Fracture Energy of Concrete by Maturity Method

(ProQuest Information and Learning: ... denotes formulae omitted.)

INTRODUCTION

Compressive strength f'^sub c^ of concrete is employed as a fundamental parameter in design and, to some extent, as a measure of material quality. It influences the design stresses, and for this reason, building

codes specify the cylinder tests for the determination of f'^sub c^. Because the standard cylinder test is only representative of potential strength and makes no allowance for the effects of placing, compaction, and curing, safe progression of construction operations, such as removal of forms and shoring or release of prestressing force, requires the determination of in-place properties at early ages. Nondestructive in-place test methods have been developed for estimating the compressive strength of concrete in structures.1 In this respect, the maturity method has been effectively employed during construction for in-place estimation of gain in compressive strength.2 The maturity index combines the effects of time and temperature on the rate of compressive strength development of concrete from early to later ages. The procedures for application of this test method have been documented in ASTM C 1074.3

Concrete fails by cracking and, for this reason, a considerable amount of research has been done for the characterization of concrete failure in terms of fracture mechanics parameters. In general, cracking is preceded by formation of microcracks and their coalescence into macrocracks. The energy stored in concrete during this process is released at the time of fracture. The released energy is the fracture energy G^sub f^, which is considered a fundamental material parameter. It is defined as the amount of energy absorbed within the damaged zone during the fracturing process. Several methods are available for the determination of fracture parameters in concrete including the fictitious crack4 (RILEM standard), size effect,5 and the two-parameter6 fracture models. An example of such is the RILEM three-point bend test,7 through which the fracture energy can be obtained simply by computing the area under the load displacement relationship of a standard specimen.

Whereas laboratory-based techniques provide a measure of G^sub f^, a more effective approach is to determine in-place fracture energy. Yu et al.8 expanded the application of the maturity method to in-place determination of fracture energy in concrete structures. His work was based on the assumption that the hyperbolic relationship between the compressive strength and maturity index would also hold for the fracture energy and maturity. His experimental work involved cylinder tests for compressive strength as well as three-point-bend beam tests for fracture energy. He was able to demonstrate that for the same concrete, both types of tests yield the same activation energy.

The objective for the work presented herein is two fold: (1) to analytically develop and validate the hyperbolic relationship used in Yu et al.'s estimation of fracture energy; (2) to study the effect of curing temperature on the early and later age fracture energy development in concrete. The experimental work is based on the extensive study conducted in a PhD dissertation and documented in detail.9 It involved testing of cylinders and beams as well as mortar cubes for the determination of datum temperature for the mixture and the range of curing temperatures involved in the experiments. A brief explanation of experiments is given later in this article for completeness. The analytical formulations are given in the following.

RESEARCH SIGNIFICANCE

In this study, the applicability of the hyperbolic relationship for the prediction of concrete fracture energy gain based on the maturity method is investigated. The basic assumptions used for the establishment of the hyperbolic relationship are validated by experimentation. Results from the research presented provide the basis for the application of the maturity method for in-place determination of fracture energy in concrete during construction.

ANALYTICAL INVESTIGATION

Bernhardt10 studied the influence of temperature on the hardening of concrete and established the hyperbolic relationship between the compressive strength and the rate of hardening of the material as a function of age and curing temperature. The theoretical foundation for his formulations was based on the assumption that the rate of compressive strength gain at any time during the curing period is a function of temperature and current strength. In functional form, he proposed the following relationship

... (1)

where T denotes temperature, f^sub c^ is the current concrete compressive strength, and t represents time. The values g(f^sub c^) and K^sub T^ are functions that are only dependent on fc and T, respectively. K^sub T^ is often referred to as the rate constant function. Tank and Carino11 have previously developed rate constant functions for estimation of concrete compressive strength.

Based on this assumption, Bernhardt established that the compressive strength could be related to the maturity function by

... (2)

where f^sub cu^ is the ultimate compressive strength of the concrete at infinite time, t = 8 and M(t,T) is a function of time and temperature and can be computed for different mixture proportions by experimentation.

