# Stress-Strain Relationships of Fiber Reinforced Phosphogypsum under Uniaxial Compression

Authors: Li Zhou; Bo Chen; Yong Huang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

To investigate the stress-strain relationship of fiber reinforced phosphogypsum (PG) under uniaxial compression, a total of twenty-seven PG prism specimens were fabricated and tested. The influences of the content of admixture, content of fiber, and water-solid ratio on the stress-strain curve of the specimens were investigated. Three kinds of failure modes were summarized by observing the experimental phenomena; they were called “compaction failure,” “tension failure,” and “mixed stress failure,” respectively. Through in-depth analysis of the test data, it was found that decreasing the water-solid ratio can lead to increasing the peak stress and secant modulus of specimens; increasing the fiber content can lead to improving the mechanical property of PG mixture specimens. However, adjusting the contents of the cement and quicklime has no significant effect on the mechanical property of specimens. In addition, according to the test data and the characteristics of stress-strain curves, the stress-strain curve of PG specimens was divided into four parts, and a mathematical model was developed to predict the stress-strain curve of PG specimens. The validations of the model showed that the curves calculated by the proposed model were well in agreement with the test data of this study and previous studies.


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## Body

## 1. Introduction

Phosphogypsum (PG) is an industrial by-product of the wet production of phosphoric fertilizer and acid. Every year, the global PG production is approximately 200–300 million tons, and these productions are mainly concentrated in developing countries, such as China, India, and Brazil [1]. However, it should be noted that the low utilization of PG leads to a huge inventory. The huge amounts of stored PG not only occupy considerable land resource but also cause serious pollution and ecological hazards of soil, water system, and atmosphere by harmful impurities leaching with rainwater to produce acidic wastewater [2]. Thus, it can be seen that improving the effective utilization of PG is a significant issue for the sustainable development of phosphorus chemical industry.In order to improve the comprehensive utilization of PG, many scholars all over the world have carried out a lot of fruitful research. Over the last few decades, PG has been widely used as raw material for the preparation of cement [3–5], cement retarder [6, 7], binder [8–10], and composites [11]. Although the above applications are mature, the consumption of PG is still limited. Therefore, more and more researchers have attempted to use the PG waste in various new fields, such as soil amendments [12, 13], backfill materials in goafs [14–16], road base materials [17, 18], and building materials [19, 20]. Recently, with the improvement of treatment technologies of PG and environmental protection awareness, more and more researchers focused on the utilization of PG waste in the field of the construction engineering. Various building products produced by using PG waste were proposed, for instance, the bricks [21–24], self-leveling mortars [25, 26], insulation materials [27], sound absorbing materials [28], and walls [29, 30]. Nowadays, researchers strive to make PG waste become a new building material with bearing capacity, light weight, thermal insulation, and fire prevention.With the continuous expansion of the application of PG in construction engineering, researchers have tried to use the waste to build the load-bearing components of low-rise buildings [30, 31]. However, it should be noted that the key issue of PG used in load-bearing components is its mechanical properties. To address that, a few studies focused on the mechanical properties of PG waste. Değirmenci [32] prepared mixtures with fly ash, PG, and hydrated lime. For all test groups of mixtures, the PG content ranged from 0 to 50%. The results depicted that higher compressive strength could be obtained with increasing calcined PG content. Wu et al. [33] investigated the mechanical properties and stress-strain curves of cast-in-situ PG. The percentages of PG, slag powder, hydrated lime, and cement fluctuated from 75% to 85%, from 9% to 20%, from 3% to 6%, and from 2% to 10%, respectively. The studied results showed that the test specimens presented brittle failure after reaching their ultimate loads, and the strength of cast-in-situ PG specimens is in the range of 1.9 MPa to 9.5 MPa. Besides, the measured elastic modulus and Poisson's ratio of the cast-in-situ PG are approximately 2000 MPa and 0.19, respectively. Zhang et al. [34] investigated the mechanical performance of undisturbed PG, the fly ash, and blast furnace slag and cement was added to PG mixtures at proportions ranging from 10% to 40%, respectively. The results showed that the addition of blast furnace slag and cement produced a positive effect on the mechanical strength of PG, and fly ash led to decreasing mechanical strengths of PG mixtures. Hua et al. [35] studied the effects of different fibers on mechanical properties of PG-slag based cementitious materials. Mineral, glass, and polypropylene (PP) fibers with various dosages are incorporated into the PG-slag system, respectively. The research showed that all those fibers can significantly increase the flexural strength of PG-slag mixtures. In particular, the PP fiber showed an excellent dispersion, high bonding behavior, and good strength contribution.Although the mechanical behavior of PG has been investigated by some scholars, only simple mechanical indexes such as the compressive strength, flexural strength, and elastic modulus have been measured before. In the analysis of a structure, the stress-strain relationship of materials is the basis for elastic-plastic analyses of structures, and the data of the stress-strain relationship are directly related to multiple key performance indicators of structures, such as hysteretic behavior, energy dissipation, and structural ductility. Up to now, there have been few studies focusing on the stress-strain relationship of PG waste, especially for that of PG reinforced with fibers. To address that, an in-depth study on the stress-strain relationship of PG reinforced with fibers is conducted in this paper. Twenty-seven PG prism specimens were fabricated with the consideration of the content of admixture, content of fiber, and water-solid ratio. Then, all specimens were tested under uniaxial compression, and the failure process, failure mode, stress-strain curve, ultimate strength, and elastic modulus were analyzed in detail. The conclusion of this paper can provide some references for structural design of the structure with phosphogypsum; the test data and established model can be used for dynamic analysis, as mentioned by Modano et al. [36].


## 2. Experimental Program

### 2.1. Materials

β-hemihydrate PG, cement, quicklime, polypropylene fiber, and retarder were used for the preparation of mixtures in this experimental program, as shown in Figure 1. Among them, the PG used in this study was obtained from a phosphate fertilizer plant in Guizhou Province, and the main mineral components of PG are listed in Table 1. The mineral contents can meet the requirements of the Chinese code for phosphogypsum (GB/T 23456-2009) [37].Figure 1
Compositions of mixtures: (a)β-hemihydrate PG, (b) cement, (c) quicklime, (d) polypropylene fiber, and (e) retarder.
(a)(b)(c)(d)(e)Table 1
Mineral components of PG.
CompositionsCaSO4·0.5H2ONa2OMgOP2O5K2OContent (%)68.50.220.171.050.06As for the cement used for the preparation of mixtures, the ordinary Portland cement with the grade of P.O 32.5 was selected. The grade of the quicklime used in this study is CL75 according to the Code of Building quicklime (JC/T 479–2013) [38], and the content of CaO and MgO exceeds 75% of the total weight. Polypropylene fiber has characteristics of good dispersion and high strength contribution. The diameter and length of the fiber used are 25 μm and 10 mm, respectively. The measured tensile strength of the fiber is approximately equal to 450 MPa. Special retarder for gypsum is used to extend the setting time of the mixture, and the dosage of the retarder is determined according to the instruction manual. Before the formal test, the trial mixing was carried out to ensure that the setting time reached 30 minutes.


