Journal of the Mechanics and Physics of Solids 91 (2016) 311–333

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Journal of the Mechanics and Physics of Solids

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Coupled hydro-mechanical effects in a poro-hyperelastic material

A.P.S. Selvadurai n, A.P. Suvorov

Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke Street West, Montréal, QC, Canada H3A 0C3 article info abstract

Article history: Fluid-saturated materials are encountered in several areas of engineering and biological Received 17 December 2015 applications. Geologic media saturated with water, oil and gas and biological materials Received in revised form such as bone saturated with synovial fluid, soft tissues containing blood and plasma and 7 March 2016 synthetic materials impregnated with energy absorbing fluids are some examples. In Accepted 9 March 2016 many instances such materials can be examined quite successfully by appeal to classical Available online 15 March 2016 theories of poroelasticity where the skeletal deformations can be modelled as linear Keywords: elastic. In the case of soft biological tissues and even highly compressible organic geolo- Poro-hyperelasticity gical materials, the porous skeleton can experience large strains and, unlike rubberlike Fluid-saturated media materials, the fluid plays an important role in maintaining the large strain capability of the Canonical analytical solutions material. In some instances, the removal of the fluid can render the geological or biological Large deformations Time-dependent phenomena material void of any hyperelastic effects. While the fluid component can be present at Calibration of computational results various scales and forms, a useful first approximation would be to treat the material as hyperelastic where the fabric can experience large strains consistent with a hyperelastic material and an independent scalar pressure describes the pore fluid response. The flow of fluid within the porous skeleton is defined by Darcy's law for an isotropic material, which is formulated in terms of the relative velocity between the pore fluid and the porous skeleton. It is assumed that the form of Darcy's law remains unchanged during the large strain behaviour. This approach basically extends Biot's theory of classical poroelasticity to include finite deformations. The developments are used to examine the poro-hyperelastic behaviour of certain one-dimensional problems. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction

The mathematical theory of finite elasticity represents a defining development of modern non-linear continuum me- chanics, a position achieved partly due to contributions of lasting value made by a number of individuals and partly due to its ability to provide meaningful and applicable results to many problems of technological interest. The history of the subject of finite elasticity can be traced back to the works of Cauchy, Green, Piola and others but the modern development of the subject commences with the seminal works of R.S. Rivlin, which have been compiled by Barenblatt and Joseph (1997). The contributions to the theory of elastic materials exhibiting large strain phenomena, subsequent to Rivlin's work, are far too numerous to be cited individually. Complete accounts of these developments are given in review and survey articles and volumes by Doyle and Ericksen (1956), Rivlin (1960), Adkins (1961), Spencer (1970), Beatty (1987) and in the volumes by

n Corresponding author. E-mail address: [email protected] (A.P.S. Selvadurai). http://dx.doi.org/10.1016/j.jmps.2016.03.005 0022-5096/& 2016 Elsevier Ltd. All rights reserved. 312 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

Murnaghan (1951), Eringen (1962), Green and Zerna (1968), Green and Adkins (1970), Wang and Truesdell (1973), Carlson and Shield (1980), Marsden and Hughes (1983), Ogden (1984), Hanyga (1985), Valent (1988), Lur’e (1990), Truesdell and Noll (1992), Antman (1995), Drozdov (1996), Carroll and Hayes (1996), Holzapfel (2000), Taber (2008) and Selvadurai (2015a). The pedagogical value of the theory of finite elasticity is highlighted in many excellent texts (Murnaghan, 1951; Eringen, 1962; Jaunzemis, 1967; Green and Zerna, 1968; Green and Adkins, 1970; Wang and Truesdell, 1973; Carlson and Shield, 1980; Marsden and Hughes, 1983; Ogden, 1984; Hanyga, 1985; Valent, 1988; Lur’e, 1990; Truesdell and Noll, 1992; Antman, 1995; Drozdov, 1996; Carroll and Hayes, 1996; Holzapfel, 2000; Taber, 2008) and in the compact volumes by Chadwick (1976), Gurtin (1981), Atkin and Fox (1980) and Spencer (2004). A survey of recent developments in the theory of non-linear elasticity is given by Beatty (2001a), Fu and Ogden (2001), Hill (2001) and the lecture series and volumes organized by Zubov (1997), Libai and Simmonds (1998), Dorfmann and Muhr (1999), Hayes and Saccomandi (2001), Saccomandi and Ogden (2004) and Dorfmann and Ogden (2004). The experimental aspects of hyperelastic materials date back to the work of Mooney (1940), Rivlin and Saunders (1951), Gent and Rivlin (1952a,b), Ogden (1972), Bell (1973) and Treloar (1975, 1976). Recent experimental investigations of hyperelastic rate-sensitive and non-rate sensitive materials are given by Pamplona and Bevilacqua (1992), Gent and Hua (2004), Selvadurai (2006) and Selvadurai and Shi (2012). The articles by Selvadurai (2006) and Selvadurai and Shi (2012) also provide extensive references to several other experimental investigations, notably those involving inflation and indentation of membranes, which have important applications in the identification of the constitutive behaviour of hyperelastic materials. The study of the mechanics of fluid-saturated porous media has its origins in the modelling of consolidation of soils. A theory for the analysis of one-dimensional soil consolidation was first proposed by Terzaghi (1923). The model considers the linear deformability of the porous skeleton, Darcy flow of the saturating fluid through the pore space and an effective stress relationship between the stresses in the porous skeleton and the pore fluid pressures. The classical theory of poroelasticity proposed by Biot (1941) is a complete theory that accounts for three-dimensional elasticity and fluid transport effects in the formulation. The developments in the classical theory of Biot poroelasticity are quite extensive and these are documented in the review articles and volumes by Scheidegger (1960), Paria (1963), Rice and Cleary (1976), Schiffman (1984), Whitaker (1986), Coussy (1995), de Boer (2000), Wang (2000), Cowin (2001), Selvadurai (2007, 2015b), Verruijt (2015) and Cheng (2015). The classical theory has been extended by Selvadurai and Suvorov (2012, 2014) to include elasto-plastic behaviour of the porous skeleton, which constitutes an important aspect in the modelling of geomaterial behaviour. Applications of thermo-poroelasticity in the area of geosciences have also been discussed in a recent volume by Selvadurai and Suvorov (2016). The consideration of large strain effects in the modelling of saturated geomaterial behaviour is important to model the consolidation of soft sediments, although such materials can display irreversible phenomena in the constitutive be- haviour of the porous skeleton. The influence of large strain phenomena on the one-dimensional consolidation of geo- materials was first examined by Gibson et al. (1967) who formulated the problem with reference to a Lagrangian approach for the description of the strains. The analysis takes into consideration the alterations in the void fraction in the porous skeleton, which in turn alters the permeability of the porous medium and its one-dimensional deformability properties. These alterations are also suggested by experimental evidence on saturated sediments. In the approach adopted by Gibson et al. (1967), the one-dimensional consolidation problem is reduced to a non-linear partial differential equation for the void ratio, which is converted to a linearized form and applied to examine consolidation problems of geotechnical interest. An extension of the classical theory of poroelasticity (Biot, 1941) to include finite deformations was also proposed by Biot (1972), although the developments were not adopted for the solution of problems in poro-hyperelasticity. The finite strain one-dimensional soil consolidation problem has been extensively studied in the literature on soil me- chanics and developments in this area are discussed by Gibson et al. (1981), Cargill (1984), Townsend and McVay (1990), Morris (2002, 2005), Ichikawa et al. (2010) and Ichikawa and Selvadurai (2012). The approach proposed in these studies is largely restricted to one-dimensional problems and suitable only for situations where there is no unloading of the con- solidating soil; i.e. the deformability characteristics of the consolidating soil do not account for large strain irreversible processes in the porous skeleton. This can be addressed by implementing finite strain plasticity effects into the skeletal behaviour but such advances cannot be made solely through analytical approaches (Lee, 1969; Lubarda, 2001; Selvadurai and Yu, 2006a, 2006b, 2008; Yu and Selvadurai, 2007). The work of Uzuoka and Borja (2012) deals with the computational modelling of finite deformations of poro-hyperelastic behaviour of soils with a neo-Hookean form of a strain energy function, which again, does not account for irreversible deformations of the soil skeleton undergoing finite deformations. Alternative perspectives of poromechanics have been considered in the context of the theory of mixtures by Green and Steel (1966), Crochet and Naghdi (1966), Mills and Steel (1970), Green and Naghdi (1970), Atkin and Craine (1976), Bowen (1976), Bedford and Drumheller (1983), Dell’Isola and Romano (1987), Coussy (1995), Murad et al. (1995), Rajagopal and Tao (1995), Bennethum and Cushman (1996), Drumheller (2000), Huyghe (2015a) and others. Of particular interest are the approaches presented by Shi et al. (1981), Dai et al. (1991), Baek and Srinivasa (2004), Gajo (2010) and Pence (2012). The article by Pence (2012) also contains a systematic treatment of the application of the mixture theory approach to fluid- saturated media and a comprehensive exposition of the current status. Also of particular interest to the discussion are the articles by Duda et al. (2010), Chester and Anand (2010) and Baek and Pence (2011) who examine the application of coupled theories in the context of polymer swelling. The mixture theory approaches are certainly elegant and complete from the point of view of continuum formulations, and are particularly relevant when a number of species saturating the porous space are encountered. The mixture theory-based formulations may have advantages when dealing with non-linear theories of material behaviour, where the porous skeleton can contain multi-species of pore fluids and the porous skeleton can A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 313 experience large strains or large strain non-linear viscoelastic and viscoplastic phenomena that have corresponding influ- ences on the fluid transport problem. The application of a mixture theory approach to a ground water flow problem and a comparison with predictions based on Darcy's law are given in (Munaf et al., 1993). The application of the theories of hyperelasticity to describe the behaviour of biological soft tissues has been quite extensively studied since the seminal studies by Spilker and Simon (1988), Fung (1985, 1993), Humphrey (2002), Taber (2008) and Huyghe (2015b). Computational aspects of hyperelastic modelling as applied to biological materials are also discussed by Spilker and Simon (1988), Holzapfel and Ogden (2006) and Zhurov et al. (2007). Early applications of poro- hyperelasticity to the study of soft tissue biomaterials are given by Simon and Gaballa (1988), Simon (1992), and Simon et al. (1996) who take into consideration the influence of large elastic deformations of the porous skeleton, Darcy flow of the saturating fluid through the pore space and an effective stress relationship between the porous skeletal stresses and the pressure of the fluid saturating the pore space. A recent application of poro-hyperelastic modelling to abdominal aortic aneurysms is given by Ayyalasomayajula et al. (2010), who also provide references to other studies in the area. The al- terations to the fluid transport properties due to pore closure that can result from large deformations of the porous skeleton are also considered in (Ayyalasomayajula et al., 2010). In this paper we examine the class of poro-hyperelasticity problems where the porous skeleton exhibits hyperelasticty effects characterized by a constitutive relationship with a specified form of the strain energy function. The hyperelastic behaviour is assumed to be isotropic. The fluid flow through the porous skeleton is characterized by an isotropic form of Darcy's law and the permeability of the porous skeleton is assumed to remain isotropic and constant during the finite deformation of the fluid-saturated porous medium. The paper describes the constitutive relationships governing the poro- hyperelastic material and applies the developments to the study of one-dimensional problems involving consolidation of a column of finite thickness and consolidation of a poro-hyperelastic sphere that is subjected to a radial compressive stress while maintaining the boundary fully drained. The mathematical analysis associated with these problems employs a finite difference approach for the solution of the non-linear partial differential equations with the skeletal strain energy function in the form of a reduced polynomial that depends on the strain invariants, and also accommodates a strain energy function of the neo-Hookean type. The results of the present approach are compared with results derived from a finite element method incorporated in a standard computational platform.

