtain a system of equations in the primary unknowns ū(s) and ū(f) directly comparable with Equations (6.7) obtained by Biot in [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF].

In particular, the system of ( 6), [START_REF] Hashin | Analysis of composite materials -a survey[END_REF], the second of [START_REF] Gray | Unsaturated flow theory including interfacial phenomena[END_REF], and ( 10) can be written in the following form:

   -ks kf φ (s) φ (f )       ê(s) ê(f)    =    kr ks φ (s) ē(s) φ (s) ē(s) + φ (f ) ē(f)    (50)
Hence we have:

   ê(s) ê(f)    = - 1 ksφ (f ) + kf φ (s)    φ (f ) -kf -φ (s) -ks       kr ks φ (s) ē(s) φ (s) ē(s) + φ (f ) ē(f)    =       -φ (f ) kr ks φ (s) ē(s) + kf φ (s) ē(s) + φ (f ) ē(f) ks φ (f ) + kf φ (s) kr ks ē(s) + ks φ (s) ē(s) + φ (f ) ē(f) ks φ (f ) + kf φ (s)       (51)
Substitution of ( 3) and ( 38) into ( 8), and substitution of ( 51) in ( 9) yield equations having a ū(s) -ū (f ) form easily comparable with equations (6.7) in [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range[END_REF]:

μ (∇ • ∇) ū(s) + λ + μ ē(s) ∇ + + φ (s) + kr 2 ksf ē(s) ∇ + φ (s) + kr φ (f ) ksf ē(f) ∇ + + b(sf) = ρ(s) ü (s) (52)
φ (s) + kr φ (f ) ksf ē(s) ∇ + φ (f ) 2 ksf ē(f) ∇ + -b(sf) = ρ(f) ü (f ) (53)
where ksf is a modulus defined as:

1 ksf = φ (s) ks + φ (f ) kf (54)
and which can be interpreted as a series-coupling of solid and fluid intrinsic stiffnesses ks and kf .

It is interesting to observe that the general structure of Biot's PDEs (6.7) is recovered, with one important difference in the particular expressions of the elastic coefficients. Actually, upon conveniently arranging the coefficients of the differential terms ē(s) ∇, ē(f) ∇, entering ( 52) and [START_REF] Biot | General theory of three-dimensional consolidation[END_REF] in the matrix form:

   λ + μ 0 0 0    + ksf    φ (s) + kr 2 φ (s) + kr φ (f ) φ (s) + kr φ (f ) φ (f ) 2    (55)
these can be compared with the coefficients entering Biot's equation (6.7) which are function of λ, μ, and of two elastic coefficients, Q and R introduced by Biot proceeding from the synthetic consideration of a single strain energy for the whole mixture depending on both solid and fluid strains. In VMTPM, we proceed istead from the consideration of individual strain energies of the solid and fluid phases, and explicit relations are obtained for Q and R as functions of elastic coefficients of individual phases:

Q = φ (s) + kr φ (f ) φ (s) ks + φ (f ) kf -1 , R = φ (f ) 2 φ (s) ks + φ (f ) kf -1 (56)
The important difference with Biot's equation (6.7) concerns the coefficient multiplying the term ē(s) ∇ in [START_REF] Lee | Elastic properties of hollow-sphere-reinforced composites[END_REF]. In Biot's formulation, this is equal to λ + μ. Conversely, in [START_REF] Lee | Elastic properties of hollow-sphere-reinforced composites[END_REF], this term turns out to be equal to:

λ + μ + φ (s) + kr 2 ksf ( 57
)
with an added stiffness coupling term equal to φ (s) + kr 2 ksf . Also, in contrast with Biot's formulation, no added mass terms are present in ( 52) and ( 53).

PDE for static and quasi-static interaction

For static or quasi-static problems, the sum of ( 11) and ( 12) yields:

∂ σ(s) ij ∂x j - ∂p ∂x i = 0 (58)
Substituting ( 3) and ( 38) into (58) yields:

μ + λ (∇ ⊗ ∇) ū(s) + μ (∇ • ∇) ū(s) -1 + kr ∇p = 0 (59)
Moreover, considering the following position introducing the relative solid-fluid velocity:

w (f s) = ū(f) -ū(s) (60)
according to which the following substitution can be performed:

(∇ ⊗ ∇) ū(f) = (∇ ⊗ ∇) w (f s) + (∇ ⊗ ∇) ū(s) . (61)
Upon excluding inertia terms, equation ( 53) provides:

ksf φ (f ) 2 (∇ ⊗ ∇)w (f s) + ksf φ (f ) 1 + kr (∇ ⊗ ∇)ū (s) -b(sf) = 0 (62)
The system of PDE ( 59) and ( 62) governing this class of quasi-static problems requires a specification of the particular solid fluid interaction. Hereby, as a simplest possible choice, a linear Darcy law is considered for b(sf) :

b(sf) = -b(fs) = K ∂w (f s) ∂t ( 63
)
where the proportionality coefficient K, in N s/m 4 , can be expressed as follows [START_REF] Markert | A constitutive approach to 3-d nonlinear fluid flow through finite deformable porous continua[END_REF]:

K = (φ (f ) ) 2 µ (f ) κ ( 64
)
where µ (f ) is the coefficient of effective fluid viscosity, in N s/m 2 , and κ is the intrinsic permeability of the porous material, measured in m 2 .

Account of ( 63) and of zero external volume forces in [START_REF] Coussy | Mechanics of porous continua[END_REF] provides the equation completing the system of ( 59) and ( 62):

∇p = b(fs) φ (f ) = - K φ (f ) ∂w (f s) ∂t (65)
Inclusion of (65) into ( 59) and (62) yields the system of governing equations for the isotropic problem with Darcy interaction in the so-called u-w form:

μ + λ (∇ ⊗ ∇) ū(s) + μ (∇ • ∇) ū(s) + 1 + kr K φ (f ) ∂w (f s) ∂t = 0 (66) ksf φ (f ) 2 (∇ ⊗ ∇)w (f s) + ksf φ (f ) 1 + kr (∇ ⊗ ∇)ū (s) -K ∂w (f s) ∂t = 0 (67)

CSA estimates of elastic moduli

Estimates of the constitutive moduli ks and kr have been derived 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], based on the simple Composite Spheres Assemblage (CSA) homogenization technique established by Hashin [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF][START_REF] Hashin | Analysis of composite materials -a survey[END_REF][START_REF] Lee | Elastic properties of hollow-sphere-reinforced composites[END_REF]. Derivation of CSA estimates is based on the assumption that the microstructural realization of the solid medium consists of hollow spherical cells filling out space up to the limit of zero volume of unfilled space [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF][START_REF] Hashin | Analysis of composite materials -a survey[END_REF].

It is important to remark that the peculiarity of the CSA assumption on the microstructural realization lay aside the sought feature of medium independence for the poroelastic theory since, besides isotropy, a specific hypothesis is introduced on the realization of the microstructure of the medium. For this reason, all relations making use of these estimates will be marked as '(Obtained with CSA)'. In the following, these relations will be only prudently invoked to have subsidiary correlations between macroscopic and microscopic moduli, once microstructureindependent isotropic laws of more general validity are first obtained.

However, CSA provides relations of practical use between the macroscopic moduli ks , kr , and kV , and the elastic parameters which define the isotropic response at the microscale of the material constituting the solid phase. These parameters are the microscale shear modulus µ, the microscale bulk modulus k s , and the microscale Poisson ratio ν, related to k s and µ by the usual relation ν = (3k s -2µ)/[2 (3k s + µ)] holding for isotropic materials.

The relations provided by CSA between macroscopic and microscale elastic moduli are:

kr = - φ (s) 4 3 µ 4 3 µ + k s (1 -φ (s) ) , ks = 1 1 -φ (s) 4 3 µ + k s (1 -φ (s) ) (68) kV = φ (s) 4 3 µk s 4 3 µ + (1 -φ (s) )k s (69)
An alternate equivalent expression for kr as function of φ (s) and ν is [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]:

kr = - 2(1 -2ν)φ (s) 3 -3ν -φ (s) (1 + ν) (70)
In view of the bounds 0 ≤ ν ≤ 0.5 and 0 ≤ φ (s) ≤ 1, the following bounds apply for kr :

-1 ≤ kr ≤ 0 (71)
Specifically, the upper bound 0 is achieved in the limit of vanishing solid volume fraction φ (s) = 0, and when the solid constituent material is volumetrically incompressible (ν = 0.5); the lower bound is attained at φ (s) = 1.

