ORIGINAL PAPER

L. Sanavia Æ F. Pesavento Æ B. A. Schrefler

Finite element analysis of non-isothermal multiphase geomaterialswith application to strain localization simulation

Received: 28 May 2004 / Accepted: 3 February 2005 / Published online: 31 May 2005� Springer-Verlag 2005

Abstract Finite element analysis of non-isothermal el-asto-plastic multiphase geomaterials is presented. Themultiphase material is modelled as a deforming porouscontinuum where heat, water and gas flow are taken intoaccount. The independent variables are the solid dis-placements, the capillary and the gas pressure and thetemperature. The modified effective stress state is limitedby the Drucker-Prager yield surface for simplicity. Smallstrains and quasi-static loading conditions are assumed.Numerical results of strain localization in globally un-drained samples of dense, medium dense and loose sandsand isochoric geomaterial are presented. A biaxialcompression test is simulated assuming plane straincondition during the computations. Vapour pressurebelow the saturation water pressure (cavitation) devel-ops at localization in case of dense sands, as experi-mentally observed. A case of strain localization inducedby a thermal load where evaporation takes place is alsoanalysed.

Keywords Strain localization Æ Multiphase porousmaterials Æ Cavitation Æ Elasto-plasticity Æ Finiteelements

Nomenclature

Cp effective specific heat of porous medium[Jkg)1K)1]

Cgp specific heat of gas mixture [Jkg)1K)1]

Cwp specific heat of liquid phase [Jkg)1K)1]

(qCp)eff

effective thermal capacity of porous medium[JL)3K)1]

Dgwg effective diffusivity tensor of water vapour in dry

air [m2s)1]

Dgag effective diffusivity tensor of dry air in water

vapour [m2s)1]e emissivity of the interface [)]g gravity acceleration [ms)2]k intrinsic permeability tensor [m2]krp

relative permeability of p-phase (p=ga, gw) [)]Ma molar mass of dry air [kg kmol)1]Mw molar mass of water [kg kmol)1]Mg molar mass of gas phase [kg kmol)1]_mvap rate of mass due to phase change [kgm)3 s)1]n total porosity (pore volume/total volume) [)]pc

capillary pressure [Pa]pg

gas pressure [Pa]pw

liquid water pressure [Pa]pga

dry air partial pressure [Pa]pgw

water vapour partial pressure [Pa]pgws

saturated water vapour pressure [Pa]q stress-like internal variable [Pa]R gas constant (8314.41Jkmol)1K)1)RH relative humidity [-]Sw liquid phase volumetric saturation (liquid vol-

ume/pore volume) [)]Sg gas phase volumetric saturation (gas volume/

pore volume) [)]T absolute temperature [K]Tcr critical temperature of water [K]t time [s]u displacement vector of solid matrix [m]vps

relative velocity of p-phase (p=g, w) withrespect to the solid phase [ms)1]

vgag diffusionvelocity ofwater vapour in dry air [m s)1]vgag diffusionvelocity of dry air inwater vapour [m s)1]1 second order identity tensor [)]

Greek symbols

ac convective heat exchange coefficient[Wm)2K)1]

bc convective mass exchange coefficient [ms)1]

Comput Mech (2006) 37: 331–348DOI 10.1007/s00466-005-0673-6

Dedicated to Professor S. Valliappan in occasion of his retirement

L. Sanavia (&) Æ F. Pesavento Æ B. A. SchreflerDepartment of Structural and Transportation Engineering,University of Padua, via F. Marzolo 9,35131 Padova, Italy

bs cubic thermal expansion coefficient of solid[K)1]

bswg combine (solid + liquid + gas) cubic thermalexpansion coefficient [K)1]

bsw combine (solid + liquid) cubic thermalexpansion coefficient [K)1]

bw thermal expansion coefficient of liquid water[K)1]

veff effective thermal conductivity of the porousmedium [Wm)1K)1]

DHvap enthalpy of vaporization per unit mass [Jkg)1]Dt time step [s]ee elastic strain tensor [)]_c continuum consistency parameter [)]lp dynamic viscosity of the constituent p-phase

(p = ga, gw) [lPas]q apparent density of porous medium [kgm)3]qg gas phase density [kgm)3]qw liquid phase density [kgm)3]qs solid phase density [kgm)3]qga mass concentration of dry air in gas phase

[kgm)3]qgw mass concentration of water vapour in gas

phase [kgm)3]r Cauchy stress tensor [Pa]r¢ effective stress tensor [Pa]ro Stefan-Boltzmann constant

[5.670 · 10)8Wm)2K)4]n equivalent plastic strain (strain-like internal

scalar hardening variable) [)]oG=oX Jacobian matrixwc water potential [Jkg)1]

Operators

o=ot partial time derivativegrad gradient operatordiv divergence operator� dyadic product

1 Introduction

In recent years, increasing interest in thermo-hydro-mechanical analysis of saturated and partially saturatedmaterials has been observed, because of its wide spec-trum of engineering applications. Typical examples be-long to environmental geomechanics, where somechallenging problems are of interest for the researchcommunity. A step in the development of a suitablephysical, mathematical and numerical model for thesimulation of geoenvironmental engineering problems ispresented in this work, where applications to soil failure

situations due to strain localization in plane straincompression tests are considered.

In this paper we deal with a mathematical and finiteelement model for non-isothermal elasto-plastic multi-phase geomaterials. To this end, classical elasto-plastic-ity has been introduced in the geometrically linear finiteelement code Comes-Geo [1]. The multiphase material ismodelled as a deforming porous continuum where heat,water and gas flow are taken into account [1], [2], [3], asrecalled in Sect. 2. In particular, the gas phase is mod-elled as an ideal gas composed of dry air and watervapour, which are considered as two miscible species.Phase changes of water (evaporation-condensation,adsorption-desorption) and heat transfer through con-duction and convection, as well as latent heat transferare considered. The primary variables are the solid dis-placements, the capillary and the gas pressure and thetemperature. The governing equations written at mac-roscopic level in Sect. 2 are based on averaging proce-dures (Hybrid Mixture Theory), following the generalThermo-Hydro-Mechanical model developed in [1]. Theelasto-plastic behaviour of the solid skeleton is assumedhomogeneous and isotropic; the effective stress state islimited by a temperature independent Drucker-Prageryield surface for simplicity, with linear isotropic hard-ening and non associated plastic flow, as described inSect. 3. For such yield surface a particular ‘‘apex for-mulation’’ is advocated. The macroscopic balanceequations are discretised in space and time within thefinite element method in Sect. 4. In particular, a Galer-kin procedure is used for the discretisation in space andthe Generalised Trapezoidal Method is used for the timeintegration. Small strains and quasi-static loading con-ditions are assumed. Numerical results of a biaxialcompression test where strain localization develops insamples of initially water saturated dense, medium denseand loose sands and isochoric geomaterial are presentedin Sect. 5. Vapour pressure below the saturation waterpressure (cavitation)1 appears in the shear bands atlocalization in case of dense sands (dilatant material)with globally undrained conditions, as experimentallyobserved in [4] and [5]. This is not the case whenparameters for a loose sand (contractant material) andgranular material exhibiting isochoric plastic flow areselected. In the last example, evaporation of the porewater is simulated in the domain and strain localizationinduced by a thermal load is analyzed. These exampleshave been simulated to emphasize the importance of anon-isothermal multiphase model for the simulation ofthe hydro-thermo-mechanical behavior of saturated/partially saturated soils.