2. Constitutive modelling

The total Cauchy stress σij in the fluid-saturated poro-hyperelastic material is assumed to be composed of the Cauchy stress in the porous hyperelastic fabric denoted by σi′j and the isotropic stress in the interstitial fluids denoted by p, such that

σσij = ij′ − p δij, ()2.1 where δij is the Kronecker delta. The effective stress relationship for the fluid-saturated hyperelastic material given above is identical to that proposed by Terzaghi (1923) and is a limiting case of the relationship developed by Biot (1941), both of which were developed in connection with the mechanics of fluid-saturated media describing infinitesimal deformations of the porous skeleton. The result (2.1) implies that the compressibility of the skeletal material composing the porous solid is larger than the compressibility of the pore fluid, which is a satisfactory assumption for materials with a highly deformable skeletal fabric. The effective stress is further represented in terms of its deviatoric component si′j and an isotropic stress (effective pressure) p′ such that

σδij′ = sp ij′ −′ij,/3 p ′=− σij′ ()2.2

Consider the initial coordinates of a point in the deformable porous skeleton with coordinates Xi (=i 1, 2, 3) (with X123===XX;; YX Z), which moves to the new position with coordinates defined by xii (=1, 2, 3) (with xxxyxz123===;;). The deformation gradient tensor F is given by

∂xi F ==fij . ∂Xj ()2.3

The strain tensor B¯ is defined by

T BFF¯ = ¯¯ , ()2.4 where

FF¯ ==JJ−1/3 ;det. F ()2.5

The strain energy function U for the isotropic hyperelastic material is taken to be of the form

UUIIJ=(¯¯12,,) ()2.6 ¯ ¯ ¯ where I1 and I2 are, respectively, the first and second invariants of the strain tensor B. The deviatoric component of the 314 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 effective stress s′ is defined as

2 ⎧⎛ ∂U ∂U ⎞ ∂U ⎫ sBBB′= dev⎨⎜ + I¯ ⎟ ¯ − ¯ ⋅ ¯ ⎬. ¯ 1 ¯ ¯ J ⎩⎝ ∂I1 ∂I22⎠ ∂I ⎭ ()2.7

The isotropic component of the effective stress is defined as ∂U p′= − . ∂J ()2.8

The forms of strain energy functions are many and varied (Murnaghan, 1951; Doyle and Ericksen, 1956; Rivlin, 1960; Adkins, 1961; Eringen, 1962; Green and Zerna, 1968; Green and Adkins, 1970; Spencer, 1970; Wang and Truesdell, 1973; Carlson and Shield, 1980; Marsden and Hughes, 1983; Ogden, 1984; Hanyga, 1985; Beatty, 1987; Valent, 1988; Lur’e, 1990; Truesdell and Noll, 1992; Antman, 1995; Drozdov, 1996; Carroll and Hayes, 1996; Barenblatt and Joseph, 1997; Holzapfel, 2000; Taber, 2008; Selvadurai, 2015a) and discussions of the development and application of various forms of strain energy functions are given in (Mooney, 1940; John, 1960; Treloar, 1975, 1976), see also (Varga, 1966; Hart-Smith and Crisp, 1967; Alexander, 1968; Varley and Cumberbatch, 1980; Ogden, 1982; Gent, 1996; Hill and Arrigo, 1999; Beatty, 2001b, 2008; Ru, 2002; Selvadurai, 2006; Wang and Schiavone, 2012, 2014; Selvadurai and Shi, 2012; Horgan, 2015; Mihai et al., 2015) and others. If the strains experienced by the hyperelastic material are not large, the theory of finite deformations can be ap- proximated by the second-order theory developed by Signorini (1942), Rivlin (1953), Rivlin and Topakoglu (1954), Green and Spratt (1954), Selvadurai and Spencer (1972), Selvadurai (1973a, 1973b, 1974, 1975), Capriz and Podio-Guidugli (1974), Lindsay (1985, 1992), Selvadurai et al. (1988) and strain energy functions of the Mooney-Rivlin type can provide adequate descriptions of moderately hyperelastic behaviour. Considering finite deformations, a second-order reduced polynomial can be introduced to take the form

2 1 2 1 4 UCI=(10¯ 1−)+33 CI 20 (¯ 1 −)+ (−)+J 1 (−)J 1, D1 D2 ()2.9 where C10, C20, D1 and D2 are material constants. The constants C10 and D1 can also be defined in terms of the linear elastic shear modulus (G) and the bulk modulus (K) as 2 2,CGD10== 1 . K ()2.10 Consequently, for this particular material the deviatoric component of the effective stress is given by

2 T sFF′= [CCIdev10 +23 20 (¯ 1 −)](¯¯ ) , J ()2.11 and the isotropic component of the effective stress (effective pressure) can be expressed in the form 2 4 p′=− (−)−J 1 (−)J 1.3 D12D ()2.12