### 2.2. Preparation Process of Specimens

To investigate the effect of different material content on the physical and mechanical properties of the mixture, nine kinds of mix proportions were studied in this paper and numbered as A1 to A9. The detailed mix proportion of each kind of mixtures is listed in Table2. For each kind of mixtures, three prism specimens were molded and labeled as A∗-1, A∗-2, and A∗-3.Table 2
Mix proportion of mixtures.
Mix typePG (kg)Cement (kg)Quicklime (kg)Fiber (g)Retarder (g)Water-solid ratioA19.00.50.510100.8A29.00.50.510100.7A39.00.50.510100.6A48.510.510100.7A59.500.510100.7A68.50.5110100.7A79.50.5010100.7A89.00.50.50100.7A99.00.50.520100.7During the preparation process of specimens, water was firstly poured into a mixing drum. Then, the fiber was evenly scattered into the drum and stirred for 1 minute to ensure that the fiber was well dispersed. Finally, all powder materials were poured into the drum with stirring for 2 minutes. After stirring, the slurry was molded into 100 mm × 100 mm × 300 mm prism specimens. Then, the weight of each specimen was measured after demolding. The specimens after demolding are shown in Figure2. According to the code of gypsum plasters-determination of mechanical properties (GB/T 17669.3–1999) [39], the test specimens were dried to constant weight at 40 ± 4°C before the formal test, as shown in Figure 3.Figure 2
Specimens after demolding.Figure 3
Drying process of specimens.


### 2.3. Test Setup

The electric-hydraulic servo universal test machine, which has a loading capacity of 10 tons, was adopted as a loading device for specimens. The test setup is shown in Figure4. During the test, the specimen was tested under axial compression load with a loading rate of 0.03 mm/s. To obtain the test data of specimens, the displacement sensor and load sensor, which were installed on the servo universal test machine, were adopted to measure the corresponding response values. Further, the whole failure processes of the specimens were recorded by the camera.Figure 4
Test setup.


## 2.1. Materials

β-hemihydrate PG, cement, quicklime, polypropylene fiber, and retarder were used for the preparation of mixtures in this experimental program, as shown in Figure 1. Among them, the PG used in this study was obtained from a phosphate fertilizer plant in Guizhou Province, and the main mineral components of PG are listed in Table 1. The mineral contents can meet the requirements of the Chinese code for phosphogypsum (GB/T 23456-2009) [37].Figure 1
Compositions of mixtures: (a)β-hemihydrate PG, (b) cement, (c) quicklime, (d) polypropylene fiber, and (e) retarder.
(a)(b)(c)(d)(e)Table 1
Mineral components of PG.
CompositionsCaSO4·0.5H2ONa2OMgOP2O5K2OContent (%)68.50.220.171.050.06As for the cement used for the preparation of mixtures, the ordinary Portland cement with the grade of P.O 32.5 was selected. The grade of the quicklime used in this study is CL75 according to the Code of Building quicklime (JC/T 479–2013) [38], and the content of CaO and MgO exceeds 75% of the total weight. Polypropylene fiber has characteristics of good dispersion and high strength contribution. The diameter and length of the fiber used are 25 μm and 10 mm, respectively. The measured tensile strength of the fiber is approximately equal to 450 MPa. Special retarder for gypsum is used to extend the setting time of the mixture, and the dosage of the retarder is determined according to the instruction manual. Before the formal test, the trial mixing was carried out to ensure that the setting time reached 30 minutes.


## 2.2. Preparation Process of Specimens

To investigate the effect of different material content on the physical and mechanical properties of the mixture, nine kinds of mix proportions were studied in this paper and numbered as A1 to A9. The detailed mix proportion of each kind of mixtures is listed in Table2. For each kind of mixtures, three prism specimens were molded and labeled as A∗-1, A∗-2, and A∗-3.Table 2
Mix proportion of mixtures.
Mix typePG (kg)Cement (kg)Quicklime (kg)Fiber (g)Retarder (g)Water-solid ratioA19.00.50.510100.8A29.00.50.510100.7A39.00.50.510100.6A48.510.510100.7A59.500.510100.7A68.50.5110100.7A79.50.5010100.7A89.00.50.50100.7A99.00.50.520100.7During the preparation process of specimens, water was firstly poured into a mixing drum. Then, the fiber was evenly scattered into the drum and stirred for 1 minute to ensure that the fiber was well dispersed. Finally, all powder materials were poured into the drum with stirring for 2 minutes. After stirring, the slurry was molded into 100 mm × 100 mm × 300 mm prism specimens. Then, the weight of each specimen was measured after demolding. The specimens after demolding are shown in Figure2. According to the code of gypsum plasters-determination of mechanical properties (GB/T 17669.3–1999) [39], the test specimens were dried to constant weight at 40 ± 4°C before the formal test, as shown in Figure 3.Figure 2
Specimens after demolding.Figure 3
Drying process of specimens.


## 2.3. Test Setup

The electric-hydraulic servo universal test machine, which has a loading capacity of 10 tons, was adopted as a loading device for specimens. The test setup is shown in Figure4. During the test, the specimen was tested under axial compression load with a loading rate of 0.03 mm/s. To obtain the test data of specimens, the displacement sensor and load sensor, which were installed on the servo universal test machine, were adopted to measure the corresponding response values. Further, the whole failure processes of the specimens were recorded by the camera.Figure 4
Test setup.


