computational mechanics of multiphase problems - scale … · 2011-04-21 · cmm-2011 – computer...

CMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, PolandCMM-2011 – Computer Methods in Mechanics 9–12 May 2011, Warsaw, Poland

Computational Mechanics of Multiphase Problems - Scale-bridging Modeling Strategies

Günther Meschke, Dirk Leonhart, Jithender J. Timothy and Meng Meng ZhouInstitute for Structural Mechanics, Ruhr University Bochum

44780 Bochum, Germanye-mail: guenther.meschke rub.de

Abstract

The paper addresses various scale-bridging modeling and discretization strategies for multiphase materials, starting with a novel cascademicromechanics model for ion transport in porous materials, covering multiple scales involved in the pore structure of cementitiousmaterials, to generate homogenized diffusion coefficients. Using homogenized macroscopic properties, the theory of poromechanicsprovides a suitable modeling framework for the macroscopic representation of transport and phase change processes as is demonstratedfor freezing of porous materials using a three-field formulation. The theory of poromechanics is again employed as an appropriaterepresentation of more or less intact porous materials, in conjunction with a two-field Extended Finite Element model to describecoupled hygro-mechanical processes in cracked porous materials.

Keywords: micromechanics, poromechanics, extended finite element method, homogenization, multiphase models, diffusion, durability,soil freezing

1. Introduction

Computational simulations of structures made of heteroge-neous porous materials such as concrete or soil, when subjectedto combined mechanical, thermal, hygral or chemical loading,require an adequate representation of the multiphase characterof the problem, considering the highly interacting multi-physics(mechanical, hygric, thermal and chemical) processes actingwithin the microstructure. These processes are closely related tothe changing pore structure of the material and to micro-defectseventually evolving to macro-cracks. Since, for structural simula-tions, the model description eventually is formulated on a macro-scopic level, methods of scale transition must be applied in con-junction with adequate discretization strategies for the solutionof the governing equilibrium and balance equations. With re-gards to the first aspect, two modeling frameworks suitable forthe homogenization of microscopic or submicroscopic quantitiesare discussed: poromechanics as a classical averaging methodand methods of micromechanics for the up-scaling of transportproperties within the highly tortuous pore structure, ranging fromthe nano- to the micrometer scale, and eventually within (dis-tributed) microcracks. As far as the second aspect is concerned,the finite element method for the analysis of intact materials aswell as the Extended Finite Element Method (X-FEM) for theanalysis of cracked multiphase materials will be addressed.

The paper is organized within three parts as follows: The firstpart is concerned with the determination of up-scaled transportproperties for ion diffusion in cementitious materials based upona cascade micormechanics model. Homogenized material prop-erties such as diffusion coefficients or liquid permeabilities can bedirectly used in macroscopic models for multiphase solids usingmore classical up-scaling concepts such as the theory of porome-chanics. This theory is used as the framework for a three-phasemodel for soil freezing presented in the second part of the paper.In the third part, the incorporation of cracks - constituting distur-bances at a small scale compared to the larger uncracked part - isaccomplished using the Extended Finite Element Method in thecontext of a two-field hygro-mechanical formulation.

2. Multiscale cascade micromechanics model for ion trans-port in solids with complex pore structure

Estimations of homogenized transport properties such as dif-fusivity, conductivity and permeability to be used in macroscopicstructural analyses of porous materials (e.g. for durability analy-ses) need to account for the complex structure of the pore space,its tortuosity and the large range of pore sizes, ranging from nano-to millimeter. To circumvent large efforts connected with the ex-perimental determination of the effective diffusivity and to obtainmore insight into the role of the micro-structure w.r.t ion trans-port, micromechanics is used as the modeling framework to com-pute the the macroscopic diffusion coefficient considering the dis-tribution of pores and micro-cracks, respectively. A novel multi-level micromechanics concept, denoted as cascade micromechan-ics model, is proposed, in which micro inclusions with specificcharacteristics are embedded in a matrix with another character-istics reflecting again a porous material with a micostructure. Thediffusion equation describing diffusive ion transport within a rep-resentative elementary volume (REV) of a body Ω (x) readsc (z) = ∇xc · z ∀z ∈ ΓREV

