# Mechanical Attributes of Uniaxial Compression for Calcium Carbonate Whisker Reinforced Oil Well Cement Pastes

Authors: Yuanyi Yang; Shun Fu; Xingkui Li
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

It is crucial for design and safety of the cementing sheath to develop better understanding of the CaCO3 whisker reinforced oil well cement pastes. The uniaxial compression curve, mechanical constitutive relation, and reinforcing mechanism of the CaCO3 whisker reinforced oil well cement pastes are studied in this script. The results indicate that the CaCO3 whisker under the 10% dosage could improve the tensile strength of the cement paste significantly. The peak stress, elasticity modulus, and the energy at different stages of the stress-strain curve of the CaCO3 whisker reinforced cement paste are reinforced with the increasing of CaCO3 whisker. Afterward, the constitutive model of stress-strain curve, the toughness index, and capability coefficients index of the CaCO3 whisker reinforced cement paste are established. A physical model of the interface layer is also established and the micromechanical reinforcement is related to the double film layer between the CaCO3 whisker and cement matrix which could be bonded with much more fastness to the cement surface. The development of this script provides new ways to analyze the toughening mechanism of CaCO3 whisker and establishes a correlation between basic material structure and the physical properties.


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## Body

## 1. Introduction

With many advantages such as excellent mechanical properties, modest price, easy producing process, and natural compatibility with the cement, the application of inorganic crystal whiskers is developing rapidly in the industry. The application of inorganic crystal whiskers in the cement based composite not only is with high worth for academic research and good prospect of engineering employment but is also important for the energy conservation and sustainable development in the manufacture. The researches focused on the inorganic crystal whiskers reinforced cement pastes are gradually improved as a hotspot. Cao et al. [1, 2] have firstly explored the calcium carbonate whiskers in the cement based composite as reinforcing material, and the result of the study shows that the compressive strength, flexible strength, and tensile strength of the cement pastes with calcium carbonate whisker are increased. Cao et al. [3] have researched the cement pastes with hybrid fibers of calcium carbonate whisker and basalt fiber which displays that the hybrid effect with different kinds of fibers could arise in the cement pastes and the mechanical property of the cement pastes would be improved further than with just single fiber. Simultaneously, with the perfect performance of high strength, high elasticity modulus, excellent heat-resisting, and heat insulation, the calcium carbonate whisker as a kind of desired filling material could fit the requests of the cementing and geological engineering in petroleum industry well [1, 4]. Li et al. [5, 6] have explored the workability of the cement slurry and the mechanical performance and the microstructure of the oil well cement pastes with hybrid fibers of calcium carbonate whisker and carbon fiber. The strengthening mechanism with hybrid fibers is elaborated as hybrid reinforcement. Ming et al. [7] have found that calcium carbonate whisker could not only reinforce mechanical performance of the oil well cement pastes but also develop the static gel strength conspicuously that could enhance the gas channeling prevention of the oil well cement slurry.From the above, almost all the studies are focused on the traditional mechanical performances of cement pastes and workability of the cement slurry. Meanwhile, the constitutive relation is based on the mechanical properties of cement pastes and the macroscopic constitutive model of cement paste could be established through the experimental data as a kind of continuous state material. As a key basic property, the uniaxial compression of cement pastes is important to research of the bearing capacity and the deformability of cement sheath. And the stress-strain diagram is also a reflection of comprehensive capability of the cement material. The arising of the plastic deformation, producing and extending of the microcrack, peak strength, and the ultimate deformation of the cement pastes all appeared in the stress-strain diagram as the most important factors account for the compressive capacity and nonlinear process [8–11]. Nevertheless, the researches appearing in studies of the constitutive relation and stress-strain diagram of uniaxial compression focused on the CaCO3 whisker reinforced oil well cement pastes are quite insufficient.The purpose of this study is to gain a better comprehending of the physical properties as well as the strengthening and toughening mechanism of the CaCO3 whisker reinforced oil well cement pastes. Establishing the constitutive relation between the CaCO3 whisker and oil well cement pastes, analyzing the stress-strain diagram, and calculating the physical model of interface layer are helpful for further research on the oil well cement based composite material. Therefore, the uniaxial compression curve, mechanical constitutive relation, and microreinforcing mechanism of the CaCO3 whisker reinforced oil well cement pastes are tested and discussed in this script to elaborate the fundamental physical effect of CaCO3 whisker reinforced oil well cement paste, which could lead to a better design and more safer cementing sheath, to achieve desirable toughing effects and improve wellbore integrity.


## 2. Materials and Methods

The materials applying in this script were cement (G high sulfur resistant oil well cement, Sichuan, China), CaCO3 whisker, water reducer (phenol and formaldehyde condensation polymer, Chengdu, China), defoaming agent (dibutyl phosphate, Chengdu, China), and pure water. The basic properties and chemical constituents of CaCO3 whisker were shown in Table 1, respectively.Table 1
Properties of CaCO3 whisker.
Material
Density ratio
Relative molecular weight
Draw ratio
Appearance
Size
Origin
CaCO3 whisker
2.8 g/cm3
100.09
20–30
20–80μm
China, Shandong Runxing Chemical Industry Co., Ltd.The cement slurry of CaCO3 whisker was produced based on the Chinese standard “GB10238-2005 Oil Well Cement” and “GB/T19139-2012: The Application Performance and Test Methods of Oil-Well Cement.” Thereafter, the mix design was divided into three steps which were elucidated as follows.Step 1.
The water reducer was mixed with the cement powder for 5 min by cement mortar mixer as predispersing powder.Step 2.
The CaCO3 whisker powder was mixed with the cement predispersing powder according to the quality fraction of the cement (as shown in Table 2) for 5 min before being agitated with water.Table 2
Mix proportions of cement pastes.
CaCO3   whisker dosage
Cement (g)
Defoaming agent(g)
Water reducer(g)
Water(g)
CaCO3 whisker (g)
Water-cement ratio
0
800
1
8
352
0
0.44
3%
776
1
8
352
24
0.44
5%
760
1
8
352
40
0.44
8%
736
1
16
352
64
0.44
10%
720
1
16
352
80
0.44Step 3.
All the mixed powder was transported to a high speed mixer for blending with water in 2 min and producing the cement slurry, whereby the mixing procedures were illustrated in Figure1.Figure 1
The mixing procedure for fresh cement paste mixtures.Afterwards, the curing condition of the cement slurry was under the relative humidity 95% and 30°C for 7 days as preparation.The split tensile strength of the cement pastes was test based on the Chinese standard “GB/T50081-2002: The Standard of Concrete Mechanics Performance Test Method” and the test specimen size of the cement pastes was 70.7 mm × 70.7 mm × 70.7 mm. The uniaxial compression experiment of cement pastes was tested by a CMT-300 universal testing machine with 300 kN maximum test load and 0.5 accuracy index, which was produced by the Shandong Jinan Lian Engineering Testing Technology, Co., Ltd. The test specimen size of the cement pastes for uniaxial compression experiment was 70.7 mm × 70.7 mm × 70.7 mm as well. The control mode of this experiment was displacement and the loading rate of the experiment was 0.06 mm/s; additionally the test data was collected by the computer software and the diagram of experimental facility as in Figure2.Figure 2
(a) Experimental facility. (b) Uniaxial compression fixture for cement pastes.
(a)
(b)