## 3. Test Results and Analysis

### 3.1. Failure Process

Three kinds of failure modes were observed in the specimens after the tests, arising from the change of the water-solid ratio and content of fiber, in which specimens A1, A2, A4, A5, A6, A7, and A9 have similar failure modes. Take specimen A2-1 for example; the failure process of the specimen is shown in Figure5. During the initial stage of the loading process, no cracks were visually observed on the body of the specimen. When the compressive stress was increased to 2.87 MPa (approximately as 67% of the peak stress), a visible horizontal crack appeared on the side surface of the upper part of the specimen. As the applied vertical load increases, the horizontal crack keeps propagating. When the compressive stress reached the peak stress, the horizontal crack ran through the whole section of the specimen. Then, the compressive stress decreased rapidly, and several vertical cracks appeared along the horizontal crack. Overall, the micropore of this kind of specimens is abundant, and the skeletons between the pores will collapse under the action of compressive stress; this is the reason why the horizontal cracks appeared first. After the specimen is compressed and compacted, vertical cracks will appear and the specimen will be divided into several small size struts. Thus, this kind of failure mode can be called compaction failure.Figure 5
Failure process of specimen A2-1.Specimen A3 with low water-solid ratio showed a different failure mode. Figure6 shows the failure process of specimen A3-2. It can be seen that the crack distribution of the specimen is quite different from that of specimen A2-1. During the loading process, no noticeable deformation of the specimen was visually observed in the initial stage. When the compressive stress was increased approximately to 63% of the peak stress, the first vertical crack appeared on the side surface of the prism. Thereafter, the crack propagated along the vertical direction and the crack width increased. When the stress reached the peak state, two vertical through cracks were formed on the side of the specimen. Then, the values of the compressive stress decreased gradually. During the subsequent loading process, new cracks propagated along the horizontal direction from the positions of the vertical cracks, while a bulge was observed due to the buckling of the small size struts. In general, the specimen with low water-solid ratio showed a higher compressive strength. When the specimen suffered an axial compressive load, the tensile stress formed by compression-expansion effect reached the limit value before the compression failure of the specimen. Hence, several vertical cracks were observed on the surfaces of the specimen due to tension effect. Thus, this kind of failure mode can be defined as tension failure.Figure 6
Failure process of specimen A3-2.Specimen A8 without polypropylene fiber also showed a different failure mode. The representative sample A8-1 was selected to show the failure process, as illustrated in Figure7. During the initial loading stage, no cracks were observed on the surface of the specimen. When the compressive stress increased to 55% of the peak stress, both horizontal and vertical cracks were observed on the middle part of the side surface. Subsequently, some new cracks were observed and the existing cracks kept propagating with the increase in the vertical load. When the specimen reached the peak state, a slight cracking sound was heard and then the stress dropped. Thereafter, the edge of the middle part of the specimen was crushed gradually. The failure mode of specimens without fiber is similar to that of the concrete prisms, and the failure surface of the specimen presented a pyramidal shape. This is due to the least restraint effect in the middle part of the specimen. Under the compression and expansion effects, the stress reached the ultimate strength of the material first. Thus, this kind of failure can be defined as a mixed stress failure.Figure 7
Failure process of specimen A8-1.For the above three failure modes, the corresponding failure mechanism was investigated in detail. Figure8 shows the damage evolution mechanism of specimens with compaction failure. For this kind of PG specimen, a certain amount of fiber and a large amount of water were used during pouring. Higher water consumption leads to the larger porosity of micropores of the PG specimen. The compaction failure started with the collapse of micropores. Thus, a horizontal crack was first observed on the body of the specimen. The collapse region kept expanding with the increase in the applied load. Finally, all the micropores at the weak section collapsed and formed a penetrating horizontal crack. Before the peak state, the tensile stress produced by the compression-expansion effect is weaker than the tensile resistance of materials due to the restraint effect of fibers. Therefore, no vertical cracks were formed. After the peak state, the rapid growth of the transverse deformation of the specimen leads to the failure of some fibers. Thus, several vertical cracks were observed. For specimens with tension failure, the damage evolution mechanism is illustrated in Figure 9. It is well known that the lower the water-solid ratio, the smaller the porosity of specimens. Consequently, even as the applied load keeps increasing, none of the collapse phenomena of micropores were observed. It should be noted that the specimen with low water-solid ratio has a higher compressive strength. When the specimen suffered a higher load, the tensile stress produced by the compression-expansion effect was also relatively high. Hence, vertical cracks were formed after the tensile stress exceeded the tensile resistance of materials. Figure 10 shows the damage evolution mechanism of specimens with mixed stress failure. Due to the high porosity of the specimen, the collapse of micropores occurred in the early loading stage. Besides, the tensile resistance of the material is significantly reduced due to the lack of the restraint effect of fibers, which leads to the rapid formation of vertical cracks. Finally, the specimen was crushed under tensile and compressive stresses.Figure 8
Damage evolution mechanism of specimens with compaction failure.Figure 9
Damage evolution mechanism of specimens with tension failure.Figure 10
Damage evolution mechanism of specimens with mixed stress failure.


### 3.2. Stress-Strain Curve

The stress-strain curves of fiber reinforced PG not only reflect the mechanical characteristics of PG hardened materials but also provide basis data for studies on the bearing capacity and deformation of the PG structural components, which would be an important reference for analyzing the linear and nonlinear behavior of PG structural components. Based on the measured values of applied loads and the corresponding displacements, the nominal stressσ and nominal strain ε were calculated according to (1), in which N is the applied load, A stands for the cross-sectional area of specimens, Δl is the measured displacement, and L is the total height of specimens.(1)σ=NA;ε=ΔlL.After analyzing the changing trend of all stress-strain curves, some common characteristics of the curves can be obtained, as illustrated in Figure11. It can be seen from the figure that both the ascending stage and descending stage of the curve can be divided into two parts. For the ascending stage, the shape of the first part has the characteristic of concave; the reason is that the specimen has a compaction process in the early loading stage, and this is consistent with the findings of Wu et al. [33]. The second part of the ascending stage has a convex shape; this indicates that the damage of the specimen is accumulating gradually. As for the descending stage, the curve first presents a convex shape. This is due to the accelerated damage of the specimen. Then, the residual bearing capacity of the specimen is obtained gradually under the condition of new internal microscopic structures. That is why the second part of the descending stage is concave.Figure 11
Characteristics of stress-strain curves.Figure12 shows the nominal stress-strain curves of all specimens under uniaxial compression. After comparative analysis, some conclusions can be drawn as follows: (1) Specimens A1, A2, and A3 have different water-solid ratios. It can be found from their stress-strain curves that the curve of a specimen with higher water-solid ratio shows a gentle slope, and the stress value under the same strain level is relatively low. The reason is that expanding the water-solid ratio can increase the porosity of the specimen, and the higher porosity leads to the decrease of the stress value. (2) The difference between specimens A4, A2, and A5 is the content of the cement and PG. It can be seen that the stress-strain curve of each specimen has little change within the range of change set in this study. Previous studies have also found that the strength of modified PG was improved slowly by low amounts of cement [34]. (3) Comparison of the stress-strain curves of specimens with different quicklime contents showed that the influence of quicklime on the curves cannot be ignored. However, it failed to show a certain regularity. (4) Comparing specimens A8, A2, and A9 (fiber content = 0, 10 g, and 20 g), the impact of the fiber content on the stress-strain curve of specimens is highly remarkable. Increasing the fiber content can not only improve the bearing capacity of the specimen but also improve the ductility of the materials.Figure 12
Stress-strain curves of specimens.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 3.3. Peak Stress