1 Cavitation of water (i.e., rapid phase change at ambient temperature)may occur when the absolute value of the water pressure is equal or lessthan the saturation water pressure at the considered temperature(neglecting the surface tension of the interface of the arising vapourbubbles). For thermodynamic equilibrium state, the vapour pressure atthe interface between saturated and partially saturated porous materialis equal, at T = 20�C, to the saturation value of 2:338; 8 Pa (i.e.,�98:986; 2 Pa, with reference to the atmospheric pressure).

332

The present model improves the isothermal mono-species approach and the isothermal two phase flow modeldeveloped in [6] and [7] to study cavitation observedduring localization in dense geomaterials, as explainedat the end of Sect. 5. This paper follows [8], where theabove mentioned simplified models are critically dis-cussed and analyzes strain localization within a threephase approach ([9] and [10]) taking into account non-isothermal conditions (some preliminary numerical re-sults on strain localization in non-isothermal three phasegeomaterials were presented in [11]).

For the aspects of the regularization properties of themultiphase model at localization, due to the presence ofa Laplacian operator in the mass balance equation of thefluids when Darcy’s law is used, the interested reader cansee [6] and [12]. The internal length scale lw contained inthe model is presented in [13]. Further interesting aspectsof the effectiveness of lw are presented in [14] and [15].However, if mesh-independence in all situations is aimedat, some sort of regularization should be introducedsuch as, e.g., gradient dependence [14]. Here the mainemphasis is given to the fact that for all the above de-scribed numerical examples a single THM model can beused successfully ([15], [16]) and in a straightforwardmanner.

A review of non-isothermal thermo-hygro-mechani-cal models is beyond the scope of this paper; the inter-ested reader can find it in [17] and [18].

2 Macroscopic balance equations

The full mathematical model necessary to simulatethermo-hydro-mechanical transient behaviour of fullyand partially saturated porous media is developed in [1]and [3] using averaging theories following Hassanizadehand Gray [19]–[21]. The underlying physical model isbriefly summarised in the present Sect. for sake ofcompleteness.

The partially saturated porous medium is treated asmultiphase system composed of p ¼ 1; . . . ; k constitu-ents with the voids of the solid skeleton ðsÞ filled withwater ðwÞ and gas ðgÞ. The latter is assumed to behaveas an ideal mixture of two species: dry air (noncon-densable gas, ga) and water vapour (condensable one,gw). Using spatial averaging operators defined over arepresentative elementary volume R.E.V. (of volumedvðx; tÞ in the deformed configuration, Bt � R3, see,Fig. 1, where x is the vector of the spatial coordinatesand t is the current time), the microscopic equationsare integrated over the R.E.V. giving the macroscopicbalance equations [1], [3].

At the macroscopic level the porous media materialis modelled by a substitute continuum of volume Btwith boundary oBt that fills the entire domain simul-taneously, instead of the real fluids and the solidwhich fill only a part of it. In this substitute contin-uum each constituent p has a reduced density which isobtained through the volume fraction gpðx; tÞ ¼ dvp

ðx; tÞ=dvðx; tÞ. In the general model [1] inertial forces,heat conduction, vapour diffusion, heat convection,water flow due to pressure gradients or capillary ef-fects and water phase change (evaporation and con-densation) inside the pores are taken into account.The solid is deformable and non-polar, and the fluids,the solid and the thermal fields are coupled. All fluidsare in contact with the solid phase. The constituentsare assumed to be isotropic, homogeneous, immiscibleexcept for dry air and vapour, and chemically nonreacting. Local thermal equilibrium between solidmatrix, gas and liquid phases is assumed, so that thetemperature is the same for all the constituents. Thestate of the medium is described by capillary pressurepc, gas pressure pg, absolute temperature T and dis-placements of the solid matrix u. In the partially sat-urated zones water is separated from its vapour by aconcave meniscus (capillary water). Due to the cur-vature of this meniscus the sorption equilibriumequation (e.g., [22]) gives the relationship between thecapillary pressure pcðx; tÞ and the gas pgðx; tÞ andwater pressure pwðx; tÞ [22]pc ¼ pg � pw ð1ÞPore pressure is defined as compressive positive for thefluids, while stress is defined as tension positive for thesolid phase. For a detailed discussion about the chosenprimary variables see, Appendix 1.

Moreover, in multiphase materials theory it iscommon to assume the motion of the solid as a ref-erence and to describe the fluids in terms of motionrelative to the solid. This means that a fluids relativevelocity with respect to the solid is introduced. Thefluids relative velocity vpsðx; tÞ or diffusion velocity isgiven by

vpsðx; tÞ ¼ vpðx; tÞ � vsðx; tÞ with p ¼ g;w ð2Þand will be described by the Darcy law.

The macroscopic balance equations of the imple-mented model are now summarized. These equations areobtained introducing the following assumptions in themodel developed in [1]

Fig. 1 Typical averaging volume dvðx; tÞ of a porous mediumconsisting of three constituents [1]

333

� at the micro level, the porous medium is assumed tobe constituted of incompressible solid and waterconstituents, while gas is considered compressible;

� the process is considered as quasi-static and developedin the geometrically linear framework.

2.1 Linear momentum balance equations

The linear momentum balance equation of the mixturein terms of total Cauchy stress rðx; tÞ assumes the form

div rþ qg ¼ 0 ð3Þwhere qðx; tÞ is the density of the mixture,

q ¼ ½1� n�qs þ nSwqw þ nSgqg ð4Þ

with nðx; tÞ the porosity and Swðx; tÞ and Sgðx; tÞ thewater and gas degree of saturation, respectively. Thetotal Cauchy stress can be decomposed into the effectiveand pressure (equilibrium) parts following the principleof effective stress

r ¼ r0 � pg � Swpc½ �1 ð5Þwhere r0ðx; tÞ is the modified effective Cauchy stresstensor and 1 is the second order identity tensor. Thisform using saturation as weighting functions for thepartial pressures was first introduced in [23] using vol-ume averaging (see, also [24] and [25]) and is thermo-dynamically consistent ([22], [40] and recently also [9]).The form of Eq. (5) assumes the grain incompressible (asopposed to the skeleton), which is a common assump-tion in soil mechanics. (To account for compressiblegrains, the Biot coefficient a should appear in front ofthe solid pressure. This becomes important in rock andconcrete).

2.2 Mass balance equations

The mass conservation equation for the solid skeleton,the water and the vapour is

n qw� qgw½ � oSw

oToTotþ oSw

opc

opc

ot

� �

þ qwSw þ qgw 1� Sw½ �½ �div ou

ot

� �

þ 1� Sw½ �n oqgw

oToTotþ oqgw

opc

opc

ot

� �

� div qg MaMw

M2g

Dgwg grad

opgw

opc

� � !

þ div qw kkrw

lw�grad pgð Þ þ grad pcð Þ þ qwg½ �

� �

þ div qgw kkrg

lg�grad pgð Þ þ qgg½ �

� �� bswg

oTot¼ 0 ð6Þ

where, in particular, kðx; tÞ is the intrinsic permeabilitytensor, krwðx; tÞ the water relative permeability, lwðx; tÞ

the water viscosity and bswg ¼ bs½1� n� Sgqgw þ Swqw� �

þnbwSwqw. The inflow and outflow fluxes have been de-scribed using the Fick law (13) for the diffusion of thevapour in the gas phase and by the Darcy law for thewater and gas flows.

Similarly, the mass balance equation for the dry air is

� nqga oSw

oToTotþ oSw

opc

opc

ot

� �

� bsqga½1� n� 1� Sw½ � oT

ot

þ 1� Sw½ �qga divou

ot

� �

þ ½1� Sw�noqga

oToTotþ oqga

opc

opc

otþ oqga

opg

opg

ot

� �

� div qg MaMw

M2g

Dgag grad

pga

pg

� � !