## 3. Results and Discussion

In this part, the macroscopic mechanical properties of split tensile strength, uniaxial compression strength, stress-strain curve, constitutive model, and fracture energy were studied and elucidated. Meanwhile the micrographs of cement pastes fracture with CaCO3 whiskers and the reinforcing effect were also shown and elaborated as follows.


### 3.1. The Split Tensile Strength

Under the oil well, the cement sheath would be easily destroyed by the tensile stress. Therefore the split tensile strength of the cement pastes with different dosage of the CaCO3 whisker was presented in Figure 3. With the increasing of the CaCO3 whisker, the improving trend of cement pastes tensile strength has taken place. Thereafter, the dosage exceeded 10%; the tensile strength started to decrease significantly. At the 10% point, the growth of the tensile strength was increased by 54% as the highest comparing to the control sample. With high mechanical strength, the CaCO3 whisker could fill in the cement and restrict the formation and propagation process of the microcrack [12, 13]. At the dosage of CaCO3 whisker of 15%, the tensile strength of the cement pastes has fallen down seriously. Hereby, over this dosage, the CaCO3 whisker could not be dispersed well and the cement material was decreased so that the CaCO3 whisker could not be embraced homogeneously by the cement slurry. As in Figure 4, the white dots were marked by red circles, which were the CaCO3 whiskers aggregating on the test-piece failure surface. Thus the added quantity of the CaCO3 whisker was confined from 1% to 10% in the next test part of the uniaxial compression.Figure 3
Tensile strength of CaCO3 whisker.Figure 4
The failure surface of test-piece reinforced cement pastes.


### 3.2. The Stress-Strain Curve

Incorporating with different dosages of CaCO3 whisker, the stress-strain curves of the cement pastes were presented in Figure 5, used to assess the contribution of different dosage of CaCO3 whisker. By increasing the CaCO3 whisker, the peak compressive stress was improved effectively and each deforming phase was still obvious; hereby that was the possibility of establishing the mathematical model to understand the complete stress-strain process. By increasing the CaCO3 whisker, the stiffness of the cement paste was also improved observably at the elastic stage comparing to the control sample; this improvement was caused by the curve slope and the elasticity modulus rising. Since the peak stress and peak strain were enhanced with the CaCO3 whisker increasing, the deformation energy of the cement pastes could be improved at the elastic-plastic deformation stage and the energy analysis would be elucidated in Section 3.4 adequately. Afterward, beyond the ultimate high dosage, the dispersity of CaCO3 whisker in cement slurry had been hindered for further improving. However, as the filling effect, the peak stress of cement paste with high CaCO3 whisker quality could be still higher than the control sample but with reduction of the deformation energy.Figure 5
The stress-strain curve of different CaCO3 whisker dosage cement pastes.The results of uniaxial compression test were shown in Table3 and the mathematical fitting relationship between CaCO3 whisker and the peak stress was expressed as formula (1). Thereby the fitted curve was shown in Figure 6 and the relationship between CaCO3 whisker and the peak stress was approximately a linear function. This means that the peak stress of the cement paste was increasing with the CaCO3 whisker. The fitting formula between CaCO3 whisker and the elasticity modulus was shown as formula (2), and the fitted curve was shown in Figure 7. The relationship between the CaCO3 whisker and elasticity modulus was likely monomial function; thus the CaCO3 whisker had reinforced the elasticity modulus of the cement paste well. The relationship between the CaCO3 whisker and peak strain was exhibited in Figure 8, which showed that the peak strain was fluctuated from 0.6 to 0.8 with different dosage of CaCO3 whisker. Thereby it could not be ruled and the CaCO3 whisker could hardly impact the peak strain of cement pastes.(1)Fc=2.86Nf+128.32(2)E=18.19Nf-0.10,where the Fc, Nf, and E were with respect to the peak stress, CaCO3 whisker dosage, and elasticity modulus.Table 3
The result of uniaxial compression test.
Dosage ofCaCO3 whisker
Peak stress(kN)
Peak strain(mm)
Elasticity modulus (N/mm2)
0
128.80
1.89
1356.97
3%
136.50
1.41
1564.12
5%
140.40
1.69
1473.30
8%
155.70
1.87
1418.84
10%
154.50
2.03
1387.21Figure 6
Plot of peak stress against CaCO3 whisker.Figure 7
Plot of elasticity modulus against CaCO3 whisker.Figure 8
Plot of peak strain against CaCO3 whisker.