Equation (2) deals with maturity concept for the prediction of in-place strength. Based on this, Yu et al.8 assumed that a similar relationship exists for the fracture energy of concrete G^sub f^

... (3)

where G^sub fu^ is the ultimate fracture energy of concrete at infinite time t = 8. Based on the experimental results, Yu et al. concluded that this relationship is also valid for fracture energy. Their work, however, did not include any theoretical basis for the hypothesis and the applicability of the hyperbolic equation was solely validated based on the fitted experimental results.

Development of the theoretical basis for Eq. (3) will also require establishment of functional relationships between the rate of fracture energy gain at any age t during the curing period, temperature, and current fracture energy in the following format

... (4)

Rewriting Eq. (4)

... (5)

and integrating Eq. (5) leads to

... (6)

where t^sub 0^ represents the induction period or the starting time for hardening of concrete. If g(G^sub f^) is a known function of G^sub f^, it will then be possible to evaluate G^sub f^ as a function of ... if necessary by numerical integration.

One possible way to satisfy Eq. (4) is to assume a linear relationship between G^sub f^ and ... as shown in Fig. 1. Then G^sub f^ can be written as

... (7)

At time t = 8, no further gain in fracture energy is possible, or when G^sub f^ = G^sub fu^, the rate of fracture energy gain ... becomes zero. Therefore, b = G^sub fu^, and Eq. (7) becomes

... (8)

When concrete is fresh, that is, at time zero, fracture energy is zero, and therefore, slope of the line in Eq. (8) can be obtained from

... (9)

Assuming that G^sub fu^ is independent of T, it can be said that G^sub fu^ is a constant material property for a certain concrete mixture proportion irrespective of the curing temperature. Therefore, as shown in Fig. 1, all lines corresponding to different curing temperatures intersect at (0, G^sub fu^).

Because the value of ... is dependent on the curing temperature T, slope of the lines in Fig. 1 are temperature dependant, that is, m = m(T).

Therefore, Eq. (8) can be rewritten as

... (10)

or

... (11)

Integrating both sides of Eq. (11) leads to

... (12)

For constant curing conditions, m(T) is also a constant and independent of time t; therefore

... (13)

At this point, rate constant functions will be developed for estimation of concrete fracture energy. The validity of results, however, is limited to the temperature range 14 to 35 ?C (57.2 to 95 ?F) for which experiments are available. Due to the fact that hydration is an exothermic reaction, a reasonable portrayal of the dependency of the rate constant to a wider temperature range is through the Arrhenius equation. Applicability of the Arrhenius function is, however, verified at a later section of this paper.

Integration of the left side of Eq. (12) yields

... (14)

Equating Eq. (13) and (14) results in

... (15)

and Eq. (15) can be rewritten in the following form to yield the hyperbolic maturity relationship

... (16)

The validity of the assumptions leading to the hyperbolic relationship will be examined by experimentation.

EXPERIMENTAL PROGRAM

The major assumptions used for the establishment of Eq. (16) were: 1) insensitivity of G^sub fu^ to curing temperatures; and 2) existence of linear relationship between G^sub f^ and ....

The experimental program to validate these assumptions included: (a) preparation and testing of three-point-bend beams for the evaluation of fracture energy according to the RILEM standard; (b) curing the beams under three different isothermal curing conditions; and (c) three-point-bend tests to determine fracture energy of the beams at different ages varying from a few hours to full maturity. Experimental details including preparation and testing of cylinders for establishment of the maturity relationships for compressive strength and mortar cubes for the determination of datum temperature are given elsewhere.9 The cylinder strength test data and maturity parameters, however, are compared with the fracture energy results when appropriate. Key elements of the testing program are briefly described in the following paragraphs.

The RILEM fracture test method7 was adopted to obtain the fracture energy. The experiments involved single-edge notched beams subjected to loads in three-point-bend configuration. G^sub f^ values were evaluated by using the following relationship

... (17)

where w^sub 0^ equals the area under load deflection curve from three-point bend tests; m equals the mass of the specimen between supports; g equals the acceleration due to gravity; w^sub l^ equals the deformation at total failure of the beam; and Alig equals the projection of fracture zone on a plane perpendicular to beam axis.