þ div qga kkrg

lg�grad pgð Þ þ qgg½ �

� �¼ 0 ð7Þ

The quantities Swðx; tÞ, Sgðx; tÞ, krwðx; tÞ and krgðx; tÞ aredefined at the constitutive level, as described in Sect. 3.

2.3 Energy balance equations

The energy balance equation of the mixture is

qCp� �

eff

oTotþ qwCw

p

"kkrw

lw

�� grad pgð Þ

þ grad pcð Þ þ qwg�#� grad Tð Þ

þ qgCgpkkrg

lg�grad pgð Þ þ qgg½ �

� �� grad Tð Þ

� div veff grad Tð Þ� �

¼ � _mvapDHvap ð8Þwhere, in particular, _mvapDHvap considers the contribu-tion of the evaporation and condensation. This balanceequation takes into account the heat transfer throughconduction and convection as well as latent heat transfer(see, [1]) and neglects the terms related to the mechanicalwork induced by density variations due to temperaturechanges of the phases and induced by volume fractionchanges. (A more general balance equation is developedin [18]).

3. Constitutive equations

The pressure pgðx; tÞ is given in the sequel. For a gaseousmixture of dry air and water vapour, the ideal gas law isintroduced because the moist air is assumed to be aperfect mixture of two ideal gases. The equation of stateof perfect gas (the Clapeyron equation) and Dalton’slaw applied to dry air ðgaÞ, water vapour ðgwÞ and moistair ðgÞ, yields

334

pga ¼ qgaRT =Ma pgw ¼ qgwRT=Mw ð9Þ

pg ¼ pga þ pgw qg ¼ qga þ qgw ð10ÞIn the partially saturated zones, the equilibrium watervapour pressure pgwðx; tÞ can be obtained from theKelvin-Laplace equation

pgw ¼ pgwsðT Þ exp � pcMw

qwRT

� �ð11Þ

where the water vapour saturation pressure pgwsðx; tÞ,depending only upon the temperature T ðx; tÞ, can becalculated from the Clausius-Clapeyron equation orfrom an empirical correlation. The saturation Spðx; tÞand the relative permeability krpðx; tÞ are experimentallydetermined function of the capillary pressure pc and thetemperature T

Sp ¼ Sp pc; Tð Þ; krp ¼ krp pc; Tð Þ; p ¼ w; g ð12ÞFor the binary gas mixture of dry air and water vapour,Fick’s law gives the following relative velocitiesvp

g ¼ vp � vg (p ¼ ga; gw) of the diffusing species

vgag ¼ �

MaMw

M2g

Dgag grad

pga

pg

� �

¼ MaMw

M2g

Dgag grad

pgw

pg

� �¼ �vgw

g ð13Þ

where Dgag is the effective diffusivity tensor and Mg is the

molar mass of the gas mixture

1

Mg¼ qgw

qg

1

Mwþ qga

qg

1

Mað14Þ

The elasto-plastic behaviour of the solid skeleton is as-sumed to be described within the classical rate-inde-pendent elasto-plasticity theory for geometrically linearproblems. The yield function restricting the effectivestress state r0ðx; tÞ is developed in the form of Drucker-Prager for simplicity, to take into account the dilatant/contractant behaviour of dense or loose sands, respec-tively. The return mapping and the consistent tangentoperator is developed, solving the singular behaviour ofthe Drucker-Prager yield surface in the zone of the apexusing the concept of multisurface plasticity, following anidea suggested in [26] in case of perfect plasticity anddeviatoric non-associative plasticity. The return map-ping algorithm and the consistent tangent moduli usedfor the numerical simulations are developed in [27] forisotropic linear hardening/softening and volumetric-de-viatoric non-associative plasticity in case of large strainelasto-plasticity. Here, the geometrically linear case isconsidered.

The mechanical behaviour of the solid skeleton isassumed to be governed by the Helmholtz free energy wfunction in the form

w ¼ w ee; nð Þ ð15Þdependent on the small elastic strain tensor, eeðx; tÞ, andthe internal strain-like scalar hardening variable, nðx; tÞ,

i.e., the equivalent plastic strain. The second law ofthermodynamic yields, under the restriction of isotropy,the constitutive relations

r0 ¼ owoee

; q ¼ � owon

ð16Þ

and the remaining dissipation inequality

r0 : _ee � q _n � 0 ð17Þwhere qðx; tÞ is the stress-like internal variable account-ing for the evolution of the yield locus in the stress space.The evolution equations for the rate terms of the dissi-pation inequality (17) can be derived from the postulateof the maximum plastic dissipation in the case of asso-ciative flow rules [28]

_ee ¼ _e� _coFor0

and _n ¼ _coFoq

ð18Þ

subjected to the classical loading-unloading conditionsin Kuhn-Tucker form

_c � 0; F ðr0; qÞ 0; _cF ¼ 0 ð19Þwhere _c is the continuum consistency parameter andF ¼ F ðr0; qÞ the isotropic yield function.

For the computation the classical elastoplastic modelof the Drucker-Prager [29] yield function with linearisotropic hardening has been used in the form

F ðp; s; nÞ ¼ 3aF p þ ksk � bF

ffiffiffi2

3

rc0 þ hn½ � ð20Þ

in which p ¼ 13 r0 : 1½ � is the mean effective Cauchy

pressure, ksk is the L2 norm of the deviator effectiveCauchy stress tensor r0, c0 is the initial apparent cohe-sion, aF and bF are two material parameters related tothe friction angle / of the soil

aF ¼ 2

ffiffi23

qsin/

3� sin/bF ¼

6 cos/3� sin/

ð21Þ

and h the hardening/softening modulus.

Remarks: In the present contribution, the effect of thecapillary pressure pc and of the temperature on theevolution of the yield surface is not take into account.The interested reader can refers e.g., to [30], [31] and [9]for a constitutive relationship function of the effectivestress and of the capillary pressure and to [32] for thenumerical implementation of constitutive law proposedin [31] and its application to strain localization simula-tion.

3.1 Initial and boundary conditions

For the model closure the initial and boundary condi-tions are needed. The initial conditions specify the fullfields of primary state variables at time t ¼ t0, in thewhole analysed domain B and on its boundary oB,oB ¼ oBp [ oBq

p; p ¼ g; c; T ; u� �

335

pg ¼ pg0; pc ¼ pc

0; T ¼ T0; u ¼ u0; on B [ oB; ð22ÞThe boundary conditions (BCs) can be of Dirichlet’s

type on oBp for t � t0:

pg ¼ pg on oBg; pc ¼ pc on oBc;

T ¼ T on oBT ; u ¼ u on oBuð23Þ

or of Cauchy’s type (the mixed BCs) on oBqp for t � t0:

nSgqgavgs þ J

gad

� �� n ¼ qga; on oBq

g

nSwqwvws þ nSgqgwvgs þ J

gwd

� �� n

¼ qgw þ qw þ bc qgw � qgw1

� �; on oBq

c

nSwqwvwsDHvap � veff grad T� �

� n¼ qT þ ac T � T1ð Þ þ ero T 4 � T 4

1� �

; on oBqT

r � n ¼ �t on oBqu ð24Þ

where nðx; tÞ is the unit normal vector, pointing to-ward the surrounding gas, qgaðx; tÞ; qgwðx; tÞ; qwðx; tÞand qT ðx; tÞ are respectively the imposed fluxes of dryair, vapour, liquid water and the imposed heat flux,and �tðx; tÞ is the imposed traction vector related to thetotal Cauchy stress tensor rðx; tÞ, qgw

1 ðx; tÞ and T1ðx; tÞare the mass concentration of water vapour and thetemperature in the far field of undisturbed gas phase,eðx; tÞ is emissivity of the interface, ro the Stefan-Boltzmann constant, while acðx; tÞ and bcðx; tÞ areconvective heat and mass exchange coefficients. Theboundary conditions with only imposed fluxes arecalled Neumann’ BCs. The purely convective boundaryconditions for heat and moisture exchange are alsocalled Robin’ BCs.

