A general approach is proposed for defining the macroscopic free-energy function of a saturated porous medium subjected to finite deformations, under isothermal conditions, in the case of compressible fluid and solid constituents. Reference is made to an elementary volume treated as an ‘open system’, moving with the solid skeleton. The strain-like variables are characterized by a suitable strain measure and the variation of fluid-mass content. Although the free energy is obtained by assembling the contributions of the single constituents, the resulting free energy is not the simple sum of the free energies of the single constituents. It is shown that the simplified approaches previously proposed in the literature are recovered as particular cases of the more general framework proposed in the present article and, in addition, new simplified constitutive frameworks are investigated. The proposed approach paves the way to the consistent hyperelastic–plastic and thermo-elastic modelling of saturated porous media with compressible fluid and solid constituents.
Although the small-strain theory of the coupled behaviour of linear elastic saturated porous media under quasi-static and dynamic conditions was formulated by Biot (1941, 1956, 1962) more than 50 years ago, the analysis of saturated porous media is still a challenging problem due to the strong non-linearities (resulting both from nonlinear and irreversible material responses and from nonlinear geometrical effects due to the large strains typically occurring in these media) and to the wide range of applications involving the behaviour of multi-phase porous media, such as environmental problems, petroleum engineering, geophysics, geotechnical engineering, earthquake engineering, industrial processes (involving polymer and metal foams, sponges or powder compaction), concrete structures (Wang et al. 2009) and bio-engineering (e.g. Loret & Simoes 2005; Franceschini et al. 2006).
A comprehensive description of the historical development of porous media theories has been provided by De Boer (1996). Two apparently different approaches have been used: mixture theories and the so-called ‘purely macroscale theories’. In mixture theories, the porous media are represented by superposed and interacting continua, the field equations of each constituent are derived from averaging processes and are, in many cases, written using a Eulerian description. Reference is usually made to the energy of each single phase and the energy of the whole porous medium results as the sum of the energies of the single constituents. It is generally accepted that mixture theories of n-phase media lack (n−1) equations (e.g. Svendsen & Hutter 1995), and several approaches have consequently been proposed to overcome this difficulty: Morland (1972) introduced a constitutive equation for the stress tensor acting on the solid and fluid constituents; Drumheller (1978), Bowen (1982) and Bennethum & Cushman (1996) adopted an additional evolution equation for the volume fractions; Nunziato & Walsh (1980) formulated additional balance equations for the so-called ‘equilibrated forces’; Svendsen & Hutter (1995), De Boer (1996) and Bluhm & De Boer (1996) incorporated the saturation condition in the entropy inequality using a Lagrangian multiplier; Bluhm et al. (1995) introduced an additional balance equation for the density; and Wilmanski (1996) considered the balance equation for porosity.
In contrast, ‘purely macroscale theories’ assume that the standard concepts of continuum mechanics are still relevant on a macroscale (Coussy et al. 1998). Biot (1972) was the first to assume the existence of a macroscopic thermodynamic potential—the free energy of the saturated porous medium—according to which the saturated porous medium is conceived as an open system, and reference is made to an elementary volume moving with the solid skeleton described in Lagrangian terms. This macroscopic free energy is a potential for the stresses in the solid and fluid phases, so its derivatives furnish the generalized stresses acting on the saturated elementary volume. Biot (1973) also provided a semilinear formulation, in which the volume changes of the solid constituent are assumed to be small (and therefore more or less linearly dependent on stress and pore pressure), while a strong nonlinear behaviour may be associated with the solid skeleton (due to grain rearrangements and changes in pore microstructure). Biot (1977) and later Coussy (1989) extended this general framework to non-isothermal conditions. These formulations are very general and include the compressibility of all constituents, although no suggestion has been made as to how to deduce the free energy of the saturated porous medium at finite strains starting from the free energies of the single constituents.
Coussy et al. (1998) proved that the mixture-theory equations imply the existence of a single macroscopic thermodynamic potential (as assumed in ‘purely macroscale theories’), so the two formulations are related to each other and the choice is only a matter of personal preference (Detournay & Cheng 1993).
It is worth emphasizing that the structure of constitutive equations changes profoundly when the compressibility of the solid constituent is taken into account (the effects of the compressible solid constituent on the definition of the effective stress in the small-strain theory are examined by Suklje (1969), Nur & Byerlee (1971), Zimmermann et al. (1986), Loret & Harireche (1991) and Lade & De Boer (1997)), and it is generally agreed that the assumption of incompressibility of the solid grains immensely simplifies the relationships (Bennethum 2006) because the coupling between the solid and fluid phases is much weaker in this case. That is why the compressibility of the solid constituent has mostly been neglected in the engineering applications and constitutive models proposed so far for considering porous media at finite strain (e.g. Borja & Alarcon 1995; Diebels & Ehlers 1996; Larsson & Larsson 2002), or it has been considered by means of simplifying assumptions: Advani et al. (1993) and De Boer & Bluhm (1999) adopted the semilinear theory of Biot (1973); Coussy (1995) proposed a linear extension of Biot’s classical poroelasticity to small (not necessarily infinitesimal) deformations; Armero (1999) treated the compressibility of the constituents in a simplified way considering Biot’s parameter as constant, as in linear theory; whereas Larsson et al. (2004) identified the solid constituent with the fibre bundles being wetted with resin in processing fibre-composite materials and assumed that the solid constituent may compact as a result of pore pressure and wetting, neglecting the role of the effective stress. These simplifications contrast with small-strain formulations in which the compressibility of the solid constituent is usually taken into account.