### 3.3. The Constitutive Model of Stress-Strain Curve

Figure5 showed that although the stress-strain curves of CaCO3 whisker were at different dosage, the basic geometrical characteristics and processes of the uniaxial compression test were similar. A test result of experiment was chosen to illustrate the typical fiber reinfoced cement uniaxial stress-strain curve in Figure 9. Meanwhile the constitutive model of CaCO3 whisker cement pastes stress-strain curve was established and analyzed in this part. A new mathematic relation was proposed in this part, which was enlightened by Zhenhai Guo from Tsinghua University [11].Figure 9
Typical fiber reinforced cement uniaxial stress-strain curve.(
1
) The stress-strain curve of CaCO3 whisker reinforced cement paste was split into two parts as ascent stage and descent stage for fitting and analyzing as follows:(3)y=a0+a1x+a2x5+a3x6x≤1the  equation  of  ascent  stage(4)y=xb0+b1x+b2x2x≥1the  equation  of  descent  stage,where x=ε/εp, y=δ/δp, and ε and δ were the stress and strain. εp, δp were the peak strain and peak stress and a0,a1,a2,a3,b0,b1,b2 were the calculating parameters of the curve.(
2
) The ascent stage (0≤x≤1) is as follows.As the stress-strain curve showed,a0,a1,a2, and a3 should fit the geometrical feature as follows:(A) The curve went through the origin and that meant whenx=0, then y=0.(B) When0≤x<1,∂2y/∂x2<0, the slope (dy/dx) of ascent stage decreased monotonously and with no inflection point.(C)x=1,dy/dx=0,y=1 was the necessary condition of the peak point C on the curve.Taking (A), (B), and (C) conditions into (3), calculate the results as a0=0, a2=6-5a1 and a3=4a1-5. Afterwards, to solve the problem was subject to calculating the independent parameter a1.Then (3) was turned into (5)y=a1x+6-5a1x5+4a1-5x6.From (3), when x=0, then dy/dx=a1.Thus the following could be obtained:(6)a1=dydxx=0=dδ/δpdε/εcx=0=dδ/dεx=0δp/εc=E0Ep,where the E0=(dδ/dε)|x=0 (N/mm2) was the elasticity modulus of initial tangent of the cement pastes and Ep=δp/εc (N/mm2) was the ratio between the peak strength and peak strain as the secant modulus of the peak point of the curve, while a1 was the ratio of the initial tangent modulus and secant modulus. The ascent stage of the curve could be calculated by Ec and Ep. Meanwhile the curve also fit condition (B), when 0≤x<1 and ∂2y/∂x2<0; then the following could be obtained:(7)whenx=1as∂2y∂x2=206-5a1x4+304a1-5x5≤0x=1⟶206-5a1+304a1-5≤0⟶20a1≤30⟶a1≤1.5also  thea1=E0EpEc>0,Ep>0,thena1>0,0<a1≤1.5.The plots of the calculated curves at the ascent stage with differenta1 were shown in Figure 10. When a1>1.5, at the top part of the curve that was violated the actual test results were caused by y>1. Simultaneously, the calculated results were contrasted with the actual test curves as Figure 11 (where the solid lines showed the calculation and the dashed lines showed the test samples). When a1<0.6, the calculating curve could not fit the test result without an inflection point. Therefore, the data range of the a1 could be suggested as (8)0.5<a1≤1.5.Figure 10
a
1 and the calculated curve at the ascent stage.Figure 11
The calculated curve contrast to the test curve.(
3
) The descent stage (x≥1) is as follows.As the stress-strain curve showed,b0, b1, and b2 should fit the geometrical feature as follows:(A) Whenx=1, dy/dx=0, and y=1 the curve should reach peak point C.(B) When∂2y/∂x2=0, xD>1.0, hereby there was another inflection point as D.(C) When∂3y/∂x3=0, xE(>xD), then the maximum curvature point of the descent stage was E.(D) Whenx→∞, y→0 and dy/dx→0, the descent stage of the curve would be extended indefinitely and converged to coordinate axis without intersects; thus the whole curve should fit x≥0, 0<y≤1.To take (A) into (4) this would obtain the result as b1=1-2b0, b2=b0 that could left the only parameter as b0. Then (4) would change as (9)y=xb0x-12+xx≥1.Equation (9) should fit the conditions of (C) and (D). When b0=0, then y≡1.As the curve presented whenb0→∞, then y→0 and after peak point C, the residual stress of the cement paste was approaching 0 and at this time the cement paste would be considered as fragile material completely.Based on condition (B), it could be calculated as(10)∂2y∂x2=2b0x3-3x+2-1/b0b0x-12+x3=0.Afterward, the value of point D could be calculated (whenx>1) based on condition (C) and that would be(11)∂3y∂x3=-6b0b02x4-6b02x2+8b02-4b0x-3b02-4b0+1b0x-12+x3=0.Then the maximum curvature point E could be also calculated (xE>xD).The descent stage of the curves against differentb0 was shown in Figure 12. When b0=0.2 or b0=0.5, the curve of the descent stage almost was linear, and inflection point D and maximum curvature point E were nonsignificant. Simultaneously, though the test-piece was deformed seriously, the spice was still with high strength which could not fit the actual test result.Figure 12
b
0 and the calculated curve at the descent stage.The calculated curves were contrasted with the actual test curves in Figure13. The shape of descent stage of the CaCO3 whisker reinforced cement paste was complicated and the shape was effectively impacted by increasing the CaCO3 whisker dosage. Figure 13 showed the descent stage was with a large range of residual stress, when the residual stress of the cement paste had fallen below 60% that would determine that the cement paste was complete failure in this script, which was because the cement paste was covered with macroscopic crack and destroyed seriously at this moment. From the above, the range of b0 of the descent stage curve was suggested to belong to 5<b0<500.Figure 13
Calculating curve in contrast to the test curve.The above calculating equations of the stress-strain curve were focused on the CaCO3 whisker reinforced cement paste and were under a definitive water-cement ratio. When the conditions were changed the equations should be amended as well.(
4
) The complete curve is as follows.To sum up, the equations of (5) and (9) were combined together to represent the constitutive model of the stress-strain curves in Figure 14, and the constitutive model was contrasted to the test curves in Figure 15. The results of the experiment curves were all contained by the calculating curves, so the fitted equation would be effective, especially at the ascent stage and the descent stage before the stress below 60%. After the stress had fallen below 60%, the cement paste would be destroyed completely, which was considered to lose the research value.Figure 14
The calculating curve of constitutive model.Figure 15
The integrated calculating curve in contrast to the complete test curve. The colored dotted lines referred to the incorporation of calculated results in Figures11 and 13.The constitutive model of stress-strain curve of the CaCO3 whisker reinforced cement paste could be described by (5) and (9) as(12)y=a1x+6-5a1x5+4a1-5x6,0≤x≤1y=xb0x-12+x,x≥1,where a1 and b0 were undecided parameters and the range of a1 and b0 of the CaCO3 whisker reinforced cement paste was elaborated as 0.5<a1≤1.5, 5<b0<500 in this script.


