The Pennsylvania State University. The Graduate School. College of Engineering. A Dissertation in. Civil Engineering.

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering CHARACTERIZING SATURATED MASS TRANSPORT IN FRACTURED CEMENTITIOUS MATERIALS A Dissertation in Civil Engineering by Alireza Akhavan Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2012

2 The dissertation of Alireza Akhavan was reviewed and approved* by the following: Farshad Rajabipour Assistant Professor of Civil and Environmental Engineering Dissertation Adviser Chair of Committee John Earl Watson Professor of Crop and Soil Science Maria Lopez de Murphy Associate Professor of Civil and Environmental Engineering Tong Qiu Assistant Professor of Civil and Environmental Engineering William Burgos Professor of Civil Engineering Chair of Graduate Program of the Department of Civil and Environmental Engineering *Signatures are on file in the Graduate School.

3 ABSTRACT Concrete, when designed and constructed properly, is a durable material. However in aggressive environments concrete is prone to gradual deterioration which is due to penetration of water and aggressive agents (e.g., chloride ions) into concrete. As such, the rate of mass transport is the primary factor, controlling the durability of cementitious materials. Some level of cracking is inevitable in concrete due to brittle nature of the material. While mass transport can occur through concrete s porous matrix, cracks can significantly accelerate the rate of mass transport and effectively influence the service life of concrete structures. To allow concrete service life prediction models to correctly account for the effect of cracks on concrete durability, mass transport thru cracks must be characterized. In this study, transport properties of cracks are measured to quantify the saturated hydraulic permeability and diffusion coefficient of cracks as a function of crack geometry (i.e.; crack width, crack tortuosity and crack wall roughness). Saturated permeability and diffusion coefficient of cracks are measured by constant head permeability test, electrical migration test, and electrical impedance spectroscopy. Plain and fiber reinforced cement paste and mortar as well as simulated crack samples are tested. The results of permeability test showed that the permeability of a crack is a function of crack width squared and can be predicted using Louis formula when crack tortuosity and surface roughness of the crack walls are accounted for. The results of the migration and impedance tests showed that the diffusion coefficient of the crack is not dependent on the crack width, but is primarily a function of volume fraction of cracks. The only parameter that is changing with the crack width is the crack connectivity. Crack connectivity was found to be linearly dependent on crack width for small crack and constant for large cracks (i.e.; approximately larger than 80 μm). The results of this study can be used to predict diffusion and permeability coefficients of fractured concrete. iii

4 TABLE OF CONTENTS List of Figures... viii List of Tables... xiii Chapter 1: Introduction Introduction Research objectives Organization of contents References... 4 Chapter 2: Mechanisms of Deterioration and Mass Transport in concrete Concrete durability problems Corrosion of steel reinforcement Carbonation Chloride attack Freeze/thaw damage Alkali silica reaction (ASR) Sulfate attack Mechanisms of mass transport in concrete Saturated permeation iv

5 2-2-2 Unsaturated permeation Diffusion Other transport mechanisms Service-life prediction models Life-365 Service Life Prediction Model STADIUM SIGHT Summary References Chapter 3: Quantifying the Effects of Crack Width, Tortuosity, and Roughness on Water Permeability of Cracked Mortars Introduction Quantifying the geometric properties of cracks Effective crack width Crack tortuosity and surface roughness Materials and experiments Sample preparation Permeability measurement Measuring crack dimensions Results and discussion v

6 3-4-1 Comparison between average, effective, and LVDT crack measurements Saturated permeability as a function of crack width Crack tortuosity and surface roughness Effect of tortuosity and roughness on crack permeability Conclusions References Chapter 4: Evaluating Ion Diffusivity of Cracked Cement Paste Using Electrical Impedance Spectroscopy Introduction Methods for Measuring the Diffusion Coefficient of Concrete Theory Materials and Experiments Results and Discussion Conclusion References Chapter 5: Permeability, Electrical Conductivity, and Diffusion Coefficient of Simulated Cracks Introduction Methods Theory vi

7 5-3-1 Hydraulic Permeability of Cracks Ion Diffusivity of Cracks Experimental Methods Results and Discussion Hydraulic Permeability Ion Diffusivity Conclusion References Chapter 6: Summary and Conclusion Summary of Research Approach Conclusion Suggested Future Research Appendix A Appendix B vii

8 LIST OF FIGURES Figure 1-1: Rapid corrosion of steel due to cracking...2 Figure 2-1: Deterioration processes in reinforced concrete...6 Figure 2-2: The corrosion of steel begins with the rust expanding on the surface of the bar and causing cracking near the steel/concrete interface. As the corrosion products build up, more extensive cracking develops until the concrete breaks away from the bar, eventually causing spalling... 8 Figure 2-3: Cracking and spalling due to corrosion of steel reinforcement in a concrete beam... 8 Figure 2-4: Corrosion of steel reinforcement... 9 Figure 2-5: Relative volume of iron and its oxides Figure 2-6: Bryant Patton Bridges in Florida displays a reinforced concrete pile with significant corrosion induced damage Figure 2-7: Typical example of concrete deteriorated from freeze thaw actions. Non-air-entrained concrete railing Figure 2-8: Saturated area adjacent to crack Figure 2-9: Map cracking due to ASR Figure 2-10: Effect of reactive silica content on concrete expansion due to ASR Figure 2-11: Pipeline support chair damaged after 19 years due to external sulfate attack Figure 2-12: Illustration of fluid flow under pressure gradient Figure: 2-13: Influence of w/c ratio on the permeability of (a) cement paste and (b) concrete Figure 2-14: Unidirectional unsaturated flow Figure 2-15: Typical water retention curves for a sand and a clay loam Figure 2-16: Unsaturated hydraulic conductivity K θ versus volumetric water content θ viii

9 Figure 2-17: Illustration of solute transport due to concentration gradient Figure 3-1: Schematic illustration of the splitting tension setup used to fracture mortar disk specimens (after modification of the setup used by Wang et al.) Figure 3-2: A thru-thickness crack in a mortar disk specimen showing: (a) crack width variability and crack tortuosity, (b) crack wall roughness Figure 3-3: Cumulative distribution function showing the variability of crack profile along the surface of a disk specimen Figure 3-4: Method for calculation of the effective crack width (adopted from Dietrich et al.).. 50 Figure 3-5: Correlation between the effective surface and thru crack widths Figure 3-6: (a) Digitized profile of an actual thru crack; (b) Schematics of a crack profile to illustrate surface metrology procedures Figure 3-7: Permeability test Figure 3-8: Correlation between (a) average and effective crack widths, (b) average crack width and LVDT readings Figure 3-9: Theoretical and experimental values of crack permeability as a function of effective crack width Figure 3-10: Effective crack length as a function of sampling length scale: (a) Fiber-reinforced crack fitted by a fractal power function; (b) Comparison between plain and fiber-reinforced cracks Figure 3-11: Crack surface roughness as a function of sampling length scale Figure 3-12: Estimation of crack permeability based on Eq. 3-19; data points show experimental results Figure 4-1: Steady-state diffusion test ix

10 Figure 4-2: Salt ponding test Figure 4-3: Bulk diffusion test Figure 4-4: Electrical migration tests Figure 4-5: Rapid migration test Figure 4-6: rapid chloride permeability test (RCPT) Figure 4-7: Parallel law for ion diffusion in cracked concrete Figure 4-8: Schematics of (a) smooth, and (b) constricted crack Figure 4-9: Resistance vs. reactance for a cracked fiber reinforced cement paste sample Figure 4-10: Splitting tension setup used to fracture cement paste disks Figure 4-11: Crack patterns for dual and single cracked samples Figure 4-12: EIS test setup Figure 4-13: Variation of the electrical conductivity of cracked cement paste samples (σ Composite ) versus crack volume fraction ( Cr ) Figure 4-14: Estimated diffusion coefficient of cracked samples (D Composite ) as a function of crack volume fraction ( Cr ) Figure 4-15: Variation of the electrical conductivity of cracked samples (σ Composite ) versus the average crack width (w Cr ) Figure 4-16: The calculated crack connectivity ( Cr ) as a function of average crack width (w Cr ) Figure 5-1: Plexiglas test sample used to simulate cracks in concrete Figure 5-2: Noncontact optical profilometer Figure 5-3: Topography map of the test samples surfaces, (a): Rough (b): Smooth Figure 5-4: Test samples installed between two test cells x

11 Figure 5-5: Permeability test setup Figure 5-6: Migration test configuration Figure 5-7: Migration test setup Figure 5-8: Variation of chloride concentration over time in downstream cell (migration test). 123 Figure 5-9: Electrical impedance test setup Figure 5-10: Typical result of electrical impedance test Figure 5-11: Measured and predicted Permeability coefficient Figure 5-12: Permeability test results (data point for mortar samples was obtained from) Figure 5-13: Diffusion coefficient of crack vs. crack width Figure 5-14: Normalized conductivity vs. crack width Figure 5-15: Crack connectivity coefficient, obtain from EIS Figure 5-16: Diffusion coefficient of crack normalized by crack connectivity obtained from EIS Figure A-1: Vacuum impregnation of disk samples with epoxy Figure A-2: A polished epoxy impregnated sample Figure A-3: A vertically sectioned specimen at the mid-point perpendicular to the surface crack Figure A-4: A thru crack detected and segmented to measure crack width Figure A-5: Crack width distribution of the surface crack Figure A-6: The portion of surface crack between and points that was assumed to correspond with the middle thru section is establishing the correlation between effective surface and thru crack widths Figure A-7: Correlation between the effective surface and thru crack widths Figure A-8: (a) A detected thru crack. (b) The thru crack, sectioned every 1 mm xi

12 Figure A-9: Roughness is calculated by averaging the crack variation within the sampling length Figure B-1: Splitting tension setup used to fracture cement paste disks Figure B-2: Schematic illustration of the splitting tension setup used to fracture mortar disk specimens Figure B-3: Variation of applied vertical load and lateral deflection of the sample during splitting tension test xii

13 LIST OF TABLES Table 3-1: Mixture proportions for mortar specimens...56 Table 3-2: Average tortuosity and roughness measured using different values of λ...68 Table 4-1: Mixture proportions...95 Table 4-2: Pore solution composition...95 Table 4-3: Parameters used in eqs and Table A-1: Summary for the surface crack Table A-2: Summary for the thru crack Table A-3: Summary for the surface crack in the mid quarter Table A-4: Crack width measurement for the surface crack Table A-5: Crack width measurement for the thru crack Table A-6: Crack width measurement for the surface crack in the mid quarter xiii

14 Acknowledgements First, I would like to thank my advisor Dr. Farshad Rajabipour for his guidance, suggestions, and support. I would also like to thank my parents for unconditionally loving and supporting me. xiv

15 CHAPTER 1: INTRODUCTION 1-1 Introduction Concrete is the most widely used man made material in the world. The United States uses about 400 million cubic yards of ready mixed concrete each year [1]. Worldwide, 12 billion tones ( 6.5 billion cubic yards) of concrete are manufactured annually [2]. Most of the transportation infrastructure is made of concrete with a design service life of 50 to 100 years. Long lasting materials play a major role in building durable and cost effective structures. Durability is a problem especially when concrete is exposed to aggressive environment such as deicing salts, marine structures, or severe freezing and thawing environment. The need to design long lasting concrete structures requires knowledge of parameters affecting the durability and service life of concrete and steel reinforcement. Some of the most common durability problems of concrete are freeze/thaw damage, alkali-silica reaction (ASR), sulfate attack and corrosion of reinforcing steel [3]. The primarily factor governing the durability of concrete is mass transport. Deterioration of concrete due to the previously mentioned mechanisms is significantly influenced by the rate of moisture, ion, and gas/vapor transport in concrete[4]. This is further discussed in chapter 2. A number of durability models have been developed that can predict service life of concrete structures by considering the physical and chemical phenomena that influence concrete s longterm performance and service life expectancy. Most of the existing service life models (e.g., STADIUM, Life-365) consider concrete as a continuum porous media and do not account for the presence of localized or distributed cracks. Cracking on the other hand, is inevitable in concrete. Humidity and temperature changes and the resulting volume changes can cause tensile stress 1

16 development and cracking if concrete is restrained against such movements [4]. Load induced cracking also occurs when tensile stress (e.g., at negative moment regions in a bridge deck) exceeds the tensile strength of concrete[5]. Such cracks can widen over time due to creep and further cracks could develop by fatigue (e.g., due to repeated traffic load). Cracking can increase the deterioration rate of concrete significantly by accelerating transport of moisture and aggressive agents into concrete and to the level of reinforcement. Figure 1-1 shows a submarine pile that was cracked during driving. Signs of rust are visible on the surface of concrete only 6 months after installation. Figure 1-1: Rapid corrosion of steel due to cracking 1-2 Research objectives The goal of the presented study is to characterize mass transport in saturated fractured concrete. The results will provide the much needed material/crack transport property inputs that can be incorporated into service-life prediction models to allow simulation of the effect of cracks on 2

17 durability of concrete. This is especially significant for prediction of the remaining life of structures in service and selection of the best maintenance strategies for concretes that have experienced some level of cracking (e.g., early age shrinkage cracking). The results will quantify saturated transport properties (permeability, diffusivity) as a function of crack geometry (width, length, tortuosity, surface roughness). This will allow one to determine if there is a safe crack width that has negligible impact on durability of concrete. Safe crack width can be prescribed as the maximum allowable crack width in codes and specifications such as ACI-318 or AASHTO Bridge Design Manual. This will further enable weighing the benefits of crack mitigation strategies (e.g., use of fiber reinforcement or shrinkage reducing admixtures) against their costs. 1-3 Organization of contents The following provides a brief description of the contents of this thesis. Chapter 2 addresses the most common durability problems of concrete. The mechanism of each problem is explained and the theory behind it is briefly discusses. Various modes of mass transport in concrete are reviewed with focus on fluid permeation and ion diffusion. Finally, some of the existing service life prediction models are introduced. In chapter 3, water permeability of cracked mortars in saturated conditions is studied. Effect of cracking on permeation rate of water into concrete is experimentally determined. Geometry of cracks is characterized with the use of digital image analysis and relationships between crack geometry parameters (e.g., width, roughness, tortuosity) and permeability are established. These relationships are evaluated against the theory of laminar flow inside parallel-plate gaps. 3

18 Chapter 4 uses electrical impedance spectroscopy to measure electrical conductivity and saturated diffusion coefficient of cracked cement paste samples. The relation between diffusion coefficient and crack geometry is studied. Crack connectivity (e.g., inverse tortuosity) is also measured by electrical impedance spectroscopy. Chapter 5 introduces a Plexiglas setup that was designed to simulate cracks in concrete. Saturated permeability, diffusion coefficient (using electrical migration test) and electrical connectivity (using electrical impedance spectroscopy) are measured on sample cracks with a broader range of crack widths. Using this setup allows simulation of parallel-plate cracks with desired width and surface roughness. The results are used to evaluate four hypotheses regarding permeability, diffusivity, connectivity and surface effects of cracks in concrete. Finally, chapter 6 provides a summary of the findings in this study and discusses the main conclusions. Suggestions for future work are also provided. 1-4 References [1] Portland Cement Association., Design and control of concrete mixtures. Engineering bulletin, Skokie, Ill. etc.: Portland Cement Association, [2] J.P. Broomfield, Corrosion of steel in concrete : understanding, investigation and repair. 2nd ed., London ; New York: Taylor & Francis. xvi, 277 p., [3] M.G. Richardson, Fundamentals of durable reinforced concrete. Modern concrete technology., London ; New York: Spon Press. xii, 260 p., [4] S. Mindess, J.F. Young, D. Darwin, Concrete, 2nd Ed., Prentice Hall, Upper Saddle River, New Jersey,

19 [5] M.N. Hassoun, A.A. Al-Manaseer, Structural concrete : theory and design. 4th ed., Hoboken, NJ: J. Wiley,

20 CHAPTER 2: MECHANISMS OF DETERIORATION AND MASS TRANSPORT IN CONCRETE Concrete is a durable material if exposure conditions are properly predicted and considered during the design phase and the structure is subsequently constructed according to quality standards and specifications. Concrete is exposed to various environment conditions and may deteriorate due to physical and chemical causes. Designing durable concrete that exhibit satisfactory performance over its designed service life requires knowledge of the mechanisms that deteriorate concrete [1][2]. Figure 2-1 shows the most common deterioration processes in concrete. Figure 2-1: Deterioration processes in reinforced concrete, adopted from [3] 6

21 In this chapter, some of the most common concrete durability problems are discussed and the process of deterioration is explained for each problem. Afterwards, mass transport as the primary factor controlling durability of concrete is explained. Mechanisms of mass transport in concrete and their effect on different deterioration processes are also discussed in this chapter. Finally some of the existing service-life prediction models are briefly explained and their assumptions are discussed. 2-1 Concrete durability problems Corrosion of steel reinforcement Corrosion of reinforcing steel is one of the primary causes of deterioration in concrete. In 2002 the cost of corrosion on US highway bridges was estimated as $8.3 billion [4]. The loss of cross section of rebar and bond between concrete and rebar result in reduction in load bearing capacity of reinforced concrete elements and may lead to collapse or at least serviceability problems (e.g., cracking). However, the loss of rebar cross section is not the only problem caused by corrosion of steel in concrete. The volume of resulting rust is greater than the volume of steel by a factor of up to 7. The resulting expansion applies tensile stresses to the concrete, which can eventually cause cracking and spalling [5]. This is illustrated in figures 2-2 and 2-3. The process of corrosion of steel reinforcement is shown in figure 2-4. This can be divided into anodic and cathodic reactions. At the anode, iron oxidizes which results in release of two electrons: The anodic reaction:

22 Figure 2-2: The corrosion of steel begins with the rust expanding on the surface of the bar and causing cracking near the steel/concrete interface. As the corrosion products build up, more extensive cracking develops until the concrete breaks away from the bar, eventually causing spalling [6]. Figure 2-3: Cracking and spalling due to corrosion of steel reinforcement in a concrete beam. 8

23 The released electrons must be consumed elsewhere; otherwise large amount of electrical charge will build up at one place on the steel. At the cathode, the electrons reduce water and oxygen and generate hydroxyl ions: The cathodic reaction: The generated hydroxyl ions must travel within the pore network of concrete and react with the ferrous ions ( ) and form ferrous hydroxide ( : Ferrous hydroxide in presence of water and oxygen forms ferric hydroxide ( which spontaneously changes to hydrated ferric oxide (rust). Figure 2-4: Corrosion of steel reinforcement [5] 9

24 rust Unhydrated ferric oxide ( ) is about twice the volume of steel. As it hydrates, it swells even more. This is illustrated in figure 2-5. The resulting expansion is a primarily factor in corrosion damage in concrete [7]. Fe(OH)3.3H2O Fe(OH)3 Fe(OH)2 Fe2O3 Fe3O4 FeO Fe Figure 2-5: Relative volume of iron and its oxides [8] After initiation of corrosion, the process of corrosion rapidly decelerates inside concrete to a negligible rate. In the alkaline environment of concrete (ph commonly greater than 12.5), a thin but dense oxide layer (known as passive layer) forms on the surface of steel that prevents further corrosion by limiting access of oxygen and water to the metal [1]. The passive layer will preserve and repair itself if damaged in the presence of an alkaline environment [7]. However, if 10