Stress partitioning in ideal compression tests

Stress partitioning is hereby investigated for the cases of four ideal static infinitesimal compression tests in oedometric conditions. The macroscopic physical domain Ω (M ) of the boundary value problems is the mixture contained inside a cylindrical compression chamber. The boundaries of the mixture are the walls of the compression chamber and the compressive plug, see For all the tests considered, isotropy and homogeneity of the initial configuration is assumed together with hypotheses of negligible gravitational forces. Accordingly, the domain equations and the boundary conditions hereby applied are those of Section 3, and the mixture is assumed to have initially uniform porosity φ (f ) . A simple short-range solid-fluid interaction of Darcy type is also considered.

Boundary conditions

For all the tests investigated, boundary conditions on the bottom surface ∂Ω traction t (ext) is normal to the confining walls:

t (ext) = σ (ext) n n.
On the top boundary surface ∂Ω

(M )
u , four different ideal contact and loading conditions are considered for each of the four tests, whose descriptions are reported in the following.

Due to the quasi-static nature of the loads applied, equations (66) and (67) can be used as domain equations. Herein, the analysis is limited to the final stationary equilibrium configuration at time t = ∞ when consolidation phenomena have fully developed. Accordingly, time rates of ū(f) and ū(s) are set to zero together with the rate of w (f s) . Altogether, vanishing of w (f s) , the cylindrical symmetry of the system, and the uniformity of boundary conditions, ensure that the macroscopic displacement field of the solid, solving (66) and (67) is linear [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].

Such linearity implies that the strain and stress states inside the mixture are macroscopically homogeneous. Accordingly, the partial differential problem turns out to be conveniently converted into an algebraic one, where the unknowns are the uniform stress quantities σ(s) h , p h , and strains ε(s)

h and ê(s) . Owing to space homogeneity of stresses, and in light of the medium-independent stress partitioning laws in Section 2, a physically meaningful external stress tensor σ (ext) can be introduced, which is related to internal stresses by the Terzaghi-like relation [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF].

On account of the cylindrical symmetry and homogeneity of the stress field in Ω (M ) , the stress and strain matrices in the (x, r, θ) reference system all have the transversely isotropic form:

σ(s) =       σ(s) xx 0 0 0 σ(s) tt 0 0 0 σ(s) tt       , σ (ext) =       σ (ext) xx 0 0 0 σ (ext) tt 0 0 0 σ (ext) tt       (72) ε(s) =       ε(s) xx 0 0 0 ε(s) tt 0 0 0 ε(s) tt       (73)
with (•) tt = (•) rr = (•) θθ being the transverse normal components of the above tensors.

The quantities that can be directly measured during the compression tests are:

• the fluid pressure inside the specimen p h

• the external normal traction t ext x applied by the superior plate over ∂Ω

(M )
u , and related to the force F o measured by the load cell by:

t ext x = σ (ext) xx = F o A ≤ 0 with F o ≤ 0 ( 74
)
where t ext x and σ

(ext) xx
are equated on account of [START_REF] Mow | Basic orthopaedic biomechanics & mechano-biology[END_REF], and where A is the area of ∂Ω (M ) u , being t ext x negative for compressive tractions.

• the longitudinal strain of the specimen ε(ext)

x , which is the only nonzero component of the externally applied macroscopic strain ε(ext) , and turns out to be related to the displacement applied at the plate U o by:

ε(ext) x = U o L , ε(ext) =       ε(ext) x 0 0 0 0 0 0 0 0       (75)
Taking advantage of the recognized algebraic nature of the problem at hand, the operative criterion used in the following examples to cope with unilateral contact is the following: as a trial step, bilateral undrained contact conditions ( 14) are first applied directly to strain components with:

ε(s) trial = ε(ext) (76)
which corresponds to setting

(ε (s) xx ) trial = U o L (77) (ε (s) tt ) trial = 0 ; (78)
Next, a trial solution of the stress tensor ( σ(s) ) trial is computed from (ε (s) ) trial by applying the isotropic stress-strain relation, and the sign of

σ(s) n,trial = σ(s) n • n is checked: if σ(s) n > 0,
boundary conditions are switched to unilateral ones (i.e., relation [START_REF] Oloyede | Complex nature of stress inside loaded articular cartilage[END_REF]).

Although the response of the system is measured in terms of primary measured quantities When loading conditions are such that contact is preserved everywhere across the container walls, the stress path can be analyzed exclusively in terms of spherical extrinsic-intrinsic (EI) coordinates. Actually, in such a case, one has ε(s) = ε(ext) , and by virtue of volumetric-deviatoric uncoupling ( 27)-( 28) and of the Terzaghi-like variational partitioning law for homogeneous stresses relation [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF], the following relations hold:

p h ,
ε(s) dev = ε(ext) dev , σ (ext) dev = σ(s) dev = K(s) dev       2 3 ε(ext) x 0 0 0 -1 3 ε(ext) x 0 0 0 -1 3 ε(ext) x       (79)
Thus, the deviatoric part of stresses is immediately related to the external applied strain by (75).

For this reason, for each of the four tests examined, the response of the porous medium will be analyzed in terms of both primary measurable quantities and EI coordinates, postponing a separate evaluation of deviatoric stresses only in case of violation of closed-contact conditions.

For what concerns spherical coordinates, an external pressure p ext can be standardly defined on account of the cylindrical symmetry by taking one third of the trace of (20):

p ext = p(s) h + p h (80)
with

p ext = - 1 3 tr σ (ext) = - 1 3 σ (ext) xx + 2σ (ext) tt (81)
The extrinsic and intrinsic constitutive laws for the solid phase provided by ( 36) and ( 37) are respectively written:

σ(s) h = 2με (s) h + λē (s) h I -kr p h I (82) φ (s) ks p h = -kr ē(s) h -φ (s) ê(s) h (83)
By taking one third of the trace of (82), one obtains:

p(s) h = -kV ē(s) h + kr p h (84)
When closed contact is preserved at the boundaries, it is inferred from (76) that ē(s) h is also directly observable, being ē(s) h = ε(ext)

x . Since the strain applied by the impermeable plate (ē

(s) h )
and the fluid pressure (p h ) are quantities that can be measured more easily than the extrinsic solid pressure (p

(s)
h ), it is convenient to recast the previous equations in a form conveniently involving only the directly observable quantities p ext , p h , and ē(s) h . Accordingly, equation (80) can be solved for p h and substituted into (84). This yields:

p ext = -kV ē(s) h + 1 + kr p h (85)
Notably, relation ( 85) is in full agreement with the well known p ext -ē (s) -p relations obtained by experimental measures on sandstone [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]. In such contribution, Nur and Byerlee have experimentally investigated the optimality of several strain-pressure relations of the form:

p ext = -kV ē(s) h + αp h ( 86
)
where α is a fitting coefficient generally referred to as the Biot's coefficient [START_REF] Biot | General theory of three-dimensional consolidation[END_REF]. Validated by direct measurements of p ext , ē(s) and p on sandstone specimens, the best fitting expression for α is reported in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]: α = 1 -kV /k s . The combination of this expression with (86) (which corresponds to the combination of equations ( 3), ( 4), ( 6) and ( 7) in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]) hence turns out to be equal to

p ext = -kV ē(s) h + 1 - kV k s p h . ( 87
)
On the other hand, relation (87), which is of experimental origin in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], turns out to be the result of a pure theoretical deduction in VMTPM. Actually, it is noteworthy exactly the one inferred from (85) when minimal CSA homogenization estimates are exploited to relate kr to the microscale solid bulk modulus k s . As a matter of fact, (68) and (69) yield:

1 + kr = 1 - kV k s (Obtained with CSA) (88)
which substituted in (85) provides (87). Hence, for CSA-microstructured media, such an identification also yields that kr is related to Biot's coefficient by kr = α -1.

