Pore pressure development model of fiber-reinforced calcareous sand based on shear strain characteristics

In this study, the pore pressure buildup characteristics of fiber-reinforced calcareous sand were investigated by examining the influences of fiber contents and fiber length through a series of cyclic simple shear tests. The test results indicate that the generation mechanisms of pore pressure ratio and shear strain of fiber-reinforced calcareous sand are interrelated under cyclic loading. The fiber content and fiber length have a significant influence on the relationship of pore pressure ratio versus number of cycles. Nevertheless, the correlation of pore pressure ratio versus shear strain is independent of fiber content and fiber length. According to the unique relationship, a pore pressure development model based on shear strain was established, exhibiting excellent predictive accuracy in simulating the pore pressure generation of fiber-reinforced calcareous sand with various fiber content and fiber lengths under cyclic loading. Moreover, the proposed model is also applicable to clean calcareous sand and siliceous sand.


Introduction
The liquefaction of sand has resulted in enormous losses of human life and property, and massive infrastructure has also been damaged [1].Consequently, mitigating the liquefaction sensitivity of sand under cyclic loading (e.g., waves and earthquakes) is necessary.Fiber-reinforced soil technique as one of the common soil improvement approaches has excellent application prospects in enhancing the liquefaction resistance of soils [2].
The accumulation of excess pore water pressure in saturated sand under cyclic loading is the fundamental cause of liquefaction.Over the past decade, massive effort has been devoted to studying the pore pressure built-up characteristics and liquefaction behaviors of sandy soils [3].Compared to common siliceous sand, few studies have been done to date in terms of investigating the pore pressure generation characteristics and corresponding prediction models of calcareous sand.Calcareous sand with high angularity and intra-particle void is highly crushable [4], resulting in a significant difference in the pore generation pressure mechanisms between siliceous sand and calcareous sand under external load [5].
In recent years, following the framework of the Seed model, some researchers have established several modified models to simulate the pore pressure development trend of calcareous sand through a series of undrained cyclic tests [6].However, as noted by Polito et al. [7], the Seed model and the corresponding modified models always have a calibration parameter NL (i.e., the number of cycles to reach liquefaction), which exhibits dynamic variation with the change of the soil properties and test conditions.Additionally, the use of NL has another drawback, i.e., it cannot be applied to nonliquefiable soils e.g., dense sand.To avoid the defect of the Seed model in assessing the pore pressure generation of sand, seeking another form of pore pressure model is essential.
Consequently, some researchers have attempted to establish the relationship between pore pressure and shear strain in sand under undrained cyclic loading conditions [8].Yu et al. [9] conducted undrained cyclic shear tests on calcareous sand, and found that the bilinear model can effectively simulate the relationship of the pore pressure ratio versus the generalized strain under cyclic loading.Wu et al. [10] discussed the influences of nonplastic fines content, relative density, and cyclic stress ratio on the generation characteristics of pore water pressure in saturated calcareous sand through a hollow-cylinder torsional shear apparatus, and a pore water pressure model based on shear strain was proposed.
The aforementioned research indicated that some effort has been developed to the investigating the pore pressure accumulation characteristics of calcareous sand and establishing a prediction model based on shear strain.Nevertheless, a study on the assessment of the pore pressure behaviors of fiberreinforced sand is limited [11], and so far, the excess pore water pressure prediction model based on the shear strain of the fiber-reinforced calcareous sand is completely unknowable and yet to be tested, thus requiring further investigation.Therefore, investigating the pore pressure behaviors of fiberreinforced calcareous sand and establishing a pore water pressure model based on shear strain is necessary.
In this study, a series of cyclic simple shear tests were conducted to investigate the effects of the fiber content and fiber length on the pore pressure generation characteristics of fiber-reinforced calcareous sand.Moreover, by analyzing available test data on fiber-reinforced calcareous sand, a pore pressure development model based on shear strain characteristics was established.

Test materials and apparatus
The employed calcareous sand with irregular shapes and numerous intraparticle pores was sampled from the South China Sea.The basic index properties of the calcareous sand are summarized in Table 1.The discrete polypropylene fibers with a diameter of 0.025 mm were used as reinforcement materials for calcareous sand.The used fibers have a tensile strength of 550 MPa, an elasticity modulus of 4150 MPa, and a specific gravity of 0.91.
A multi-direction dynamic simple shear apparatus was used in this study to investigate the cyclic behaviors of unreinforced and fiber-reinforced calcareous sand.The apparatus consists of three encoder-controlled high-precision actuators, including one axial actuator responsible for exerting a vertical force on specimens and two horizontal actuators responsible for exerting horizontal forces on specimens.The detailed information and photograph of the apparatus can be found in Zhou et al. [12].