25 the ph drops below 11.5, the passive layer is destroyed [1]. Two mechanisms can break down the protective passive layer in concrete, carbonation and chloride attack [7] Carbonation If penetrated into concrete, carbon dioxide gas ( ) interact with calcium hydroxide ( ) in the concrete. Carbon dioxide dissolves in water and forms carbonic acid ( ) which neutralizes the alkalies in the concrete pore solution and generates calcium carbonate ( ): The ph drop due to consumption of calcium hydroxide destroys the passive layer and allows the corrosion to restart. However, the volume of solid calcium hydroxide in concrete is a lot more than the amount dissolvable in pore solution. When the dissolved calcium hydroxide is consumed by carbonic acid in the neutralization reaction, the solid calcium hydroxide starts to dissolve in the pore solution which keeps the ph at its normal level (i.e., >12.5). As the carbonation reaction proceeds, eventually all the solid calcium hydroxide is consumed and eventually the ph drops and corrosion initiates [7]. Carbonation damage occurs more rapidly when concrete has high permeability and diffusivity. The rate of ingress controls the rate of corrosion due to carbonation. Cracks in concrete (by increasing permeability and diffusivity) enhance the transport of and increase the rate of deterioration. This is especially significant in the vicinity of cracks. 11

26 Chloride attack Chloride penetration is generally known as the most significant threat to reinforced concrete structures. The source of chlorides can be deicing salts, seawater and admixtures (e.g., CaCl 2 accelerating admixture). Bridges, pavements, and near shore structures are exposed to chloride through deicing salt and seawater. Due to lack of oxygen, Concrete submerged deep inside seawater may not experience corrosion [2]. Splash zones on the other hand are recognized as high corrosion risk areas (figure 2-6). Figure 2-6: Bryant Patton Bridges in Florida displays a reinforced concrete pile with significant corrosion induced damage [9] Chlorides ions are capable of destroying the protective layer even at high alkalinities. When iron ions react with chloride ions, the reaction product serves to carry Fe 2+ away from the metal surface resulting in an unstable and porous passive layer [1]. The process starts with oxidation of iron: 12

27 2 2 8 Then the iron ions combine with chloride ions to form ferrous chloride (and FeOCl): Ferrous hydroxide and ferric hydroxide are subsequently formed in presence of water. Chloride ions are recycled and the process continues Since the chloride ions are recycled, the attack continues even in low chloride contents. But there is a chloride threshold for corrosion since the passive layer can effectively re-establish itself when damaged at high ph values. This threshold is suggested to be 0.6 [10] in terms of chloride/hydroxyl ion ratio. Buenfield [11] suggested the range of 0.2 to 0.4 percent (total chloride as a percentage of the mass of concrete) for different exposure climates. The time to the onset of corrosion is commonly called the initiation period. In other words, the initiation period is the time it takes for chlorides to penetrate the concrete cover and reach a certain threshold at the level of reinforcements, sufficient to initiate the corrosion. Propagation period is the time for corrosion to reach an unacceptable level, in which the corrosion products build up on the surface of the steel reinforcement and cause damage to concrete. 13

28 Permeability and ion diffusivity of concrete significantly affect its durability in chloride rich environments. In a dry condition, sorptivity and permeability of concrete control the rate of chloride ingress; the penetration rate is initially dominated by convectional flow of the moisture containing chlorides in concrete. Transport inside the concrete will also be by diffusion. Chloride ions can diffuse through concrete porosity or reach the steel through cracks [2]. The diffusion mechanism is dominant at high saturation degrees. In the presence of oxygen and water, the intrusion of chloride ions into reinforced concrete is the major cause of corrosion. Corrosion accelerates in concrete with higher water permeability and ion diffusivity [5]. Cracking accelerate corrosion rate by increasing permeability and diffusion coefficient of concrete Freeze/thaw damage Freeze/thaw damage is due to expansion of water ( 9%) in concrete pores when the temperature drops below freezing point of pore solution. This expansion applies tensile stresses inside the concrete matrix resulting in cracking if the stresses developed exceed the tensile strength of concrete. In saturated concrete, most of the water in cement paste will not freeze at 0⁰ C. Depending on the pore diameter, water only freezes when the temperature drops well below 0⁰ C. For example water in pores of 10-nm diameter will not freeze until -5⁰ C (23⁰ F). Also, the presence of ions further depresses the freezing point [1]. Dilated pores and developed microcracks resulting from frost attack may increase water content of concrete and lead to more severe expansions. Therefore the rate of deterioration increases and can eventually cause failure 14

29 or serviceability problems. [2]. Figure 2-7 shows an example of concrete deterioration due frost attack. Figure 2-7: Typical example of concrete deteriorated from freeze thaw actions. Non-air-entrained concrete railing [12] Powers [13] explained the mechanism of frost attack. He explained that formation of ice and consequent expansion apply pressure to residual water and water tends to move out and escape from capillary pores to a free space to relieve the pressure. Water needs to move through unfrozen pores to reach an escape boundary. If the distance to free space is too far or enough unfrozen pores are not available due to low temperature, the hydraulic pressure causes microcracking and pores enlargement [1]. 15

30 Generation of hydraulic pressure by ice formation was believed to be the major contributor of dilation for many years, but this change is insufficient to account for all of the dilation observed in concrete during freeze-thaw cycles. Later, Powers [13] found that expansion of ice and the accompanying hydraulic pressure could not be the major cause of damage. He observed expansion even in partially dry pastes which have enough empty pores to accommodate the 9% expansion. Also damage was observed in concrete saturated with liquids that do not expand upon freezing. Powers observed that when paste starts to freeze, it first shrinks and then expands. One explanation of this behavior is based on osmotic pressure. Freezing results in an increase of solute concentration in the unfrozen liquid adjacent to the freezing sites. Because of this concentration difference and through the process of osmosis, water is drawn from surrounding pore solution toward freezing sites which causes the paste away from the freezing sites to shrink and crack. If the osmotic and hydraulic pressures are to be relieved, water needs to travel to reach free spaces. This distance must not be too large. 200 μm is recommended as the maximum distance [2]. Air entrained concrete provides free spaces for water to expand and can mitigate frost attack. Partially dry concrete is less vulnerable to freeze-thaw damage since the empty capillaries provide free space for water [1]. Concrete in a saturated condition is highly susceptible to frost attack. In addition to total air Figure 2-8: Saturated area adjacent to crack 16

31 content and air void spacing, the degree of saturation and permeability are the major controlling factors in freeze-thaw damage. Cracks are larger in size than capillary pores and if filled with water, will greatly contribute to deterioration due frost attack. The resulting expansion causes further propagation of cracks and the process accelerates as the volume fraction of cracks increases. In cold and wet climates, cracks are often water saturated, and stresses generated by freezing can deteriorate the surrounding concrete. Figure 2-8 shows how cracks can keep the surrounding area saturated which further enhances the freeze/thaw damage in surrounding concrete Alkali silica reaction (ASR) ASR is due to the presence of reactive silicious aggregate in the alkaline environment of concrete. The amorphous silica in aggregates can react with hydroxyl ions in concrete and form an alkali-silica gel that expands as it adsorbs water. Similar to the freeze/thaw process, the expansion of gel causes tensile stresses in concrete and induces distributed cracks (figure 2-9) which reduce strength and elasticity modulus and ultimately destroys concrete. Several factors control the alkali-silica expansion. The nature of reactive silica (degree of Figure 2-9: Map cracking due to ASR (photograph, courtesy of the Federal Highway Administration) amorphousness), amount of reactive silica and available alkali, particle size of reactive material, 17

32 and amount of moisture are the major affecting factors. Smaller aggregates and porous particles are more reactive due to their increased surface area [1]. The effect of amount of reactive silica and available alkali on ASR expansion is interdependent. Following is the further explanation of this effect. The products of alkali silica reaction are divided into two components: calcium-alkali-silicatehydrate (C-N-S-H) gels and alkali-silicate-hydrate (N-S-H) gels. Only the second component is a swelling gel and the first component which has calcium is a nonswelling gel and harmless to concrete [1]. The source of calcium ions in pore solution is calcium hydroxide which is a byproduct of cement hydration. Since the solubility of calcium hydroxide decreases with an increase in alkali concentration, in a high ph environment, fewer calcium ions are available to form calcium-alkali-silicate-hydrate gels. Consequently the high ph results in the formation of swelling products. The swelling gel (N-S-H) absorbs water and expands which applies local tensile stresses in concrete and can cause cracking. 2 Expansion (%) Silica content (%) Figure 2-10: Effect of reactive silica content on concrete expansion due to ASR 18

33 In a high alkaline environment, as the reactive silica content increases, the amount of swelling product (N-S-H) increases which result in greater expansion. After a certain point, further increase in silica content reduces the ph of the environment and consequently the solubility of calcium hydroxide increases which means more available calcium ions (Ca 2+ ). Availability of calcium ions leads to the formation of nonswelling C-N-S-H instead of N-S-H gel. Figure 2-10 shows expansion in concrete due to alkali silica reaction as a function of silica content. Expansion increase with increases in silica content and at a certain percentage of reactive silica (known as the pessimum content), maximum expansion occurs. After the peak, increase in silica content results in decrease in expansion. The percentage at peak which causes maximum deleterious expansion depends on the water to cement ratio, nature of reactive silica, and the degree of alkalinity [1]. Besides a high ph environment and the presence of amorphous silica, alkali silica reaction requires water to proceed. Water serves as the carrier of alkali and hydroxyl ions which attack the silica structure in aggregates. Also, the expansion of the resulting ASR gel is due to water which is absorbed by the gel and causes swelling of the gel. The developed pressures due to the expansion will eventually cause cracking in concrete. ASR accelerates in concrete with higher water permeability. Cracks in concrete enhance the penetration of water and can increase deterioration due to ASR. In addition, when the source of alkalis is external (e.g., deicing salts) cracking accelerates and enhances the availability of alkali ions. Supplementary cementitious materials have been used to mitigate ASR. If cement is replaced, in proper proportions, with silica fume, fly ash, or slag, expansion due to alkali-silica reactivity 19

34 reduces considerably [1][14][15]. Supplementary cementitious materials decrease the permeability and diffusion coefficient of concrete and can control ASR by limiting the supply of water required to cause expansion of the ASR gel. In addition, partial replacement of cement reduces the ph of the pore solution through alkali dilution and binding which mitigates attack on silicious aggregates and promotes formation of nonswelling gel. Using low alkali cement, avoiding highly reactive aggregates, and using low water to cement ratio concrete are some other ways to control ASR [1] Sulfate attack Sulfate ions are present in sea water and often in groundwater when high proportion of clay are present in the soil. Also groundwater in the vicinity of industrial wastes and municipal wastewater may contain sulfates. Aggregates can be a source of sulfates as well. Sulfate is found in the form of a variety of salts such as sodium sulfate, calcium sulfate, magnesium sulfate and potassium sulfate. Reaction of sulfates with hydration products of concrete (mainly with calcium aluminates) generates expansive products. Similar to the other mentioned deterioration processes, the resulting expansion can destroy concrete (figure 2-11). Sulfate ions react with calcium hydroxide (Ca(OH) 2 ) and calcium aluminate hydrates (e.g., 3CaO.Al 2 O 3.CaSO 4.12H 2 O also known as monosulfate) which are the products of cement hydration in concrete [2]. Gypsum (calcium sulfate CaSO 4.2H 2 O) is the product of reaction of sulfate with calcium hydroxide. The reaction is accompanied with solid volume expansion of 120% [1]. The reaction of sulfate with monosulfate and other calcium aluminates forms ettringite (calcium tri-sulfoaluminate hydrate: 3CaO.Al 2 O 3.3CaSO 4.32H 2 O). If sulfate is totally consumed 20

35 by hydration of tricalcium aluminate (C 3 A), the remaining C 3 A reacts with Ettringite and form monosulfate: If sulfate is reintroduced after setting of concrete (e.g., due to penetration of external sulfates), ettringite forms again [2]. Conversion of monosulfoaluminate to ettringite is accompanied by 55% increase in solid volume [1]. Figure 2-11: Pipeline support chair damaged after 19 years due to external sulfate attack [16] In the processes described so far the source of sulfate was external (e.g., seawater). Sulfate may also be supplied internally. The process is known as internal sulfate attack or delayed ettringite formation (DEF). Ettringite is unstable at high temperatures. During early hydration of Portland 21

36 cement, if temperature goes above 70⁰ C, ettringite will not form. High temperature may occur because of heat of hydration of cement especially in mass concrete; i.e., large concrete members such as concrete dams or drilled shafts. It may also be experienced in accelerated curing when concrete is kept in a warm environment or heated by steam. Accelerated curing is used when concrete strength gain is desired at early ages. In such conditions, reaction of sulfate with calcium aluminates (C 3 A) forms monosulfoaluminate instead of ettringite. When concrete subsequently cools down, ettringite crystals start to form if the source of sulfate (gypsum interground with Portland cement) is not depleted, which causes expansion and cracking. Formation of etrringite is harmless when concrete is in a plastic state and can accommodate volume increases. But after the concrete sets, the expansion due to DEF cause internal stresses and may cause cracking if these stresses exceed the tensile strength of concrete [2]. Transport of sulfate ions into the pores of concrete is controlled by diffusion and permeability coefficients. Cracking considerably increases both permeability and diffusion coefficient and can enhance sulfate attack [1]. 2-2 Mechanisms of mass transport in concrete The majority of concrete durability problems, such as those discussed in section 2-1, are due to penetration of moisture and aggressive agents into concrete. The rate of this transport is one of the primary controlling factors of concrete deterioration rate. Water permeation in saturated and unsaturated condition as well as ion diffusion, as two major transport mechanisms, are discussed in this section. 22

37 2-2-1 Saturated permeation Permeation is the convectional mode of transport. In presence of a pressure difference, fluid (e.g., water containing solutes) moves from higher to lower pressure regions. The pressure difference in concrete is generally due to capillary suction (for unsaturated concrete) or gravity (for saturated concrete). Under certain assumptions (Newtonian fluid, laminar flow, inert nonswelling media), fluid flux in porous media is proportional to the pressure gradient according to Darcy s law. Δh J Figure 2-12: Illustration of fluid flow under pressure gradient Figure 2-12 shows a sample with different water heads at two sides which generates a pressure gradient and causes water to flow from the high pressure region to the low pressure region. Darcy s law is the fundamental convection transport equation which relates fluid flux (may be referred as flux density in other fields) J (m/s) in a porous media to the pressure gradient that drives the flow ( / ), permeability coefficient (m 2 ), and fluid viscosity (Pa.s) [17]

38 Where Q (m 3 /s) is discharge rate and A (m 2 ) is the cross section of the specimen perpendicular to direction of flow. Darcy s law is more commonly stated in terms of pressure head (m) where, (kg/m3 ) is fluid density and (m/s 2 ) is gravitational acceleration: 2 14 The coefficient K (m/s) in eq is known as the hydraulic conductivity and is related to the permeability coefficient [17]. The saturated permeability coefficient of concrete can be measured by forcing flow through a specimen of uniform cross-sectional area with the lateral surface of the specimen sealed to ensure unidirectional flow. The quantity of the fluid flowing through the specimen is measured and Darcy s law (eq. 2-13) is used to obtain the permeability coefficient. Both the US Army standard [18] and ASTM standard [19] can be used to measure the permeability of concrete. The saturated permeability coefficient of concrete is strongly dependent on the porosity and pore size distribution of cement paste which are primarily controlled by water to cement ratio (w/c). Higher w/c reflects a higher porosity and larger pore sizes which result in a higher permeability (figure 2-13). Hagen-Poiseuille law describes flux in a cylindrical tube with length of L (m) and radius of r (m) due to pressure difference Δ (Pa) [20]. 8 Δ

39 Figure: 2-13: Influence of w/c ratio on the permeability of (a) cement paste and (b) concrete [1] By combining Hagen-Poiseuille law and Darcy s law one can show that permeability is a linear function of porosity but a square function of pore size Cracking can increase the permeability of concrete by several orders of magnetite depending on crack width (i.e., aperture). This is because crack apertures are much wider than size of typical pores in concrete. A detailed analysis of the saturated permeability of cracks is presented in chapter 3. 25

40 2-2-2 Unsaturated permeation In unsaturated condition, the permeability coefficient is a non-linear function of the moisture content (κ κ θ where θ (-) is volumetric water content). The continuity equation for a representative volume element of the unsaturated zone states that the change in total volumetric water content with time is equal to the sum of any change in the flux of water into and out of the representative volume element (figure 2-14) [20] J(x) J(x+dx) dx Figure 2-14: Unidirectional unsaturated flow Where (-) is volumetric moisture content, t (s) is time and J (m/s) is water flux. By combining Darcy s law and the continuity equation, a partial differential equation known as the Richards equation is obtained which governs moisture transport in unsaturated media. In case of one dimensional flow, Richards equation would be simplified as [20]:

41 Where is the pressure (i.e., capillary suction) gradient which is a function of moisture content. In order to solve Richards equation, both permeability as a function of moisture content κ θ and capillary pressure as function of moisture content must be known. The later is known as the water retention function of porous material [20]. A number of equations have been developed to describe unsaturated hydraulic conductivity (which is more commonly used than unsaturated permeability). In most of these equations unsaturated hydraulic conductivity is related to saturated hydraulic conductivity. Eq. 2-19, developed by Mualem [21], is one of the most widely used in soil physics: 2 19 Where (m/s) is hydraulic conductivity as a function of reduced water content, (-) is the reduced or normalized water content which is defined as: θ / with (-) and (-) being practical saturated and dry moisture contents. (m/s) is the saturated hydraulic conductivity, (-) is a fitting parameter and (m) is the capillary head as a function of water content. The relation between capillary head and water content is known as the water retention curve. A typical water retention curve is shown in figure An example of the variation of unsaturated hydraulic conductivity K θ versus volumetric water content θ for sand, clay and loam is shown in figure

42 Figure 2-15: Typical water retention curves for a sand and a clay loam [20]. Similar to unsaturated hydraulic conductivity, a number of equations have been developed for water retention curve in soil physics. One of the most widely used water retention functions is that developed by van Genuchten (1980) [22]: where (1/m), (-) and (-) are fitting parameters. With the assumption of 1, Mualem (eq. 2-19) and van Genuchten (eq. 2-20) equations are coupled to give eq [20]: 1 1 /

43 Eq describes the unsaturated hydraulic conductivity as a function of normalized moisture content and can be used to numerically solve Richards equation (eq. 2-18). Figure 2-16: Unsaturated hydraulic conductivity K θ versus volumetric water content θ [20]. The measurement of unsaturated flow properties of concrete is complex and not common. Instead, concrete professionals have adopted a simplified sorptivity test [23]. Sorptivity is a parameter that describes the rate of penetration of water into unsaturated concrete. To measure sorptivity concrete is exposed to water from one end and the weight gain due to water absorption is monitored over time. The depth of penetration of water into unsaturated concrete is measured as a function of time (m). Sorptivity is calculated according to eq [24]:

