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Coupled material modellingand multi®eld structural

analyses in civil engineeringJosef Eberhardsteiner

Institute for Strength of Materials, Vienna University of Technology,Vienna, Austria

GuÈnter HofstetterInstitute for Structural Analysis and Strength of Materials,

University of Innsbruck, Innsbruck, Austria

GuÈnther MeschkeInstitute for Structural Mechanics, Ruhr-University Bochum,

Bochum, Germany

Peter Mackenzie-HelnweinInstitute for Strength of Materials, Vienna University of Technology,

Vienna, Austria

Keywords Wood, Soils, Concrete, Shrinkage, Stress

Abstract In this paper, three research topics are presented referring to different aspects ofmulti®eld problems in civil engineering. The ®rst example deals with long term behaviour of woodunder multiaxial states of stress and the effect of moisture changes on the deformation behaviourof wood. The second example refers to the application of a three-phase model for soils to thenumerical simulation of dewatering of soils by means of compressed air. The soil is modelled as athree phase-material, consisting of the deformable soil skeleton and the ¯uid phases ± water andcompressed air. The third example is concerned with computational durability mechanics ofconcrete structures. As a particular example of chemically corrosive mechanisms, the materialdegradation due to the dissolution of calcium and external loading is addressed.

1. PrologueThis paper is dedicated to Professor Herbert A. Mang, Head of the Institutefor Strength of Materials at Vienna University of Technology, Austria,

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

http://www.emeraldinsight.com/researchregister http://www.emeraldinsight.com/0264-4401.htm

Several parts of the presented development have been supported by the APART Program of theAustrian Academy of Sciences, the Deutsche Forschungs-Gesellschaft (DFG), and the FWFAustrian Science Fund, which is thankfully acknowledged. Moreover, the authors wish to thanktheir co-workers F. Bangert, S. Grasberger, A. HanhijaÈrvi, Dr Kuhl, G. Oettl, and R.F. Stark fortheir collaboration in the development of the models and the numerical examples presentedin this paper. Special thanks to R.L. Taylor for his support on the implementation andcomputations with FEAP.

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Received May 2002Revised December 2002Accepted January 2003

Engineering ComputationsVol. 20 No. 5/6, 2003pp. 524-558

q MCB UP Limited0264-4401DOI 10.1108/02644400310488754

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and Secretary General of the Austrian Academy of Sciences, on the occasion ofhis 60th birthday. Herbert A. Mang is active in many research areas related tothe broad ®eld of computational mechanics. His interests cover non-linear ®niteelement (FE) analyses of reinforced and prestressed concrete structures,adaptive FE procedures, the numerical simulation of tunnelling by both the FEand the boundary element (BE) methods as well as coupled BE-FE models, thenumerical modelling of the behaviour of tires, non-linear stability theory, andexperimental methods in the ®eld of strength of materials. He is the author ofmore than 130 papers in international scienti®c journals, more than 100 papersin proceedings of international scienti®c conferences and about 30contributions to scienti®c books. Many of these papers have been publishedjointly with PhD students working under his guidance. Several of his formerPhD students remain active in research ®elds, to which they have been inspiredduring their earlier work under the guidance of Professor Mang.

With the present paper, the authors express their great personal delight andgratitude for the privilege to belong to Professor Mang’s former PhD students.His care and great support for the scienti®c development of his studentsand his enthusiasm for scienti®c progress in computational mechanics hadand still has a fruitful and positive in¯uence on their own scienti®cdevelopments.

2. IntroductionThis paper is concerned with three research topics currently carried on by(partially former) co-workers of Professor Mang all of which represent differentaspects of multi®eld problems in civil engineering. The overall basis of theseresearch activities was laid during their time as PhD students under theguidance of Professor Mang at Vienna University of Technology. Hence, thepaper brings together three individual branches having, however, commonscienti®c roots. Although the presented research projects refer to such differentmaterials as wood, soils and concrete, they illustrate the relevance of coupledmodels for numerical structural simulations in the context of various civilengineering applications. It will be shown that the consideration of interactionsbetween the mechanical behaviour and damage, respectively, andnon-mechanical constituents affecting the short- and/or long-term behaviourof porous materials, such as the transport of moisture, liquid and gaseousphases or concentrations of aggressive chemical substances, allows for a morerealistic modelling of the complex behaviour of the different materialsconsidered in the paper. It is hoped that, at least partially, multi®eld analyseswill allow for the substitution of more or less heuristic approaches oftenpursued in classical, purely mechanical analyses.

The ®rst example deals with the long-term behaviour of wood undermultiaxial states of stress and the effect of moisture changes on thedeformation behaviour of structures. Within this paper, a brief overview of

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the theoretical formulation of the orthotropic creep model and the relateddescription of moisture transport are given. The basic steps of thegeneralisation of the one-dimensional model for arbitrary three-dimensionalstates of stress on the basis of a generalised Maxwell model are shown and afully three-dimensional formulation is presented. Proof of applicability of themodel is given by means of two representative three-dimensional examples,analysed using the non-linear ®nite element method (FEM). These examplescover bending of beams and the analysis of the long-term deformationbehaviour of a joint both under cyclically varying humidity.

The second topic refers to the application of a three-phase model for soils tothe numerical simulation of dewatering by means of compressed air. The soil ismodelled as a three-phase material, consisting of the deformable soil skeletonand the ¯uid phases ± water and compressed air. Employing a coupledapproach allows to take into account the interactions between the ¯ow of the¯uids ± water and compressed air in the soil and the deformations of the soil ina physically consistent manner. Moreover, many other geotechnical problemscan be treated by special cases of the three-phase formulation. Requiring the airpressure to be equal to the atmospheric one yields a model for dewatering ofsoils under atmospheric conditions and assuming a water saturated soil yieldsa model for consolidation analysis.

The third part is concerned with computational durability mechanics ofconcrete structures. As a particular example of chemically corrosivemechanisms, the material degradation due to the dissolution of calcium andexternal loading is addressed. Chemically induced weakening of concrete mayconsiderably affect the long-term durability of structures such as wastecontainments exposed to ground water. A coupled chemo-mechanical model(Bangert et al., 2001) is proposed in which chemically and mechanically induceddamage as well as the interacting effects concerning the changing transportand the material properties are taken into account. As a representativenumerical example, a coupled chemo-mechanical numerical analysis of aconcrete beam subjected to combined chemical and mechanical loading over aperiod of several 100 years is presented.

3. Long-term behaviour of wood under multiaxial states of stressand simulation of structural detailsAs a consequence of the geometric shape of trees, and also because of the highperformance of wood in the longitudinal direction, most wooden structuralmembers are designed for uniaxial loading in grain direction. Nevertheless, alltypes of joints ± traditional as well as modern design ± locally inducemultiaxial states of stress. The deformation behaviour of joints may have acrucial in¯uence on the response of statically indeterminate structures.Three-dimensional structural analysis by means of the FEM offers onepossibility for systematic studies on the performance of joints in wooden

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structures. For this purpose, material models for two- and three-dimensionalstates of stress are required.