∇zc (z) = 0 ∀z ∈ Ω (x) (1)j = −D (z)∇zc (z) ∀z ∈ Ω (x)

The localization tensor A provides the relation between the con-centration gradient on the macroscopic and the microscopic scale:

∇zc (z) = A (z)∇xc (x) (2)

Inserting the relations between scales into the governing equa-tion, the macroscopic flux now reads

J = −∇xc (x)1

Ω (x)

∫Ω(x)

D (z)A (z) dΩ (x) . (3)

The up-scaled diffusion coefficient is obtained as

Dmacro =

np∑i=1

DiAiφi, (4)

with φi as the volume fraction and the over-bar indicates aver-aged quantities over individual phases. To obtain homogenized


diffusivity for ion transport in porous materials with a complex,tortuous pore structure, a two phase material composed of a fluidphase and the solid phase is considered. The localization ten-sor, which provides information on the tortuous micro-structureis written as a function of a cascade variable n and the porosityφ.

Af = Af (n, φf ) . (5)

For the isotropic case we have obtained the localization tensor asfollows

Af (n, φf ) =3n (3φf − 1)

1φf

n−1

−

1φf

n+ 2φf · 3n

, n ∈ [0,∞) .

(6)

Furthermore, it is postulated that the average localization tensor

Af = Af (n, φ) , n→∞. (7)

limn→∞

3n (3φf − 1)1φf

n−1

−

1φf

n+ 2φf · 3n

=

H

(φf −

1

3

)· 3φf − 1

2φf, φf ∈ [0, 1] (8)

H is the HEAVISIDE function.Thus, for a 2-phase material with an isotropically distributedmicro-structure, the macroscopic diffusion coefficient is

Dmacro = Df3φf − 1

2H

(φf −

1

3

). (9)

The tortuosity which characterizes the resistance to transport dueto the micro-structure obtained from the proposed model is ob-tained as

τ =2φf

3φf − 1, φf ∈

[1

3, 1

]. (10)

0.0 0.2 0.4 0.6 0.8 1.0

2

3

4

5

Tortu

osity

Porosity

a

b

c

d

e

f

gh

Boudreau (1996) Model

Figure 1: Plot of the porosity vs. relative diffusivity for theproposed model and comparison with experimental results andother analytical models [3] : a-Mackie and Mears, b-Beekman, c-Maxwell, d-Rayleigh, e-Weissberg, f-Millington, g-Bruggemann,h-Tsai and Streider.

Fig.1 contains comparisons of the results from the proposedcascade micromechanics model with results from laboratory testsand 8 analytical models. It shows, that the multi-level strategyused in the micromechanics model, in contrast to the existing an-alytical models, is able to efficiently capture the micro-structuralcharacteristics affecting ion diffusion and reflects fairly well thecharacteristics of the experimental results. The effect of a prefer-ential orientation of the micro-structure (such as induced by dis-tributed micro-cracks) on the transport properties can be realizedby solving for prolate or oblate spheroids [9]. This will also beelaborated in the presentation.

3. Three-phase modeling of phase change in freezing solids

The modeling of the freezing process of the pore water con-tained in the pore structure of porous materials on a macro-scopic scale requires consideration of coupled thermo-hydro-mechanical processes acting on various (lower) scales of the porestructure such as the multi-scale physics of confined crystalliza-tion of ice phase and the cryo-suction mechanism driving the liq-uid towards the frozen sites. For the macroscopic description ofthe mechanical behavior of water-infiltrated materials upon freez-ing the theory of poromechanics [5] provides a sound up-scalingframework and allows for a physics-oriented understanding of theunderlying mechanisms. Within this framework, the behavior offreezing porous materials is described by the constitutive equa-tions, the liquid-crystal equilibrium relation and liquid saturationcurve in addition to the balance equations.