If C20 = 0, D2 →∞we recover the strain energy function from (2.9) for the neo-Hookean material. It should also be emphasized that the resulting neo-Hookean material is not incompressible. The constitutive developments will be complete when a relationship is developed to account for the flow of fluid through a poro-hyperelastic material. We assume that the entire pore space of the hyperelastic skeleton is saturated with a fluid and that the flow takes place as a result of a gradient in the Bernoulli potential, which consists of the pressure head, the velocity head and the datum head. In the case of slow flows through the pore space, the velocity head can be neglected in com- parison with the other contributions and the datum head can also be neglected provided the potential is measured with reference to a fixed datum (Selvadurai, 2000). Considering the principle of conservation of mass we have ∂u −∇⋅φ (vvfs − ) = ∇⋅ ∂t ()2.13 where φ is the porosity, vf is the velocity of the fluid in the pore space and vs is the velocity of the solid skeleton of the porous material, ∇ is the gradient operator referred to the coordinates of a particle of fluid in the deformed configuration, and u is the displacement vector defined as uxX=−. Derivation of the Eq. (2.13) is presented in Appendix A. We assume that flow of the fluid through the isotropic hyperelastic skeleton can be described by an isotropic form of Darcy's law, which takes the form k φ(−)=−∇vvfs p, η ()2.14 where k is the permeability, which is assumed to be a constant, and η is the dynamic fluid viscosity. One advantage of the mixture theory approach is that non-linear laws of the Darcy-type governing flow in porous media emerge through A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 315 consideration of momentum transfer between the solid and the fluid phases. The linearization of such generalized re- lationships leads to Darcy's result. Darcy's law, as used in the present context, is a phenomenological assumption that can be reconciled with simplifications of relationships derived from the mixture theory approach. The modelling can be made more complex if permeability heterogeneity is also given due consideration (Du and Ostoja-Starzewski, 2006; Selvadurai and Selvadurai, 2010; Selvadurai and Selvadurai, 2014). It is important to note that when porous hyperelastic media are sub- jected to stresses, the porosity will change with the stress state and the permeability will evolve with porosity alteration since there is a correlation between the porosity and permeability (Federico and Herzog, 2008; Federico and Grillo, 2012). Even in the case of geological materials such as sandstone and granite that experience small strain behaviour, the appli- cation of isotropic compression to the rocks can result in pore closure and the concomitant reduction in permeability (Selvadurai, 2004; Selvadurai and Głowacki, 2008; Selvadurai et al., 2011). The consideration of the dependence of the permeability on the stress state is outside the scope of the present paper.

3. The one-dimensional poro-hyperelasticity problem

We now apply the developments presented in the preceding section to examine the poro-hyperelastic behaviour of a one-dimensional column. In the ensuing, we shall use a coordinate description that will dispense with the indicial notation.

We consider a column of length L which is constrained to deform in the x1-direction. A total normal stress σ0 is applied at the upper surface X = L of the one-dimensionally constrained poro-hyperelastic column (Fig. 1). The exposed surface of the column is allowed to drain, thereby maintaining the pore fluid pressures at the upper surface as zero. The normal fluid velocity is set to zero on the constrained surfaces. The one-dimensional problem requires the solution of the deformation and fluid flow fields in the poro-hyperelastic column. In one-dimensional consolidation, the motion of the skeleton of the hyperelastic column can be described by the function ξ (Xt, ), such that

xXtyYzZ=(ξ ,; ) = ; = ()3.1 where (XYZ,,) are the coordinates of a particle in the undeformed configuration and (xyz,,) are the coordinates of a particle in the deformed configuration. The deformation gradient is given by

Fig. 1. One-dimensional compression of a poro-hyperelastic column. 316 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

⎛ ∂ξ ⎞ ⎜ 00⎟ ∂X F = ⎜ ⎟, ⎜ ⎟ ⎜ 010⎟ ⎝ 001⎠ ()3.2 and J = ddXξ/ . The transformed deformation gradient matrix F¯ is given by ⎛ ∂ξ ⎞ ⎜ 00⎟ ⎛ ∂ξ ⎞−1/3 ∂X F̅ = ⎜⎟ ⎜ ⎟, ⎝ ⎠ ⎜ ⎟ ∂X ⎜ 010⎟ ⎝ 001⎠ ()3.3 and the strain matrix is given by

⎛ ⎛ ⎞2 ⎞ ⎜ ⎜⎟∂ξ ⎟ −2/3 00 T ⎛ ⎞ ⎝ ⎠ ⎜⎟∂ξ ⎜ ∂X ⎟ BFF̅ = ̅ ̅ = ⎝ ⎠ ⎜ ⎟. ∂X ⎜ 010⎟ ⎝ 001⎠ ()3.4 ¯ ¯ The first invariant I1 of the strain matrix B is given by

⎛ 2 ⎞ ⎛ ⎞ ⎛ ∂ξξ⎞−−2/3 ⎛ ∂ ⎞ ⎛ ∂ ξ⎞4/3⎛ ∂ ξ⎞ 2/3 ¯ ⎜⎟⎜ ⎜⎟⎟ ⎜ ⎜⎟ ⎜⎟⎟ I1 = ⎝ ⎠ ⎜ ⎝ ⎠ +=22,⎟ ⎝ ⎠ + ⎝ ⎠ ∂XX⎝ ∂ ⎠ ⎝ ∂ X∂ X⎠ ()3.5 and its deviatoric component is

⎛ 2 ⎞ ⎛ ∂ξ ⎞ 2 ⎜ ⎜⎟− κ 00⎟ ∂ξ ⎛ ∂ξ ⎞−2/3⎜ ⎝ ⎠ ⎟ + 2 ¯ ⎜⎟ ∂X ()∂X dev(B)=⎝ ⎠ ⎜ ⎟; κ = ∂X ⎜ 01− κ 0⎟ 3 ⎝ 001− κ⎠ ()3.6

Consequently, from (2.11) and (2.12) the axial component of the deviatoric effective stress tensor and the effective pressure can be expressed in the forms ⎛ ⎞ 4 ⎛ ∂ξξ⎞1/3⎛ ∂ ⎞− 5/3 ¯ ⎜ ⎜⎟ ⎜⎟⎟ sCCIxx′ =[10 +23 20 ( 1 −)]⎝ ⎠ − ⎝ ⎠ 3 ⎝ ∂XX∂ ⎠ ()3.7 ⎛ ⎞ ⎛ ⎞3 2 ⎜⎟⎜⎟∂ξξ4 ∂ p′= − ⎝ −−1⎠ ⎝ − 1⎠ DX12∂ DX∂ ()3.8

According to the decomposition (2.2), the axial effective stress can be expressed in the form ⎧ ⎡ ⎛ ⎞ ⎤⎫⎛ ⎞ 4 ⎪⎪⎛ ∂ξξ⎞4/3⎛ ∂ ⎞−− 2/3⎛ ∂ ξξ⎞ 1/3⎛ ∂ ⎞ 5/3 ⎨ ⎢ ⎜ ⎜⎟ ⎜⎟⎟ ⎥⎬⎜ ⎜⎟ ⎜⎟⎟ σxx′ =+CC10223 20 + − − 3 ⎩⎪⎪⎣⎢ ⎝ ⎝ ∂XX⎠ ⎝ ∂ ⎠ ⎠ ⎦⎥⎭⎝ ⎝ ∂ XX⎠ ⎝ ∂ ⎠ ⎠ ⎛ ⎞ ⎛ ⎞3 2 ⎜⎟⎜⎟∂ξξ4 ∂ + ⎝ −+1⎠ ⎝ − 1⎠ DX12∂ DX∂ ()3.9

Considering the definition of the displacement uX( , t) ξ ()=+()Xt,,, X uXt ()3.10 the axial effective stress can also be written as ⎧ ⎡ ⎛ ⎞ ⎤⎫⎛ ⎞ 4 ⎪⎪⎛ ∂u ⎞4/3⎛ ∂u ⎞−− 2/3⎛ ∂u ⎞ 1/3⎛ ∂u ⎞ 5/3 ⎨ ⎢ ⎜ ⎜⎟⎜⎟⎟ ⎥⎬⎜ ⎜⎟⎜⎟⎟ σxx′ =+CC1021 20 +++ 21−+ 31−+ 1 + 3 ⎩⎪⎪⎣⎢ ⎝ ⎝ ∂X ⎠ ⎝ ∂X ⎠ ⎠ ⎦⎥⎭⎝ ⎝ ∂X ⎠ ⎝ ∂X ⎠ ⎠ ⎛ ⎞3 24∂u ⎜⎟∂u + + ⎝ ⎠ D12∂XD∂X ()3.11

Considering the equilibrium for the one-dimensional column in the deformed configuration, we have ∂σ ∂(σ′ −)p xx = xx = 0. ∂x ∂x ()3.12

