Experimental Asymptotic Analysis of Expansion of Concrete Exposed to Sulfate Attack

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

Although the overall chemical equations for the deterioration of concrete exposed to sulfate solutions are well established,1-3 the simplicity of the equations somewhat obscures the complexity of the processes. For instance, most researchers agree that expansion is primarily related to the formation of secondary ettringite; however, there is no agreement on how the formation of ettringite leads to the expansion of concrete. Several competing theories are discussed in the literature: a) formation of ettringite generates a large increase in solid volume; b) ettringite is formed by a topochemical reaction leading to expansion; c) crystal growth leads to pressure; and d) small ettringite crystals imbibe water.2-3 Another complication is that the effect of gypsum formation on the stiffness and expansion of the matrix is not well understood.

Models that successfully account for the diffusion of sulfates in concrete exist using computer simulations to predict the distribution of the various phases that precipitate and dissolve within the sulfate penetration front. The coupling of these zones containing different crystals to the micromechanics of the expansion of the matrix is more challenging. Naturally, a function that relates the microstructure and the resulting expansion is possible, but requires experimental or field results to validate the predictions. Unfortunately, only very limited expansion data are available for sufficiently long periods of time (decades).

One of the most comprehensive experimental programs ever undertaken was a 40-year study (completed in 1991) conducted under the auspices of the U.S. Bureau of Reclamation (USBR), where concrete samples were immersed in sulfate solution and the expansion measured over time. Recent literature4-6 used different methodologies to analyze the comprehensive data set. Kurtis et al.4 used panel data methods to identify some of the key variables that influence the expansion of concrete. As expected, the data confirmed that expansion increases with time, water-cement ratio (w/c), and increasing C3 A (tricalcium aluminate) content, particularly when it is greater than 8%. Corr et al.5 used these statistical equations for cements with C3 A content less than 8% to conduct a reliability analysis, which showed that the uncertainty in the w/c is the most influential parameter in the reliability model, roughly twice as important as exposure time. The simulation also indicated that the influence of w/c is roughly one order of magnitude larger than the influence of the C 3 A content. Monteiro and Kurtis6 analyzed the time of failure of these samples as influenced by their w/c, cement composition, and percent replacement of cement with fly ash. The analysis showed a safe-zone for concrete made with a w/c lower than 0.45 and cement with a C3 A content lower than 8%; failure did not occur over the 40-year exposure period for concrete specimens with these specifications.

This study expands upon and gives details for the method of scaling and intermediate asymptotics applied to sulfate attack in concrete.7 It presents a brief introduction to the concept of similitude and how it can be used to analyze experimental data. It is important to emphasize that scaling laws are not an expedient method of curve fitting, but rather the determination of a self-similar intermediate asymptotic behavior where the phenomenon repeats itself on changing scales. Intermediate asymptotic methods are then used to analyze the long-term USBR expansion data, which considered a broad number of variations in concrete composition, to study the formation of self-similar stages and the sharp transition of the self-similar stages to subsequent ones. The present work will show how the w/c and cement composition influences the initiation of the scaling domain and how these parameters influence the scaling exponent of the scaling law. Such information allows for the definition of a damage parameter that can provide a guideline on the response of a given concrete mixture exposed to sulfate solutions.

RESEARCH SIGNIFICANCE

Sulfate attack of concrete may lead to cracking, spalling, increased permeability, and strength loss. Presently, there is no complete mathematical formulation of the problem, which may include diffusion and chemical and physical interaction between ingressing sulfate ions and the cement paste; formation of expansive products; progressive damage; and, in rarer cases, healing of the matrix. Based on experimental asymptotic analysis, a series of scaling and saturation laws are proposed for concrete exposed to sulfate attack. These laws are critical for designing durable structures and for developing sulfate-resistant cements. The intermediate asymptotic analysis allows for determining the effect of w/c and cement composition on the scaling laws. The results can then be used by civil engineers to develop more precise life-cycle models and by cement chemists to improve the performance of sulfate-resistant cements.

BRIEF REVIEW OF MATHEMATICAL FORMULATION

This section addresses some of the similarity laws that are relevant to sulfate attack of concrete. Starting with the one-dimensional diffusion equation

... (1)

with diffusion coefficient D and initial condition

... (2)

where l is the width of the initial distribution and Ao is the area of the initial distribution. Note that as l . 0, the initial condition becomes

... (3)

The solution of the diffusion equation at time t is

... (4)

As l [arrow right] 0, the previous equation reduces to

... (5)

The same result is obtained for very long periods of time where t >> l2/D.