Test procedures
During the preparation of the fiber-reinforced specimen, the fibers with the required mass were added to and mixed with calcareous sand using an electric mixer.Subsequently, the sand-fiber mixture was carefully poured into a mold in three layers using the dry deposition method.Slightly tamping the side of the mold in a uniform and consistent way was adopted to prepare specimens with a relative density of Dr=50%.
The prepared specimens were installed on the base of the multi-directional dynamic simple shear apparatus.Afterward, an initial vertical consolidation stress of Vcv0=150 kPa was applied to calcareous sand specimens, and then a series of stress-controlled cyclic simple shear tests were conducted.The frequency of 0.1 Hz has extensively been used to investigate the liquefaction behavior of sand under earthquakes [13].Thus, a sinusoidal cyclic stress at a constant frequency of 0.1 Hz was also adopted in this study.
To discuss the effect of various factors (i.e., fiber contents Fc and fiber length FL), twenty cyclic simple shear tests were conducted in this study.The specific information of the conducted tests is summarised in Table 2.In this table, Fc is the ratio of fiber mass to the calcareous sand mass, and the cyclic stress ratio CSR can be defined as: whereWxmax is the amplitude of the applied shear stress in the x-direction, and maintains constant 10 kPa in this test.

Generation characteristics of pore pressure and shear strain for fiber-reinforced calcareous sand
In this study, pore pressure ratio (ru) is defined as the ratio of the generated excess pore pressure under cyclic loading to the initial vertical consolidation stress (Vcv0).Figure 1 depicts the development trend of pore pressure ratio (ru) and cyclic shear strain (J) versus the number of cycles (N) of fiberreinforced calcareous sand with various Fc.As can be observed, the ru gradually increases with increasing N, and its generation trend can be classified into three stages, i.e., (1) presents a sharp increase; (2) linearly increases; and (3) exhibits a rapid increase until reaching liquefaction.Similar to ru, the development mode of J can also be defined in two stages, i.e., (1) slowly increases with the increase in N; (2) presents a dramatic increase when the liquefaction is imminent (i.e., the J suddenly increase during the last few number of cycles).Generally, the generation mode of ru and J versus N of fiber-reinforced calcareous sand under cyclic loading shows a similarity, especially for the stage in which liquefaction is imminent to the time when liquefaction occurs.Consequently, it can be concluded that the generation mechanisms of ru and J of fiber-reinforced calcareous sand are interrelated under cyclic loading.
To further investigate the correlation of ru and J, figure 2 compares the relationship between the ru and N, as well as the relationship between the ru and J, where the results of tests performed on the reinforced calcareous sand with the same Vcv0=150 kPa, and under different levels of Fc and FL.It is evident that both the Fc and FL have a significant influence on the correlation of ru versus N.With the increase of Fc and FL, the built-up rates of ru gradually decrease.Thus, it is difficult to unify the pore pressure accumulation characteristics of calcareous sand with various Fc and FL.Nevertheless, as illustrated in figures 2 (c) and (d), the correlation of ru versus J is independent of Fc and FL.The results demonstrate the potential for unifying the pore pressure accumulation of calcareous sand with various Fc and FL from the viewpoint of shear strain.

