Mechanical properties tests and multiscale numerical simulations for basalt fiber reinforced concrete

Construction and Building Materials 202 (2019) 58–72

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Mechanical properties tests and multiscale numerical simulations for basalt fiber reinforced concrete Xinjian Sun a,b, Zhen Gao a, Peng Cao a,c,⇑, Changjun Zhou d a

School of Hydraulic and Electric Engineering, Qinghai University, Xining, People’s Republic of China State Key Laboratory of Plateau Ecology and Agriculture, Xining, People’s Republic of China c College of Architecture and Civil Engineering, Beijing University of Technology, Beijing, 100124 China d School of Transportation & Logistics, Dalian University of Technology, Dalian, People’s Republic of China b

h i g h l i g h t s
A damage constitutive model was developed for BFRC.
FE model with the damage model was utilized to simulate BFRC mechanical properties.
Effects of basalt fiber on the BFRC mechanical properties was investigated.
Parabolic equation proposed for describing the effect of the aggregate and fiber volume contents.
Regression formula obtained for size effect of mechanical properties of the optimum BFRC.

a r t i c l e

i n f o

Article history: Received 18 July 2018 Received in revised form 8 November 2018 Accepted 3 January 2019

Keywords: Basalt fiber reinforced concrete Mechanical properties Numerical simulation Size effect

a b s t r a c t Basalt fiber reinforced concrete (BFRC) can be regarded as a composite of cement mortar, aggregate and basalt fiber. In this paper, the influences of the basalt fiber’s length and content on the fundamental mechanical properties of concrete were investigated by multi-scale simulation. At the mesoscopic scale, a damage constitutive model was developed in accordance with the Mori-Tanaka homogenization theory and progressive damage theory to predict the composited material properties of BFRC. At the macroscopic scale, the obtained material properties of BFRC from mesoscopic were input into the finite element specimen model to simulate the mechanical performance of BFRC. By coupling the mesoscopic material model with the finite element macroscopic model, the effects of basalt fiber on the mechanical performance of BFRC specimen at macroscopic scale can be investigated. The compressive, splitting tensile, and bending performances of BFRC were studied by both experiments and numerical simulation. The predicted results from the proposed multiscale model show a good agreement with the experimental results. It is also found that with the increase of fiber content, the compressive and splitting tensile strengths of concrete increase first followed by decrease while the bending strength keeps increase. For the different lengths of basalt fiber, the BFRCs with 6 mm basalt fiber display a better compressive and splitting tensile performance than BFRCs with 12 mm fiber, whereas the differences in bending strength is slight. Results shows that BFRCs with 2‰ 6 mm basalt fiber can achieve the maximum strength. Moreover, the size effect on BFRC’s basic properties of BFRC with 2‰ 6 mm basalt fiber was further explored by regression. Those regression formula has very high R2 values. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Due to the brittle fracture characteristics of cement concrete, there is a potential fracture risk in engineering construction under the effect of load and environment [1–3]. Especially, in the huge dam structure. Due to the influence of load and environment is ⇑ Corresponding author. E-mail address: [email protected] (P. Cao). https://doi.org/10.1016/j.conbuildmat.2019.01.018 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

more complicated, the plain concrete with poor toughness and brittleness cannot satisfy the requirement of mass concrete structure for better durability and safety [4,5]. In recent years, due to harsh natural conditions such as arid and cold, large temperature difference, high radiation and salt, fracturing phenomenon in concrete have arisen on many large hydraulic structures on the Qinghai Tibet Plateau [6]. It should be notable that this kind of issue is being gradually solved with the development and application of high strength concrete materials in the past thirty years. By