4 Finite element formulation

The finite element model is derived by applying theGalerkin procedure for the spatial integration and theGeneralised Trapezoidal Method for the time integra-tion of the weak form of the balance equations of Sect. 2(see, e.g., [1]).

In particular, after spatial discretisation within theisoparametric formulation, the following non-symmet-ric, non-linear and coupled system of equation is ob-tained

Cgg Cgc Cgt Cgu

0 Ccc Cct Ccu

0 Ctc Ctt Ctu

0 0 0 0

26664

37775

o

ot

�pg

�pc

�T

�u

26664

37775

þ

Kgg Kgc Kgt 0

Kcg Kcc Kct 0

Ktg Ktc Ktt 0

Kug Kuc Kut Kuu

26664

37775

�pg

�pc

�T

�u

26664

37775 ¼

fg

fc

ft

fu

26664

37775 ð25Þ

where the solid displacements uðx; tÞ, the capillary andthe gas pressure pcðx; tÞ and pgðx; tÞ and the temperatureT ðx; tÞ are expressed in the whole domain by global

shape function matrices NuðxÞ, NcðxÞ, NgðxÞ, NT ðxÞ andthe nodal value vectors �uðtÞ, �pcðtÞ, �pgðtÞ and �TðtÞu ¼ Nu�u; pc ¼ Nc�p

c; pg ¼ Ng�pg; T ¼ NT �T ð26ÞIn a more concise form Eq. (25) is written as

CoX

otþ KX ¼ F ð27Þ

Finite differences in time are used for the solution ofthe initial value problem over a finite time stepDt ¼ tnþ1 � tn. Following the Generalised TrapezoidalMethod as shown for instance in [1], Eq. (27) is rewrittenat time tnþ1 using the relationships

oX

ot nþhj ¼ Xnþ1 � Xn

Dtð28Þ

Xnþh ¼ ½1� h�Xn þ hXnþ1; with h ¼ ½0; 1�; ð29Þwhere Xn and Xnþ1 are the state vectors at times tn andtnþ1, thus obtaining

G Xnþ1ð Þ ¼ Cþ hDtK½ �nþhXnþ1

� C� ½1� h�DtK½ �nþhXn

� DtFnþh ¼ 0 ð30ÞLinearised analysis of accuracy and stability suggest theuse of h � 1

2. In the examples Sect., implicit one-step timeintegration has been performed (h ¼ 1).

After time integration the non-linear system ofequation is linearised, thus obtaining the equationssystem that can be solved numerically (in compact form)

oG

oX

Xi

nþ1

�DXiþ1nþ1 ffi �G Xi

nþ1� �

ð31Þ

with the symbol ð�Þiþ1nþ1 to indicate the current iterationðiþ 1Þ in the current time step ðnþ 1Þ and where theJacobian matrix has the following form

oG

oX

Xi

nþ1

¼

oGg

o�pgoGg

o�pcoGg

o�ToGg

o�uoGc

o�pcoGc

o�pcoGc

o�ToGc

o�uoGT

o�pgoGT

o�pcoGT

o�ToGT

o�uoGu

o�pgoGu

o�pcoGu

o�ToGu

o�u

266664

377775 ð32Þ

Details concerning the matrices and the residuumvectors of the linearised equations system can be foundin the Appendix 2. Owing to the strong coupling be-tween the mechanical, thermal and the pore fluidsproblem, a monolithic solution of (31) is preferred usinga Newton scheme.

Finally, the solution vector X ¼ �pg; �pc; �T; �u½ �T is thenupdated by the incremental relationship

Xiþ1nþ1 ¼ Xi

nþ1 þ DXiþ1nþ1 ð33Þ

4.1 Algorithmic formulation for elasto-plasticity

The problem of the calculation of ee; n and r0 is typicallysolved by an operator split into an elastic predictor and

336

plastic corrector [33]. The calculation of the trial elasticstate ð�Þtr is based on freezing the plastic flow at timetnþ1. The ee

nþ1� �tr

is hence obtained from the load step bymeans of ee

nþ1� �tr¼ enþ1 The corresponding trial elastic

state is obtained from the hyperelastic free energyfunction as

r0trnþ1 ¼owoee

h iee¼ ee

nþ1½ �tr

qtrnþ1 ¼ �

owon

h in¼ntr

nþ1

ð34Þ

If this trial state is admissible, it does not violate theinequality F 0trnþ1 ¼ F r0trnþ1; q

trnþ1

� � 0 and the stress state

is hence already computed. Otherwise the return map-ping or plastic corrector algorithm is applied to computeDcnþ1 satisfying the consistency condition Fnþ1 ¼ 0.

From the knowledge of Dcnþ1 the equivalent plasticstrain is computed by the backward Euler integration ofEq. (16)2

nnþ1 ¼ nn þ Dcnþ1oFoq

nþ1

ð35Þ

The Cauchy stress components are then computed bythe hyperelastic constitutive law Eq. (16)1 with the freeenergy w ¼ w ee; nð Þ written as function of the principalelastic strain components and the equivalent plasticstrain (for isotropic linear hardening) is

w ¼ L2

e1e þ e2e þ e3e½ �2þG e21e þ e22e þ e23e

� �þ 1

2hn2 ð36Þ

where L and G are the elastic Lame’ constants and h thelinear hardening modulus.

4.2 Return mapping algorithm for theDrucker-Prager model with linear isotropichardening and apex solution

Originally the return mapping algorithm was developedfor J2-plasticity. Extension of this method to theDrucker-Prager model can be made taking into accounta special treatment of the corner region using theconcept of multi-surface plasticity, as developed in [26]in case of perfect plasticity and deviatoric non-asso-ciative plasticity. In this paper the return mapping andthe algorithmic tangent moduli will be obtained forisotropic linear hardening/softening and volumetric-deviatoric non-associative plasticity.

To this end, a plastic potential function

Qðp; s; nÞ ¼ 3aQp þ ksk � bQ

ffiffi23

qc0 þ hn½ � similar to (20)

is defined, where the dilatancy angle u is introduced inEq. (21) instead of the friction angle /.

The key idea is based on the fact that the returnmapping algorithm developed without any specialtreatment of the apex region leads to physically mean-ingless results (i.e., jjsnþ1jj < 0Þ for a certain range oftrial elastic stress. Once the plastic consistency parame-ter Dcnþ1 is computed by the return mapping, this

happens when the following relationship obtained fromthe updated deviatoric components of the stress tensor

jjsnþ1jj ¼ jjstrnþ1jj � 2GDcnþ1 � 0 ð37Þ

is violated. Without going into details, violation ofinequality (37) and the consistency condition Fnþ1 ¼ 0yields the inequality for which the return mapping needsto be modified, i.e.:

ptrnþ1 >

3aQK2G jjstr

nþ1jj þbF

ffiffi23

p3aF

� jjstrnþ1jj2G h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 3a2Q

qþ cn

h i ð38Þ

where the indexes F and Q of a and b are referred to theyield and the plastic potential surface, respectively.