Ideal jacketed drained test

In a jacketed drained test, compression occurs via a porous plate allowing for fluid exchange between the specimen and the environment, see Figure 1a. Hence, when mechanical equilibrium is reached, the fluid pressure in the sample is null (p = 0). Accordingly, the equation [START_REF] Wilmanski | A thermodynamic model of compressible porous materials with the balance equation of porosity[END_REF] recovers the Navier law for a single continuum. Moreover, upon considering closed contact and accounting for relation (74), compressive (negative) normal stress is recognized to exist on the whole ∂Ω (M ) :

σ(s) xx < 0, σ(s) yy = σ(s) zz = λ 2μ + λ σ(s) xx < 0 (89)
The condition of closed contact is thus confirmed to hold everywhere in ∂Ω (M ) , so that the stress states can be simply analyzed in EI coordinates, being the deviatoric stress σ x by (79). Hence, relation (85) recovers the following expression as a special condition:

p ext = -kV ē(s) (90)
and ( 13) provides:

p(s) = 0 (91)
so that the stress path in the plane of normal spherical coordinates (p (s) , p(s) ) is a horizontal straight line, see Figure 2a. The inclination of the volumetric strain paths shown in Figure 2b is inferred from equation ( 37) considering zero fluid pressure:

ê(s) ē(s) = - kr φ (s) (92)
The CSA estimates (68) and (70) provide expressions for kr in terms of φ (s) , µ and k s , and in terms of φ (s) and ν, respectively. Accordingly, the strain ratio reads:

ê(s) ē(s) = 4 3 µ 4 3 µ + k s (1 -φ (s) ) = 2(1 -2ν) 3 -3ν -φ (s) (1 + ν) (Obtained with CSA) (93)
Note that in the Limit of Vanishing Porosity (LVP) (i.e., φ (s) = 0), expression (93) achieves a unit value. Hence, the slope of the strain vector in Figure 2 is 1 : 1. Also, in the Limit of Incompressible Constituent Material (LICM) (i.e., ν = 0.5), the ratio is zero.

Ideal unjacketed test

In the unjacketed test, the compressing plug is impermeable, and the chamber is fully occupied by the specimen and the fluid, see Figure 1b. The space between the plug and the specimen is occupied by the fluid phase: there is no direct contact between the plug and the upper boundary

∂Ω (M )
u . Under these mechanical conditions, the plug induces a stress state directly over the fluid phase which, in its turn, compresses the porous specimen. Accordingly, ∂Ω solid-fluid macroscopic interface of type S (sf ) , where the surface condition ( 16) applies:

σ(s) xx = 0, over ∂Ω (M ) u ( 94
)
On the upper boundary of Ω (M ) , equilibrium between the plug and the fluid is expressed considering a null extrinsic stress tensor in [START_REF] Pietruszczak | On the mechanical response of saturated cemented materialspart i: theoretical considerations[END_REF]: u , and p can be regarded as the stress input for the specimen.

σ (ext) xx = -p (

Response under bilateral contact

For the boundary ∂Ω

(M ) l
, if bilateral contact conditions are considered, we have:

ε(s) tt = 0 (96)
Hence, specialization of the equation ( 36) for the xx component yields:

2μ + λ ε(s) xx -kr p = 0 (97)
It follows that:

ē(s) = ε(s) xx = kr 2μ + λ p < 0 (98)
Given the above relations, the transverse normal stress component reads:

σ(s) tt = -kr 2μ 2μ + λ p > 0 (99)
The positive sign of σ(s) tt indicates that, when bilateral contact is ensured, VMTPM predicts that a tensile increment of extrinsic stress (or, in presence of prestress, a decrease of compressive extrinsic stress) can be even induced as the effect of external compressive loadings. This prediction of the onset of tensile extrinsic stress increments in response to compressive loading is peculiar of VMTPM, as previously pointed out [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].

Moreover, since (98) and (39) yield:

ê(s) = - k2 r φ (s) 2μ + λ + 1 k s (100)
from ( 2), the variation of solid volume fraction dφ (s) turns out to be:

dφ (s) = -φ (s) 1 k s + kr 2μ + λ kr φ (s) + 1 (101)
which can be negative depending on the relative values of the elastic moduli inside the square brackets.

Both the insurgence of positive increments of extrinsic normal stresses, shown by (99), and the possibility of negative dφ (s) are particularly significant in cohesionless mixtures. In these materials where friction plays a primary role in the overall stability, these features can be respectively put in direct relation with decrease of confining (effective) stress and with the (relative) increment of intergranular space dφ (f ) = -dφ (s) > 0. Such two features determine a decrease in friction which can be put in relation with the insurgence of phenomena of liquefaction occurring in low density saturated soils [START_REF] Kramer | Initiation of soil liquefaction under static loading conditions[END_REF][START_REF] Youd | Liquefaction resistance of soils: summary report from the 1996 NCEER and[END_REF]. In this respect, it is important to remark that, although liquefaction is mostly known to be associated with laboratory and in situ conditions as an effect essentially induced by deviatoric undrained loading and excitations, there exist experimental evidences indicating that sands can be also liquefied by isotropic compressive stress applied under quasistatic drained conditions [START_REF] Fragaszy | Undrained compression behavior of sand[END_REF].

Response under unilateral contact

When the closed contact condition is violated according to [START_REF] Oloyede | Complex nature of stress inside loaded articular cartilage[END_REF], open contact has to be considered also on ∂Ω (M ) l

. Consequently, open contact conditions (18) 2 apply across the whole

∂Ω (M ) : σ(s) n = o, over ∂Ω (M ) (102)
As a result, recalling that σ(s) is uniform, one infers σ(s) = O. Accordingly, one has:

p(s) = 0 (103)
In this case, due to (103), the normalized spherical stress path is a vertical line, as shown in Figure 3a.

It is important to remark that, when contact is lost, (79) no longer holds since, due to equations ( 46) and ( 24), one has ε(s)

dev = O and hence ε(s) dev = ε(ext) dev .
The configuration of the unjacketed compression test is characterized via (84) in terms of primary measured quantities by the following condition:

-kV ē(s) + kr p = 0 (104) which yields:

p = kV kr ē(s) (105)
Substituting ( 105) into [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF], the intrinsic-to-extrinsic strain ratio for the U test is:

ê(s) ē(s) = - 1 ks kV kr + kr φ (s) (106)
For a medium with a CSA microstructure, using (68) and ( 69), the special form achieved by ( 106) is:

ê(s) ē(s) = 1 (Obtained with CSA) (107)
Also, for such a medium, the stiffness coefficient in (105) coincides with the microscale solid bulk modulus:

-ē (s) = 1 k s p (Obtained with CSA) (108)
since ( 68) and (69) yield:

- kr kV = 1 k s (Obtained with CSA) (109)

Ideal jacketed undrained test

Stress partitioning in the jacketed undrained (JU) test has been previously described [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]. Hereby, the stress partitioning solution is recalled and expanded with considerations on the consequences of unilateral contact, and analyzed in terms of volumetric strain and stress EI paths. An impermeable plug compresses the sample by displacing of U 0 (< 0), see Figure 1c.

Response under bilateral contact

The trial condition of bilateral contact along the whole ∂Ω (M ) is initially considered. Accordingly, as a first step, the trial boundary conditions expressed by the first of (18) are applied. In particular, these conditions for displacements and stresses over ∂Ω (M ) u are respectively:

ū(s) x = ū(f) x = U 0 (110) σ (ext) xx = σ(s) xx -p (111)
As shown 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], strains in the mixture for a jacketed undrained test with closed contact at the boundaries are such that:

ē(f) = ē(s) (112)
with ē(s) = ε(s) xx = Uo L in the cylindrical configuration. Denoting by i the unit vector of the x axis and by (i ⊗ i) the associated projector, the corresponding trial stress solution to the system composed of ( 6), ( 38), ( 39) and ( 112) is:

p = -1 + kr ksf U o L (113) σ(s) = 2μ U o L (i ⊗ i) + λ + ksf kr 1 + kr U o L I (114) p(s) = -φ (s) 1 + kr ksf U o L (115)
U o L (117)
For bilateral boundary conditions, and when σ(s) xx < σ(s) tt < 0, closed contact is preserved all through ∂Ω (M ) . In this last case, the true stress state in the mixture is defined by ( 113)-(115). The corresponding expression of Skempton's coefficient B [START_REF] Skempton | The pore-pressure coefficients a and b[END_REF], defined as the ratio of the induced fluid pressure p to the applied stress t ext x , has been computed 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]:

B = p σ (ext) xx = - 1 + kr ksf 2μ + λ + 1 + kr 2 ksf (118)
A similar coefficient B iso can be defined in terms of pressure ratio as:

B iso = p p ext = 1 + kr ksf 2 3 μ + λ + 1 + kr 2 ksf (119)
The intrinsic-to-extrinsic pressure ratio can then be computed by recalling ( 13), (80), and (111):

p(s) p(s) = φ (s) B iso 1 -B iso = φ (s) 1 + kr ksf 2 3 μ + λ + kr 1 + kr ksf (120)
For non-liquefying mixtures, the volumetric strain ratio is obtained substituting (115) and ē(s) = Uo L into (37):

ê(s) ē(s) = 1 + kr ksf ks - kr φ (s) (121)
Figure 4 illustrates the volumetric stress and strain paths for the JU test with unilateral contact.