X. Sun et al. / Construction and Building Materials 202 (2019) 58–72

introducing toughening materials such as fiber, the compressive strength, shear and fracture resistance of fiber reinforced concrete can be significantly improved, being widely introduced by many papers [7–9]. Meanwhile, fiber reinforced concrete materials have been widely fabricated in hydraulic and civil buildings around the world in past decades [10,11]. In the research and application of fiber reinforced concrete materials, carbon fiber is the most common reinforced concrete material which has many advantages such as lightweight, high strength, good stability, excellent combination with cement and so on. However, due to its high cost, the application of carbon fiber reinforced concrete has been greatly limited since more and cheaper alternative fibers were used [12,13]. On the other hand, in the field of civil engineering, due to high strength and good cement combination performance, steel fiber is commonly used which can improve the fracture resistance of concrete significantly [14–17]. By the mean of numerical calculation and experimental study, the toughening mechanism of steel fiber for concrete under dynamic and static conditions has been studied by Su. It is suggested that a proper amount of steel fiber can increase the fracture resistance of concrete by several times [18]. Nieuwoudt has found that the strength of the interface between the steel fiber and cement matrix is at least equivalence to that of the concrete, indicating that the steel fiber concrete material is more suitable for resistant fracture behavior [19]. However, as a hydraulic material, the concrete will be exposed to water perennial. For steel fiber reinforced concrete materials, water convection and diffusion happens and eventually, the steel fiber is eroded. In terms of this reason, the application of steel fiber reinforced concrete in hydraulic structures is strictly limited [20,21]. Polyethylene fiber (short for PE fiber) reinforced concrete is also known as a common high toughness concrete material [22]. By considering double-K fracture criterion, the fracture energy of polythene reinforced concrete was studied by Cao. It has been found that the increase of PE fiber content has insignificant influence on fracture-initiation toughness, whereas it can improve the unstable fracture toughness obviously [23,24]. According to the research published by Bilodeau and Zhu [25,26], with a certain amount of PE fiber, the fire resistance and freeze-thaw resistance of polyethylene fiber reinforced concrete material can be greatly improved and prompted. However, the industrial polyethylene fiber materials appear in cluster form usually which will result in difficulties in mixing procedure. Due to the wide use of cement and fiber in hydraulic concrete, if the mixing effect of polyethylene fiber and cement cannot be guaranteed, there will be a great potential risk on the safety and integrity of the structure [27]. As a substitute fiber, regarding on its excellent performance, basalt fiber has drawn great attention world-widely. With the industrialization of basalt fiber, it has been widely applied in aviation, aerospace and automobile fields. In civil engineering and hydraulic engineering, basalt fiber is often cut into short pieces and mixed with concrete material enhance the fracture strength of concrete [28,29]. Compared with polythene fiber reinforced concrete, reinforced concrete with basalt fiber is not only conducive to cost saving [30], but also has better performance on binding strength with cement due to its chemical property is more similar with cement [31,32]. Whereas the polyethylene fiber needs to be modified from both Chemical and physical property to enhance the binding strength of the interface between PE fiber and cement [33,34]. In the use of basalt fiber, it is usually weaved into specific form [35] and the knitted basalt fabric has good performance on shock resistant and fracture toughness [36,37]. However, it should be noticed that the knitted basalt fiber cannot be widely used in the massive concrete structure. On the other hand, when using long basalt fiber as enhanced phase, the concrete material has more capability on fracture resistant and it has been used in the field of civil construction and hydraulic construction [38].

59

Although it has been proved that the mechanical properties of concrete can be greatly improved by adding fibers, the research on the mechanical model of fiber reinforced concrete is relatively backward. In the available model related studies, the fracture model of fiber reinforced concrete was mentioned firstly [39]. For instance, in the paper published by Zhang Jun [40], the increase of fracture strength factor was used to present the increase of fracture strength caused by fiber. On the other hand, based on the Hillerberg cohesive method [41], some studies were focused on the fracture strength of fiber reinforced concrete by simulation and analyzing. According to the research published by Enfedaque, a fracture toughness cohesive force model of glass fiber reinforced concrete was predicted with the use of ABAQUS [42]. In addition, because the robustness and applicability of concrete can be verified by the elastoplastic damage model [43–45], in consequence, this model with various kinds of fiber content was applied by some scholars to study the fracture and damage behavior of fiber reinforced concrete [46]. Based on the research conducted by Chira and Kumar, the damage behavior of concrete with different fiber contents was simulated by using the elastoplastic damage model of concrete, it had been found that the numerical simulation had good agreement with the experimental results [47]. With the development of computer technology, more concrete models from mesoscopic have been initiated to study the fracture and stiffness degradation behavior of fiber reinforced concrete under the ultimate load [48–50]. For example, in the paper presented by Xu, a mesoscopic concrete model was established. In which, twodimensional finite element method was applied to simulate the aggregate distribution and the long and narrow unit was recognized as fiber. Based on this simulation, the failure behavior of fiber reinforced concrete under indirect tension and impact compression were simulated successfully [51,52]. By employing the beam element, the steel fiber in the steel fiber reinforced material was simulated and a finite element model of three-dimensional steel fiber reinforced concrete was established by Liang and Wu. In this model, the fiber was distributed randomly and a dense of network structure was formed. Meantime, the concrete was treated as an equivalent isotropic medium and the embedded contact was simulated for the fiber and concrete. Through which, the secondary hardening phenomenon of steel fiber reinforced concrete under the ultimate bearing capacity was explained rationally [53]. On the other hand, by applying CT scanning and three-dimensional finite element method, the anisotropic behavior of fiber reinforced concrete was predicted and analyzed by Qsymah [54]. From the mesoscopic point of view, the aggregate and fiber distribution should be considered in the process of finite element modeling. Due to huge amount of calculation should be taken, this kind of model is not suggested for large-scale structure simulation and analyzing. Therefore, it is urgent to develop a more competitive computing method in terms of accuracy and efficiency. At present, the research of basalt fiber reinforced concrete (BFRC) in water conservancy is just in its start-up step. Especially, for the models of short-cut basalt fiber-reinforced concrete, there is no comprehensive experimental study and simulation yet. To fill this gap, based on experimental and finite element methods (FEM), this study aims to investigate the mechanical properties of BFRC by multi-scale coupling analyses.