Therefore, the total axial stress σxx is independent of the coordinate x. It is more convenient to adopt a Lagrangian description and take the initial coordinate Xof a material point as an independent variable instead of the coordinate x. Using A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 317 the relation ∂x = ∂ξ and familiar chain rule for differentiation of a composite function, we can now express (3.12) in terms of ∂XX∂ the derivatives with respect to coordinates in the undeformed configuration ∂σ′ ∂p xx − = 0. ∂X ∂X ()3.13 Using (3.11), we have ⎧ ⎧ ⎡ ⎛ ⎞ ⎤⎫ ⎫ 4 ⎪⎪⎛ ∂u ⎞4/3⎛ ∂u ⎞− 2/3 ⎪ ⎨ ⎢ ⎜⎟⎜⎟ ⎥⎬ ⎪ CC10++21 20 ⎜ ++ 21⎟ − 3 ⎪ 3 ⎪⎪⎣⎢ ⎝ ⎝ ∂X ⎠ ⎝ ∂X ⎠ ⎠ ⎦⎥ ⎪ ⎪ ⎩ ⎭ ⎪ 2 ⎪ ⎛ ⎞ ⎪ ∂σ′ ∂ u 1 ⎛ ∂u ⎞−−2/35 ⎛ ∂u ⎞ 8/3 xx = ⎨ ×+⎜ ⎜⎟1 ++ ⎜⎟1 ⎟ + ⎬ ∂X ∂X2 ⎪ ⎝ 3 ⎝ ∂X ⎠ 3 ⎝ ∂X ⎠ ⎠ ⎪ ⎪ ⎪ ⎛ ⎞2 2 ⎪ 32 ⎛ ∂u ⎞1/3⎛ ∂u ⎞− 5/3 212⎛ ∂u ⎞ ⎪ ⎪ ++C ⎜ ⎜⎟⎜⎟11−+ ⎟ ++ ⎜⎟⎪ 20 ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ ⎠ ⎩ 9 ∂X ∂XDD12∂X ⎭ ()3.14

The laws governing fluid flow through the deformable porous medium, (2.13) and (2.14), can now be applied to the one- dimensional case giving kp∂ ∂u = . η ∂x ∂t ()3.15

Next, using the equilibrium condition (3.13) we can obtain

∂u ⎛ ∂u ⎞ k ∂σ′ − ⎜ 10.+ ⎟ + xx = ∂t ⎝ ∂X ⎠ η ∂X ()3.16

Eliminating ∂′σ xx / ∂X between (3.14) and (3.16), we obtain the equation of one-dimensional poro-hyperelastic deforma- tion in terms of the displacement uX( , t) in the form ⎧ ⎧ ⎡ ⎛ ⎞ ⎤⎫ ⎫ 4 ⎪⎪⎛ ∂u ⎞4/3⎛ ∂u ⎞− 2/3 ⎪ ⎨ ⎢ ⎜⎟⎜⎟ ⎥⎬ ⎪ CC10++21 20 ⎜ ++ 21⎟ − 3 ⎪ 3 ⎪⎪⎣⎢ ⎝ ⎝ ∂X ⎠ ⎝ ∂X ⎠ ⎠ ⎦⎥ ⎪ ⎪ ⎩ ⎭ ⎪ 2 ⎪ ⎛ ⎞ ⎪ ⎛ ∂u ⎞ ∂u ku∂ 1 ⎛ ∂u ⎞−−2/35 ⎛ ∂u ⎞ 8/3 ⎜⎟1 + = ⎨ ⎜ ⎜⎟ ⎜⎟⎟ ⎬ ⎝ ⎠ 2 ×+1 ++1 + ∂X ∂t η ∂X ⎪ ⎝ 3 ⎝ ∂X ⎠ 3 ⎝ ∂X ⎠ ⎠ ⎪ ⎪ ⎪ ⎛ ⎞2 2 ⎪ 32 ⎛ ∂u ⎞1/3⎛ ∂u ⎞− 5/3 212⎛ ∂u ⎞ ⎪ ⎪ ++C ⎜ ⎜⎟⎜⎟11−+ ⎟ ++ ⎜⎟⎪ 20 ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ ⎠ ⎩ 9 ∂X ∂XDD12∂X ⎭ ()3.17

Another result of importance to the modelling of the poro-hyperelasticity problem relates to the non-linear partial differential equation governing the pore fluid pressure in the medium. In the preceding developments, the displacement uX( , t) cannot be eliminated from (3.15) and (3.17) to yield a non-linear partial differential equation for the fluid pressure in a convenient form. From (3.13) it is, however, evident that, for the one-dimensional case, the pore fluid pressure p(Xt, ) can be obtained by an integration of (3.13) as

pX()=,,. tσσxx′ ()− X t xx ()3.18

The total axial stress σxx is a constant of integration and equal to the stress applied at the upper surface of the column, as a result of the traction boundary condition (see, for example, (3.22)). The effective stress σx′x can be found from (3.11) once a solution for the displacement uX( , t) is developed.

3.1. Limiting cases

(i) In the case where the deformations of the poro-hyperelastic medium are small,

∂u ⎛ ∂u ⎞2 =()o 1;⎜⎟→ 0 ∂X ⎝ ∂X ⎠ and (3.17) reduces to

∂u k ⎛ 8 2 ⎞ ∂2u =+⎜ C ⎟ , ⎝ 10 ⎠ 2 ∂t η 3 D1 ∂X ()3.19 which can be derived by considering the classical Biot poroelasticity problem of a fluid-saturated porous medium. Using 318 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

definitions of the elastic constants (2.10), the Eq. (3.19) can be rewritten as

∂u ∂2u kE(−)1 ν = c ; c = ∂t ∂X2 η (+)(−112νν ) ()3.20

where c is a constant of consolidation.

(ii) For the case where the poro-hyperelastic material is impermeable, k → 0 and the deformations of the poro-hyperelastic material will be time-independent.

(iii) In the case when the porous skeleton is incompressible or v = 1/2, the displacements are governed by the partial differential equation

∂2u = 0, ∂X2 ()3.21 and the solution of (3.21) can be obtained for a specific initial boundary value problem.

3.2. The initial boundary value problem

We shall consider the solution of the non-linear partial differential equation (3.17) as applied to the one-dimensional poro-hyperelastic consolidation problem where the column, which is constrained from movement at its base, is subjected to

an axial stress σ0 in the form of a Heaviside step function of time (Fig. 1). The base of the one-dimensional column is maintained impervious and during consolidation the fluid is allowed to be expelled from the loaded surface. The boundary conditions governing the displacement field uX( , t) in the one-dimensional poro-hyperelastic problem are as follows:

()i,for0σσxx (XLt = )=0 ∀ t ≥ ()3.22 ()iiuX ( = 0, t )= 0 for ∀ t ≥ 0 ()3.23

The initial condition governing the displacement field is: ‘()( iiiuX , 0 )= 0 for all X ∈( 0, L ) . ()3.24

The boundary conditions governing the pore pressure p(Xt, ) field in the one-dimensional poro-hyperelastic problem are: ()(ivpL , t )= 0 for ∀ t > 0 ()3.25 ⎛ ⎞ ⎜⎟∂p ()v0for0⎝ ⎠ =∀>t ∂X X=0 ()3.26 The initial condition governing the pore pressure field is

()(vipX , 0 )=−σ0 for all X ∈( 0, L ) . ()3.27

The solution of the initial boundary value problem posed by (3.17) with boundary and initial conditions (3.22) to (3.24) will be discussed in a subsequent section.

4. Numerical solution of the initial boundary value problem for 1D column

4.1. Finite difference solution

It is unlikely that the nonlinear initial boundary value problem can be solved in an exact closed form and recourse needs to be made to suitable numerical techniques to obtain results of interest. We employ a finite difference scheme with explicit integration in time. In this approach the column of length L is subdivided into N equal segments of length ΔXL= /N and the time step Δt is chosen to be sufficiently small to satisfy the stability criterion Δt ≤ C, ()ΔX 2 ()4.1

where C is a fixed constant that depends on the particular problem being examined (Mitchell and Griffiths, 1980). Other criteria that guarantee unconditional stability of the time integration scheme are available and a comparison of such schemes is given in (Khoshghalb et al., 2011). The partial derivatives with respect to time and the spatial coordinate are approximated by the following differences A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 319

∂u uX(+)−(),, tΔ t uX t ()=Xt, , ∂t Δt ∂u uX(+ΔΔ X,, t )−(− uX X t ) ()=Xt, , ∂X 2ΔX ∂2u uX(+ΔΔ X,2, t )−( uX t )+(− uX X , t ) ()=Xt, . ∂X22()ΔX ()4.2

The first spatial derivative at the upper surface of the column X = L is found from the traction boundary condition (3.22). Suppose that the first spatial derivative at X = L is ∂u ()=Lt,. s0 ∂X ()4.3 Then, uL(+ΔΔ X,, t )−(− uL X t ) s0 = 2ΔX ()4.4 and thus,

uL(+ΔΔΔ X,2 t )= s0 X + uL (− X , t ) ()4.5

Therefore, the second derivative at the upper surface can be approximated by the result

∂2u 22,2,sXΔΔ−()+(− uLt uL Xt ) ()=Lt, 0 . ∂X2 ()ΔX 2 ()4.6

In the approximation of the first and second spatial derivatives at X = 0 we use the fact that ut()=0, 0. Consequently, the first derivative at the lower surface of the column X = 0 must be approximated by ∂u uXt()Δ , ()=0, t , ∂X ΔX ()4.7 and the second derivative at X = 0 takes the form

∂2u uXt()Δ , ()=0, t . ∂X22()ΔX ()4.8

4.2. A numerical example

The suitability of the finite difference procedure for the solution of the poro-hyperelastic problem described in Section 4.1 is illustrated by appeal to a specific problem. We consider a one-dimensionally constrained column of length 1 m, which

is subjected to an axial compressive stress σ0 =−300000 Pa. Three types of material behaviour are considered:

(i) the porous skeleton exhibits linear elastic behaviour (LE) consistent with the skeletal properties described previously, (ii) the porous skeleton exhibits neo-Hookean hyperelastic behaviour (NH) and (iii) the skeletal deformation behaviour corresponds to the behaviour described by the reduced polynomial hyperelastic model (RP).

