Modeling of cementitious materials exposed to isothermal calcium leaching, considering process kinetics and advective water flow. Part 2: Numerical solution

International Journal of Solids and Structures 45 (2008) 6241–6268

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Modeling of cementitious materials exposed to isothermal calcium leaching, considering process kinetics and advective water flow. Part 2: Numerical solution Dariusz Gawin a,*, Francesco Pesavento b, Bernhard A. Schrefler b a b

Department of Building Physics and Building Materials, Technical University of Łódz´, Al. Politechniki 6, 93-590 Łódz´, Poland Department of Structural and Transportation Engineering, University of Padua, via Marzolo 9, 35131 Padova, Italy

a r t i c l e

i n f o

Article history: Received 2 March 2008 Received in revised form 18 June 2008 Available online 7 August 2008 Keywords: Numerical solution Finite element method Calcium leaching kinetics Calcium advection

a b s t r a c t The second part of the paper presents numerical solutions of the mathematical model of hydro-chemo-mechanical behavior of cementitious materials exposed to contact with deionized water of part 1. The model defines kinetics of the calcium leaching process instead of a direct application of a curve describing equilibrium between solid calcium in the material skeleton and the calcium dissolved in the pore solution. It further takes into account the advective flux of calcium ions. Both aspects are new as compared to previous models. The weak form of the governing equations of the model is derived first using the Galerkin method. Then, the equations are discretized in space with finite elements and in time domain with finite differences, and finally the procedures used for numerical solution of their discretized form are presented. Three numerical examples are solved to test the numerical solution procedure proposed and demonstrate its robustness for solution of 1D and 2D problems concerning fast and slow leaching of cement-based materials. The effect of various factors on the results concerning chemical degradation of structures made of cementitious materials is analyzed as well. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction In part 1 of the paper (Gawin et al., 2008), a novel mathematical model of hydro-chemo-mechanical behavior of cementitious materials exposed to contact with deionized water was developed. Unlike to the previous models, the present one defines kinetics of the calcium leaching process instead of direct application of a curve defining equilibrium between the solid calcium in the material skeleton and the calcium dissolved in the pore solution. Then, it takes into account the advective flux of calcium ions which was not considered previously. Finally, it allows solving problems with physically more realistic boundary conditions, i.e. of the convective-type for an element immersed in deionized water, or those corresponding to the situation when water flows through the elements due to water pressure gradient, carrying away the calcium ions dissolved from the skeleton of a cement-based material to the pore solution. In this paper we present a procedure for numerical solution of the governing equations of the mathematical model proposed in the companion paper (Gawin et al., 2008). As shown by Kuhl et al. (2004), during numerical solution of non-linear reaction–diffusion problems, like that of calcium leaching of cementitious materials, serious numerical problems arise due to the pronounced reaction front moving through the structure and the related strong mass sources. Application of the appropriate numerical methods solves the problems * Corresponding author. Tel.: +48 42 631 35 60; fax: +48 42 631 35 56. E-mail addresses: [email protected] (D. Gawin), [email protected] (F. Pesavento), [email protected] (B.A. Schrefler). 0020-7683/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2008.07.023

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only in part, as demonstrated by Kuhl and Meschke (2007). In this paper we investigate numerical performance of the applied numerical solution procedure, by solving the same model problems as in Kuhl and Meschke (2007), and analyze if the proposed description of the calcium leaching process kinetics, based on non-equilibrium thermodynamics, improves numerical performance of the solution obtained. The paper is organized as follows. First, the weak form of the governing equations of the model will be derived using the Galerkin method. Then, the equations will be discretized in space with finite elements and in time domain with finite differences. Finally, a procedure used for numerical solution of the resulting system of non-linear equations will be presented. Three numerical examples will be solved and discussed in the paper to demonstrate the robustness of the code and analyze the effect of various factors on the chemical degradation of structures made of cementitious materials. The first example is based on a test reaction–diffusion problem, solved previously by Kuhl et al. (2004) and Kuhl and Meschke (2007), and concerning calcium leaching from a cement paste wall (1D problem) exposed to one-side action of deionized water, modeled with Dirichlet’s BCs. The effect of the advective flow due to a water pressure difference, the rate of decrease of liquid calcium concentration on the wall surface, and the characteristic time of leaching on the simulation results will be analyzed. The second example is similar to the first one, but solved for physically more realistic boundary conditions. For the case without advection, Robin’s BCs with two different values of mass exchange coefficient are assumed, and for the case with external pressure gradient it is assumed that pure water enters one wall surface, and water containing calcium dissolved from the skeleton, with a concentration corresponding to the surface value, flows out through the opposite side. The third 2D example, concerning the concrete square sample immersed in deionized water, will be solved for Robin’s BCs to demonstrate application of the model for predicting the progress in time of the sample chemical degradation and compare it with the 1D case.

2. Weak formulation of the model equations The mathematical model of chemo-hydro-mechanical phenomena in cementitious materials has been formulated in the companion paper (Gawin et al., 2008). To solve its governing equations numerically with FEM, their weak form is necessary. This can be obtained by means of the weighted residual method following (Lewis and Schrefler, 1998). Applying Green’s theorems (for the details of necessary transformations, see Lewis and Schrefler, 1998), the model equations derived in part 1 of the paper (Gawin et al., 2008), can be rewritten in the following form:  Dry air mass conservation equation:

   oSw opc oSw ocCa wg nqga þ dX c op ot ocCa ot X     Z Z _ 1n oqs oCleach T mdiss d dX  wg S X  w S g qs g oCleach ot qs g X X   Z ou dX þ wg ð1  Sw Þqga mT L ot X   ga c  Z oq op oqga opg oqga ocCa þ wg ð1  Sw Þn þ þ dX opc ot opg ot ocCa ot X    Z rg kk  ðrwg Þ  qga g ðrpg þ qg gÞ dX l X ("  gw #) Z M p a M w gw g dX  ðrwg Þ  q Dg r 2 pg M X g Z þ wg qga dC ¼ 0

Z

Cqg

 Water species mass conservation equation:

  gw c  oq op oqgw ocCa wc ð1  Sw Þn þ dX opc ot ocCa ot X    Z oSw opc oSw ocCa þ wc nðqw  qgw Þ þ dX opc ot ocCa ot X   Z _ diss gw m  wc ðq Sg þ qw Sw Þ dX qs X   Z 1n oqs oCleach gw w  wc ðS q þ S q Þ dX g w qs oCleach ot X