In this case, the stress region characterised by (38)may be treated like a corner region in non-smooth multi-surface plasticity. To this end a second yield condition F2

is introduced in addition to (20) as

F2ðp; nÞ ¼ 3aF p � bF

ffiffiffi2

3

rc0 þ hn½ � ð39Þ

which is derived from (20) with the condition jjsjj ¼ 0and the plastic evolution equations need to be modifiedfollowing Koiter’s generalisation introducing a secondplastic consistency parameter _c2 related toQ2ðp; nÞ ¼ 3aQp � bQ

ffiffi23

qc0 þ hn½ �. Hence the evolution

Eq. (18) will be substituted by the generalised plasticevolution laws

_e p ¼X

i

_cioQi

or0; and _n ¼

Xi

_cioQi

oq; i ¼ 1; 2 ð40Þ

In particular, the following implicit equation is obtainedenforcing the consistency condition F2 pnþ1; nnþ1ð Þ = 0

ptrnþ1 � 3aQK Dc1 þ Dc2½ �nþ1¼

bF

ffiffi23

q3aF

� cn þ hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDc1

2nþ1 þ 3a2Q Dc1 þ Dc2½ �2nþ1

q� �ð41Þ

Equation (41) has been solved iteratively via a Newtonscheme to compute Dc2nþ1, while Dc1 is given asDc1nþ1 ¼ jjstr

nþ1jj=2G from the condition snþ1 ¼ 0.Once the two plastic multipliers have been computed,the equivalent plastic strain n can be updated as

nnþ1 ¼ nn þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDc1

2nþ1 þ 3a2Q Dc1 þ Dc2½ �2nþ1

qð42Þ

as well as the components of the Cauchy stress tensor r0,the elastic strain tensor ee and the plastic strain tensor ep.In particular, for the Cauchy stress tensor

r0nþ1 ¼ ptrnþ1 � 3aQKDcnþ1 þ 1� 2GDcnþ1

kstrnþ1k

� �str

nþ1 ð43Þ

for the non corner zone and

r0nþ1 ¼ ptrnþ1 � 3aQK Dc1 þ Dc2½ �nþ1 ð44Þ

for the corner zone.

337

4.3 Algorithmic tangent moduli with apex solutionfor the Drucker-Prager model with linearisotropic hardening

The algorithmic tangent moduli are computed by lin-earisation of the computed Cauchy stress tensor. Twotangent moduli are obtained, the first one valid for thestress state where the Drucker-Prager model is satisfied,i.e., for the stress for which equation (38) is violated, thesecond one for the stress state which belongs to thecorner region. The computed moduli for the two casesare respectively:

� for the non corner zone:

aepnþ1 ¼ c1K1� 1þ 2G I� 1

31� 1

� �1� 2GDcnþ1

jjstrnþ1jj

� �

� 6aQKGc2

1� ntrnþ1 �

6aF KGc2

ntrnþ1 � 1

� 4G2 1

c2� Dcnþ1jjstr

nþ1jj

� �ntr

nþ1 � ntrnþ1

where ntrnþ1 and the coefficients c1 and c2 are

ntrnþ1 ¼

strnþ1jjstr

nþ1jj; c1 ¼ 1� 9aF aQK

c2

� �;

and c2 ¼ 9aF aQK þ 2Gþ bF h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

31þ 3a2Q

h ir

� for the corner zone:

aepnþ1 ¼ c3K1� 1þ c3K

2aQG Dc1 þ Dc2½ �nþ11� str

nþ1 ð46Þ

where the coefficient c3 is

It can be observed that the moduli (46) are nonsymmetric even for associated plasticity, while (45) arenon symmetric only for non associated plasticity. In caseof perfect plasticity (h ¼ 0) the coefficient c3 and hencethe moduli of equation (46) vanish and thus, stablesolutions are achieved only if a small number of pointsare in the corner region. Moreover, the moduli (45) arereduced to those of the von Mises model by selectingaF ¼ 0 and bF ¼ 1.

5 Numerical results

The first example deals with the simulation of the planestrain compression test of [4] in case of dense sand,where strain localization and cavitation of the porewater were experimentally observed. A rectangularsample of homogeneous soil of 34 cm height and 10 cmwidth has been discretised using a regular grid of340 quadrilateral elements, as depicted in Fig. 2. The

material is initially water saturated (hydrostatic distri-bution of water pressure is assumed as initial condition)and the boundaries of the sample are impervious andadiabatic. Imposed vertical displacements are applied onthe top surface with the constant velocity of 1:2 mm/suntil strain localization is observed. Vertical and hori-zontal displacements are constrained at the bottom

surface. Plane strains and quasi-static loading conditionsare assumed.

The initial temperature in the sample is constant andfixed at the ambient value. Gravity acceleration is takeninto account. The mechanical behavior of the solidskeleton is simulated using the elasto-plastic Drucker-Prager constitutive model of Sect. 3, with isotropic linearsoftening behavior as phenomenological description ofdamage effects and non associated plastic flow. Thematerial parameters used in the computation are listedin Table 1.

The constitutive relationships for the water degree ofsaturation Sw pcð Þ and the water relative permeabilitykrw Swð Þ are of the type of Safai and Pinder in isother-mal condition, as plotted in Fig. 3 and Fig. 4 respec-tively. For the gas relative permeability krg Swð Þ, therelationship of Brooks and Corey in isothermal condi-tion has been selected, as depicted in Fig. 4. Theserelationships have been used because of the lack ofexperimental results.

c3 ¼aQbF

ffiffi23

qh Dc1 þ Dc2½ �nþ1

3aF KffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDc21nþ1 þ 3a2Q Dc1 þ Dc2½ �2nþ1

qþ aQbF

ffiffi23

qh Dc1 þ Dc2½ �nþ1

34 c

m

10 cm

Imperviousboundary

Imposed vertical displacements (1.2mm/s)

Fig. 2 Description of the geometry, boundary and loading conditions

338

In the analysis, the dilatant behaviour of dense sandsis simulated selecting a positive value of the angle ofdilatancy (u ¼ 20�). For this material under globallyundrained conditions, the increment of the void ratiodue to the volumetric plastic deformations will cause thewater pressure to drop. Cavitation will develop ifthe saturation water pressure at the temperature of thesample is reached. A detailed analysis is described inthe following with the help of contours plots in the entiredomain and of time histories in two nodal points insideand outside the shear bands (the position of these pointsare marked in Fig. 8). The numerical results indicate thepronounced accumulation of inelastic strains in narrowzones after 32 s, as can be observed in Fig. 5 where theequivalent plastic strain contour is depicted. These shearbands develop from the bottom surface because of thehigher stress state due to gravity load and the con-strained conditions and propagate versus the top sur-face, as can be observed in Figs. 6 and 7, where thecontour lines of the equivalent plastic strain are depictedat 6.5 s and 12.5 s respectively. The contour of the vol-umetric strain (Fig. 8) emphasizes the dilatant behaviorof the shear bands, because only positive values developinside the plastic zones, while negative values are ob-served in the elastic domain (see, also Figs. 15 and 16).As a consequence, water pressure decreases inside theplastic zones up to the development of negative waterpressure, as depicted in Figs. 9 and 17. At these values ofpressures, a vapour phase appears because the satura-

tion water pressure at ambient temperature of 2338,8 Pais reached. Cavitation of water is hence described by themodel2, as shown in Fig. 10 where it can be observedthat the vapour phase appears only inside the dilatantplastic zones (see, also the plot of Fig. 18). At the sametime, the shear bands become partially saturated, as itcan be seen in Fig. 11 and 19, and the temperatureslightly decreases inside the plastic zones (see, Figs. 12and 20), due to the water phase change (evaporation)induced by mechanical effects. After cavitation, capillarypressure continues to develop because of a rise in posi-tive volumetric strain due to the external load in dilatantmaterial (this effect is captured because of the coupling