2. Research objective and scope The purpose of this study is to explore the effect of basalt fiber on the fundamental mechanical properties of concrete based on experimental investigation and finite element modeling. The novelty of this study is that the multi-scale modeling method have been employed to simulate the compressive, indirect tensile and

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bending strength of BFRC with different fiber contents. In mesoscale, the BFRC is regarded as a composite of plain concrete with basalt fiber in which the basalt fiber is distributed to the matrix of plain concrete. This composite is looked as a new equivalent medium and its material properties is calculated by Mori-Tanada method. Besides, to take the damage behavior and strength under different loading modes, Hashin method is adopted to define the damage strength of matrix, i.e. plain concrete, under tensile, compressive and shear load. By this way, the damage strengths under different load modes will not impact on each other, which is difficult to be realized in conventional elastoplastic models. In the transformation between the meso-scale and macro-scale constitutive models, firstly, the meso-scale properties of concrete are calculated by MT method. Then, an applicable Jacobian matrix is generated by MT method from the calculated results. After that, the generated Jacobian matrix is transferred to the UMAT subroutine of ABAQUS to analyze the mechanical performance of concrete in macro-scale. Through this process, the multiscale modeling analysis can be realized by coupling the meso-scale properties with macro-scale performances of basalt fiber reinforced concrete. Using this method, the compressive, splitting and bending strengths of BFRC can be analyzed by coupling fiber contents with fiber lengths. Based on that, the change trends of compressive, splitting and bending strengths of BFRC with the fiber content and the fiber length can be derived and the optimum combination of fiber content and fiber length in the engineering application can be determined. In the end, the size effect of aggregate, as well as the influence of aggregate content, on mechanical properties of BFRC, were investigated by finite element method based on energy releasing theory. Additionally, the regression formula were proposed for the size effect of BFRC with determined basalt content on mechanical properties.

the particle gradation was shown in Table 2. On its saturated surface, the dry apparent density of gravel stone as well as middle stone were 2680 kg/m3 and 2660 kg/m3, respectively. In addition, water reducing agent (water reduction rate of 17% and gas content of 1.3%) and Air entraining agent (water reduction rate of 6% and gas content of 5.2%) were included as well. The basalt fibers with 6 mm and 12 mm in length [55], manufactured by Shanghai Chen Qi Chemical Technology Co., Ltd, were selected in this research as shown in Fig. 1 and its performance indexes were presented in Table 3. 3.2. Concrete mix proportion The strength grade C25, the waterproofing grade W8 and the frost resistance grade F300, named C25W8F300, were applied as the control group for the benchmark concrete. The mix proportion of C25W8F300 is listed in Table 4. The water reducer and air entraining agent was added by the volume of cementitious materials, i.e., 1% and 0.04%, respectively. The 15–30 min slump was measured as 90– 110 mm while the density of fresh concrete is 2380 kg/m3. 3.3. Experiments During the preparation of basalt fiber reinforced concrete specimens, the dry mixing of fiber method was proposed to improve the dispersion. According to literatures [56,57], 0‰, 1‰, 2‰, 3‰, 4‰

3. Raw material and experimental method 3.1. Raw material P.O 42.5 ordinary Portland cement was selected and its performance indexes were shown in Table 1. The involved grade 2 fly ashes were produced by Yongdeng Lian Electric Fly Ash Co., Ltd. The fine and coarse aggregates were collected from the natural aggregate and crushed gravel on site. Through tests, for fine aggregate, its fineness modulus was 3, and the apparent density was 2680 kg/m3. On the other hand, for the second gradation coarse aggregate, its particle size in the range from 5 mm to 40 mm and

Fig. 1. Short chopped basalt fibers.

Table 1 Measured physical and mechanical properties of cement.

1 Code value

Cement kind

Strength rank

P.O P.O

42.5 42.5

Bending strength

Compressive strength

3d

28 d

3d

28 d

5.2 3.5

7.8 6.5

20.1 17.0

45.2 42.5

Setting time

2.35 45 min

4.15 10 h

Standard consistency

Stability

29 –

1.5 5.0

Table 2 The coarse aggregate grain composition. Small aggregate

Medium aggregate

Size (mm)

Percentage Content (%)

Cumulative percentage (%)

Size (mm)

Percentage content (%)

Cumulative percentage (%)

<4.75 4.75–9.5 9.5–16.0 16.0–19.0 >19.0

2.0 32.8 55.5 6.6 3.2

2.0 34.8 90.2 96.8 100.0

<19.0 19.0–26.0 26.0–31.5 31.5–37.5 >37.5

2.8 29.3 44.4 23.5 0

2.8 32.1 76.5 100.0 100.0

61

X. Sun et al. / Construction and Building Materials 202 (2019) 58–72 Table 3 The basalt fiber performance parameters. Diameter (lm)

Length (mm)

Tensile strength (MPa)

Elasticity modulus (GPa)

Fracture elongation (%)

Density (kg/m3)

17.4

6 (12)

2000

85

2.5

2699

Table 4 Mix proportion of the control group for the benchmark concrete (kg). Water