Unilateral boundary conditions

The

Cohesionless granular materials

For cohesionless media, conditions (123) and ( 124) achieve an even stronger mechanical significance. Actually, in such materials, vanishing of extrinsic stress determines vanishing of intergranular (effective) stress. Consequently the loss of frictional interaction produced by opening contact is not limited to ∂Ω (M ) , but it affects all surfaces interior to the specimen. On these surfaces, condition σ(s) n = o applies with n being the normal to the interior surface. As a consequence, friction is prevented across these internal surfaces, and this determines potential sliding in a way similar to liquids. Hence, conditions (123) and (124) discriminate the proneness of a given cohesionless mixture to liquefaction. For this reason we define a cohesionless mixture to be full liquefying when its elastic moduli are such that the stronger condition (123) holds, and to be partially liquefying cohesionless mixture when only the weaker condition (124) is verified.

When neither condition is verified, the cohesionless medium is denominated a non-liquefying cohesionless mixture. In particular, condition (123) is expected to be attained for mixtures such as water-saturated loose sands. In such mixtures, the moduli ks and kf are expected to have magnitude much higher than the macroscopic aggregate modulus 2μ+ λ. Hence, ksf >> 2μ+ λ and, when kr retains a nonvanishing value, the second negative term in (123) prevails over the first one.

For fully liquefying mixtures, even in the JU test, primary measured quantities comply with the unjacketed relation (104), and ê(s) ē(s) recover the other corresponding unjacketed relations also reported in Section 4.3.

Media with CSA microstructure

All previously reported relations hold for generic isotropic media, since no assumptions for their specific microstructural realization has been made. Special expressions, holding for media with CSA microstructure, can be obtained substituting relations (68), (69), and ( 54) in (120) and (121), and considering that, for the fluid phase, the macro-and microscale bulk moduli kf and k f coincide:

p(s) p(s) = k f 4 3 µ + k s µ (k s -k f ) (Obtained with CSA) (125) ê(s) ē(s) = 4 3 µ + k f 4 3 µ + φ (s) k f + (1 -φ (s) )k s (Obtained with CSA) (126)
Although the relations (125) and ( 126) are less general, they allow examining some limit behaviors of the system in relation to the microscale fluid stiffness. In particular, when the fluid stiffness is zero, the JD response is recovered. When the microscale bulk stiffnesses of the two materials coincide (i.e., when k s = k f ), the stress and strain ratios recover the response characteristic of the unjacketed compression test. Also, at LVP (i.e., when φ (s) 1), the strain ratio achieves unity as expected. This implies that, when porosity is low, the volumetric strain path stays in close proximity of the LVP line.

Creep test with controlled pressure

In the CCFP test, an external pressure p ext 0 is kept constant via an impermeable plug. The fluid pressure in the biphasic medium is quasi-statically decreased by controlling the fluid outflow through a valve until reaching equilibrium (i.e., zero fluid pressure), see Figure 1d. The stress path in volumetric coordinates is shown in Figure 5, and it is composed of two stages: the first one consists of an unjacketed path up to p 0 = p ext 0 , and ends up with a stress state in the solid defined as σ0 = O and p(s) 0 = φ (s) p 0 ; the second stage is determined by quasi-statically decreasing p from p 0 to 0 (allowing for controlled fluid exudation), while keeping constant the external pressure p ext 0 .

In the transition between the first and the second stage, closed contact conditions between the plug and ∂Ω (M ) u are restored with unaltered stress state in the mixture (i.e., σ = O, p(s) 0 = φ (s) p 0 , and p = p 0 ). This condition ensures that no fluid is interposed between the specimen and the compressive plug. During the second stage, the volumetric stress components read (see relations (80) and ( 13)):

p ext 0 = p(s) + p, p(s) = φ (s) p = φ (s) p ext 0 -p(s) (127)
Differentiating relation (127), the relevant increments read:

dp (s) = -dp, dp (s) = φ (s) dp (128)
The sign of the extrinsic stress increments σ(s) xx and σ(s) tt are evaluated to check the open/closed contact conditions. During the second loading stage, pressure reduces and strain variations are related by (84):

dp (s) = -kV dē (s) + kr dp (129)
and, accounting for (128) 1 , one infers:

dp = kV (1 + kr ) dē (s) (130)
having both dp < 0 and dē (s) < 0. Due to the zero condition for σ(s) 0 , the extrinsic stress increments dσ

(s)
xx coincide with their overall value, viz.:

σ(s) xx = σ(s) 0xx + dσ (s) xx = dσ (s) xx (131)
Similarly, we have dσ

(s) tt = σ(s) tt .
Variations of trial normal stresses can then be computed applying [START_REF] Wilmanski | A thermodynamic model of compressible porous materials with the balance equation of porosity[END_REF] to strain and stress increments accounting for the property dε

dσ (s) xx = 2μ + λ - kr kV (1 + kr ) dē (s) , dσ (s) tt = λ - kr kV (1 + kr ) dē (s) (133)
Since dē (s) is negative and the terms in round brackets are positive, it is recognized that closed contact conditions are never violated during the CCFP test. Hence, an account of linear bilateral boundary constraint is sufficient for the analysis of this test.

The ratio dê (s) dē (s) is similarly computed by substituting (130) into [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF], upon writing the latter for strain and stress increments:

φ (s) ks dp = -kr dē (s) -φ (s) dê (s) (134)
Substitution yields:

dê (s) dē (s) = - 1 ks kV (1 + kr ) + kr φ (s) (135)
Media with CSA microstructure

The CSA estimates for relation (135) yield:

dê (s) dē (s) = 4 3 μ 4 3 μ + k s = 2 (1 -2ν) 3 (1 -ν) (Obtained with CSA) ( 136
)
This ratio is always positive, so that the corresponding vector in the EI volumetric strain space has a positive slope, albeit bounded by the LVP line, as indicated by the arrow in Figure 5b.

Analysis of Nur and Byerlee experiments

Hereby, VMTPM is applied to the analysis of the kinematics and mechanical state of water saturated sandstone specimens as tested by Nur and Byerlee [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]. Specifically, based on an analysis in EI coordinates of the reported experimental data, the hydro-mechanical conditions effectively applied during experiments are identified. Subsequently, it is shown that EI coordinate analysis also makes possible interpreting and inferring predictions on the nonlinear mechanical response exhibited by this class of poroelastic media.

The tests reported in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF] were carried out by jacketing full water-saturated sandstone specimens of porosity φ (f ) = 0.06 in a copper sleeve, and compressing them by a steel plug at controlled flow and pressure. The experimental data set consisted of the confining pressure p ext , the apparent macroscopic volumetric strain of the specimens ē(s) , and the fluid pressure p. Table 1 reports a numerical digitalization of the data reported in Figure 2 of [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]: labels have been added to reference each record of measurements, and the corresponding values of the extrinsic and the intrinsic pressures in EI coordinates have been included operating the coordinates changes p(s) = p extp and p(s) = φ (s) p, respectively. The corresponding plot of measured extrinsic strain vs. confining pressure is shown in Figure 6. It can be observed that the experimental points are lined up vertically by groups characterized by the same confining pressure (groups are identified by the same letter).