The peak stress is one of the most important indexes for assessing the material strength of the PG mixture specimen. In this study, three specimens were prepared for each type of mixture. According to the measured values of the peak stresses, the variability of the peak stresses for each kind of specimens was analyzed. The standard deviation (SD) and coefficient of variation (CV) are shown in Figure13. It can be found from the figure that the maximum values of SD and CV are only 0.32 MPa and 0.089, respectively. This indicates that the dispersion of the peak stress of each group is small.Figure 13
Variability of peak stresses.In this paper, the symbolfc is used to denote the peak stress of the prism specimen. Figure 14 shows a comparison between the peak stresses for specimens with different variable parameters. To better reflect the changes of the peak stress among different kinds of specimens, the average value is selected as the representative value of the peak stress. In Figure 14(a), the average peak stresses of specimens A1, A2, and A3 with different water-solid ratios are compared. It can be found that the average peak stresses of specimens gradually increased with the decrease of water-solid ratios. When the water-solid ratio decreases from 0.8 to 0.7 and 0.6, the average peak stress is found to increase by 30.8% and 108.2%, respectively. This is due to the fact that the decrease of the water-solid ratio increases the compactness of the specimen. Figure 14(b) shows the influence of the content of the cement and PG on the peak stress by comparing specimens A4, A2, and A5. It can be noted that no significant difference was found among the peak stresses of specimens. Among these three kinds of specimens, the maximum average peak stress is only 16.1% higher than the minimum one. This is consistent with the findings of Zhang [34]. In Figure 14(c), the average peak stresses of specimens A6, A2, and A7 with different quicklime contents are compared. When the quicklime content decreases from 1.0 kg to 0.5 kg, the average peak stress increases by 5.9%. Thereafter, the changing trend of the peak stress reversed with further decrease of the quicklime content. When the quicklime content decreases from 0.5 kg to 0, the average peak stress decreases by 14.7%. Previous studies on the strength of PG mixtures also showed similar mechanical characteristics [40]. In Figure 14(d), the average peak stresses of A8, A2, and A9 specimens with different fiber contents were compared. It can be seen that the peak stresses of specimens increased with the increase of fiber contents. Increasing the fiber content from 0 to 10 g and 20 g increases the average peak stress by 3.1% and 27.5%, respectively. Therefore, the fiber content is the key factor affecting the peak stress of the specimen.Figure 14
Comparison of peak stresses.
(a)(b)(c)(d)


### 3.4. Peak Strain

The strain corresponding to the maximum stress is called peak strain, which is an important parameter in analyzing the mechanical behavior of materials. According to the measured test data, the peak strains of all specimens are plotted in Figure15. As can be seen from the figure, the peak strains of all the tested specimens are within the range of 0.0055∼0.0091, and the average value of which is approximately 0.0072.Figure 15
Peak strains of specimens.


### 3.5. Secant Modulus

In this study, the initial part of the stress-strain curve is concave due to the compaction effect of the specimen in the initial loading stage. Hence, it is difficult to determine the initial elastic modulus of the specimen. For the convenience of the study, the secant modulusE0, which is the slope of the line connecting the origin and peak point on the uniaxial stress-strain curve, was adopted to evaluate the ability to resist deformation. Based on the test data of specimens, the relationship between the secant modulus and the peak stress is illustrated in Figure 16. It can clearly be seen that the secant modulus of specimens gradually increased with the increase of peak stresses, and the corresponding fitting formula is as follows:(2)E0=126.1×fc1.073.Figure 16
Relation betweenE0 and fc.The comparisons between the secant moduli for specimens with different variable parameters are illustrated in Figure17. For specimens with different water-solid ratios, the secant modulus increased with the decrease of the water-solid ratio. When the water-solid ratio decreases from 0.8 to 0.7 and 0.6, the average secant modulus is found to increase by 38.9% and 124.4%, respectively. The comparison results are shown in Figure 17(a). Specimens A4, A2, and A5 have different contents of the cement and PG. It can be found from Figure 17(b) that no obvious change in secant modulus was found among those three kinds of specimens. Combined with the peak stress comparison results, it can be concluded that changing the content of the cement and PG has no significant impact on the mechanical property of PG mixture specimens. Figure 17(c) shows the influence of the content of the quicklime on the secant modulus by comparing specimens A6, A2, and A7. When the quicklime content decreases from 1.0 kg to 0.5 kg, the average secant modulus increases by 32.3%. However, the secant modulus will decrease as the content of quicklime continues to decrease. When the quicklime content decreases from 0.5 kg to 0, the average secant modulus decreases by 12.4%. In Figure 17(d), the average secant moduli of specimens A8, A2, and A9 with different fiber contents are compared. When the fiber content increases from 0 to 10 g, the strength of the specimen changes a little, which is consistent with the comparison result of the peak stress. However, the strain increases with the increase of the fiber content. Thus, the secant modulus exhibits a decrease. When the fiber content increases from 10 g to 20 g, the peak stress of the specimen increases significantly, but the strain increases a little. Therefore, the increase of the secant modulus is more significant. Thus, it can be seen that the content of the fiber is the key to the mechanical properties.Figure 17
Comparison of secant modulus.
(a)(b)(c)(d)


