Dynamic Properties of Lightweight Cellular Concrete for Geotechnical Applications

The LCC materials used in this study were provided by the Cell-Crete Corporation (Monrovia, California). Two concurrent processes are used by the Cell-Crete Corporation to cast the LCC samples tested in this study. First, one part of a specific biodegradable protein-based surfactant (a by-product of the food industry) and 40 parts of water were mechanically agitated through a small nozzle to produce a foam, and subjected to compressed air action at a high air pressure. The use of different protein-based surfactants will produce LCC materials with varying unit weights. Second, on the basis of a specific mix design, cement and water were blended together to produce a neat cement slurry. This mixing process took place in a customized concrete mixer, which was coupled with a progressing cavity pump. Then, the neat cement slurry was used to produce an air-filled cellular concrete in a proprietary blending system with the addition of the preformed foam. The unit weight of the LCC was determined on the basis of the quantity of the preformed foam added to the neat cement slurry, and ranged from 3.14 to $18.9\mathrm{kN}/{\mathrm{m}}^3$. The LCC mixture then was poured into Styrofoam molds with the required dimensions to begin the curing process. In this study, five sets of samples were cast to a height of approximately 38.1 mm and a diameter of approximately 66.0 mm. Samples representing four different unit weights were cast. The details of the samples will be discussed subsequently.

After the LCC mixture was poured into the Styrofoam molds, the samples were allowed to set for 4 days. Between the 5th and 7th day after the pour date, the Styrofoam molds were cut carefully and the cast samples were removed. During this process, careful attention was paid to ensure that the samples were not accidently broken or cut. Each removed sample was wrapped in wet towels. These towels were soaked in deionized water for approximately 30 min prior to use. The wrapped LCC samples then were placed in an air-tight container, in which they were stored for 25 days after the pour date. The towels were moistened on a daily basis during this curing period. The wet towels and the lid to the air-tight container were removed on the 26th day after the pour date. For the next 3 days, that is, until the 28th day after the pour date, the LCC samples were allowed to air dry and continue curing. The trimming and testing of the samples took place on the 29th and 30th day after the pour date.

For each cured sample, the height, weight, and diameter of the sample first were recorded. Using a Vernier Caliper (Aerospace, South El Monte, California), height measurements were taken at three different locations approximately 120° apart. A total of nine (three each at the top, middle, and bottom of the sample) diameter measurements were taken. Each sample also was weighed three times. These measurements were used to determine the moist unit weight of the specimen prior to trimming. After obtaining all of these measurements, the samples were carefully twisted into a ring with an inside diameter of 63.5 mm, such that the 25.4-mm tall ring was located at central 25.4 mm of the specimen. This twisting process trimmed the LCC sample to the required diameter. The portion of the specimen extruding outside the ring was carefully trimmed, on either side, using a frosted knife to obtain a specimen with a height of 25.4 mm. A typical trimmed LCC sample prior to testing is presented in Fig. 1. The height, weight, and diameter of the specimen was measured again following the procedure outlined previously. These measurements were used to compute the test unit weight of the specimen following the trimming procedures. The measured unit weights are outlined in the next section. The dry unit weight for each specimen was computed after the completion of cyclic simple shear testing using the weight measurements obtained after the specimen was placed in an oven for at least 24 h at a constant temperature of 110°C.

Although the same the LCC batch mixture was used to cast each set of specimens, some differences in the unit weights of the specimens was expected. In this study, the unit weight of each specimen was determined prior to trimming, after trimming just prior to testing (or the test unit weight), and after 24 h of oven drying following the cyclic simple shear testing (or the dry unit weight). The range and average unit weights for each set of samples tested in this study are presented in Table 1. Table 1 shows that the average unit weight of the specimen prior to and after the trimming process was nearly the same; hence, it is safe to assume that the results obtained in this study were not affected by the trimming procedures followed.

In this study, a Norwegian Geotechnical Institute (NGI)-type CDSS device, manufactured by GeoComp (Acton, Massachusetts), was used to conduct the cyclic direct simple shear (CDSS) testing (Dyvik et al. 1987; Bjerrum and Landva 1966). This automated, computer controlled apparatus uses a stack of 31 Teflon rings to confine the specimen laterally. Each Teflon ring is 0.94-mm thick. The horizontal loads were applied via a servo-motor; whereas, a microstepper motor was used to apply the vertical loads on the specimen. The CDSS device has both horizontal and vertical load capacities of 4,448 kN. In the horizontal direction, $\pm 12.5\mathrm{mm}$ displacement was permitted. Similarly, 12.5 mm of deformation was permitted in the vertical direction. Both the horizontal and vertical displacement readings were resolved to 0.0013 mm.