### 3.4. Toughness of Uniaxial Compression

The toughness of the cementitious material was focused on the ability of absorbing energy from initial loading to materials failure; the more the energy was absorbed, the more tough the cementitious material would be. The absorbing energy of the material under uniaxial compression was calculated by the area of the stress-strain curve in Figure16. The work of the load could be calculated by (13)W=∫Fdσ,where W, F, and σ were the load work, load, and the strain capacity.Figure 16
The toughness of the material.The toughness of the cementitious material was related not only to the bearing capacity but also to the mechanical deformation capability. In this script, the toughness of the cementitious material which was calculated referring to the Chinese standard of “Steel Fiber Reinforced Concrete Testing Methods (CECS13: 2009)” and was elucidated as follows.(
1
) The compression curve of the cement was divided into 4 parts in Figure 17, drawing a straight line though the x-axis at the point of Fcra (where Fcra = 0.85∗Fmax; Fmax was the peak stress), which was parallel to the y-axis and intersected the load curve at the critical point A. The x-axis of point A was the critical elastic deformation as σcra and the area of SOAB was defined as critical toughness of SOAB.Figure 17
The stress-strain curve and toughness index.(
2
) Calculate the values of 1.2, 1.5, and 2.0 times of σcra, respectively, which were determined as the points D, F, and H on the load curve. The areas of SOAB, SOACD, SOAEF, and SOAGH were also calculated. Hereby, the toughness indexes were defined as follows:(14)ηc1.0=SOFmaxSOABηc1.2=SOACDSOABηc1.5=SOAEFSOABηc2.0=SOAGHSOAB.(
3
) The capability coefficients index was defined as follows: (15)ζ=ηc,n-aa-1,where a was the deformation value as 1.2, 1.5, and 2.0 and ηc,n was the toughness index relating to the deformation value in Table 4.Table 4
The toughness index and capability coefficients index.
CaCO3 whisker   quality
η
c
,
n
η
c
1.0
η
c
1.2
η
c
1.5
ζ
c
1.2
ζ
c
1.5
0%
1.35
1.40
1.67
0.99
0.34
3%
1.37
1.42
1.66
1.11
0.32
5%
1.45
1.51
1.92
1.55
0.84
8%
1.43
1.49
1.96
1.44
0.92
10%
1.43
1.51
2.06
1.54
1.11The relationships between the toughness index or capability coefficients index and the dosage of the CaCO3 whisker were shown in Figures 18 and 19. The increasing trend of ηc1.0, ηc1.2, and ηc1.5 was shown in Figure 18. Additionally, the peak damage energy and the compressive toughness of the cement pastes were developed observably. The growth of ηc1.5 was most remarkable; at the point 10% it was improved 1.23 times the control sample. The increasing trend of capability coefficients index of the cement paste was also shown in Figure 19. ζc1.2 and ζc1.5 of point 5% were improved 1.57 and 2.47 times the control sample. ζc1.2 and ζc1.5 of point 10% were improved 1.56 and 3.26 times the control sample. Thus the improvement of toughness of the cement paste was further characterized by the energy index as in Table 4.Figure 18
The toughness index of CaCO3 whisker reinforced cement paste.Figure 19
Capability coefficients index of CaCO3 whisker reinforced cement paste.


### 3.5. Micromechanical Mechanisms

(
1) The Microstructure. The CaCO3 whiskers with sharp regularity were inset in the cement pastes which could be identified easily. From Figure 20(a), the cement paste was a kind of inhomogeneity material with small opening where the CaCO3 whiskers with small size could fill in the microcracks and holes. Because of the CaCO3 whiskers with rather higher mechanical strength, when the cement paste was under the stress damage, the CaCO3 whiskers could support the stress as a bridge between the cracks in Figure 20(b), which could expend the destroying energy of the cement matrix by bridging toughening effect [14–16]. In Figure 20(c), caused by the rising stress, the collapsing strength of CaCO3 whiskers far exceeded the cement paste cohesion strength; the crack could not break down the CaCO3 whiskers; instead the CaCO3 whiskers could be pulled out to consume the energy for rubbing action and stripping effect [15, 16]. Also the crack would be impeded by CaCO3 whiskers [3, 17–20] as in Figure 20(d). This was marked by the red circle in Figure 20(e); over the ultimate high dosage, the CaCO3 whiskers could be scattered unevenly; in reverse the whisker would ease the mechanical property of cement paste, whereby the strengthening effort of the CaCO3 whiskers would be summed up as the destroying energy conservation and microcrack inhibiting and a physical model was stated in Figure 20.Figure 20
(a) The CaCO3 whisker filling the crack. (b) The bridging effect of CaCO3 whisker. (c) Pull-out effect of CaCO3 whisker. (d) Confined crack developing. (e) the CaCO3 whiskers scattered unevenly.
(a)
(b)
(c)
(d)
(e)(
2) Physical Model of Interface Layer. With high draw ratio, the CaCO3 whiskers were short fibers with different granulate material which could bear the longitudinal load. And that was confirmed in mechanical experiment above in this script. A physical interface layer model [21–23] was established to elaborate this mechanism in Figure 21. With a large specific surface area; when the CaCO3 whiskers were blended with the cement slurry, a water molecule layer could be adhered to the surface of CaCO3 whisker, which could promote the cement hydration. Furthermore the CH crystals would gather in this layer and form a specific fiber-matrix layer between the fiber and cement matrix; this layer consisted of double film layer, CH gathering layer, and porous layer. The thickness of the double film layer was about 1~2 μm, which was composed of CH crystals and C-S-H. Due to the small size of the CaCO3 whiskers, the double film layer could be bonded with much more fastness to the cement surface compared with the organic fiber which could raise the pull-out energy and inhibit the crack initiation and coalescence [21, 22]. With orienting CH gathering, the CH gathering layer was a weak link of the fiber-matrix layer for the loose structure. Another weak link of the fiber-matrix layer was the porous layer with abundant micropores and loose structure which consisted of CH crystals and C-S-H. The thickness of the fiber-matrix layer was about 10~100 μm and CH gathering layer and porous layer could firmly impact the mechanical strength of the cement paste as the loose structure and constitution. As the fiber spacing theory, the anticracking of the fiber was consanguineously related to the average distance between the reinforcing fiber and the cement matrix; thereby the distance was smaller and the anticracking was stronger. Since the distance was smaller, the thickness of the weak link of the fiber-matrix layer was thinner which indicated that the fiber could adhere the matrix with much fastness; this meant that the mechanical strength was well developed.Figure 21
Physical model of interface layer.As assumed, the CaCO3 whiskers were well dispersed in the three-dimensional direction in the cement paste and the equation of the average distance of the CaCO3 whiskers was shown as follows [24–26]:(16)S-=4.88VfFSS,where S-, Vf, and FSS were the average distance (μm), fiber volume fraction (%), and the surface area of the fiber in the unit volume cement matrix (mm2).From (16), when the value of the FSS was increased, S- would be seriously reduced and the shape of the CaCO3 whiskers was assumed as a uniform cylinder to calculate the FSS:(17)FSSCaCO3whisker=NPflf=πdfπ/4df2Vf=4dfVf,where (18)N=VfAflf,Pf=πdf, where N was the number of fibers in unit volume cement matrix, Pf was the cross-sectional area of the CaCO3 whiskers (mm2), and df was the diameter of the reinforcing fiber (mm).Under ideal dispersing, the parameters of different dosage of CaCO3 whisker were calculated (where L = 50 μm and df = 1 μm) in Table 5. S- was over 14 μm at 1% CaCO3 whiskers and the weak link of the fiber-matrix layer was at a relative large scale and the reinforcement of the CaCO3 whiskers was not significant. With the CaCO3 whisker increasing, S- was decreasing remarkably. And at 10% CaCO3 whisker, S- was decreased to 4.75 μm; hereby the fiber-matrix layer was compressed conspicuously and the tensile strength of the cement paste was well improved under the ideal condition. S- of the 15% was the lowest but the strength had still fallen down, because the CaCO3 whiskers could not evenly be dispersed at this dosage; the CaCO3 whiskers could be reunited and would decrease the cementation between the CaCO3 whisker and cement matrix. Thus the fiber spacing theory would be invalid under this mixing condition.Table 5
The parameters of the physical model.
CaCO3 whiskers (%)
1
5
10
15
V
f (%)
0.67
3.40
6.60
14.80
FSS (μm2⋅μm−3)
0.027
0.136
0.264
0.592
S
- (μm)
14.84
6.16
4.75
3.17(
3) XRD. The results of XRD under 0%, 5%, and 10% of CaCO3 whisker were shown in Figure 22. The types of the hydration products with CaCO3 whiskers were not changed, whereas with the increasing trend of CaCO3 whisker, the diffraction peaks of C3S and the C2S had fallen down; besides the diffracted intensity of the C-H-S and CH was increased significantly compared with the control sample. Because of the small size and large specific surface area, the CaCO3 whisker was considered as a submicron material; the water molecule layer could be easily adhered to the whisker surface and formed a water firm. The water on the surface of the CaCO3 whisker could stimulate the cement hydration and the hydration products would fix the defect and strengthen the structure of the interface layer. Therefore, the physical model of interface layer had been demonstrated when the C-H-S and CH were increased significantly after adding CaCO3 whisker.Figure 22
The XRD results of CaCO3 whisker reinforce cement pastes.