3.1. Model constituents

To describe the mechanical behavior of freezing porous ma-terials, the thermo-poroelastic constitutive equations [5] providerelations between the total stress tensor σ and the partial poros-ity change ϕJ, the strain tensor ε, the pore pressures pJ and thetemperature T , respectively [10]:

σij = (K − 2G/3) ε δij + 2Gεij

− (bL pL + bC pC + 3 aK (T − T0)) δij ; (11)

ϕL = bL ε+pL

NLL+

pC

NCL− aL (T − T0); (12)

ϕC = bC ε+pL

NCL+

pC

NCC− aC (T − T0); (13)

where ε = εkk is the volumetric dilation, K and G denote thebulk modulus and the shear modulus, a, aL and aC are the ther-mal volumetric dilation coefficient related to the porous solid, liq-uid water and crystal ice and bJ andNJK are the generalized BIOTcoefficient and coupling moduli, respectively.

Exploring the thermodynamic equilibrium between the liquidpore water (L) and the adjacent crystal ice (C) a relation betweenthe crystal pressure pC and liquid pressure pL can be formulated:

pC − pL = Σf (Tf − T ) , with Σf =ρCLf

Tf, (14)

where T is the current temperature, while Tf, Σf, Lf and ρC arethe bulk freezing temperature, the freezing entropy, the latent heatof freezing and the ice mass density, respectively. The mechani-cal equilibrium of the liquid-crystal interface is governed by theYOUNG-LAPLACE law

pC − pL =2 γCL

R, (15)

where γCL is the liquid-crystal interface energy andR is the meancurvature radius of the interface. Combining Eq. (14) and (15)yields the GIBBS-THOMSON relation

R =2 γCL

Σf (Tf − T ), (16)

which implies that all pores having an entry radius larger than Rwill freeze at the given temperature T .


As a further ingredient of scale transition the liquid saturationcurve for a porous material saturated with liquid water and air isinferred from the relation between the degree of liquid saturationand a macroscopic measure of interface forces acting betweenthe pore water and the solid particles, generally denoted as thecapillary pressure. Incorporating the VAN GENUCHTEN capillaryfunction into the YOUNG-LAPLACE law at the gas-liquid inter-face provides a relationship between the liquid saturation and thepore size

SL =

(1 +

(2 γGL

N R

) 11−m

)−m= S(R) , (17)

Consequently, the remaining liquid saturation SL equals the cu-mulative fraction S(R) of pore volume occupied by pores havinga pore entry smaller than R, where γGL is the liquid-air inter-face energy and m is a constant representing the shape of thecapillary curve. Replacing the pore size R in (17) by the GIBBS-THOMSON law (16) gives a relationship between the liquid satu-ration and the temperature

SL =

(1 +

(Tf − T∆Tch

) 11−m

)−m, (18)

with ∆Tch = N γCLΣf γGL

as the characteristic cooling temperature.Both parameters ∆Tch andm are related to the pore size and poresize distribution [4].

The mechanical behavior of water-infiltrated materials uponfreezing is not solely related to the expansion of liquid watertransforming into ice crystals. Under freezing conditions, the re-maining liquid water in pores can be drawn towards the alreadyfrozen sites via a cryo-suction process which has been identifiedas the driving force of the frost heave phenomenon. At a temper-ature below the bulk freezing point, confined water can partiallyremain liquid provided that it de-pressurizes relative to the adja-cent ice crystals, provoking, in turn, a cryo-pumping of the distantliquid water.

3.2. Thermo-hydro-mechanical finite element formulation

By choosing the solid displacement, the liquid water pressureand the mixture temperature as state variables, (1) mass balanceof liquid water and crystal ice, (2) overall momentum balanceand (3) overall entropy balance are set up as the governing setof balance equations. In the context of small deformations, thefully-coupled thermo-hydro-mechanical formulation involves thetime integration and the simultaneous solution of the governingbalance equations.