The aim of this work is to propose a general approach to deducing the macroscopic thermodynamic potential of the saturated porous medium at finite strains, in the case of compressible solid and fluid constituents, starting from the free energies of the solid and fluid phases (which can be measured independently of the porous solid) on the one hand, and from the free energy of the solid skeleton (which can be deduced from experimental tests performed on the dry porous solid or using homogenization procedures by averaging the microstructure of the solid phase) on the other (§§2 and 3). This gives rise to a rigorous method for defining Biot’s macroscopic potential at finite strains. In contrast with mixture theory (e.g. Bowen 1982), the resulting free energy of the whole medium is not the simple sum of the free energies of the single constituents, and the saturation condition is taken into account in the single thermodynamic potential thus defined.
The method proposed in this work can be applied to any form of free-energy density function for the pore fluid and solid skeleton, providing the latter can be split into separate volume and isochoric contributions (as assumed also by Bluhm & De Boer 1996). As for the solid constituent, the proposed method needs a particular form of the free-energy density, which it must be possible to split into separate volume and isochoric contributions, and the former must also imply a linear behaviour in the logarithmic volume strain. This makes it possible to match the additive stress decomposition (effective stress plus pore pressure) with the multiplicative decomposition of the deformation gradient (instead of the additive strain decomposition adopted in small-strain theory). In contrast with Wilmanski (2006), the free energy proves to be a potential for the stresses in the solid and fluid phases, and—extending Biot’s linear theory—the volume fractions are obviously not considered constant. As a by-product, the proposed method provides the extension to large strains of the effective stress formulated by Nur & Byerlee (1971) for small strains and compressible solid constituents. The main limitation of the method coincides with the one typically involved in Biot’s theory, i.e. the microstructure must be such that an isotropic macroscopic stress induces an isotropic microscopic strain equivalent to the one induced by the pore pressure (see §3c).
In §4, the following particular cases are analysed in detail: (i) geometrically linearized theory, (ii) semilinear theory (in which the solid constituent is assumed to be nearly incompressible), (iii) incompressibility of the solid constituent, and (iv) incompressibility of both the fluid and the solid constituent. As a result, the simplified constitutive approaches previously proposed in the literature (and involving the incompressibility of some constituents) are recovered as special cases of the proposed unified framework. Finally, the rate forms of the hyperelastic constitutive equations are given in §5, whereas the different compressibilities of porous media at finite strains (depending on whether the pore pressure is allowed to drain or not) are discussed in the electronic supplementary material, appendix A.
It is worth emphasizing that this work is devoted only to the method for formulating hyperelastic constitutive models at finite strains within the framework of ‘purely macroscale theories’. As a consequence, momentum and mass-balance equations and energy dissipation due to relative motion between the phases (Darcy law) are not considered here and can be found elsewhere (e.g. Coussy (1989) or Gajo & Denzer (submitted), who consider the finite-element implementation of the proposed constitutive model for the analysis of boundary-value problems under dynamic conditions). The proposed approach paves the way to the hyperelastic–plastic modelling of saturated porous media with compressible constituents and to the consistent definition of hyperelastic-constitutive models under non-isothermal conditions.
Four tensorial products will be used, which are defined as follows: for every second-order tensor A, B and C. Moreover, let dev A be the deviator component of the second-order tensor A, namely , with and the symmetric fourth-order unit tensor (Miehe 1995).
2. Basic kinematics
Let x=φ(X,t) denote the current position (with in the current configuration ) of the solid-skeleton particle having the initial position in the reference configuration, . Moreover, let F=grad φ and J=det F denote the deformation gradient and its Jacobian, respectively, whereas is the spatial gradient of the spatial velocity field (i.e. grad v). The current infinitesimal volume dΩ of the solid skeleton is obviously related to its initial material volume dΩ0, through dΩ=J dΩ0.