44 Sorptivity of concrete is dependent on both the characteristics of concrete (e.g., porosity, permeability) and the liquid (e.g., viscosity) being absorbed. In addition, the degree of saturation of concrete prior to the test has a large impact on its sorptivity. As such, sorptivity, although easy to measure, is not a material property and cannot be directly used to model unsaturated flow inside concrete. Using a sharp front model (assuming that the penetrating water front in concrete is sharp), the saturated permeability coefficient k (m 2 ) of concrete can be obtained from sorptivity measurements if porosity (-), viscosity of water (Pa.s) and capillary suction (Pa) are known [24] can be determined from the equilibrium internal relative humidity of concrete based on Kelvin s equation [25][26] Diffusion Solute transport occurs through diffusion and convection. Diffusion takes place due to a concentration gradient. Ions and other solutes travel within the pore solution of concrete from higher to lower concentration regions (figure 2-17). Diffusion occurs through interconnected moisture-filled pores and fractures. Cracks, when saturated, can enhance the process of diffusion by providing wide pathways filled with a large volume of pore fluid. Fick s first law [27] relates 30

45 the diffusion flux J (mol/m 2.s) to the concentration gradient and a material property known as the diffusion coefficient D (m 2 /s) C1 C2 J Figure 2-17: Illustration of solute transport due to concentration gradient By combining Fick s first law and a mass balance equation, Fick s second law can be derived. The change in concentration over time inside a representative volume element is equal to the difference between the flux entering and exiting the element in that period of time. The mass balance equation in one dimensional form (z) is written as: 2 25 where C is concentration (mol/m 3 ). Fick s 2 nd law predicts how diffusion causes the concentration to change with time

46 It can be shown that [1]: 2 27 Where (-) is porosity of the concrete, D 0 (m 2 /s) is diffusion coefficient of the ion of interest inside the concrete s pore solution and β (-) is pore connectivity which is a measure of tortuosity and constrictions of the pore network [28]. The inverse of parameter β is known as the material s formation factor (-) and accounts for the physical resistance of the pore network [1][27] In the presence of a convective flow, the total ionic flux (J) is the summation of ionic diffusion flux (J d ) and flux due capillary suction (J c ) [7] Concrete pore solution has a relatively high ionic strength (approximately 0.5 mol/kg). At this strength, idealized transport, which considers that each ionic species behaves independently from others, is a poor assumption for modeling concrete pore solution. The interaction between different ionic species is related to (a) the chemical activity of the species which accounts for a reduction in ion mobility due to high ion concentrations (i.e., crowdedness) of pore solution, and 32

47 (b) the charge imbalances stemming from differences in self-diffusion coefficients of various ions. In other words, in a multi-ion solution, diffusion of positive and negative ions creates an electrical field which influences ion transport. The complete flux equation which considers interaction between ions and also accounts for diffusion, conduction (i.e., ion diffusion due to an electrical field), and permeation is known as the electro-diffusion or Nernst-Plank equation [27]: Where (m 2 /s) is self diffusion coefficient of i th ionic species, (-) is the formation factor, (-) is the ion activity coefficient which varies between 0 and 1, (mol/m 3 ) is concentration of i th ionic species, is the valency of i th ion, F is Faraday constants (=96485 J/V.mol), R is gas constant (= J/mol.K), T (K) is absolute temperature, (V) is electrical voltage created by charge imbalance, (m 2 ) is the bulk permeability coefficient and (Pa.s) is the fluid viscosity [27]. Similar to the permeability coefficient, the diffusion coefficient in concrete is dependent on porosity. However, eq suggests that is independent of pore size, at least when pore surface interactions are not dominant. It would be interesting to investigate whether the diffusion coefficient of cracked concrete is primarily dependent on volume fraction and tortuosity of cracks or if the effects aperture and surface roughness are also significant. A research study that addresses this question is provided in chapter 4. 33

48 2-2-4 Other transport mechanisms There are other transport mechanisms that have effect on service life of concrete. Gas/vapor transport are important especially in corrosion of steel reinforcement in concrete where carbon dioxide and oxygen penetrate into concrete. Gas/vapor transport is mostly through diffusion. The effects of chemical reactions on the mechanisms of ionic transport in concrete should also be studied. Chemical reaction, in which ions are attracted to the solid surface of the pores under the influence of electrostatic forces, dissolution and precipitation effect ion transport in concrete. 2-3 Service-life prediction models for concrete Despite advances in service-life prediction of concrete structures (e.g., Life-365 and STADIUM software), most of the existing models do not account for the effect of cracks in accelerating transport and deterioration of concrete. Some service life models are briefly described below Life-365 Service Life Prediction Model Life-365 TM software was funded by American Concrete Institute (ACI) Strategic Development Council (SDC) and the first version was released in 2001 to be used to evaluate corrosion protection strategies in order to increase service life of reinforced concrete. In Life-365, it is assumed that corrosion of steel reinforcement due to chloride attack is the primary mode of deterioration [29]. In Life-365, service life is defined as the sum of time to initiate the corrosion and the propagation time required for corroding steel to cause sufficient damage to require repair. Initiation time represents the time required for the critical threshold concentration of chlorides to 34

49 reach the depth of reinforcing steel. Life-365 uses an approach based on Fick s second law (eq. 2-26) to model diffusion and predict initiation time. The chloride age-dependent diffusion coefficient is calculated by the software from eq [29]: 2 31 Where (m 2 /s) is diffusion coefficient at age (s) and is diffusion coefficient at reference age which is 28 days in Life-365, and is a constant depending on mixture proportions. Considering the information on mix design inputted by the user and based on an incorporated experimental data obtained from bulk diffusion tests, the software selects and and calculates up to 25 years. After 25 years, the diffusion coefficient is assumed to be constant and equal to D(25 years ). Eq. 2-32, suggested by Stanish [30], shows the relationship between and water to cement ratio (w/c) for concrete exposed to chloride at early age (28 days or less) The software also accounts for temperature-dependent changes in diffusion coefficient. Eq is used to calculate diffusion coefficient as a function of temperature [29]:

50 Where U is activation energy of diffusion process (35000 J/mol), R is gas constant ( J/mol.K), Tref is 293 k (20⁰ C), and T is absolute temperature. The user inputs required to predict initiation period are geographic location, type of structure, nature of exposure, thickness of concrete cover, water to cement ratio, type and quantity of mineral admixtures, and type of steel reinforcement and coatings. Supplementary cementitious materials (such as silica fume, fly ash and slag) reduce permeability and diffusivity of concrete. Their subsequent effect on corrosion initiation period is considered in Life-365. The software applies a reduction factor to concrete diffusion coefficient to account for the effect of silica fume [29]: Where D SF is reduced diffusion coefficient due to use silica fume and SF (%) is the level of silica fume replacement in term of cement weight. Eq is only valid up to replacement level of 15% and for the higher percentages, the software assume diffusion coefficient equal to D 15%. The effect of fly ash and slag on early age diffusion coefficient (D 28 ) is assumed to be negligible but their effect on long-term reduction in diffusivity is considered. The parameter in eq is modified by eq [29]:

51 Where FA and SG are level of fly ash and slag replacement respectively in term of cement weight. The relationship is only valid for FA up to 50% and SG up to 70% and thus the maximum is 0.6 [29]. In Life-365, the propagation period is assumed to be fixed and equal to 6 years. This time is extended to 20 years if epoxy-coated steel is used. As a result, the time to repair predicted by the software is simply equal to initiation period plus 6 (or 20) years [29]. Concrete in Life-365 is modeled as saturated and uncracked STADIUM STADIUM TM, developed by SIMCO Technologies is able to model unsaturated multi-ionic transport in concrete [31]. Information on two sets of data is used as input parameters for the software: material properties and environmental conditions. Based on parameters such as the concrete cover and the type of rebar, the software can estimate the service life of the structure. Information on geometry of the concrete element, mixture proportion (such as type, quantities and densities of cement, supplementary cementitious materials, and aggregate), transport property (such as porosity, diffusivity, and conductivity) and exposure condition can be inputted by the users. Material properties can be measured experimentally and inputted. An example is measurement of diffusion coefficient using a migration test [32]. Alternatively, database on material properties from 24 different mixture proportions as well as different exposure conditions are available in the software. The environmental conditions are composed of the temperature, relative humidity, and exposure level. Eight ionic species are considered: OH -, Na +, K +, SO 2-4, Ca 2+, Al(OH) - 4, Mg 2+, and Cl - [31]. 37

52 Richards equation (eq. 2-18) is used to simulate water flow in unsaturated condition. The extended Nernst-Planck equation is used to describe ionic transport in unsaturated media [33] SIGHT The computer model 4SIGHT, developed by the U.S. National institute of standards and technology (NIST), allows durability assessment of buried concrete structure [27]. The program simulates multi-species ion transport and chemical reaction in unsaturated concrete (similar to STADIUM). 4SIGHT has a simplified module to account for moisture flow inside cracks and its impact on service life prediction. Crack spacing, crack width, and crack depth can be inputted by the user. Alternatively the software has a module to predict flexural and drying shrinkage cracks based on simple structural analysis. Porosity, permeability, w/c, formation factor, cement properties, hydraulic pressure, and exposure condition (OH -, Na +, K +, SO 2-4, Mg 2+, and Cl - ) are some of the other inputs of the software. 4SIGHT provides concentration at any depth over specified period of time as output [27]. Diffusion coefficient and porosity can be estimated by the software if not entered by user. Knowing water to cement ratio and degree of hydration (α, 4SIGHT uses the following equations to estimate diffusion coefficient (m 2 /s) of chloride ion and porosity of concrete [27]: log D C 6.0 w/c α w/c

53 Formation factor is defined as ratio of pore solution conductivity ( ) to bulk conductivity ( ) for a nonconductive porous solid saturated with conductive solution [27]: This is similar to eq described previously. Cracks in 4SIGHT are approximated by smooth parallel walls with a gap equal to the observed crack width. This assumption is conservative in case of flexural crack which are V shape and the observed width is the maximum width. Also, neglecting tortuosity and roughness of cracks results in overestimating of permeability. The permeability of a crack is assumed to be a function of crack width square [27]: This is the upper limit of permeability for crack with width of b and is further explained in chapter Summary This chapter provided a review of the concrete durability problems and the transport mechanisms associated with them. Chapter 3 focuses on saturated permeability as one of the transport mechanisms. 39

54 2-5 References [1] S. Mindess, J.F. Young, D. Darwin, Concrete, 2nd Ed., Prentice Hall, Upper Saddle River, New Jersey, [2] M.G. Richardson, Fundamentals of durable reinforced concrete. Modern concrete technology, London ; New York: Spon Press. xii, 260 p., [3] P.K. Mehta, P.J.M. Monteiro, Concrete: Structure, Properties, and Materials., Prentice- Hall, [4] G.H. Koch, P.H. Brogers, N. Thompson, Y.P. Virmani, J.H. Payer, Corrosion Cost and Preventive Strategies in the United States, FHWA Report; FHWA-RD , Federal Highway Administration, Washington, DC, [5] Corrosion of Embedded Metals. Portland Cement Association [cited 2011 June]; Available from: [6] Corrosion Cycle of Steel Rebar. [cited 2011 June]; Available from: [7] J.P. Broomfield, Corrosion of steel in concrete : understanding, investigation and repair. 2nd ed., London ; New York: Taylor & Francis. xvi, 277 p., [8] F.Mansfield, Recording and Analysis of AC Impedance Data for Corrosion Studies, Corrosion, 37 (1981) [9] A. Sohanghpurwala, W.T. Scannell, Repair and Protection of Concrete Exposed to Seawater, Concrete Repair Bulletin, Merritt Island, FL, [10] D.A. Hausmann, Steel Corrosion in concrete: How Does it Occur? Materials Protection, 6 (1967)

55 [11] G.K. Glass, N.R. Buenfeld, Chloride threshold levels for corrosion induced deterioration of steel in concrete, Chloride Penetration into Concrete, (Ed. L.-O. Nilsson and J. Ollivier), 1995, pp [12] Freeze - Thaw Deterioration of Concrete. [cited 2011 June]; Available from: [13] T. C. Powers, Freezing Effects In Concrete, American Concrete Institute SP 47, Detroit, MI, 1975, pp [14] R.N. Swamy, The Alkali-silica reaction in concrete, Glasgow, New York: Blackie ;Van Nostrand Reinhold. xv, 336 p., [15] B. Lothenbach, K. Scrivener, R.D. Hooton, Supplementary cementitious materials, Cement and Concrete Research (2011) /j.cemconres [16] CEMENTAID Company Profile. [cited 2011 June]; Available from: [17] H.W. Reinhardt, RILEM Technical Committee 146-TCF., Penetration and permeability of concrete : barriers to organic and contaminating liquids : state-of-the-art report prepared by members of the RILEM Technical Committee 146-TCF. 1st ed. RILEM report 16, London, New York, E & FN Spon. x, 331 p., [18] CRD-C48-92, Standard Test Method for Water Permeability of Concrete, Handbook of Cement and Concrete, US Army Corps of Engineers, [19] ASTM D , Standard Test Methods for Measurement of Hydraulic Conductivity of Saturated Porous Materials Using a Flexible Wall Permeameter. [20] D.E. Radcliffe, J. Simunek, Soil physics with HYDRUS: modeling and applications, Boca Raton, FL: CRC Press/Taylor & Francis. xiii, 373 p.,

56 [21] Y. Mualem, A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resources Research, 12 (1976) [22] M.T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America, 44 (1980) [23] ASTM C e1 Standard Test Method for Measurement of Rate of Absorption of Water by Hydraulic-Cement Concretes [24] S. Kelham, A water absorption test for concrete, Magazine of Concrete Research, 40 (1988) [25] C. Hall, W.D. Hoff, Water transport in brick, stone and concrete. 2nd. ed., New York: Taylor & Francis, [26] F. Rajabipour, J. Weiss, Electrical conductivity of drying cement paste. Materials and Structures, 40 (2007) [27] K.A. Snyder, Validation and Modification of the 4SIGHT Computer Program, NIST-IR 6747, National Institute of Standards and Technology, Department of Commerce, [28] F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure, Academic Press, New York, [29] Life-365 Service Life Prediction Model, Computer Program for Predicting the Service Life and Life-Cycle Costs of Reinforced Concrete Exposed to Chlorides, Silica Fume Association, [30] K. Stanish, Predicting the Diffusion Coefficient of Concrete from Mix Parameters, University of Toronto Report, [31] J. Marchand, Modeling the behavior of unsaturated cement systems exposed to aggressive chemical environments. Materials and Structures, 34 (2001)

57 [32] E. Samson, J. Marchand, K. A. Snyder, Calculation of Ionic Diffusion Coefficients on the Basis of Migration Test Results. Materials and Structures, 35 (2003) [33] E. Samson, J. Marchand, Numerical Solution of the Extended Nernst-Planck Model. Journal of Colloid and Interface Science, 215 (1999)

58 CHAPTER 3: QUANTIFYING THE EFFECTS OF CRACK WIDTH, TORTUOSITY, AND ROUGHNESS ON WATER PERMEABILITY OF CRACKED MORTARS The existing service-life prediction models rarely account for the effect of cracks on mass transport and durability of concrete. To correct this deficiency, transport in fractured porous media must be studied. The objective of this chapter is to quantify the water permeability of localized cracks as a function of crack geometry (i.e., width, tortuosity, and surface roughness). Plain and fiber-reinforced mortar disk specimens were cracked by splitting tension; and the crack profile was digitized by image analysis and translated into crack geometric properties. Crack permeability was measured using a Darcian flow-thru cell. The results show that permeability is a function of the square of the crack width. Crack tortuosity and roughness reduce the permeability by a factor of 4 to 6 below what is predicted by the theory for smooth parallel plate cracks. Although tortuosity and roughness exhibit fractal behavior, their proper measurement is possible and results in correct estimation of crack permeability. 3-1 Introduction The permeability of concrete has an important impact on its durability since permeability controls the rate of penetration of moisture that may contain aggressive solutes and also controls moisture movement during heating and cooling or freezing and thawing [1]. While permeability of concrete is commonly measured using uncracked laboratory specimens [2,3], in real structures, the existence of cracks (induced by restrained shrinkage or mechanical loading) can 44

59 significantly increase the penetration of moisture and salts into concrete. This can especially be significant for high strength concretes which are known to have a higher tendency for cracking due to a larger autogenous and thermal shrinkage and a lower capacity for stress relaxation [4,5,6]. As such, for service-life predictions, it is important to account for the effect of cracks on accelerating the transport of moisture and aggressive agents inside concrete. Unfortunately, the present generation of service-life models largely overlooks the effect of cracks on durability. Research on the water permeability of crack-free concrete has been extensive [7,8,9,10,11,12] and has led to a general understanding that the saturated water permeability of concrete is a function of its porosity, pore connectivity, and the square of a threshold pore diameter [10,11,12]. In addition to the classical flow-thru permeability measurements [2,3], new methods (e.g., thermal expansion kinetics [13], beam bending [14], and dynamic pressurization [15]) have been offered that allow a more rapid and repeatable measurement of the saturated permeability. In comparison, research on the permeability of cracked concrete has been limited. The pioneering works of Kermani [16], Tsukamoto and Wörner [17], and Gérard et al. [18] explored changes in permeability of concrete caused by the application of compressive or tensile stress. Wang et al. [19] measured the permeability of concrete disks fractured using a splitting tensile test, and correlated the crack opening displacement (COD) with the permeability coefficient of a crack. Their results suggested that for COD smaller than 25μm, there is no significant increase in permeability beyond the matrix permeability. For larger cracks, permeability increases exponentially. It should be noted that in this study (as well as some future studies [20,21,22]), crack width was not directly measured; but assumed to be equal to the lateral displacement of the 45

60 disk specimen which was measured using an LVDT setup (figure 3-1). This assumption could result in inaccuracies due to crack branching, variability of crack width along its length, and inelastic deformation of the matrix; as discussed later in this chapter. Frame holding LVDTs Diametric crack LVDT LVDT Y Z Disk specimen Figure 3-1: Schematic illustration of the splitting tension setup used to fracture mortar disk specimens (after modification of the setup used by Wang et al. [19]) For use in service-life prediction models, it is important to establish a quantitative correlation between crack geometry and its permeability. Using the theory of laminar flow of incompressible Newtonian fluids in a smooth parallel-plate gap, equation 3-1, often referred to as the Poiseuille law, can be derived showing that the cumulative water flow through a crack, Q (m 3 /s), is related to the cube of crack width, b (m) [23]:

61 where Lb (m 2 ) is the crack cross sectional area perpendicular to the direction of flow, η (Pa.s) is dynamic viscosity of fluid, and (Pa/m) is the pressure gradient that drives the flow. This equation can be combined with Darcy s law: 3 2 and alternatively presented in terms of the permeability coefficient of a crack (m 2 ), as a function of the square of the crack width [24]: Equations 3-1 and 3-3 are strictly valid for a smooth, straight, and parallel plate crack. Real cracks in concrete never have such characteristics. As shown in figure 3-2, the crack width often varies along the length of a crack; cracks are tortuous meaning their actual length is larger than their nominal length; and crack wall surfaces are rough. These features reduce the permeability of a crack, sometimes significantly. To account for this reduction in permeability, in equations 3-1 and 3-3, an empirical reduction factor ξ has been included; the values of ξ = to 0.1 have been reported for plain and fiber reinforced concrete [21,25]. Unfortunately, these values are uncertain (vary several orders of magnitude), purely empirical, and have not been correlated to the geometric properties of cracks. For implementation in service-life models, it is important to improve the estimation of crack permeability (and other transport properties) as a function of 47