From the mechanical point of view, wood exhibits a pronounced orthotropicbehaviour with large ratios of the mechanical properties, such as Young’smodulus or strength, between the respective values parallel and transverse tothe grain direction. Furthermore, wood exhibits a pronounced time dependentdeformation behaviour. Those time dependent deformations are usually splitinto viscoelastic creep at constant moisture content and mechano-sorptive creepdue to varying moisture content. See HanhijaÈrvi (1995), MaÊrtensson (1992),Ranta-Maunus (1975) and Toratti (1992) for further discussion and materialmodels for mechano-sorptive creep.

The aim of the present paper is to study the effect of moisture contentchanges on the deformation behaviour of structural details subjected to staticload. These studies are performed by means of a three-dimensional extension ofthe model by HanhijaÈrvi (1995) and its application within the FEM. The currentimplementation and the computations of the example problems are performedusing the research code FEAP [1].

3.1 Material modelThe material model used in the subsequent numerical simulations is based onthe non-linear model for viscoelastic and mechano-sorptive creeps byHanhijaÈrvi (1995). The original formulation is restricted to one-dimensionalstates of stress which are coaxial to the preferred material directions. Thismodel has been reformulated for three-dimensional states of stress. It has norestrictions on the stress state such as coaxiality with the material orientation.

An important part of the model is the description of the sorption behaviourof wood. Due to availability within the FEM software, a simple Fickean modelwill be used. The coupled mechano-sorptive analysis will be carried out on thebasis of a partially decoupled algorithm for the sorption problem. Themechano-sorptive creep problem requires coupling of the mechanical andsorption problems.

Within the following subsections, a short overview of the model formulationwill be given. For detailed discussion of the material formulation the reader isguided to HanhijaÈervi (1995).

3.1.1 Mechanical formulation of the material model. The generic modelbehaviour is characterised by linear elastic response under rapid loading, stressrelaxation under long-term loading, and increasing creep rates due to moisturecycling. The latter phenomenon is known as mechano-sorptive creep. It was®rst addressed by Ranta-Maunus (1975). The present model can be describedby the rheological model shown in Figure 1.

The ®gure shows a generalised Maxwell model built by parallelarrangement of springs and spring-dashpot elements enriched by

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hygroexpansion devices. These devices introduce the coupling between thesorption process and the mechanical deformation behaviour.

The total strain in any of the considered Maxwell chains is obtained as

e = e i = eei + e in

i = J i;0si +Z t

2 1Çe ini dt; (1)

where eei and e in

i are, respectively, the elastic and the inelastic parts of thetotal strain, J i;0 = 1=E i is the elastic or initial compliance, and Çe in

i the rate of theinelastic strain. Solving equation (1) for the acting stress si yields

si = gis0 2 qi; (2)

where gi = E i=E0 is a material parameter, E0 is the initial or short-termstiffness, s0 = E0[e 2 au(u 2 u0)] is the stress response under rapid loadingand qi is the creep-induced back-stress (Simo, 1987). The value of au(u 2 u0)describes the hygroexpansion strain with respect to a state with referencemoisture content u0. Introduction of the back-stress enables a description of thecreep law in stress space. The relation between the inelastic strain rate Çe in

i andthe back-stress qi is obtained from equations (1) and (2) as

qi = giE0

Z t

2 1Çe ini dt 2 au(u 2 u0)

µ ¶: (3)

Based on chemical kinetics, HanhijaÈrvi’s creep law describes the evolution ofthe inelastic strain as

Figure 1.Rheological model forcoupled viscoelastic andmechano-sorptive creepof wood

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Çeini = Ai sinh[fisi + Bi tanh(Di Çu)] + au;i Çu: (4)

Ai, fi, Bi, Di and au,i are material parameters satisfying AiBiDi + au;i = au:Time differentiation of equation (3) and substitution of Çe in

i according toequation (4) yields the rate equation for the back-stress as

Çqi = giE0 Ai sinh[fisi + Bi tanh(Di Çu)] 2 giE0(au 2 au;i) Çu: (5)

Introduction of a new set of parameters as

ti :=1

g iE0Aifi

; b i := 1 2au;i

au;

*su;i := tib igiE0au (6)

and the assumption of stress-free hygroexpansion at the limit Çu ! 0 yields thealternative description of equation (5) as

Çqi =1

tif2 1

i sinh fi si + *su;itanh(Di Çu)

Di

³ ´µ ¶2 *su;i Çu

» ¼: (7)

Even though equations (5) and (7) are equivalent descriptions of theone-dimensional evolution equation, equation (7) leads to a more compactthree-dimensional extension and less material parameters than equation (5).Assuming that, for a three-dimensional extension of equation (7), thecharacteristic creep time ti and the moisture scaling parameter Di remain scalarquantities, the evolution law (7) for each of the back-stress tensors qi formallybecomes

Çq i =1

ti

F 2 1i sinh F i s i + *su;i

tanh(Di Çu)Di

³ ´µ ¶2 *su;i Çu

» ¼: (8)

The mechanical part si and the sorptive part*su;i of the driving stresses,

respectively, become second-order symmetric tensors. The scalar parametersfi have to be extended to symmetric fourth-order tensors F i. Thethree-dimensional extension of the driving stresses follows directly fromequations (2) and (6) as

s i = gi#0 : [e ­ au(u 2 u0)] 2 q i = gi s 0 2 q i (9)

and

*su;i = tig ib i #0 : au; (10)

where #0 and au are the fourth-order elastic stiffness tensor and thesecond-order hygroexpansion tensor, respectively.

The second-order tensor au and the fourth-order tensors #0 and F i have tobe invariant under transformations of the orthotropic symmetry group. This is

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achieved by constructing them as linear combinations of the second-order andfourth-order structural tensors

{Ma}ij := {Aa ^ Aa}ij = {Aa}i{Aa}j; a = 1; 2; 3; (11a)

{-a}ijkl :=1

4(dikMa;jl + djlM a;ik + djkM a;il + dilMa; jk); a = 1; 2; 3; (11b)

which by de®nition are invariant under orthotropic symmetry transformations(Betten, 1987; Spencer, 1987; Zheng, 1994). The vectors {A1 = AL; A2 = AR;A3 = AT} are unit normal vectors to the symmetry planes of the material andthus form an orthonormal frame. These vectors are aligned with the preferredmaterial directions. Their orientation within the stem is shown in Figure 2.

It can be proved that the three structural tensors Ma and the tensors -a undMa ^ Mb ; a; b = 1; 2; 3; form a complete set of second-order and fourth-ordertensor generators for constant orthotropic tensors, respectively (Zheng, 1994).Expressing the hygroexpansion tensor as

au :=X3

a= 1

a(au)a Ma (12)

reveals three independent parameters. Constructing

F i :=X3

a= 1

X3

b= 1

b(F )ab Ma ^ Mb +

X3

a= 1

c(F )a -a (13)

as the general form of an orthotropic fourth-order tensor reveals 12independent parameters. Assuming Fi to possess full symmetries withrespect to the indices a and b reduces the number of independent parametersto 9. Obviously, the elastic stiffness tensor #0 can be expressed analagously toF i and due to its standard symmetries can be described by 9 independentelastic parameters b(#)

ab = b(#)ba and c(#)

a :This discussion demonstrates the intrinsic problem of parameter

identi®cation. While the one-dimensional formulation required ®ve

Figure 2.De®nition of theorthonormal frame{AL, AR, AT} and itsalignment to the grain(L), radial (R) andtangential (T) directionsof the stem

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independent parameters per mechano-sorptive chain in the rheological model,the three-dimensional formulation requires 13 independent parameters for thesame chain. The most severe problem is that uniaxial creep tests for all threepreferred directions permit unique identi®cation of just three out of the nineparameters of Fi. The parameters for the following numerical examples areobtained after introduction of signi®cant simplifying assumptions.