The volume occupied by the pore fluid in the reference configuration is n0 dΩ0, where n0 is the initial porosity in the reference configuration; let n denote the current porosity. If ρw and ρw0 denote the current and the initial density of the pore fluid, respectively, let Jw denote their ratio, namely Jw=ρw0/ρw (which can be interpreted as a measure of volume deformation, due to the conservation of fluid mass). The initial pore-fluid-mass content in the infinitesimal volume of solid skeleton dΩ0 is 2.1 whereas the fluid-mass content in the current configuration is 2.2 which can obviously be rewritten as 2.3 Let mw0 and mw denote the initial and the current fluid-mass content, both expressed per unit volume of the solid skeleton in the reference configuration, namely and , then from equation (2.3), the ratio Jw is 2.4 Finally, the variation of fluid-mass content (obviously coinciding with the exchanged mass of pore fluid) per unit volume of the undeformed solid skeleton, , is given by the difference between the current and the initial fluid-mass content (e.g. Coussy 1989) 2.5
Likewise, let ρs0 and ρs denote the initial and the current density of the solid constituent, respectively, and let Js define their ratio 2.6 Note that, due to the conservation of the mass of the solid constituent, Js also represents the local ratio of the deformed to the undeformed volume of the solid constituent, namely Js=dΩs/dΩs0, where dΩs=(1−n) dΩ is the volume of the solid constituent in the current configuration and dΩs0=(1−n0) dΩ0 is the volume of the solid constituent in the reference configuration. As a result, the current porosity n is obviously given by (Morland 1972; Coussy 1989; Advani et al. 1993; Bluhm 1997) 2.7 Note that equation (2.7) expresses a purely geometrical relationship with n=n(J,Js), whereas possible alternative expressions involving easily measurable quantities, such as , can be obtained after introducing the constitutive relationships for the single constituents and the solid skeleton, as described below.
3. The hyperelastic model taking into account the solid-phase compressibility
Let us consider a cubic sample of porous material, saturated with an inviscid compressible fluid and wrapped in a soft jacket, so that the pore fluid can flow in or out of a tube penetrating through the jacket. According to Biot (1972), the free energy ψ0 of the saturated porous medium at finite strains under isothermal conditions depends on a ‘strain measure and on the total mass of fluid added in the pores’. If the left Cauchy–Green strain tensor B=FFT is considered as a strain measure, and the behaviour is elastic, then the free energy of the saturated porous solid per unit volume of the undeformed solid skeleton can be expressed as 3.1 where is the variation of fluid-mass content per unit volume of the undeformed solid skeleton (equation (2.5)). The time differentiation of the free energy is given by 3.2 where D=(L+LT)/2 is the rate of deformation tensor and is the rate of change of pore-fluid-mass content. In equation (3.2), the scalar-valued tensor function ψ0 has been assumed to be isotropic, so B and ∂ψ0/∂B commute (Miehe 1995; Haupt 2000).
In an open system, where substances can enter and exit under isothermal conditions, the stress power W0 is given by the sum of the work done by the total stress in the corresponding work–conjugate strain rate and by the chemical potential of the pore fluid μw during the addition/subtraction of fluid mass , namely 3.3 where K is the total Kirchhoff stress (with K=JT and T the Cauchy stress tensor), and the chemical potential of the pore fluid μw is defined as follows: 3.4 where pw is the (positive) pore-fluid pressure. The chemical potential μw (also denoted in the literature as the pressure function) takes into account that, before a fluid can be injected into the porous solid at a pressure pw, it must first be compressed from the reference pore pressure, pw0, to the current pore pressure, pw, so the chemical potential μw represents the free enthalpy per unit mass of injected fluid.
Using standard arguments (Coleman & Noll 1963), the comparison of the stress power (equation (3.3)) with the differential of the free-energy function (equation (3.2)) yields the following constitutive functions: 3.5
Using the chemical potential to express the free energy of an open system in which there is an exchange of mass is well accepted in thermodynamics, and this was also recently adopted by Loret et al. (2002) to model the chemo-mechanical coupling of saturated active clays.
Thus, a macroscopic free energy for the whole mixture ψ0 is assumed to exist, and reference is made to an elementary volume that is considered an open system moving with the solid skeleton. Note that this approach is based on the most reasonable choice of strain-like quantities, since they are the easiest to evaluate in laboratory tests on saturated soil samples, namely a strain measure (deduced from the change in the sample’s dimensions, in the same way as for single-phase media) and the change in sample mass resulting from the in-flow/out-flow of pore fluid (which can easily be assessed from the sample’s weight).
To obtain the expression of the free-energy density function of the whole saturated porous medium, it is necessary to define the constitutive equations of each constituent and of the solid skeleton as a whole, and then assemble the single contributions. The presentation will be organized in exactly the same way, starting from the contributions of the single constituents and then assembling them. In this way, the free-energy density function is not given directly, it is built up step by step, thus illustrating a method for defining the free-energy density function that can be used with any constitutive assumption for the single constituents. Given the selected reference elementary volume (based on the solid skeleton), the resulting total free energy is not the simple sum of the contributions of the single constituents, as in the mixture theory.
(a) The pore fluid
The pore fluid is assumed to have null viscosity and to be barotropic (i.e. its behaviour is independent of temperature), namely 3.6 where Tw is the Cauchy stress in the fluid and pw(ρw) is the pore pressure (positive in compression), which depends only on fluid-mass density, ρw, and I denotes the second-order unit tensor. Let us assume that the reference pressure coincides with the atmospheric pressure (pw0=0), so the pore-fluid density ρw0 is the fluid-mass density measured at pw0=0 (and Jw=1). If denotes the free energy of the pore fluid per initial unit volume of the fluid, since the stress power per initial unit volume of the fluid is , then the pore pressure is related to the fluid density by 3.7 and the rate expressions are 3.8 The symbol for the partial derivative (∂/∂Jw) has been used in equations (3.7) and (3.8) to emphasize that would generally also depend on the pore-fluid temperature θw, namely . Equation (3.82) can be integrated to obtain the chemical potential of the pore fluid 3.9 where Jw is given by equation (2.4) in terms of J and , and k is an integration constant that can be evaluated by imposing that the chemical potential of pore fluid is null (μw=0) at atmospheric pore pressure (pw0=0 and Jw=1). Thus, from equations (3.4) and (3.9), μw turns out to be a known function of pw.