The initial values for the elastic constants of the porous skeleton for all three cases are taken as follows and they re- present typical values applicable to tissues experiencing abdominal aortic aneurysms (Polzer et al., 2011): E ==600000 Pa,ν 0.3.

Consequently, initial values for the shear and bulk moduli are GK==230769 Pa, 500000 Pa.

Thus, from (2.10),

−6 CD10==×115384 Pa, 1 4 10 1/Pa.

For the case of a reduced polynomial hyperelastic material we take

CD20==∞300000 Pa, 2 .

In case of a neo-Hookean material C20 = 0. The permeability of the porous skeleton and the fluid viscosity are taken as follows: 320 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

k =×3 10−14 m2 ;η = 0.001 Pa s.

Initial values of the permeability and viscosity are the same for all three materials. For a linear elastic material (LE) the strains are assumed to be small whereas for hyperelastic materials (NH and RP models) the strains are assumed to be finite. In the graphs, the solution obtained using the finite difference method described above is indicated in a solid line. The problem was also examined with the finite element approach, using the ABAQUS™ code. The application of the finite element code ABAQUS™ to modelling the poro-hyperelasticity problem presented in Section 3 is relatively straightforward and adequate calibrations have been performed in the literature to verify the accuracy in terms of the mesh refinement and time-stepping needed to establish unconditional stability of the computational results. The finite element discretization uses 4-node plane strain quadrilateral elements with 12 degrees of freedom per element (for the two displacement com- ponents and the fluid pressure). The displacements and fluid pressure fields are assumed to be bilinear within the element. Altogether 40 elements were used to discretize the one-dimensional region and the finite element mesh used in the computations is shown in Fig. 2. In Fig. 3 the displacement at the upper surface of the one-dimensional column as a function of time is given; it can be observed that, for the hyperelastic materials, the displacement of the upper surface is smaller (in absolute value) than that for the linear elastic material, i.e., the response of the hyperelastic materials appears to be “stiffer” in comparison to the linear elastic material. We also observe that the surface displacement for the neo-Hookean material is larger (in absolute terms) than that for the reduced polynomial material. Fig. 4 shows the fluid pressure evolution at the lower surface of the column (=)X 0 . Although the initial and long term values for the fluid pressure are the same for the linear elastic and hyperelastic materials, we can see that consolidation is more rapid for hyperelastic materials. In addition, the fluid pressure for the neo-Hookean material dissipates at a slower rate than for the reduced polynomial material.

5. Consolidation problem for a poro-hyperelastic sphere

We now consider the spherically symmetric poro-hyperelasticity problem related to the consolidation of a sphere of finite radius with a free draining boundary, which is subjected to a radial stress (Fig. 5). In the field of geomechanics, the sphere consolidation problem has been examined by Mandel (1950), de Josselin de Jong (1953, 1957), Cryer (1963), Gibson et al. (1963, 1989, 1990) and by Selvadurai and Shirazi (2004), within the framework of the classical theory of poroelasticity proposed by Biot (1941) and accounting for the effects of damage of the porous skeleton. More recently, Selvadurai and

Fig. 2. The finite element modelling of the one-dimensional domain. A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 321

Fig. 3. Displacement of the upper surface of a column subjected to a normal stress of À0.3 MPa at the upper surface. Material of the column is either linear elastic, neo-Hookean hyperelastic, or of the reduced polynomial hyperelastic type. The solutions were obtained using a finite difference scheme and the finite element method.

Suvorov (2012, 2014) examined both the sphere and spherical cavity consolidation problems in relation to thermo-por- oelasto-plasticity where elasto-plasticity effects are accommodated through the application of a Cam-Clay theory based on the incremental theory of plasticity (Davis and Selvadurai, 2002; Pietruszczak, 2010). In these developments, attention is focused on evaluation of the response of the sphere, with boundary fluid drainage, to either a radial stress or uniform boundary temperatures. In the current research application, the theory of poro-hyperelasticity presented in Section 2 is applied to examine the spherically symmetric behaviour of the radially compressed sphere. Referring to systems of spherical polar coordinates, we denote the coordinates of a point in the undeformed configuration by (R,,ΘΦ) and the coordinates in the deformed configuration by (r,,θϕ).

Consider a fluid-saturated poro-hyperelastic sphere of radius a0, which is subjected to a normal stress σ0. The de- formation of the sphere can be described by a function ζ (R) such that

rRt=(ζθΘϕΦ,; ) = ; = ()5.1

The deformation gradient matrix applicable to the spherically symmetric problem can be written as

Fig. 4. Fluid pressure at the lower surface of a column subjected to a normal stress of À0.3 MPa at the upper surface. Material of the column is either linear elastic, neo-Hookean hyperelastic, or of the reduced polynomial hyperelastic type. The solutions were obtained using a finite difference scheme and the finite element method. 322 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

Fig. 5. Porous sphere subjected to normal stress of prescribed magnitude.

⎛ ∂ζ ⎞ ⎜ 00⎟ ⎜ ∂R ⎟ ⎜ ζ ⎟ F = ⎜ 00⎟, ⎜ R ⎟ ⎜ ζ ⎟ 00 ⎝ R ⎠ ()5.2 and the Jacobian is J =()∂ζζ2. Also, from (2.5) and (2.4), we have ∂RR

⎛ ∂ζ ⎞ ⎜ 00⎟ ⎜ ∂R ⎟ ⎛ ∂ζζ⎞−−1/3⎛ ⎞ 2/3⎜ ζ ⎟ ¯ ⎜⎟ ⎜⎟ F = ⎝ ⎠ ⎝ ⎠ ⎜ 00⎟, ∂RR⎜ R ⎟ ⎜ ζ ⎟ 00 ⎝ R ⎠ ()5.3 and

⎛ ⎞ ⎛ ∂ζ ⎞2 ⎜ ⎜⎟ 00⎟ ⎜ ⎝ ∂R ⎠ ⎟ ⎜ ⎟ −−2/3 4/3 2 T ⎛ ∂ζζ⎞ ⎛ ⎞ ⎛ ⎞ ¯ ⎜⎟ ⎜⎟ ⎜ ζ ⎟ BFF= ̅ ̅ = ⎝ ⎠ ⎝ ⎠ 00⎜⎟ . ∂RR⎜ ⎝ R ⎠ ⎟ ⎜ ⎟ ⎜ ⎛ ζ ⎞2⎟ ⎜ 00⎜⎟⎟ ⎝ ⎝ R ⎠ ⎠ ()5.4

¯ ¯ The first invariant I1 of the tensor B can be found as

⎛ ⎞ ⎛ ∂ζζ⎞−−2/3⎛ ⎞ 4/3 ⎛ ∂ ζ⎞22⎛ ζ⎞ ¯ ⎜⎟ ⎜⎟⎜ ⎜⎟ ⎜⎟⎟ I1 = ⎝ ⎠ ⎝ ⎠ ⎜ ⎝ ⎠ + 2.⎝ ⎠ ⎟ ∂RR⎝ ∂ R R⎠ ()5.5

Deviator of the tensor B¯ can be found as A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 323

⎛ ⎞ ⎛ ∂ζ ⎞2 ⎜ ⎜⎟− κ 00⎟ ⎜ ⎝ ∂R ⎠ ⎟ ⎜ ⎟ ∂ζζ22 ⎛ ∂ζζ⎞−−2/3⎛ ⎞ 4/3 ⎛ ⎞2 + 2 ¯ ⎜⎟ ⎜⎟ ⎜ ζ ⎟ ()∂RR () dev(B)=⎝ ⎠ ⎝ ⎠ 00⎜⎟− κ , κ = ∂RR⎜ ⎝ R ⎠ ⎟ 3 ⎜ ⎟ ⎜ ⎛ ζ ⎞2 ⎟ ⎜ 00⎜⎟− κ⎟ ⎝ ⎝ R ⎠ ⎠ ()5.6