Cement

Fly ash

Sand

Small aggregate

Medium aggregate

Water reducer

Air entraining agent

130

217

54

694

708

708

2.71

0.108

and 5‰ of each type of basalt fiber in volume of fresh concrete were added. Therefore, there are totally 11 types of concrete were prepared and tested. For each test, 3 duplicated specimens were prepared and measured. 150 mm cubic specimens prepared for compression and splitting tests while 100 mm  100 mm  400 mm rectangle specimens were prepared for three-point bending strength test. The total specimens were cured in standard curing room for 28 days before testing. 4. Results and analyses on BFRC mechanical properties

was no obvious change. However, once the maximum failure load was reached, a sudden rupture happened as shown in Fig. 2c which shows obvious brittle. 4.2. Experiment results 4.2.1. Experiment result calculation According to elastic theory, the compressive strength of concrete cube was calculated regarding on formula (1):

f cc ¼

4.1. BFRC failure mode The failure modes of BFRC with different fiber volumes and lengths were similar to that of ordinary concrete which were presented in Fig. 2. During the compression experiment, the surface and corner of specimen cube began to bulge and peel off. With the increase of stress load, cracks appeared gradually and extended throughout the whole specimen finally as shown in Fig. 2a. In the splitting test, a micro crack appeared in the middle of the specimen. After releasing the load, with slightly knock, the test specimen was broken from the middle as shown in Fig. 2b. The failure section was mainly focused on the binding intersection between coarse aggregate and cement base. Occasionally, the coarse aggregate might be split as well. For bending test, during the load, there

F A

ð1Þ

In Eq. (1), f cc stands for compressive strength (MPa), Fis the maximum load when specimen failed (N) and A is the loading area (mm2). The calculated result presented in Table 5. The splitting tensile strength of concrete cube was calculated regarding on formula (2):

f ts ¼

2F

pA

¼ 0:637

F A

ð2Þ

In Eq. (2), f ts stands for splitting tensile strength (MPa), F is the maximum load when specimen failed (N) and A is the area of splitting interface (mm2). The calculated result presented in Table 5. The bending strength of concrete cube was calculated regarding on formula (3):

Fig. 2. Concrete failure mode.

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Table 5 28-day concrete compression, splitting tensile and bending strength test results. Specimen

Compression strength (MPa)

Standard deviation

Splitting tensile strength (MPa)

Standard deviation

Bending strength (MPa)

Standard deviation

PC B-6-0.1 B-6-0.2 B-6-0.3 B-6-0.4 B-6-0.5 B-12-0.1 B-12-0.2 B-12-0.3 B-12-0.4 B-12-0.5

29.9 31.4 33.4 37.2 35.9 33.9 30.5 32.4 33.5 26.8 25.5

0.258 0.852 1.875 1.462 1.042 0.726 0.497 2.139 1.289 0.589 0.497

2.29 2.36 2.38 2.50 2.51 2.40 2.28 2.27 2.26 2.28 2.20

0.036 0.121 0.092 0.108 0.051 0.045 0.096 0.057 0.036 0.057 0.057

3.83 4.21 4.02 3.89 4.32 4.91 4.00 4.10 3.95 4.78 4.94

0.036 0.057 0.110 0.093 0.042 0.094 0.083 0.059 0.139 0.064 0.054

ff ¼

3Fl 2

2bh

ð3Þ

In Eq. (3), f f stands for bending strength (MPa), F is the maximum load when specimen failed (N), b represents the section width of specimen (mm), h is the section height of specimen (mm) and l stands for the distance between supports which equals to 3 times of section height (mm). The calculated result presented in Table 5 as well. The compressive, splitting and bending test results and their standard deviations were all listed in Table 5. The serial number PC means the ordinary concrete specimen and the serial number of basalt fiber reinforced concrete specimen were named in the form of B-x-y in which B stands for basalt fiber reinforced concrete, x represents for the length of fiber and y means fiber volume ratio. For instance, B-6-0.1 means the specimen was made of basalt fiber with 6 mm in length and the total fiber volume was 1‰ of the total specimen. 4.2.2. Compressive strength The compressive strength of the standard cube is an important indicator for dividing the strength grade of concrete. From Table 5, no matter the length of fiber was 6 mm or 12 mm, with the fiber volume increasing, the compressive strength increased first then decreased. On the other hand, the compressive strength of mixed concrete with 6 mm basalt fiber was higher than that with 12 mm. In addition, it can be seen that the maximum compressive strength was reached in the test specimen containing 2.5‰–3‰ of basalt fiber in volume. By analyzing the results listed in Table 5, the influence of fiber length and content on concrete was suggested as that: due to the excellent mechanical properties and good compatibility with cement base, basalt fiber reinforced concrete with 6 mm fiber had higher compressive strength over that of ordinary concrete. Secondly, because of uneven mixing and fibrous mass phenomenon, when test specimen containing basalt fiber with 12 mm in length or with relatively high volume, the compressive strength was relatively low. 4.2.3. Splitting tensile strength Splitting tensile strength was also measured which reflect the tensile strength of concrete indirectly. According to Table 5, for specimens with 6 mm basalt fiber, the tensile strength increased first then decreased with the increasing of fiber volume and its splitting tensile strength was greater than that of ordinary concrete. For specimen with 12 mm basalt fiber, the splitting tensile strength decreased gradually with fiber volume increasing. By means of analysis and calculation, it has been found that the splitting tensile strength was proportional to the compressive strength and the tension-compression ratio was in the range of 0.062– 0.086.