3.3. Model validation

For the validation of the model performance related to thecoupled thermo-hydro-mechanical behavior a cubic cement pastespecimen (length: 0.1 m), initially fully saturated by liquid waterand exposed to uniform cooling, is analyzed. The specimen isassumed as stress-free and undrained, i.e. no external load is ap-plied and the total mass of water existing either in solid or liquidform remains constant. The modeling conditions are chosen ac-cording to experiments performed by Beaudoin & MacInnis [1].

In Fig. 2 the numerical results obtained for the evolution ofthe volumetric strain and the principal strains during freezing arecompared to the analytical solutions presented in [6]. The analy-sis shows that there are three contributions to the volumetric di-latation during freezing (Fig. 2 left). The main contribution ε∆ρ

accounts for the hydraulic effect because the excess of liquid wa-ter expelled from the freezing sites, as a result of the liquid-crystaldensity difference, cannot escape from the sample. The other twocontributions εth and εΣf are consequences of the thermal contrac-tion of the mixture and the micro-cryo-suction process caused bythe difference between the liquid pressure and crystal pressure,

respectively. Only small difference caused by the discretizationerror between the numerical and analytical solution is observed.It has been concluded in [6] that εΣf is always positive even forliquids that usually contract when they solidify, and would vanishonly for a zero freezing entropy. This explains why dilation wasobserved in the experiments by Beaudoin & MacInnis, in whichbenzene, which, unlike water, contracts when it solidifies, wasused as the saturating liquid.

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ç numerical solution

analytical solution

0 5 10 15 20

0.000

0.002

0.004

0.006

0.008

0.010

0.012

cooling: T f -T @°CD

volu

met

ric

dila

tion:

¶

¶th<0

¶DΡ>0

¶S f>0

¶>0

Figure 2: Validation of freezing model: Evolution of dilatationand principal strains during freezing

4. Two-phase Extended Finite Element Method

For the numerical analysis of undamaged porous materials,considering the transport of ions and liquid substances (water ormoisture) through the pore space in addition to the mechanicalloadings, the theory of poromechanics as described in Section 3is, by now, a well established approach. In computational struc-tural analyses involving discontinuities such as evolving cracksin concrete structures, joints in rocks or shear bands in soft soils,however, the highly accelerated moisture transport in the openingdiscontinuities has to be taken into account.

Figure 3: Moisture transport in porous materials across cracks

Discontinuities such as cracks are characterized by a distur-bance of the overall distribution of the field variables within avery small length. To resolve the small and the large scale portionof the solution, the Extended Finite Element method [8], exploit-ing the Partition of Unity property of shape functions, offers thepossibility to include arbitrary enhancement functions into the fi-nite element approximation to improve the local approximationof non-smooth distributions of field variables, such as disconti-nuities of the displacement field or the fluid pressure field acrosscracks [7, 2]. For an element fully crossed by a crack, the Signfunction is used for the representation of the discontinuous dis-placement field across the crack:

u = u + SSu ≈nr∑i=1

Niunri + SS

nc∑i=1

Niunci , (19)

where nr is the number of nodes used for the spatial discretiza-


tion of the regular displacement field (superscript r). In cou-pled hygro-mechanical problems involving porous materials withcracks, the flow of the pore fluid across the crack will be discon-tinuous (Fig.3).