Z

ð1Þ

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  ou dX wc ½qgw ð1  Sw Þ þ qw Sw mT L ot X " #  gw  Z M a M w gw p dX  ðrwc Þ  qg D r g 2 pg Mg X   Z rg kk  ðrwc Þ  qgw g ðrpg þ qg gÞ dX þ

Z

X



Z

X

þ

ð2Þ

l

  rw kk ðrwc Þ  qw w ðrpg þ rpc þ qw gÞ dX

l

Z

Cqc

wc ½qw þ qgw þ bc ðqgw  qgw 1 ÞdC ¼ 0

 Dissolved calcium mass conservation equation:

 oCleach ou dX þ acCa qw Sw mT L ot q oCleach ot X     Z oSw opc oSw ocCa ocCa þ ws ncCa qw þ þ nqw Sw dX opc ot ocCa ot ot X Z  ðrws Þ  ðqw DCa d rc Ca ÞdX 

Z

 1n ws cCa qw Sw s

X



Z

X



Z

X



ðrws Þ  cCa qw

kk

oqs

rw

lw

ð3Þ

 ðrpg  rpc þ qw gÞ dX

  Z _ m _ diss  cCa qw Sw diss dX þ ws m s

q

CqCa

^w dC ¼ 0 ws ^cCa q

Table 1 Scheme of the BCs assumed in Section 4.1

BC type

Dirichlet’s

Case

1 1b 2 3 4

Leaching

BC on side d

1 /g[mol/(J s)]

T leach (s)

pw (bar)

BC on side b pw (bar)

5.40  108 1.08  108 5.40  108 5.40  108 5.40  108

109 109 1.25  108 109 1.25  108

1 1 1 6 6

1 1 1 1 1

Figures with results

1, 2, 5–8, 11, 12 11, 12 3–6, 9, 10 7, 8 9, 10

Table 2 Main material properties assumed in our simulations Material properties

Symbol

Unit

Cement paste

Concrete

Water/cement ratio Aggregate/cement ratio Porosity (initial) Intrinsic permeability (initial) Young’s modulus Poisson’s ratio Compressive strength

w/c a/c n k E

(–) (–) (%) (m2 ) (GPa) (–) (MPa)

0.50 – 20 0:5  1018 24 0.20 26

0.45 5.97 12.2 0:5  1018 30.0 0.18 34.5

m fc

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Fig. 1. The time histories of liquid calcium concentration (a) and its space distributions at different time stations (b), obtained from simulations of the slow reaction–diffusion problem (Tleach = 109 s).

 Linear momentum conservation equation:

    ou oec oech oDc ðLwu ÞT Dc L   þ ðLu  ec  ech Þ dX ot ot ot ot X   g  Z ws op o v opc T  ðLwu Þ amT  s pc  vws dX s ot ot ot ZX Z oq þ wTu wTu tdC ¼ 0 gdX þ ot X Cqu

Z

ð4Þ

where wg, wc, ws are weighting functions of class C0 and wu is the vector of weighting functions of the same continuity class. The weighting functions have been chosen is such a way that they satisfy the following conditions on boundary C ¼ Cj [ Cqj of the integration domain X:

wj ¼ 0;

x 2 Cj

where j = g, c, s, u.

 j ¼ wj ; and w

x 2 Cqj

ð5Þ

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Fig. 2. The time histories of chemical damage (a) and space distributions of solid calcium content at different time stations (b), obtained from simulations of the slow reaction–diffusion problem (Tleach = 109 s). Thin lines show the sCa distributions corresponding to chemical equilibrium.

3. Numerical solution Discretization in space of the governing equations is carried out by means of the finite element method (Zienkiewicz and Taylor, 2000a,b; Lewis and Schrefler, 1998). The unknown variables are expressed in terms of their nodal values as

 g ðtÞ; pc ðtÞ ¼ Np p  c ðtÞ; pg ðtÞ ¼ Np p  ðtÞ: cCa ðtÞ ¼ Ns cCa ðtÞ; uðtÞ ¼ Nu u

ð6Þ

Galerkin’s method is then applied to the model equations in weak form, (1)–(4). The details of transformations to be performed to Eqs. (1), (2) and (4) in order to solve them with FEM are explained in (Lewis and Schrefler (1998)). Here we present the result in extensive form only for the dissolved calcium mass conservation, which was not dealt with by Lewis and Schrefler (1998).

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Fig. 3. The time histories of liquid calcium concentration (a) and its space distributions at different time stations (b), obtained from simulations of the fast reaction–diffusion problem (Tleach = 1.25  108 s, solid markers) and compared to the reference case 1 (Tleach = 1  109 s, empty markers).

Eq. (3) results in the following form:

!  oT 1  n oqs oCleach w w T ou dX  q bsw Nt þ q Sw s þ aq Sw m B ot ot q oCleach ot X     Z  c oSw oSw op ocCa ocCa w þ NTs cCa nqw N N q S N þ þ n dX p s w s opc ot ocCa ot ot X Z   ðrNs ÞT ðqw DCa d rNs cCa ÞdX   ZX rw kk  g  rNp p  c þ qw gÞ dX  ðrNs ÞT cCa qw w ðrNp p l X    Z Z cCa qw Sw T ^ w dC ¼ 0 _ diss 1   Ns m NTs ^cCa q dX þ s Z

NTs cCa

X

w

q

CqCa

ð7Þ

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Fig. 4. The time histories of chemical damage (a) and space distributions of solid calcium content at different time stations (b), obtained from simulations of the fast reaction–diffusion problem (Tleach = 1.25  108 s, solid markers) and compared to the reference case 1 (Tleach = 1  109 s, empty markers).

where B = LNu. For all four discretized governing equations the terms related to different state variables and their time derivatives can be grouped together and the model equations can be written in matrix form as follows:

Þ Cij ðx

 ox  Þx  ¼ f i ðx Þ þ Kij ðx ot

ð8Þ

with

2

Cgg 60 6 Cij ¼ 6 40 Cug

Cgc

Cgs

Ccc

Ccs

Csc Cuc

Css Cus

2 Kgg 7 6K Ccu 7 6 cg 7; Kij ¼ 6 4 Ksg Csu 5 Cuu 0 Cgu

3

Kgc

Kgs

Kcc

Kcs

Ksc

Kss

Kuc

0

8 9 fg > > > > > = 7 0 7 c 7; f i ¼ > 0 5 fs > > > > ; : > Kuu fu 0

3

ð9Þ

g ; p c;   g and the non-linear matrix coefficients Cij(x), Kij(x) and fi(x) are defined in detail in where xT ¼ fp cCa ; u Appendix. The time discretisation is carried out by means of a fully implicit finite difference scheme (backward difference):

Wi ðxnþ1 Þ ¼ Cij ðxnþ1 Þ

xnþ1  xn þ Kij ðxnþ1 Þxnþ1  f i ðxnþ1 Þ ¼ 0 Dt

where superscript i (i = g,c,s,u) denotes the state variable, n is the time step number and Dt the time step length.