4.2.4. Bending strength To a certain extent, the bending strength can reflect the toughness of concrete. Regarding on Table 5, it can be noted that no matter what the fiber lengths are, the bending strength of BFRC increase with the incorporation of fibers and reaches the maximum value at a 5‰ content. The slight decrease found at 3‰ content may be due to the dispersion problem, i.e., the fibers may be entangled together for some specimens. 5. Numerical simulation on BFRC mechanical properties In order to theoretically analyze the mechanical properties of BFRC under the mechanism of fracture, a model for predicting the mechanical properties of basalt fiber reinforced concrete was established based on multiscale theory. From mesoscopic level, the fiber reinforced concrete material is equivalent to plain concrete and basalt fiber, of which the plain concrete is the matrix and basalt fiber is the inclusion. Moreover, from more mesoscopic level, the plain concrete is equivalent to cement mortar, aggregate and interface, of which cement mortar is the matrix phase and the aggregate wrapped by the interface is the inclusion phase. By applying the Mori-Tanaka homogenization method, the mechanical properties of each equivalent medium under different volume composition can be predicted. 5.1. Mean-field homogenization (MFH) for Two-Phase composites 

In the multiscale method, for every macro point x, the macro 



strain e is given and the macro stress r can be derived by solving boundary value problem (BVP), and vice versa. In mesoscale level, the macro point is regarded as the center of the representative vol

ume element (RVE), i.e. x 2 x, and the boundary condition is @ x. This multiscale method is illustrated in Fig. 3. As the energy equivalence in both scales, and therefore the strain-stress relationship in

Fig. 3. Schematic diagram of the multiscale method.

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X. Sun et al. / Construction and Building Materials 202 (2019) 58–72

macroscale can be transferred to the relationship of average strain 



hei and average stress hri, namely e ¼ hei and r ¼ hri. The definition of stress mean field f on RVE is:

hf i ¼

1

Z

v

x

  f x; x dv

ð4Þ 

In the equation, f is the internal stress/strain field, x is the macro point, x is the meso point. The definition of mean stress field on both matrix and inclusion phases is:

hf ixr ¼

1

vr

Z

xr

  f x; x dv r ;

r ¼ 0; 1

ð5Þ

Where 0 and 1 represent for matrix and inclusion, respectively. Then Eq. (6) stands for the relation of average strains of RVE, the matrix, and the inclusion, while Eq. (7) stands for the relation of average stress.

heix ¼ v 0 heix0 þ v 1 heix1

ð6Þ

hrix ¼ v 0 hrix0 þ v 1 hrix1

ð7Þ

where v 0 and v 1 are the volume of matrix and inclusion, respectively, v0 þ v1 ¼ 1. When a linear displacement occurred on the RVE’s boundary, the average strain of each inclusion can be defined by the strain concentration tensor, Be :

heix1 ¼ Be : heix0

ð8Þ

Then there exists a relation between heix1 , heix0 , and hei: 1 heix1 ¼ Be : ½v 1 Be þ ð1  v 1 ÞI : hei

ð9Þ

where I represents the symmetric equivalent tensor, changing with Be . Since MFH is based on Eshelby tensor, according to Eshelby solution, the strain inside inclusion is evenly distributed and is related to the remote strain.

eðxÞ ¼ He ðI; C 0 ; C 1 Þ : E; 8x 2 I

ð10Þ

e

where H represents the strain concentration tensor in a single inclusion, which is defined 其中, as:

    He ðI; C 0 ; C 1 Þ ¼ ½I þ nðI; C 0 Þ : C 1 : C 1  C 1  0 0

ð11Þ

where nðI; C 0 Þ is Eshelby tensor, related to the shape and properties of material. Then the macro equivalent stiffness tensor of concrete can be obtained: 

C¼

h





v 1 C 1 : Be þ ð1  v 1 ÞC 0

i  1 : v 1 Be þ ð1  v 1 ÞI

ð12Þ

And the relationship between stress and strain in macro scale is: 





r ¼ C:e

Fig. 4. Schematic diagram for double inclusion model.

model, the fiber of stiffness C1 is surrounded with the plain concrete of stiffness C0 in its close surroundings while the outside those areas are assumed as an unknown reference medium of stiffness Cr. By applying the Mori-Tanaka homogenization method with multi-level inclusions in DIGIMAT software, the constitutive model of BFRC was predicted. Firstly, the elastic mechanical parameters of the plain concrete were determined. The elastic mechanical properties of each phase were shown in Table 6. The elastic moduli and Poisson’s ratios of cement mortar, aggregate, and interface were cited from Stock et al. [58] while the densities of aggregate and cement mortar were obtained in laboratory. It is noted that the density of interface was assumed to be the same as cement mortar. Therefore, the basic elastic parameters of the plain concrete can be calculated by Mori-Tanaka homogenization method, which is shown in Table 7. Secondly, combining Tables 7 and 3, the elastic parameters of plain concrete and fiber can be utilized in the Mori-Tanaka homogenization method to predict the mechanical parameters of BFRC.

5.2.2. Progressive damage model For unidirectional fiber reinforced composites, the threedimensional Hashin failure criterion is widely applied [59]. In order to describe the strength of fiber concrete, the three-

Table 6 Elastic mechanical parameters of plain concrete in numerical simulation.