3.1.2 Moisture transport. The description of moisture transport will be doneby means of Fick’s law (Coussy, 1995). It relates the vapour mass ¯ux q to thegradient of the vapour partial pressure as follows:

q = 2 D grad pv

isotropic= 2 D grad(hps)

isothermal= 2 D* grad h: (14)

ps is the water vapour saturation pressure and the ratio h = pv=ps is therelative humidity. As outlined in equation (14), for the isotropic case thediffusion tensor D degenerates to a spherical tensor with only one diffusioncoef®cient D. For simplicity, the vapour transport in the cell lumen isconsidered isotropic despite the fact that the diffusion coef®cient is slightlylarger in grain direction. For isothermal conditions, Fick’s law may beformulated in terms of the relative humidity h instead of the vapour pressure.For this purpose, the diffusion coef®cient has to be scaled by the saturationpressure ps.

Mass balance for vapour (relative humidity, h) and adsorbed water (moisturecontent, u) yields the differential equation of state as

R0cÇh = div(D* grad h) in V : (15)

R0 is the mass density of perfectly dry wood and

c :=1

h2 2 h1

Zh2

h1

­ uemc

­ hdh (16)

is the average moisture storage capacity. Equation (15) is based on theassumption of local equilibrium between h in the cell lumen and u in thewooden skeleton according to the equilibrium moisture content in desorption.No hysteresis is considered in the present analysis. The functional relationuemc(h) has been taken from (Avramidis, 1989).

FE analysis is performed by means of analogy between equation (15) and thedifferential equation for transient thermal problems. Thus, a standard thermalelement may be used and the computed temperature is treated as relativehumidity h in the mechanical model. Since the standard version of FEAP doesnot permit surface resistance terms, the following set of boundary conditionshas been applied:

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h(t; x) = h(t; x) = 0:60 + 0:30 sin 2p (t 2 1:75 days)7 days

± ²on free surfaces

qn(t; x) = q ´ n = 0 on symmetry planes

(17)

The moisture content u required by the mechanical model is obtained fromthe computed relative humidity h and the equilibrium moisture content indesorption as u = uemc(h):

3.2 Numerical examplesThe following example computations are used to demonstrate theapplicability of the three-dimensional model. Furthermore, it remains tobe proven whether dropping the strain dependency of the parameters Ai,i = 1; . . .; n; as suggested by HanhijaÈrvi (1995) causes any de®ciencies in themodel’s response.

Both examples use the same history of the environmental humidity as givenby equation (17). Equation (16) and similar averaging for the diffusioncoef®cient D* has been evaluated for T = + 108 C; h1 = 0:30 and h2 = 0:80:Parameter values for both problems have been shown in Table I. Stiffnessparameters are taken from Holmberg et al. (1999) and hygroexpansioncoef®cients from MaÊrtensson (1992). Creep parameters are partially extractedfrom parameters given by HanhijaÈrvi (1995), but are for the mere purpose of afeasibility study reduced to two Maxwell chains and thus will not ®t well with

EL

(MPa)ER

(MPa)ET

(MPa)GLR

(MPa)GRT

(MPa)GLT

(MPa) nLR nRT nLT

Initial stiffness (short-term response) for both examples13,000 1,100 600 650 35 550 0.05 0.20 0.020

aL aR aT

u0 [1](ref. moisture)

T0

( 8 C) R0 (kg/m3) c [1]D*

(kg/cm d)

Sorption and hygroexpansion for both examples0.00540 0.19000 0.10000 0.063179

= uemc(0.30)+10 500 0.200 5.1073£ 10 2 05

i ti (d) g i [1] bi [1] Di (d)

b(f)ab

(MPa2 1)

c(f)L

(MPa 2 1)

c(f)R

(MPa2 1)

c(f)T

(MPa2 1)

Creep parameters for Example 11 21.0 0.375 2.4 £ 10 2 3 4.0 0.000 1.6667 1.6667 1.66672 210.0 0.375 2.4 £ 10 2 4 4.0 0.000 1.5000 1.5000 1.5000

Creep parameters for Example 21 21.0 0.375 0.6 £ 10 2 3 4.0 0.000 2.6667 2.6667 2.66672 210.0 0.375 0.6 £ 10 2 4 4.0 0.000 2.0000 2.0000 2.0000

Table I.Material parametersfor the examplecomputations

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experimental data for long time tests. Hence, the following examples have to beviewed with respect to their general tendency only. Numerical experiments (notpresented) showed a strong sensitivity of the numerical stability with respect tothe choice of creep parameters. This is a consequence of the exponentialcharacter of the sinh-function in the evolution law which causes numericalinstabilities at high stress levels. This shortcoming has to be removed byconsidering plasticity and local failure as well as better parameteridenti®cation. This will be the topic of upcoming research, but is not coveredin this paper.

In the literature, the terminology free hygroexpansion is used as a synonymfor stress-free states of the material throughout the whole process of absorptionand desorption, respectively. Nevertheless, there may appear fairly highinternal stresses due to time-dependent variation of the moisture distribution.In this paper, we will adopt the terminology free hygroexpansion for thecomputed deformations obtained under cyclically varying environmentalconditions, but without external loading.

For both examples, three loadcases are computed:

(1) constant external load and constant moisture content, u = u0 (denoted asviscoelastic creep),

(2) constant external load and cyclic humidity (a combination of viscoelasticand mechano-sorptive creep), and

(3) zero external load and cyclic humidity (denoted as free hygroexpansion).

Only the difference between (2) and (3) and the results of (1) are used in thedeformation over time plots.

3.2.1 Example 1: bending of a beam. Most experimental investigations onmechano-sorptive creep are performed as bending of beams. In order to study thecapability of the present model to reproduce the structural behaviour of beams,the three-dimensional model of a wooden beam is analysed by means of theproposed model. Figure 3 shows the geometric shape, the grain orientation,loading conditions and geometric boundary conditions. The boundaryconditions on the left side of the beam enforce a plane cross-section of thebeam. The loading conditions on the right end represent a bending moment ofM = 26:667 kN cm ( p = 5 MPa): As a consequence of the symmetry of theproblem, only half of the structure is discretised and symmetry conditions areapplied on the x-y-plane (z = 0): Due to the non-homogeneous moisturedistribution at cyclic humidity, the right end cross-section doesnot remain plane.

Figure 4(a) shows the computed distribution of relative humidity h of the airenclosed in the cell lumen as a snapshot at t = 10 days. The correspondingmoisture content u is assumed to be the equilibrium moisture content uemc(h).

Figure 4(b) shows the related distribution of the axial component of stress.The cross-section in front corresponds to the left end of the beam (Figure 3).As can be easily veri®ed, the non-homogeneous moisture distribution causes

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a signi®cant deviation from the linear stress distribution expected for plainbending at homogeneous moisture content. The linear distribution on theloaded (right) side of the beam induces the slightly non-uniform state of stressalong the beam axis (x-direction).

