Modeling of Stiffness Degradation and Expansion in Cement Based Materials Subjected to External Sulfate Attack Barzin Mobasher Department of Civil and Environmental Engineering, Arizona State University

Transport Properties & Concrete Quality Workshop Arizona State University October 10-12, 2005, Tempe, AZ

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Outline of Presentation  Overview

of the model  Governing mechanisms  Mathematical model  System Parameters  Parametric studies  Comparison with experimental results  Summary and Conclusions

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Overview of the Model Initial Material Parameters Cement composition w/c ratio Degree of hydration Mineral admixtures Initial sulfate content

Size & Shape of Members Specimen geometry Reinforcement type Boundary conditions

Computed Material Parameters Mechanical properties Capillary porosity Diffusivity, D Reactive calcium aluminate content

Exposure & Loading Sulfate exposure conditions Time & humidity

Service Life Expansion-Time History Degradation levels

Tixier, Mobasher, ASCE J. of Materials, 2003 You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)

Schematics of 1-D Diffusion of Sulfates

Penetration of Sulfates, Chemical reactions of ettringite formation or gypsum crystallization, expansion induced damage causing both decalcification or microcracking of the CSH paste.

tetracalcium aluminate hydrate, C4 AH 13 calcium aluminate monosulfate hydrate C 4 A S H 12 residual tricalcium aluminate

C3 A

ettringite formation

SO4-Aggressive solution

unreacted calcium aluminate sites

SO4--

Sulfates ingress front

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expansion

Sulfate Attack - Simplified approach 

1st step: formation of gypsum Ca(OH)2 + Na2SO4.10H2O  CaSO4.2H2O



2nd step: formation of ettringite. C4 AH 13  3 C S H 2  14H   C6 A S 3 H 32  CH C4 A S H 12  2 C S H 2  16 H   C6 A S 3 H 32 C3 A  3C S H 2  26 H   C6 A S 3 H 32

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Sulfate Attack - Simplified approach All reactions lumped in one:

CA  q S   C6 A S 3 H 32 with CA equivalent of C4AH13, monosulfate and residual C3A combined, and q equivalent stoichiometric coefficient obtained from the proportions and molar volume. Phase

Ettringite C3A C4AF Gypsum

Molar Volume 3 (cm /mole) 735.0 89.1 128.0 74.2

Molecular Weight (g/mole) 1249.5 269.9 477.4 172.1

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Specific Gravity 3 (g/cm ) 1.70 3.03 3.73 2.32

Solution of diffusion Equations: 1-D diffusion with first order reaction U=U0 @ T

U  2U D  kU 2 T X

U=U0 @ T

U=0 at T=0 reaction, k

Solution:

U 4  1  n X   1  sin  ( k   exp T (k   ) U0  m  0 n(k   )  L  n  2m  1

 n  ,   D   L 

2

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Diffusion & Reaction in Time & Space Standard Fick’s law Diffusion

f(p)= f(porosity, gel space ratio,..)

calcium aluminates

f(p)*[SO4--] Sulfate ingress ettringite front

SO4-D, [CA], p

[SO4--]

Present Model reacted calcium aluminates

calcium aluminates

Bound SO4--

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2nd Order Reaction Equation CA  q S   C6 A S 3 H 32 

Rates of reaction :

d U SO

4

dt

 - k U SO U CA 4

k U SO U CA d U CA dt q 4

with k rate constant, U molar concentration

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Diffusion Reaction equation for 2nd Order Reaction Equation   



Fick’s second Law- Diffusion as a function of time is driven by the concentration gradient. Assume no advection takes place, and no diffusion term for Calcium aluminates (immobile). Moving Boundary condition problem since sulfates are consumed, and the diffusion coefficient may be a function of location depending on the crack percolation. Let:

CA  q S   C6 A S 3 H 32 T time,

U  U SO and C  U CA

X distance,

4

U  2U D - kUC 2 T X

C  T

kUC q

D diffusion coefficient

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Concentration profiles for k=10-8 m3 /(mol.s), CA0=8.15 mol/m3 concentration/U0

1 0.9

SO4

0.8

CA

0.7 0.6 0.5

time

0.4 0.3 0.2

time

0.1 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

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0.4

0.45

0.5

Modeling of expansion: stress-strain response stress

damage 1

B



f’t

A

0 

w   log(sec  / 2)     1  w0 f 't  log(sec  0 / 2) 

0



O e th

• OA: linear elastic E= E0,  = 0 • AB: crack initiation E=E0(1-) • BC: crack propagation, E derived from expression:

ep

C strain Karihaloo and Fu, 1990

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Damage Evolution Law 0  i       (    ) 1 um  0



f (1 )  1

, 0  i  1  f (  1 )  1 1.00

Karihaloo and Fu, 1990 a= 0.16, b= 2.3 eum= ultimate strain for uniaxial tension

1

E( ) / E0

Em

Em 

b = 0.8

0.80 b = 0.6 b = 0.4

0.60

16  i (1 -  m2 ) 3

Nemat Nasser and Hori, Micromechanics: overall properties of Heterogeneous Materials, 1993

0.40 0.000

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0.002 0.004 Axial Strain, mm/mm

0.006

Simplified Approach- Fick’s second law of diffusion with a first order reaction 

U is the concentration of sulfates, k is a rate constant, D is the diffusivity m2/s, X is depth and T is time, n=2m+1.