Table 1 Material parameters used in the computation of the firstexample

Solid density qs 2000 kg/m3

Water density qw 1000 kg/m3

Young modulus E 3.00E+07 PaPoisson ratio m 0.4Initial apparent cohesion c0 5.00E+05 PaLinear softening modulus h �1.00E+06 PaAngle of internal friction / 30�

Angle of dilatancy u 20�

Initial porosity n 0.2Intrinsic water permeabilityin water saturated conditions

k 5.00E)14 m2

(initial water conductivity) kw 5.00E)7 m/s

0

0,2

0,4

0,6

0,8

1

1,2

0,00E+00 2,00E+05 4,00E+05 6,00E+05 8,00E+05

Capillary pressure [Pa]

Wat

er s

atu

rati

on

[-]

Fig. 3 Water degree of saturation

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Saturation [-]

Rel

ativ

ep

erm

eab

ility

[-]

watergas

Fig. 4 Water and gas relative permeability

DEFP

1.783E-01

1.540E-01

1.297E-01

1.054E-01

8.104E-02

5.673E-02

3.242E-02

8.104E-03

Fig. 5 Equivalent plastic strain contour at 32 s using Drucker-Prager’law with u ¼ 20�

2 The gaseous phase at cavitation is the water vapour alone only inglobally undrained samples and if the dissolved air in the water phasecan be neglected. In the nodal point considered inside the shear band(see, Fig. 8), cavitation was recorded at 27; 50 s with a vapour pressureof 2.336,34 Pa and a water pressure of �142.077 Pa (while at 27 svapour pressure was of 2.338,8 Pa and water pressure of 2.533,57 Pa).

339

between the thermo-hygral and the mechanical part) andalso because of a drop in vapour pressure with thetemperature.

Also permeability decreases inside the shear bands,due to the adopted constitutive law of Brooks and Corey(Fig. 4), as plotted in Fig. 13. The water flow directionsof Fig. 14 indicate that water flows into the dilatantplastic zones.

The symmetric localization patterns of Fig. 5 isobtained because of the symmetry of the analyzedproblems (geometry, boundary and initial conditions,applied load and the use of homogeneous and isotropicmaterials). A single localization plane can be obtainedfor example by using a weak element to trigger the shearband or using a small horizontal external load in addi-tion to the vertical one.

In a second analysis, the behaviour of loose sands ([4]) issimulated. The contractant behaviour of this geomate-rial is analyzed with the Drucker-Prager material modelchoosing a negative value of the angle of dilatancy(u ¼ �5�). The other material parameters are those ofTable 1 except for the angle of internal friction(/ ¼ 20�). Also boundary and loading conditions are thesame as for the previous example. In this case, negativevolumetric plastic strain will develop, causing theincrement of the water pressure, as experimentally ob-served in [4]. Vertical displacements are applied untilshear band formation has been observed (Fig. 21). Inthis example, water pressure increases everywhere in thedomain and in particular inside the shear bands, as

Fig. 7 Equivalent plastic strain contour at 12.5 s using Drucker-Prager’ law with u ¼ 20�

PWAT

3.097E+04

–3.795E+04

–1.069E+05

–1.758E+05

–2.447E+05

–3.136E+05

–3.826E+05

–4.515E+05

Fig. 9 Water pressure contour [Pa] using Drucker-Prager’ law withu ¼ 20�

Fig. 6 Equivalent plastic strain contour at 6.5 s using Drucker-Prager’law with u ¼ 20�

Fig. 8 Volumetric strain contour using Drucker-Prager’ law withu ¼ 20�

340

plotted in Figs. 22 and 24 and the material remains fullysaturated with water. In this example, the water flowsout of the plastic zone (Fig. 23).

Figures 25 and 26 show the equivalent plastic strainand the water pressure distribution at the end of the loadhistory in case of zero dilatancy (which means isochoricplastic flow). In this case, the plastic zones have noinfluence on the pattern of the water pressure. (More-over, it can be observed that the resulting shear bandand the deformation pattern are very similar to thoseobtained using the von Mises material law).

In the last analysis, a sample of initially water satu-rated medium-dense sand is heated with a uniform

thermal load of 5� K/s from the ambient temperature of273:15� K up to the temperature of 383:15� K. Theboundary is impervious to any flux and, unlike in theprevious example, the vertical displacements are con-strained also at the top surface. The material parametersused in the computation are listed in Table 2.

In this case, the heat supply becomes a mechanicalload for the solid skeleton because of the constrainedvertical displacements. In this analysis, two symmetricshear bands develop in the upper and lower part of thedomain (Fig. 27) and the domain becomes partiallysaturated because of the evaporation of the water ini-tially contained in the sample, as it can be observed in

PVAP

2.338E+03

2.337E+03

2.336E+03

2.335E+03

2.334E+03

2.333E+03

2.332E+03

2.331E+03

Fig. 10 Vapour pressure contour [Pa] using Drucker-Prager’ law withu ¼ 200

SATU

9.063E-01

8.127E-01

7.190E-01

6.253E-01

5.317E-01

4.380E-01

3.443E-01

2.506E 01

Fig. 11 Water degree of saturation contour using Drucker-Prager’law with u ¼ 20�

Fig. 12 Temperature contour using [0K] Drucker-Prager’ law withu ¼ 20�

Fig. 13 Water permeability contour ½m2� using Drucker-Prager’ lawwith u ¼ 20�

341

Figs. 28 and 29 where a vapour phase appears in all thedomain, with the lower values inside the dilatant shearbands.

5.1 Remarks on the developed multiphase model forcavitation modelling

The mathematical model for multiphase geomaterialsdeveloped in this work improves the isothermal mono-species approach and the isothermal two phase flowmodel introduced in [6] and [7] to study cavitation ob-served during localization in dense geomaterials. This isbecause it is a non-isothermal three phase model wherewater phase change is taking into account. In particu-lar, the isothermal monospecies approach [6] is an iso-thermal model which neglects the dry air density and

Fig. 14 Water flow directions using Drucker-Prager’ law withu ¼ 20�

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 5 10 15 20 25 30 35

Time [s]

Vo

lum

etri

cst

rain

[-]

Outside shear band

Inside shear band

Fig. 15 Volumetric strain vs. time inside and outside shear band

-0.03

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0 5 10 15 20 25 30 35

Time [s]

Eq

uiv

alen

tp

last

icst

rain

[-] Outside shear band

Inside shear band

Fig. 16 Equivalent plastic strain vs. time inside and outside shearband

-6.E+05

-4.E+05

-2.E+05

0.E+00

2.E+05

4.E+05

6.E+05

8.E+05

0 5 10 15 20 25 30 35

Time [s]

Wat

erp

ress

ure

[Pa]

Outside shear band

Inside shear band

Fig. 17 Water pressure vs. time inside and outside shear band

2330.00

2331.30

2332.60

2333.90

2335.20

2336.50

2337.80

2339.10

2340.40

0 5 10 15 20 25 30 35

Time [s]

Vap

ou

rp

ress

ure

[Pa]

Outside shear band

Inside shear band

Fig. 18 Vapour pressure vs. time inside and outside shear band

0.E+00

2.E-01

4.E-01

6.E-01

8.E-01

1.E+00

1.E+00

0 5 10 15 20 25 30 35

Time [s]

Wat

er s

atu

rati

on

[-]

Outside shear band

Inside shear band

Fig. 19 Water saturation vs. time inside and outside shear band

342

the vapour density qgw with respect to the water densityqw. From the state equation of vapour (9) follows thatalso the vapour pressure pgw is negligible. From thedefinition of capillary pressure (1) and taking theatmospheric pressure as the reference pressure, it fol-lows that pc ffi �pw for the desaturated zones whencavitation takes place. Hence, conditions of partialsaturation may develop once the relative negative waterpressure exceeds the bubbling pressure (the bubblingpressure, see, [34], is the minimum value of pc on theSw � pc curve at which a continuous gas phase existsin the void space, which corresponds to Sw < 1, see,Fig. 3).