Figure 5 shows the total de¯ection for the central ®ber on the right end of thebeam as a function of time (also Figure 3). Free hygroexpansion does notproduce lateral displacements along the beam’s axis. Furthermore, the ®gurecontains the relative humidity of the surrounding air during the consideredinterval of 60 days. The diagram shows signi®cant mechano-sorptive creepduring the ®rst humidity cycle. The time delay of approximately 2 daysbetween the strongest humidity gradient and the increased creep rate resultsfrom the sorption process and thus from the delayed moisture change in theinterior of the beam. Unfortunately, the subsequent moisture cycles did notcause remarkable mechano-sorptive creep, which is known to occur inexperiments. This can be caused either by incomplete representation of thematerial behaviour by applying only two Maxwell chains, or the need for straindependency of the creep parameters as considered in the underlyingone-dimensional material model (HanhijaÈrvi, 1995).

3.2.2 Example 2: long-term deformation behaviour of a wooden joint. Jointsin wooden structures are a major reason for three-dimensional states of stressin naturally grown wood. Such a joint is shown in Figure 6. The two woodenparts are assumed to be homogeneous with respect to grain orientation andmechanical properties. No imperfections are considered in the followingcomputations. No gap is modelled between the two pieces. The contact surfaceis assumed to be perfectly connected and does not limit moisture transferbetween both parts. The diagonal rod is loaded by an axial compressive load of15

���2

p= 21:213 kN: Displacements perpendicular to the diagonal rod’s axis are

®xed in the loaded cross-section. Symmetry conditions are applied at z = 0:Figure 7 shows the computed distribution of relative humidity, h, after

21 days and 25 days. The ®gures are related to a minimum and close tothe following maximum environmental humidity, respectively. The relatedmoisture content u is assumed to be the equilibrium moisture content uemc(h).

Comparison of both ®gures shows that the centre part of both the rodsslowly approaches the mean value of the environmental humidity (h = 0:60)

Figure 3.Geometric description ofthe beam. Loading (rightend) and geometricboundary conditions (leftend and x-y-symmetryplane)

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Figure 4.Snapshot of the FEM

analysis after 10 days.(a) Distribution of

relative humidity, h; and(b) distribution of axial

stress in (MPa)

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and does not change much during the 7 days humidity cycles. Only the outerzones of the structure undergo remarkable humidity/moisture changes andthus only those zones are the source of mechano-sorptive creep deformations.

As a measure for the deformation of the joint, the vertical displacement ofpoint A on the symmetry plane is used (Figure 6).

Figure 8(a) shows the total displacements, which are dominated byhygroexpansion in R-direction during the humidity cycles. Figure 8(b) shows

Figure 5.Viscoelastic andmechano-sorptivede¯ection of a beamunder constant bendingmoment. De¯ections arenormalised by theshort-term response

Figure 6.Geometric description ofthe joint model. Loadingand geometric boundaryconditions

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Figure 7.Distribution of relative

humidity h after(a) 21 days ± structure

surrounded by air of30 per cent relative

humidity, and(b) 25 days ± structure

surrounded by air of87 per cent relative

humidity

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the development of viscoelastic creep deformations as well asmechano-sorptive creep deformation perpendicular to the grain direction ofthe horizontal rod. The displacements are normalised by the elasticdeformations. Figure 8 shows that the effect of load-induced deformations injoints is of minor importance compared to the shape changes due tohygroexpansion. Furthermore, the effect of mechano-sorptive creep comparedto the viscoelastic creep points out to be of at least equal importance. The lattercomment has to be viewed with special care since the choice of parameters forcreep is so far not critically veri®ed against the experimental data. This will bethe topic of an upcoming paper.

Figure 8.y-displacement of thebeam for point A onthe symmetry plane(see Figure 6). (a) Totaldisplacement (cm), and(b) total deformationminus hygroexpansion(normalised)

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4. Numerical simulation of dewatering of soils by compressed airFor a tunnel, which is driven below the groundwater table, the use ofcompressed air for displacing the groundwater from the working area at thetunnel face is motivated by the advantage of smaller ground settlementscompared to lowering the groundwater table with pumping wells and drivingthe tunnel under atmospheric conditions. This favourable effect of smallersettlements, which may be important for shallow tunnels in urban areas,results from the air pressure and drag forces of the air¯ow in the soil which, tosome extent, counteract the deformations caused by the dewatering processand the tunnel excavation (Kramer and Semprich, 1989).

4.1 Governing equations of the three-phase formulationFor the coupled numerical simulation of the dewatering of soils by means ofcompressed air, a model treating the soil as a three-phase medium consistingof a deformable soil skeleton (superscript s) and the two ¯uid phases waterand compressed air (superscripts w and a) is employed (Figure 9). Themathematical description of the three-phase medium relies on the application ofthe averaging theory and basically follows the approach proposed in Lewis andSchre¯er (1998) and Schre¯er and Simoni (1995).

Introducing the porosity n of the soil, de®ned as the ratio of the pore volumeto the total volume of the representative elementary volume, and the degree ofsaturation of the respective ¯uid phase S f, f = w; a; indicating the part of thevoid space ®lled by the particular ¯uid, the volume fractions are obtained as(1 2 n) for the solid phase and nS f for each ¯uid phase f = w; a: For thedegrees of saturation the condition S w + S a = 1 holds.

Assuming that each constituent ®lls the entire volume dV of the soil elementaccording to its volume fraction yields the averaged density r of thethree-phase mixture,

r = (1 2 n)rs + nS wrw + nS ara; (18)

where r s, r w and r a denote the intrinsic densities of the solid and the ¯uidphases water and air.

Figure 9.Representative

elementary volume ofthe soil. (a) Before, and

(b) after averaging

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By means of area fractions of the individual constituents which are assumed tobe equal to the volume fractions, the averaged stress tensor s for thethree-phase material can be de®ned similar to the averaged density in equation(18) using the partial stress tensors s s, s w and s a of the solid, water and airphase, respectively. In a soil mechanics context, it is convenient to decomposethe total averaged stresses into the averaged effective stresses s 0 and thehydrostatic ¯uid stresses p w and p a. This partitioning results in the followingform of the total averaged stress tensor s;

s = (1 2 n) s s + nS wsw + nS as a = s 0 + (S wpw + S ap a) I; (19)

where I describes the second order unit tensor. For both the soil skeleton and¯uid phases, tensile stresses are de®ned as positive quantities.

The arti®cial velocity v fr; i.e. the relative velocity between the ¯uid phase fand the soil skeleton averaged over dA, is obtained as

v fr = nS f(vf 2 v); (20)

where v f and v denote the velocities of the ¯uid phase and soil skeleton,respectively. The velocity vfr is actually measured in laboratory or ®eld testsand is contained in Darcy’s law which in the present model is assumed to bevalid for the transport of both ¯uids in the soil.

For dewatering of soils by means of compressed air the assumption of smalldisplacements and small strains seems to be justi®ed. Hence, the relationshipbetween the displacements u of the soil skeleton and the strains e in the soilskeleton is given as

e =1

2[grad u + (grad u)T]: (21)

The mass balance equations are derived by making use of the fact that in a®xed volume in space, mass changes of the respective constituent only occurdue to the in¯ow or out¯ow of mass through the surface of the control space.Since all phases are conceived to ®ll the entire volume element underconsideration simultaneously, the mass balance equations for the individualconstituents are referred to the same volume element. Considering theseassumptions the mass balance law for a ¯uid phase, f, reads as

2 div(rf vfr) = rfS f Çevol + nS f Çr f + nrf ÇS f : (22)

Equation (22) re¯ects the fact that the in¯ow of ¯uid mass into a given volumeelement (term on the left hand side) can be stored in the volume element eitherby an increase of the volumetric strain of the soil skeleton, Çevol; or by anincrease of the mass density Çr f or the degree of saturation ÇS f of the ¯uid phase,or by a combination of the latter three terms.