Stress points in EI pressure coordinates p(s) and p(s) are reported in Figure 7a, and follow a pattern similar to that theoretically deduced in Section 4.5 and reported in Figure 5a. Such pattern similarity and the constant value of the confining pressure suggest that these experiments are identifiable as CCFP compression tests. The identification of a CCFP test is important since it confirms that the solid stress can be analyzed in terms of simple EI coordinates. Actually, in this test unilateral phenomena have been shown to be not relevant so that deviatoric strains of the solid are easily obtained from their coincidence with the homogeneous deviatoric strain produced in the compressive chamber (ε Figure 6: Plot of measured extrinsic strain vs. confining pressure for Weber sandstone specimens (data taken from [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF])

(s) dev = ε(ext) dev ),
In order to represent the corresponding volumetric strain path in EI coordinates, it should be considered that:

• the strain-to-stress response of sandstone exhibits a non negligible nonlinearity. Nur and Byerlee recognized this to be an effect of crack closure, which is a typical feature of compressed sandstones [START_REF] Zimmerman | Compressibility of sandstones[END_REF]. As originarily observed by the authors, the nonlinearity is specifically pronounced in response to changes of p ext when p is kept fixed, and it is almost absent in response to variations of p alone. Accordingly, this nonlinear response can be described as a secant bulk modulus kV [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], varying as function of the quantity p extp, viz.: kV = kV p(s) . Volumetric strain path in EI coordinates (ē (s) , ê(s) ), as estimated by relation (143) from data in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF].

• the intrinsic strain ê(s) is not among the data reported in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF].

To address such deviations from linearity in EI coordinates, volumetric compliance functions

ē(s) = ē(s) p(s) , p(s) , ê(s) = ê(s) p(s) , p(s) (137)
are considered, which generalize to the nonlinear range the linear volumetric compliance relations in EI coordinates represented by equations ( 40) and [START_REF] Travascio | Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach[END_REF]. The experimental data in Table 1 are used to curve-fit the function ē(s) = ē(s) p(s) , p(s) accounting for the above mentioned nonlinear dependence on variable p(s) . Moreover, since no measurement of ê(s) is reported in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], its values are extrapolated assuming that the missing information about ê(s) can be obtained via CSA estimates.

Determination of ē(s)

An expression of ē(s) is provided by the first of (41), and reads:

ē(s) = - 1 kV p(s) + kr φ (s)k V p(s) (138)
Aimed at capturing the nonlinear behavior of the rock material with the simplest interpolation, a single quadratic term in the extrinsic pressure is added. Accordingly, the employed interpolating function for ē(s) reads:

-ē (s) = a q p(s) 2 + b q p(s) + c q p(s) (139)
where a q , b q , and c q are three coefficients. Curve-fitting of the above expression with the experimental data provides:

a q = -0.00188 [kb -2 ], b q = 0.00782 [kb -1 ], c q = 0.00168 [kb -1 ], (140)
with a coefficient of determination R 2 = 0.9979. The proximity of R to unity indicates the agreement of the experimental data with the proposed model ( 139), and confirms that p(s) is the sole stress variable regulating the stiffness changes of the specimens, as originarily observed in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF].

By comparing (138) and ( 139), the following nonlinear secant expressions for kV and kr are computed: kV = kV p(s) = 1 a q p(s) + b q , kr = kr p(s) = -φ (s) c q a q p(s) + b q (141) and, as expected, they turn out to be both nonlinear functions of p(s) alone, and independent from p(s) . Within the range of stresses investigated by Nur and Byerlee, kV increases by a ratio of 42 % from 127.9 [kb] to 181.6 [kb], while kr changes from -0.2019 to -0.2867 as p(s) increases. In particular, it can be observed that (141) yields:

kr φ (s)k V = -c q (142)

Estimates of ê(s)

Function ê(s) = ê(s) p(s) , p(s) is similarly computed from the second scalar compliance equations provided by [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] Travascio | Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach[END_REF]. Accordingly, we have:

ê(s) = kr φ (s)k V p(s) - 1 φ (s) 1 ks + kr 2 φ (s)k V p(s) (143)
where the previously evaluated nonlinear secant interpolations (141) of the experimental data are employed for coefficients kr and kV .

Lack of experimental data on ê(s) for the determination of the remaining macroscopic modulus ks is supplied by a computation via CSA estimates. Accordingly, ks is related to kr and to the microscale shear modulus µ using the estimates in (68):

1 ks = - 1 -φ (s) φ (s) 1 4 3 µ kr (Obtained with CSA) (144)
Regarding the evaluation of µ in (144), it has been reported that the mineralogical components of sandstone (i.e., quartz, calcite and feldspar) are characterized by an almost constant value of the shear modulus [START_REF] Zimmerman | Compressibility of sandstones[END_REF] for which µ = 4.6 • 10 6 [psi]= 317. [START_REF] Hellmich | Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation[END_REF] [kb] is taken as an average representative value. Hence, based also on the results of ( 141) and ( 140), the magnitude of the term 1 ks is estimated to be in the lower range of 10 -5 [kb -1 ], making it negligible when compared to the other terms in (143). This is expected since ks diverges in proximity to zero porosity.

The other term kr φ (s) kV in (143) has been already computed in (142) to be equal to the constant value -c q , which is independent from p(s) . For this reason its computation does not require subsidiary homogenization estimates. However, it is interesting to observe that its CSA estimate provided by ( 68) and (69) turns out to be:

kr φ (s)k V = - 1 φ (s) k s (Obtained with CSA) (145)
The above estimate corroborates the property found in (142) that kr φ (s) kV is a constant parameter independent of p(s) and hence unaffected by the nonlinearities of the deformation history during the experiments. Actually, the two terms φ (s) and k s are negligibly affected by the nonlinearity of the strain-stress response. In particular, the microscale bulk modulus k s is a constant microscale parameter which is unaffected by crack closure, a nonlinear phenomenon taking place at a mesoscale level. Also, for the loading conditions applied and the material considered, φ (s) can be assumed to be constant.

The paths given by (143) are reported in Figure 7 in the volumetric EI coordinates: the experimental data are all aligned along the same curve. This is due to the low porosity of the sandstone specimens: in the limit of low porosity, the matrix C(s) iso 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] becomes singular at LVP.

The alignment of the data highlights the difficulty in recognizing intrinsic and extrinsic strains to be two independent coordinates from the only kinematic point of view in systems having low porosity such as the sandstone specimens. On the other hand, the necessity of addressing the description of sandstone specimens in terms of two independent volumetric stress and strain coordinates is undeniably recognized. Actually, the examined experimental data have shown that, in these poroelastic systems, strain and stiffness depend on both p(s) and p(s) (see (138), and (141)). Such dependency cannot be lumped into one single stress variable governing both strain and stiffness. Moreover, the convenience of analyzing the strain response in terms of two strain variables (ē (s) , ê(s) ) can be appreciated in Figure 7b: as the deformation increases, the trend of the strain path is to incline towards the direction of the LVP line, which represents an insuperable upper bound for the inclination of the curves in the space (ē (s) , ê(s) ) as discussed in Section 4. Such tendency of the EI volumetric strain path to increase its slope towards the direction of the LVP line can be interpreted as the effect of crack closures and, more generally, as a characteristic nonlinear behavior of compacting materials.

Domain of validity of Terzaghi's principle according to VMTPM

In the light of the results presented in Sections 4 and 5, it is possible to conduct an analysis in EI coordinates on the range of validity of Terzaghi's effective stress principle as a trustworthy stress partitioning law generally applicable to fluid saturated porous media. The notion of single effective stress is of great historical importance in application of porous media theories to geomechanics [START_REF] Terzaghi | Die berechnung der durchlassigkeitsziffer des tones aus dem verlauf der hydrodynamischen spannungserscheinungen[END_REF][START_REF] Skempton | Effective stress in soils, concrete and rocks, Selected papers on soil mechanics[END_REF][START_REF] Lancellotta | Geotechnical engineering[END_REF]. Besides, it is also of substantial practical engineering convenience for general constitutive modelling purposes since, when applicable, it allows to connect back the behavior of a porous medium in a multiphase environment to the response of the solid phase alone, treated as a single-phase continuum.