The experimental program encompassed fabrication and testing of three different size beams, as given in Table 1. Beam dimensions for specimens designated as Type A conformed to exact RILEM specifications based on the maximum aggregate size of 3/8 in. (9.5 mm). Type B (smaller) and Type C (larger) beams were tested to examine the effect of specimen size on the evaluated parameters. Compression tests were performed on 3 x 6 in. (76 x 152 mm) cylinders made from the same concrete mixtures as the fracture specimens. The mixture proportion by weight of concrete mixture was designed to be 1.0:2.0:2.7:0.50, corresponding to cement:sand:aggregate:water using Type I portland cement for all specimens. The fine aggregate was natural river sand, and the coarse aggregate was angular granite with a nominal maximum size of 3/8 in. (9.5 mm). Cylindrical specimens were prepared using plastic molds. All beams were pre-notched with a notch to depth ratio of 0.5.

One-hundred and fifty beams were cured and tested at different ages ranging from very young to later ages. The test ages of the specimens are given in Table 2. Approximately 1 to 4 hours after mixing, specimens were lowered in water baths in special curing chambers for curing at the desired temperatures. Actual temperatures inside the specimens were constantly monitored by embedded thermocouples of Type T. Temperature data were transferred to a microcomputer via a data acquisition board. Data were collected at 2-minute intervals during the first 12 hours after mixing, and every 30 to 60 minutes after that.

TESTING PROCEDURE

Specimens were tested in a stiff frame closed-loop servo-hydraulic system. The control parameter was chosen to be the rate of increase in crack mouth opening displacement (CMOD) of 50 ?m./second (1.27 ?m/second). A clip-gauge extensometer was employed for measuring CMOD across the notch. Deflection of the beams at their midspan was measured by a linear variable differential transformer (LVDT). All the test data were collected using a multi-channel data acquisition board. The test setup for the beam is shown in Fig. 2.

EXPERIMENTAL RESULTS

Experimental results pertaining to the gain in fracture energy as a function of the curing period are given in Tables 3 through 5. Determination of the limiting fracture energy G^sub fu^ followed the approach employed for the determination of limiting compressive strengths.2,12 This procedure involves analysis of the fracture energy gain data considering the later ages only. By using the approximation t [asymptotically =] t - t^sub 0^, Eq. (16) can be rewritten in the following form

... (18)

Typical results from regression analysis of data corresponding to the reciprocals of fracture energy and age are given in Fig. 3.

Examination of results shown in Table 6 reveals that the limiting fracture energies are basically independent of curing temperatures. This is in contrast to the results obtained for limiting compressive strengths.2,8,10,13-15 Whereas more experiments are needed to consider even wider range of curing temperatures, the curing temperatures studied herein basically cover the range experienced in practice. It can be concluded at this point, however, that it is reasonable to assume that the limiting fracture energy is independent of the curing temperature.

Assessment of the second assumption pertaining to the linearity of the relationship between G^sub f^ and ... requires graphical determination of the slope dG^sub f^/dt, for the fracture energy-age relationships. One way to accomplish this is to parametrically fit a curve to the experimental data (fracture energy gain as a function of age). This approach, however, will induce large errors in the computation of slopes. A more powerful approach is to use a nonparametric fit to the data points. This is possible by applying a piecewise cubic Hermite interpolating polynomial as the interpolant to the data points resulting in very smooth fits covering all the data points.16 This enables accurate estimation of the slopes dG^sub f^/dt along the smooth curve. Typical results from this analysis in terms of fracture energy as a function of curing period are given in Fig. 4. Finally, linearity of the relationship between G^sub f^, and ... is demonstrated through regression fits to the data points. As shown in Fig. 5, the data points are fairly linear for the Type C specimens (minimum R^sup 2^ = 0.96). Similar results are obtained for other specimens (R^sup 2^ = 0.96 to 0.99), and this establishes the rationality of the second assumption.

The rate constant K^sub T^ represents the intensity of concrete hardening at the given temperature. The linear form of Eq. (16) is again employed for the computation of rate constant, this time in terms of K^sub T^ and t^sub 0^

... (19)

where values of K^sub T^ and t^sub 0^ are computed from the slopes and y-intercepts of the linear regression relationships. A typical plot of Eq. (19) is given in Fig. 6, and Table 7 corresponds to the KT and t^sub 0^ values for all the specimens. The rate constants obtained based on fracture energy are, in general, larger than the ones computed based on the compressive strengths. The rate constants based on the cylinder test data are also shown in Table 7.