Therefore, for the neo-Hookean material the radial component of the deviatoric effective stress tensor can be expressed from (2.11) in the form ⎛ ⎞ 4 ⎛ ∂ζζ⎞1/3⎛ ⎞−−− 10/3⎛ ∂ ζ⎞ 5/3⎛ ζ⎞ 4/3 ⎜ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎟ sCrr′ = 10 ⎝ ⎠ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ . 3 ⎝ ∂RR∂ R R⎠ ()5.7

The effective pressure takes the form ⎛ ⎞ ζζ⎛ ⎞2 2 ⎜ ∂ ⎜⎟ ⎟ p′= − ⎜ ⎝ ⎠ − 1.⎟ DRR1 ⎝ ∂ ⎠ ()5.8

Therefore, the radial effective stress is found using the decomposition (2.2) as

⎛ ⎞ ⎛ 2 ⎞ 4 ⎛ ∂ζζ⎞1/3⎛ ⎞−−− 10/3⎛ ∂ ζ⎞ 5/3⎛ ζ⎞ 4/3 2 ∂ ζζ⎛ ⎞ ⎜ ⎜ ⎟ ⎜⎟ ⎜ ⎟ ⎜⎟⎟ ⎜ ⎜⎟ ⎟ σrr′ = C10 ⎝ ⎠ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ + ⎜ ⎝ ⎠ − 1.⎟ 3 ⎝ ∂RR∂ R R⎠ DRR1 ⎝ ∂ ⎠ ()5.9

Similarly, the circumferential effective stress can be found as

⎛ ⎞ ⎛ 2 ⎞ 2 ⎛ ∂ζζ⎞−−5/3⎛ ⎞ 4/3⎛ ∂ ζζ⎞ 1/3⎛ ⎞ − 10/3 2 ∂ ζζ⎛ ⎞ ⎜ ⎜ ⎟ ⎜⎟ ⎜ ⎟ ⎜⎟⎟ ⎜ ⎜⎟ ⎟ σθθ′ = C10 ⎝ ⎠ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ + ⎜ ⎝ ⎠ − 1.⎟ 3 ⎝ ∂RR∂ RR⎠ DRR1 ⎝ ∂ ⎠ ()5.10

The single equation of equilibrium in the radial direction requires that

∂σrr 2 +(σσrr −θθ )=0. ∂rr ()5.11

As before, it is more convenient to use a Lagrangian description and take the initial coordinate R of a material point as an independent variable instead of the coordinate r. Using the relation ∂r = ∂ζ and familiar chain rule for differentiation of a ∂RR∂ composite function, and also the definition of effective stress given by (2.1), we can rewrite (5.11) as

∂σrr′ 2 ∂ζ ∂p + ()σσrr′ − θθ′ = . ∂RRζ ∂ ∂R ()5.12

From the expression for the radial effective stress (5.9) we can show that

⎛ ⎞ 2 ∂σ′ 4 1 ⎛ ∂ζζ⎞−−2/3⎛ ⎞ 10/35 ⎛ ∂ ζζ⎞ −− 8/3⎛ ⎞ 4/3 ∂2 ζ2 ∂2 ζζ⎛ ⎞ rr = C ⎜ ⎜ ⎟ ⎜⎟+ ⎜ ⎟ ⎜⎟⎟ + ⎜⎟+ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 2 ⎝ ⎠ ∂R 3 ⎝ 3 ∂RR3 ∂ RR⎠ ∂R D1 ∂R R ⎛ ⎞ 4 10 ⎛ ∂ζ⎞1/3⎛ ζ⎞−−− 13/34 ⎛ ∂ ζ⎞ 5/3⎛ ζ⎞ 7/3 14⎛ ∂ ζζ⎞ ζζζζ∂ ⎛ ∂ ⎞ +−C ⎜ ⎜⎟⎜⎟+ ⎜⎟ ⎜⎟⎟ ⎜−+ ⎟ ⎜− ⎟ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 ⎝ ⎠ 3 ⎝ 3 ∂RR3 ∂ R R⎠ RRRD∂ 1 R ∂RR∂ R ()5.13

Therefore, the equilibrium Eq. (5.12) can be written as

⎛ ⎞ 2 4 1 ⎛ ∂ζζ⎞−−2/3⎛ ⎞ 10/35 ⎛ ∂ ζζ⎞ −− 8/3⎛ ⎞ 4/3 ∂2 ζ2 ∂2 ζζ⎛ ⎞ C ⎜ ⎜ ⎟ ⎜⎟+ ⎜ ⎟ ⎜⎟⎟ + ⎜⎟+ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 2 ⎝ ⎠ 3 ⎝ 3 ∂RR3 ∂ RR⎠ ∂R D1 ∂R R ⎛ ⎞ 4 10 ⎛ ∂ζ⎞1/3⎛ ζ⎞−−− 13/34 ⎛ ∂ ζ⎞ 5/3⎛ ζ⎞ 7/3 14⎛ ∂ ζζ⎞ ∂ ζζζζ⎛ ∂ ⎞ +−C ⎜ ⎜⎟⎜⎟+ ⎜⎟ ⎜⎟⎟ ⎜−+ ⎟ ⎜−+ ⎟ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 ⎝ ⎠ 3 ⎝ 3 ∂RR3 ∂ R R⎠ RRRDR∂ 1 ∂ R ∂RR ⎛ ⎞ 4C ⎛ ∂ζζ⎞4/3⎛ ⎞−−− 10/3⎛ ∂ ζ⎞ 2/3⎛ ζ⎞ 4/3 ∂p 10 ⎜ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎟ + ⎝ ⎠ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ = ζ ⎝ ∂RR∂ R R⎠ ∂R ()5.14

We now define the displacement function ρ(Rt, ) in the form

ζρ()=+()Rt,, R Rt ()5.15

Considering Darcy's law (2.14) applicable to problems of spherical symmetry we obtain, using (2.13), 324 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

kp∂ dζρ∂ = . η ∂R dR∂ t ()5.16

Thus, the equilibrium Eq. (5.14) can be rewritten as

⎛ ⎞ 2 4 1 ⎛ ∂ζζ⎞−−2/3⎛ ⎞ 10/35 ⎛ ∂ ζζ⎞ −− 8/3⎛ ⎞ 4/3 ∂2 ζ2 ∂2 ζζ⎛ ⎞ C ⎜ ⎜ ⎟ ⎜⎟+ ⎜ ⎟ ⎜⎟⎟ + ⎜⎟+ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 2 ⎝ ⎠ 3 ⎝ 3 ∂RR3 ∂ RR⎠ ∂R D1 ∂R R ⎛ ⎞ 4 10 ⎛ ∂ζζ⎞1/3⎛ ⎞−−− 13/34 ⎛ ∂ ζ⎞ 5/3⎛ ζ⎞ 7/3 14⎛ ∂ ζζ⎞ ∂ ζζζζ⎛ ∂ ⎞ +−C ⎜ ⎜⎟⎜⎟+ ⎜⎟ ⎜⎟⎟ ⎜−+ ⎟ ⎜−+ ⎟ 10 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 ⎝ ⎠ 3 ⎝ 3 ∂RR3 ∂ R R⎠ RRRDR∂ 1 ∂ R ∂RR ⎛ ⎞ 41C ⎛ ∂ζζ⎞4/3⎛ ⎞−−− 10/3⎛ ∂ ζ⎞ 2/3⎛ ζ⎞ 4/3 ∂ζρ∂ 10 ⎜ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎟ + ⎝ ⎠ ⎝ ⎠ − ⎝ ⎠ ⎝ ⎠ = ζ ⎝ ∂RR∂ R R⎠ () kRt/η ∂ ∂ ()5.17

where ∂ζ =+1 ∂ρ and ζ =+1 ρ . Eq. (5.17) must be solved for the unknown displacement function ρ(Rt, ). ∂RR∂ RR After obtaining the displacement field ρ(Rt, ), the fluid pressure in the sphere can be found by integrating (5.16); i.e.