## 3.1. Failure Process

Three kinds of failure modes were observed in the specimens after the tests, arising from the change of the water-solid ratio and content of fiber, in which specimens A1, A2, A4, A5, A6, A7, and A9 have similar failure modes. Take specimen A2-1 for example; the failure process of the specimen is shown in Figure5. During the initial stage of the loading process, no cracks were visually observed on the body of the specimen. When the compressive stress was increased to 2.87 MPa (approximately as 67% of the peak stress), a visible horizontal crack appeared on the side surface of the upper part of the specimen. As the applied vertical load increases, the horizontal crack keeps propagating. When the compressive stress reached the peak stress, the horizontal crack ran through the whole section of the specimen. Then, the compressive stress decreased rapidly, and several vertical cracks appeared along the horizontal crack. Overall, the micropore of this kind of specimens is abundant, and the skeletons between the pores will collapse under the action of compressive stress; this is the reason why the horizontal cracks appeared first. After the specimen is compressed and compacted, vertical cracks will appear and the specimen will be divided into several small size struts. Thus, this kind of failure mode can be called compaction failure.Figure 5
Failure process of specimen A2-1.Specimen A3 with low water-solid ratio showed a different failure mode. Figure6 shows the failure process of specimen A3-2. It can be seen that the crack distribution of the specimen is quite different from that of specimen A2-1. During the loading process, no noticeable deformation of the specimen was visually observed in the initial stage. When the compressive stress was increased approximately to 63% of the peak stress, the first vertical crack appeared on the side surface of the prism. Thereafter, the crack propagated along the vertical direction and the crack width increased. When the stress reached the peak state, two vertical through cracks were formed on the side of the specimen. Then, the values of the compressive stress decreased gradually. During the subsequent loading process, new cracks propagated along the horizontal direction from the positions of the vertical cracks, while a bulge was observed due to the buckling of the small size struts. In general, the specimen with low water-solid ratio showed a higher compressive strength. When the specimen suffered an axial compressive load, the tensile stress formed by compression-expansion effect reached the limit value before the compression failure of the specimen. Hence, several vertical cracks were observed on the surfaces of the specimen due to tension effect. Thus, this kind of failure mode can be defined as tension failure.Figure 6
Failure process of specimen A3-2.Specimen A8 without polypropylene fiber also showed a different failure mode. The representative sample A8-1 was selected to show the failure process, as illustrated in Figure7. During the initial loading stage, no cracks were observed on the surface of the specimen. When the compressive stress increased to 55% of the peak stress, both horizontal and vertical cracks were observed on the middle part of the side surface. Subsequently, some new cracks were observed and the existing cracks kept propagating with the increase in the vertical load. When the specimen reached the peak state, a slight cracking sound was heard and then the stress dropped. Thereafter, the edge of the middle part of the specimen was crushed gradually. The failure mode of specimens without fiber is similar to that of the concrete prisms, and the failure surface of the specimen presented a pyramidal shape. This is due to the least restraint effect in the middle part of the specimen. Under the compression and expansion effects, the stress reached the ultimate strength of the material first. Thus, this kind of failure can be defined as a mixed stress failure.Figure 7
Failure process of specimen A8-1.For the above three failure modes, the corresponding failure mechanism was investigated in detail. Figure8 shows the damage evolution mechanism of specimens with compaction failure. For this kind of PG specimen, a certain amount of fiber and a large amount of water were used during pouring. Higher water consumption leads to the larger porosity of micropores of the PG specimen. The compaction failure started with the collapse of micropores. Thus, a horizontal crack was first observed on the body of the specimen. The collapse region kept expanding with the increase in the applied load. Finally, all the micropores at the weak section collapsed and formed a penetrating horizontal crack. Before the peak state, the tensile stress produced by the compression-expansion effect is weaker than the tensile resistance of materials due to the restraint effect of fibers. Therefore, no vertical cracks were formed. After the peak state, the rapid growth of the transverse deformation of the specimen leads to the failure of some fibers. Thus, several vertical cracks were observed. For specimens with tension failure, the damage evolution mechanism is illustrated in Figure 9. It is well known that the lower the water-solid ratio, the smaller the porosity of specimens. Consequently, even as the applied load keeps increasing, none of the collapse phenomena of micropores were observed. It should be noted that the specimen with low water-solid ratio has a higher compressive strength. When the specimen suffered a higher load, the tensile stress produced by the compression-expansion effect was also relatively high. Hence, vertical cracks were formed after the tensile stress exceeded the tensile resistance of materials. Figure 10 shows the damage evolution mechanism of specimens with mixed stress failure. Due to the high porosity of the specimen, the collapse of micropores occurred in the early loading stage. Besides, the tensile resistance of the material is significantly reduced due to the lack of the restraint effect of fibers, which leads to the rapid formation of vertical cracks. Finally, the specimen was crushed under tensile and compressive stresses.Figure 8
Damage evolution mechanism of specimens with compaction failure.Figure 9
Damage evolution mechanism of specimens with tension failure.Figure 10
Damage evolution mechanism of specimens with mixed stress failure.


## 3.2. Stress-Strain Curve

The stress-strain curves of fiber reinforced PG not only reflect the mechanical characteristics of PG hardened materials but also provide basis data for studies on the bearing capacity and deformation of the PG structural components, which would be an important reference for analyzing the linear and nonlinear behavior of PG structural components. Based on the measured values of applied loads and the corresponding displacements, the nominal stressσ and nominal strain ε were calculated according to (1), in which N is the applied load, A stands for the cross-sectional area of specimens, Δl is the measured displacement, and L is the total height of specimens.(1)σ=NA;ε=ΔlL.After analyzing the changing trend of all stress-strain curves, some common characteristics of the curves can be obtained, as illustrated in Figure11. It can be seen from the figure that both the ascending stage and descending stage of the curve can be divided into two parts. For the ascending stage, the shape of the first part has the characteristic of concave; the reason is that the specimen has a compaction process in the early loading stage, and this is consistent with the findings of Wu et al. [33]. The second part of the ascending stage has a convex shape; this indicates that the damage of the specimen is accumulating gradually. As for the descending stage, the curve first presents a convex shape. This is due to the accelerated damage of the specimen. Then, the residual bearing capacity of the specimen is obtained gradually under the condition of new internal microscopic structures. That is why the second part of the descending stage is concave.Figure 11
Characteristics of stress-strain curves.Figure12 shows the nominal stress-strain curves of all specimens under uniaxial compression. After comparative analysis, some conclusions can be drawn as follows: (1) Specimens A1, A2, and A3 have different water-solid ratios. It can be found from their stress-strain curves that the curve of a specimen with higher water-solid ratio shows a gentle slope, and the stress value under the same strain level is relatively low. The reason is that expanding the water-solid ratio can increase the porosity of the specimen, and the higher porosity leads to the decrease of the stress value. (2) The difference between specimens A4, A2, and A5 is the content of the cement and PG. It can be seen that the stress-strain curve of each specimen has little change within the range of change set in this study. Previous studies have also found that the strength of modified PG was improved slowly by low amounts of cement [34]. (3) Comparison of the stress-strain curves of specimens with different quicklime contents showed that the influence of quicklime on the curves cannot be ignored. However, it failed to show a certain regularity. (4) Comparing specimens A8, A2, and A9 (fiber content = 0, 10 g, and 20 g), the impact of the fiber content on the stress-strain curve of specimens is highly remarkable. Increasing the fiber content can not only improve the bearing capacity of the specimen but also improve the ductility of the materials.Figure 12
Stress-strain curves of specimens.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 3.3. Peak Stress