Although any constitutive assumption can be adopted for , to illustrate the proposed method, the free energy of the pore fluid per initial unit volume of the fluid is assumed to be expressed as 3.10 so a linear elastic behaviour is obtained in the logarithmic volume strain 3.11 where is a constitutive parameter acting as a bulk stiffness of the pore fluid and the integration constant becomes k=0.
Finally, it is worth adding that, in the case of an incompressible pore fluid (i.e. ρw=ρw0 and Jw=1), the pore pressure is indeterminate and, according to Biot (1972), the chemical potential μw comes down to 3.12
(b) The solid skeleton
In geomechanics, reference is often made to the behaviour of the solid skeleton, which is intended as the mechanical response obtained in an ideal test (denoted from now on as test A), in which the pore-fluid pressure of the saturated porous medium is maintained at constant atmospheric pressure (pw0=0). According to Biot (1973), if F′ denotes the deformation gradient of such an ideal test and T′ and K′=JT′ are the corresponding Cauchy and Kirchhoff stress tensors, then from the standard arguments adopted for equation (3.5), the existence of a free-energy density function ψsk0(B′) (under atmospheric pore pressure) per unit volume of the reference configuration implies that 3.13 where B′=F′F′T. Here again, the symbol for the partial derivative (∂/∂B′) has been used to emphasize that the dependence on the solid temperature θs should generally be taken into account, i.e. .
Alongside the ideal test A, let us consider a second ideal test, test B, in which a jacketed sample is first immersed in a fluid at a pressure p=pw (thus T=−pwI and K=−JpwI), while increasing the (uniform) pore pressure by the same amount (pw). Let us assume an isotropic behaviour and denote the corresponding initial deformation of the sample. The sample is subsequently distorted to arrive at the final deformation gradient F, so the final deformation gradient of test B can be multiplicatively decomposed as 3.14 Let test B be performed so that F*=FF−1s−f in equation (3.14) is equal to F′ of test A. The stress applied in test B can be additively decomposed into 3.15 where K* is the stress increment applied in the second phase of test B. Note that K (and T) denotes the total Kirchhoff (and Cauchy) stress tensor (and is defined in equation (3.5)), whereas K+JpwI represents the so-called effective Kirchhoff stress tensor (and T+pwI is likewise the effective Cauchy stress tensor).
It is worth recalling that a similar multiplicative splitting to the one adopted in equation (3.14) was employed by Morland (1972), Bluhm & De Boer (1996) and De Boer (1996). The relationships between the two different assumptions are examined below.
The initial distortion Fs−f (of test B), and consequently the relationship between F′ (of test A) and F (of test B), depend on the compressibility of the solid constituent and will be examined later on.
(i) The incompressible solid constituent
Let us consider first the case of an incompressible solid constituent. This case is the simplest and the most well known. In fact, the initial phase of test B is ineffective and the deformation gradients of test A and test B consequently coincide, so F′=F. In other words, a change of pore pressure does not affect the mechanical response of the solid skeleton, providing the effective stress is constant, and the pore fluid consequently just flows through the solid skeleton. In an elastic granular medium, F′ is due to the reversible grain rearrangements and the grain distortions induced by intergranular contact stress (which is due to the applied stress K′, see §3c).
In this case, the free-energy density function per initial unit volume of the solid skeleton, , depends only on the deformation of the solid skeleton (B in this case), so the effective Kirchhoff stress is 3.16 As a result, the principle of effective stress is derived from the existence of the free-energy density function and the incompressibility of the solid constituents (this is equivalent to when Coussy (2004) deduced the effective stress from the dissipation inequality, admitting the existence of ψsk0(B) and the incompressibility of the solid constituents). The saturated porous medium thus behaves like a single-phase medium, providing the effective stress (e.g. K′) is considered instead of the total stress (e.g. K), and any hyperelastic constitutive model proposed for single-phase media can consequently be adopted for two-phase media as well.