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Since the air pressure is increased slowly up to the desired value, quasi-staticconditions can be assumed and, consequently, the linear momentum balanceequations degenerate to the equilibrium equations for the three-phase mixture

div s + rg = 0; (23)

where g denotes the vector of gravitational acceleration.For quasi-static conditions the linear momentum balance equation for a ¯uid

phase yields Darcy’s law for the arti®cial velocity (Lewis and Schre¯er, 1998;Schre¯er and Simoni, 1995)

vfr =kofk rf

rfg(H p f + rfg); (24)

where kof denotes the permeability tensor of the fully saturated soil withrespect to the particular ¯uid phase f and k rf is the relative permeabilitycoef®cient. The latter is introduced in order to account for the presence of two¯uid phases in the partially saturated soil and is therefore a function of thedegree of water saturation. The values for k rf vary between 0 and 1.

Since the solid phase is assumed to be incompressible, only the effectivestresses cause deformations of the soil skeleton. Thus, the non-linearconstitutive law for the soil skeleton is formulated relating the rate of theaveraged effective stresses to the rate of the strains in the soil skeleton as

sÇ 0 = CT : Çe: (25)

In equation (25), CT = ds=de denotes the tangent material matrix of the soilskeleton. More complex constitutive models for unsaturated soils contain twoindependent stress state variables, e.g. the total stress in excess of pore airpressure, denoted as net stress, and the capillary stress (Coleman, 1962). Theuse of two stress state variables allows to consider the effect of the capillarystress on the shear strength of unsaturated soils.

The constitutive equation for a compressible barotropic ¯uid phase isgiven as

Çrf

rf= 2

Çpf

K f; (26)

with K f denoting the bulk modulus of the respective ¯uid phase, f. The minussign on the right hand side of equation (26) results from the de®nition ofcompressive stresses being negative quantities.

The constitutive relationship between the degree of water saturation S w andthe capillary stress pc, which is de®ned as the difference between thehydrostatic stress in the air and water phases, p c = p a 2 pw; can be written ina general form as

S w = S w( p c): (27)

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Equation (27) is usually determined from experiments for the relationshipbetween matric suction, s, and the degree of water saturation, Sw. If the stressesof the air phase differ from the atmospheric pressure then the matric suction,s, has to be replaced by the capillary stress pc. In the literature, severalapproximations of the relationship (27) have been suggested. In the presentmodel, the relationship

S w = Swr + Sw

s 2 Swr

¡ ¢1 +

p c

pcb

³ ´nµ ¶ 2 m

; (28)

proposed by van Genuchten and Nielsen (1985) is employed. In equation (28),pc

b denotes the air entry value, also referred to as the bubbling pressure, whichcan be viewed as a characteristic pressure that has to be reached before theair actually enters the pores; m and n are empirical constants to ®t the curvesto experimental data.

4.2 FE formulationFor the numerical solution of problems involving three-phase media, weakformulations of the mass balance equations for the individual ¯uid phases(equation (22)) and of the equilibrium equations for the three-phase mixture(equation (23)) are required. Multiplying the respective equations with virtualhydrostatic stresses and virtual displacements, respectively, and integratingover the domain under consideration yields these weak formulations.

Within the framework of the FEM the spatial discretization of the weakforms of the governing equations involves subdivision of the particular domaininto FEs. In the current model, the displacements of the soil skeleton, ue, andthe hydrostatic stresses in the ¯uid phases water and air, pw

e and pae ; are chosen

as primary variables. Since time derivatives of the primary variables occur inthe equations, a numerical integration in the time domain has to be performed.To this end, the implicit, unconditionally stable Euler backward integrationscheme is employed.

Finally, the coupled set of non-linear equations

K Cwn+ 1 Ca

n+ 1

(Cwn+ 1)

T 2 Swn+ 1 2 Dtn+ 1 Hw

n+ 1 Cwan+ 1

(Can+ 1)

T Cwan+ 1 2 Sa

n+ 1 2 Dtn+ 1 Han+ 1

2

6664

3

7775

DUn+ 1

DPwn+ 1

DPan+ 1

8>><

>>:

9>>=

>>;

=

fexn+ 1 2 fin

n 2 Cwn+ 1 Pw

n 2 Can+ 1 Pa

n

Dtn+ 1Çfw

n+ 1 + Hwn+ 1 Pw

n

± ²

Dtn+ 1Çfa

n+ 1 + Han+ 1 Pa

n

± ²

8>>>>><

>>>>>:

9>>>>>=

>>>>>;

(29)

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is obtained, which has to be solved for the incremental nodal values of thedisplacements of the soil skeleton, DU, and for the incremental nodal values ofthe hydrostatic stresses in the ¯uid phases, DPw and DPa, e.g. by a directiteration procedure. In equation (29), K denotes the stiffness matrix of the soilskeleton, the matrices Cw, C a and Cwa describe the coupling between theindividual constituents, Sw and Sa contain the compressibility of the particular¯uid and the constitutive relationship between the degree of water saturationand the capillary stress and Hw and Ha are the permeability matrices. On theright hand side f in and f ex are the internal and the external forces of the soilskeleton and the vectors Çfw and Çfa are associated with the ¯ow of the respective¯uid due to gravity and the ¯ow through the surface of the domain underconsideration. A detailed description is contained in Oettl (2002) and Oettl et al.(2002).

4.3 Applications: numerical simulations of a lab-test and of a full-scale in situ testWithin the framework of the Austrian Joint Research Initiative on NumericalSimulation in Tunnelling (Beer and Plank, 1999) laboratory tests have beenperformed at the Institute for Soil Mechanics and Foundation Engineering atGraz University of Technology (Kammerer, 2000) in order to investigate the airloss through cracks in the shotcrete lining and the ¯ow of compressed air in theadjacent soil. The numerical simulation of one of these experiments isdescribed subsequently.

In Figure 10 a schematic diagram of the test set-up is shown. A steelcontainer ®lled with a partially saturated sand was placed on top of a shotcrete

Figure 10.Set-up of the laboratorytests conducted at Graz

University ofTechnology

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element containing a crack of de®ned width. To simulate the overburden abovethe crown of the tunnel, a constant surface load of p = 250 kN=m2 was appliedon top of the soil using a grid system of girders and beams, which wasvertically prestressed and ®xed by four spindle rods. At the bottom ofthe shotcrete element, an excess air pressure was applied causing compressedair to ¯ow through the crack in the slab as well as through the adjacent soil.During the experiment the development of the air pressure in the soil wasmeasured at selected points.

The material parameters for the sand and the ¯uid phases water and air aswell as the parameters for the relationship between the degree of watersaturation and the capillary stress according to equation (28) are given in Oettl(2002). Young’s modulus of the sand has been determined by considering theoverburden pressure of the soil specimen. Since the changes of the soil stressesduring the experiment are small compared to the initial soil stresses due to theapplied overburden, linear elastic material behaviour of the soil skeleton is anadequate assumption for the numerical simulation.