ð10Þ

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Fig. 5. The space distributions of solid calcium mass source intensities at different time stations, obtained from simulations of the leaching process with two different values of Tleach: (a) Tleach = 1  109 s, (b) Tleach = 1.25  108 s. In the latter case the results (empty markers) are superimposed to the results of reference case 1 (Tleach = 1  109 s, solid markers).

The equation set (10) is solved by means of a monolithic Newton–Raphson iterative procedure (Lewis and Schrefler, 1998): i

W

ðxknþ1 Þ

 oWi  ¼  ox 

Dxknþ1 ;

k k xkþ1 nþ1 ¼ xnþ1 þ Dxnþ1

ð11Þ

Xknþ1

where k is the iteration index and the Jacobian matrix is defined as

2

 oWi   ox 

xknþ1

oWg g 6 op 6 6 oWc 6 op 6 g ¼6 6 oWs 6 op g 6  4 oWu g op

oWg c op oWc c op oWs c op oWu c op

oWg ocCa oWc ocCa oWs ocCa oWu ocCa

3 oWg   7 ou 7  oWc 7   7 ou 7 s 7 oW 7   7 ou 7 u 5 oW   x¼xk ou

nþ1

ð12Þ

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Fig. 6. Comparison of the results concerning progress of the portlandite dissolution front (sCa ffi 9 kmol/m3) in function of the square root of time, obtained for the analyzed cases of the wall leaching process: (a) with no advection (Dpw = 0), (b) with advection (Dpw = 5 bar). The results for the slow processes (Tleach = 109 s) are marked with triangles, and for the fast ones (Tleach = 1.25  108 s) with diamonds.

The linearized equation system (11) is solved with a solver of frontal-type (Bianco et al., 2003). Alternatively, a parallel version of the solver, using multi-frontal method can be used (Wang et al., 1996). The same method has been previously successfully used for solving such difficult coupled multi-physics problems like damaging of concrete at high temperature (Gawin et al., 2003) and autogeneous, shrinkage and creep strains of concrete at early ages (Gawin et al., 2006). The numerical procedure described above has been implemented in the research computer code COMES-LEACH. This will be used for all the simulations described in next section. 4. Numerical examples Three numerical examples are solved and discussed in this section to demonstrate the robustness of the code and analyze the effect of various factors on the results concerning chemical degradation of structures made of cementitious materials. The first example is based on a test reaction–diffusion problem solved previously by Kuhl et al. (2004) and Kuhl and Meschke (2007). It concerns calcium leaching from a cement paste wall (1D problem) exposed to one-side action of deionized water which is modeled by means of Dirichlet’s BCs, the same as in (Kuhl and Meschke (2007)).

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Fig. 7. Comparison of the simulation results of the slow leaching process (Tleach = 1  109 s), obtained for the liquid calcium concentration from simulations with 1/g = 5.4  108 mol/(J s) for two different values of water pressure difference (Dpw = 0 – bold lines with solid markers, Dpw = 5 bar – thin lines with empty markers): (a) space distribution, (b) time histories.

This example is solved for two different rates of decrease of liquid calcium concentration on the surface, with the same water pressure at both wall surfaces (case of leaching due only to diffusion of calcium ions) or with higher values of the pressure at the surface exposed to deionized water action (not considered by Kuhl and Meschke (2007)) what causes additional advective flow of the pore solution. The effect of the characteristic time of leaching on the simulation results is also analyzed for the case of a slower process without a gradient of water pressure. Then, numerical performance of the computer code is tested using the fast reaction–diffusion problem, but with a 2.5-times higher rate of calcium concentration decrease on the surface. The second example is similar to the first one, but with physically more realistic boundary conditions considered. For the case without external pressure gradient, diffusive exchange of mass on the surface of a concrete wall, due to the difference of calcium concentration on the surface and in the surrounding liquid is assumed (Robin’s BCs for calcium described by the second term on the rhs of Eq. (83) in Gawin et al., 2008). The example is solved for two

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Fig. 8. Comparison of the simulation results of the slow leaching process (Tleach = 1  109 s), obtained from simulations with 1/g = 5.4  108 mol/(J s) for two different values of water pressure difference (Dpw = 0 – bold lines with solid markers, Dpw = 5 bar – thin lines with empty markers): (a) space distributions of the solid calcium content, (b) time histories of the chemical damage at different positions.

different values of mass exchange coefficient dc. The case with external pressure gradient and resulting advective flow of water is solved assuming that pure water enters the surface of a cement paste wall with a higher pressure, and water containing calcium with a concentration corresponding to the surface value, flows out through the other side. The third, 2D example concerns chemical degradation of a concrete square sample immersed in deionized water. For description of mass exchange of calcium on the external surfaces of sample, convective-type boundary conditions (Robin’s BCs) are assumed. The analyzed problems are solved with the assumption that the characteristic time of the dissolution process of different solid skeleton components is the same, and equal to that of the portlandite. The latter process kinetics influences to the greatest extent the overall material performance, as can be seen in Figs. 6 and 7 of the companion paper (Gawin et al., 2008).

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Fig. 9. Comparison of the simulation results of the fast leaching process (Tleach = 1.25  108 s), obtained for the liquid calcium concentration from simulations with 1/g = 5.4  108 mol/(J s) for two different values of water pressure difference (Dpw = 0 – bold lines with solid markers, Dpw = 5 bar – thin lines with empty markers): (a) space distribution, (b) time histories.

4.1. Leaching of a cement paste wall – test problem with Dirichlet’s BCs This example deals with a 1D calcium leaching problem where chemical action of deionized water on the surface of a 16-cm cement paste wall is modeled with Dirichlet BCs, the same as in (Kuhl and Meschke (2007)). The liquid calcium concentration on the wall surface exposed to chemical degradation is decreasing linearly during the time span Tleach, from the initial value c0Ca ¼ 20:6 mol=m3 , i.e. the value typical for a sound cementitious material at T = 25 °C, to the final value c1Ca ¼ 1:0 mol=m3 , and then kept constant. This process results in a gradual dissolution of calcium contained in the material skeleton and related transport phenomena of the Ca2+ ions. The characteristic time of leaching sleach = 2 h, corresponding to 1/g = 5.40  108 mol/(J s), which is in a possible range of values, see Table 1 of the companion paper (Gawin et al., 2008), is assumed. This test problem is solved for four different cases of boundary conditions (Table 1):

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Fig. 10. Comparison of the simulation results of the fast leaching process (Tleach = 1.25  108 s), obtained from simulations with 1/g = 5.4  108 mol/(J s) for two different values of water pressure difference (Dpw = 0 – bold lines with solid markers, Dpw = 5 bar – thin lines with empty markers): (a) space distributions of the solid calcium content, (b) time histories of the chemical damage at different positions.