62 crack geometric parameters; i.e., average or effective width, tortuosity, and roughness. This chapter pursues this objective. (a) (b) 0.5mm 10mm Figure 3-2: A thru-thickness crack in a mortar disk specimen showing: (a) crack width variability and crack tortuosity, (b) crack wall roughness 3-2 Quantifying the Geometric Properties of Cracks Effective crack width In a fractured disk specimen (figure 3-1), the actual crack profile is highly variable in both parallel and perpendicular dimensions with respect to the direction of the flow. In other words, the crack widths are variable both on the surface and through the thickness of the disk. For example, figure 3-3 shows the cumulative distribution function of crack widths on the surface of a mortar disk specimen. For comparison, the horizontal permanent displacement (after 48

63 unloading), measured by LVDTs (figure 3-1), is also shown. It is clear that the LVDT reading is not a good measure of the actual crack profile or even the average crack width. 100% Cumulative distribution function of crack widths 80% 60% 40% 20% 0% Avg. crack width = 84.8μm LVDT reading = 204μm Crack width (µm) Figure 3-3: Cumulative distribution function showing the variability of crack profile along the surface of a disk specimen Using the digitized crack profile, an effective thru crack width, b eff-thru, can be calculated that results in the same permeability coefficient as the actual variable crack. This is done by extension of a technique originally suggested by Dietrich et al. [26] for fractured rocks. The crack profile is discretized into a series of local parallel plates, which are further combined into a global parallel plate (figure 3-4). In figure 3-4 (b), dimensions X, Y, and Z represent respectively the direction of the flow (e.g., thru thickness), diametric direction parallel to crack, and diametric direction perpendicular to crack (also see figure 3-1). If b ij represent the crack width for the i th element in direction X and j th element in direction Y, the first row of elements can be represented by b 1j. To calculate an effective thru crack width, b eff-thru, first, the effective 49

64 crack for each row of elements is obtained (b 1,eff, b 2,eff, etc.). According to Darcy s law, for the first row of elements, the volumetric discharge rate (Q 1,T ) is described as: (a) Actual crack profile Set of local parallel plates Global parallel plate b eff (b) L 14 =L Y Z X (direction of flow) d 11 =d Figure 3-4: Method for calculation of the effective crack width (adopted from Dietrich et al.), 1 ΔP 3 4 where n is the number of elements in each row, and and ΔP 1j represent the permeability and pressure loss for each element. Assuming that the elements length and thickness are chosen constant: L ij =L and d ij =d, and that the flow is 1-dimensional (ΔP 11 = ΔP 12 = = ΔP 1 ): 50

65 , 1 12 ΔP 3 5 Combining Eq. 3-5 and Darcy s law results in:, 1 12 ΔP b, 3 6, Eq. 3-7 can be used to determine the effective surface crack width at the top and bottom faces of each disk specimen (shown as b eff-surf in figure 3-5). To determine the effective thru crack width, b eff-thru, a similar summation procedure is performed in the X direction. For a column of m crack elements with length, width, and thickness nl, b i,eff, and d:,,, 1 12 ΔPb 1 12 ΔP b, 1 12 ΔP b, Where Q T is the total discharge rate, ΔP = ΔP ΔP m is the total pressure loss across the specimen, and m is the number of rows. Simplification of Eq. 3-9 results in: 51

66 ΔP ΔP b, b, 3 10 ΔP ΔP ΔP b, 1 b, ΔP b, 1 b 3 11 And ultimately: b, Fiber-reinforced Plain b eff-thru (μm) y = x R² = b eff-surf (μm) Figure 3-5: Correlation between the effective surface and thru crack widths An example of calculation steps to obtain effective crack width is given in Appendix A. 52

67 3-2-2 Crack tortuosity and surface roughness Figure 3-6 (a) and (b) show an actual thru-thickness crack profile and a schematic sketch of a crack to illustrate surface metrology procedures. The crack profile is wavy (i.e., not straight) resulting in an effective crack length (L e ) larger than the nominal crack length X max. The ratio: mm (a) Z reference line Slope=α λ x o x o +λ (b) X max X Figure 3-6: (a) Digitized profile of an actual thru crack; (b) Schematics of a crack profile to illustrate surface metrology procedures 53

68 is known as the tortuosity factor. It has been shown [27] that permeability is reduced proportionally with (X max /L e ) 2 and not with (X max /L e ) since the larger effective length affects both pressure gradient and fluid velocity. In addition to tortuosity, the crack surfaces are rough which creates additional friction against the flow. Louis [28][29] suggested the following equation to estimate the permeability of a parallel-plate crack with rough walls in laminar flow: where /2 is the relative surface roughness, and R a (m) is the absolute roughness defined as the mean height of the surface asperities. To quantify tortuosity and roughness, surface metrology techniques [30] can be employed. First, the surface profile is digitized and the x and z coordinates of all pixels on the crack surface are identified. The profile is then divided into brackets of length λ. Within each bracket, a reference line is drawn connecting the beginning and end points where the bracket intersects the crack profile. The entire nominal length (X max ) is covered by n brackets (note that n does not have to be integer) and the lengths of the reference lines are determined. By summation of the lengths, the effective length (L e ) is obtained and used for calculation of tortuosity. Note that L e depends on the sampling length λ; smaller λ results in a longer L e (more on this in section 4.3). 54

69 Roughness is determined in two steps. First the bracket x=0 to x=λ is selected and its roughness is determined by calculating the average height of surface asperities with respect to its reference line:, where R a,l (m) is the local roughness over this bracket, and the quantity in front of Σ is the absolute value of the difference between crack profile and the reference line in the direction perpendicular to the reference line. Next, the bracket is shifted one pixel to the right (x=1 to x=λ+1) and the local roughness is recalculated. The bracket is swept over the entire assessment length (x=0 to x=x max ) and the corresponding R a,l values calculated. A total of (X max -λ) number of R a,l values are averaged to determine the global surface roughness:, Avg.(R a,g, over entire assessment length) 3 16 In addition, R a,l values obtained can be used to construct a probability density function for the surface roughness of the crack. The roughness can be measured using the top, bottom or both crack surfaces. Note that R a,l and R a,g will depend on the sampling length λ (more on this in section 4.3). In this study, the procedures for measurement of tortuosity and roughness, as described above, were executed automatically through a MATLAB programming code. An example of the procedure used to obtain tortuosity and roughness is given in Appendix A. 55

70 3-3 Materials and Experiments Sample preparation Disk-shape plain and fiber-reinforced mortar specimens were prepared, diametrically fractured, and tested for permeability. The mortar mixture proportions are provided in Table 1. Type I/II portland cement (per ASTM C150-07), natural glacier sand (meeting the gradation requirements of ASTM C33-07), and polypropylene fibers (8mm length, 39μm diameter, vol. fraction 1%) were used. Disks (8.9cm diameter 2.5cm thickness) were cut from 17.8cm tall mortar cylinders Table 3-1: Mixture proportions for mortar specimens Component Proportions (kg/m 3 ) Plain Fiber-Reinforced Cement Sand Water Fiber Stabilizing Admixture Water Reducing Admixture after 28 days of moist curing. The disks were fractured using a deformation controlled splitting tensile test (figure 3-1). Vertical load was applied using a Universal Testing Machine by maintaining a constant rate of vertical deformation at 1 µm/s. The horizontal displacement was continuously monitored using two LVDTs positioned at the opposite sides of the specimen. As each specimen approached its peak load, a localized vertical crack formed starting from the middle section of the disk and growing outwards. After reaching a desired horizontal displacement, each specimen was unloaded at a vertical displacement rate of 5 µm/s. Various average crack widths in the range 10 to 200µm were generated using this procedure. More details on the fracture inducing method is given in Appendix B. After fracturing, specimens were wrapped in plastic covers and kept in a moist room until they were due for permeability 56

71 test. Each disk was vacuum saturated inside saturated Ca(OH) 2 solution for 24 hours prior to the permeability test Permeability measurement The saturated permeability was measured using a Darcian flow-thru cell (figure 3-7) and according to the procedure of CRD-C Inside a stainless steel cell, a disk specimen was securely seated on a retainer ring bonded to the specimen using a layer of high strength plaster. The circumferential surface of the specimen was sealed using a 70/30 mixture of paraffin and rosin. A layer of silicone sealant was applied on the top to seal the steel-wax interface. The silicon was allowed to cure for 4 hours while the top surface of the specimen was kept wet to prevent drying of the mortar. The permeability test was performed using a pressure gradient of 68.9 kpa (10psi). This resulted in a laminar flow with Reynolds numbers smaller than 118. The input water was pressurized by air inside a bladder, and this pressure was constantly monitored during the test. The output water was at atmospheric pressure. The outflow was collected inside a volumetric flask placed on top of a digital balance with accuracy 0.01g. Weight measurements were performed automatically by a computer at 10sec intervals. To prevent evaporation of outflow water, the mouth of the volumetric flask was sealed with adhesive plastic with a small puncture to allow pressure equilibrium. Further, the water inside the flask was covered with a thin layer of oil. Past research has shown that due to a self-healing phenomenon, permeability of cracks continuously decreases during the test [31,32,33]. The crack healing during the permeability test has been attributed to carbonation of concrete and formation of calcite (CaCO 3 ), renewed 57

72 hydration of cement, and/or dissolution and re-deposition of portlandite (Ca(OH) 2 ). The results of the current study show up to 85% reduction in crack permeability during the first 24 hours of the experiment, with narrower cracks showing a higher reduction than wider cracks. To maintain consistency, it was decided to use the outflow rate at 15 minutes to determine the permeability of cracks. The 15-min water flux inside cracks of various sizes was measured as 3 to 53 cm/sec. Considering the specimens thickness (2.5cm), the measured flux values suggest that cracks are fully saturated within the first few seconds of the test. In addition, the entire specimen had been vacuum saturated in Ca(OH) 2 solution before the test was initiated. Outflow collected and weighted Pressure Control board Flow thru cell Concrete disk Figure 3-7: Permeability test 58

73 3-3-3 Measuring crack dimensions Immediately after permeability measurement, the specimen was removed from the cell, cleaned and air dried for 24 hours (23 o C, 50%RH). It should be noted that some changes in the crack width may be inevitable due to drying shrinkage. The crack dimensions were measured using digital image analysis. To reach higher contrast between the crack and the matrix, specimens were vacuum impregnated with a low viscosity black epoxy for 15 minutes. After the epoxy hardened, the specimen s top and bottom faces were polished to remove the surface layer of epoxy and obtain flat surfaces (see Appendix A for more details). Next, the crack profile on the top and bottom faces was scanned using a digital scanner with resolution 9600dpi (i.e., pixel size 2.65μm). This resulted in a crack detection limit of approximately 5.3μm (i.e, 2 pixels wide). The surface crack width was measured every 200μm along the diametric crack, and the results were used to obtain the effective surface crack width b eff-surf (Eq. 3-7). In addition to crack width measurements along the top and bottom surfaces of each disk, three plain and five fiber-reinforced specimens were vertically sectioned at the mid-point along a diameter perpendicular to the surface crack and the crack profile through the specimen s thickness was scanned (figure 3-5). The thru crack widths were measured every 50μm, and the results were used to obtain the effective thru crack width b eff-thru using Eq To be able to calculate the effective thru crack width for the entire specimen, the possibility of establishing a correlation between the effective surface and the effective thru crack widths was explored. For the eight specimens vertically sectioned, the effective thru crack width was calculated along each section. Also, the effective surface crack width corresponding to each section was calculated. The portion of surface crack between and points was assumed to correspond with 59

74 the middle thru section (figure 3-5). Figure 3-5 shows a linear correlation between the effective surface and thru crack widths obtained for both plain and fiber-reinforced specimens. Using this correlation, for all specimens, the effective surface crack width was calculated by scanning the crack at top and bottom surfaces and this value was translated into an effective thru crack width. It should be noted that alternatively, 3D tomography techniques (e.g., X-ray CAT) can be used to obtain the three dimensional crack profile. However, the resolution of such measurements can be a limiting factor. For commonly available X-ray tomography instruments, the resolution is on the order of 1/1000 of the sample dimension (e.g., 89μm for 89mm diameter specimens). 3-4 Results and Discussion Comparison between average, effective, and LVDT crack measurements A total of 20 plain and fiber-reinforced disk specimens were fractured and tested in this study. Figure 3-8 shows comparisons among the average and effective crack widths and LVDT measurements. The average and effective crack widths are closely correlated with the effective thru crack widths approximately 13% larger than the average surface crack widths. This may suggest that when the average crack width is properly determined from the specimens surfaces, the effective crack width can be estimated with a reasonable accuracy without the need to slice the specimens or perform calculations described by equations 3-7 and In comparison, the LVDT measurements show a significant scatter while they are consistently over-estimating the crack widths, approximately by a factor 2.5. This again suggests that horizontal LVDT measurements must not be used to estimate crack widths in a splitting tensile test. 60

75 y = x R² = b eff-thru (μm) b avg (μm) (b) y = x R² = b LVDT (μm) Line of equality b avg (μm) Figure 3-8: Correlation between (a) average and effective crack widths, (b) average crack width and LVDT readings 61

76 3-4-2 Saturated permeability as a function of crack width The results of experimental measurements of crack permeability for plain and fiber-reinforced mortars are presented in figure 3-9. For comparison, the values predicted by the parallel plate theory (κ = b 2 /12) are also included. The curves present the best fit of Eq. 3-3 to the experimental and theoretical data. For the experimental data, the best ξ corresponding to the least error was determined. Several important observations can be made. (1) The permeability of cracks is more than 6 orders of magnitude larger than the matrix permeability. (2) The experimental results agree with the trend predicted by the theory. In other words, crack permeability is a function of square of crack width. (3) However, the experimental values of permeability are smaller than the theory by a factor of 4 to 6. The best fit for the plain specimens results in ξ = and for the fiber-reinforced specimens ξ = This could be due to crack tortuosity, and the friction caused by the crack s surface roughness and the presence of fibers. (4) The experimental results exhibit considerable scatter. While a coefficient of variation of 65% has been reported for single-operator permeability measurement of uncracked concrete [34], the existence of cracks can further contribute to scattering of results due to crack branching and variability of crack profile in three dimensions. Future research can explore the precision in permeability measurement of less variable cracks (e.g., manufactured gaps with certain thickness and surface roughness). Measurement of the crack permeability for very narrow crack is relatively difficult since the lowspeed water flux through the narrow crack is hard to accurately measure. Also crack healing during the permeability test may significantly change the crack geometry in small cracks. In addition, measurement of crack profile for very narrow cracks is difficult due to resolution 62

77 limitation of image capturing devices. Therefore the results shown in figure 3-9 for small cracks (less than 30 μm) may contain errors both in the measured permeability coefficient and effective crack width. To address these difficulties, artificial cracked samples were used in this study. More details are provided in chapter 5. 1.E-08 Crack permeability, κ (m 2 ) 1.E-10 1.E-12 1.E-14 1.E-16 1.E-18 Matrix permeability Theory Experiment: Plain Experiment: Fiber-reinforced b eff-thru (μm) Figure 3-9: Theoretical and experimental values of crack permeability as a function of effective crack width Crack tortuosity and surface roughness It is known that fracture surfaces exhibit fractal behavior [35]. This means that crack profile looks similarly tortuous and jagged at different scales of magnification (a property called selfsimilarity). Examples of fractal functions are numerous in nature including mountains, coastlines, clouds, plants, and natural and manufactured surfaces. The fractal nature of cracks in concrete materials has been recognized by earlier researchers [36,37,38,39] who attempted to 63

78 link the surface area and roughness of cracks to the fracture toughness of the material. Lange et al [37] found a correlation between roughness and fracture toughness, but no correlation to compressive strength, total porosity, and effective pore diameter (derived from mercury porosimetry). Ficker et al [38] found roughness to be closely related to water-to-cement ratio and, as a consequence, to compressive strength. Issa et al [39] suggested an exponential equation to quantify fracture toughness as a function of fractal dimension and stress intensity factor. A similar approach can be adopted to relate the tortuosity and roughness of cracks to their transport properties. Crandall et al [40] tried to find a quantitative relationship between the roughness of rock fracture and how this wall roughness affect the fluid flow through the fractures. They used Computational tomography scanning to obtain a three dimensional mesh from Rock fractures. They characterized the tortuosity and wall roughnesses of the obtained meshes and used Navier-stokes numerical model to relate roughness to the effective flow through the fractures. They calculated tortuosity (τ) for differenct fracture with different wall roughness. The Permeability coefficient from their numerical model showed close relationship to the rock fracture roughness In this study, tortuosity and roughness are measured for different sampling length with the procedure explained in section Figure 3-10(a) shows the effective length (L e ) of a thruthickness crack, in a fiber-reinforced specimen, measured using significantly different values of sampling length scale (λ) per section

79 100 Effective length, L e (mm) y = x R² = Sampling length, λ (μm) 100 (b) Effective length, L e (mm) Fiber-reinforced Plain Sampling length, λ (μm) Figure 3-10: Effective crack length as a function of sampling length scale: (a) Fiber-reinforced crack fitted by a fractal power function; (b) Comparison between plain and fiber-reinforced cracks 65

80 The crack had a nominal length X max = 20.66mm. It is observed that the measured values of L e depend strongly on λ and increase from 22.40mm at λ=12,755μm to 49.19mm at λ=3.8μm. This represents a change in the tortuosity factor from τ = 0.85 to 0.18 (i.e., becoming considerably more tortuous at smaller λ s). Despite its significant dependence on λ, L e can be considered a statistically self-similar fractal only if it follows the power function [35]: 3 17 where D (-) is the fractal dimension and F (m) is a constant. This power function shows as a straight line on a log-log scale which fits well to the data reported in figure 3-10(a), and results in a fractal dimension D = A comparison between the measured L e values from two cracks in a plain and a fiber-reinforced specimen is provided in figure 3-10(b). The plain crack shows similar or slightly smaller L e values (i.e., less tortuous crack) depending on the measurement s length scale (λ). The fractal dimension of the plain crack was determined as D = Similar results were obtained by analyzing other thru-thickness cracks in plain and fiber-reinforced specimens. In addition, the global roughness of the plain and fiber-reinforced thru cracks was measured based on the procedure of section 2.2. The results are presented in figure Unlike the effective length (L e ), which increases for smaller λ s, the crack roughness decreases monotonically as λ decreases. This is anticipated since at smaller sampling length scales, crack shows smaller surface features. Except for very large values of λ, the crack roughness shows a strong self-similar fractal behavior that can be represented by the power function [41]: 66

81 , Fiber-reinforced Plain Crack roughness, R a,g (μm) Sampling length, λ (μm) Figure 3-11: Crack surface roughness as a function of sampling length scale The fractal dimensions of D = and D = were obtained for the plain and fiberreinforced cracks. Further, the presence of fibers does not show a measurable impact on the roughness of cracks Effect of tortuosity and roughness on crack permeability Table 2 shows the average values (between plain and fiber-reinforced specimens) of crack tortuosity factor and surface roughness measured using different values λ. These values can be used, along with the effective or average crack width, to estimate crack permeability using Louis Eq that has been modified by adding the tortuosity factor: 67