## 3.1. The Split Tensile Strength

Under the oil well, the cement sheath would be easily destroyed by the tensile stress. Therefore the split tensile strength of the cement pastes with different dosage of the CaCO3 whisker was presented in Figure 3. With the increasing of the CaCO3 whisker, the improving trend of cement pastes tensile strength has taken place. Thereafter, the dosage exceeded 10%; the tensile strength started to decrease significantly. At the 10% point, the growth of the tensile strength was increased by 54% as the highest comparing to the control sample. With high mechanical strength, the CaCO3 whisker could fill in the cement and restrict the formation and propagation process of the microcrack [12, 13]. At the dosage of CaCO3 whisker of 15%, the tensile strength of the cement pastes has fallen down seriously. Hereby, over this dosage, the CaCO3 whisker could not be dispersed well and the cement material was decreased so that the CaCO3 whisker could not be embraced homogeneously by the cement slurry. As in Figure 4, the white dots were marked by red circles, which were the CaCO3 whiskers aggregating on the test-piece failure surface. Thus the added quantity of the CaCO3 whisker was confined from 1% to 10% in the next test part of the uniaxial compression.Figure 3
Tensile strength of CaCO3 whisker.Figure 4
The failure surface of test-piece reinforced cement pastes.


## 3.2. The Stress-Strain Curve

Incorporating with different dosages of CaCO3 whisker, the stress-strain curves of the cement pastes were presented in Figure 5, used to assess the contribution of different dosage of CaCO3 whisker. By increasing the CaCO3 whisker, the peak compressive stress was improved effectively and each deforming phase was still obvious; hereby that was the possibility of establishing the mathematical model to understand the complete stress-strain process. By increasing the CaCO3 whisker, the stiffness of the cement paste was also improved observably at the elastic stage comparing to the control sample; this improvement was caused by the curve slope and the elasticity modulus rising. Since the peak stress and peak strain were enhanced with the CaCO3 whisker increasing, the deformation energy of the cement pastes could be improved at the elastic-plastic deformation stage and the energy analysis would be elucidated in Section 3.4 adequately. Afterward, beyond the ultimate high dosage, the dispersity of CaCO3 whisker in cement slurry had been hindered for further improving. However, as the filling effect, the peak stress of cement paste with high CaCO3 whisker quality could be still higher than the control sample but with reduction of the deformation energy.Figure 5
The stress-strain curve of different CaCO3 whisker dosage cement pastes.The results of uniaxial compression test were shown in Table3 and the mathematical fitting relationship between CaCO3 whisker and the peak stress was expressed as formula (1). Thereby the fitted curve was shown in Figure 6 and the relationship between CaCO3 whisker and the peak stress was approximately a linear function. This means that the peak stress of the cement paste was increasing with the CaCO3 whisker. The fitting formula between CaCO3 whisker and the elasticity modulus was shown as formula (2), and the fitted curve was shown in Figure 7. The relationship between the CaCO3 whisker and elasticity modulus was likely monomial function; thus the CaCO3 whisker had reinforced the elasticity modulus of the cement paste well. The relationship between the CaCO3 whisker and peak strain was exhibited in Figure 8, which showed that the peak strain was fluctuated from 0.6 to 0.8 with different dosage of CaCO3 whisker. Thereby it could not be ruled and the CaCO3 whisker could hardly impact the peak strain of cement pastes.(1)Fc=2.86Nf+128.32(2)E=18.19Nf-0.10,where the Fc, Nf, and E were with respect to the peak stress, CaCO3 whisker dosage, and elasticity modulus.Table 3
The result of uniaxial compression test.
Dosage ofCaCO3 whisker
Peak stress(kN)
Peak strain(mm)
Elasticity modulus (N/mm2)
0
128.80
1.89
1356.97
3%
136.50
1.41
1564.12
5%
140.40
1.69
1473.30
8%
155.70
1.87
1418.84
10%
154.50
2.03
1387.21Figure 6
Plot of peak stress against CaCO3 whisker.Figure 7
Plot of elasticity modulus against CaCO3 whisker.Figure 8
Plot of peak strain against CaCO3 whisker.