The trimmed LCC specimens were placed in a rubber membrane, confined by the stack of Teflon rings and secured in the cyclic simple shear apparatus. Then, the specimen was subjected to the desired consolidation pressure. At the end of the primary consolidation, determined from a real-time logarithm of time versus vertical deformation curve, the cyclic loading phase began. In this study, cyclic loading consisted of a series of different strain-controlled sinusoidal waves applied to the sample in undrained conditions. Each specimen was subjected to five cycles, each of 0.5-Hz frequency, of sinusoidal waves with strain amplitudes of 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50%. The test progressed through each of these amplitudes with no pauses between each step. The loading function applied is presented in Fig. 2. At the end of the cyclic loading phase, the specimen was removed from the cyclic simple shear device and placed into an oven for at least 24 h to determine its moisture content and dry unit weight. A total of 11 specimens were tested for each batch of LCC under four different consolidation pressures, i.e., 25 kPa (three specimens), 50 kPa (three specimens), 100 kPa (three specimens), and 350 kPa (two specimens).

A limitation of this study was the use of specimens that were 25.4 mm in height in all of the testing conducted. In the case of soft rock, surface fracture depends on the size of the specimen. These tests were conducted with the sample size specific to the testing equipment; larger size samples may yield slightly different results than those presented in the subsequent sections. A separate study is needed to evaluate the influence of the sample size on the dynamic behavior of the LCC materials.

A typical set of stress-strain hysteresis loops obtained from the CDSS test is presented in Fig. 3. The results in Fig. 3 are for a Class II–Batch 1 specimen under a consolidation pressure of 25 kPa. However, all of the specimens exhibited behavior similar to that shown in Fig. 2. The area of the hysteresis loop increased as the amplitude of the cyclic loading increased. The samples tested at higher consolidation pressures also exhibited similar behavior, but the area enclosed by the hysteresis loop at the same strain amplitude decreased. None of the hysteresis loops for any sample became open or banana-shaped.

The results obtained from the hysteresis loops were used to develop backbone curves for the LCC materials, as provided in Fig. 4, which represents a Class II–Batch 1 specimen tested at a consolidation stress of 25 kPa. In Fig. 4, the data points represent the peaks and troughs, corresponding to the points of stress reversal in the stress-strain hysteresis loops from Fig. 3. Also presented in Fig. 4, is a hyperbolic function as expressed in Eq. (1), which was fitted to these data points. In Eq. (1), $\tau$ is the shear stress, $\gamma$ is the shear strain, whereas $a$ and $b$ are curve-fitting parameters. The CDSS apparatus is limited in measuring the cyclic shear stresses for shear strains less than 0.02%. However, as presented in Fig. 4, the hyperbolic function fitted well into the results provided by the CDSS for all samples. The maximum shear modulus ($G_{\text{max}}$) value was calculated by measuring the slope of the curve at axial strain of 0% by first calculating the derivative of the hyperbolic function and substituting $\gamma$ as 0%. The shear wave velocities obtained from the CDSS for five samples was calculated on the basis of the value of $G_{\text{max}}$, obtained with the process outlined previously, one for each type of LCC material at 25-kPa consolidation pressure, matched well with the shear wave velocities obtained through bender element tests (Tiwari and Ajmera 2015). Fig. 5 contains several backbone curves for Class II–Batch 1 specimens tested with different consolidation stresses. This figure shows that the backbone curves shift upward as the consolidation pressure increased. Fig. 6 depicts the backbone curves for all types of LCC materials used in this study, in which upper and lower bounds referred to the Class II and $7.1\text{-}\mathrm{kN}/{\mathrm{m}}^3$ cast unit weight LCC material, respectively, measured at the consolidation stress of 50 kPa. Close observation of all test data showed that the unit weight of the LCC had a very small influence on the backbone curves. Therefore, the results presented in Fig. 6 can be considered as the maximum and minimum ranges, and as average backbone curves for the LCC materials

The values of parameters $a$ and $b$ in Eq. (1) can quantitatively provide additional insight into the behavior of the backbone curves. Eq. (1) shows that the parameter $a$ scales the hyperbolic function such that an increase in $a$ corresponds to an upward shift in the position of backbone curve. In this study, an increase in $a$ would imply that a larger stress was required to cause the same amount of deformation in the LCC specimen. In contrast, slope of the hyperbolic function at a strain value of zero; thus, the curvature of the hyperbolic function is controlled by the parameter $b$. Specifically, an increase in the value of $b$ implies that the curvature decreases. A higher value of $b$ will imply that a lower shear stress is required to cause the same shear deformation.