82 Table 3-2: Average tortuosity and roughness measured using different values of λ λ (μm) τ (-) R a,g (μm) E-08 1.E-10 Permeability (m 2 ) 1.E-12 1.E-14 1.E-16 1.E-18 Matrix Permeability Plain Mortar Fiber Reinforced Mortar Smooth Parallel Plates Theory Roughnes= 8.9 μm, Tortuosity=0.21 Roughnes= 70 μm, Tortuosity=0.27 Roughnes= 637 μm, Tortuosity= b eff-thru (μm) Figure 3-12: Estimation of crack permeability based on Eq The results are presented in figure 3-12 which compares the estimated permeability from Eq with values measured by experiment. Among the three estimate curves, the one corresponding to λ = 10μm (τ =0.21, R a,g =8.9μm) matches the best to experimental data. This underlines the significance of choosing a proper sampling length for estimation of crack tortuosity and roughness. The observations from figure 3-12 suggest that the sampling length 68

83 must be several times smaller than the width of the examined crack. Further, Eq can provide a good quantitative estimate of crack permeability, at least for the effective crack widths in the range 35 to 100μm. Future research should examine the applicability of this equation for cracks of different size and in concrete materials other than the specific mortars studied in this work. 3-5 Conclusions Based on the results of this research, the following conclusions can be drawn: Using a digitized crack profile, an effective crack width can be calculated that results in the same permeability as the actual crack whose width is variable along its length. The effective crack width shows a reasonably good correlation with the arithmetic average of crack widths. On the other hand, horizontal displacement of disk specimen during the splitting tensile test (i.e., LVDT reading) does not correlate well with average or effective crack width and should not be used to estimate crack dimensions. Experimental measurements show that crack permeability coefficient is a function of crack width squared. While this trend agrees with the theory of laminar flow in smooth parallel plate gaps, the measured permeability values are smaller than the theory by a factor 4 to 6 likely due to tortuosity and surface roughness of cracks. Tortuosity and surface roughness of cracks exhibit fractal behavior. In other words, the numerical values of these parameters depend significantly on the magnification of length 69

84 scale. In this work, plain and fiber-reinforced cracks were examined at several different length scales from μm to mm. Both tortuosity and roughness show a statistically selfsimilar fractal behavior across these length scales, with fractal dimensions measured in the range to Towards the main objective of this work, a modification of the Louis equation by adding a tortuosity factor was found to be capable of quantifying crack permeability as a function of crack geometry (i.e., width, tortuosity, and surface roughness). Tortuosity and roughness of crack must be measured using a sampling length scale that is several times smaller than crack width. 3-6 References [1] S. Mindess, J.F. Young, D. Darwin, Concrete, 2nd Ed., Prentice Hall, Upper Saddle River, New Jersey, [2] CRD-C48-92, Standard Test Method for Water Permeability of Concrete, Handbook of Cement and Concrete, US Army Corps of Engineers, [3] ASTM D : Standard Test Methods for Measurement of Hydraulic Conductivity of Saturated Porous Materials Using a Flexible Wall Permeameter, American Society for Testing and Materials, West Conshohocken, Pennsylvania, [4] P.D. Krauss, E.A. Rogalla, Transverse Cracking in Newly Constructed Bridge Decks, NCHRP Report No. 380, Transportation Research Board, Washington, D.C [5] D. Darwin, J. Browning, W.D. Lindquist, Control of cracking in bridge decks: Observations from the field, Cement Concrete and Aggregates, 26 (2004)

85 [6] ACI 231R-10 Report on Early-Age Cracking: Causes, Measurement and Mitigation, American Concrete Institute, Farmington Hills, Michigan, [7] A.S. El-Dieb, R.D. Hooton, Water-permeability measurement of high performance concrete using a high-pressure triaxial cell, Cement and Concrete Research, 25(1995) [8] D. Ludirdja, R.L. Berger, J.F. Young, Simple method for measuring water permeability of concrete, ACI Materials Journal, 86(1989) [9] T.C. Powers, L.E. Copeland, J.C. Hayes, H.M. Mann, Permeability of portland cement paste, Journal of the American Concrete Institute, 51 (1954), [10] A.J. Katz, A.H. Thompson, Quantitative prediction of permeability in porous rock, Physical Review B, 34(1986) [11] P. Halamickova, R.J. Detwiler, D.P. Bentz, E.J. Garboczi, Water permeability and chloride ion diffusion in portland cement mortars: Relationship to sand content and critical pore diameter, Cement and Concrete Research, 25 (1995) [12] M.R. Nokken, R.D. Hooton, Using pore parameters to estimate permeability or conductivity of concrete, Materials and Structures, 41(2008) [13] H. Ai, J.F. Young, G.W. Scherer, Thermal expansion kinetics: Method to measure permeability of cementitious materials: II, Application to hardened cement pastes, Journal of the American Ceramic Society, 84 (2001) [14] G.W. Scherer, Measuring permeability of rigid materials by a beam-bending method: I, Theory, Journal of the American Ceramic Society, 83 (2000)

86 [15] Z.C. Grasley, G.W. Scherer, D.A. Lange, J.J. Valenza, Dynamic pressurization method for measuring permeability and modulus: II. Cementitious materials, Materials and Structures, 40 (2007) [16] A. Kermani, Permeability of stressed concrete, Building Research and Information, 19 (1991) [17] M. Tsukamoto, J.-D. Wörner, Permeability of cracked fibre-reinforced concrete, Darmstadt Concrete: Annual Journal on Concrete and Concrete Structures, 6 (1991), [18] B. Gérard, D. Breysse, A. Ammouche, O. Houdusse, O. Didry, Cracking and permeability of concrete under tension Materials and Structures, 29 (1996) [19] K. Wang, D.C. Jansen, S.P. Shah, Permeability study of cracked concrete, Cement and Concrete Research, 27 (1997) [20] C-M Aldea, S.P. Shah, A. Karr, Effect of cracking on water and chloride permeability of concrete, ASCE Journal of Materials in Civil Engineering, 11 (1999) [21] V. Picandet, A. Khelidj, H. Bellegou, Crack effect on gas and water permeability of concrete, Cement and Concrete Research, 39 (2009) [22] S.Y. Janga, B.S. Kimb, B.H. Oh, Effect of crack width on chloride diffusion coefficients of concrete by steady-state migration tests, Cement and Concrete Research, 41 (2011), [23] B. Massey, J. Ward-Smith, Mechanics of Fluids, 8th Ed., Taylor & Francis, London, [24] D. Snow, Anisotropic permeability of fractured media, Water Resources Research, 5 (1969)

87 [25] J-P Charron, E. Denarié, E. Brühwiler, Transport properties of water and glycol in an ultra high performance fiber reinforced concrete (UHPFRC) under high tensile deformation, Cement and Concrete Research, 38 (2008) [26] P. Dietrich, R. Helming, M. Sauter, H. Hötzl, J. Köngeter, G. Teutsch, Flow and Transport in Fractured Porous Media, Springer, Berlin, [27] J. Bear, Dynamics of Fluids in Porous Media, Dover Publications, New York, [28] G. de Marsily, Quantitative Hydrogeology, Academic Press, San Diego, [29] C. Louis, Section III, Introduction à l'hydraulique des roches, Bull BRGM Série 2, vol. 4, 1974, pp , (in French). [30] D. Whitehouse, Surfaces and Their Measurement, Taylor and Francis, New York, [31] N. Hearn, Self-sealing, autogenous healing, and continuous hydration: What is the difference?, Materials and Structures, 31 (1998) [32] C. Edvartsen, Water permeability and autogenous healing of cracks in concrete, ACI Materials Journal, 96 (1999) [33] H-W Reinhardt, M. Jooss, Permeability and self-healing of cracked concrete as a function of temperature and crack width, Cement and Concrete Research, 33 (2003) [34] A. Bhargava, N. Banthia, Permeability of concrete with fiber reinforcement and servicelife predictions, Materials and Structures, 41 (2007) [35] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, New York, [36] V.E. Saouma, C.C. Barton, N.A. Gamaleldin, Fractal characterization of fracture surfaces in concrete, Engineering Fracture Mechanics, 35 (1/2/3) (1990)

88 [37] D.A. Lange, H.M. Jennings, S.P. Shah, Relationship between fracture surface roughness and fracture behavior of cement paste and mortar, Journal of the American Ceramic Society, 76(3) (1993) [38] T. Ficker, D. Martišek, H.M. Jennings, Roughness of fracture surfaces and compressive strength of hydrated cement pastes, Cement and Concrete Research, 40 (2010) [39] M.A. Issa, A.M. Hammad, A. Chudnovsky, Correlation between crack tortuosity and fracture toughness in cementitious materials, International Journal of Fracture, 60 (1993) [40] Dustin Crandall, Grant Bromhal, Zuleima T. Karpyn, Numerical simulations examining the relationship between wall-roughness and fluid flow in rock fractures, International Journal of Rock Mechanics and Mining Sciences, 47 (2010) [41] J.C. Russ, Fractal Surfaces, Plenum Press, New York,

89 CHAPTER 4: EVALUATING ION DIFFUSIVITY OF CRACKED CEMENT PASTE USING ELECTRICAL IMPEDANCE SPECTROSCOPY Cracking can significantly accelerate mass transport in concrete and as such, impact its durability. This chapter is aimed at quantifying the effect of saturated cracks on ion diffusion. Electrical conductivity, measured by electrical impedance spectroscopy (EIS), was used to characterize the diffusion coefficient of fiber-reinforced cement paste disks that contained one or two through-thickness cracks. Crack widths in the range 20 to 100μm were generated by a controlled indirect tension test. Crack profiles were digitized and quantified by image analysis to determine crack volume fraction and average crack width. Crack connectivity (e.g., inverse tortuosity) was also measured by EIS. The results suggest that the diffusion coefficient of cracked samples is strongly and linearly related to the crack volume fraction; but is not directly dependent on crack width. Crack tortuosity does reduce the ion diffusion through cracks, but its impact is not very significant. Overall, the most important parameter governing ion diffusion in saturated cracked concrete is the volume fraction of cracks. Theoretical justifications of these observations are also provided. 4-1 Introduction Corrosion of steel is a major durability problem in reinforced concrete structures. Penetration of chloride ions is known to be the primary cause of steel corrosion in concrete exposed to deicing salts and in marine environments [1]. In saturated concrete, the major chloride transport mechanism is ionic diffusion. The rate of chloride penetration is primarily a function of diffusion coefficient of concrete, which is known to be dependent on the porosity and 75

90 connectivity of the pores in the concrete matrix [2][3]. While literature on the measurement techniques for the diffusion coefficient of undamaged concrete is extensive [2][4][5][6][7][8][9], cracked concrete has received much less attention [10][11][12]. In practice, concrete is often cracked due to restrained shrinkage and/or mechanical loading. Cracking can significantly accelerate mass (e.g., moisture, ion) transport in concrete and can reduce the service life of concrete structures in aggressive environments. In the present work, ion diffusion in cracked cement paste is studied using electrical impedance spectroscopy. The main objective is to quantify how cracking affects ion transport in saturated concrete and how diffusion in a crack is related to crack geometry. A number of past researchers attempted to quantify diffusion in cracked concrete simply as a function of the level of stress that the concrete had experienced. For example, Locogne et al [13] found that microcracks caused by hydrostatic pressure up to 200 MPa have no influence on the effective diffusion coefficient of concrete; while Konin et al. [14] reported a linear correlation between apparent diffusion coefficient of concrete and the applied load. This approach, although simple, may not be accurate since it makes an implicit assumption that cracking density (e.g. volume fraction) and crack geometry (e.g., crack width, length, and tortuosity) is only a function of the applied stress level. It is now well known that geometry of cracks is not solely dependent on the stress level, but also on loading patterns and material properties (e.g., fracture toughness, aggregate content, presence of fibers and other reinforcement, etc.). As such, a more 76

91 fundamental approach is needed to quantify the transport properties of cracks based on their geometry. Other researchers studied the effect of cracks on the ion diffusion coefficient in concrete. Jacobsen et al. [10], Aldea et al. [15], and Gerard and Marchand [16] reported a linear correlation between crack width and the ion diffusion coefficient in cracked concrete. Meanwhile, Gagne et al. [17] and Jang et al. [18] reported that below a threshold crack width (e.g., 80µm), the ion diffusion coefficient in concrete is not affected by cracking; while above this threshold, the ion diffusion coefficient increases linearly with crack width. A number of researchers tried to relate the ion diffusion coefficient in crack, D cr, to its geometry. Rodriguez and Hooton [19] suggested that D cr should be independent of both crack width and crack wall roughness and is equal to the diffusivity of ions in bulk pore solution (D o ). Others [11][20][21], suggested a threshold crack width of 53 to 80 μm, in which cracks wider than the threshold have D crack =D o ; while for smaller cracks, D crack is related to the crack width. Ismail et al. [11] attributed this crack width dependence to mechanical interaction between the closely spaced fracture surfaces, as well as self-healing and deposition of hydration products in the crack path. It should be noted that these researchers did not account for the connectivity (e.g., tortuosity and constrictedness) of cracks. As will be discussed in this chapter, crack connectivity could be affected by crack width, resulting in a reduction in the apparent diffusivity of the crack; while the actual diffusion coefficient of ions in the solution saturating the crack is not affected by the crack width (i.e., = D o independent of the crack width). 77

92 To address contradictions in the existing literature and improve the understanding of ion diffusion in cracked concrete, this research measures the diffusivity of cement paste disks that contain one or two through-thickness cracks. The measured diffusivity is related to the crack volume fraction, width, and tortuosity. Specifically, the following hypothesis is evaluated: The diffusion coefficient of a cracked cementitious matrix (D composite ) can be properly described based on the modified parallel law, which relates D composite to the diffusion coefficient (D cr ), volume fraction ( cr ), and connectivity (β cr ) of cracks. Among these parameters, only β cr is directly influenced by crack width, while D cr is independent of crack width and is equal to the ion diffusivity of pore solution, D o. 4-2 Methods for Measuring the Diffusion Coefficient of Concrete Before describing the theory and experimental procedures, a review of common methods for measuring the ion diffusion coefficient in concrete is helpful to justify the validity of using impedance spectroscopy for diffusivity measurements. These include the steady-state diffusion test [22], salt ponding test [23], the bulk diffusion test [24][25], the steady-state migration test [5][26], the rapid migration test [27], the rapid chloride permeability [28] and other DC resistivity tests, and the AC electrical impedance spectroscopy [29][30]. In the steady-state diffusion test (figure 4-1), a concrete disk is placed between two compartments; one filled with saturated Ca(OH) 2, and the other with saturated Ca(OH) 2 and 1M 78

93 NaCl. The time dependent changes in the concentration of Cl - in the two solutions (due to chloride ion diffusion from high to low concentrations) are determined by titration or using ionselective electrodes. This data is used to calculate the diffusion coefficient of Cl - through concrete. Concrete Sample Ca(OH) 2 Ca(OH) 2 1M NaCl Figure 4-1: Steady-state diffusion test This test can nicely duplicate ion diffusion in saturated concrete. However, similar to other diffusion tests, it is time consuming as it may take a few weeks (depending on concrete porosity) before the test reaches the steady-state condition. An alternative is the salt ponding test [23], which attempts to duplicate the non-steady-state diffusion. Figure 4-2 shows the test setup. Figure 4-2: Salt ponding test 79

94 Three duplicate concrete slabs are prepared, moist cured for 14 days, and then allowed to dry at 50% relative humidity for 28 day. The slabs are then exposed to 3% NaCl solution, ponded on their top surface. Each slab s sides are sealed and the bottom is exposed to drying in ambient air. After 90 days of continuous ponding, the solution is removed, the slabs are milled, and Cl concentration is determined in 12.5mm increments from the exposed surface to calculate the concrete s effective diffusion coefficient. This too is a long-term test, as it takes 132 days to complete. In addition, slab samples are never saturated; as such, chloride transport due to a combination of ion diffusion and moisture flow is measured. To address this drawback, the bulk diffusion test [24][25] has been offered. In this test, a concrete sample is saturated with Ca(OH) 2 solution and then exposed to 16.5% NaCl solution from one surface (other surfaces are sealed). After 35 days exposure, the sample is removed and milled in thin layers. The Cl profile is determined and used to calculate the apparent diffusion coefficient of concrete. Figure 4-3 shows the test setup. Figure 4-3: Bulk diffusion test 80

95 To shorten the test duration, migration tests (figure 4-4) accelerate Cl transport by application of a constant DC voltage. The steady-state migration test [5][26] is performed in a two-chamber cell with the pre-saturated concrete sample in between (similar to the cell used for the steadystate diffusion test). The upstream chamber is filled with 5% NaCl solution and the downstream chamber is filled with 0.3N NaOH solution [31]. A DC voltage in the range 10~12V is applied and the chloride concentration change in the downstream chamber is monitored over time. After establishing the steady-state condition, the results are used to calculate concrete diffusivity. - + ΔE Concrete sample Cathode Upstream 5% NaCl Downstream 0.3N NaOH Anode Figure 4-4: Electrical migration tests Figure 4-5: Rapid migration test 81

96 An alternative and faster test is the rapid migration test [27][32], which is based on non-steady state ion migration. The test setup is shown in figure 4-5. Here, the applied voltage is higher (up to 60V) and there is no need for monitoring the Cl concentration in the chambers solutions. Instead, after the test (6 to 96 hours depending on concrete electrical resistivity), the concrete sample is split and sprayed with AgNO 3 solution to determine the chloride penetration depth. This result is used in a formula derived from non-steady-state migration theory to back calculate the diffusion coefficient [33]. In comparison with direct diffusion methods, migration tests are faster and easier to perform. Figure4-6: Rapid chloride permeability test (RCPT) An even faster and easier method of assessing ion transport in concrete is based on electrical conductivity (i.e., inverse of resistivity) measurements. Since the solid skeleton of concrete is electrically insulating [34], the electrical current passes exclusively via ionic conduction through the liquid filled pores. In parallel, ionic diffusion is also limited to the liquid phase. As such, electrical conductivity measurements can quantify the resistance of microstructure against the 82

97 movement of ions. The most common conductivity-based test is the rapid chloride permeability test (RCPT) [28]. The test setup is shown in figure 4-6. In this test, a 60V DC voltage is applied to a concrete sample that is sandwiched between two electrolyte cells. The electric charge passed through concrete is measured over 6 hours and correlated to concrete s diffusion coefficient [1]. An even faster version of this test has been recently adopted by ASTM [35] in which the electrical current passing 1 minute after the application of 60V voltage is measured and used for calculation of concrete s electrical conductivity. These are rapid and commonly used tests but are prone to a number of problems. Mainly, the tests do not account for the electrical conductivity of pore solution and its effect on concrete conductivity. As such, the magnitude of charge passed may not truly reflect the microstructural diffusion coefficient. This problem can be especially acute when concrete contains some mineral or chemical admixtures (e.g., fly ash, or ionic accelerators) that significantly alter the ionic strength of pore solution. A second drawback of RCPT is that the high voltage can cause considerable temperature rise, which would result in erroneously high currents. In addition, the direct current (DC) results in developing electric polarization, which causes the actual voltage to be reduced [33]. Other DC resistivity measurements, such as the 4- point surface resistivity test [36], are prone to similar problems. In addition, it is critical to account for the significant effect of the concrete s moisture content on its conductivity [37][38]. An alternative method for measuring electrical conductivity is electrical impedance spectroscopy (EIS). EIS is a powerful tool for measuring the dielectric properties of materials and interfaces. 83