K^sub T^ is a function of the curing temperature; and in the traditional maturity concept, the relationship between the rate constant and the temperature is assumed to be linear18,19

K^sub T^ = B(T-T^sub 0^) (20)

where T^sub 0^ is the Datum temperature and it is the minimum temperature at which chemical reactions in concrete could occur, and B is constant of linearity. Linearity of the relationship between K^sub T^ and T is investigated through regression fits to the data points. Results shown in Fig. 7 indicate that the linear regression fits data points well (R^sup 2^ = 0.89 to 0.92). The linear relationship, however, is chosen to simplify computations and it lacks theoretical basis. There is no assurance that this works for temperatures outside of the range of the experiments performed herein (57.2 to 95 ?F [14 to 35 ?C]). The Arrhenius function is a more reasonable assumption for a wider variation in temperature ranges because the activation energy is directly related to the hydration that is an exothermic chemical reaction. Therefore, the variation of rate constant with temperature according to the Arrhenius equation is presented as follows where ? equals a constant, day-1; Tk equals temperature, ?K; E equals activation energy (KJ/mole); and R equals the universal gas constant ....

To consider whether the Arrhenius equation describes the variation of the rate constant with curing temperatures, Eq. (21) is transformed to a linear form by taking the natural logarithm of both sides of the equation as

... (22)

Linearity of the relationship between K^sub T^ and 1/Tk is investigated through regression fits to the data points. Results shown in Fig. 8 indicate that the Arrhenius function is a better fit to the data points (R^sup 2^ = 0.92 to 0.96). Therefore, it can be concluded that the Arrhenius equation provides a better representation of K^sub T^ over the temperature range 57.2 to 95 ?F (14 to 35 ?C). This conclusion could have also been reached by fitting the Arrhenius function in its original nonlinear form to the data points. This is demonstrated in Fig. 9 comparing the nonlinear Arrhenius function and linear regression fit to Type B specimens. Moreover, as shown in Table 8, the activation energy values evaluated from fracture tests are essentially unique and equal to the one obtained from the compressive strength tests.9 The results provide a one-to-one correspondence between the activation energy involved in the hydration of cement paste and the fracture energy.

From Eq. (18), it should be noted that rate constant K^sub T^ mostly affects the value of G^sub f^ at early ages, and when t [arrow right] 8, K^sub T^ effect would be negligible. In fact, K^sub T^ shows the intensity of hardening at a given temperature. Because G^sub fu^ is the limiting fracture energy, in direct analogy with Bernhardt's explanation for gain in compressive strength, the term (1 - G^sub f^ /G^sub fu^) may then be regarded as an expression for the concentration of the cementitious material that has not yet gone into the reaction. Therefore, a general probable assumption for the speed of the reaction dG^sub f^/dt might be

... (23)

The value of n = 2 results in the same hyperbolic formula presented in Eq. (16). For n = 1, however, and for a constant curing temperature, the solution can be easily obtained as

... (24)

Equation (24) is similar to the relation that Nykanen17 presented for the case of compressive strength. To obtain the values of K^sub T^ and t^sub 0^, Eq. (24) is rewritten in the following linear form as

... (25)

Table 9 shows the regression analysis results for Type A (RILEM standard) specimens.

As shown in this table, values of K^sub T^ and t^sub 0^ are totally different from the ones obtained from the hyperbolic formula. For the curing temperatures of 73.4 and 95 ?F (23 and 35 ?C), initial time of hardening t^sub 0^ is negative, which is physically incorrect. At this point, the experimental results indicate that the hyperbolic relationship is the most reasonable form for presentation of the fracture energy gain during the curing period.

Using Arrhenius function, Eq. (23) can be written as

... (26)

For n = 2, rearranging terms, and integrating both sides of Eq. (26) will result in the following general Arrhenius fracture energy-maturity relationship

... (27)

Also, for n = 2, inserting Eq. (20) into Eq. (23) will lead to the following general fracture energy-maturity relationship

... (28)

Most often, the concept of equivalent age is employed in representing the effects of time and temperature on the concrete mixture.15 Equivalent age represents the age at a reference curing temperature that results in the same relative strength gain as under the actual temperature history. Mathematically, equivalent age is defined as

... (29)

where K^sub r^ equals the value of rate constant at reference temperature T^sub r^ ; and K^sub T^ equals the value of rate constant at temperature during time interval ?t.