As often assumed in the finite-strain constitutive modelling of single-phase compressible hyperelastic materials, the constitutive response is split into separate volumetric and isochoric contributions (according to the original proposal by Flory (1961)). The basis for this decomposition is the multiplicative split 3.17 with Fvol=J1/3I, thus 3.18 where J=det Fvol=(det B)1/2 and is a volume-preserving tensor (with ). The same assumption was also made by Bluhm & De Boer (1996) for saturated porous media, within the context of mixture theory. A key aspect for the applicability of the proposed method is that the isotropic hyperelastic response of the solid skeleton can be expressed in a decoupled form, namely the free energy per unit reference volume can be expressed as the sum of a volumetric and an isochoric term 3.19 where the isochoric contribution, , generally depends on the first two invariants of , namely
The effective Kirchhoff stress can be obtained from equation (3.16) by taking into account that ∂J/∂B=1/2JB−1, thus 3.20
Although any arbitrary constitutive assumption can be adopted for and , for the purpose of illustrating the proposed method, the free energy is assumed to equate to 3.21 where and μ are the constitutive parameters describing the bulk and the shear stiffness of the solid skeleton. Thus, 3.22 One result of equation (3.22) is that the volume component of the effective Kirchhoff stress admits a linear behaviour in the logarithmic volumetric strain. The isochoric contribution of the Kirchhoff stress K′iso=JT′iso is the deviatoric component of K′ in the Eulerian description, namely 3.23
It is worth recalling that, although the volumetric term of the free energy, , adopted in equation (3.21) has been used by many authors (e.g. Hencky 1933; Valanis & Landel 1967; Simo 1992), as observed by Doll & Schweizerhof (2000) equation (3.21) does not make sense in the case of very large volumetric expansions (ln J>1, i.e. J>2.718), because polyconvexity is lost.
(ii) The compressible solid constituent
In the case of a porous medium with a compressible solid constituent subjected to test B, the initial immersion of the jacketed sample in a fluid at a pressure pw, so T=−pwI in the initial phase, and the associated increase of the (uniform) pore pressure to the same amount (pw) induce a volume compression of the solid constituent (i.e. the solid grains in a granular medium) and a consequent deformation of the whole sample. This case is the most difficult and challenging.
Concerning the volumetric compression of the whole sample, it is important to note that the initial volume compression of the solid constituent has the same effect as a temperature decrease in the solid constituent. So, for an unconstrained sample, the initial volumetric compression of the whole sample, Js−f, equates to the volumetric compression of the solid constituent (that is defined in §3c). As a result, the initial volume compression of the solid constituent is treated in analogy to a temperature drop in a single-phase thermoelastic solid, and the multiplicative decomposition of equation (3.14) corresponds to the one adopted for single-phase thermoelastic solids (Lu & Pister 1975). In particular, the multiplicative decomposition of F implies the introduction of a new intermediate configuration, denoted as in figure 1, which is generally incompatible and locally effective-stress-free.
From equation (3.13), and from the requirement that F*=F′, the effective stress K+JpwI (which is the stress producing the deformation B*=F*F*T and which is measured from the configuration ) applied in test B is equal to the stress K′ applied in test A, namely K′=K+JpwI.
It is noteworthy that the coincidence of F*=F′ between the distortions of test B and test A (in which the pore pressure is null) under the same effective stress state (K*=K′) implies that, in an elastic granular medium with a compressible solid constituent, F′ is due to the reversible grain rearrangements and the isochoric grain distortions (as in the case of incompressible solid constituents), and also to the volume compression of the solid grains caused by intergranular contact stress (which is induced by the applied stress K′, see §3c). In order to clarify this aspect, it is worth comparing the multiplicative split of F adopted here with the one used by Morland (1972), Bluhm & De Boer (1996) and De Boer (1996), who considered that F can be split into one part due to the microscopic deformations of the material of the solid constituent FR and the remaining part due to the change in size and shape of the pores FN, thus F=FNFR, where both FN and FR may be multiplicatively decomposed into separate volumetric and isochoric contributions. As compared to FR, Fs−f is only a fraction of the volumetric part of FR (namely the volume compression due to pore pressure), so the remaining part of FR (namely ) is due to the volumetric and isochoric deformations of the solid constituent caused by intergranular contact stress. In this work, the part FRF−1s−f is assumed to be included in F′, thus .
According to equation (3.17), the distortion F*=F′ can in turn be multiplicatively decomposed into , with F′vol=J′1/3I. Thus, since , we get Fvol=F′volFs−f and .
It is worth noting that, for the special choice of given in equation (3.21), namely 3.26 from equation (3.25), the effective Kirchhoff stress equates to 3.27 implying a linear behaviour of the solid skeleton in the logarithmic volumetric strain.
Let us emphasize once more that the free-energy density function ψsk0(B′) (and consequently the elastic stiffness coefficients and μ appearing in equations (3.21) and (3.26)) incorporates two sources of deformation: the deformation due to the reversible grain rearrangements on the one hand and, on the other, the deformation due to the distortion and volume compression of the solid constituent induced by intergranular contact stresses (which will be defined in §3c).
(c) The solid constituent
In order to describe the constitutive behaviour of the solid constituent, the intrinsic average stress within the solid constituent needs to be defined. In terms of Cauchy stress in the current configuration, the intrinsic average stress within the solid constituent Ts is generally identified (e.g. Coussy 1995; De Boer 2005) as 3.28 The physical meaning of the two terms is obvious: the first one represents the stress in the solid constituent () induced by intergranular contact forces, whereas the second term takes into account the effects of the pore-fluid pressure. Using equation (2.7), the total Kirchhoff stress within the solid constituent is 3.29 where K′s=JsT′s is the effective Kirchhoff stress within the solid constituent (which is induced by intergranular contact forces).