The peak stress is one of the most important indexes for assessing the material strength of the PG mixture specimen. In this study, three specimens were prepared for each type of mixture. According to the measured values of the peak stresses, the variability of the peak stresses for each kind of specimens was analyzed. The standard deviation (SD) and coefficient of variation (CV) are shown in Figure13. It can be found from the figure that the maximum values of SD and CV are only 0.32 MPa and 0.089, respectively. This indicates that the dispersion of the peak stress of each group is small.Figure 13
Variability of peak stresses.In this paper, the symbolfc is used to denote the peak stress of the prism specimen. Figure 14 shows a comparison between the peak stresses for specimens with different variable parameters. To better reflect the changes of the peak stress among different kinds of specimens, the average value is selected as the representative value of the peak stress. In Figure 14(a), the average peak stresses of specimens A1, A2, and A3 with different water-solid ratios are compared. It can be found that the average peak stresses of specimens gradually increased with the decrease of water-solid ratios. When the water-solid ratio decreases from 0.8 to 0.7 and 0.6, the average peak stress is found to increase by 30.8% and 108.2%, respectively. This is due to the fact that the decrease of the water-solid ratio increases the compactness of the specimen. Figure 14(b) shows the influence of the content of the cement and PG on the peak stress by comparing specimens A4, A2, and A5. It can be noted that no significant difference was found among the peak stresses of specimens. Among these three kinds of specimens, the maximum average peak stress is only 16.1% higher than the minimum one. This is consistent with the findings of Zhang [34]. In Figure 14(c), the average peak stresses of specimens A6, A2, and A7 with different quicklime contents are compared. When the quicklime content decreases from 1.0 kg to 0.5 kg, the average peak stress increases by 5.9%. Thereafter, the changing trend of the peak stress reversed with further decrease of the quicklime content. When the quicklime content decreases from 0.5 kg to 0, the average peak stress decreases by 14.7%. Previous studies on the strength of PG mixtures also showed similar mechanical characteristics [40]. In Figure 14(d), the average peak stresses of A8, A2, and A9 specimens with different fiber contents were compared. It can be seen that the peak stresses of specimens increased with the increase of fiber contents. Increasing the fiber content from 0 to 10 g and 20 g increases the average peak stress by 3.1% and 27.5%, respectively. Therefore, the fiber content is the key factor affecting the peak stress of the specimen.Figure 14
Comparison of peak stresses.
(a)(b)(c)(d)


## 3.4. Peak Strain

The strain corresponding to the maximum stress is called peak strain, which is an important parameter in analyzing the mechanical behavior of materials. According to the measured test data, the peak strains of all specimens are plotted in Figure15. As can be seen from the figure, the peak strains of all the tested specimens are within the range of 0.0055∼0.0091, and the average value of which is approximately 0.0072.Figure 15
Peak strains of specimens.


## 3.5. Secant Modulus

In this study, the initial part of the stress-strain curve is concave due to the compaction effect of the specimen in the initial loading stage. Hence, it is difficult to determine the initial elastic modulus of the specimen. For the convenience of the study, the secant modulusE0, which is the slope of the line connecting the origin and peak point on the uniaxial stress-strain curve, was adopted to evaluate the ability to resist deformation. Based on the test data of specimens, the relationship between the secant modulus and the peak stress is illustrated in Figure 16. It can clearly be seen that the secant modulus of specimens gradually increased with the increase of peak stresses, and the corresponding fitting formula is as follows:(2)E0=126.1×fc1.073.Figure 16
Relation betweenE0 and fc.The comparisons between the secant moduli for specimens with different variable parameters are illustrated in Figure17. For specimens with different water-solid ratios, the secant modulus increased with the decrease of the water-solid ratio. When the water-solid ratio decreases from 0.8 to 0.7 and 0.6, the average secant modulus is found to increase by 38.9% and 124.4%, respectively. The comparison results are shown in Figure 17(a). Specimens A4, A2, and A5 have different contents of the cement and PG. It can be found from Figure 17(b) that no obvious change in secant modulus was found among those three kinds of specimens. Combined with the peak stress comparison results, it can be concluded that changing the content of the cement and PG has no significant impact on the mechanical property of PG mixture specimens. Figure 17(c) shows the influence of the content of the quicklime on the secant modulus by comparing specimens A6, A2, and A7. When the quicklime content decreases from 1.0 kg to 0.5 kg, the average secant modulus increases by 32.3%. However, the secant modulus will decrease as the content of quicklime continues to decrease. When the quicklime content decreases from 0.5 kg to 0, the average secant modulus decreases by 12.4%. In Figure 17(d), the average secant moduli of specimens A8, A2, and A9 with different fiber contents are compared. When the fiber content increases from 0 to 10 g, the strength of the specimen changes a little, which is consistent with the comparison result of the peak stress. However, the strain increases with the increase of the fiber content. Thus, the secant modulus exhibits a decrease. When the fiber content increases from 10 g to 20 g, the peak stress of the specimen increases significantly, but the strain increases a little. Therefore, the increase of the secant modulus is more significant. Thus, it can be seen that the content of the fiber is the key to the mechanical properties.Figure 17
Comparison of secant modulus.
(a)(b)(c)(d)


## 4. Stress-Strain Model

### 4.1. Establishment of Model

The whole stress-strain curve can be divided into four parts, as illustrated in Figure11. For the ascending stage, an important issue is to determine the dividing point A between the concave and convex segments. In this study, the intersection of the curve and the line passing through the peak and origin points is defined as the dividing point, as shown in Figure 18(a). For the descending stage of the curve, the slope of the curve first increases and then decreases, and the process of slope reduction can be regarded as a curve convergence. Therefore, the point with the maximum slope can be defined as the dividing point between the convex and concave segments, as shown in Figure 18(b).Figure 18
Determinations of dividing points.
(a)(b)Based on the test data, the dividing points of the curves are illustrated in Figure19. In the figure, η1 and η2 are used to reflect the dividing points of the ascending and descending stages of the curves, respectively. The expressions of η1 and η2 are given as follows:(3)η1=εaεo,η2=εcεo.Figure 19
Distributions ofη1 and η2.In the above equation,εo is the peak strain of the specimen; εa and εc are the strains corresponding to dividing points A and C, respectively.It can be found from Figure19 that the distributions of η1 and η2 have a certain regularity to follow. The values of η1 are within the range of 0.08∼0.32, and the average value of which is approximately 0.19. Previous studies have found that the strain of the dividing point of the ascending stage is about 20% of the peak strain [33]. This is basically consistent with the value of η1 calculated in this study. For the convenience of establishing the stress-strain model, the value of η1 is taken as 0.2 in this study. The value of η2 ranges from 1.13 to 1.35, and the average value is approximately 1.21. The value of η2 is taken as 1.2 while establishing the model.Based on the determinations of the dividing points of the curves, the stress-strain curve can be clearly divided into four segments, as shown in Figure20. These four segments can be expressed by functions f1(x), f2(x), f3(x), and f4(x), respectively. For the convenience of function calculation, the whole stress-strain curve of specimens with different parameters can be normalized through the following relations:(4)x=εεo,fix=σfci=1,2,3,4,where σ and ε are the stress and strain at any point on the curve.Figure 20
Segmentations of curves.For the proposed piecewise functions, the following relations should be satisfied:(5)f10=0,f1″x>00<x≤0.2.Note that the slope of the first part of the ascending stage increases monotonically.(6)f10.2=f20.2,f1′0.2=f2′0.2,f2″x<00.2<x≤1.Note that the slope of the second part of the ascending stage decreases monotonically.(7)f21=f31=1,f2′1=f3′1=0,f3″x<01<x≤1.2.Note that the slope of the first part of the descending stage increases monotonically.(8)f31.2=f41.2,f3′1.2=f4′1.2,f4″x>0x>1.2.Note that the slope of the second part of the descending stage decreases monotonically.(9)f4′x⟶0,f4x⟶0whilex⟶∞.According to the above relations (5)∼(9), the functions f1(x), f2(x), f3(x), and f4(x) can be expressed as follows:(10)f1x=24a−25x2+10−8ax0≤x≤0.2,f2x=−ax2+2ax+1−a0.2<x≤1.0,f3x=−bx2+2bx+1−b1.0<x≤1.2,f4x=0.576bx+1−0.52bx>1.2.In the above equation,a and b are the control parameters of the ascending stage and descending stage of the curve, respectively.