1. The same water pressure, pw = 1 bar, is maintained on both wall surfaces (no advective water flow), and liquid calcium concentration on the wall exposed to chemical action decreases during a time span of Tleach = 109 s (the highest value considered by Kuhl and Meschke (2007)). The second wall surface remains impermeable for water and dissolved calcium. This case will be called later on ‘case 1’ or ‘slow reaction–diffusion problem’; 2. The same as case 1, but with Tleach = 1.25  108 s (the lowest value considered by Kuhl and Meschke (2007)), called later on ‘case 2’ or ‘fast reaction–diffusion problem’. To analyze numerical performance of the code and perform some comparisons, the example was also solved for Tleach = 5  107 s (called ‘very fast reaction–diffusion problem’). The latter case was not considered in (Kuhl and Meschke (2007)) and its results will be not thoroughly analyzed here; 3. The water pressure on the surface exposed to chemical action with Tleach = 109 s is kept constant at p1w ¼ 6 bar, while at the other surface, which is now permeable for water and dissolved calcium, a constant water pressure p2w ¼ 1 bar is maintained. This case will be called later on ‘case 3’ or ‘slow reaction–advection–diffusion problem’; 4. The same as case 3, but with Tleach = 1.25  108 s, called later on ‘case 4’ or ‘fast reaction–advection–diffusion problem’.

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Fig. 11. Comparison of the space distribution changes of liquid calcium concentration (a) and solid calcium content (b), obtained from simulations of the slow reaction–diffusion problem (Tleach = 1  109 s) for two different values of characteristic time of leaching: (1/g = 5.4  108 mol/(J s) – bold lines with solid markers, 1/g = 1.08  108 mol/(J s) – thin lines with empty markers, chemical equilibrium values – thin lines without markers).

In order to analyze the effect of the characteristic time of leaching sleach on the results obtained, simulations for case 1 are additionally performed, but assuming the sleach-value five-times larger, i.e. five times slower calcium dissolution process for any pair of values, (cCa, sCa). This will be called later on ‘case 1b’. The wall is modeled by means of 100 isoparametric eight-noded finite elements of equal size. The material properties assumed in our simulations are presented in Table 2. The simulations are performed for all the analyzed cases with constant time step length Dt = 0.1 day, for time span of 2  109 s, the same as that considered by Kuhl and Meschke (2007). The results of the simulations, concerning time histories indicated with (a) and distributions at different time stations (b) of liquid calcium concentration and solid calcium content, obtained for the slow reaction–diffusion problem, are presented in Figs. 1 and 2, respectively. For all the considered numerical examples, the time history of solid calcium content is presented

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Fig. 12. Comparison of the results concerning progress of the portlandite dissolution front (sCa ffi 9 kmol/m3) in function of the square root of time, obtained from simulations of the slow reaction–diffusion problem (Tleach = 1  109 s), with two different 1/g-values: 1/g = 5.4  108 mol/(J s) – marked with diamonds and 1/g = 1.08  108 mol/(J s) with triangles.

Table 3 Scheme of the BCs assumed in Section 4.2

BC type

Neumann’s

a

Case

1 2 3

1/g

BC on side d

(1/s)

Calcium

5.4108 5.4108 2.16108

dc ¼ 106 kg=m s, cCa1 ¼ 0 mol=m 2 3 dc ¼ 105 kg=m s, cCa1 ¼ 0 mol=m 2 qCa ¼ 0 kg=m s

BC on side b

2

3

Figures with results

Water

Calcium

Water

pw ¼ 1 bar pw ¼ 1 bar pw ¼ 6 bar

– – qCa ¼ cCa qw ; cCa ¼ cCa surf a

pw ¼ 1 bar pw ¼ 1 bar pw ¼ 1 bar

15–17 13, 14, 17 18–21

cCa surf is the value of calcium concentration on the boundary ‘‘b”.

as the chemical damage evolution. The corresponding results for the fast reaction–diffusion problem are presented in Figs. 3 and 4. Comparison of the (cCa, sCa) points, corresponding to the equilibrium state at temperature T = 298.15 K, sCa ¼ seq Ca ðc Ca Þ, and those obtained from the simulations after every 2000 time steps in all the Gauss points, for the abovementioned cases, is presented in Fig. 6a and b in the companion paper (Gawin et al., 2008). The simulation results concerning liquid calcium concentration, obtained here for the both cases are practically the same to those presented in (Kuhl and Meschke (2007)). However, due to different models of calcium mass sources, non-equilibrium state here, while equilibrium was assumed in (Kuhl and Meschke (2007)), the progress of solid calcium dissolution is visibly slower in our results, especially at initial stages of the process when portlandite is dissolved. This is shown in Fig. 2b, where the computed calcium content profiles are compared to the profiles of equilibrium values, seq Ca ðc Ca Þ, corresponding to the liquid calcium concentrations obtained from the simulations at the same time instants. The effect is also visible in Fig. 6a and b in the part 1 paper (Gawin et al., 2008), especially at the portlandite dissolution front. As it could be expected, there is a visible effect of the rate of liquid calcium outflow through the wall surface (modeled in this example with Dirichlet’s BCs) upon the process evolution and in particular the decalcification front progress, Fig. 2a and Fig. 4a. For the faster decrease of liquid calcium concentration on the wall surface, progress of the portlandite dissolution front is faster and the related mass sources of liquid calcium are greater, see Fig. 5a and b, especially at initial stages of the process. This is caused by a faster calcium outflow from the wall during the initial period: Tleach = 109 s for case 1 and

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Fig. 13. The time histories of liquid calcium concentration (a) and its space distributions at different time stations (b), obtained from simulations of the natural reaction–diffusion problem with dc = 105 kg/(m2 s).

pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffi Tleach = 1.25  108 s for case 2 ( 109 ffi 3:16  104 and 1:25  108 ffi 1:12  104 on the t axis in Fig. 6), when the Dirichlet BCs with decreasing cCa value are assumed on the wall surface. Then the difference is gradually decreasing due to slightly smaller gradients of liquid calcium concentration for the process with Tleach = 1.25  108 s, Fig. 1a and Fig. 3a. To analyze those phenomena more profoundly, the graphs of the portlandite front position (assuming that it corresponds pffiffi to the value sCa ffi 9 kmol/m3) in function of the square root of time, x ¼ f ð t Þ, are shown in Fig. 6a for the two reaction–diffusion problems. Usually such a graph is linear as for all processes governed by a diffusion-type transport. This is perfectly the case for the period (indicated in Fig. 6 with empty markers) when the constant value of liquid calcium concentration cCa = 1 mol/m3 is assumed on the wall surface and the correlation coefficients for an linear relationship have values very close to one. The simulation results, obtained for the fast and slow reaction–advection–diffusion problems, are quite similar to those of the corresponding cases without the advective water flow, especially at initial stages of the wall decalcification. For this reason, only the comparison of the results with those for the corresponding reaction–diffusion problems are presented in Fig. 7a and b and Fig. 8a and b for case 3 (Tleach = 109 s), and in Fig. 9a and b and Fig. 10a and b for case 4 (Tleach = 1.25  108 s). As it

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Fig. 14. The time histories of chemical damage at different positions (a) and space distributions of solid calcium content at different time stations (b), obtained from simulations of the natural reaction–diffusion problem with dc = 105 kg/(m2 s).

can be observed for the both cases, the difference of the results is gradually increasing with time, showing that the advective water transport has increasing influence on the results obtained, in particular at the advanced stages of the process, when the calcium mass flux caused by diffusion gradually decreases due to decreasing gradients of liquid calcium concentration, see Fig. 7a and Fig. 9a. However, despite of the application of a considerable water pressure gradient with Dpw = 5 bar, causing inflow of water of lower calcium concentration, only a slight acceleration of the wall decalcification is observed both for the fast and slow processes, Fig. 10a and Fig. 8a. This can also be seen by comparing Fig. 6b and a, presenting progress of the portlandite dissolution front in the wall for all the considered cases. The latter figures also show that the overall form of the pffiffi x ¼ f ð t Þ graphs for the fast and slow processes remained similar as for the cases without advection, showing the dominant role of the diffusive calcium transport. Only a small trend towards faster front progress can be observed for higher time values, confirming the increasing role of the advective mass transport. To analyze the effect of the characteristic time of leaching upon the simulation results, the slow reaction–diffusion problem has been solved for a characteristic time of leaching five-times longer than in case 1, corresponding to a

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Fig. 15. The time histories of liquid calcium concentration (a) and its space distributions at different time stations (b), obtained from simulations of the natural reaction–diffusion problem with dc = 106 kg/(m2 s) (empty markers) compared to the case with dc = 105 kg/(m2 s) (solid markers).

five-times smaller value of the 1/g-parameter, 1/g = 1.08  108 mol/(J s). Results of the simulations (empty markers) are compared in Fig. 11a and b to those of case 1 with 1/g = 5.40  108 mol/(J s) (full markers). In general, the differences are relatively small, hence only the results concerning space distributions of liquid calcium concentration and solid calcium content, at five different time stations, are presented here. As can be observed, assumption of a five-times greater value of characteristic time of leaching has a relatively small effect on the liquid calcium solution and the solid calcium content. Only a slightly slower leaching for the case of 1/g = 1.0  108 mol/(J s) is visible and limited to the period or wall zone where the portlandite dissolution takes place. The assumed value of 1/g parameter gives the solutions which are very close to chemical equilibrium state (see thin lines in Fig. 11b). Very small differences can also be observed in Fig. 12 where the progress of the portlandite dissolution front is compared for the two considered 1/g-values, showing that the parameter has a small effect on the long-term assessment of chemical degradation extent for the wall. However, assumption of a lower value of the 1/g-parameter results in visibly better numerical performance of the computer code due to lower intensity of the calcium mass sources. Finally, to analyze the numerical performance of the code, and in particular the effect of the time step length on the solution convergence and the results obtained, a very fast reaction–diffusion problem, with Tleach = 5  107 s, has been solved. It should be underlined, that due to the mathematical model used for description of calcium leaching, a time step length applied in the simulations cannot be too great, not only for numerical (accuracy) reasons (the fully explicit finite difference scheme is applied here for integration of the solid calcium mass conservation equation) but also for physical reasons. In Eq. (44) of the companion paper (Gawin et al., 2008), the calcium mass source intensity is dependent on the ‘thermodynamic distance’ from the equilibrium curve, increasing rapidly with an increasing ‘imbalance’, see Fig. 4 of the part 1 paper (Gawin et al., 2008). Hence, the greater the time step length Dt used in simulations and the

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Fig. 16. The time histories of chemical damage at different positions (a) and space distributions of solid calcium content at different time stations (b), obtained from simulations of the natural reaction–diffusion problem with dc = 106 kg/(m2 s) (empty markers) compared to the case with dc = 105 kg/ (m2 s) (solid markers).

related change of liquid calcium concentration DcCa at a single time step, the greater is the mass source intensity. As a result, an increase of dissolved calcium mass calculated for one longer time step Dt can be much higher than that obtained during k consecutive time steps with the length of Dt/k, what additionally can also cause some numerical problems. To analyze this problem more in detail, the simulations have been performed with the same time step as in the all previous cases, Dt = 0.1 days, and with a value two-times smaller, as well as 2- and 4-times higher. The results for the case with Dt = 0.05 days, are indistinguishable from those with Dt = 0.1 days (for this reason they are not presented here), and numerical performance (number of iterations at every time step) has been very similar. A somewhat worse solution convergence (i.e. number of iterations higher by a factor 1 or 2) as well as slightly lower liquid calcium concentrations and practically the same results for the solid calcium content have been obtained for Dt = 0.2 days. For the greatest considered value of time step, Dt = 0.4 days, it was impossible to obtain converging solution already at initial stages of the process evolution, when the portlandite dissolution front started to form. Concluding, one can state that the model of calcium leaching kinetics proposed in part 1 paper (Gawin et al., 2008) and the method for numerical solution of the related initial boundary problem allowed for effective numerical solution of the test reaction–diffusion–advection problem, up to a limit value of time step length, even with a very fast decrease of liquid calcium concentration on the wall surface (which was not possible to solve by using the equilibrium description of the leaching process in our model). At the same time, when applying for the same examples the calcium mass sources defined by the equilibrium formulation, Eq. (35) of the companion paper (Gawin et al.,

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Fig. 17. Progress of the portlandite dissolution front (sCa ffi 9 kmol/m3) in function of the square root of time, obtained from the simulations of natural reaction–diffusion problem for two different values of mass exchange coefficient: dc = 105 kg/(m2 s) and dc = 106 kg/(m2 s).