With specific reference to soils, Terzaghi refers to the effective stress as: "All the measurable effects of a change of stress, such as compression, distortion and a change in the shearing resistance are exclusively due to changes in effective stresses" [START_REF] Terzaghi | The shearing resistance of saturated soils and the angle between the planes of shear[END_REF]. Later on, the concept of effective stress has been extended and examined in relation to specific observable effects such as strain [START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF][START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Rice | Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents[END_REF] and strength [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][START_REF] Pietruszczak | On the mechanical response of saturated cemented materialspart i: theoretical considerations[END_REF][START_REF] Pietruszczak | On the mechanical response of saturated cemented materials -part ii: experimental investigation and numerical simulations[END_REF] properties; also, it has been used for describing the behavior of rocks and partially saturated soils [START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF][START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF][START_REF] Fredlund | Stress state variables for unsaturated soils[END_REF], as well as bone [START_REF] Hellmich | Drained and undrained poroelastic properties of healthy and pathological bone: a poro-micromechanical investigation[END_REF] and cartilage [START_REF] Oloyede | Complex nature of stress inside loaded articular cartilage[END_REF] tissues. In the following, it is shown how Terzaghi's statement can be actually inferred as a prediction of VMTPM for cohesionless frictional granular materials such as soils. Subsequently, the implications of the recovery of the Terzaghi's principle in relation to the mechanical response observed in media different from soils (e.g., rocks) is discussed.

Recovery of Terzaghi's law for cohesionless frictional granular materials

In granular materials, compressing deformation typically occurs at the expense of intergranular pore space [START_REF] Lancellotta | Geotechnical engineering[END_REF]. This implies that the extrinsic strain is much larger than the intrinsic one (i.e., ē(s) >> ê(s) ). Hence, evaluation of kr by a JD test yields kr = -φ (s) ê(s) ē(s) 0, see (92). In the condition of a vanishing kr , the extrinsic and intrinsic strains become decoupled. This is inferred from (43) for ē(s) , and from ( 40) and ( 41) for ê(s) (see also ( 143)). So we have:

kr = 0, ε(s) = 1 + ν Ē σ(s) -ν Ē tr σ(s) I , ê(s) = - 1 φ (s)k s p(s) (146)
Most importantly, the extrinsic strain, which is normally the primary strain object of measure, becomes dependent exclusively on the extrinsic stress tensor σ(s) .

Furthermore, the results shown in this study indicate that, for those materials such as cohesionless granular materials whose strength and/or stiffness properties are determined by contact over interior surfaces, strain (ε (s) ), strenght and stiffness are solely governed by the stress variable σ(s) , with p(s) having a negligible effect.

An experimental evidence of the insensitiveness of interior contacting interfaces to changes of p(s) can be deduced from the stiffening in sandstone specimens. As shown in Section 5.2, p(s)

has no effect on stiffness. Accordingly, since stiffening in sandstones is known to be an effect of crack closure [START_REF] Zimmerman | Compressibility of sandstones[END_REF], it is deduced that p(s) is the only stress variable responsible for increase or decrease of contact across interior surfaces. This property finds also a rational justification in light of the results of Section 4.3 concerning the response to an unjacketed test, where it is deduced that an increment in the intrinsic stress alone induces a homothetic strain response (ē (s) = ê(s) ) in a two-phase medium. Since the homothety preserves a similitude between the shapes of the solid domain before and after deformation takes place, it does not alter the quota of interior surfaces in open-contact or closed-contact conditions.

Hence, for cohesionless frictional granular materials for which both condition (146) and insensitivity to p(s) apply (e.g., soils), the observed effects comply with Terzaghi's statement if ê(s) is excluded from the measured effects of stress changes. The corresponding relations recovered by VMTPM between primary observed volumetric quantities and external spherical loadings p ext and p are collected from the previous sections (see [START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF], ( 32), ( 80), (141) 1 and the trace of ( 146)) in synoptic form:

p(s) = p ext -p, p(s) = - 1 3 tr σ(s) (147)
ē(s) = - 1 kV p(s) , kV = kV p(s) (148)
∂ σ(s) ij ∂x j -φ (s) ∂p ∂x i + b(sf) i + b(s,ext) i = 0 (149)
To point out the significance of relations ( 147)-(149) within VMTPM, three important properties must be remarked:

• The extrinsic stress tensor σ(s) in equation ( 147) is not just a stress variable introduced for constitutive purposes, but it is actually the same stress quantity entering the linear momentum balance of the solid phase (see equation [START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF]), recalled by the second equation in (149) for statics.

• Relations (147)-(149) hold for media with finite volumetric compressibility of the solid and fluid constituent materials. In this respect, it is important to recall that, as classically demonstrated (see for instance [START_REF] Biot | Theory of elasticity and consolidation for a porous anisotropic solid[END_REF][START_REF] Coussy | Mechanics of porous continua[END_REF][START_REF] Rice | Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents[END_REF]), recovery of compliance with Terzaghi's statement can be also put in connection with the stronger hypothesis that the constituent solid material is volumetrically incompressible. In such a limit of volumetrically incompressible constituent solid material (LICM) (corresponding to µ k s → 0, or equivalenty to ν → 0.5, it can be inferred from estimates (68)-(70) that 1 ks → 0 and kr → 0, while kV retains a finite value. Under this hypothesis, a specialization even stronger than (146) is obtained, which yields ê(s) = 0. Accordingly, for frictional granular materials, when an ad-hoc LICM hypothesis is added, all macroscopically observable effects become truly dependent only on p(s) , since ê(s) is strictly zero and p(s) has a negligible effect on shear strength and stiffness. In contrast, within VMTPM, the property of low kr is completely independent from the microscale stiffness parameters µ and k s , which can retain finite values even when kr = 0. This makes possible an ordinary employment of VMTPM for wave propagation analysis also in cohesionless granular materials, since singularities in the wave speed are avoided.

• From a formal point of view, relation (147) exactly matches with Terzaghi's stress partitioning law. Nevertheless, while this formula was originarily introduced by Terzaghi with the purpose of condensing all observed experimental mechanical effects over soils, the significance of (147) in VMTPM is much more general and not confined to soils alone.

Actually, in VMTPM, equation (147) stems directly from [START_REF] Biot | The elastic coefficients of the theory of consolidation[END_REF], which is obtained based on a derivation which is variationally consistent and medium-independent (i.e., in absence of any hypothesis on the microstructure or of constitutive type) [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF]. As such, relation (147) is also independent from the microstructural realizaton of a given medium, and from any constitutive hypothesis such as intrinsic volumetric incompressibility. Hence, as observed in [START_REF] Serpieri | Variationally consistent derivation of the stress partitioning law in saturated porous media[END_REF], compliance with (147) reflects a fundamental medium-independent equilibrium property rooted in the least Action principle, and thus holding regardless of the microstructure. In this respect, a parallel can be drawn with the property of symmetry of the stress tensor, which similarly reflects the fundamental rotational equilibrium and also holds irrespective of the microstructure.

Extensibility of Terzaghi's effective stress and Terzaghi's principle beyond cohesionless granular materials

For classes of media other than cohesionless granular materials, the veracity of Terzaghi's principle, as expressed by ( 148)-( 149), has to be assessed case by case. Similarly, the extensibility of Terzaghi's principle (i.e. the possibility of condensing the dependence of all observed effects upon a single stress variable) has to be assessed considering the specific nature of the mechanical response of the class of media examined.

For saturated sandstone rocks, as observed in Section 5.2, a significant deviation from (148) is found. This is because the strain-confinement pressure-fluid pressure relation measured in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF] results into the law p ext = -kV ē(s) b + αp b with α far from a unit value. Such a deviation is theoretically confirmed as an ordinary prediction of VMTPM, see relation (85) when kr is not zero. Furthermore, this study reveals an even stronger agreement with the estimate obtained by Nur and Byerlee for α, which they found to be experimentally well fitted by α = 1 -kV /k s . Actually, as shown in Section 4, the combination of these last two relations can be also theoretically inferred from (85) when CSA estimates are employed to relate microscale and macroscopic stiffness moduli (see Eqs.( 85) and ( 88)).

Turning to the search for a single effective stress regulating all observable effects in water saturated sandstones, it can be recognized that a single volumetric stress coordinate regulating both strain and stiffness in these media cannot be found. The choice most frequently encountered in the literature (and also adopted in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]) for generalizing the notion of effective stress in order to accomodate the deviation from Terzaghi's law is to replace it with the isostrain stress p ē(s) .