U  2U D  kU 2 T X  n X  U 4  1  1  sin  k   exp T ( k   )    L  U0  m0 n ( k   )

D=1.14xx10-11

D=2.24x10-13

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Simplified solution algorithm     

 

Solve for sulfate distribution at a given time with a given diffusivity and rate of reaction. (D, and k= finite amount) Solve the PDE with k=0 for the amount of sulfates that would have penetrated if there were no reactions with C3A. Difference between the two levels is the amount of reacted sulfates. The average values were found by integrating the U/U0 values over the entire depth of the specimen The average value of U/U0 must be multiplied by the initial sulfate concentration at the surface (U0= 0.35moles/Lit, input) to find the concentration of reacted sulfates. Stoichiometric and molar volume relations are used to convert reacted sulfates to reacted aluminates, and ettringite formed. Volumetric changes are related to linear expansion values.

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Modeling of expansion: volumetric changes Reaction

P  aS   ettringite

P: expansive CA

Volume change in paste due to unit change in vol. of Hyd. Products 1

mv 

mv ettringite  VP     1 1 a  VP   mv P mv gypsum

d M

Volumetric strain due to Volumetric change of Paste in concrete  VP   VP  V  CAreacted    V  P   VC 

CA reacted = Ca – Cremaining

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Modeling of expansion: Effect of porosityvolumetric change V corrected   V  f  f 

fraction of capillary porosity being filled capillary porosity     max  f c  

w   0.39  c ,0 w  0.32  c 

fc = cement content w/c = water/ cement ratio  = degree of hydration

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Cracking – Diffusivity relationship finite difference approach plane of symmetry

D

D

cracked zone: D=D1

uncracked zone: D=D2

D1 D2

D2

XS (0)=0

At T=0

L/2

X

0

XS (T)

At T>0

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L/2

X

Adaptation of the 1D solution to a 2D problem- finite difference approach concentric layers perimeters uncracked zone Cracked zone          

Average Stiffness= Weight-averaged at nodes

    

     

      

       

        

: point

where U is computed

         

X

X

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Strain generation across a 1/4 portion of the sample exposed on both sides 0.0165

Strain, mm/mm

0.0165 0.016

0.016 0.0155

0.0155

0.015 0.015

0.0145 0.014

0.0145

0.0135 0.013

0.014

5

25

10

20 15

15

X coordinate, mm

10

20 25

5

Y coordinate, mm

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D D1 D=D1

Effect of Concrete Diffusivity

uncracked zone: D=D2

D2 0 X

HPC

Control Concrete 3

3

D1/D2=1000 D2=1x10-13 m2/s

D1/D2=10

2

Expansion, %

Expansion, %

D1/D2=1000

D1/D2=100

1

D2=1x10-12 m2/s

D1/D2=10

2 D1/D2= 100

D1/D2=1

1

D1/D2=5

0

0

500

1000 1500 Time, days

2000

2500

0

0

200 400 Time, Days

600

As time progresses, an increased diffusivity due to cracking results in faster rates of expansion. You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)

Effect of Initial Sulfate Content on Expansion and Degradation 0.4

20 C3A = 8% Gypsum= 0-10% Degree of hyd. of C3A =0.5 -2

-2

SO4 = 4-10%

Modulus, GPa

Expansion, %

0.3

C3A = 8% Gypsum= 0-10% Degree of hyd. of C3A =0.5

16 SO4 = 2%

0.2

12 -2

SO4 = 0%

8

-2

0.1

SO4 = 2%

-2

4 SO4 = 4-10% -2

SO4 = 0% 0

0

100 200 Time, Days

300

0

0

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100 200 Time, Days

300

Effect of Internal porosity on the Dissipation of Reaction Products 0.4

Linear Expansion, %

f=0.15 0.3

f=0.35 0.2

f=0.45

0.1

f=0.55 0

0

200

400

Time, Days

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600

Validation of model Accuracy of prediction of experimental results depends on availability of values for parameters such as: • sulfate diffusivity, • rate constant, • capillary porosity, • hydrated calcium aluminates content and form. • Experimental conditions may not match model hypotheses, as for pH monitoring.