## 3.3. The Constitutive Model of Stress-Strain Curve

Figure5 showed that although the stress-strain curves of CaCO3 whisker were at different dosage, the basic geometrical characteristics and processes of the uniaxial compression test were similar. A test result of experiment was chosen to illustrate the typical fiber reinfoced cement uniaxial stress-strain curve in Figure 9. Meanwhile the constitutive model of CaCO3 whisker cement pastes stress-strain curve was established and analyzed in this part. A new mathematic relation was proposed in this part, which was enlightened by Zhenhai Guo from Tsinghua University [11].Figure 9
Typical fiber reinforced cement uniaxial stress-strain curve.(
1
) The stress-strain curve of CaCO3 whisker reinforced cement paste was split into two parts as ascent stage and descent stage for fitting and analyzing as follows:(3)y=a0+a1x+a2x5+a3x6x≤1the  equation  of  ascent  stage(4)y=xb0+b1x+b2x2x≥1the  equation  of  descent  stage,where x=ε/εp, y=δ/δp, and ε and δ were the stress and strain. εp, δp were the peak strain and peak stress and a0,a1,a2,a3,b0,b1,b2 were the calculating parameters of the curve.(
2
) The ascent stage (0≤x≤1) is as follows.As the stress-strain curve showed,a0,a1,a2, and a3 should fit the geometrical feature as follows:(A) The curve went through the origin and that meant whenx=0, then y=0.(B) When0≤x<1,∂2y/∂x2<0, the slope (dy/dx) of ascent stage decreased monotonously and with no inflection point.(C)x=1,dy/dx=0,y=1 was the necessary condition of the peak point C on the curve.Taking (A), (B), and (C) conditions into (3), calculate the results as a0=0, a2=6-5a1 and a3=4a1-5. Afterwards, to solve the problem was subject to calculating the independent parameter a1.Then (3) was turned into (5)y=a1x+6-5a1x5+4a1-5x6.From (3), when x=0, then dy/dx=a1.Thus the following could be obtained:(6)a1=dydxx=0=dδ/δpdε/εcx=0=dδ/dεx=0δp/εc=E0Ep,where the E0=(dδ/dε)|x=0 (N/mm2) was the elasticity modulus of initial tangent of the cement pastes and Ep=δp/εc (N/mm2) was the ratio between the peak strength and peak strain as the secant modulus of the peak point of the curve, while a1 was the ratio of the initial tangent modulus and secant modulus. The ascent stage of the curve could be calculated by Ec and Ep. Meanwhile the curve also fit condition (B), when 0≤x<1 and ∂2y/∂x2<0; then the following could be obtained:(7)whenx=1as∂2y∂x2=206-5a1x4+304a1-5x5≤0x=1⟶206-5a1+304a1-5≤0⟶20a1≤30⟶a1≤1.5also  thea1=E0EpEc>0,Ep>0,thena1>0,0<a1≤1.5.The plots of the calculated curves at the ascent stage with differenta1 were shown in Figure 10. When a1>1.5, at the top part of the curve that was violated the actual test results were caused by y>1. Simultaneously, the calculated results were contrasted with the actual test curves as Figure 11 (where the solid lines showed the calculation and the dashed lines showed the test samples). When a1<0.6, the calculating curve could not fit the test result without an inflection point. Therefore, the data range of the a1 could be suggested as (8)0.5<a1≤1.5.Figure 10
a
1 and the calculated curve at the ascent stage.Figure 11
The calculated curve contrast to the test curve.(
3
) The descent stage (x≥1) is as follows.As the stress-strain curve showed,b0, b1, and b2 should fit the geometrical feature as follows:(A) Whenx=1, dy/dx=0, and y=1 the curve should reach peak point C.(B) When∂2y/∂x2=0, xD>1.0, hereby there was another inflection point as D.(C) When∂3y/∂x3=0, xE(>xD), then the maximum curvature point of the descent stage was E.(D) Whenx→∞, y→0 and dy/dx→0, the descent stage of the curve would be extended indefinitely and converged to coordinate axis without intersects; thus the whole curve should fit x≥0, 0<y≤1.To take (A) into (4) this would obtain the result as b1=1-2b0, b2=b0 that could left the only parameter as b0. Then (4) would change as (9)y=xb0x-12+xx≥1.Equation (9) should fit the conditions of (C) and (D). When b0=0, then y≡1.As the curve presented whenb0→∞, then y→0 and after peak point C, the residual stress of the cement paste was approaching 0 and at this time the cement paste would be considered as fragile material completely.Based on condition (B), it could be calculated as(10)∂2y∂x2=2b0x3-3x+2-1/b0b0x-12+x3=0.Afterward, the value of point D could be calculated (whenx>1) based on condition (C) and that would be(11)∂3y∂x3=-6b0b02x4-6b02x2+8b02-4b0x-3b02-4b0+1b0x-12+x3=0.Then the maximum curvature point E could be also calculated (xE>xD).The descent stage of the curves against differentb0 was shown in Figure 12. When b0=0.2 or b0=0.5, the curve of the descent stage almost was linear, and inflection point D and maximum curvature point E were nonsignificant. Simultaneously, though the test-piece was deformed seriously, the spice was still with high strength which could not fit the actual test result.Figure 12
b
0 and the calculated curve at the descent stage.The calculated curves were contrasted with the actual test curves in Figure13. The shape of descent stage of the CaCO3 whisker reinforced cement paste was complicated and the shape was effectively impacted by increasing the CaCO3 whisker dosage. Figure 13 showed the descent stage was with a large range of residual stress, when the residual stress of the cement paste had fallen below 60% that would determine that the cement paste was complete failure in this script, which was because the cement paste was covered with macroscopic crack and destroyed seriously at this moment. From the above, the range of b0 of the descent stage curve was suggested to belong to 5<b0<500.Figure 13
Calculating curve in contrast to the test curve.The above calculating equations of the stress-strain curve were focused on the CaCO3 whisker reinforced cement paste and were under a definitive water-cement ratio. When the conditions were changed the equations should be amended as well.(
4
) The complete curve is as follows.To sum up, the equations of (5) and (9) were combined together to represent the constitutive model of the stress-strain curves in Figure 14, and the constitutive model was contrasted to the test curves in Figure 15. The results of the experiment curves were all contained by the calculating curves, so the fitted equation would be effective, especially at the ascent stage and the descent stage before the stress below 60%. After the stress had fallen below 60%, the cement paste would be destroyed completely, which was considered to lose the research value.Figure 14
The calculating curve of constitutive model.Figure 15
The integrated calculating curve in contrast to the complete test curve. The colored dotted lines referred to the incorporation of calculated results in Figures11 and 13.The constitutive model of stress-strain curve of the CaCO3 whisker reinforced cement paste could be described by (5) and (9) as(12)y=a1x+6-5a1x5+4a1-5x6,0≤x≤1y=xb0x-12+x,x≥1,where a1 and b0 were undecided parameters and the range of a1 and b0 of the CaCO3 whisker reinforced cement paste was elaborated as 0.5<a1≤1.5, 5<b0<500 in this script.