This last stress quantity is defined as the confining stress that should be externally applied to the same specimen, upon drying it, to induce the same macroscopic strain produced by the concomitant action of p and p(s) , viz:

p ē(s) : ē(s) = - 1 kV p ē(s) (150)
This definition yields, on account of ( 10) and (138), the following expression for p ē(s) :

p ē(s) = p(s) -kr p = p ext -1 + kr p (151)
which, as opposite to (147) 1 , is manifestly driven by the microstructure, since it contains kr .

While this modification of effective stress into p ē(s) is capable of restoring the dependency of volumetric strain upon a single stress variable, as already pointed out by Nur and Byerlee and confirmed by the analysis carried out in Section 5.2 (see equation ( 141)), the stiffness of the saturated speciments does not depend on p ē(s) alone. In fact, it is solely regulated by the other volumetric stress coordinate p(s) , being insensitive to p(s) . It is thus concluded that all the macroscopically observable mechanical effects in sandstone cannot be regulated by p ē(s)

alone. In this respect, the inadequacy of the isostrain stress for assessing the strength of several classes of saturated porous materials has been previously observed, and it has been specifically discussed (see for instance [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]).

In conclusion, for rock materials and other classes of biphasic porous media which do not belong to the family of cohesionless granular materials, neither the extrinsic stress p(s) nor the isostrain stress p ē(s) can adequately be employed to lump the dependence of all mechanically significant observable effects into a single effective stress variable.

A generalization of Terzaghi's principle urges a dependency upon two volumetric stress coordinates at least. Within the framework of VMTPM, the natural two-coordinate extension of the classical statement of Terzaghi's principle, encompassing all theoretical and experimental results herein discussed, is represented by an extension of the dependency of all observable mechanical effects upon all DEI stress coordinates σ(s) dev , p(s) and p(s) . Actually, these coordinates represent the privileged choice for defining a volumetric stress state in VMTPM, being p(s) and p(s) the stresses work-associated with primary volumetric strain variables of the solid phase (ē (s) and ê(s) ). Accordingly, the following generalized statement is proposed: Generalized Terzaghi's principle:

"All the measurable mechanical effects over the solid porous phase produced by a change of stresses in a multi-phase porous medium are exclusively due to changes in DEI coordinates, i.e., in the deviatoric stress tensor and in the extrinsic and intrinsic pressures."

The classical statement for soils, represented by (147)-(149), is only recovered as a specialization of the above statement for the subclass of cohesionless granular materials. In this case the insensitivity to the intrinsic pressure makes the observable effects solely driven by the extrinsic stress tensor.

Discussions and Conclusions

Stress partitioning between solid and fluid phases in fluid saturated porous media is complex and yet not completely understood. In this study, the variational continuum poroelastic framework VMTPM was adopted to derive general procedures and operative formulas for the analysis of stress partitioning in biphasic systems subjected to compression in both drained and undrained conditions. VMTPM provides a suitable ground for stress partitioning analysis since it only employs stress variables which are ordinarily defined in terms of explicit work-association with primary strain variables, with no use of Lagrange multipliers.

In Part I [START_REF] Serpieri | A purely-variational purely-macroscopic theory of two-phase porous media -part i: Derivation of medium-independent governing equations and stress partitioning laws[END_REF], local strong form of boundary conditions and conditions holding at solid-fluid interfaces were derived based on a variational analysis. These conditions have been combined in Section 2 to address unilateral contact over a binary solid-fluid mixture (by Equations ( 18)) via a simple extension of the use of a set-valued law and a gap function ordinarily employed in singlephase contact theories. Such boundary conditions allow addressing, in a variationally-consistent and medium-independent macroscopic continuum treatment, a large class of boundary value poroelastic problems without invoking further mechanical arguments.

In Section 3.2 it is shown that, under hypotheses of isotropy, VMTPM momentum balance equations recover the structure of Biot's equations. Such a recovery is notable since VMTPM proceeds from a purely variational statement of the poroelastic two-phase problem while previous works, see, e.g., [START_REF] Wilmanski | A few remarks on Biot's model and linear acoustics of poroelastic saturated materials[END_REF], have argued the difficulty of obtaining Biot's equations proceeding from a variational principle, motivated by the presence of a nonequilibrium variable, the increment of fluid content, in Biot's derivation of governing equations. The variational statement of the problem considered in VMTPM does not contain, however, nonequilibrium variables in the kinematic description. Explicit expressions for the elastic coefficient entering these Biot-like equations, as function of the elastic moduli of individual phases, are also obtained in Section 3.2.

However, an important difference with Biot's framework is the retrieval of a positive increment of a stiffness coefficient entering the solid linear momentum balance with respect to the corresponding term of Biot's theory (added stiffness). The magnitude of this coefficient which seems to have been overlooked in Biot's theory, appears to be non-negligible in most applications. In contrast, in the medium-independent equations herein considered, no added mass coefficients appear. A further element of reciprocal corroboration between VMTPM and Biot's theory is the identification of the relationship kr = α -1 between the constitutive dimensionless ratio kr of VMTPM and Biot's coefficient α.

The capability of VMTPM of addressing a comprehensive variety of boundary and surface conditions was exploited to perform a comprehensive analysis of the mechanical responses of biphasic specimens during JD, U, JU and CCFP static compression tests, see Section 4. This was carried out with no limitation on porosity and/or other material properties (e.g. compressibility, permeability, etc.). The four examined testing conditions offered a broad scope of compression histories, which can be used for the mechanical identification of a wide range of experimental conditions. Hence, the deriving results can be deployed for investigating various porous media applications.

These elemental responses were subsequently employed in Section 5 for identifying the hydro-mechanical conditions determined by the experimental set-up of Nur and Byerlee during their compression tests on Weber sandstone specimens [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], which is generally regarded as a proof of deviation from Terzaghi's effective stress principle. It was shown that the experimental conditions in which the specimen were tested corresponded to those of a CCFP compression test, and that the peculiar behavior of this soil can be actually predicted by VMTPM. In particular, the Weber sandstone response to CCFP compression was analyzed in terms of primary observed quantities (p ext , p, and ē(s) ) via relations (85)-( 88), and also accounting for the nonlinear response of volumetric stress and strain paths, see Figures 7.

An appropriate selection of independent primary stress variables or single effective stress is of special relevance for constitutive modeling of unsaturated soils [START_REF] Nuth | Effective stress concept in unsaturated soils: clarification and validation of a unified framework[END_REF][START_REF] Fredlund | Stress state variables for unsaturated soils[END_REF]. Within VMTPM, privileged stress coordinates are naturally identified in DEI coordinates. Specifically, DEI strain coordinates consist of the deviatoric strain ε(s) dev plus the volumetric extrinsic and intrinsic strains (ē (s) ), (ê (s) ), while DEI stress coordinates consist of the corresponding work-associated variables σ(s) dev , p(s) and p(s) .

DEI coordinates were used to track the poroelastic response to ideal JD, U, JU, CCFP compression tests, and for predicting and interpreting the experimental data reported in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF], see Section 5.2. In particular, when DEI coordinates were adopted to analyze the JU test, it was shown how the phenomenon of compression-induced liquefaction, experimentally observed in sands [START_REF] Fragaszy | Undrained compression behavior of sand[END_REF] and deemed responsible for blast-induced liquefaction, results to be a natural theoretical prediction of VMTPM for cohesionless materials. Such prediction emerges naturally (i.e., without adding ad-hoc constitutive features to describe compression-induced liquefaction) when the unilateral character of cohesionless materials is taken into account, see Section 4.4. Actually, discriminated by the specific values of stiffness moduli of the solid phase and of the interstitial fluid in a given cohesionless mixture, three distinct behaviors were identified and classified into nonliquefying, partially liquefying and full liquefying cohesionless mixtures. In this respect, it is important to remark that, although liquefaction is mostly known to be associated with laboratory and in situ conditions, as an effect essentially induced by deviatoric undrained loading and excitations, there exist experimental evidences indicating that sands can be also liquefied by isotropic compressive stress applied under quasistatic drained conditions. Moreover, DEI coordinates were also deployed to predict the effective stress law experimentally determined in [START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF],

and when interpreting the nonlinear features of the experimental data set reported therein, see Section 5.2. Specifically, the stiffening in sandstone specimen due to crack closure was recognized to be reflected by a reorientation of the strain curve in EI coordinates, which tends to align along the LVP line.