2008), we observed serious numerical problems with convergence, especially at initial stages of leaching process, when osCa/ ocCa ffi 0. 4.2. Leaching of a wall modeled with Neumann’s BCs This example is similar to the first one, but here a more realistic description, from a physical point of view, of calcium mass exchange on the interface between an element made of cementitious material and surrounding water is assumed. The geometry of the wall is the same as in the previous example. Full control of liquid calcium concentration on external surfaces of the element immersed in water solution (corresponding to Dirichlet’s BCs) is possible only in laboratory conditions, especially in the case of rapid concentration changes. In field conditions, the calcium concentration value on an element surface results from the mass exchange of calcium ions between the element and surrounding water. In absence of water pressure gradients, this process is governed by ions’ diffusion due to the difference of calcium concentration on the surface and in the surrounding liquid and it can be described by Robin’s boundary conditions (the second term on the rhs of Eq. (96) for calcium in the companion paper Gawin et al., 2008). When water advective flow through the element surface due to existing water pressure gradients is observed, the calcium mass flux is strictly related to the water mass flux, as mentioned at the end of Section 5 in the part 1 paper (Gawin et al., 2008). Two different physical problems are analyzed here. The first one concerns leaching of a concrete wall immersed in pure water (cCa = 0 mol/m3), where the whole process is controlled by diffusion. It will be called later on ‘natural reaction–diffusion problem’. The example is solved for two different values of mass exchange coefficient dc = 106 kg/(m2 s) and dc = 105 kg/(m2 s), cases 1 and 2 in Table 3. The second value is rather high and corresponds to the forced-convection case, while the first one to the natural convection. The simulations have been performed with constant time step length Dt = 0.1 day, for the time span of 105 days. The second problem concerns leaching of a cement paste wall exposed to external pressure gradient of 5 bar, causing advective flow of water through the element pores. We assume that pure water (cCa = 0 mol/m3) enters the wall surface with higher pressure, and water containing calcium in concentration corresponding to the value at surface with lower pressure, flows through the other side out, case 3 in Table 3. This problem will be called later on ‘natural reaction–advection–diffusion problem’. This example has been solved with constant time step length Dt = 0.1 day for the time span of 2  105 days. The main material properties of a concrete and a cement paste assumed in the simulations are given in Table 2. Figs. 13 and 14 show the results of simulations, concerning time histories (a) and distributions at different time stations (b), for liquid calcium concentration and solid calcium content (shown in terms of chemical damage), obtained for the natural reaction–diffusion problem with an intensive calcium mass exchange on the wall surface, dc = 105 kg/(m2 s). The corresponding results for the case of dc = 106 kg/(m2 s) are presented in Figs. 15 and 16. Comparison of the (cCa, sCa) points, corresponding to the equilibrium curve sCa ¼ seq Ca ðcCa Þ at temperature T = 298.15 K, and those obtained from the simulations every 5000 time steps in all the Gauss points, for the two abovementioned cases, is presented in Fig. 7a and b in the companion paper (Gawin et al., 2008). For the case of natural convective mass exchange, dc = 106 kg/(m2 s), all the graph points, even those related to the wall surface, are very close to the equilibrium curve. For the other case, i.e. with dc = 105 kg/(m2 s),

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a greater dispersion of the (cCa, sCa) points can be noticed, especially for those corresponding to the surface zone where a very fast change of liquid calcium concentration occurs, Fig. 13b. As can be observed, for the convective-type boundary conditions, even with very high value of calcium mass exchange coefficient, dc = 105 kg/(m2 s), the evolution of leaching process is much slower than that obtained for the slow test reaction–diffusion problem (i.e. with Dirichlet BCs and for Tleach = 109 s), compare Figs. 1 and 2 with Figs. 13 and 14. After the time span of DT = 105 days, the progress of the portlandite dissolution front for convective-type boundary conditions with dc = 105 kg/(m2 s) corresponds to that obtained for the Dirichlet BCs after DT1 ffi 2.0104 days, and for the case with dc = 106 kg/(m2 s) even after DT2 ffi 4.0  103 days. For both the considered cases, the graphs showing time-evolution of liquid calcium concentration on the wall surface have a non-linear character, i.e. different from the linear variation of the quantity assumed in Dirichlet’s BCs in previous section. This clearly shows that boundary conditions of the latter type are rather far from physical reality and should not be used during analysis of practical engineering problems. pffiffi The progress of the portlandite dissolution front position in function of the square root of time, x ¼ f ð t Þ, for the two analyzed cases of dc-value are presented in Fig. 17. Both graphs have almost perfectly linear character (the values of R2 are very

Fig. 18. The time histories of liquid calcium concentration (a) and its space distributions at different time stations (b), obtained from simulations of the natural reaction–advection–diffusion problem with Dpw = 5 bar.

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Fig. 19. The time histories of chemical damage of solid skeleton (a) and the space distributions of solid calcium content at different time stations (b), obtained from simulations of the natural reaction–advection–diffusion problem with Dpw = 5 bar.

close to one) what results from the fact that the analyzed process is fully controlled by diffusive mass transport, both inside the wall and on its surface. The simulations results of the natural reaction–advection–diffusion problem (case 3 in Table 3) are presented in Figs. 18– 21. Due to some convergence problems in the surface zone, observed at initial stages of the process evolution (visible in Fig. 18a), a 2.5-times lower value of the 1/g parameter, 1/g = 2.16  108 mol/(J s), is assumed for this example. The problems are caused by a local and intermittent increase of liquid calcium concentration above an equilibrium value and resulting ceasing of the mass sources due to the advective transport of calcium dissolved closer to the surface, see Fig. 7c in the companion paper (Gawin et al., 2008). The space distribution, at different time stations, of liquid calcium concentration and solid calcium content are shown in Fig. 18b and Fig. 19b, while their time evolutions at seven different distances from the surface exposed to higher water pressure – in Fig. 18a and Fig. 19a, respectively. Additionally, the space distributions of calcium mass source intensity are presented in Fig. 20. First of all, a very slow development of the calcium leaching process, considered in this example, should be underlined. After 200,000 days (i.e. about 548 years) the zone, where the portlandite is fully leached, is of only about 33 mm.

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Fig. 20. The space distributions of liquid calcium source intensity at different positions of the wall surface, obtained from simulations of the natural reaction–advection–diffusion problem with Dpw = 5 bar.