Adopting poromechanics as the modeling framework, and as-suming pores to be filled by an ideal mixture of water vapor anddry air (subscript g), the two-field problem is characterized bythe displacement field u and the capillary pressure pc as degreesof freedom. Considering a jump of the moisture flux across thediscontinuity denoted by ∂SΩ leads to the weak form of the massbalance∫Ω

δpcml

ρldV −

∫Ω

δ∇pc · ql dV +

∫∂SΩ

δpc [[ql]] · nS dA

=

∫Γq

δpc q∗l dA . (20)

Considering the local mass balance at the discontinuity (i.e. onthe small scale level) provides the normal component of the mois-ture flux across the crack as

q∗n = nS · ql,∂SΩ

=w

ρlM−1 ∂pc

∂t− ∂

∂xt(Dt

c∂pc∂xt

w) +Df∂pc∂xt

∂w

∂xt, (21)

where the index n and t stands for the normal and tangential di-rection w.r.t the crack, respectively,w is the crack width,M is theBIOT modulus,Dt

c is the hydraulic permeability within the crack,and Df is the permeability in the bulk material. The capillarypressure is discretized using a local enhanced partition of unityapproximation of the C1 discontinuity of the capillary pressureacross the crack:

pc = pc + pc

≈NN∑i=1

Ni peic +

NE∑j=1

Nj ψj pec︸︷︷︸

Nenrj

= Np pec + Nenrp pec (22)

In (22) the function ψ is related to the distance function. Fig. 4illustrates the functions used for the enhanced approximations ofthe displacement field u and the capillary pressure pc.

Figure 4: Two-phase Extended Finite Element Method: En-hanced approximations of displacements (left) and capillary pres-sure (right)

5. Concluding remarks

The paper has provided a synthesis of scale-bridging numeri-cal modeling and discretization strategies for multiphase mate-

rials. Using a multi-level strategy, a novel cascade microme-chanics model was proposed to obtain macroscopic diffusivitiesfor ion transport based upon to the pore size distribution. Themodel allows to incorporate essential topological information ofthe microstructure, ranging from the nano- to the meso-level, intomacroscopic models, e.g. using poromechanics. The theory ofporomechanics, using a three-field formulation, was employed asthe up-scaling modeling framework for the description of crystal-lization of the ice phase and the cryo-suction mechanism involvedin freezing processes of porous materials. Finally, poromechan-ics was employed in conjunction with a two-field Extended FiniteElement method to describe coupled hygro-mechanical processesin cracked porous materials. Exploiting the potentials of eachsub-model opens the perspective to further increase the ability ofnumerical simulations of multiphase materials to capture the es-sential physical mechanisms and to reduce the phenomenologicalcharacter of numerical models.

References

[1] J.J. Beaudoin and C. MacInnis. The mechanism of frostdamage in hardened cement paste. Cement and ConcreteResearch, 4:139–147, 1974.

[2] C. Becker, S. Jox, and G. Meschke. 3D higher-orderX-FEM model for the simulation of cohesive cracks incementitious materials considering hygro-mechanical cou-plings. Computer Modeling in Engineering and Sciences,57(3):245–276, 2010.

[3] P. Bernard Boudreau. The diffusive tortuosity of fine-grained unilithified sediments. Geochimica et cosmochim-ica acta, 60:3139–3142, 1996.

[4] N. Boukpeti. One-dimensional analysis of a poroelasticmedium during freezing. Int. J. Numer. Anal. Meth. Ge-omech., 32:1661–1691, 2008.

[5] O. Coussy. Poromechanics of freezing materials. J. Mech.Phys. Solids, 53:1389–1718, 2005.

[6] Oliver Coussy and Paulo J.M. Monteiro. Poroelastic modelfor concrete. Cement and Concrete Research, 38:40 – 48,2008.

[7] G. Hofstetter and G. Meschke, editors. Numerical Model-ing of Concrete Cracking. CISM International Centre forMechanical Sciences. Springer, 2011. in print.

[8] N. Moës, J.E. Dolbow, and T. Belytschko. A finite elementmethod for crack growth without remeshing. InternationalJournal for Numerical Methods in Engineering, 46:131–150, 1999.

[9] Jithender J. Timothy, J Kruschwitz, and G Meschke. De-termination of homogenized diffusion properties in micro-cracked porous materials. PAMM, 10(1):429–430, 2010.

[10] M. Zhou and G. Meschke. Numerical modelling of cou-pling mechanisms during freezing in porous materials. InPAMM, 2011.

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