### 4.2. Determination of Control Parameter

According to the fitting results, the relations between the control parametera and the peak stress fc are shown in Figure 21. It can be seen that parameter a generally increases with an increase of peak stress of specimens. On this basis, the nonlinear function relationship is established by performing regression analysis, as shown below.(11)a=−0.01fc2+0.135fc+0.822.Figure 21
Relations between parametera and fc.The fitting results indicated that the control parameterb of the descending stage is related not only to the peak stress of the specimen but also closely to the fiber content. Figure 22 illustrates the relations between parameter b and the peak stress fc. The selected data in Figure 22 were obtained from the specimens with the fiber content of Cf = 0.1. After regressing the selected data, the corresponding functional relation can be expressed as follows:(12)bCf=0.1=0.057fc+1.793.Figure 22
Relations between parameterb and fc.The fiber content has a significant effect on the descending stage of the stress-strain curve. Taking the fiber content ofCf = 0.1 as a reference, the control parameter b decreases with an increase of the fiber content, as shown in Figure 23. Through analyzing the variation of the parameter, the corresponding functional relation can be obtained, as shown below.(13)bbCf=0.1=1.554−5.577Cf.Figure 23
Relations between parameterb and Cf.Based on (12) and (13), the function of the control parameter b with the considerations of the peak stress and fiber content can be obtained, as shown below.(14)b=1.554−5.577Cf0.057fc+1.793.


### 4.3. Verification of Stress-Strain Model

According to the stress-strain model proposed in this paper, stress-strain curves of specimens with different parameters were modeled. Then, the predicted curves were compared with the ones measured from the test, as illustrated in Figure24. It can be clearly seen from the figure that the curves predicted by the proposed model well conform to the results obtained from the test, which verifies the correctness of the proposed model. On the whole, the predicted curves are in good agreement with the ascending stages of the experimental curves. However, the accordance rate in descending stages between the predicted curve and the experimental curve is relatively poor. The possible reason is that the experimental data of specimens have relatively large discreteness after specimens reach their peak stresses. This is consistent with the experimental results reflected by the uniaxial compression tests of other materials, such as concrete [41].Figure 24
Comparison between the predicted and measured stress-strain curves.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Wu et al. [33] investigated the stress-strain relations of PG prism specimens without fiber, and the corresponding curves were obtained through a series of uniaxial compression tests. To further verify the correctness of the proposed model, some of the measured curves reported by Wu et al. [33] were selected as validation data. The comparison results are shown in Figure 25. It can be found from the comparison results that the predicted curves are slightly different from the measured curves in the ascending stages. This is due to the fact that the test equipment led to a relatively long compaction process, as pointed out by Wu et al. [33]. However, the predicted curves well conform to the measured curves in the descending stages. Overall, the predicted curves are in good agreement with the test data. Thus, it can be proved that the model proposed in this paper has strong applicability.Figure 25
Further verification of the proposed model.
(a)(b)(c)(d)(e)(f)The yield point, peak point, and failure point are the key characteristic points of the stress-strain curve. Previous studies showed that the yield strength in ascending stage is close to 70% of the peak strength, and the failure strength in descending stage is commonly determined as the strength falls to 85% of the peak strength [42, 43]. In the proposed stress-strain model, the measured peak stress and peak strain need to be substituted into the model; thus the peak point of the model curve is the same as that of the measured curve. As for the yield point and the failure point, the predicted strain to measured strain ratios (ΔM/ΔT) are shown in Figure 26. It can be seen from Figure 26(a) that the predicted strain values (ΔM) were slightly lower than the measured strain values (ΔT), yet the prediction errors are all within 20%. That is to say, the yield strain predicted by the model is relatively small; this may be due to the underestimation of the compaction deformation of the PG specimen. In addition, the predicted failure strains are almost consistent with the test data, as illustrated in Figure 26(b). On the whole, the proposed stress-strain model showed a good prediction ability, and the calculated values can meet the accuracy requirements of engineering applications.Figure 26
Comparison between the predicted and measured strain values.
(a)(b)


## 4.1. Establishment of Model

The whole stress-strain curve can be divided into four parts, as illustrated in Figure11. For the ascending stage, an important issue is to determine the dividing point A between the concave and convex segments. In this study, the intersection of the curve and the line passing through the peak and origin points is defined as the dividing point, as shown in Figure 18(a). For the descending stage of the curve, the slope of the curve first increases and then decreases, and the process of slope reduction can be regarded as a curve convergence. Therefore, the point with the maximum slope can be defined as the dividing point between the convex and concave segments, as shown in Figure 18(b).Figure 18
Determinations of dividing points.
(a)(b)Based on the test data, the dividing points of the curves are illustrated in Figure19. In the figure, η1 and η2 are used to reflect the dividing points of the ascending and descending stages of the curves, respectively. The expressions of η1 and η2 are given as follows:(3)η1=εaεo,η2=εcεo.Figure 19
Distributions ofη1 and η2.In the above equation,εo is the peak strain of the specimen; εa and εc are the strains corresponding to dividing points A and C, respectively.It can be found from Figure19 that the distributions of η1 and η2 have a certain regularity to follow. The values of η1 are within the range of 0.08∼0.32, and the average value of which is approximately 0.19. Previous studies have found that the strain of the dividing point of the ascending stage is about 20% of the peak strain [33]. This is basically consistent with the value of η1 calculated in this study. For the convenience of establishing the stress-strain model, the value of η1 is taken as 0.2 in this study. The value of η2 ranges from 1.13 to 1.35, and the average value is approximately 1.21. The value of η2 is taken as 1.2 while establishing the model.Based on the determinations of the dividing points of the curves, the stress-strain curve can be clearly divided into four segments, as shown in Figure20. These four segments can be expressed by functions f1(x), f2(x), f3(x), and f4(x), respectively. For the convenience of function calculation, the whole stress-strain curve of specimens with different parameters can be normalized through the following relations:(4)x=εεo,fix=σfci=1,2,3,4,where σ and ε are the stress and strain at any point on the curve.Figure 20
Segmentations of curves.For the proposed piecewise functions, the following relations should be satisfied:(5)f10=0,f1″x>00<x≤0.2.Note that the slope of the first part of the ascending stage increases monotonically.(6)f10.2=f20.2,f1′0.2=f2′0.2,f2″x<00.2<x≤1.Note that the slope of the second part of the ascending stage decreases monotonically.(7)f21=f31=1,f2′1=f3′1=0,f3″x<01<x≤1.2.Note that the slope of the first part of the descending stage increases monotonically.(8)f31.2=f41.2,f3′1.2=f4′1.2,f4″x>0x>1.2.Note that the slope of the second part of the descending stage decreases monotonically.(9)f4′x⟶0,f4x⟶0whilex⟶∞.According to the above relations (5)∼(9), the functions f1(x), f2(x), f3(x), and f4(x) can be expressed as follows:(10)f1x=24a−25x2+10−8ax0≤x≤0.2,f2x=−ax2+2ax+1−a0.2<x≤1.0,f3x=−bx2+2bx+1−b1.0<x≤1.2,f4x=0.576bx+1−0.52bx>1.2.In the above equation,a and b are the control parameters of the ascending stage and descending stage of the curve, respectively.