Finally, In Section 6, on the basis of the results of Sections 4 and 5, it was shown that:

• when VMTPM is specialized to describe media with granular microstructure and cohesionless behavior of the solid phase, the resulting poroelastic theory is characterized by full compliance with Terzaghi's stress partitioning law: this theory actually predicts that both macroscopic strain and stiffness are solely dependent on p(s) , and insensitive to p(s) , so that σ(s) assumes the role of the effective stress (i.e., a stress quantity regulating all macroscopically observable effects on the solid phase);

• such specialized poroelastic theory (i.e., under the hypotheses of granular microstructure and cohesionless behavior) entails no incompressibility constraint; therefore, it can be applied to wave propagation analysis in saturated granular solids, such as soils;

• when the hypotheses of granular microstructure and cohesionless behavior are removed, as in the case of sandstone rocks, VMTPM predicts that a generalization of Terzaghi's principle, with a single stress coordinates regulating all observable effects, is not feasible; consequently, a generalization of dependence upon more than one stress coordinate becomes mandatory;

• a generalized statement of Terzaghi's principle was proposed for multiphase problems, postulating that changes of DEI coordinates are responsible for all macroscopically measurable mechanical effects on the solid phase produced by loading.

Altogether the applications shown in this contribution have exemplified the capability of VTMPM to describe and predict a large class of linear and nonlinear mechanical behaviors observed in two-phase saturated materials. Future applications of VMTPM will be directed towards the assessment of its capability to describe, upon introducing suitable constitutive hypotheses, responses stemming from volumetric-deviatoric coupling such as dilatancy in geomaterials, as well its predictive features in combination with elastoplastic constitutive laws.

List of Symbols

Vector and tensor quantities are notated in bold. Null vector quantities and null tensor quantities are indicated by o and O, respectively. Gibbs nabla vector ∇ is used, when necessary, for implicit shorthand representation of differential operations. For all microscale quantities, of kinematic, stress and stiffness type, no accent is used. In general, an overbar is used to denote macroscopic kinematic quantities, except for kinematic quantities of intrinsic type which are denoted by a hat accent. Macroscopic stress quantities are all notated with accents different from the overbar, with the only exceptions of macroscopic Cauchy stresses and of the fluid pressure which have no accent. Intrinsic macroscopic stresses are denoted by a hat accent while extrinsic macroscopic stresses are denoted by a check accent (a reversed hat). In general, subscripts indicate the Cartesian indexes of scalar components. The phase to which a given quantity belongs is denoted by bracketed superscripts s or f , if not otherwise stated. Specific items used in the paper are listed in the following tables.

Symbol Description

Ω (M ) macroscopic domain of the mixture Ω (s) macroscopic subset domain of the mixture with nonvanishing solid phase Ω (f ) macroscopic subset domain of the mixture entirely occupied by the fluid 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 χ(s) macroscopic placement vector of the solid phase χ(f) macroscopic placement vector of the fluid phase Φ (s) 0 reference solid volume fraction Φ (f ) 0 reference fluid volume fraction φ (s) current solid volume fraction φ (f ) current fluid volume fraction Ĵ(s) finite macroscopic intrinsic volumetric strain of the solid phase J(s) finite macroscopic extrinsic volumetric strain of the solid phase J (f ) finite microscale volumetric strain of the fluid phase ρ(s) Cauchy stress tensor p interstitial fluid pressure t (ext) external boundary traction vector (current configuration) 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 ε(s) dev deviatoric part of the infinitesimal extrinsic strain tensor ε(s) sph spherical part of the infinitesimal extrinsic strain tensor K (s) iso extrinsic-intrinsic coupling stiffness matrix for isotropic linear materials C (s) iso extrinsic-intrinsic coupling compliance matrix for isotropic linear materials ψ(s) dev deviatoric part of the strain energy of the solid phase in isotropic materials ψ(s) sph spherical part of the strain energy of the solid phase in isotropic materials K ē(s) ē(s) ē(s) -associated entry of K (s) iso K ê(s) ē(s) off-diagonal entry of K (s) iso K ê(s) ê(s) ê(s) -associated entry of K (s) iso K(s)

  10) where b(sf) = -b(fs) are the volume forces representing the internal short-range solid-fluid interaction, b(f,ext) and b(s,ext) are external volume forces, ρ(s) and ρ(f) are solid and fluid apparent mass densities.

Figure 1 .

 1 Figure 1. Cylindrical coordinates are introduced over Ω (M ) with x being the direction of the axis of the cylinder, and with r and θ being the radial and angular coordinates, respectively. The origin of the reference frame is set at the bottom center of the specimen of length L, directed upward along the radial axis. A compressive plug is positioned on the upper side of the specimen at x = L, see Figure 1. Four experimental setups are considered: a jacketed drained test (JD); an unjacketed test (U); a jacketed undrained test (JU); and a creep compression test with controlled fluid pressure and constant stress at the plug (CCFP). It should be noted that no viscous creep effects for the individual solid phase are considered in in CCFP.

Figure 1 :

 1 Figure 1: Schematics of the four compression tests analyzed: a) Jacketed Drained (JD) test; b) Unjacketed test (U); c) Jacketed Undrained (JU) test; d) Creep test with controlled Fluid Pressure (CCFP).

Figure 2 :

 2 Figure 2: Representation in EI coordinates of the volumetric mechanical response during a Jacketed drained test. a) EI pressure path in the (p (s) , p(s) ) plane; b) EI volumetric strain path in the (ē (s) , ê(s) ) plane. Dotted lines indicate the LVP and LICM limits.

  95) Hence, relations (94) and (95) correspond, from a practical point of view, to an open contact condition over ∂Ω (M )

Figure 3 :

 3 Figure 3: Representation in EI coordinates of the volumetric mechanical response during an ideal unjacketed compression test. a) EI pressure path in the (p (s) , p(s) ) plane; b) EI volumetric strain path in the (ē (s) , ê(s) ) plane. Dotted lines indicate the LVP and LICM limits.

Figure 4 :

 4 Figure 4: Representation in EI coordinates of the volumetric mechanical response during an ideal jacketed undrained compression test. a) EI pressure path in the (p (s) , p(s) ) plane; b) EI volumetric strain path in the (ē (s) , ê(s) ) plane. Dotted lines indicate the LVP and LICM limits.

  xx = dē (s) : dσ (s) xx = 2μ + λ dē (s) -kr dp, dσ (s) tt = λdē (s) -kr dp (132) Moreover, in consideration of relation (130), one has:

Figure 5 :

 5 Figure 5: Representation in EI coordinates of the volumetric mechanical response during an ideal creep test with controlled fluid pressure. a) EI pressure path in the (p (s) , p(s) ) plane; b) EI volumetric strain path in the (ē (s) , ê(s) ) plane. Dotted lines indicate the LVP and LICM limits.

Figure 7 :

 7 Figure 7: a) Volumetric stress points plotted in the EI pressure coordinate space (p (s) , p(s) ) b) Volumetric strain path in EI coordinates (ē (s) , ê(s) ), as estimated by relation (143) from data in [21].

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) apparent solid density ρ(f) apparent fluid density Table 2: List of symbols Symbol Description 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 infinitesimal extrinsic strain tensor ψ(s) apparent solid potential energy density ψ(f) apparent fluid potential energy density ψ (f ) microscale fluid potential energy density ψ(f) true fluid strain energy density b(s,ext) solid external volume forces b(f,ext) fluid external volume forces b(fs) , b(sf) drag volume forces associated with solid-fluid interaction σ(s) extrinsic solid stress tensor p(s)intrinsic solid pressure σ(s)

dev

  deviatoric stiffness modulus in isotropic materials σ(s) dev deviatoric part of the extrinsic stress tensor σ(s) sph spherical part of the extrinsic stress tensor

Table 1 :

 1 Confining pressure, p ext , volumetric strain, ē(s) , fluid pressure, p, measured in jacketed compression tests on water saturated Weber sandstone specimens ([START_REF] Nur | An exact effective stress law for elastic deformation of rock with fluids[END_REF]) and corresponding EI pressure coordinates.

	(ext) dev .

and σ(s) dev = σ

Table 3 :

 3 List of symbols

Skempton's coefficient (pressure ratio) a q , b q , c q coefficients for curve fitting the nonlinear extrinsic compliance of sandstone specimens