## 3.4. Toughness of Uniaxial Compression

The toughness of the cementitious material was focused on the ability of absorbing energy from initial loading to materials failure; the more the energy was absorbed, the more tough the cementitious material would be. The absorbing energy of the material under uniaxial compression was calculated by the area of the stress-strain curve in Figure16. The work of the load could be calculated by (13)W=∫Fdσ,where W, F, and σ were the load work, load, and the strain capacity.Figure 16
The toughness of the material.The toughness of the cementitious material was related not only to the bearing capacity but also to the mechanical deformation capability. In this script, the toughness of the cementitious material which was calculated referring to the Chinese standard of “Steel Fiber Reinforced Concrete Testing Methods (CECS13: 2009)” and was elucidated as follows.(
1
) The compression curve of the cement was divided into 4 parts in Figure 17, drawing a straight line though the x-axis at the point of Fcra (where Fcra = 0.85∗Fmax; Fmax was the peak stress), which was parallel to the y-axis and intersected the load curve at the critical point A. The x-axis of point A was the critical elastic deformation as σcra and the area of SOAB was defined as critical toughness of SOAB.Figure 17
The stress-strain curve and toughness index.(
2
) Calculate the values of 1.2, 1.5, and 2.0 times of σcra, respectively, which were determined as the points D, F, and H on the load curve. The areas of SOAB, SOACD, SOAEF, and SOAGH were also calculated. Hereby, the toughness indexes were defined as follows:(14)ηc1.0=SOFmaxSOABηc1.2=SOACDSOABηc1.5=SOAEFSOABηc2.0=SOAGHSOAB.(
3
) The capability coefficients index was defined as follows: (15)ζ=ηc,n-aa-1,where a was the deformation value as 1.2, 1.5, and 2.0 and ηc,n was the toughness index relating to the deformation value in Table 4.Table 4
The toughness index and capability coefficients index.
CaCO3 whisker   quality
η
c
,
n
η
c
1.0
η
c
1.2
η
c
1.5
ζ
c
1.2
ζ
c
1.5
0%
1.35
1.40
1.67
0.99
0.34
3%
1.37
1.42
1.66
1.11
0.32
5%
1.45
1.51
1.92
1.55
0.84
8%
1.43
1.49
1.96
1.44
0.92
10%
1.43
1.51
2.06
1.54
1.11The relationships between the toughness index or capability coefficients index and the dosage of the CaCO3 whisker were shown in Figures 18 and 19. The increasing trend of ηc1.0, ηc1.2, and ηc1.5 was shown in Figure 18. Additionally, the peak damage energy and the compressive toughness of the cement pastes were developed observably. The growth of ηc1.5 was most remarkable; at the point 10% it was improved 1.23 times the control sample. The increasing trend of capability coefficients index of the cement paste was also shown in Figure 19. ζc1.2 and ζc1.5 of point 5% were improved 1.57 and 2.47 times the control sample. ζc1.2 and ζc1.5 of point 10% were improved 1.56 and 3.26 times the control sample. Thus the improvement of toughness of the cement paste was further characterized by the energy index as in Table 4.Figure 18
The toughness index of CaCO3 whisker reinforced cement paste.Figure 19
Capability coefficients index of CaCO3 whisker reinforced cement paste.