98 EIS is very fast (e.g., <1min depending on the voltage frequency) and allows insitu, nondestructive, and continuous measurements. EIS avoids heating of the specimen since the potential difference is low (<1V) and polarization is not a concern as an alternating voltage (AC) is applied [30]. In addition, measurements are obtained over a wide range of frequencies, which allows frequency-dependent responses to be properly characterized. The history of EIS goes back to late 19th century through the work of Oliver Heaviside who defined the terms impedance and reactance that are still being used. However the application of EIS to cementitious materials was developed mostly in the last 30 years [30][39][40][41][42]. By coupling EIS with the measurement or estimation of pore solution conductivity, the microstructural formation factor and diffusion coefficient can be determined [37][43][44]. In this chapter, EIS is used to measure the diffusion coefficient of Cl - through cracked cement paste. 4-3 Theory Solute transport in concrete occurs through a combination of diffusion and convection. In saturated concrete and in the absence of a pressure gradient, diffusion is the sole transport mechanism. Diffusion takes place as a result of a concentration gradient. Ions and other solutes travel within the pore solution of concrete from higher to lower concentration regions. Diffusion occurs through interconnected moisture-filled pores and fractures. Cracks, when saturated, can enhance the process of diffusion by providing wide pathways filled with large volume of pore fluid. The complete ionic flux equation through bulk aqueous solutions, which considers the interaction between multiple ions and also accounts for diffusion and migration (i.e., ion 84

99 movement due to an electrical field) is known as the electro-diffusion or Nernst-Plank equation [44][45]:, Where subscript i represents the i th ionic specie, (mol/m 3.s) is the ionic flux in bulk solution, (m 2 /s) is the self diffusion coefficient of ion in bulk solution, (-) is the ion activity coefficient 0 1, (mol/m 3 ) is the ion concentration in pore solution, is the ion valency, F is Faraday constants (=96485 J/V.mol), R is universal gas constant (= J/mol.K), T (K) is absolute temperature, and (V) is the electrical voltage (imposed externally or created by charge imbalance). The term 1 accounts for the non-ideality of high ionic strength solutions where ion-ion interactions are not negligible [44]. The convective transport due to a pressure gradient is not considered in eq. 4-1 but can be simply added as a separate term. For a composite material containing parallel solid and liquid phases:,,.,. 4 2 where subscripts S and L represent solid and liquid phases, respectively; and (-) is the volume fraction of each phase. For porous materials where ion transport occurs only in the liquid phase, 85

100 the first term on the right hand side of eq. 4-2 is eliminated. In a more general case where the liquid phase is tortuous:,,., 4 3 where β L (-) is the pore connectivity that is a measure of tortuosity and constrictions of the pore network, and is the microstructural formation factor which represent the resistance of microstructure to movement of ions [46]. Combining eqs. 4-3 and 4-1 results in:,, 4 4 Similarly, the electrical conductivity of composite, (S/m), can be related to the electrical conductivity of pore solution, (S/m):

101 Eqs. 4-4 and 4-5 provide a theoretical basis to use electrical conductivity measurement for calculation of ion diffusivity for porous materials with insulating solid skeleton. Combining eqs. 4-4 and 4-5 results in eq. 4-6, which is known as the Nernst-Einstein equation [43]:,, Electrical conductivity measurements have been used by a number of researchers to estimate the diffusion coefficient of rocks and concrete [29][47]. Their results showed a good agreement with other diffusion measurement techniques. It should be noted that eq. 4-6 accounts for the geometric restriction effect of pore structure on ion motion and neglects the interaction between ions and the pore walls [2]. In most solid-liquid interfaces, an electrical double layer forms inside the solution adjacent to the solid surface due to the surface being electrically charged [48]. As such, hydrated ions with opposite charge are attracted to pore walls. This electrical double layer interferes with ionic movement and reduces the velocity of ions near the walls [49]. Within the double layer near the surface, in a so-called Stern layer, the ions are immobile. To account for such surface effects, one should consider the change in the electrical field near pore surfaces. The electrical potential is maximum at the surface, (V), and decreases as one proceeds out into the bulk solution. The electrical potential ( at distance (m) from the surface is given by eq. 4-7 [50]: 87

102 4 7 The effective thickness of the double layer is defined as m, which is also known as Debye length. The Debye length is controlled by the type of electrolyte, ionic strength of the solution, and ion valences [49]. For pore solution in concrete, the Debye length is in the range of a few nanometers. Similarly, Rajabipour and Weiss [37] showed that surface conduction in cement paste is only significant within approximately 15 nm of pore surfaces. As such, when studying ion transport through cracked concrete, for cracks that are at least few tens of μm wide, the electrical effect of cracks walls can be ignored. This means that cracks only have a geometric effect on ion transport that can be properly characterized by eq Diffusion through a cracked concrete occurs as diffusion through the concrete matrix plus diffusion through the cracks. A concrete specimen containing through-thickness cracks can be defined as a composite containing the concrete matrix in parallel with one or more cracks (figure 4-7). A modified parallel law (eq. 4-2), which also include phase connectivity terms) can be used to quantify the diffusion coefficient of this composite as: D Composite D Matrix D Cr = + Figure 4-7: Parallel law for ion diffusion in cracked concrete 88

103 4 8 Similarly, the electrical conductivity of the composite can be described as: 4 9 Here, and are the volume fractions of the matrix and crack ( 1. An important question is how crack density and crack geometry affect eqs. 4-8 and 4-9; more specifically, the parameters,, and. As discussed above, for μm-wide cracks, electrical interactions with crack walls can be neglected and as such, and. The crack density can be simply represented by the crack volume fraction,. The impact of crack aperture on is only through changing the volume occupied by cracks. The connectivity factor (β) is defined as the reciprocal of tortuosity ( ) multiplied by the constrictedness factor ( ) [46]: Tortuosity is the square of the effective length (L e ) divided by the nominal length (L) of a crack, while constrictedness represents the effect of change in the crack aperture over its length (figure 4-8): 89

104 (a) L e > L L (b) w 1 w 2 L/2 L/2 Figure 4-8: Schematics of (a) smooth, and (b) constricted crack The parameter S should be calculated for every single sharp change in the crack aperture and averaged over the length of the crack to determine an effective constrictedness S eff = Avg(S 1,, S n ). Eq suggests S to be dependent on crack width for a similar crack surface profile. For example, a 20μm change in crack aperture from w 1 =50μm to w 2 =30μm results in S=1.284; while the same 20μm change in aperture but from w 1 =220μm to w 2 =200μm results in a much less significant S= The impact of crack aperture on tortuosity factor T might be less pronounced. 90

105 Eqs. 4-8 and 4-9 can be further simplified by combining with the Nernst-Einstein equation for the matrix phase : Where σ o and D o are the electrical conductivity and ion diffusion coefficient of the solution saturating the crack and the matrix phase. It should be noticed here that is the connectivity of the matrix phase in the cracked sample, which is different than the connectivity of the liquid-filled pores inside the matrix,. In a two-phase composite specimen of cracked concrete, the cracks are considered as one phase, and the concrete matrix (together with its pores, air voids, and other constituents) is considered as a continuum second phase. For specimens containing through-thickness cracks (as those studied here), it is reasonable to assume 1. Also, in eqs and 4-14,. Eqs and 4-14 suggest that the diffusion coefficient of cracked concrete ( ) can be determined simply be measuring σ Composite and σ o, and making a justified approximation of D o based on the type of ionic species and the ionic strength of pore solution. Further, the only parameter in eq that is directly affected by crack aperture is 1/. Where 91

106 changes in with respect to crack width are insignificant, eq suggests a linear relationship between diffusion coefficient of cracked concrete and the volume fraction of cracks,. The validity of these conclusions will be examined in this study. In this study electrical impedance spectroscopy was use to measure σ Composite. The theoretical basis of this method is further explained here. Electrical Impedance spectroscopy consists of multi-frequency alternating (AC) measurement of concrete s impedance. A sinusoidal voltage is applied over a broad range of frequencies and the generated current is measured. The current has the same frequency of the corresponding voltage but with a phase shift of θ (rad). Electrical impedance Z (Ω) can be calculated as follows: cos cos 4 15 Where is voltage (V) at time t (s), 2 is the angular frequency (rad/s), is frequency (Hz), is current (A) as a function of time and (V) and (A) are the amplitude of voltage and current respectively. Alternatively, current and voltage can be described in polar coordinates and electrical impedance can be written as: exp exp exp

107 Where impedance amplitude Z 0 (Ω) is V 0 /I 0 and 1 (unitless). Electrical impedance (Z) in equation 4-16 is composed of real and imaginary components. The real term is known as resistance ( and the imaginary term is known as reactance (. The typical experimental result of resistance vs. reactance for a cracked fiber reinforced cement paste sample is shown in figure 4-9 which is known as Nyquist plot. As it is shown in figure 4-3 at a particular frequency, the imaginary component of electrical impedance becomes zero and the total impedance (Z) becomes equal to real impedance (Z ). This value is called bulk resistance which is used in eq to calculate electrical conductivity. Z" (Ω) Bulk resistance (R) Z' (Ω) Figure 4-9: Resistance vs. reactance for a cracked fiber reinforced cement paste sample In this study bulk resistance R (Ω) was measured for all samples and used to calculate electrical conductivity. Then electrical conductivity of the samples was used in combination with image analysis results of the crack profile to calculate pore connectivity (β) and finally the diffusion coefficient of cracked samples was calculated using equation

108 4-4 Materials and Experiments Fiber reinforced cement paste samples were tested in this study. Table 1 shows the mixture proportions. PVA fibers were used (at 5.8 % volume fraction) to increase the ductility of the samples and to prevent their sudden fracture and ensure stable formation of cracks during splitting tensile test. Water reducing admixture was added to improve the workability. ASTM C150 type I portland cement was used. The paste was mixed according to ASTM C305 and cast in 10 20cm cylindrical molds in three layers and consolidated on a shaker table. After 3 days moist curing at 22⁰C, one sample was demolded for measurement of the pore solution composition. This sample was broken into pieces and its pore solution was extracted using a pore fluid expression die with capacity of 550 MPa [51]. The pore solution chemical composition was determined using inductively coupled plasma atomic emission spectroscopy (ICP-AES). The results are provided in Table 4-2. The knowledge of pore solution composition was needed for saturating cracks with a similar synthetic pore solution with known D o and σ o, as discussed later. Pore solution conductivity was measured using a commercially available conductivity meter. After one week moist curing, the remaining cylinders were demolded and disks of mm (diameter thickness) were cut from the cylinders using a diamond blade saw. The disks were then submerged in synthetic pore solution at 60⁰C for one more week to reach an equivalent age (maturity) of 26 days at ambient temperature (22⁰C) (assuming an approximate datum temperature 0⁰C). The disk samples were then cracked in a deformationcontrolled indirect tension test. Figure 4-10 shows the indirect tension test setup. Two LVDTs were used to control the lateral displacement of the sample during loading. This lateral displacement was used as a rough estimate of average crack width during the test. Actual crack widths were later quantified using digital image analysis, as discussed below. The fracture tests 94

109 were conducted using displacement-control method with the actuator displacement rate of 1μm/s and 5 μm/s for the loading and unloading phases, respectively. Cracks in the approximate range of 20 to 100μm were induced, by unloading the test at desired values of lateral displacements. Figure 4-11 shows typical crack patterns induced. Samples with both single crack and dual cracks were created. The dual cracked samples were produced by reloading the single cracked samples in a direction perpendicular to the first crack. After cracking, the circumferential surface of the samples was sealed using an epoxy-based paint. Samples were then re-saturated (under vacuum) in the synthetic pore solution before diffusion testing. Table 4-1: Mixture proportions Component Proportions (/m3 paste) Cement (Kg) 1480 Water (Kg) Fiber (Kg) 5.3 Water Reducing Admixture (Lit) Table 4-2: Pore solution composition Element Concentration (ppm) Al 7.1 Ca 43.8 K Na 6900 S 2800 Fe 0.59 Si

110 Direction of load LVDTs Figure 4-10: Splitting tension setup used to fracture cement paste disks Figure 4-11: Crack patterns for dual and single cracked samples 96

111 The setup for measurement of the diffusion coefficient using electrical impedance spectroscopy (EIS) is shown in figure A cracked disk sample was installed between two fluid compartments using a test cell similar to those used in the rapid chloride permeability test [28]. The joint between the sample and the fluid compartments was sealed with silicone sealant. Care was taken to prevent drying. The synthetic pore solution was introduced in the two compartments 3 hours prior to the test and was renewed immediately before the test time to minimize the potential carbonation effect. Two stainless steel electrodes (8 155mm diameter length) were immersed into solutions to establish the electrical connectivity. The bulk resistance (R b ) of the test cell containing the sample was measured by applying a 500mV AC voltage in the sweep frequency range of 40 Hz to 10 MHz. Frequency sweep were performed in a logarithmic mode with 150 measurements recorded per frequency decade. A nice review of EIS measurements and data interpretation for cement-based materials is provided by [30]. Here, the electrical conductivity of a cracked sample (σ Composite ) was determined based on its bulk resistance: 4 17 where the geometry factor, k (1/m), was determined experimentally according to the method of [34]. 97

112 After EIS measurements, the sample was taken out of the cell, air dried, and vacuum impregnated with a low viscosity black epoxy to fill the cracks and increase the optical contrast between cracks and the cement paste matrix. After the epoxy had set, samples were surface polished (grit #220), and scanned using a digital scanner at the resolution of 4800 dpi (pixel size = 5.3μm). Crack width was measured by image analysis on the two surfaces of each disk sample at 0.85mm intervals along the length of each crack. The results were arithmetically averaged to obtain the mean crack width (w Cr ). For dual cracked samples, the second crack was induced to be approximately the same size as of the first crack. The width of both cracks were quantified by image analysis and averaged to determine w Cr. The length of cracks was measured as well. The volume fraction of cracks in each sample ( ) was obtained by multiplying the mean crack width by the crack length. Synthetic pore solution Cement paste sample Figure 4-12: EIS test setup 98

113 4-5 Results and Discussion The results of EIS electrical conductivity measurements (σ Composite, ) were used in combination with the image analysis results (w Cr, ) in eqs and 4-14 to calculate the values for crack connectivity ( ) and the diffusion coefficient of the cracked sample (D Composite ). The constant parameters used in eqs and 4-14 are listed in Table 3. For D o, the self diffusion coefficient of NaCl in water is used [52]; however, this value can be determined more accurately by accounting for the activity and ionic strength of the solution. Table 4-3: Parameters used in eqs and 4-14 Solution electrical conductivity σ o (S/m) Solution diffusivity D o (m 2 /s) Matrix electrical conductivity σ Matrix (S/m) Matrix formation factor μ Matrix (-) Matrix connectivity factor β Matrix (-) 1.0 Figure 4-13 shows the measured electrical conductivity of cracked samples as a function of volume fraction of cracks. σ Composite exhibits an approximately linear relationship with the crack volume fraction, in agreement with the theoretical prediction (eq. 4-14). The vertical intercept corresponds to the conductivity of the crack-free cement matrix; σ Matrix =0.061 S/m. The slope of this line equals (note that 1), which suggests approximately The conductivity measurements can be further translated into the diffusion coefficient of the cracked samples (D Composite ), which is presented in figure Again, 99

114 D Composite shows a linear relationship with (in agreement with eq. 4-13); having a slope of and a vertical intercept D Matrix = (m 2 /s) w Cr = 64.8 μm σ composite (S/m) w Cr = 38.2 μm R 2 =0.91 Double Cracked 0.02 Single Cracked Crack volume fraction ( Cr ) Figure 4-13: Variation of the electrical conductivity of cracked cement paste samples (σ Composite ) versus crack volume fraction ( Cr ) Figures 4-13 and 4-14 also show that single- and dual-cracked samples with similar but significantly different mean crack width (w Cr ) show similar σ Composite and D Composite. For example, a single-cracked sample with w Cr =64.8μm and a dual-cracked sample with w Cr =38.2μm have = and , respectively. These samples show σ Composite = and S/m, respectively, and D Composite = and m 2 /s, respectively. This suggests that conductivity and ion diffusivity of cracked cementitious materials are dictated by the volume fraction of cracks and not by crack widths. When the results are graphed as a function of crack width (figure 4-15), the (or ) correlation is weak and 100

115 primarily due to the fact that w Cr indirectly impacts. These observations may suggest that, strictly speaking, for fully saturated systems, crack width has minor impact on the diffusivity of cracked cementitious materials. This might further suggest that methods such as fiber reinforcement, which are designed to control and reduce crack width, may have less than anticipated benefits for saturated cracks if they do not reduce the volume fraction of cracks. 1.2E E-11 D Composite (m 2 /s) 8.0E E E E-12 R 2 =0.91 Double Cracked Single Cracked 0.0E Crack volume fraction ( Cr ) Figure 4-14: Estimated diffusion coefficient of cracked samples (D Composite ) as a function of crack volume fraction ( Cr ) The connectivity of crack path ( ) is the only term in eqs and 4-14 that could be dependent on crack width. Using eq and the EIS measurements, at each data point was calculated (figure 4-16). The results show considerable scatter in agreement with the nature of cracks in cementitious materials. The average is 0.547, which is slightly less than the 0.619, obtained from the line slope in figure A weak correlation between w Cr and is observed; with modestly increasing for wider cracks. This could be due to an increase in the 101

116 0.12 σ Composite (S/m) R² = Double Cracked Single Cracked Avg. crack width; w Cr (μm) Figure 4-15: Variation of the electrical conductivity of cracked samples (σ Composite ) versus the average crack width (w Cr ) 1 Crack connectivity; β Cr (-) R² = Ave. crack width; w Cr (μm) Figure 4-16: The calculated crack connectivity ( Cr ) as a function of average crack width (w Cr ) constrictedness factor S Cr for smaller cracks as suggested by eq Meanwhile, Cr is primarily dictated by its tortuosity factor (T Cr ) which is largely dependent on concrete properties 102

117 (such as aggregate size, type, and volume fraction; presence of fibers or other reinforcement) as well as loading patterns. The effect of crack width (w Cr ) on T Cr could be small. Most importantly, crack connectivity is relatively large; meaning that it does not significantly reduce the ion diffusion in saturated cracks. 4-6 Conclusions Based on testing of fiber-reinforced cement paste disk samples that contained one or two through-thickness cracks, it was found that: Ion diffusion coefficient and electrical conductivity of cracked samples are strongly (and approximately linearly) related to the volume fraction of cracks. This is in agreement with the modified parallel law. Diffusivity and conductivity are not significantly influenced by crack width. Crack connectivity ( Cr ) in the range of 0.37 to 0.69 was measured, suggesting that Cr does not significantly reduce ion diffusion in cracks (i.e., beyond a factor =2.70). Cr decreases modestly by reducing crack width. 4-7 References [1] A. Bentur, S. Diamond, N.S. Berke, Steel Corrosion in Concrete: Fundamentals and Civil Engineering Practice, Taylor & Francis, London, UK, [2] HW. Reinhardt, Penetration and permeability of concrete: barriers to organic and contaminating liquids, RILEM Technical Committee, London, UK,