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Experimental Program         

Sulfate attack on blend of mortars under controlled environment. Length change of mortar bars, dynamic modulus, SEM. C3A content ranging from 2 % to 11.6 % The w/c ratio used was 0.485 Sand/cement ratio of 2.75. 5% solution of Sodium Sulfate Specimen size: 25 mm x 25 mm x 275 mm and 10 mm x 10 mm x 40 mm. Constant pH and variable pH environments Dimensions recorded on a weekly basis for periods of up to three years

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Effect of C3A 1

0.25 Experiment, Initial C3A= 6% Simulation

f=0.3; -C3A=0.9 E0=28000 MPa f 't= r = 3 MPa D1/D2=12 D1 = 1x10 -13

0.15

0.1

6% C3A

0.05

0

0.8 Expansion, %

Expansion, %

0.2

0

200

400 600 Time, Days

Experiment, C3A=11.6% Simulation

800

=0.9; f=0.25 -C3A=0.9 E0=28000 MPa f 't=3 MPa r = 10 MPa D1/D2=20 D2 = 1x10 -13

0.6

0.4

0.2

1000

0

0

40

finite difference approach You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)

80 120 Time, Days

11.6% C3A

160

200

Simulation of Effect of C3A content from 4% to 11.6% 0.6

Expansion, %

11.6% 0.4 8% 6% 0.2

4% 0

finite difference approach

0

200

400 600 Time, Days

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800

Effect of Specimen Size 1

Same variables for same materials, Only input Variable: specimen size

Experiment Simulation

0.6 0.4 0.2

25.4x25.4x279.4 mm

0 -0.2

finite difference approach

10x10x40mm

0.8

Expansion, %

w/c=0.485; C3Ai=0.06%; Gypsum=0.06%; -C3A=0.9; E0=28000 MPa; ft'=3 MPa; =0.9; f=0.45;

0

40

80 120 Time, Days

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160

200

Comparison of Expansion and stiffness degradation 40

0.6

30

0.4

20

PCA-174Asodium

0.2

0

0

100 200 Time, Days

Modulus, GPa

Expansion, %

0.8

Expansion (Exp) Modulus (Exp) Expansion (simulation) Modulus (simulation)

10

0 300

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finite difference approach

Effect of Curing Duration on Expansion 2.5 D2 = 1x10-13 D1/D2 = 38 f = 0.15 

Modeled By three variables:

Expansion, %

1) Degree of Hydration, j 2) Internal porosity, f 3) Diffusivity changes due to cracking D1/D2

2

1.5

D2 = 1x10-13 D1/D2 = 25 f = 0.25 

LTS-2 1 year curing

TXI-18 1 day curing

1

0.5

0

finite difference approach

0

40

80 Time, Days

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120

160

Reduction of diffusivity due to the use of class F flyash 0.16

Expansion, %

0.12

w/c=0.44; f=0.55; C3A=8%

C100FA0 C45FA55 Simul. Control Simul. , FA%55

0.08 D = 1.14x10-11 D = 2.0 x10 -13

0.04

SO4 -- = 6% -C3A=0.9 E0=28000 MPa f 't= 3 MPa r = 10 MPa

Class F Flyash effect

0 0

100

200

300

Time, Days

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Effect of W/C ratio and C3A content 0.25

0.8

C3A=12% C3A=7% Simul., C3A=12% Simul., C3A=7%

W/C Effect, C3A=8% w/c = 0.45 w/c = 0.6 Simul., w/c=0.45 Simul., w/c=0.6

0.15

D= 1 x 10-12 D1/D2 = 10 f= 0.35

0.1

D= 5 x 10-13 D1 /D2 = 5 f= 0.5

0.05 0

0

40

0.6

Expansion, %

Expansion, %

0.2

80

Time, Days

120

160

0.4

C3AEffect, w/c=0.6

0.2

0 0

100

200

300

400

Time, Days -C3A=0.9 E0=28000 MPa f 't= 3 MPa finite difference approach r = 10 MPa

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Simulation of Effect of C3A using the simplified approach 1

Expansion, %

0.8 0.6 0.4

w/c=0.6; Exp.C3A=4.3% f=0.45; Exp.C3A=7% -C3A=0.9 Exp.C3A=8.8% E0 =28000 MPa Exp.C3A=12% f 't= 3 MPa model, k=2.2e-8 r = 10 MPa k=3.5e-8 D = 1.14x10-11 k=6e-8 k=1e-7

0.2 0 0

100

200

300

400

Time, Days You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)

Series solution

Simulation of Tertiary mixes with Flyash and Silica fume 1 Control 3%SF+40%FA Finite Difference, Control Finite Difference, 3%SF+40%FA Series Solution, Control Series Solution, 3%SF+40%FA

Expansion, %

0.8 0.6

Pozzolan Effect

0.4 M.D.A. Thomas, et. al. CCR, 29 (1999)

0.2 0

0

100

200

300

Time, Days You created this PDF from an application that is not licensed to print to novaPDF printer (http://www.novapdf.com)

400

Series solution

Conclusions     

A Predictive model is presented based on cement chemistry, concrete physics and mechanics. important properties: C3A content, internal porosity, use of flyash, and diffusivity. input parameters compatible with experimental data. Effects of degradation, specimen size, curing duration, and material ingredients can be verified using the model. Simplified solution approach requiring a spreadsheet approach.

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Acknowledgement  Financial

Support of Salt River Project, SRP in the Program on Blended cements is greatly appreciated

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