ð13Þ

For a two-phase composite with moderate fiber volume, this method is quite accurate. All the phases in mesoscale are assumed ideally homogenous and isotropic for BFRC.

Cement mortar Aggregate Interface

Elasticity modulus (GPa)

Poisson’s ratio

Density (kg/m3)

13.4 74.5 5.36

0.25 0.15 0.30

2000 2650 2000

5.2. Mesoscale model for BFRC 5.2.1. Double inclusion model for BFRC In this study, the BFRC is regard as a composite that basalt fiber embedded into the plain concrete. The double inclusion model is applied to simulate the plain concrete and basalt fiber composited matrix and the schematic diagram is presented in Fig. 4. In this

Table 7 Elastic mechanical parameters of plain concrete calculated from Mori-Tanaka homogenization method. Elasticity modulus (GPa)

Poisson’s ratio

Density (kg/m3)

23.76

0.22

2260

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X. Sun et al. / Construction and Building Materials 202 (2019) 58–72

dimensional Hashin failure criterion, failure indicator and exponential damage evolution methods were applied to simulate the damage process of BFRC. (1) Three Dimensional Hashin Failure Criterion and Failure Indicator Generally, the three-dimensional Hashin criterion can be applied to simulate the failure behavior of fiber composite materials. With the purpose to ensure the integrity of materials in different failure modes, four failure indicators were introduced: fiber tensile failure indicator F 1 , fiber compression failure indicator F 2 , matrix tensile failure indicator F 3 and matrix compression failure indicator F 4 . The value of each failure indicators can be determined   by the failure function F rf ¼ 1 and if any failure indicator value was equal or greater than 1, failure on composite material happened then. In Table 8, direction 1 was defined as the direction of fiber by all failure criteria. The direction perpendicular to the fiber in the plane was direction 2 treated as the matrix direction and the normal direction of the plane was direction 3. X t and X c are the tensile strength and compressive strength in fiber direction of composite materials; Y t and Y c are the tensile strength and compressive strength in matrix direction of composite materials, respectively; S and SI are the shear strength between composites. The detailed strength properties of BFRC were shown in Table 9.

Table 10 Damage variable value of MLT model. Damage mode

Damage variable value

Fiber damage Matrix damage Transverse shear damage of Composite materials Transverse shear damage of Composite materials

D1 D2 D3 D4

damage evolution function, each damage variables were determined. In damage evolution Eqs. (14) and (15), a and b are material response parameters separately. In this paper, the value of a was set to 1 and the constitutive model can be obtained by changing b which can result in stress-strain curve softening following Weibull distribution. The value of b can be estimated from the position with the maximum stress. When the value of b tended to be infinity, the exponential damage evolution model was turned into instantaneous damage mode. In this paper, from the trail calculation, we found when b was set to 0.5, the numerical results matched well with experimental results. Therefore, the value of b was set to 0.5 [60]. Fig. 5 shows the relationship between the failure indicator f and the damage variable D during the damage evolution process when the response parameter of the material was changed.

D ¼ /ðf Þ; 0  D  1

(2) Damage Evolution

(

In accordance with the Matzenmiller-Lubliner-Taylor (MLT) model, the damage evolution in this paper was finalized by assuming each phase material of concrete subjecting to the Weibull distribution in meso-scopic level. In fact, the damage evolution is the strain energy releasing process. Thus, the material is believed to be softened. In other words, it is the degradation of elastic modulus and decreasing of bearing capacity from the macroscopic point. The mentioned soften form in this paper was expressed in exponential form. In the period of damage evolution, the damage variable was treated as the key issue due to it could reflect the damage mechanism of materials directly. The damage variable D in this research was expressed by the damage evolution function /ðf Þ which can explain the relationship between failure indicator f and damage variable D more accurately. As Table 10 shows, using

Table 8 Three-dimensional Hashin failure criterion. Failure mode Fiber tensile failure

¼ /ðmaxff 1 ; f 2 gÞ ¼ /ðmaxff 3 ; f 4 gÞ ¼ 1  ð1  D1 Þð1  D2 Þ ¼ D3

uðf Þ ¼

ð14Þ f  f min

0 Dmax  ð1 

f ab f ab expð eb min ÞÞ

f  f min

ð15Þ

Finally, according to the definition of failure criterion, damage variables and the law of damage evolution, the progressive damage model of BFRC in the mesoscopic was obtained. Moreover, the damage model was used as material property and imported into ABAQUS to simulate the basic mechanical properties of BFRC. 6. Simulation and verification 6.1. Comparison on experimental and numerical results of mechanical properties of BFRC 6.1.1. Compressive, splitting, and bending strengths of BFRC The geometric size of the simulated model was the same as the experimental size as shown in Fig. 4. After doing concrete model mesh, the hexahedral mesh was generated. After calculation, the