Fig. 21. Progress of the portlandite dissolution front (sCa ffi 9 kmol/m3) in function of the time, obtained from the simulations of natural reaction– advection–diffusion problem with Dpw = 5 bar.

Due to this, in almost all the points and during the whole analyzed time span, thermodynamic state of the material is very close to the equilibrium (except of a limited number of situations mentioned in previous paragraph), see Fig. 7c in the part 1 paper (Gawin et al., 2008). Then, the results of this example, in which the whole process is controlled by the advective water flow, have a visibly different character from that in the previous examples. Here, outflow of liquid calcium from the zone close to the surface, where pure water enters the wall, and dissolution of calcium from the skeleton takes place, is caused only by advection. The leached calcium is not fully removed from the wall (due to the BCs) and it accumulates close to the ‘outflow’ surface, causing an increasing gradient of liquid calcium concentration, Fig. 18a. Due to the gradient, diffusive mass transport of calcium ions, towards the surface with higher water pressure (i.e. in opposite direction than the advective mass flux) is observed. As a result, after an initial period (the longer, the greater the distance of a point from the ‘inflow’ surface is) the

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Table 4 Scheme of a concrete sample with square cross-section, showing the BCs and positions of the selected points

Side

Variables

Values and coefficients

BC type

a, b c

cCa uy cCa ux cCa

cCa1 ¼ 0:0 ½mol=m3 ], dc ¼ 105 (kg m2 s1 ) uy ¼ 0 qCa ¼ 0 ux ¼ 0 qCa ¼ 0

Robin’s Dirichlet’s Neumann’s Dirichlet’s Neumann’s

d

content of liquid calcium is decreasing gradually with a constant rate, practically the same in all the points, see Fig. 18b. This causes a linear-in-time progress of the portlandite dissolution front, Fig. 19a and b, and resulting calcium mass sources of practically constant intensity (except of an initial period when the front is formed near the ‘inflow’ surface), as can be seen in Fig. 20. A progress of the portlandite dissolution front, strictly proportional to time is confirmed by the graph presented in Fig. 21. 4.3. Leaching of a concrete square sample This 2D example deals with calcium leaching of a concrete sample with square cross-section (4 cm  4 cm, i.e. dimensions typically used in material laboratories), exposed to Robin’s boundary conditions. Boundary conditions of the convective-type, the same as in Section 4.2 for the natural reaction–diffusion problem with dc = 105 kg/(m2 s), and the analyzed time span of 1000 days are considered. The same material properties as in previous sections, given in Table 2, and 1/g = 5.4  108 mol/(J s) are assumed. The sample is modeled with 400 (20  20) isoparametric eight-noded FEs of equal size. Simulations are performed with constant time step length of Dt = 0.1 day. The scheme of the sample with the assumed BCs and the positions of five characteristic points, where time evolutions of physical quantities are analyzed, are shown in Table 4. The results of computations, concerning the space distribution of liquid calcium concentration and solid calcium content at the end of the simulations, are shown in Fig. 22a and b, respectively. The time evolutions of liquid calcium concentration and solid calcium content in the points: A, B, C, D and E (see Table 4 for their positions), are presented in Fig. 23a and b, respectively. Similarly as for the 1D example in Section 4.2, a very slow progress of decalcification process is observed. After 5000 days, the front of portlandite leaching has reached the core part of sample, with a radius of about 13 mm, and leaching of the CSH phases has not started yet, Fig. 22b. The progress of calcium leaching in the 2D sample is visibly faster than in the wall (1D structure). The solid calcium content sCa ffi 9.0 kmol/m3 is reached in the point D on the sample diagonal, in about 1-cm distance from the surfaces, after about 5000 days (versus 6500 days in the wall), see Fig. 23b and Fig. 16b. Faster evolution of calcium leaching in the 2D structures, as compared to the wall (1D) exposed to the same BCs, is caused by easier outflow of calcium ions from the corner zone, due to 2D character of mass transport towards two surfaces of the sample, as compared to 1D calcium outflow in the wall. 5. Conclusions The weak form of the equations of mathematical model of hygro-chemo-mechanical processes in cementitious materials, proposed in part 1 of the companion paper (Gawin et al., 2008), has been obtained using Galerkin’s method. Discretization in space of the equations by means of finite element method and in time domain using finite differences has been performed. The obtained non-linear equation set has been solved with a Newton–Raphson procedure. Three numerical examples concerning leaching of 1D and 2D structures, made of cement-based materials, exposed to various boundary conditions have been solved and discussed. They show numerical robustness of the non-equilib-

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Fig. 22. Space distributions of liquid calcium concentration (a) and solid calcium content (b) obtained after t = 5000 days for the concrete sample leaching process modeled with Robin’s BCs.

rium model of calcium leaching as compared to the traditional model based on thermodynamic equilibrium assumption, what allows for analysis of very fast processes. They show also significant differences of the results concerning calcium leaching modeled with Dirichlet’s and Robin’s boundary conditions. The latter boundary conditions describe much better the phenomenon’s physics in real situations and they result in much slower progress of the calcium leaching process. Elements made of a cement based material, exposed to direct contact with deionized water from more than one direction, like for example columns, groins or corners, are jeopardized by much faster progress of chemical degradation than 1D structures, due to easier outflow of calcium ions from the leaching zone where calcium concentration is elevated.

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Fig. 23. The time histories of liquid calcium concentration (a) and solid calcium content (b), obtained from simulations of the concrete sample leaching process modeled with Robin’s BCs.

One can also state that the faster leaching process is, the more distant from the equilibrium state is calcium contained in the material skeleton and ions in pore solution, what influences the leaching process rate. The numerical model presented allows for analyses of durability of structures made of cement composites, in various conditions, also those which were not modeled before, like for example leaching due to existing water pressure gradient. Acknowledgements This research was carried out as part of the project ‘Mathematical-Numerical modeling of the short-long term behavior of cement based materials’ (prot.60A09-8287/07) at the University of Padova, Italy. The research was also partly supported by the Polish Ministry of Science and Higher Education, project No. 4 T07E 032 30 ‘Modeling stresses and degradation of external layers of building walls exposed to variable conditions of internal and external climate’ at the Technical University of Lodz, Poland.