Because Figs. 5 and 6 showed that there was a significant influence of the consolidation pressure and very small effect of the LCC unit weight on the backbone curves, the variation in the parameters $a$ and $b$ with the consolidation pressure was calculated and presented in Figs. 7 and 8, respectively. The bars in these figures represent the range of values for the parameters $a$ and $b$ across all specimens tested at any particular consolidation pressure. It shows that an increase in the consolidation pressure (${\sigma }_v^{\prime }$) resulted in an increase in the value of $a$, which can be modeled by Eq. (2). Similarly, an increase in the consolidation pressure also resulted in an increase in the value of the parameter $b$. The best-fit regression line for the values of the parameter $b$ with respect to the consolidation pressure is provided in Eq. (3). In Eqs. (2) and (3), the consolidation pressure should be expressed in kPa. Figs. 7 and 8 with Eqs. (2) and (3), respectively, suggest that as the consolidation pressure increased, the backbone curves for the LCC specimens shifted upward and tended to become less curved. Vertical solid lines presented in both Figs. 7 and 8, and in all subsequent figures, are the ranges of data and the solid circles are the average values

The maximum shear modulus was calculated as the maximum slope of the backbone curve, which occurred at zero shear strain. The variation in the maximum shear modulus with the consolidation pressure is presented in Fig. 9. To compare the values of $G_{\text{max}}$ for LCC materials with those for other geomaterials, AASTHO (1996) suggested that $G_{\text{max}}$ range from 69 to 345 MPa for dense sands and gravels, from 27.6 to 138 MPa for silty sands, from 6.9 to 34.5 MPa for medium stiff clays, and from 27.5 to 137.5 MPa for soft clays. For all types of the LCC materials tested, the maximum shear modulus increased as the consolidation pressure increased. The increase in the maximum shear modulus with consolidation pressure was constant regardless of the unit weight of the material. The maximum shear modulus was highly dependent on the unit weight of the LCC materials and nearly independent of the consolidation stress, as presented in Fig. 10. The regression equations obtained from Figs. 9 and 10 are presented in Eqs. (4) and (5), respectively. The values of $G_{\text{max}}$ and ${\sigma }_v^{\prime }$ are in kPa, and $\gamma$ is in $\mathrm{kN}/{\mathrm{m}}^3$

Fig. 11 contains a typical set of modulus reduction curves for LCC samples from Class II–Batch 1 at different consolidation pressures. At a constant shear strain, the ratio of the shear modulus to the maximum shear modulus ($G/G_{\text{max}}$) decreased as the consolidation pressure increased. Samples from different unit weights behaved in a manner similar to that shown in Fig. 11, although the results are not presented in this paper, but are available in Tiwari and Ajmera (2015). Similar to the behavior observed in the backbone curves, close observation of the data showed that the unit weight of the LCC materials appeared to have little influence on the reduction in the shear modulus with respect to the shear strain experienced by the material. The ranges of modulus reduction curves for all the specimens tested at a consolidation pressure of 100 kPa and the curve for average values are presented in Fig. 12.

The results obtained from the CDSS test also were used to compute the damping ratios for the LCC specimens. Figs. 13(a–e) contains the variation in the damping ratio with shear strain separated on the basis of the consolidation pressure for each of the five sets of LCC specimens tested. Broken lines with arrows in Fig. 13 show the points of maximum curvature in each curve. The damping ratio was dependent on both the consolidation pressure and the shear strain. Fig. 13 shows that except for the samples consolidated at the effective stress of 25 kPa, there was a slight decrease in the damping ratio with an increase in the shear strain up to certain shear strains, generally between the shear strains of 0.25 and 0.35%, beyond which there was a significant increase in damping ratio with an increase in shear strain. The shear strain values corresponding to when the change in the mode of damping occurred was dependent on the consolidation pressure. Fig. 14 shows the variation of damping ratio with shear strain for all types of LCC materials tested at the consolidation stress of 50, 100, and 350 kPa. Samples behaved in a similar manner at other consolidation pressures as well. Broken lines with arrows in Fig. 14 show the points of maximum curvatures. These locations exhibited the strain in which the mode of damping ratio-shear strain relationship changed. Fig. 14 shows that except for one sample, the damping ratios of all samples at shear strain of 0.5% were similar, despite the unit weight of the LCC material. The cause for the difference in behavior in the Class II–Batch 2 sample was not obvious. Moreover, Fig. 14 shows that the majority of the samples exhibited a similar range of shear strains in which the damping ratio changed the mode, i.e., from a slight reduction in damping ratio to a significant increase in damping ratio with shear strain.