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120 [20] E. Kato, Y. Kato, T. Uomoto, Development of simulation model of chloride ion transportation in cracked concrete, Journal of Advanced Concrete Technology, 3 (2005) [21] A. Djerbi, S. Bonnet, A. Khelidj, V. Baroghel-bouny, Influence of traversing crack on chloride diffusion into concrete. Cement and Concrete Research, 38 (2008) [22] CL. Page, NR. Short, A. El Tarras, Diffusion of chloride ions in hardened cement pastes, Cement and Concrete Research, 11 (1981) [23] AASHTO T259, Standard method of test for resistance of concrete to chloride ion penetration, American Association of State Highway and Transportation Officials, Washington D.C., USA, [24] ASTM C , Standard test method for determining the apparent chloride diffusion coefficient of cementitious mixtures by bulk diffusion, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA, [25] NT BUILD 443, Concrete, hardened: Accelerated chloride penetration, Nordtest, Esbo, Finland, [26] NT BUILD 355, Chloride diffusion coefficient from migration cell experiments, Nordtest, Tekniikantie 12, FIN-02150, Espoo, Finland, [27] NT BUILD 492, Concrete, mortar and cement-based repair materials: chloride migration coefficient from non-steady-state migration experiments, Nordtest, Esbo, Finland, [28] ASTM C , Standard test method for electrical indication of concrete s ability to resist chloride ion penetration, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA,

121 [29] A. Atkinson, AK. Nickerson, The diffusion of ions through water-saturated cement, Journal of Materials Science, 19 (1984) [30] BJ. Christensen, T. Coverdale, RA. Olson, SJ. Ford, EJ. Garboczi, HM. Jennings, TO. Mason, Impedance spectroscopy of hydrating cement-based materials: Measurement, interpretation, and application, Journal of the American Ceramic Society, 77 (1994) [31] L. Tang, HE. Sørensen, Precision of the Nordic test methods for measuring the chloride diffusion/migration coefficients of concrete, Materials and Structures, 34 (2001) [32] L. Tang, LO. Nilsson, Chloride Diffusivity in High Strength Concrete, Nordic Concrete Research, 11 (1992) [33] KD. Stanish, RD. Hooton, MDA. Thomas, Testing the Chloride Penetration Resistance of Concrete: A Literature Review, FHWA Contract DTFH61-97-R-00022, US Federal Highway Administration, Washington, D.C., USA, [34] F. Rajabipour, In situ electrical sensing and material health monitoring of concrete structures, Ph.D. Dissertation, Purdue University, West Lafayette, Indiana, USA, [35] ASTM C , Standard test method for bulk electrical conductivity of hardened concrete, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA, [36] R. Polder, C. Andrade, B. Elsener, Ø. Vennesland, J. Gulikers, R. Weidert, M. Raupach, Test methods for onside measurement of resistivity of concrete, Materials and Structures, 33 (2000) [37] F. Rajabipour, WJ. Weiss, Electrical conductivity of drying cement paste, Materials and Structures, 40 (2007)

122 [38] J. Weiss, K. Snyder, J. Bullard, D. Bentz, Using a saturation function to interpret the electrical properties of partially saturated concrete, Journal of Materials in Civil Engineering (in peer review), (2012). [39] WJ. McCarter, S. Garvin, and N. Bouzid, Impedance Measurements on Cement Paste, Journal of Materials Science Letters, 7 (1988) [40] K. Brantervik, GA. Niklasson, Circuit models for cement based materials obtained from impedance spectroscopy, Cement and Concrete Research, 21 (1991) [41] CA. Scuderi, TO. Mason, HM. Jennings, Impedance spectra of hydrating cement pastes, Journal of Materials Science, 26 (1991) [42] P. Gu, P. Xie, JJ. Beaudoin, R. Brousseau, A.C. Impedance Spectroscopy (I): A New Equivalent Circuit Model for Hydrated Portland Cement Paste, Cement and Concrete Research, 22 (1992) [43] EJ. Garboczi, Permeability diffusivity and microstructural parameters: A critical review, Cement and Concrete Research, 20 (1990) [44] KA. Snyder, Validation and Modification of the 4SIGHT Computer Program, NIST-IR 6747, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland, USA, [45] L. Dresner, Some remarks on the integration of the extended Nernst-Planck equations in the hyperfiltration of multicomponent solutions, Desalin., 10 (1972) [46] FAL. Dullien, Porous Media; Fluid Transport and Pore Structure, 2nd ed. Academic Press, New York, USA, [47] WF. Brace, Permeability from resistivity and pore shape. Journal of Geophysical Research, 82 (1977)

123 [48] JOM. Bokris, AKN. Reddy, M. Gamboa-Aldeco, Modern Electrochemistry: Fundamentals of Electrodics, Kluwer, New York, USA, 2000 [49] T. Zhang, OE. Gjørv, Diffusion behavior of chloride ions in concrete, Cement and Concrete Research, 26 (1996) [50] AW. Adamson, Physical Chemistry of Surfaces, 6th ed. Chapter V, Wiley, New York, USA, [51] RS. Barneyback, S. Diamond, Expression and analysis of pore fluid from hardened cement pastes and mortars, Cement and Concrete Research, 11 (1981) [52] RC. Weast, MJ. Astle, WH. Beyer, CRC Handbook of chemistry and physics, 66th Ed. CRC Press, Boca Raton, Florida, USA,

124 CHAPTER 5: PERMEABILITY, ELECTRICAL CONDUCTIVITY, AND DIFFUSION COEFFICIENT OF SIMULATED CRACKS In this chapter, transport properties of simulated cracks are measured to quantify the permeability and diffusion coefficient of cracks based on crack geometry (crack width, and crack wall roughness). Saturated permeability and diffusion coefficient of cracks are measured using constant head permeability test, electrical migration test, and electrical impedance spectroscopy. A Plexiglas rough parallel plate is used to simulate cracks in concrete. The results of permeability test showed that permeability of a crack can be predicted using Louis equation; which determines permeability based on crack width and surface roughness of the crack walls. The result of migration and impedance tests proved that the diffusion coefficient of cracked samples is linearly related to the crack volume fraction. When crack connectivity is correctly accounted for, diffusion coefficient of cracks is independent of crack width and is equal to the diffusion coefficient of the solution contained in the cracks. Crack connectivity increases with increase in crack width up to a threshold value (~ 80 μm) where the connectivity value reaches its theoretical maximum (β=1). Cracks larger than the threshold width can be assumed to be fully connected. 5-1 Introduction As discussed in chapter 1, the main objective of this dissertation is qualifying the transport properties of cracked cementitious materials (more specifically, saturated permeability, and ion 110

125 diffusivity). In the two previous chapters, permeability and diffusion coefficient of cracked mortar and cement paste samples were studied. One of the major difficulties that exists in the study of transport properties of fractured concrete is measurement of the crack geometry. Crack profile of natural cracks in concrete varies both in the surface and through the depth of the fracture. Precise measurement of the crack characteristics is very important and failing to do so will result in unreliable and inaccurate conclusions. In addition to this problem, it is often not easy to obtain and measure very narrow and very wide cracks with real concrete samples due to equipment limitations and brittleness of concrete. To address these difficulties, in this chapter a test which setup is designed and built to simulate straight cracks in concrete is introduced. The setup is used to quantify transport properties (saturated diffusion and saturated permeability coefficients) of crack as a function of crack geometry (crack width and surface roughness). The advantages of using an artificial cracked sample are 1- measuring crack geometry more accurately, and 2- achieving wider range of crack width. In this research a setup was designed and built from Plexiglas to simulate cracks in concrete. Using a simulated crack sample, crack width and connectivity were accurately measured. Permeability and diffusion coefficient of simulated cracks were measured using constant head permeability test, electrical impedance spectroscopy and electrical migration test. Since Plexiglas materials are non-porous and inert, the measured values only reflect the crack properties. Permeability and diffusivity of cracks in concrete was studied in Chapter 3 and 4. In this chapter theses properties are further studied to fully address the research questions in two previous chapters. The result of this study was used to evaluate the following hypotheses: 111

126 I. Permeability of cracks can be qualitatively predicted based on crack width, tortuosity and crack wall roughness. II. Diffusion coefficient of crack is independent from crack width and is equal to the diffusivity of the solution saturating the crack. III. IV. The surface effect on diffusion coefficient of cracks is insignificant and can be ignored. Crack connectivity is dependent on crack width for small cracks and constant for large cracks. 5-2 Methods There are several developed method that are used to measure coefficients of permeability [1][2] and diffusion [3][4][5][6][7][8][9][10]. In this study constant head permeability test was used to measure permeability coefficient. Migration test [7][9] and electrical impedance test [10] was used to measure diffusion coefficient. More details on these methods are provided in section 5-4. Description of various methods for measurement of ion diffusivity of concrete is provided in chapter Theory Hydraulic Permeability of Cracks As discussed in chapter 3, using the theory of laminar flow for incompressible Newtonian fluids inside a smooth parallel-plate gap, the permeability coefficient of a crack (m 2 ), can be written as a function of crack width square [11]: 112

127 Where b (m) is crack width. Eq. 5-1 is valid for straight thru cracks with parallel and smooth wall surfaces. Such perfect cracks are rarely observed in concerete whose cracks are tortuous, have rough surfaces, and are continuously narrowing and widening along the crack path (see figure 4-8). Crack tortuosity and roughness reduce permeability by introducing friction and energy loss. Chapter 3 discussed an empirical equation to account for the reduction in crack permeability due to crack tortuosity and surface roughness (Eq. 5-2): Where is the tortuosity factor which is defined and square of nominal crack length to effective crack length. /2 is the relative surface roughness and (m) is the absolute roughness defined as the mean height of the surface asperities. In this chapter, the measured values of permeability are compared to estimated values obtained from eq. 5-2 for wider range of crack widths using the simulated crack sample. The simulated crack sample models straight cracks with rough surfaces. Therefore the tortousity factor is assumed to be one in this chapter. 113

128 5-3-2 Ion Diffusivity of Cracks Ions can travel through cracks filled with pore fluid. This occurs through a combination of convection, diffusion, and migration (i.e., ion movement due to an electrical field or voltage gradient). The electro-diffusion or Nernst-Plank equation [12][13] gives the complete ionic flux through bulk aqueous solutions: 1,, 5 3 Where subscript i represents i th ionic species, (mol/m 3.s) is the ionic flux in bulk solution, (m 2 /s) is the self diffusion coefficient of ion in bulk solution, (-) is the ion activity coefficient 0 1, (mol/m 3 ) is the ion concentration in pore solution, is the ion valency, F is Faraday constants (=96485 J/V.mol), R is gas constant (= J/mol.K), T (K) is absolute temperature, (V) is the electrical voltage (imposed externally or created by charge imbalance), and (m/s) is the convective velocity. The term 1 accounts for the non-ideality of high ionic strength solutions. The ion-ion interactions are accounted for by the change rate of the logarithm of chemical activity per unit change in ion concentration [12]. The convective transport term in eq. 5-3 is equal to zero if there is no pressure gradient. Diffusion coefficient of cracked fiber reinforced cement paste samples were studied in chapter 4, using electrical impedance spectroscopy (EIS). The relationship between diffusion coefficient of 114

129 crack and crack width was not fully covered in chapter 4. In this chapter diffusion coefficient of crack is measured for the simulated crack samples with wider range of crack widths. In addition to EIS, electrical migration test was conducted on the samples to measure ionic flux. Crack diffusion coefficient was calculated and its relation with crack width is studied. 5-4 Experimental Methods The test setup shown in figure 5-1 is designed to simulate a cracked concrete sample. The setup is made of Plexiglas with adjustable crack width. The test setup is composed of two half cylinders which can move towards or away from each other by means of two adjustment rods and provide a broad range of gaps in between. Two LVDTs is mounted in two sides of the setup to measure and monitor the gap between to cylinders. Cracks width in the range of 10 to 220 μm were produced and tested with this method. LVDTs Plexiglas half cylinders Adjustment rods & nuts Figure 5-1: Plexiglas test sample used to simulate cracks in concrete 115

130 Duplicates of this setup were built with exactly the same dimensions. The only difference was surface roughness of gap walls which were treated with machining with different grit sizes. The surface roughness of the setups was measured using noncontact optical profilometry. Figure 5-2 shows the profilometer. Noncontact optical profilometry is a surface metrology technique in which light from a lamp is split into two paths by a beam splitter. One path directs the light onto the surface under the test, the other path directs the light to a reference mirror. Reflections are recombined to generate an interface which contains information about the surface contours of the test surface. Vertical resolution can be on the order of several angstroms. Figure 5-2: Noncontact optical profilometer Figure 5-3 shows the topography map of the surface of both setups measured using noncontact optical profilometry. For each test sample (smooth and rough), three rectangles of µm was scanned. Within each rectangle, five linear sections (with the length of 4000 µm) were 116

131 +15 μm (a) -15 μm +4 μm (b) -4 μm Figure 5-3: Topography map of the test samples surfaces, (a): Rough (b): Smooth analyzed to measure roughness. Each section was divided into five segments and the height of highest peak and the lowest valley was averaged for each segment. The total absolute roughness was calculated by averaging the values obtained from all 75 segments (total of 150 peaks and valleys). The absolute roughness obtained with this method for the rough Plexiglas test sample with rough and smooth gap wall were µm and 1.70 µm respectively. 117

132 The test setup was installed between two compartments as shown in figure 5-4. These compartments can contain water or other aqueous solution and facilitates the measurements of hydraulic permeability and ion diffusivity coefficients. The interface of the Plexiglas cylinders with the compartments was sealed using epoxy sealants. Each compartment is equipped with an stainless steel mesh which is used to apply an electrical potential. Two water values are attached to each compartment which are used to apply pressurized water and collect outflow in permeability test (as shown in figure 5-5). Two cylindrical holes were provided near the sample surfaces which were used as electrodes point of contact with the solutions. The electrodes are used to measure chloride concentration and voltage across the sample. These holes were blocked by brass caps when permeability test (figure 5-5) was conducted. Brass cap Water valve Cylindrical compartment (725 ml) Plexiglas test sample Stainless steel mesh and rod Figure 5-4: Test samples installed between two test cells 118

133 The saturated permeability was measured using a constant head method. A layer of silicone sealant was applied on the compartments-sample interface. The silicon was allowed to cure for 24 h. After silicon was cured, the two compartments were filled with water under -30 psi vacuum pressure to remove entrapped air. The permeability test was performed using a constant pressure gradient ranging from 2 to 10 psi (varying depending on the gap width) which resulted in a laminar flow with Reynolds number smaller than 186. The inflow water was pressurized by air inside a bladder, and this pressure was constantly monitored during the test. The outflow water was at atmospheric pressure. The outflow was collected inside a volumetric flask placed on top of a digital balance with accuracy 0.01 g. Weight measurements were performed automatically by a computer at 10 s intervals. To prevent evaporation of the outflow water, the mouth of the volumetric flask was sealed with adhesive plastic with a small puncture to allow pressure equilibrium. Further, the water inside the flask was covered with a thin layer of oil. Water pressure gauges Balance & flask Test sample Pressure tank Inflow compartment Outflow compartment Figure 5-5: Permeability test setup 119

134 The steady-state migration test was adopted in this study to measure diffusion coefficient of cracks. Figure 5-6 shows the migration cell configuration. The two compartments were filled with different concentrations of sodium chloride solution (20000 ppm chloride in upstream and 100 to 300 ppm in downstream, the concentration variation in downstream solution is due to filling procedure. The gap was also initially filled with low concentration solution). A vacuum pressure of -30 psi was applied to the setup during filling of the solutions to remove the entrapped air. An electrical potential difference (i.e., voltage) was applied to accelerate ion transport. The driving forces in this case are both concentration gradient and potential difference. Two Ag-AgCl reference electrodes were used to monitor electrical potential during the test. Chloride concentration in both cells was measured periodically during the test using a chloride ion selective electrode. Chloride concentration variation was used to calculate ionic flux (J). - + ΔE Plexiglas sample Cathode Upstream NaCl ppm Downstream NaCl 100 ppm Anode Figure 5-6: Migration test configuration For the test setup shown in figure 5-6, the diffusion coefficient D (m 2 /s) within the Plexiglas sample can be determined from eq. 5-4 (assuming dilute solution, i.e. 1) [14]. 120

135 5 4 Where J(x) is the flux of chloride ions (mol/m 2 s), is ions valency ( =1 for chloride ions), F is Faraday constant (F=96485 (J/V.mol)), (Volts) is the potential drop measured (ΔE in figure 5-6), R is gas constant (R= (J/mol.K)), T is the absolute temperature (K), C is the average chloride concentration in upstream compartment during the test, and is the concentration gradient across the sample. The term is electrical potential gradient across the sample which assumed to be linear and equal to where L is sample thickness (m). The applied electrical potential (E) in this test was 13 volts. Because of the electrode solution interaction, when an external voltage is applied, the solution experiences a lower potential. Reference electrodes were used to determine the exact voltage that is applied to the solution. The value of 12.3 volts was measured adjacent to the sample surface. The later value was used in the calculation of the diffusion coefficient. At this electrical potential difference, the effect of concentration gradient on ionic flux (J) is very small compared with migration flux due to electrical potential gradient. The ratio of the two is about Therefore it is reasonable to assume that 0. With this assumption eq. 5-4 may be written as: 5 5 The ionic flux (J) can be determined by eq. 5-6: 121

136 5 6 Where V (m 3 ) is volume of the cell, A (m 2 ) is the cross section area of the ionic flux (gap area in this test), and is the rate of chloride concentration change which is obtained by monitoring chloride concentration in downstream cell over time. Combining eq. 5-5 and 5-6, the diffusion coefficient can be calculated by eq. 5-7: 5 7 Plexiglas Sample Reference Electrodes Volt Meter Ion Meter Power Source Magnetic Stirrers Ion Selective Electrode Figure 5-7: Migration test setup 122

137 The test setup used in this study to perform migration test is shown in figure 5-7. Figure 5-8 shows typical results of migration test. For the sample with crack width of 60 μm, there was no significant change in the concentration of downstream cell in the first couple of hours. This is the time required for the chloride ions to travel through the sample thickness and establish a steadystate condition. After this initial unsteady-state period, the rate of increase in chloride concentration of downstream compartment becomes constant. The steady state diffusion coefficients D (m 2 /s) can be calculated from eq. 5-7, knowing the rate of chloride concentration change in the downstream cell. Concentration (ppm) Gap width=60 μm y = 1.056x R² = Time (hr) Figure 5-8: Variation of chloride concentration over time in downstream cell (migration test) Electrical conductivity of all test samples was also measure using electrical impedance spectroscopy. After migration test, both cells were filled with ppm chloride solution under -30 psi vacuum pressure. Bulk resistance of the test cells containing the sample were measured (figure 5-9) by applying 500 mv alternating voltage with frequency ranging from 40 Hz to

138 MHz. For each frequency decade, about 150 measurements were recorded. A typical result of electrical impedance test is shown in figure In this curve the intersection between the half circle and the horizontal axis is considered the bulk resistance. Other data in this figure is used to determine the capacitance of the composites which is not covered in this dissertation. More details on this are given in [15]. Figure 5-9: Electrical impedance test setup 124

139 Reactance (Ω) Bulk resistance (R) Angular frequency (ω) Resistance (Ω) Figure 5-10: Typical result of electrical impedance test 5-5 Results and Discussion Hydraulic Permeability The result of permeability test is shown in figure Permeability coefficients obtained from smooth setup (average roughness of 0.43 µm) closely match the theoretical values from theory of smooth parallel plate. There is a reduction in permeability coefficient for the rough setup (average roughness of 5.43 µm). This reduction is due to friction caused by the features on crack wall surface. The result of this test is compared to the equation suggested by Louis (eq. 5-2) to estimate the permeability of a parallel-plate crack with rough walls in laminar flow. The dashed line shows the predicted values from Louis equation (both lines intersect vertical axis at zero). The predicted values are fairly close to measured values. This indicates that the Louis equation is able to properly predict the permeability coefficient of cracks with rough surfaces and proves the first hypothesis of this chapter. 125