For the numerical simulation of the experiment, the soil specimen isdiscretized by a regular mesh of quadrilateral FEs for plane strain conditionswith quadratic interpolation of the displacements and bilinear interpolation ofthe ¯uid stresses. At the top of the soil specimen, a row of FEs of 4 cm heightwas added to consider a small layer of water saturated soil as observed in theexperiment.

The boundary conditions for the soil skeleton and the ¯uid phases areassumed as follows: with respect to the displacements the domain is fullyconstrained at the bottom, vertically constrained at the top and horizontallyconstrained at the vertical boundaries. For the two ¯uids water and air, thevertical boundaries and the lower horizontal boundary are assumed to beimpermeable, except for the region at the middle of the lower horizontalboundary, where the excess air pressure of 17.5 kN/m2 enters the soil. Along theupper horizontal boundary of the soil element atmospheric air pressureprevails.

The initial conditions are chosen as follows: the overburden of the soilspecimen is accounted in the primary stress state. At the beginning of theexperiment there are no excess air stresses in the soil and the water stresses pw

are computed from equation (28) for the initial degree of water saturation of thepartially saturated soil of 52 per cent.

The distributions of the excess air stresses in the soil due to the air injectionare shown in Figure 11 shortly after the excess air pressure is applied and at theend of the experiment after 45 min. Considering horizontal sections of the soilspecimen, the distribution of the excess air pressure at a ®xed height is then quiteuniform in the upper part of the soil specimen. For comparison, the values of theexcess air pressure, which have been measured at the end of the test, are included(dots and printed values in Figure 11(b)). It follows from this comparison that

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the numerical model captures the distribution of the hydrostatic stresses in theair phase quite well.

In addition to the development of the air pressure, the displacements of thesoil skeleton at the end of the test are shown in Figure 12. As one would expect,the ¯ow of the ¯uids from the bottom to the top of the soil specimen causesdisplacements of the soil in the opposite direction of gravity. Due to theassumed displacement constraint at the top of the soil specimen, the maximumvalues of the displacements are obtained about half-way between the lower andupper horizontal boundaries.

The second application of the three-phase formulation refers to thenumerical simulation of a full-scale in situ air permeability test, which wascarried out in connection with the application of compressed air for the subwayconstruction in Essen, Germany (Kramer and Semprich, 1989).

Figure 13 shows the test set-up. In total, three different sets of tests wereperformed, in which compressed air was injected into the ground via a large

Figure 11.Computed excess air

pressure distribution inthe soil specimen.

(a) Shortly after theexcess air pressure isapplied, and (b) at theend of the experiment

Figure 12.Computed displacements

of the soil specimen atthe end of the experiment

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bore hole of 1.5 m diameter at different depths below the ground surface. Inexperiment 1B, which is considered here, compressed air was injected into theground at a depth between 18 and 21 m below the ground surface. The excessair pressure was applied in three steps (1.60, 2.20 and 2.35 bar) keeping itconstant for about 1 day at each of the three levels.

From the documentation of this experiment in the literature (Kramer andSemprich, 1989) it follows that, in contrast to the schematic diagram inFigure 13, the soil pro®le at the test location actually consists of four distinctlayers (Figure 14).

At the top a ®ll layer of about 3 m thickness is encountered, which is mainlycomposed of sand and silt; next is a 7-8 m thick silt layer, characterized by arather low permeability both with respect to water and air. In this layer also thegroundwater table is located at a depth of about 4.7 m below the groundsurface. Underneath the silt there is a thin, highly permeable sand-gravel layer;

Figure 13.Diagram of the in situair permeability testin Essen (dimensionsin (m))

Figure 14.Employed FE modelwith different soil layersand boundary conditions

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at the depth where the compressed air is injected a thick layer of marl isencountered, for which averaged values of the permeabilities are usedaccording to Kramer and Semprich (1989). Details about the materialparameters of the various layers can also be found in Kramer and Semprich(1989).

The material parameters for the four soil layers and the ¯uid phases waterand compressed air as well as the parameters of the relationship between thedegree of water saturation and the capillary stress according to equation (28),which have been employed in the numerical analysis, are given in Oettl (2002).Linear elastic material behaviour was assumed for all soil layers.

The FE-discretization is shown in Figure 14 together with the appliedboundary conditions. For the numerical simulation, advantage of axialsymmetry has been taken. The FEs mesh consists of axisymmetricquadrilateral FEs with quadratic interpolation of the displacements andbilinear interpolation of the hydrostatic stresses in the ¯uid phases. The leftvertical boundary of the domain to be discretized is given by the wall of thebore hole whereas the right vertical boundary and the lower horizontalboundary have to be chosen such that the error induced by these arti®cialboundaries remains small.

The boundary conditions are given as follows: as indicated in Figure 14,with respect to the displacements, the domain is fully constrained at the bottomand horizontally constrained at the vertical boundaries. With respect to the two¯uid phases the following boundary conditions are applied: the wall of the borehole, except the ®lter part where the compressed air is injected, as well as thelower horizontal boundary and the right vertical boundary are assumed to beimpermeable. Along the ®lter part of the bore hole the excess air pressure,which has been applied in the experiment, is prescribed. At the upperhorizontal boundary, i.e. at the ground surface, atmospheric air pressureprevails.

The initial conditions are given as follows: the primary soil stresses arecomputed from the self-weight of the soil taking into account the buoyancyforces below the groundwater level. The hydrostatic stresses in the two¯uid phases are determined according to the given groundwater table,whereas the excess air stresses are equal to zero at the beginning of theexperiment.

The results of the numerical simulation of the ®eld test 1B concerning the®rst pressure level of 1.60 bar are presented in Figures 15 and 16. In these®gures only the part of the discretized domain in the vicinity of the bore hole isshown.

Figure 15 shows the computed development of dewatering of the soil,characterized by a decreasing degree of water saturation from the initially fullysaturated state. Emanating from the location of the air injection in the bore holeis a cone of soil mass with a considerably lower degree of water saturation

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compared with the initial conditions, indicating the extent of the air ¯ow ®eld.It follows from Figure 15 that the applied excess air pressure of 1.60 bar is notlarge enough to dewater the silt layer, located above the highly permeablesand-gravel layer.

Figure 16 shows vector plots of the displacements of the soil skeleton atthe beginning and at the end of the ®rst pressure level. These plots clearlyshow the effect of the ¯ow of compressed air on the deformations of the soilskeleton. Shortly after the application of the compressed air only verysmall displacements of the soil occur primarily in the region close to thelocation of the air injection (Figure 16(a)), whereas at the end of the ®rst

Figure 15.Computed developmentof the degree of watersaturation in the soil

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pressure level, larger displacements in a considerably extended region areobtained (Figure 16(b)).

5. Computational analysis of long-term degradation of concrete dueto combined chemical dissolution and external loadingAs a particular example for chemical dissolution processes, the dissolution anddiffusion of calcium and the interaction with damage mechanisms incementitious materials is addressed in this section.

The dissolution of calcium is connected with an increase of the porosity,which may considerably affect the long-term durability of structures such aswaste containments or pipelines exposed to water (GeÂrard, 1996). Therefore,experimental investigations and numerical modelling of the matrix dissolutionand the transport of calcium ions (Ca2+) within the pore liquid of cementitiousmaterials has received considerable attention, in particular, in the recent years(Carde et al., 1996, 1997; GeÂrard, 1996; Grasberger et al., 2002; Heukamp et al.,2001; Kuhl et al., 2000; TraÈgaÊrdh and Lagerblad, 1998; Ulm et al., 1999).