A dimensional analysis of the diffusion equation yields the following dimensionless parameters

... (6)

and

Π = f(Π1,Π2) (7)

As pointed out by Goldenfeld,8 l does not show in the similarity solution; therefore, Π = fs(Π1). This type of solution, where the formulation can be derived using dimensional analysis, is referred to as self-similarity of the first kind; however, there are many examples in the mathematical physics9 where it is not possible to determine the self-similar variables using only dimensional analysis. Problems of this kind are referred to as self-similar solutions of the second kind. Barenblatt9 describes an insightful problem of groundwater flow in a porous medium where self-similar solutions of the first kind are obtained by assuming that the medium does absorb the fluid; however, once this assumption is no longer valid, dimensional analysis cannot predict the parameters.

Previously, Mainguy and Coussy10 and Coussy11 have used poromechanics models to study the propagation fronts during calcium leaching and chloride penetration in cementitious materials. A similar approach will be used herein to analyze the penetration of sulfate ions in concrete. Neglecting electrical activity, the diffusion of sulfate ions through the pore solution can be described by

... (8)

where n is matrix porosity, .s is the mass density of the free particles forming the solute per unit of porous volume, .b is the mass density per unit of matrix volume of the bound particles, and D is the effective diffusion coefficient, which is assumed to be constant.

As sulfate ions diffuse through the matrix, reactions can occur with unhydrated cement grains and hydration products (calcium hydroxide, monosulfate hydrate, and eventually calcium silicate hydrates), causing the sulfate ions to be strongly bound. Freundlich's law can be used to describe this process

ρb = Kρ^sup γ^^sub s^ with K>0 and 0 < γ < 1 (9)

It is insightful now to reanalyze the one-dimensional diffusion case with binding. Consider a semi-infinite concrete, where the solute and solid constituent were originally at an equilibrium condition, with its surface exposed to an aggressive solution containing sulfate ions at concentration

... (10)

Note, in the lower limit of a zero concentration (.s . 0), the term .s .dominates and a penetration front is formed. The zonation of the reaction products-reported in the microstructural observations of concrete exposed to sulfate solutions12-14-is a result of this penetration front.

Using dimensional analysis, the following dimensionless parameters can be obtained

... (11)

The functional relationship between the evolution of the penetration front and the resulting expansion is more complex because the mechanisms of crystallization pressure of the new products, the filling of the existing pores and cracks in the cement paste matrix, and the effect of water adsorption on the characteristics of the expansion are not well-defined. As the chemical reactions progress, damage accumulated in the matrix as a result of localized expansion caused by secondary ettringite formation and stiffness degradation caused by the decalcification of the existing hydration products. The traditional modeling of damage assumes that the damage rate is a function q of the current damage w, the applied stress s, and the temperature T

... (12)

where τ is a characteristic time of the damage process.

Barenblatt15 extended this concept for materials that contain microinhomogeneities, such as concrete and rocks, which lead to nonlocal models. Of particular interest is the analysis of the damage process caused by the diffusion of aggressive ions, where the presence of microinhomogeneities influence the diffusion and the damage, and the accumulated damage, in turn, influences the diffusion

... (13)

∂θc = ∂ε[A(w)∂εc] (14)

where ε = x/L, µ = γσo/kT, θ = texp (U/kT)/τ, U is the activation energy, k is the Boltzman constant, so is the bulk stress, . is a kinetic parameter, . is the length scale of the aggregates containing coherent microstructural elements, and L is the characteristic length (length of the sample).

Ideally, experimental results should define the functions A(w) and B(c); but, unfortunately, such experimental data are not available. To further complicate matters, self-healing of cracks has been widely observed in high performance concrete in the field.16 Although the mathematical formulation could easily account for the healing, controlled laboratory tests of healing of concrete exposed to sulfate solutions have not been reported. In summary, a complete mathematical formulation for sulfate attack should include diffusion-absorption and subsequent filtration-absorption flow in a reactive porous medium that generates expansion, cracking, and subsequent healing for a low-porosity matrix.

To date, a complete self-sustained theory of expansion and damage of concrete exposed to sulfate attack does not exist; therefore, the most efficient approach is to carefully analyze the existing expansion data and to develop mathematical models for special classes of problems. Note that the ultimate goal is to extract similitude laws for the phenomena and not construct simple curve-fitting models. For instance, based mainly on experimental results and using incomplete similarity, Barenblatt et al.17 were able to replace the Van Karman-Prandtl universal logarithmic law, one of the last mathematical dogmas of turbulence. Based on the success of mathematicians in obtaining similitude laws from experimental results in the field of turbulence, the scaling models presented in the following follow a similar approach.