The final key assumption of the proposed approach is that the free energy can be split into separate volumetric and isochoric contributions for the solid constituent too. Additionally, a linear behaviour in the logarithmic volumetric strain is assumed for Ks, thus the volumetric component of the free-energy density function ψsgvol0(Js), per unit volume of the solid constituent in the reference configuration equates to 3.30 where is the constitutive parameter describing the bulk stiffness of the solid constituent. As a result, the volumetric component of the Kirchhoff stress within the solid constituent is expressed by 3.31 with Ksvol=Ksvol⋅I/3, so Ksvol=KsvolI. Let us recall here too that the free-energy function ψsgvol0(Js) adopted in equation (3.30) does not make sense in the case of very large volumetric expansions (ln Js>1, i.e. Js>2.718) because polyconvexity is not ensured Doll & Schweizerhof (2000). Note that Js can also be multiplicatively decomposed into two contributions, 3.32 where Js−f is the volume compression of the solid constituent due to the pore fluid and Js−m is the volume compression of the solid constituent due to the intergranular contact forces (i.e. induced by ), so that, from equation (3.32) 3.33 Comparing equations (3.29) and (3.33), we can deduce that 3.34 This is a key property of the selected ψsgvol0(Js), through which the effects of the average effective stress in the solid constituent () and of the pore fluid (Jspw), namely Js−m and Js−f, can be decoupled from each other. This key property is a direct consequence of the assumed additive decomposition of the average stress in the solid constituent (equation (3.29)) and the assumed linear elastic behaviour in the logarithmic volumetric strain of the solid constituent (equation (3.30)). Note also that, due to equation (3.28), the macroscopic volumetric stress K′vol=Kvol+JpwI induces a microscopic volumetric strain, Js−m, equivalent to the one induced by pore pressure, Js−f. This key assumption is implicit in Biot’s parameters of linear theory, although it may not be generally applicable (for granular media, for instance, this assumption implies that the intergranular contacts are numerous and statistically distributed around each grain).
As discussed in §3b, the effects of Js−m on the volume deformation of the solid skeleton are included in the free-energy density function of the solid skeleton ψsk0(B′). The effects of the isochoric distortion of the solid constituent (induced by ) are obviously included in too, which is why only the volumetric component of the free-energy function ψsgvol0(Js) of the solid constituent is considered in the sequel.
(d) The free-energy density in terms of the left Cauchy–Green strain tensor (current configuration)
As a result of assumptions made in §3b,c, the free-energy density function per initial unit volume of the whole saturated porous medium can also be decomposed into separate volumetric and isochoric contributions, namely 3.35 Moreover, since the isochoric part of the free energy of the saturated porous solid is assumed to coincide with that of the solid skeleton, i.e. 3.36 only the volumetric part, will be considered below.
According to the approach adopted in small-strain theory, the volumetric component of the free energy of the saturated porous solid per unit volume of the undeformed solid skeleton can be deduced by combining the free energies of the solid and fluid constituents with the free energy of the solid skeleton, taking into account that, for each constituent, the free energy per unit deformed volume ψi is generally related to the free energy per unit undeformed volume ψ0i by ψi=ψ0i/Ji, where Ji is the local ratio of the deformed to the undeformed volume. We must also consider that, under large strains, is the strain rate that is work-conjugated with the volumetric Kirchhoff stress. As a result, considering the volume fractions of the single constituents in the reference configuration, we obtain the following expression of the free-energy density function per initial unit volume of the solid skeleton 3.37 where the first three terms represent the contributions of the solid skeleton and the solid and fluid constituents, respectively, while the last term is a coupling term representing the work done by the effective stress K′vol, when the solid constituent is compressed by the pore pressure, dJs−f/Js−f (see figure 2a), or, likewise, the work done by the pore pressure (−Jspw) when the solid constituent is compressed by the intergranular contact stress, dJs−m/Js−m (see figure 2b). In fact, from the assumption of linear behaviour in the logarithmic volumetric strain of the solid constituent (see equation (3.34)), the following equalities hold true: 3.38 It is worth noting that, due to the coupling term, the free-energy density function given in equation (3.37) is not the simple sum of the contributions of the single constituents. This is because reference has been made to an elementary volume deforming with the solid skeleton, in which the saturation condition is fulfilled. The term contains the integration constant k appearing in the expression of μw (equation (3.9)).
The second term in equation (3.37) includes only the contribution due to the compression of the solid constituent induced by the pore pressure (i.e. dJs−f/Js−f) because the work relating to the isochoric distortion and the volume compression (i.e. dJs−m/Js−m) of the solid constituent induced by the intergranular contact forces is fully taken into account in the free energy of the solid skeleton, ψsk0(B′) (as discussed in §3b,c). Equation (3.37) can be rearranged in the light of equations (2.7), (3.25) and (3.341), thus obtaining 3.39 where Jw is given by equation (2.4) and Js−f can be deduced from equations (3.342) and (3.7), 3.40 in which Js can be expressed in terms of n and J by equation (2.7), so ψvol0 given in equation (3.39) effectively depends only on J and .