When concrete is in contact with permanently renewed deionised water, thelower calcium ion concentration in the interstitial pore solution leads to thedissolution of the calcium bound in the skeleton as portlandite (Ca(OH)2),ettringite and calcium-silicate-hydrates (C-S-H). The reaction products of thedissolution of Ca(OH)2 are Ca2+2 and 2 OH2 -ions which are dissolved andtransported due to the concentration gradient within the pore ¯uid (Figure 17).

From experimental investigations, the following observations are made:®rst, the amount of calcium leaching in time depends almost linearly on thesquare root of time. Second, two dissolution fronts representing the dissolutionof portlandite and the C-S-H phases can be distinguished due to thediscontinuous change of the porosity. The above-mentioned observations

Figure 16.Computed soil

displacements. (a) At thebeginning, and (b) at theend of the ®rst pressurelevel (note the different

scaling of thedisplacements)

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indicate a diffusion-controlled leaching process with different chemicalequilibrium states for the dissolution of the main constituents of the cementpaste.

Since the cement matrix is a mixture of different chemical substances, thechemical equilibrium state is modelled phenomenologically. Consequently,the calcium leaching is modelled by a transient non-linear diffusion equation,assuming instantaneous matrix dissolution, whereby the calciumconcentration in the skeleton and the interstitial solution represents achemical equilibrium state, which is described by a smooth function s(c). A plotof this function proposed by GeÂrard (1996) and its derivative is show inFigure 18.

The pronounced peaks of the derivative ­ s/ ­ c at cCSHand cp indicate, that thedissolution of concrete constituents is dominated by the dissolution ofC-S-H-phases and portlandite.

The chemical and mechanical processes are fully coupled: the dissolution ofcalcium in the cement paste leads to a decrease of the mechanical material

Figure 17.Chemo-mechanicalmodel for concrete:simplifying assumptionsand couplingmechanisms

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properties, namely the stiffness and strength. Vice versa, mechanical damageresults in an increasing permeability for the calcium ions in the ¯uid phase(Figure 17). For this type of long-term degradation processes, achemo-mechanical model based on the theory of mixtures, assumingphenomenological chemical equilibrium between the calcium concentration inthe solute phase and in the cement matrix, recently developed (Kuhl et al., 2000,2002a, b), is described in this section. In this model, full coupling betweencalcium dissolution and mechanically induced damage is accomplished by thede®nition of the total porosity as the sum of the initial porosity, the porositycaused by calcium dissolution and the apparent porosity due to mechanicalloading. The porosity of the cement matrix, and, consequently, the stiffness andthe strength of concrete depends on the quantity of dissolved calcium ions. Inturn, the permeability and, consequently, the transport of the dissolved calciumions is strongly affected by the increasing porosity.

The new aspects of this model are the consistent formulation of stiffnessdegradation and increasing conductivity resulting from both deteriorationprocesses, the de®nition of an internal variable which allows for cyclic chemicalloading and an equivalent formulation of the evolution equations forchemical and mechanical induced damage as well as the formulation of thechemical reaction criterion based on thermodynamics (Kuhl et al., 2002a).Details of the numerical solution by means of an implicit second order accuratetime integration scheme and a hierarchical FE formulation are addressedin Kuhl et al. (2002b).

5.1 Porosity affected by mechanical and chemical damageThe formulation of the total porosity f of concrete, which is given by the sumof the initial porosity, f0, the porosity due to matrix dissolution, fc and theapparent mechanical porosity, fm

Figure 18.Chemo-mechanicalmodel for concrete:phenomenological

chemical equilibriumfunction, s(c), and its

derivative, ­ s/ ­ c

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f = f0 + fc + fm; (30)

allows for a consistent coupling between the chemical and mechanicaldegradation in concrete. The chemical porosity, fc is a function of the skeletoncalcium concentration s in terms of the initial concentration, s0 and the averagemolar volume, M /r of the skeleton (Kuhl et al., 2000). Since mechanical damageis restricted to the matrix material (1 2 f0 2 fc); the apparent mechanicalporosity, fm, de®nes the volume fraction of the damaged skeletoncharacterised by the damage parameter, dm

fc =Mr

(s0 2 s); fm = (1 2 f0 2 fc) dm: (31)

5.2 Constitutive laws and evolution of chemical and mechanical damageThe chemo-mechanical model is based on the chemical potential of calciumions Cc and the strain energy of the skeleton deformation Cm de®ned inthe form

Cc =f2

D0 g ´ g; Cm =1 2 f

2e : C s : e; (32)

with g = 2 Hc as the negative gradient of the concentration ®eld and e asthe linearaised strain tensor. D0 denotes the conductivity of the pore ¯uid andC s is the isotropic elasticity tensor of the solid phase. The macroscopic stresstensor, s , and calcium mass ¯ux vector, qc, are determined by thedifferentiation of the potential and the free energy function with respect to eand g, respectively, as

qc = f D0 g; s = (1 2 f) C s : e: (33)

The evolution of the porosity, fc, is described by means of the chemicaldegradation criterion formulated in terms of the calcium concentration and thechemical equilibrium threshold, kc

Fc = kc 2 c # 0: (34)

According to the Kuhn-Tucker conditions and the consistency condition

Fc # 0; Çkc # 0; Fc Çkc = 0; ÇFc Çkc = 0 (35)

the matrix dissolution process is connected with a decreasing chemicalequilibrium calcium concentration ( Çkc # 0): The dissolution threshold kc isunchanged for Fc , 0 and equal to the current calcium concentration of thepore ¯uid (kc = c) otherwise. The porosity due to chemical degradation, fc, iscalculated from equation (31), where, in contrast to existing phenomenological

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equilibrium chemistry models (GeÂrard, 1996), the change of the calciumconcentration in the skeleton is calculated as a function of the internal variablekc instead of the current concentration c.

For the description of mechanical damage of the skeleton, a strain-basedisotropic damage model (Simo and Ju, 1987) is used. According to this model,damage is characterised by the scalar damage parameter dm(km) as a functionof an internal variable, km. The evolution of km is based on the damagecriterion (Bangert et al., 2001)

Fm = Z(e) 2 km # 0: (36)

For the equivalent strain measure Z(e) a formulation proposed by Simo and Ju(1987) is used.

5.3 Initial boundary value problemThe coupled system of calcium leaching and mechanical damage incementitious materials is characterised by the concentration ®eld of calciumions, c, and the displacement ®eld, u, as primary variables and by a set ofinternal variables. The primary variables are controlled by the conservation ofthe calcium ion mass contained in the pore space and the balance of linearmomentum,

div qc + [[f0 + fc] c Ç] + Çs = 0; div s = 0: (37)

div qc describes the spatial change of the mass ¯ux vector, [[f0 + fc]c Ç]accounts for the temporal change of the calcium mass due to the change of theporosity and the concentration c, Çs is the calcium mass production and div srepresents the spatial change of the stress tensor. The system of differentialequation (37) is accompanied by related boundary and initial conditions

c(t = 0) = c*0; qc ´ n = q*

c ; c = c* ;

u(t = 0) = u*0; s ´ n = t* ; u = u* ;

(38)

where n is the normal vector on the boundary surface, qc* is the calcium ionmass ¯ux across the boundary, c* is the prescribed concentration, t* is thetraction vector and u* are prescribed displacements.