## 3.5. Micromechanical Mechanisms

(
1) The Microstructure. The CaCO3 whiskers with sharp regularity were inset in the cement pastes which could be identified easily. From Figure 20(a), the cement paste was a kind of inhomogeneity material with small opening where the CaCO3 whiskers with small size could fill in the microcracks and holes. Because of the CaCO3 whiskers with rather higher mechanical strength, when the cement paste was under the stress damage, the CaCO3 whiskers could support the stress as a bridge between the cracks in Figure 20(b), which could expend the destroying energy of the cement matrix by bridging toughening effect [14–16]. In Figure 20(c), caused by the rising stress, the collapsing strength of CaCO3 whiskers far exceeded the cement paste cohesion strength; the crack could not break down the CaCO3 whiskers; instead the CaCO3 whiskers could be pulled out to consume the energy for rubbing action and stripping effect [15, 16]. Also the crack would be impeded by CaCO3 whiskers [3, 17–20] as in Figure 20(d). This was marked by the red circle in Figure 20(e); over the ultimate high dosage, the CaCO3 whiskers could be scattered unevenly; in reverse the whisker would ease the mechanical property of cement paste, whereby the strengthening effort of the CaCO3 whiskers would be summed up as the destroying energy conservation and microcrack inhibiting and a physical model was stated in Figure 20.Figure 20
(a) The CaCO3 whisker filling the crack. (b) The bridging effect of CaCO3 whisker. (c) Pull-out effect of CaCO3 whisker. (d) Confined crack developing. (e) the CaCO3 whiskers scattered unevenly.
(a)
(b)
(c)
(d)
(e)(
2) Physical Model of Interface Layer. With high draw ratio, the CaCO3 whiskers were short fibers with different granulate material which could bear the longitudinal load. And that was confirmed in mechanical experiment above in this script. A physical interface layer model [21–23] was established to elaborate this mechanism in Figure 21. With a large specific surface area; when the CaCO3 whiskers were blended with the cement slurry, a water molecule layer could be adhered to the surface of CaCO3 whisker, which could promote the cement hydration. Furthermore the CH crystals would gather in this layer and form a specific fiber-matrix layer between the fiber and cement matrix; this layer consisted of double film layer, CH gathering layer, and porous layer. The thickness of the double film layer was about 1~2 μm, which was composed of CH crystals and C-S-H. Due to the small size of the CaCO3 whiskers, the double film layer could be bonded with much more fastness to the cement surface compared with the organic fiber which could raise the pull-out energy and inhibit the crack initiation and coalescence [21, 22]. With orienting CH gathering, the CH gathering layer was a weak link of the fiber-matrix layer for the loose structure. Another weak link of the fiber-matrix layer was the porous layer with abundant micropores and loose structure which consisted of CH crystals and C-S-H. The thickness of the fiber-matrix layer was about 10~100 μm and CH gathering layer and porous layer could firmly impact the mechanical strength of the cement paste as the loose structure and constitution. As the fiber spacing theory, the anticracking of the fiber was consanguineously related to the average distance between the reinforcing fiber and the cement matrix; thereby the distance was smaller and the anticracking was stronger. Since the distance was smaller, the thickness of the weak link of the fiber-matrix layer was thinner which indicated that the fiber could adhere the matrix with much fastness; this meant that the mechanical strength was well developed.Figure 21
Physical model of interface layer.As assumed, the CaCO3 whiskers were well dispersed in the three-dimensional direction in the cement paste and the equation of the average distance of the CaCO3 whiskers was shown as follows [24–26]:(16)S-=4.88VfFSS,where S-, Vf, and FSS were the average distance (μm), fiber volume fraction (%), and the surface area of the fiber in the unit volume cement matrix (mm2).From (16), when the value of the FSS was increased, S- would be seriously reduced and the shape of the CaCO3 whiskers was assumed as a uniform cylinder to calculate the FSS:(17)FSSCaCO3whisker=NPflf=πdfπ/4df2Vf=4dfVf,where (18)N=VfAflf,Pf=πdf, where N was the number of fibers in unit volume cement matrix, Pf was the cross-sectional area of the CaCO3 whiskers (mm2), and df was the diameter of the reinforcing fiber (mm).Under ideal dispersing, the parameters of different dosage of CaCO3 whisker were calculated (where L = 50 μm and df = 1 μm) in Table 5. S- was over 14 μm at 1% CaCO3 whiskers and the weak link of the fiber-matrix layer was at a relative large scale and the reinforcement of the CaCO3 whiskers was not significant. With the CaCO3 whisker increasing, S- was decreasing remarkably. And at 10% CaCO3 whisker, S- was decreased to 4.75 μm; hereby the fiber-matrix layer was compressed conspicuously and the tensile strength of the cement paste was well improved under the ideal condition. S- of the 15% was the lowest but the strength had still fallen down, because the CaCO3 whiskers could not evenly be dispersed at this dosage; the CaCO3 whiskers could be reunited and would decrease the cementation between the CaCO3 whisker and cement matrix. Thus the fiber spacing theory would be invalid under this mixing condition.Table 5
The parameters of the physical model.
CaCO3 whiskers (%)
1
5
10
15
V
f (%)
0.67
3.40
6.60
14.80
FSS (μm2⋅μm−3)
0.027
0.136
0.264
0.592
S
- (μm)
14.84
6.16
4.75
3.17(
3) XRD. The results of XRD under 0%, 5%, and 10% of CaCO3 whisker were shown in Figure 22. The types of the hydration products with CaCO3 whiskers were not changed, whereas with the increasing trend of CaCO3 whisker, the diffraction peaks of C3S and the C2S had fallen down; besides the diffracted intensity of the C-H-S and CH was increased significantly compared with the control sample. Because of the small size and large specific surface area, the CaCO3 whisker was considered as a submicron material; the water molecule layer could be easily adhered to the whisker surface and formed a water firm. The water on the surface of the CaCO3 whisker could stimulate the cement hydration and the hydration products would fix the defect and strengthen the structure of the interface layer. Therefore, the physical model of interface layer had been demonstrated when the C-H-S and CH were increased significantly after adding CaCO3 whisker.Figure 22
The XRD results of CaCO3 whisker reinforce cement pastes.


## 4. Conclusion

(
1
) With high elasticity modulus and strength, the CaCO3 whisker could improve the tensile strength of the cement paste significantly at the dosage of 10%.(
2
) The peak stress, elasticity modulus, and the energy of different stage of the stress-strain curve of the CaCO3 whisker reinforced cement paste were enforced with the increasing of CaCO3 whisker, and the failure characteristics of each stage were observable. The mathematic relations between the peak stress, elasticity, peak strain, and the CaCO3 whisker were discussed. The relation between the peak stress and the CaCO3 whisker was linear relation. Besides, the relation between elasticity and the CaCO3 whisker was power function and the relation between peak strain and CaCO3 whisker was not so regular that the CaCO3 whisker could impact the peak strain of the cement paste barely.(
3
) The constitutive model of stress-strain curve of the CaCO3 whisker reinforced cement paste was established in this paper which could describe the stress-strain properties and fit the experimental curve well.(
4
) The toughness index and capability coefficients index of the CaCO3 whisker reinforced cement paste were established in this paper to elucidate that the CaCO3 whisker could improve the toughness of the cement paste at different stages.(
5
) The CaCO3 whisker could develop the strength of cement pastes well and impede the microcracks initiation and coalescence in the cement paste. Caused by the pull-out and bridging effect, the CaCO3 whisker could consume the damaging energy and toughen the cement paste. A physical model of interface layer was also established to elaborate the toughing effect. CaCO3 whisker could form a water firm and the specific interface layer could compress the weak-link layer, improve the mechanical strength of the cement pastes, achieve desirable toughing effects, and improve wellbore integrity.



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*Source: 2939057-2017-11-23.xml*