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Appendix For the case of visco-elastic behavior of the solid skeleton, matrices appearing in Eq. (9) are as follows:

  Ma Np dX NTp ð1  Sw Þn RT X  Z oS Mw w Cgc ¼ NTp nqga c Np þ ð1  Sw Þn op RT X  Z oS M w w Cgs ¼ NTp nqga Ns þ ð1  Sw Þn ocCa RT X Z

 Cgu ¼ NTp ð1  Sw Þqga mT LNu dX Cgg ¼

Z

X

 kk Kgg ¼  ðrNp ÞT qa g ðrNp Þ dXþ l X " !# Z pgw T g M a M w gw Dg  r Np d X  ðrNp Þ q M 2g ðpg Þ2 X " # Z gw T g M a M w gw 1 op Kgc ¼  ðrNp Þ q Dg g r Np d X p opc M 2g X " # Z M a M w gw 1 opgw D r N Kgs ¼  ðrNp ÞT qg s dX g pg ocCa M 2g X   Z rg kk f g ¼ ðrNp ÞT qga g qg g dX l X   Z oqs oCleach T 1n  Np S dX qs g oCleach ot X   Z _ diss m S dX  NTp qs g X   Z Mw opgw w gw oSw Ccc ¼ NTp ð1  Sw Þn N þ nð q  q Þ N p p dX RT opc opc X   Z Mw opgw oSw Ns dX Ccs ¼ NTp ð1  Sw Þn þ nðqw  qgw Þ RT ocCa ocCa X Z NTp f½qgw ð1  Sw Þ þ qw Sw mT BgdX Ccu ¼ X " !# Z pgw T ga M a M w gw Dg  rNp dX Kcg ¼  ðrNp Þ q M 2g ðpg Þ2 X   Z rg rw kk kk  ðrNp ÞT qv g ðrNp Þ þ qw w ðrNp Þ dX l l X "  # Z gw M a M w gw 1 op dX D r N Kcc ¼ ðrNp ÞT qg p g pg opc M 2g X   Z rw kk  ðrNp ÞT qw w rNp dX l X "  # Z gw T g M a M w gw 1 op Kcs ¼  ðrNp Þ q Dg rNs dX pg ocCa M 2g X   Z rg rw kk kk f c ¼ ðrNp ÞT qgw g ðqg gÞ þ qw w ðqw gÞ dX l l X Z  NTp ½qw þ qgw þ bc ðqgw  qgw 1 ÞdC Z



ðA:1Þ  opga dX N p opc  opga Ns dX ocCa

ðA:2Þ ðA:3Þ ðA:4Þ

rg

ðA:5Þ ðA:6Þ ðA:7Þ

ðA:8Þ

ðA:9Þ ðA:10Þ ðA:11Þ

ðA:12Þ

ðA:13Þ ðA:14Þ

Cqc

Z

Cug

_ diss m  NTp s ðqgw Sg þ qw Sw ÞdX q X Z T T ¼ B m Np dX X

ðA:15Þ ðA:16Þ

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  ovws s oSw Np dX BT mT a vws s þ oSw opc  ws  ZX ovs oSw Ns d X Cus ¼  BT mT a oSw ocCa X Z Cuu ¼  BT Dc B dX

Cuc ¼ 

Z

ðA:17Þ ðA:18Þ ðA:19Þ

X

Kug ¼ 0

  ovws Kuc ¼  BT mT a s Np dX ot X Kus ¼ 0 Z oDc oV oCleach BT B dX Kuu ¼ oV oCleach ot X   Z oec oDc oV oCleach BT Dc ec dX fu ¼  ot oV oCleach ot X   Z oech oDc oV oCleach T þ B Dc ech dX  ot oV oCleach ot X Z Z o q ot  NTu NTu dC g dX  q ot ot X C  u Z oS w Csc ¼ NTs cCa nqw c Np dX; op X   Z oSw NTs cCa nqw þ nqw Sw Ns dX; Css ¼ ocCa X Z T w Csu ¼  Ns cCa aq Sw mT BdX; X  Z rw  kk Ksg ¼  ðrNs ÞT cCa qw w rNp dX; l X  Z rw  kk Ksc ¼  ðrNs ÞT cCa qw w rNp dX; l ZX T w Ca Kss ¼  ðrNs Þ ðq Dd ÞrNs dX Z X Z 1  n oqs oCleach ^w dC þ NTs cCa qw Sw NTs ^cCa q fs ¼  dX qs oCleach ot CqCa X   Z rw kk þ ðrNs ÞT cCa qw w qw g dX l   ZX cCa qw Sw T _ diss 1  þ Ns m dX s Z

X

ðA:20Þ ðA:21Þ ðA:22Þ ðA:23Þ

ðA:24Þ ðA:25Þ ðA:26Þ ðA:27Þ ðA:28Þ ðA:29Þ ðA:30Þ

ðA:31Þ

q

References Bianco, M., Bilardi, G., Pesavento, F., Pucci, G., Schrefler, B.A., 2003. A frontal solver tuned for fully-coupled nonlinear hygro-thermo-mechanical problems. International Journal for Numerical Methods in Engineering 57 (13), 1801–1818, doi:10.1002/nme.735. Gawin, D., Pesavento, F., Schrefler, B.A., 2003. Modelling of hygro-thermal behaviour of concrete at high temperature with thermo-chemical and mechanical material degradation. Computer Methods in Applied Mechanics and Engineering 192, 1731–1771. Gawin, D., Pesavento, F., Schrefler, B.A., 2006. Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II. Shrinkage and creep of concrete. International Journal for Numerical Methods in Engineering 67 (3), 332–363. Gawin, D., Pesavento, F., Schrefler, B.A., 2008. Modeling of cementitious materials exposed to isothermal calcium leaching, considering process kinetics and advective water flow. Part 1. Theoretical model. International Journal of Solids and Structures 45, 6221–6240. Kuhl, D., Bangert, F., Meschke, G., 2004. Coupled chemo-mechanical deterioration of cementitious materials. Part II. Numerical methods and simulations. International Journal of Solids and Structures. 41, 41–67. Kuhl, D., Meschke, G., 2007. Numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous Galerkin time integration schemes. International Journal for Numerical Methods in Engineering 69, 1775–1803. Lewis, R.W., Schrefler, B.A., 1998. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, second ed. John Wiley & Sons, Chichester. Wang, X., Gawin, D., Schrefler, B.A., 1996. A parallel algorithm for thermo-hydro-mechanical analysis of deforming porous media. Computational Mechanics 19 (2), 94–104. Zienkiewicz, O.C., Taylor, R.L., 2000a. The Finite Element Method. Volume 1. The Basis. Butterworth-Heinemann, Oxford. Zienkiewicz, O.C., Taylor, R.L., 2000b. The Finite Element Method. Volume 2. Solid Mechanics. Butterworth-Heinemann, Oxford.