The numerical solution of the coupled chemo-mechanical problem isaccomplished by a combination of common techniques in computationalmechanics, including the weak formulation, the ®nite difference timediscretisation by the second order accurate mid-point rule, the consistentlinearisation and the FE discretisation. Finally, the resulting discrete algebraicsystem of equations is solved by a Newton-Raphson procedure using theconsistent tangent operator. For a detailed description of the numerical solutionprocedure, see Kuhl et al. (2002b).

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5.4 Application: combined mechanical loading and calcium dissolution of acement beamAs a representative example for coupled chemo-mechanical degradation, abeam of unit thickness, which is subjected to cyclic mechanical loading andto cyclic chemical attack is investigated by means of the proposed materialmodel. Two-dimensional plane strain FE with bilinear shape functions areused.

Figure 19 shows the geometry, the FE discretisation and the timehistories of the mechanical and the chemical loading, respectively. The initialconditions are given by u0* = 0 and c0* = c0: A displacement driven pointload is prescribed in the hatched region at the centre of the beam. At thebottom of the beam, the calcium concentration c* of the solute is prescribed,starting from an initial value c* = c0 corresponding to the initial equilibriumstate.

Figure 20 shows the distribution of the calcium concentration c in the solute,the remaining concentration s of the calcium in the skeleton and the matrixdissolution rate Çs at ®ve stages of the loading history. As expected, a crackopening along the symmetry line starts to propagate from the bottom to the topof the beam. At t = 17 £ 103 days the chemical attack has penetratedapproximately 1/3 of the height of the beam. Two dissolution fronts, indicatedby larger values of the rate Çs and large spatial gradients of s, respectively, arevisible. The upper one corresponds to the dissolution of portlandite (kc < cp);while the second front at the bottom side of the beam is associated to thedissolution of the C-S-H phases (kc < cCSH). Between these fronts, otherconstituents like ettringite are dissolved at a much lower rate. Along the axis ofsymmetry, the chemical degradation is accelerated by the opening of a verticalcrack. The next two stages (t = 25:5 £ 103 and 34 £ 103 days) show the furtherprogress of the dissolution, while the concentration at the bottom of the beam iskept constant at c* = 0: Stages t = 42:5 £ 103 and 51 £ 103 days correspond tochemical unloading. While the calcium concentration in the solute slowlyincreases according to the prescribed boundary condition, c* = c0; the state ofthe skeleton characterised by the concentration, s, does not change. The localchemical degradation process stops ( Çs = 0); if c is increased. Consequently,no further dissolution occurs, since the initial calcium concentration, c = c0;is recovered in the whole structure.

Figure 19.Numerical analysis ofchemo-mechanicaldegradation of a cementbeam: geometry, FEmesh and loading history

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6. Concluding remarksIn this paper, three fully coupled multi®eld models suitable for multi®eldsimulations of structures made of wood, soils and concrete were described.

Regarding wood, within this paper, a brief outline of a material formulationfor viscoelastic and mechano-sorptive creep has been given. This model as wellas a simple description of sorption in wood has been generalised forthree-dimensional problems and applied within the context of the FEM. Twoexample problems were presented in order to prove the applicability of themodel to structural analysis of beams and joints. The three-dimensionalanalyses gave enhanced insight to the interaction between sorption and thetime-dependent mechanical behaviour. Furthermore, the examples showed astronger effect of mechano-sorptive creep in the R and T-directions than in theL-direction. Due to the lack of extensive veri®cation of the three-dimensionalformulation against experiments, the need for better parameter identi®cationshould be emphasised. The example studies proved the general applicability ofthe developed model, but also pointed out the direction for the ongoing researchon this topic.

The second topic focused on the numerical simulation of dewatering of soilsby means of compressed air on the basis of a three-phase model for the soil. Anappealing feature of the coupled model is the broad range of applications,

Figure 20.Chemo-mechanical

degradation of a cementbeam: calcium

concentrations c, s,matrix dissolution rate Çsand degree of damage d

at ®ve stages of theloading history

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comprising the numerical simulation of consolidation, dewatering underatmospheric conditions and tunnelling below the groundwater table by meansof compressed air. The presented examples demonstrated the potential of themodel. Further applications, such as the numerical simulation of tunnellingbelow the groundwater table, using compressed air as a means for displacingthe ground water are described in Oettl (2002).

The background of a third model described in this paper is durabilityoriented design of concrete structures. A two-phase model was developed todescribe the coupled effects of calcium leaching and mechanical loading on thelong-term behaviour of concrete structures subjected to water (Kuhl et al., 2000,2002a, b). The typical time scale of the degrading effect is in the range ofseveral 100 years. Full coupling between calcium dissolution and mechanicallyinduced damage was accomplished by the de®nition of the total porosity as thesum of the initial porosity, the porosity caused by calcium dissolution and theapparent porosity due to mechanical loading. By means of suitablemodi®cations of the relation determining the phenomenological equilibriumbetween the concentrations of calcium in the matrix and in the solute and of thepermeability, respectively, this model can be used also for other types ofdissolution processes in cementitious materials.

Note

1. Finite Element Analysis Program by Robert L. Taylor, http://www.ce.berkeley.edu/, rlt

References

Avramidis, S. (1989), ªEvaluation of `three-variable’ models for the prediction of equilibriummoisture content in woodº, Wood Science and Technology, Vol. 23, pp. 251-8.

Bangert, F., Kuhl, D. and Meschke, G. (2001), ªFinite element simulation of chemo-mechanicaldamage under cyclic loading conditionsº, in de Borst, R., Mazars, J., Pijaudier-Cabot, G.and van Mier, J. (Eds), Fracture Mechanics of Concrete Structures, Vol. 1, pp. 145-52.

Beer, G. and Plank, J. (1999), ªJoint research initiative on numerical simulation in tunnellingº,Felsbau, Vol. 17 No. 1, pp. 7-9.

Betten, J. (1987), Applications of Tensor Functions in Solid Mechanics, CISM Courses andLectures, Chapters 11-14, Vol. 292, Springer, Berlin.

Carde, C., Francois, R. and Ollivier, J-P. (1997), ªMicrostructural changes and mechanical effectsdue to the leaching of calcium hydroxide from cement pasteº, in Scrivener, K. and Young, J.(Eds), Mechanisms of Chemical Degradation of Cement-based Systems, Chapmann andHall, London, pp. 30-7.

Carde, C., Francois, R. and Torrenti, J-M. (1996), ªLeaching of both calcium hydroxide and C-S-Hfrom cement paste: modelling the mechanical behaviorº, Cement and Concrete Research,Vol. 26, pp. 1257-68.

Coleman, J. (1962), ªStress strain relations for partly saturated soilº, GeoteÂchnique, Vol. 12 No. 4,pp. 348-50.

Coussy, O. (1995), Mechanics of Porous Continua, Wiley, England.

GeÂrard, B. (1996), ªContribution des couplages meÂcanique-chimie-transfert dans la tenue a longterme des ouvrages de stockage de deÂchets radioactifsº, PhD thesis, E.N.S.Cachan.

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