1 a0 dζρ∂ pR()=−, t ∫ dR ()k/η R dR∂ t ()5.18

In particular, the fluid pressure in the center of the sphere can be found as

1 a0 dζρ∂ pt()=−0, ∫ dR. ()k/η 0 dR∂ t ()5.19

5.1. Limiting cases

(i) In the case when the deformations of the poro-hyperelastic medium are small,

∂ζρζρ∂ =+11;11≈=+≈ ∂RRRR∂

⎛ ∂ζζ⎞4/3⎛ ⎞−−− 10/3⎛ ∂ ζ⎞ 2/3⎛ ζ⎞ 4/3 ⎛ ∂ ρρ⎞ ⎛ ∂ ζζ⎞ ⎜⎟⎜⎟− ⎜⎟ ⎜⎟≈ 22 ⎜−= ⎟ ⎜− ⎟ ⎝ ∂R⎠ ⎝ R⎠ ⎝ ∂ R⎠ ⎝ R⎠ ⎝ ∂ RR⎠ ⎝ ∂ RR⎠

and (5.17) reduces to ⎛ ⎛ ⎞⎞ ∂ρ k ⎛ 8 22⎞ ∂2ρρρ∂ =+⎜ C ⎟⎜ + ⎜ − ⎟⎟, ⎝ 10 ⎠ 2 ∂t η 3 D1 ⎝ ∂R RR⎝ ∂ R⎠⎠ ()5.20

which can be derived by considering classical Biot poroelasticity of a fluid-saturated porous medium. Using definitions of the elastic constants (2.10), the Eq. (5.20) can be rewritten as ⎛ ⎛ ⎞⎞ ∂ρρρρ∂2 2 ∂ kE(−)1 ν c ⎜ ⎜ ⎟⎟, c = ⎜ 2 + −=⎟ ∂t ⎝ ∂R RR⎝ ∂ R⎠⎠ η (+)(−112νν ) ()5.21

where c is a constant of consolidation.

(ii) In the case when the poro-hyperelastic material is impermeable, k → 0 and the deformations of the poro-hyperelastic material will be time-independent.

(iii) In the case when the porous skeleton is incompressible or v = 1/2, the displacements are governed by the partial differential equation

⎛ ⎞ ∂2ρρρ2 ∂ + ⎜ −=⎟ 0 ∂R2 RR⎝ ∂ R⎠ ()5.22

and the solution of (5.22) can be obtained for a specific initial boundary value problem. A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 325

5.2. The initial boundary value problem

We shall consider the solution of the non-linear partial differential Eq. (5.17) as applied to the spherically symmetric poro-hyperelastic consolidation problem where a sphere is subjected to an axial stress σ0 in the form of a Heaviside step function of time (Fig. 5). During consolidation the fluid is allowed to be expelled from the loaded surface. The boundary conditions governing the displacement field ρ(Rt, ) in the spherically symmetric poro-hyperelastic pro- blem are as follows:

()i,for0σσrr (Rat =00 )= ∀ t ≥ ()5.23 ()(iiρ Rt = 0, )= 0 for ∀ t ≥ 0 ()5.24

The initial condition governing the displacement field ρ(Rt, ) in the spherically symmetric poro-hyperelastic problem is:

‘()( iiiρ RRa , 0 )= 0 for all ∈( 0,0 ) . ()5.25

The boundary conditions governing the pore pressure field p(Rt, ) in the spherically symmetric poro-hyperelastic pro- blem are:

()(ivpa0 , t )= 0 for ∀ t > 0 ()5.26 ⎛ ⎞ ⎜⎟∂p ()v0for0⎝ ⎠ =∀>t ∂R R=0 ()5.27

The initial condition governing the pore pressure field p(Rt, ) in the spherically symmetric poro-hyperelastic problem is

()v,0pR ( )=−σ00 forall0,. R ∈( a ) ()5.28

Since the fluid pressure is zero on the external surface, the effective stress is equal to the total stress, i.e.,

σσrr′ ()=at00,, ()5.29 and therefore we require that on Ra= 0 ⎛ ⎞ 4 ⎛ dζζ⎞1/3⎛ ⎞−10/3 ⎛ d ζ⎞−5/3⎛ ζ⎞−4/3 2 d ζζ⎛ ⎞2 ⎜ ⎜ ⎟ ⎜⎟⎜ ⎟ ⎜⎟⎟ ⎜⎟ C10 ⎜ ⎝ ⎠ ⎝ ⎠ −+(−)=⎝ ⎠ ⎝ ⎠ ⎟ ⎝ ⎠ 1,σ0 3 ⎝ dR R dR R⎠ D1 dR R ()5.30 where we have used (5.9). The traction boundary condition (5.29) allows us to find the first spatial derivative of the radial ∂ρ displacement at the surface of the sphere Ra= 0, once the radial displacement of the surface ρ(=Ra0) is known or can be ∂R approximated with a value from the previous time step. The finite difference solution of the initial boundary value problem posed by (5.17) with boundary and initial conditions (5.23) to (5.25) can be obtained in a similar way to the one-dimensional consolidation problem for a column, discussed in Section 4.1. The replacements of variables that should be made in Section

4.1 are straightforward: XR→ , La→ 0, u → ρ.

6. Numerical solution of the initial boundary value problem for a sphere

The suitability of the finite difference approach to solve the poro-hyperelastic sphere problem described in Section 5 is illustrated through an application to a specific problem. We consider a sphere of radius a0 = 1 m and analyze two cases for the initial values of the elastic constants: 1)=E 600000 Pa,ν = 0.3, 2)=E 600000 Pa,ν = 0.1,

As we see, case 2 corresponds to the smaller value of Poisson's ratio. Consequently, initial values for the shear and bulk moduli are 1)=GK230769 Pa, = 500000 Pa,

2)=GK272727 Pa, = 250000 Pa, and thus, −6 1)=CD10115384 Pa, 1 =× 4 10 1/Pa,

−6 2)=CD10136363 Pa, 1 =× 8 10 1/Pa.

For both cases, the permeability is equal to

k =×310m,−14 2 326 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

a0=1 m

Fig. 6. The finite element modelling of the spherical domain. and the viscosity η = 0.001 Pa s. The applied normal stress on the surface of the sphere is different for cases 1 and 2:

1)=−σ0 300000 Pa,

2)=−σ0 200000 Pa.

The applied load for case 2 is chosen as smaller (in absolute value) than the applied load for case 1. This choice will be justified later when we show that there is a limit on the maximum load that can be applied to the hyperelastic sphere and, in particular, for the given sphere with Poisson's ratio equal to 0.1 the stress equal to −300000 Pa cannot be applied. As before, the general purpose finite element code ABAQUS™ was used to develop computational results for the poro- hyperelasticity problem for the sphere. Due to the spherical symmetry of the problem it is possible to use an axisymmetric formulation and model only one-quarter of a plane cross-section of the sphere that passes through the center. The finite element discretization uses 4-node axisymmetric quadrilateral elements with 12 degrees of freedom per element. The displacements and fluid pressure fields are assumed to be bilinear within the element. 364 elements were used to discretize the region and this is illustrated in Fig. 6. Three types of material behaviour are considered: a linear elastic material (LE) and two nonlinear elastic materials: a neo-Hookean hyperelastic material (NH) and a material with Hooke's relationship between the Cauchy's stress and loga- rithmic strains (G). In the latter case, the skeletal behaviour of classical poroelasticity is used and the elastic constitutive

Fig. 7. Displacement of the surface of a sphere subjected to a normal stress of À0.3 MPa. Material of the sphere is either linear elastic (LE), neo-Hookean hyperelastic (NH), or nonlinear elastic with Hooke's constitutive behaviour (G). Poisson's ratio is 0.3. The solutions were obtained using a finite difference scheme (solid line) and the finite element method (dots). A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 327 relationship for the porous skeleton takes the form (Gibson et al., 1989, 1990)

⎧ ′ ⎨ σεrr =(+)+(−)KG4/3rr 2 KG 2/3 εθθ ⎪ ; ⎩ σεεθθ′ =(KG −2/3 )rr + 2 ( KG + /3 ) θθ ⎛ ⎞ ⎛ ⎞ ⎜⎟∂ζ ⎜⎟ζ εrr = ln ;εθθ = ln ⎝ ∂RR⎠ ⎝ ⎠ ()5.31

Initial values of the elastic moduli and the permeability are the same for all types of material behaviour. In particular, the values for the shear modulus and the bulk modulus are 1)=GK230769 Pa, = 500000 Pa,