## 4.2. Determination of Control Parameter

According to the fitting results, the relations between the control parametera and the peak stress fc are shown in Figure 21. It can be seen that parameter a generally increases with an increase of peak stress of specimens. On this basis, the nonlinear function relationship is established by performing regression analysis, as shown below.(11)a=−0.01fc2+0.135fc+0.822.Figure 21
Relations between parametera and fc.The fitting results indicated that the control parameterb of the descending stage is related not only to the peak stress of the specimen but also closely to the fiber content. Figure 22 illustrates the relations between parameter b and the peak stress fc. The selected data in Figure 22 were obtained from the specimens with the fiber content of Cf = 0.1. After regressing the selected data, the corresponding functional relation can be expressed as follows:(12)bCf=0.1=0.057fc+1.793.Figure 22
Relations between parameterb and fc.The fiber content has a significant effect on the descending stage of the stress-strain curve. Taking the fiber content ofCf = 0.1 as a reference, the control parameter b decreases with an increase of the fiber content, as shown in Figure 23. Through analyzing the variation of the parameter, the corresponding functional relation can be obtained, as shown below.(13)bbCf=0.1=1.554−5.577Cf.Figure 23
Relations between parameterb and Cf.Based on (12) and (13), the function of the control parameter b with the considerations of the peak stress and fiber content can be obtained, as shown below.(14)b=1.554−5.577Cf0.057fc+1.793.


## 4.3. Verification of Stress-Strain Model

According to the stress-strain model proposed in this paper, stress-strain curves of specimens with different parameters were modeled. Then, the predicted curves were compared with the ones measured from the test, as illustrated in Figure24. It can be clearly seen from the figure that the curves predicted by the proposed model well conform to the results obtained from the test, which verifies the correctness of the proposed model. On the whole, the predicted curves are in good agreement with the ascending stages of the experimental curves. However, the accordance rate in descending stages between the predicted curve and the experimental curve is relatively poor. The possible reason is that the experimental data of specimens have relatively large discreteness after specimens reach their peak stresses. This is consistent with the experimental results reflected by the uniaxial compression tests of other materials, such as concrete [41].Figure 24
Comparison between the predicted and measured stress-strain curves.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Wu et al. [33] investigated the stress-strain relations of PG prism specimens without fiber, and the corresponding curves were obtained through a series of uniaxial compression tests. To further verify the correctness of the proposed model, some of the measured curves reported by Wu et al. [33] were selected as validation data. The comparison results are shown in Figure 25. It can be found from the comparison results that the predicted curves are slightly different from the measured curves in the ascending stages. This is due to the fact that the test equipment led to a relatively long compaction process, as pointed out by Wu et al. [33]. However, the predicted curves well conform to the measured curves in the descending stages. Overall, the predicted curves are in good agreement with the test data. Thus, it can be proved that the model proposed in this paper has strong applicability.Figure 25
Further verification of the proposed model.
(a)(b)(c)(d)(e)(f)The yield point, peak point, and failure point are the key characteristic points of the stress-strain curve. Previous studies showed that the yield strength in ascending stage is close to 70% of the peak strength, and the failure strength in descending stage is commonly determined as the strength falls to 85% of the peak strength [42, 43]. In the proposed stress-strain model, the measured peak stress and peak strain need to be substituted into the model; thus the peak point of the model curve is the same as that of the measured curve. As for the yield point and the failure point, the predicted strain to measured strain ratios (ΔM/ΔT) are shown in Figure 26. It can be seen from Figure 26(a) that the predicted strain values (ΔM) were slightly lower than the measured strain values (ΔT), yet the prediction errors are all within 20%. That is to say, the yield strain predicted by the model is relatively small; this may be due to the underestimation of the compaction deformation of the PG specimen. In addition, the predicted failure strains are almost consistent with the test data, as illustrated in Figure 26(b). On the whole, the proposed stress-strain model showed a good prediction ability, and the calculated values can meet the accuracy requirements of engineering applications.Figure 26
Comparison between the predicted and measured strain values.
(a)(b)


## 5. Summary and Conclusions

Experimental and analytical investigations on the stress-strain curve of fiber reinforced PG prism specimens under uniaxial compression are presented in this study. Based on the tests and analysis results, the following conclusions can be obtained:(1)
According to experimental phenomena, three kinds of failure modes of PG prism specimens were observed in this study. The specimen with low water-solid ratio showed a tension failure, and the specimen without polypropylene fiber exhibited a mixed stress failure. The failure mode of other specimens can be identified as a compaction failure.(2)
From the characteristics of the stress-strain curve, the whole curve of the specimen can be divided into four parts, in which the first part of the ascending stage showed a concave profile; this is due to the fact that the specimen has a compaction process in the early loading stage. The shape of other parts is similar to that of the stress-strain curve of concrete under uniaxial compression.(3)
The comparison of peak stresses showed that decreasing the water-solid ratio can lead to an increase in the peak stress; increasing the fiber content can lead to enhancing the uniaxial load carrying capacity of specimens. The contents of the cement and quicklime have no significant effect on the peak stress of specimens.(4)
The variation of the peak strain of specimens is not obvious, and the values of the measured results are distributed between 0.0055 and 0.0091, with an average value of 0.0072.(5)
Decreasing the water-solid ratio led to increasing the secant moduli of specimens. Moreover, the comparison analysis also revealed that the fiber content is the key to the secant modulus. Nevertheless, the effects of the content of the cement as well as the quicklime on the elastic modulus are not significant.(6)
A stress-strain model for PG prism specimens was developed, and the proposed model can provide a good prediction for stress-strain curves of specimens. By comparison with the test data of this study and previous studies, the correctness of the proposed model was verified.



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*Source: 7221491-2022-03-20.xml*