Establishment of pore pressure development model based on shear strain characteristics
The influences of Fc and FL on the correlation of ru versus J of fiber-reinforced calcareous sand are shown in figure 3. It can be observed that the test data of ru in calcareous sand with the same FL under various Fc converges on a single line, which implies that the Fc has no effect on the relationship of ru versus J. Similar to Fc, the FL also cannot affect the development trend of ru versus J. Thus, the relationship of ru versus J of tested fiber-reinforced calcareous sand can be depicted by a single curve, irrespective of the Fc and FL.The unique curve can be explained by three different stages.In stage Ⅰ, the ru presents a sharp increase with the increasing J (approximately linearly increasing) at J1%.In stage Ⅱ, the ru development curve exhibits an upward convex shape with the increase of J, and its accumulation rate gradually decreases at 1%J3%.In stage Ⅲ, the ru basically maintains a constant 1 (i.e., corresponding to the liquefaction state) with the increase in J at J!3%.Generally, in this study, the curves of ru versus J of the tested fiber-reinforced calcareous sand show a typical hyperbolic shape.
To quantitatively analyze the ru accumulation law in calcareous sand, Eq. ( 2) was adopted by Yu et al. to simulate the relationship of ru versus J under cyclic loading.Eq. ( 2) can be presented as: where A B represent the test parameters.Consequently, to investigate the applicability of Eq. ( 2) in this study, figure 4 compares the measured data of ru versus J and the predicted trend using Eq. ( 2), where the typical performed on the fiber-reinforced calcareous sand with the FL=9 mm and different Fc are depicted.It can found that the predicted trend using Eq. ( 2) significantly deviates from the measured data of fiberreinforced calcareous sand.Correspondingly, the R-square values of nonlinear regression obtained by using Eq. ( 2) to fit the test data in figure 4 are in the range of 0.75-0.79.To better simulate the ru generation in this study, a pore pressure development model based on shear strain was developed, as follows: where a and b represent the test parameters.The comparison between measured data of ru versus J and the predicted trend using Eq. ( 3) is also shown in figure 4. It can be observed that the development trend of ru versus J of the tested fiber-reinforced calcareous sand can be well simulated by Eq. ( 3).The R-square values of nonlinear regression obtained by using Eq. ( 3) to fit the test data in figure 4 are always equal to 0.99.The results demonstrate that compared to Eq. ( 2), the proposed pore pressure development model based on shear strain (i.e., Eq. ( 3)) can better simulate the ru development of the tested fiber-reinforced calcareous sand.

Verification of pore pressure development model based on shear strain characteristics
Figure 5 compares all the measured results of ru under conditions of Fc from 0.1% to 1% and FL from 3 mm to 12 mm with the predicted values using Eq. ( 2), and Eq.(3), respectively.It is noteworthy that Eq. ( 2) is not capable of assessing the ru value of the tested fiber-reinforced calcareous sand, and an obvious defect is that the predicted value is above the measured data when the generated ru is small under cyclic loading (approximate ru0.5).Nevertheless, the predicted value is less than the measured data when the generated ru is large (0.5ru0.9).Fortunately, the proposed Eq. ( 3) always presents excellent predictive accuracy in simulating the ru value of the tested fiber-reinforced calcareous sand.
To further validate the applicability of the proposed Eq. ( 3), the test data not included in the pore pressure model development, retrieved from Yu et al. [9] for calcareous sand and Li et al. [14] for siliceous sand are utilized.Figure 6 illustrates the comparisons of the measured results of ru versus J retrieved from their studies with the predicted trend using Eq.(3).It can be concluded that the proposed pore pressure development model based on shear strain (i.e., Eq. ( 3)) can still be successfully applied to evaluate the relationship of ru versus J gathered from clean calcareous sand and siliceous sand of available studies, demonstrating its widespread applicability.
Pore pressure ratio, r

Conclusions
The pore pressure and shear strain behaviors of fiber-reinforced calcareous sand were investigated by means of cyclic simple shear tests, and the effects of fiber content and fiber length were analysed.The following conclusions can be drawn from this study: (1) The generation mode of pore pressure ratio and shear strain versus the number of cycles of fiber-reinforced calcareous sand shows a similarity, especially for the stage in which liquefaction is imminent to the time when liquefaction occurs, indicating that their generation mechanisms are interrelated under cyclic loading.(2) The correlation of pore pressure ratio versus shear strain is independent of fiber content and length, implying the potential for unifying the pore pressure accumulation of calcareous sand with various fiber content and lengths from the viewpoint of shear strain.(3) A pore pressure development model based on shear strain was established.The proposed model has a great predictive accuracy in simulating the pore pressure development of tested fiberreinforced calcareous sand under cyclic loading.Moreover, the proposed model is also applicable to clean calcareous sand and siliceous sand.

Figure 6 .
Figure 6.Validation of the proposed pore pressure development model on the test data retrieved from the published studies.

Table 1 .
Basic index properties of tested calcareous sand.

Table 2 .
Specific information of the conducted tests.