U.S. BUREAU OF RECLAMATION DATA

Figure 1 shows the broad range of cement composition studied in the USBR program. Similarly, a wide spectrum of samples with different w/c (from 0.62 to 0.40) were studied. Individual descriptions of a subset of the USBR samples are given in Tables 1 and 2.

Two cylinders, measuring 3 x 6 in. (76 x 152 mm), were cast for each mixture design with stainless steel gauge studs inserted at each end to establish a 5 in. (127 mm) gauge length.3 The cylinders were fog-cured for 2 weeks at 23 °C (73.4 °F), followed by an additional 2 weeks drying at 50% relative humidity (RH) at 23 °C (73.4 °F). After measuring an initial length (to the nearest 0.001 in. [0.025 mm]) for each specimen, samples remained submerged in 2.1% (0.15M) Na2 SO 4 solution.

Thereafter, measurements in the first years of exposure were made approximately once a month. After approximately 5 years of testing, the cylinders were measured twice a year, and after 20 years, the measurements were made annually. The sulfate solution was replaced as needed to maintain submersion of the cylinders, ensuring to some extent that sulfate ions remained in the surrounding solution over time. According to USBR Test Method 4908, expansion of 0.5% or greater indicated failure, and failed cylinders were removed from the sulfate solution. Otherwise, expansion was measured for a period of over 40 years.

EFFECT OF W/C ON EXPANSION OF CONCRETE

Figure 2 shows the expansion results for the samples shown in Tables 1 and 2 as a function of time. The dispersion of the strain evolution indicates that other parameters in addition to time influence the expansion; therefore, the expansion of the specimens with common variables are grouped together. For instance, Table 1 compares the variation of the w/c for concretes containing one of two types of cement.

As indicated in Table 1, Samples 5012 through 5026 were cast with different w/c (0.67, 0.62, 0.57, 0.47, and 0.42) with the same cement composition. Samples 7031 through 7041 were prepared with a different type of cement compared with the previous samples and also with a similar range of w/c (0.60, 0.51, 0.44, and 0.40). The expansion results are shown in Fig. 3.

It is temping to plot expansion as a function of the square root of time because the diffusion process leads to a relationship between characteristic length to square root of characteristic time. Such an approach should be avoided, however, because the experimental data should not be forced into a preconceived model. Instead, the log of the characteristic length is plotted as a function of the log of the characteristic time. If diffusion is indeed the dominant phenomenon, the linear coefficient of the line should be near 0.5. The scaling law for the expansion e of concrete exposed to sulfate attack can be expressed as

ε = K(t - t^sub o^)^sup α^ (15)

where K is a constant, to is the effective origin of time, and ais the scaling exponent.

There are examples in the physical sciences where the origin of time for a given process is not necessarily zero. Following the approach developed by Ling and Huang18 to determine the effective origin of time to after a long period of exposure to sulfate solution, the expansion should be proportional to the power law in terms of exposure time t

ε [asymptotically =]^sup 7agr;^ (16)

Therefore, a first estimation can be made for the scaling exponent a. Next, the plot of e1/a as a function of time t should be linear. The intercept with the horizontal axis provides the value of effective origin of time to , as shown in Fig. 4. Note a sharp transition at the beginning of the self-similar process. The mathematical nature of this transition is not clear, attracting the attention of the best applied mathematicians.

Lowering the w/c, thus making the concrete more impermeable, is known to make concrete more resistant to sulfate attack. This study, however, focused on identifying if lowering the w/c decreases the exponent of the scaling law or if it delays the initiation of the self-similar stage. Figures 5 and 6 present test results for concrete specimens made with high and moderate w/c. Table 1 indicates that the scaling exponent for samples showing a sharp self-similarity initiation does not depend on the w/c, but instead on the cement composition. The w/c does not affect the scaling exponent because this self-similarity reflects a state of filtration where channels percolate through the matrix such that the original porosity of the matrix is not as significant as during the earlier stage when diffusion and absorption are the dominant process. The expected scaling exponent for a case of pure filtration is 0.5, as noted previously. The degree of resulting expansion, however, depends on relative amounts of the cement minerals. This leads to the similitude problem of the second kind.