Equation (3.39) is one of the main contributions of this work. It is worth emphasizing that the method used to define in equation (3.39) is fairly general and restricted only by the present choice of the hyperelastic expressions for the solid constituent.
The proof that, according to equations (3.5) and (3.25), the expression of the free-energy density function provided in equation (3.39) is a potential for the stresses in the solid and fluid phases and effectively yields 3.41 3.42and 3.43 is straightforward, and a concise proof is given in the electronic supplementary material, appendix B.
Finally, it is worth adding that, for the special choice of ψfl0(Jw), ψskvol0(J/Js−f) and in equations (3.10) and (3.21), leading to a linear behaviour in the logarithmic volumetric strain, the hyperelastic constitutive relationships are reduced to 3.44 3.45and 3.46
For the sake of completeness, the free-energy density function in terms of the right Cauchy–Green strain tensor is given in the electronic supplementary material, appendix C, together with the constitutive relationships in terms of the first Piola–Kirchhoff stress tensor.
Definition of the ‘effective stress’ according to Nur & Byerlee (1971).
Given the special choice of ψfl0(Jw) and ψskvol0(J/Js−f) in equations (3.10) and (3.21), the hyperelastic constitutive relationship, equation (3.44), can also be rewritten as 3.47 which represents the extension to large strains of the definition of ‘effective stress’ according to Nur & Byerlee (1971), in which and 3.48 Note that, in small-strain theory, and α≈1 approximately when the solid grains are much stiffer than the solid skeleton (i.e. ; Rice & Cleary 1976). As shown in §5, for a generic choice of ψskvol0(J/Js−f), the corresponding value of α can only be defined in the rate form.
4. Particular cases
In this section, some particular cases are analysed in detail with reference to the current configuration only. However, the extension of these analyses to the reference configuration is straightforward.
(a) Incompressible fluid and solid constituents
In the case of incompressible fluid and solid constituents (Jw=1, thus ρw=ρw0, Js=1 and, from equation (3.34), Js−m=Js−f=1), the pore pressure is indeterminate and μw is deduced from equation (3.12), so the time differentiation of the free-energy density becomes 4.1 with from equations (2.5) and (2.7) and . The free-energy variations are consequently related only to the effective stress, i.e. the contribution of the solid skeleton. In fact, from equation (3.39), 4.2
(b) Incompressible solid constituents
In the case of an incompressible solid constituent (Js=1 and, from equation (3.34), Js−m=Js−f=1), equation (3.39) is largely simplified, but still contains the term relating to the compression of the pore fluid, namely 4.3 where the contributions of the solid skeleton and of the pore fluid are clearly separate.
It is worth assessing the effects of the separate contributions in the time differentiation of the free energy: from equations (2.5) and (2.7) , so equation (4.3) can be rewritten as 4.4 where the term (μw−pw/ρw) represents the work needed to compress the pore fluid from the initial pressure pw0=0 to the pressure pw (Biot 1972), 4.5 which is not negative (it is null for pw=pw0=0). This term represents the free-energy variation due to the in-flow/out-flow of pore fluid, whereas the term is the free-energy variation of the fluid contained in pores.
From equation (4.3), the free energy of the solid skeleton is obviously given by the difference between the whole free energy of the saturated porous medium (ψvol0) and the free energy of the pore fluid, i.e. 4.6 In small-strain theory, this approach is typically adopted to extract the free-energy density function of the solid skeleton from those of the whole saturated porous medium and of the pore fluid, when the solid constituent is incompressible (Loret et al. 2002). In this case, the pore fluid simply flows through the porous medium without affecting its mechanical behaviour.