In the isothermal two phase flow model [7] the vapourdensity qgw as well as the vapour pressure pgw are dif-ferent from zero. The capillary pressure pc during cavi-tation is given by pc ¼ pgw � pw. For this model, theKelvin law (11) and the state equation of perfect gas for

vapour (9)2 are needed, while the dry air density is ne-glected and isothermal conditions are assumed.

6 Conclusions

A coupled finite element formulation for the hydro-thermo-mechanical behaviour of a water saturated andpartially saturated porous material has been presented.

293.1484

293.1487

293.1490

293.1493

293.1496

293.1499

293.1502

0 5 10 15 20 25 30 35

Time [s]

Tem

per

atu

re [

K]

Outside shear band

Inside shear band

Fig. 20 Temperature vs. time inside and outside shear band

DEFP

1.304E-01

1.119E-01

9.339E-02

7.490E-02

5.641E-02

3.791E-02

1.942E-02

9.247E-04

Fig. 21 Equivalent plastic strain contour using Drucker-Prager’ lawwith u ¼ �5�

PWAT

2.555E+06

2.523E+06

2.491E+06

2.460E+06

2.428E+06

2.396E+06

2.365E+06

2.333E+06

Fig. 22 Water pressure contour [Pa] using Drucker-Prager’ law withu ¼ �5�

Fig. 23 Water flow directions using Drucker-Prager’ law withu ¼ �5�

343

This model is obtained as a result of a research in pro-gress on the thermo-hydro-mechanical modeling formultiphase geomaterials undergoing inelastic strains.Numerical results of strain localization in globally un-drained samples of dense, medium dense and loose sandsand isochoric geomaterial have been presented. Vapourpressure below the saturation water pressure (i.e., watercavitation) develops at localization in case of densesands, as experimentally observed. A case of strainlocalization induced by a thermal load in a sample wherethe displacements are constrained and evaporation takesplace is also analyzed.

The presented model improves the isothermal mono-species approach and the isothermal two phase flow modeldeveloped in [6] and [7] to study cavitation experimen-tally observed during localization in dense geomaterials.This paper clearly points out that with a sufficientlygeneral THM model very different situations can bemodeled without special assumptions.

Appendix 1

A proper choice of state variables for description of thethermo-hygral and mechanical behaviour of soils asmultiphase porous material is of particular importance.From a practical point of view, the physical quantitiesused, should be possibly easy to measure during experi-ments, and from a theoretical point of view, they shoulduniquely describe the thermodynamic state of the med-ium. They should also assure a good numerical perfor-mance of the computer code based on the resultingmathematical model. The necessary number of the statevariables may be significantly reduced if existence of localthermodynamic equilibrium at each point of the mediumis assumed. In such a case physical state of different phasesof water can be described by use of the same variable.

Having in mind all the aforementioned remarks, wewill briefly discuss now the state variables chosen for the

Table 2 Material parameters used in the computation of the heatedsample

Solid density qs 2000 kg/m3

Water density qw 1000 kg/m3

Young modulus E 3.00E+07 PaPoisson ratio m 0.4Initial apparent cohesion c0 1.00E+05 PaLinear softening modulus h �1.00E+06 PaAngle of internal friction / 20�

Angle of dilatancy u 5�

Initial porosity n 0.5Intrinsic water permeabilityin water saturated conditions

k 2.55E�13 m2

(initial water conductivity) kw 2.55E)6 m/sCubic thermal expansion coefficient of solid bs 0.9E)05 K�1

Cubic thermal expansion coefficient of water bw 2.1E)04 K�1

0.E+00

5.E+05

1.E+06

2.E+06

2.E+06

3.E+06

3.E+06

4.E+06

0 10 20 30 40 50 60 70Time [s]

Wat

erp

ress

ion

e[P

a]

0

0.1

0.2

0.3

0.4

0.5

0.6

Eq

uiv

alen

tp

last

icst

rain

[-]

pw inside shear band

pw ouside shear band

equivalent plastic strain

Fig. 24 Water pressure and equivalent plastic strain vs. time usingDrucker-Prager’ law with u ¼ �5�

DEFP

1.304E-01

1.119E-01

9.339E-02

7.490E-02

5.641E-02

3.791E-02

1.942E-02

9.247E-04

Fig. 25 Equivalent plastic strain contour using Drucker-Prager’ lawwith u ¼ 0�

PWAT

4.523E+05

4.521E+05

4.520E+05

4.518E+05

4.516E+05

4.515E+05

4.513E+05

4.511E+05

Fig. 26 Water pressure contour [Pa] using Drucker-Prager’ law withu ¼ 0�

344

present model. Use of temperature (the same for allconstituents of the medium because of the assumptionabout the local thermodynamic equilibrium state) andsolid skeleton displacement vector is rather obvious,thus it needs no further explanation. As a hygrometricstate variable various physical quantities, which arethermodynamically equivalent, may be used, e.g., volu-metric- or mass moisture content, vapour pressure, rel-ative humidity, or capillary pressure. Very differentmoisture contents may be encountered at the samemoment in a multiphase porous material, ranging from

full saturation with liquid water up to almost completelydry material. For these reasons it is not possible to use,in a direct way, one single variable for the whole rangeof moisture contents. Instead, an appropriate Stefan’sproblem could be formulated, with different state vari-ables in zones separated by moving interfaces. However,such an approach is numerically very costly, e.g., [35],and usually avoided in practical applications, as men-tioned in [36].

Apparently, the most natural choice for the statevariable seems to be mass- or volumetric moisture con-tent, which are well defined for the whole range oftemperatures and pressures. However, this quantities arenot continuous at interfaces between different materials,and are not well adapted for numerical simulations, bothin fully saturated conditions and in a range of very lowmoisture contents. Moreover, there is not any direct,physically sound (from the mechanistic point of view)relation between moisture content and stresses. Anotherpossible choice for the moisture state variable is vapourpressure, which however has no physical meaning in amedium fully saturated with water and then, it createsserious numerical problems for moisture contents closeto these conditions, as shown by our extensive tests.

The moisture state variable proposed in [36] is thecapillary pressure, which was shown to be a thermody-namic potential of the physically adsorbed water and,with an appropriate interpretation, can be also used fordescription of water at pressures higher than the atmo-spheric one, [1]. The capillary pressure has been used alsobecause it assures good numerical performance of thecomputer code, [37], [36], [38] and is very convenient foranalysis of stress state in the porous material, becausethere is a clear relation between pressures and stresses, [39]

DEFP

1.120E-02

9.611E-03

8.022E-03

6.434E-03

4.845E-03

3.257E-03

1.668E-03

7.943E-05

Fig. 27 Equivalent plastic strain contour using Drucker-Prager’ lawwith u ¼ 5�

PVAP

2.702E+05

2.701E+05

2.701E+05

2.701E+05

2.701E+05

2.701E+05

2.701E+05

2.701E+05

Fig. 28 Vapour pressure contour [Pa] using Drucker-Prager’ law withu ¼ 5�

SATU

9.902E-01

9.896E-01

9.894E-01

9.893E-01

9.892E-01

9.886E-01

9.885E-01

9.752E-01

Fig. 29 Water saturation contour using Drucker-Prager’ law withu ¼ 5�

345

[40]. Application of capillary pressure as a state variablewas avoided in some previous models because of theo-retical problems related to its definition at the macro-scale. However, some recent works in Thermodynamics,([40], [3]), resolved these theoretical problems.