The independence of the scaling exponent on the w/c became clear when the following dimensionless coordinates were defined: . = e/ei and . = (t - to)/(ti - to ). The normalizing time exponent (ti - to ) was selected because it represents the initiation of the self-similarity stage (Fig. 7). Using this new coordinate system, Fig. 8 shows the experimental tending to follow a straight line after (ti - to ), confirming the adequacy of the scaling laws for a given composition of cement.

The initiation time strongly depends on the w/c and on the type of cement. Figure 9 shows the dependency of the initiation time as a function of time. The dependency of ti on the type of cement will be discussed in the next section.

As the w/c decreases, the sharp transition indicating the beginning of the self-similar domain disappears and is replaced by a saturation process that repeats itself (refer to Fig. 10), which is defined by the following equation

ε = ε^sub lim^(1-e^sup -βt) (17)

where elim is the limiting expansion for saturation period, ß is a constant, and t is time.

The expansion evolution during the saturation process is somewhat similar to the phenomenon of stick-slip, where two opposing surfaces can move relative to each other, but friction may prevent the full relative movement. Stick-slip mechanisms have been proposed to model earthquake generation due to the movement of faults and to model fatigue cracks. In the next section, the influence of the cement composition on this saturation process will be studied.

EFFECT OF CEMENT TYPE

There is agreement that high amounts of C3 A cause a large expansion in concrete due to the formation of ettringite, which is why sulfate-resistant cements have low amounts of C3 A (less than 5%). The effect of the amount of C3 S on the degree of expansion is a more controversial issue. Moskvin et al.19 states that, as early as 1940, it has been observed that portland cements with high amounts of C2 S are more sulfate resistant than cements containing large amounts of C3 S. Later work by Mehta et al.20 reported that mortars prepared with alite cement had inferior performance compared with other cementitious systems. Monteiro and Kurtis6 showed that larger amounts of C3 S can lead to premature failure, even when moderate w/c are used. This observation is of practical technological importance as the C3 S content of portland cements has generally increased in recent years.

This section analyzes which cement compositions lend themselves to scaling laws and which ones lend themselves to saturation curves. Figure 11 shows that, similar to concrete cast with high w/c, concrete samples made with cements containing high amounts of either C3 A or C3 S will result in a clear scaling law whose beginning starts with a sharp transition. These specimens failed from sulfate exposure within a few years. The importance of reducing the total amount of C3 A in portland cements has been known since the 1930s, which led to the production of sulfate-resistant cements (usually with a C3 A content less than 5%). Figure 12 confirms that all concrete samples made with cements containing high amounts of C3 A conformed to the scaling law. The samples had premature failure, often in less than 2 years. While there is complete agreement on the deleterious nature of high-C3 A cements, there is only limited understanding on the effect of C3 S on the expansion of concrete exposed to sulfate solutions. The amount of calcium hydroxide generated by the hydration of C3 S is lower than by the hydration of C2 S; therefore, there is a compelling justification to expect that decreasing the amount of C3 S in the cement should lead to more sulfate-resistant concrete. Figure 11(b) confirms that high amounts of C3 S lead to the development of an early self-similar behavior and that failure occurred after a comparatively short exposure to sulfate solution.

Figure 13 presents the results of expansion curves for concrete made with different types of portland cement where self-similar behavior is observed. The scaling exponent and initiation time for the samples showing self-similarity are summarized in Table 1.

Figure 14 shows that many concrete samples developed a saturation curve. The scaling law indicates that the concrete will reach a percolated condition and that the deterioration process will continue at a faster pace. There is some regularity in the saturation curves, as shown in selected samples given in Fig. 15.

To design more durable structures, it is important to understand how to delay the initiation of the scaling law behavior. Earlier discussion noted that by decreasing the w/c it is possible to increase the initiation time of the percolated stage, which may lead to a saturation curve for very low w/c. This is clearly observed in Fig. 16(a). Samples 5024 and 5021 were made with the same cement, but only Sample 5024 cast with a lower w/c showed a saturation curve. Similarly, in Fig. 16(b) concrete samples cast with the same w/c but made from cements of different compositions are characterized by either scaling laws or saturation curves. The practical significance of this observation can be enormous because, as shown in Fig. 16(b), Sample 1023 had a second dormant period that lasted 15-1/2 years.