2)=GK272727 Pa, = 250000 Pa. for the cases 1 and 2, respectively. For the linear elastic material the strains are assumed to be small whereas in the case of the nonlinear elastic materials the strains are assumed to be finite. Fig. 7 shows the radial displacement of the surface of the sphere for all three types of material behaviour, with Poisson's ratio equal to 0.3 (case 1). A solution is obtained using the finite difference method (solid line in Fig. 7) and the finite element method (dotted line, Fig. 7). In the finite difference method the number of segments N, into which the sphere's radius is subdivided, was chosen equal to 10, and the time step is Δt = 12.5 sec. We can see that the surface displacement for the hyperelastic neo-Hookean material (NH) is considerably larger (in absolute value) than that for the two other materials, i.e., the response of the hyperelastic material appears “more com- pliant” in comparison to the linear elastic material and nonlinear elastic material with Hooke's constitutive behaviour. We also observe that the surface displacement for the linear elastic material is somewhat larger (in absolute value) than that for the nonlinear elastic material G, and the response of the nonlinear elastic material with Hooke's constitutive behaviour is the stiffest. Fig. 8 shows the evolution of the fluid pressure at the center of the sphere. As in Fig. 7 the value of Poisson's ratio is 0.3 (case 1). For all materials, the initial value of the fluid pressure is equal to the applied stress (in absolute value), i.e., 0.3 MPa, and the long term value of the fluid pressure is zero. Although the initial and long term values for the fluid pressure are the same for all the materials, we can see that consolidation is the swiftest for the case of a nonlinear elastic material with Hooke's constitutive behaviour (G). This can be explained by its stiff response, shown in Fig. 7, and by taking into account the decrease in the length of the drainage path during consolidation. Nevertheless, we can see that the fluid pressure for the neo-Hookean hyperelastic material (NH) dissipates at the slowest rate. The increase in the fluid pressure observed in the short term (the so-called Mandel-Cryer effect) is about the same for the linear elastic material (LE) and the nonlinear elastic material with Hooke's constitutive behaviour (G). This result was also obtained by Gibson et al. (1989, 1990). However, the Mandel-Cryer effect is obviously stronger for the neo-Hookean hy- perelastic material (NH), i.e., the initial increase in the fluid pressure for the neo-Hookean hyperelastic material is larger than that for the nonlinear elastic material with Hooke's constitutive behaviour. Moreover, unlike the linear elastic material for which the relative increase in the fluid pressure observed initially does not depend on the magnitude of the applied stress, for the nonlinear elastic materials the Mandel-Cryer effect becomes stronger as the applied stress increases.

Fig. 8. Fluid pressure at the center of a sphere subjected to a normal stress of À0.3 MPa. Material of the sphere is either linear elastic (LE), neo-Hookean hyperelastic (NH), or nonlinear elastic with Hooke's constitutive behaviour (G). Poisson's ratio is 0.3. The solutions were obtained using a finite difference scheme (solid line) and the finite element method (dots). 328 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

Fig. 9. Displacement of the surface of a sphere subjected to a normal stress of À0.2 MPa. Material of the sphere is either linear elastic (LE), neo-Hookean hyperelastic (NH), or nonlinear elastic with Hooke's constitutive behaviour (G). Poisson's ratio is 0.1. The solutions were obtained using a finite difference scheme (solid line) and the finite element method (dots).

Figs. 9 and 10 show similar results for the surface displacement and fluid pressure at the center of the sphere for the same types of materials, but with Poisson's ratio equal to 0.1 (case 2). The magnitude of the applied stress is now reduced from 0.3 MPa to 0.2 MPa in order to avoid very large displacements of the surface. In the finite difference solution, the number of segments is N = 10 and the time step is Δt = 3.125 sec. A comparison of Figs. 9 and 7 clearly shows that when the value of Poisson's ratio is decreased the surface displacement can become larger even if the magnitude of the applied stress is reduced. This is especially evident for the neo-Hookean hyperelastic material (NH). This suggests that the response of materials with smaller Poisson's ratio is more compliant and results in a larger displacement of the surface. In Fig. 10 we observe the Mandel-Cryer effect, i.e., an increase in the fluid pressure in the short term. As in Fig. 8, the magnitude of the fluid pressure increase is about the same for the linear elastic material (LE) and nonlinear elastic material with Hooke's constitutive behaviour (G). However, the Mandel-Cryer effect for the neo-Hookean hyperelastic material is stronger. Also, a careful comparison of Figs. 10 and 8 reveals that the relative increase in the fluid pressure observed initially is stronger for smaller values of Poisson's ratio. Finally, we examine the response of the hyperelastic sphere in fully drained condition, i.e., when the applied load is applied very slowly and the fluid pressure is very small or equal to zero. In this condition, the equilibrium can be achieved when the radial and circumferential stresses are equal to each other. Equating (5.9) to (5.10) and using the fact that for the fully drained condition

Fig. 10. Fluid pressure at the center of a sphere subjected to a normal stress of À0.2 MPa. Material of the sphere is either linear elastic (LE), neo-Hookean hyperelastic (NH), or nonlinear elastic with Hooke's constitutive behaviour (G). Poisson's ratio is 0.1. The solutions were obtained using a finite difference scheme (solid line) and the finite element method (dots). A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333 329

Fig. 11. Radial stress as a function of surface displacement in the sphere under fully drained condition. Material of the sphere is linear elastic (LE) or neo- Hookean hyperelastic (NH). Poisson's ratio is 0.1 or 0.3.

∂ζζ ρ ==+1, ∂RR R ()5.32 we obtain from (5.9) the radial stress in the hyperelastic sphere as ⎛ ⎞ 2 ⎛ ρ ⎞3 ⎜ ⎜⎟⎟ σrr =+−⎜ ⎝ 11,⎠ ⎟ DR1 ⎝ ⎠ ()5.33 where ρ is the radial displacement. For the linear elastic sphere the last relation is reduced to ρ σrr==33.K Kε rr R ()5.34 The radial stress as a function of the surface displacement for the linear elastic (LE) and hyperelastic Neo-Hookean (NH) spheres of radius 1 m is shown in Fig. 11. Maximum surface displacement is equal to À1 m, which corresponds to a maximum possible compression of the sphere. We observe that the response of the hyperelastic sphere is significantly softer than that of the linear elastic sphere. In addition, it is clear that the hyperelastic sphere experiences softening and there is a max maximum stress that can be applied to the hyperelastic sphere. This maximum stress is equal to σrr =−2/D1 =−K as it follows from (5.33). Thus, for the sphere with Poisson's ratio equal to 0.1, the maximum stress is equal to −250000 Pa. For the stresses that exceed the maximum stress, the solution does not exist. The response of this material could be made stiffer if the energy U had an infinite energy penalization when the Jacobian J-0. Hence, the strain energy given by (2.9) should be used with caution in situations where extreme compaction is involved.

7. Concluding remarks

The topic of poro-hyperelasticity has a wide range of applications ranging from the study of fluid-saturated geological materials to applications involving biomechanics. Alternatively, the topic can be approached using mixture theory concepts. In the classical linear theory of poroelasticity, the deformability and fluid transport characteristics of the material are as- sumed to remain constant. In a poroelasticity theory that accounts for hyperelastic effects, the fluid transport properties and the fluid-solid stress partitioning can be influenced by large strain phenomena. Such evolving properties must first be determined by recourse to experiments and the implementation of variability in the fluid-solid stress partitioning and fluid transport characteristics are most conveniently handled through the use of computational approaches. The accuracy of the computational approaches, however, needs to be verified through comparisons with analytical solutions derived for poro- hyperelastic responses that remain unchanged during large strains. Such fully developed benchmarking problems are rare and the examples presented in the paper can be used for such purposes. The analytical solutions are developed for specific forms of strain energy functions; the one-dimensional nature of the problem lends itself to simplification of the poro- hyperelasticity problem to non-linear partial differential equations of the hyperbolic type that can be solved through conventional finite difference schemes. The solutions developed in this study match very closely the results obtained from finite element techniques. 330 A.P.S. Selvadurai, A.P. Suvorov / J. Mech. Phys. Solids 91 (2016) 311–333

Acknowledgment

The authors are indebted to a reviewer for comments that led to significant improvements in the presentation. We have attempted to provide a balanced perspective of the literature applicable to both hyperelasticity and poromechanics. These fields contain substantial literature and many of the articles cited in this paper also contain extensive references to historical and current developments. The work presented in the paper was supported by a Discovery Grant awarded by the Natural Sciences and Engineering Research Council of Canada and through the support provided by the James McGill Research Chair, awarded to the first author.

Appendix A

Derivation of the mass conservation law (2.13) for a porous material with incompressible fluid and solid phases is presented here. First we define the material time derivative operator of a scalar function f Dfα ∂f = +⋅vα grad f , Dt ∂t ()A.1 where α stands for either the solid phase, α = s, or the fluid phase, α = f . Using this operator, the mass balance equation for the solid phase can be written as Ds (−)1 φ +(10, −φ )div vs = Dt ()A.2 if the solid phase is incompressible. Similarly, for the incompressible fluid phase we can write

D f φ +=φ div vf 0. Dt ()A.3 But it can be shown from the definition (A.1) that

D fsφφD =+(−)⋅vvfsgrad φ Dt Dt ()A.4 Therefore, (A.3) can be written as Dsφ +(vvfs − )⋅gradφφ + div ( vv fs − )+ ϕ div v s =0 Dt or Dsφ ++{(−)}=φφdivvvvsfs div 0. Dt ()A.5 Now adding Eqs. (A.2) and (A.5) we obtain

divvvvsfs=− div {(ϕ − )}. ()A.6

But ∂u vs = , ∂t ()A.7 where u is the displacement vector. Therefore, ∂u div =−div {(φ vvfs − )}, ∂t ()A.8 which is the required result (2.13).

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