Failure criterion

r11  0

r11 < 0 Matrix tensile failure r22 þ r33  0 Fiber compression failure

F 1 ðrÞ ¼

r211

F 2 ðrÞ ¼

r11

X 2t

þ

r212 þr313 S2

Xc

r33 Þ þ F 3 ðrÞ ¼ ðr22 þ Y2 2

t

Matrix compression failure r22 þ r33 < 0

r212 þr213 S2

þ

r2 þr2

F 4 ðrÞ ¼ ðr22 þr2 33 Þ þ 12 2 13 þ 4S S  I   2 r22 þr33 Yc þ 2S  1 Yc I 2

r223 r22 r33 S2I

r223 r22 r33 S2I

Table 9 Strength properties of basalt fiber reinforced concrete. Strength parameter (direction)

Symbol

Strength (MPa)

Transverse tensile (1) Transverse compression (1) Longitudinal tensile (2,3) Longitudinal compression (2.3) Shear (1–2,1–3) Shear (2–3)

Xt Xc Yt Yc S SI

1.49 13.70 1.49 13.70 3.39 3.39

Fig. 5. The relationship between the failure indicator f and the damage variable D during the damage evolution process.

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model presented in Fig. 4a had 4735 units and 5524 nodes, while the model in Fig. 4b had 3855 units and 4600 nodes. For the model presented in Fig. 4c, 5650 units and 6677 nodes were contained. Within the finite boundary model, the bottom surface was defined as one of the displacement boundaries with a fixed displacement limiting the horizontal and vertical displacement while the upper surface was defined as the other displacement boundary which is a free boundary. By applying displacement load, the counterforce at support was obtained. The strength parameters of the model then were calculated by the input of displacement and counterforce at support. The compression, splitting and bending experiments and the corresponding numerical models of BFRCs are shown in Fig. 6. The numerical models analyzed by finite element method reflected the strain distribution change during the stress loading. The result of each test and numerical simulation of the compression, splitting and bending are drawn in Figs. 7–9 respectively. It can be seen that the numerical simulation result had a good agreement with the experimental data, indicating that the proposed multi-scale damage model is useful for simulating the strength of BFRC. In addition, through a comprehensive evaluation of numerical simulation and experimental results, when the length of fiber was 6 mm and the volume was 2‰, the finite element calculation results of compressive strength, splitting tensile strength and flexural strength were very close to the experimental results.

6.1.2. P-CMOD of BFRC in three-point bending test on notched beams In this study, the load-crack mouth opening displacement, i.e., P-CMOD, in the three-point bending test on notched beams, as shown in Fig. 10, was utilized to describe the deformation characteristics of BFRC. The BFRC with 2‰ 6 mm basalt fiber were chosen as the test concrete. Three groups of notched beams were tested as shown in Table 11 and for each group three duplicated specimens were prepared. The proposed finite element model was used to simulate this kind of test. The numerical and experimental test data were compared in Fig. 11. It can be seen that the numerical results are very close to the experimental ones, indicating the effectiveness of the proposed method.

65

Fig. 7. Comparisons of compression experiment and numerical simulation of BFRC.

Fig. 8. Comparisons of splitting tensile experiment and numerical simulation of BFRC.

Fig. 6. Compression, splitting tensile and bending experiment and numerical model of BFRC.

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Fig. 9. Comparisons of bending experiment and numerical simulation of BFRC.

Fig. 10. The sketch of three-point bending test on notched beams.

Table 11 Geometry information of specimens for three-point bending test on notched beams. Group

Geometry (l  h  t) ðmm  mm  mmÞ

a0 ðmmÞ

S ðmmÞ

Ⅰ Ⅱ III

480  180  60 640  240  80 800  300  100

36 48 60

400 540 660

6.2. The influence of the aggregate volume in determining the strength of BFRC Furthermore, the volume contents of the aggregates in the basalt fiber reinforced concrete material were investigated and discussed by means of the multiscale simulation method. From the aforementioned analysis, it was identified that the multi-scale damage model proposed in this paper can accurately predict the tensile, compression and bending strengths of BFRC under ultimate strength. However, the effect of aggregate content on the strength of BFRC also should be concerned. Therefore, the mechanical performances of BFRCs at different aggregate content were also investigated based on the constituted model. Furthermore, the impacts of aggregate content, fiber content and the length of fiber on compressive, splitting and bending strengths were studied from mesoscopic. In this study, the mass ratios from 0.3 to 0.7 with a gap of 0.1 were selected and investigated. The compression, splitting and bending strengths tests were simulated on the built multi-scale damage model. The simulating results were illustrated in Fig. 12. Fig. 12(a) and (b) displayed the impacts of different aggregate content on the compressive strength which indicated that the compression strength of BFRC increases with the aggregate content increasing. The compressive strength of BFRC with 0.7 aggregate

Fig. 11. Numerical results versus experimental results on P-CMOD curves of threepoint bending tests on notched BFRC beams.