140 Permeability (m2) Permeability (m2) 1E-08 1E-09 1E-10 1E-11 1E-12 1E-13 1E-14 1E-08 1E-09 1E-10 1E-11 1E-12 1E-13 1E-14 Smooth Smooth parallel plate theory (eq. 5-1) Rough Louis Eq.(Ra=10.86 (Ra=10.34 μm) (eq. 5-2) Gap width (μm) Figure 5-11: Measured and predicted Permeability coefficient Smooth Rough Plain Mortar FR Mortar Louis Eq.(Ra=8.9 μm, τ= 0.21) Gap width (μm) Figure 5-12: Permeability test results (data point for mortar samples was obtained from [16]) A modified version of Louis equation which account for tortuosity (and roughness) was used to predict permeability of cracked fiber reinforced and plain mortars in chapter 3 [16]. The experimental result of that study is shown in figure 5-12 together with the result of simulated crack for comparison. The reduction in permeability coefficient is higher for mortar samples comparing to Plexiglas samples and that is because of tortuosity of natural cracks (as opposed to 126

141 straight simulated cracks). Tortuosity further reduces the flow rate and while the effect of roughness is only significant in small cracks, reductions due to tortuosity are significant for all cracks Ion Diffusivity Chloride diffusion coefficient of the crack (D cr ) was calculated from the result of migration test using eq Figure 5-13 shows the variation of D cr versus crack width. Diffusion coefficient of the crack increases linearly with increase of crack width up to a threshold (60-80 μm) and then remains constant. The value of D cr for cracks larger than the threshold is equal to diffusion coefficient of chloride in free solution ( m 2 /s). This indicates that the effect of crack width on diffusion coefficient of the crack is insignificant for large cracks (e.g. >100 μm). This result is in agreement with the result of Djebri et al. [17], Ismail et al. [18], and Kato et al. [19] although the threshold value in some cases is slightly different (80, 53, and 75 respectively). From the results shown in figure 5-13 the question arises as to why there is a drop from diffusion coefficient in free solution for small cracks. Electrical conductivity of the samples, measured using electrical impedance spectroscopy, was used to answer this question. Figure 5-14 shows the variation of electrical conductivity of the samples normalized by the volume fraction of the crack with crack width. Using eq. 4-5 crack connectivity β crack was calculated from this data. The results are shown in figure Crack connectivity is almost constant and equal to 1 (maximum connectivity) for cracks larger than the threshold (60-80 μm) and drops as the crack width decreases. This supports the hypothesis IV of this chapter. The results indicate that the 127

142 dependency of crack diffusion coefficient on crack width for small cracks can be due to variation of crack connectivity. To test this hypothesis, the values of crack diffusion coefficient from migration test ware normalized by values of crack connectivity obtained from impedance test. The results are shown in figure E-08 D crack (m2/s) 1.E-09 1.E-10 1.E Gap width (μm) Figure 5-13: Diffusion coefficient of crack vs. crack width Smooth Rough Free Solution σ/φ (S/m) 1.E+00 1.E-01 Smooth Rough Solution Gap width (μm) Figure 5-14: Normalized conductivity vs. crack width 128

143 Crack connectivity β Smooth Rough Gap width (μm) Figure 5-15: Crack connectivity coefficient, obtain from EIS 1.E-07 D/β crack (m 2 /s) 1.E-08 1.E-09 1.E-10 1.E-11 Free Solution Gap width (μm) Figure 5-16: Diffusion coefficient of crack normalized by crack connectivity obtained from EIS An interesting observation from figure 5-16 is that all the data points lie on a line that corresponds to chloride diffusion coefficient in free solution. This indicates that if the crack connectivity is accounted for, diffusion coefficient of crack is independent of crack width and is 129

144 equal to diffusion coefficient in free solution. This supports the hypotheses II and III of this chapter. 5-6 Conclusions The following conclusions can be drawn from the results presented in this study: If roughness is accounted for, permeability of cracks can be quantified based on crack width. The Louis equation showed a good ability to predict permeability coefficient for the range of cracks tested in this study. If crack connectivity is accounted for, diffusion coefficient of cracks is independent of crack width and is equal to the diffusion coefficient in the solution that the crack is saturated with. Crack connectivity increases with increase in crack width up to a threshold value (60-80 μm in this study) where the connectivity value reaches maximum value (β=1). Cracks larger than the threshold can be assumed to be fully connected. It should be noted that the discussion given in this chapter is only valid for straight cracks where the tortousity factor is assumed to be one. The effect of tortuosity can be considered by replacing nominal crack depth (or sample thickness) with effective length where it appears in calculation of permeability and diffusion coefficient (e.g.: pressure gradient, concentration gradient, electrical potential gradient, water flux, and ion flux). 130

145 5-7 References [1] CRD-C48-92, Standard test method for water permeability of concrete, Handbook of Cement and Concrete, US Army Corps of Engineers, [2] ASTM D , Standard Test Methods for Measurement of Hydraulic Conductivity of Saturated Porous Materials Using a Flexible Wall Permeameter, American Society for Testing and Materials, West Conshohocken, Pennsylvania, [3] CL. Page, NR. Short, AEl. Tarras, Diffusion of chloride ions in hardened cement pastes, Cement and Concrete Research, 11 (1981) [4] AASHTO T259, Standard method of test for resistance of concrete to chloride ion penetration, Washington D.C., USA, [5] ASTM C , Standard test method for electrical indication of concrete s ability to resist chloride ion penetration, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA, [6] ASTM C , Standard test method for determining the apparent chloride diffusion coefficient of cementitious mixtures by bulk diffusion, American Society for Testing and Materials, West Conshohocken, Pennsylvania, USA, [7] NT BUILD 355, Chloride diffusion coefficient from migration cell experiments, Nordtest, Tekniikantie 12, FIN Espoo, Finland, [8] NT BUILD 443, Concrete, hardened: Accelerated chloride penetration, Nordtest, Esbo, Finland, [9] C. Andrade, Calculation of chloride diffusion coefficients in concrete from ionic migration measurements. Cement and Concrete Research, 23 (1993)

146 [10] A. Atkinson, AK. Nickerson, The diffusion of ions through water-saturated cement, Journal of Materials Science, 19 (1984) [11] D. Snow, Anisotropic permeability of fractured media, Water Resources Research, 5 (1969) [12] KA. Snyder, Validation and Modification of the 4SIGHT Computer Program, NIST-IR 6747, National Institute of Standards and Technology (NIST), Gaithersburg, Maryland, USA, [13] L. Dresner, Some remarks on the integration of the extended Nernst-Planck equations in the hyperfiltration of multicomponent solutions, Desalin., 10 (1972) [14] S.Y. Jang, B.S. Kim, B.H. Oh, Effect of crack width on chloride diffusion coefficients of concrete by steady-state migration tests, Cement and Concrete Research, 41 (2011) [15] BJ. Christensen, T. Coverdale, RA. Olson, SJ. Ford, EJ. Garboczi, HM. Jennings, TO. Mason, Impedance spectroscopy of hydrating cement-based materials: Measurement, interpretation, and application, Journal of the American Ceramic Society, 77 (1994) [16] A. Akhavan, SMH. Shafaatian, F. Rajabipour, Quantifying the effects of crack width, tortuosity, and roughness on water permeability of cracked mortars, Cement and Concrete Research, 42 (2012) [17] A. Djerbi, S. Bonnet, A. Khelidj, V. Baroghel-bouny, Influence of traversing crack on chloride diffusion into concrete. Cement and Concrete Research, 38 (2008) [18] M. Ismail, A. Toumi, R. François, R. Gagné, Effect of crack opening on the local diffusion of chloride in inert materials, Cement and Concrete Research, 34 (2004)

147 [19] E. Kato, Y. Kato, T. Uomoto, Development of simulation model of chloride ion transportation in cracked concrete. Journal of the Advanced Concrete Technology, 3 (2005)

148 CHAPTER 6: SUMMARY AND CONCLUSIONS 6-1 Summary of Research Approach Concrete is the most widely used man made material in the world. Most of the transportation infrastructure is made of concrete with a design service life of 50 to 100 years. Long lasting materials play a major role in building durable and cost effective structures. The primary factor governing the durability of concrete is mass transport. Deterioration of concrete is significantly influenced by the rate of moisture, ion, and gas/vapor transport in concrete. A number of concrete service life prediction models currently exist that are based on simulating mass (moisture, ion, vapor/gas) transport in concrete. Despite usefulness of these models, they do not consider the presence of cracks in concrete. Since some level of cracking in concrete is inevitable, and since cracks are known to accelerate mass transport, neglecting the effect of cracks in existing models may result in inaccurate prediction of deterioration rate and expected service life. Therefore, the focus of the present study was on fractured concrete and quantifying the role of cracks in saturated mass transport in concrete. More specifically, the role of crack density (i.e. volume fraction) and crack geometry (length, width, tortuosity, surface roughness) on saturated permeability and ion diffusion coefficient of concrete was investigated. Plain mortar, fiber-reinforced mortar, and fiber-reinforced cement paste disk specimens were cracked by splitting tension; and the crack profile was digitized by image analysis and translated into crack geometric properties. A simulated crack specimen with impervious matrix (Plexiglas) 134

149 was also built and tested. Constant head permeability test, electrical migration test, and electrical impedance spectroscopy test were conducted on cracked and uncracked samples to measure saturated coefficients of permeability and diffusion. 6-2 Conclusions Based on findings of this research, the following conclusions are drawn: An effective crack width can be found from crack digitized profile that results in the same permeability as the actual crack with variable width along its length. The crack permeability coefficient is a function of the crack width square. Tortuosity and roughness of cracks reduce permeability. Tortuosity and roughness of cracks exhibit fractal behavior. In other words, the numerical values of these parameters depend significantly on the magnification of length scale chosen for measurement. A modified form of the Louis equation was found to be capable of quantifying crack permeability as a function of crack geometry (i.e., width, tortuosity, and surface roughness). Ion diffusion coefficient and electrical conductivity of cracked samples are strongly (and approximately linearly) related to the volume fraction of cracks. This is in agreement with the modified parallel law. Diffusivity and conductivity are not significantly influenced by crack width. Crack connectivity increases with increase in crack width up to a threshold value (60-80 μm in this study) where the connectivity value reaches maximum value (β=1). Cracks larger than the threshold can be assumed fully connected. 135

150 If crack connectivity is accounted for, diffusion coefficient of cracks is independent of crack width and is equal to diffusion coefficient of ions in the solution that the crack is saturated with. 6-3 Suggested Future Research An interesting area that can be investigated in future studies is transport in unsaturated fractured concrete. This will add an additional parameter (degree of saturation) to the problem and represent a more general realistic condition. Unsaturated permeability coefficient is a function of moisture content. The theoretical basis of unsaturated flow is discussed on sec of this dissertation. A number of equations have been suggested in the literature to calculate unsaturated permeability. Most of these equations relate the unsaturated permeability coefficient to saturated permeability by means of some fitting parameters. If the fitting parameters are determined, unsaturated flow can be modeled. A finite element model can be developed to calculate the moisture content of each element and use that to predict the unsaturated permeability coefficient as the moisture content changes due to the flow within concrete. Sorptivity is another parameter that is needed to be determined if existence of cracks is to be considered. X-ray tomography is a powerful tool to measure and monitor variation of moisture content and ion transport in concrete. 136

151 APPENDIX A: IMAGE CAPTURING AND ANALYSIS PROCEDURE In this study, the crack profile of the cracked sample was measured using digital image analysis methods. After tested (permeability and impedance test), samples were allowed to dry for at least 24 hours in ambient temperature. Samples were centered in plastic cylindrical molds with a dimension slightly larger than the sample dimension. Low viscosity black epoxy was introduced to the molds to fully cover the samples. A vacuum pressure of -30 psi was applied to the samples (for 15 minutes inside a dessicator) to remove entrapped air and increase the depth of penetration of the epoxy (figure A-1). Figure A-1: Vacuum impregnation of disk samples with epoxy After the epoxy hardened, the specimen were polished to remove the surface layer of epoxy and obtain flat surfaces. A polished cracked sample is shown in figure A-2. The crack profile was 137

152 scanned using a high resolution digital scanner with resolution of 9600 dpi. In addition to crack profile on the surface of the specimens, three plain and five fiber-reinforced specimens were vertically sectioned at the mid-point along a diameter perpendicular to the surface crack and the crack profile through the specimen s thickness was scanned (figure A-3). Y Z Figure A-2: A polished epoxy impregnated sample Y X Z Figure A-3: A vertically sectioned specimen at the mid-point perpendicular to the surface crack 138

153 An image analysis software package was used to detect the cracks and measure the crack width. First a curve was fitted to the crack path and then the crack was segmented perpendicular to the fitted curve every 200 μm for the surface crack and every 50 μm for the thru cracks. Figure A-4 shows a detected thru crack and the segments. For each section the width of the crack was measured. The effective crack width was calculated from the results of image analysis. Z X Z X Y Z X Figure A-4: A thru crack detected and segmented to measure crack width The following is an example of how effective crack width is calculated for a cracked sample: 139

154 The measured values on the surface of the sample for a cracked sample are given in Table A-4. Figure A-5 shows the crack width distribution on the surface of the sample. The measured values of the crack width through the thickness of the same cracked sample are shown in Table A-5. The summary of the data for surface and thru crack is given in tables A-1 and A-2. The effective surface crack width and the effective thru crack width are measured by equations A-1 and A-2: Frequency Crack Width (μm) Figure A-5: Crack width distribution of the surface crack Table A-1: Summary for the surface crack Statistics Largest Item Sec 499 Sec 499 Sec 102 Smallest Item Sec 12 Sec 12 Sec 1 Average Median Std Dev COV

155 Table A-2: Summary for the thru crack Statistics Largest Item Sec 291 Sec 291 Sec 285 Smallest Item Sec 1 Sec 1 Sec 1 Average Median Std Dev COV , 1 A 1, A 2 b, The effective surface crack width was calculated with this method for all the samples. The effective thru crack with was only calculated for the eight specimens vertically sectioned. To obtain the effective thru crack width for the entire specimen, a correlation between the effective surface and the effective thru crack widths was established. The portion of surface crack between and points was assumed to correspond with the middle thru section (figure A-6). Table A-6 shows the measured values of crack width in the mid quarter (between and points) of the cracked sample of this example. The summary of the measurements are 141

156 given in table A-3. Equation A-1 is used to calculate the effective surface crack width for the mid quarter of the sample: Figure A-6: The portion of surface crack between and points that was assumed to correspond with the middle thru section is establishing the correlation between effective surface and thru crack widths Table A-3: Summary for the surface crack in the mid quarter Statistics Largest Item Sec 38 Sec 38 Sec 3 Smallest Item Sec 194 Sec 194 Sec 1 Average Median Std Dev COV ,

157 Same procedure was repeated for other sample vertically sectioned and effective surface crack width in the mid quarter was calculated for all eight samples. the results are shown in figure A-7. The following correlation was found: Fiber-reinforced Plain b eff-thru (μm) b eff-thru =36.92 μm b eff-surf =55.65 μm y = x R² = b eff-surf (μm) Figure A-7: Correlation between the effective surface and thru crack widths And finally the effective thru crack width for the sample in this example can be calculated as shown below: This procedure was used to calculate the effective thru crack width for all the samples. 143

158 Image analysis was also used to measure tortuosity and roughness. As it is explained in chapter 3, tortousity and roughness are fractal parameters meaning that the values of these parameters are a function of sampling length. In this study a MATLAA code was used to measure tortuosity and roughness with sampling length varying in the range of 3 μm to 50 mm. The following example shows how totruosity and roughness were measured for sampling length of 1 mm. Sample thickness or nominal length of the crack = mm (a) Actual Length = mm Circle Radius = 1 mm (b) mm mm Figure A-8: (a) A detected thru crack. (b) The thru crack, sectioned every 1 mm In the example shown in figure A-8, the actual length of the crack can be calculated by counting the circles and adding the length of portion left at the end of the crack Actual length = (number of circles) X (sampling length) + length of remaining portion at the end of crack 144

159 The tortuosity factor is defined as square of nominal length to actual length: Figure A-9: Roughness is calculated by averaging the crack variation within the sampling length Within each sampling length, the variation of crack profile against a reference line (connecting the start and end of the section) were averaged to measure roughness. Figure A-9 illustrates how roughness is measured for a single section 145

160 Table A-4: Crack width measurement for the surface crack Line D value X value Y value D X Y D value X Y Line Line (µm) (µm) (µm) value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

161 Table A-4: Continued Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

162 Table A-4: Continued Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

163 Table A-4: Continued Line D X Y D X Y Line value value value value value value Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

164 Table A-5: Crack width measurement for the thru crack Line D value X value Y value D X Y D value X Y Line Line (µm) (µm) (µm) value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

165 Table A-5: Continued Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

166 Table A-5: Continued Line D X Y D X Y Line value value value value value value Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

167 Table A-6: Crack width measurement for the surface crack in the mid quarter Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

168 Table A-6: Continued Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

169 Table A-6: Continued Line D X Y D X Y D value X Y Line Line value value value value value value (µm) value value Data Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

170 Table A-6: Continued Line D X Y D X Y Line value value value value value value Data Data Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec Sec

171 APPENDIX B: CRACK INDUCING PROCEDURE Disk specimens were cut from the mortar cylinders. The thickness of the disk specimens was 25 mm. cracks were induced by indirect tension. Figure B-1 and B-2 show the indirect tension setup. Two LVDTs were used to monitor the deflection of the samples perpendicular to the direction of loading. Vertical load was applied using a Universal Testing Machine using displacement control method by a constant rate of vertical deformation of 1 µm/s. Figure B-3 shows the applied load and measured lateral displacement versus time. As the applied load increases, LVDTs show continuous increase in lateral deflection of the sample up to the point of cracking. At cracking, the load drops while there is a jump in LVDTs reading. The jump corresponds to the crack opening at the center of the disk specimen. After cracking the lateral deflection increases at higher rate. When the desired deflection is reached, samples are unloaded. The rate of unloading was 5 µm/s. As the sample is unloaded, lateral deflection decreases. However after the sample is fully unloaded all the deflection does not spring back. The residual displacement is due to cracking and can be used as a rough estimation of the crack width in the center of the sample. In the example shown in figure B-3 sample cracked at the load of 16 KN. The jump in LVDT reading at cracking was 88 µm (from 42 to 130 µm ) and sample was loaded to reach lateral deflection of 152 µm. Then sample was unloaded. The lateral deflection after unloading was 44 µm which means that 108 µm (of the maximum deflection of 152 µm) was recovered. This is mainly due to crack closing upon unloading. In this study, the samples were loaded to reach maximum deflection in the range of 80 to 600 µm and then unloaded. The residual deflection obtained with this method was in the range of 40 to 400 µm. After samples were tested, the 157

172 actual crack width was measured using image analysis (see Appendix A). The measured crack width varied in the range of 10 to 200 µm. Direction of load LVDTs Y X Z Figure B-1: Splitting tension setup used to fracture cement paste disks Frame holding LVDTs Diametric crack LVDT LVDT Y Z Disk specimen Figure B-2: Schematic illustration of the splitting tension setup used to fracture mortar disk specimens 158