(c) Nearly incompressible solid constituent (semilinear theory)
In the case of the solid constituent being nearly incompressible, where is the small-strain volume compression of the solid constituent induced by the pore pressure. As a result, equation (3.39) is simplified as follows: 4.7 where nJ≈J−(1−n0)(1+ϵs) from equation (2.7), and ϵs is the small-strain volume compression of the solid constituent, which is related to Js through Js≈1+ϵs. In this case, the following identities are valid, and
For the special choice of ψfl0(Jw), ψskvol0(J/Js−f) and given in equations (3.10) and (3.21), the hyperelastic relationships become 4.8 and thus the final expression of the stress-like quantities is brought down to 4.9 4.10and 4.11 Note that the coefficient α, defined in equation (3.48), is reduced in this case to 4.12
(d) Geometric linearization (small displacements and strains)
In the case of small displacements and strains of the solid skeleton and all constituents, where ϵ is the linearized strain tensor (ϵ=(H+HT)/2, with H=F−I the displacement gradient), then J≈1+ϵ (with ϵ=tr ϵ) and any appropriate forms of the free-energy density functions ψskvol0 and ψfl0 are reduced to the following linearized forms: 4.13 and 4.14 According to Biot (1941), the variation in pore-fluid volume content ξ can be introduced as 4.15 so mw=ρw0(n0+ξ) and Jw=nJ/(n0+ξ). Note that n≈n0+(1−n0)(ϵ−ϵs) and nJ≈n0+ϵ−(1−n0)ϵs, thus and equation (3.39) is thus reduced to 4.16 depending on ϵ and ξ; in fact, in the geometrically linear theory, Js≈1+ϵs, so 4.17 and 4.18 Let σ denote the Cauchy stress tensor (i.e. σ=T), then it can easily be demonstrated that 4.19 and 4.20 with σ′=σ+pwI as the effective stress tensor. Equation (4.20) can be rearranged in light of equation (4.17), 4.21 which can be reduced to the well-known form 4.22 with (from equation (3.48)) and 4.23 As a result, equations (4.19) and (4.16) can be rewritten in the well-known form 4.24 and 4.25
5. The rate form of the hyperelastic constitutive equations
The rate form of the hyperelastic constitutive equations can be obtained by first considering that from equation (3.43) 5.1 where 5.2 and can be deduced by time differentiating equation (2.7), 5.3 with 5.4 and 5.5 The following rate form: 5.6 can be rearranged into 5.7 with . Equation (5.7) can be substituted into equation (5.3), which can in turn be substituted into equation (5.1) leading to an implicit expression for , which can be rearranged into 5.8 where 5.9
(a) The rate form of the hyperelastic constitutive equations in the current configuration
Below, we consider the rate form of the hyperelastic equations written in terms of the rate of deformation tensor D and the objective Oldroyd rate of Kirchhoff stress 5.10 where T(2) is the second Piola–Kirchhoff stress tensor. From equation (3.44), we obtain 5.11 from which, using equations (5.3) and (5.7) with equations (3.4) and (5.8) and bearing in mind that , the Oldroyd rate of Kirchhoff stress becomes 5.12 where coincides with the corresponding tensor of single-phase media as given by Miehe (1995), while equates to 5.13
Moreover, from equation (5.8) reduces to 5.14 It is worth emphasizing that, in equations (5.12) and (5.14), there is a symmetry of the effects induced by D on with respect to the effects induced by on K∇, as follows logically from the assumption of hyperelasticity, so the tangent stiffness can be expressed in a sort of symmetric, generalized matrix form, as follows: 5.15 where is obviously endowed with the major (and minor) symmetries, and 5.16 Equation (5.15) consequently introduces the notion of generalized stress (K and μw), strain increment (D and ) and tangent stiffness.
For the sake of completeness, the rate forms of hyperelastic constitutive equations in terms of the second Piola–Kirchhoff and the right Cauchy–Green strain tensors and in terms of the first Piola–Kirchhoff stress tensor and the deformation gradient are given in the electronic supplementary material, appendix D.
Moreover the different compressibilities of porous media at finite strains (depending on whether or not the pore pressure is allowed to drain, e.g. Zimmermann et al. 1986 for small strains) are discussed in the electronic supplementary material, appendix A.
This paper is the first to indicate a general method for defining the macroscopic free-energy function of a saturated porous medium at finite deformations in the case of compressible fluid and compressible solid constituents, combining the free energies of the various constituents. The proposed method is generally applicable and can be used with any form of the free-energy density functions of the pore fluid and solid skeleton, providing the latter can be split into separate volumetric and isochoric contributions. The main requirement concerns the behaviour of the solid constituent: its free-energy density function must be suitable for splitting into separate volumetric and isochoric contributions, and the former must lead to a linear response in the logarithmic volumetric strain. In addition, an isotropic response is assumed and the macroscopic effective volumetric stress K′vol is assumed to induce a microscopic volumetric strain equivalent to the one induced by pore pressure (for granular media, for instance, this assumption implies that the intergranular contacts are numerous and statistically distributed around each grain). For the purpose of illustrating the method, the relationships are fully developed in a simple case of linear response in the logarithmic volumetric strain for the solid skeleton and the fluid constituent.
According to Biot (1972), the macroscopic free-energy density is defined for an elementary volume moving with the solid skeleton and conceived as an open system where the pore fluid can enter and exit, so the assumed independent strain-like variables are a suitable measure of the strain of the solid skeleton and the variation in the fluid-mass content of the reference elementary volume. The corresponding stress variables are a suitable stress measure (which must be work-conjugate to the selected strain measure) and the chemical potential of the pore fluid. The resulting free energy is not the simple sum of the contributions of the single constituents because the proposed expression takes into account the saturation condition.
Finally, the particular cases of geometrically linear theory, semilinear theory (in which the solid constituent is assumed to be nearly incompressible), of the incompressibility of the solid constituent, and of both the fluid and the solid constituents, are each analysed in detail, thus recovering, within the proposed unified theory, the approaches previously presented in the literature (which are based on some simplifying assumptions concerning the compressibility of the constituents) and new, simplified constitutive frameworks are investigated.
Financial support from the University of Trento is gratefully acknowledged. Special thanks go to Dr R. P. Denzer for his precious comments concerning the finite-strain modelling of single-phase media.
- Received January 12, 2010.
- Accepted April 14, 2010.
- © 2010 The Royal Society