It is worthwhile to explore in greater detail the variables that influence the formation of a saturation curve. High and moderate w/c, plus relatively high amounts of C3 A and C3 S, prevent the development of the saturation curve, directly forming a self-similar response with a sharp transition instead. A low w/c leads to dormant periods and saturation curves. For concrete with moderate w/c, there is a fine transition where small changes in cement composition can lead to different behavior. To illustrate this point, Fig. 17 compares Samples 1017 and 4054, which were made with the same w/c( 0.49) and cement containing the same amount of C3 A (4%). Figure 12(c) and Table 1 show that Sample 4054 had a self-similar behavior with a scaling exponent of 1.31. This sample failed after 20 years. Meanwhile, Sample 1017 did not fail during the experiment and showed a saturation curve. The lower C3 S content seems to be responsible for the improved behavior of Sample 1017; C3 S produces comparatively greater amount of Ca(OH)2 (calcium hydroxide) on hydration than C2 S. The greater percolation (filtration channels) in Sample 4054 may be related, at least in part, to the greater extent of leaching in the paste with the presumably higher Ca(OH)2 content during sodium sulfate exposure. No significant change was observed in the early expansion rate, as the w/c was approximately constant for the samples.

Figure 18 shows an important aspect of the expansion mechanism that is observed when cements containing relatively low amounts of C3 S and relatively high amounts of C2 S are used. Sample 1023 was produced with a cement that contained a very low amount of C3S-this type of cement is no longer commercially available because the strength of the concrete developed too slowly. The high early-age expansion (refer to Fig. 18) observed in less than half a year is evidence of the porosity of the matrix, allowing for the more rapid diffusion of sulfates. The first dormant period appeared at 4 years and lasted for approximately 6 years before the expansion began again. Then, a second dormant period occurred, which started at approximately 17 years of exposure and lasted for 14 years. Remarkably, the values of ß from the saturation expression are very similar for the two cycles. This similarity may be viewed as being somewhat similar to the stick-slip phenomenon observed in mechanical friction. The driving force for the formation of expansive products is smaller when sulfate-resistant cements are used and, when combined with a stronger, lower w/c matrix, the steady crack growth necessary for the formation of filtration does not develop. In addition, it is possible that some healing occurs and, consequently, crack tips are blunted.

Figure 19 shows saturation curves for concrete made with cements of different composition. Table 2 summarizes the results for the concrete samples that had scaling laws. The scaling exponent a depends on the percentage of C3 S, C3 A, and C4AF-the composition was determined by the Bogue equations, based on oxide analysis-initially present in the cement used. A statistical analysis indicates that the following relationship can be used to describe the experimental results

... (18)

with the compounds of the cements expressed in a percentage. Figure 20 confirms that the proposed expression provides a good estimate for the values of the scaling exponent.

The initiation time ti depends on the w/c and the percentages of C3 A, C3 S, and C4 AF, according to the following relationship

... (19)

The denominator of the previous equation can be used as an index describing the P^sub d^

P^sub d^ = C^sub 3^A^sup 1.39^C^sub 3^S^sup 2.90^ C^sub 4^AF^sup 0.77^ (w/C)^sup 7.80^ (20)

The relationship of P^sub d^ and the initiation time is shown in Fig. 21. Note that as the values of P^sub d^ decrease below 6400, a saturation curve behavior can be observed. Table 2 shows that the only exception to this rule was observed for Samples 1018 and 1019, which did not show saturation behavior, whereas Samples 1015 and 1017, which have similar composition, have strong saturation curves.

CONCLUSIONS

Applying intermediate asymptotics to a subset of an existing database that measured sulfate-induced expansion in concrete over a 40-year period produced new insights:

1. Concrete samples cast with high or moderate (that is, greater than 0.5) w/c show a region with clear self-similar behavior, where the beginning of this region is noted by a sharp transition. The scaling coefficient is not dependent on the w/c but on the cement composition;

2. It is useful to define an initiation time ti as the starting time of the self-similarity. Decreases in w/c in the concrete mixture and reductions in tricalcium silicate and the tricalcium aluminate contents in the cement in the initiation time;

3. Concrete samples cast with low w/c and sulfate-resistant cements produce a saturation curve when exposed to sulfate solutions. During the 40-year experimental study detailed previously, this saturation process did not allow the development of the self-similar response; and

4. The P^sub d^ index can provide a guideline for the sulfate resistance of a given concrete mixture proportion, which can be useful for the design of sulfate resistant concrete and the optimization of sulfate-resistant cements.

ACKNOWLEDGMENTS

P. J. M. Monteiro wishes to thank the insightful discussions with G. I. Barenblatt.