Hence, the chosen primary variables of the presentmodel are the volume averaged values of: gas pressure,pg, capillary pressure, pc, temperature, T , and displace-ment vector of the solid matrix, u.

For temperatures lower than the critical point ofwater, T < Tcr, like in soils, and for capillary saturationrange, S > SsspðT Þ (Sssp means the upper limit of thehygroscopic moisture range, being at the same time thelower limit of the capillary one, see, Fig. 30), the capil-lary pressure is defined as (1)

pc ¼ pg � pw ð47ÞThis equation is, in reality, a constitutive relationship atthermodynamic equilibrium which can be obtained froman exploitation of the entropy inequality by means of theColeman-Noll method, see e.g., [40] and [3].

When condition S < Sssp is fulfilled (there is no cap-illary water in the pores), the capillary pressure onlysubstitutes formally the water potential Wc, defined as:

Wc ¼RTMw

lnpgw

pgws

� �; ð48Þ

where the saturated vapour pressure, pgws, is in thermo-dynamic equilibrium with saturated film of physicallyadsorbedwater, [36].Having inmind theKelvin equation,valid for the equilibrium state of capillary water withwater vapour above the curved interface (meniscus):

lnpgw

pgws

� �¼ � pc

qw

Mw

RT; ð49Þ

we can note, that in the situations, where (48) is valid,the capillary pressure may be treated formally as thewater potential multiplied by the density of the liquidwater, qw, according to the relation, [36]

pc ¼ �Wcqw; ð50Þ

Thanks to this similarity, it is possible to use duringsimulations ‘‘formally’’ the capillary pressure even in the

low moisture content range, when the capillary water isnot present in the pores. However, one should remember,that in such situations capillary pressure cannot be iden-tified to a pressure in its normal physical meaning.

During computations a problem arises when themedium is fully saturated because in this case the gaspressure is undefined and the capillary pressure has nophysical meaning.

The problem is treated by setting the gas pressureequal to the atmospheric pressure and with a formalmodification of the relationship between saturation Swand capillary pressure because, when the saturation be-comes equal to one, the sign of the capillary pressure isformally set negative and the value equal to the pressurein the liquid above the gas pressure (see, Eq. (1)).Moreover, there is no gas flow for capillary pressurebelow the bubbling pressure because the gas phase isdiscontinuous and forms bubbles. Thus we perform a‘‘switching’’ from partially saturated to fully saturatedstate equations (or vice versa) when capillary pressurereaches bubbling pressure value (see, [41] for a detailedanalysis of this ‘‘switching’’).

Appendix 2

The matrices occurring in the discretised form of mass,energy and linear momentum conservation Eq. (25) aredefined as follows:

Cgg ¼ZB

NTp 1� Swð Þn Ma

RTNp

� �dV

Cgc ¼ZX

NTp

�nqga oSw

opcNp þ 1� Swð Þn Mw

RTopgw

opcNp

�dV

Cgt ¼ZB

NTp

"nqga oSw

oTNt � qgabs 1� nð Þ

� 1� Swð ÞNt

#dV þ

ZB

NTp

"1� Swð Þn

� �Mw

RTopgw

oT� pgw

T

� �� Ma

T 2R

� �Nt

#dV

Fig. 30 Hygroscopic region,non-hygroscopic region and solidsaturation point in a porousmediu

346

Cgu¼ZB

NTp 1�Swð ÞqgamTLNu� �

dV

Kgg¼�ZB

rNp� �T

qgakkrg

lg�rNp� �� �

dV

�ZB

rNp� �T�

qg MaMw

M2g

Dgwg � pgw

pgð Þ2rNp

!�dV

Kgc¼�ZB

rNp� �T�

qg MaMw

M2g

Dgwg

1

pg

opgw

opcrNp

�dV

Kgt¼�ZB

rNp� �T�

qg MaMw

M2g

Dgwg

1

pg

opgw

oTrNt

�dV

Kgu¼ 0

fg¼ZB

rNp� �T

qgakkrg

lgqgg

� �dV

Ccg¼ 0

Ccc¼ZB

NTp

"1�Swð ÞnMw

RTopgw

opc Npþn qw�qgwð ÞoSw

opc Np

#dV

Cct¼ZB

NTp

"�b�swgNtþ 1�Swð ÞnMw

RTopgw

oT�pgw

T

� �Nt

#dV

þZB

NTp n qw�qgwð ÞoSw

oTNt

� �dV

Ccu¼ZB

NTp qgw 1�Swð ÞþqwSw½ �mTB� �

dV

Kcg¼�ZB

rNp� �T"�qg MaMw

M2g

Dgwg � pgw

pgð Þ2rNp

!#dV

�ZB

rNp� �T"

qgwkkrg

lg�rNp� �

þqwkkrw

lw�rNp� �#

dV

Kcc¼ZB

rNp� �T

qg MaMw

M2g

Dgwg

1

pg

opgw

opcrNp

� �" #dV

�ZB

rNp� �T

qwkkrw

lwrNp

� �dV

Kct¼�ZB

rNp� �T��qg MaMw

M2g

Dgwg

1

pg

opgw

oTrNt

� ��dV

Kcu¼ 0

fc¼ZB

rNp� �T�

qgwkkrg

lgqggð Þþqwkkrw

lwqwgð Þ

�dV

�Zo

BqcN

Tp qwþqgwþbc qgw�qgw

1� �� �

dA

Ctg¼ 0

Ctc¼ZB

NTt DHvap �qwn

oSw

opc Np

� �dV

Ctt ¼ZB

NTt DHvap

qCp� �

effþb�swqwNt � qwn

oSw

oTNt

�dV

Ctu ¼ZB

NTt DHvap �qwSwm

TLNu� �

dV

Ktg ¼ZB

NTt qwCw

pkkrw

lw�rNp� �� �

� rT �

dV

þZB

NTt qgCg

pkkrg

lg�rNp� �� �

� rT �

dV

þZB

rNtð ÞTDHvap qw kkrw

lw�rNp� �� �

dV

Ktc ¼ZB

NTt qwCw

pkkrw

lwrNp� �� �

� rT �

dV

þZB

rNtð ÞTDHvap qw kkrw

lwrNp� �� �

dV

Ktt ¼ZB

rNtð ÞTveffrNtdV þZB

NTt

(�qwCw

pkkrw

lw

� �rpg þrpc þ qwgð Þ�� rNt

)dV

þZB

NTt qgCg

pkkrg

lg�rpg þ qggð Þ

� �� rNt

�dV

Ktu ¼ 0

ft ¼ �ZB

NTt qwCw

pkkrw

lwqwgð Þ

� �� rT

�dV

�ZB

NTt qgCg

pkkrg

lg qggð Þ� �

� rT �

dV

�ZB

rNtð ÞTDHvap qw kkrw

lwqwgð Þ

� �dV

�Z

oBqt

NTt

�qT þ ac T � T1ð Þ þ ero T 4 � T 4

1� ��

dA

Acknowledgements The authors would like to thank the ItalianMinistry of Education, University and Research (MIURCE00197531 Science and Applications of Advanced Computa-tional Paradigms) and the University of Padua (UNIPDCPDA034312 Collapse of slopes in highly heterogeneous soils) forthe financial support.

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