Alternatively, Eq. (16) can be written in terms of equivalent age in the following format

... (30)

where t^sub e^ equals the equivalent age at reference temperature; t^sub 0r^ equals the age at start of strength development at reference temperature; and K^sub r^ equals the rate constant at reference temperature.

Figure 10 shows the experimental results of all samples along with the predicted values of fracture energies from Eq. (27) and (28) using Type A (RILEM standard) parameters at the reference temperature of 73.4 ?F (23 ?C). Results are quite good considering that Fig. 10 correspond to all the beams (different sizes and curing temperatures) tested in the present study.

VARIABLE TEMPERATURE CONDITIONS

For variable temperature conditions, Eq. (23) can be rewritten in the following integral form

... (31)

The integral on the right side of Eq. (31) is the general form of the maturity function and will be denoted as M(t,T).

Assuming the relationship between the rate constant and the temperature to be linear, the maturity function would be given as

... (32)

The two terms on the right hand side of Eq. (32) are timetemperature factors. If these terms are called M and M0, the maturity function would be

M(t,T) = B(M-M^sub 0^) (33)

and for n = 2, one can obtain the following fracture energymaturity relationship from Eq. (31)

... (34)

Assuming Arrhenius function to estimate the rate constant, the maturity function can be written as

... (35)

In practice, temperatures inside concrete are measured at discrete time intervals. Therefore, the maturity function can be estimated as

... (36)

For n = 2, integrating the left side of Eq. (31) and rearranging terms, one obtains the following general fracture energymaturity relationship

... (37)

CONCLUSIONS

This study investigated the validity of the hyperbolic relationship between the fracture energy and the maturity index. Formulation of the hyperbolic model was based on the assumptions that the fracture energy is insensitive to curing temperatures; and the relationship between G^sub f^ and ... is linear. The validity of these assumptions was experimentally confirmed through standard fracture energy tests on beams. The experiments were performed for the first mode or opening mode of fracture. Fracture energy gain data for concrete beams cured at 57.2, 73.4, and 95 ?F (14, 23, and 35 ?C) were obtained at six different ages ranging nominally from one day all the way to the full curing age. Some of the conclusions from this study are:

1. The fracture energy gain of concrete under isothermal conditions can be described by a hyperbolic curve, which is defined by three parameters: 1) t^sub 0^ equals the time when fracture energy development starts to begin; 2) K^sub T^ equals the initial slope of the curve; and 3) G^sub fu^ equals the limiting fracture energy at time t = 8;

2. Results indicate that while the rate constant and t^sub 0^ are temperature dependent, the limiting fracture energy is an independent parameter, which is unique for a certain concrete mixture. The rate constant K^sub T^ increases with increasing curing temperatures, and the initial time of hardening t^sub 0^ decreases with increasing curing temperatures;

3. Rate constants computed based on fracture energy tests were larger than the same based on compressive strength test results. Because K^sub T^ pertains to the intensity of hardening at early ages, most of the gain in fracture energy occurs at early ages;

4. Linear relation between the rate constant and curing temperature is acceptable in the range of temperatures studies (57.2 to 95 ?F [14 to 35 ?C]). However, applicability of the linear relationship for temperature ranges outside of the ones studied herein should be verified by further experimentation; and

5. Fracture and compressive strength tests yielded the same value for the activation energy of concrete. Activation energy is related to the hydration of cement paste through the exothermic chemical reaction and it is the defining parameter in the Arrhenius function. Due to the wider temperature range of applicability, the Arrhenius function is a better choice for application of maturity method to estimation of fracture energy.

Considering the existence of the hyperbolic relationship for fracture energy, application of the maturity method for in-place determination of fracture energy in concrete can follow the same format as for in-place estimation of compressive strengths in structures. This will provide a valuable tool for assessment of structural health and performance based on cracking potential.