content is 1.03 times 0.3 aggregate content’s. For the BFRCs incorporated with 6 mm fiber, all the compressive strengths of BFRCs were higher than ordinary concrete. However, for the BFRCs introduced with 12 mm fiber, the compressive strengths increase first and then drop and reach the maximum value with a fiber content of 3‰. Fig. 13(c) and (d) displayed the splitting strengths of 6 mm and 12 mm length of fibers’ BFRCs with different aggregate content, respectively. The splitting strengths increase with aggregate content increasing and the splitting strength of 0.7 aggregate content is 1.06 times larger than 0.3 aggregate content’s. In comparison, the splitting strengths of BFRC with 6 mm fiber were superior to BFRC with 12 mm fiber. The bending strengths of BFRCs with different aggregate content were shown in Fig. 13(e) and (f). From the simulated results, the bending strengths increased with an increasing aggregate content for both BFRCs with 6 mm and 12 mm fibers. Moreover, the 0.7 aggregate content’s bending strength is 1.07 times of the 0.3 aggregate content’s BFRCs. The predicted results of the impacts of varied aggregate content on the BFRCs’ mechanical performances from numerical simulation are meaningful for the durability of concrete and the optimal components of concrete’s raw materials. So as to quantitatively evaluate the effects of two variables, i.e. aggregate content, fiber content and the fiber’s size, on the mechanical performances of BFRCs, regression analysis was conducted based on simulated mechanical performances’ results. Since the regression analysis is related to two variables, multiple regression analysis was adopted and conducted using STATA software. The regression formulas for compressive, splitting and bending strengths were shown below. From Eqs. (16)–(21), the results showed that all the regression equation can fit well with the simulation values. The compressive strength of BFRC with 6 mm fiber:

f ¼ 586; 571a2 þ 4027:18a þ 1:826b þ 27:81

ð16Þ

R2 ¼ 0:92 The compressive strength of BFRC with 12 mm fiber:

f ¼ 1; 733; 024a2 þ 9626:66a þ 39:45b

ð17Þ

R2 ¼ 0:96 Splitting strength of BFRC with 6 mm fiber:

f ¼ 21; 121a2 þ 137:31a þ 0:329b þ 2:07 R2 ¼ 0:93

ð18Þ

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Fig. 12. Impacts of aggregate content on BFRC’s mechanical properties.

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Fig. 13. The strength size effect of BFRC.

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Splitting strength of BFRC with 12 mm fiber:

f ¼ 6485a þ 24:98a þ 0:324b þ 2:08 2

ð19Þ

R2 ¼ 0:97 Bending strength of BFRC with 6 mm fiber:

f ¼ 30; 888a2 þ 2:93a þ 0:64b þ 3:55

ð20Þ

R2 ¼ 0:91 Bending strength of BFRC with 12 mm fiber:

f ¼ 32; 520a2 þ 46:67a þ 0:637b þ 3:49

ð21Þ

R2 ¼ 0:97 f stands for compressive, splitting and bending strengths. a is the content of fiber, whose values are 0‰, 1‰, 2‰, 3‰, 4‰ and 5‰; b is the aggregate content whose value are 0.3, 0.4, 0.5, 0.6 and 0.7. From Eqs. (16)–(21), it can be noted that the regression prediction models agree well with the simulated results for the compressive, splitting and bending strengths of BFRC, which indicates that these efficient regression models have a great significance for the application of BFRC in large hydraulic structures. 6.3. Numerical analysis of the size effect on BFRC Based on the achieved experimental data and calculated value, the multi-scale damage model proposed in this paper can predict the strength of fiber reinforced concrete accurately. Because of the structural characteristics of the concrete itself, the failure process and mechanism are quite complicated, which causes the size effect of concrete strength. In order to verify the feasibility of multi-scale damage model on predicting the strength of fiber reinforced concrete with various sizes, the finite element compression, splitting and bending models of basalt fiber reinforced concrete with different size were further established. The volumes of geometrical models were scaled to 2, 4, 6 and 8 times of the original model as shown Table 12. Through the finite element analysis, the compressive strength, splitting tensile strength and bending strength of basal fiber concrete, considering the size effect, were presented in Fig. 13. According to Fig. 13, the obvious feature of size effect had been reflected by the proposed model. From Fig. 8.a and b, when the size of the specimen was smaller than 238 mm, the compressive strength decreased relatively fast with the size of specimen scaling up. For instance, the compressive strength of plain concrete cube specimen with a length of 189 mm was decreased by 0.6 MPa over a specimen with 150 mm in length. Once the length of the specimen was larger than 238 mm, the compressive strength change became less significant with the volume increasing. For example, the compressive strength of plain concrete cube specimen with a length of 300 mm was only decreased by 0.13 MPa over a specimen with 273 mm in length. In terms of this fact, it is believed that the compressive strength of BFRC has a similar size effect change of the

plain concrete. In Fig. 8c and d, the size effect on splitting tensile strength of basalt fiber reinforced concrete was described. It can be seen that although the splitting tensile strength decreased with the increase of specimen size, the tensile strength decreasing trend became steady with the increase of specimen size decreasing. Moreover, the splitting tensile strength size effect of specimens with different basalt fiber content and fiber length was basically in the same trend. Fig. 8e and f show the bending strength size effect of BFRC, compared with the standard specimen size (100 mm  100 mm  400 mm), the bending strength of nonstandard sized specimen was significantly lower. It can be seen that the larger specimen size the smaller bending strength and the size effect on bending strength became less significant. Firstly, this phenomenon can be explained by the size effect of the concrete itself. On the other hand, because the length and the amount of basalt fiber were relatively small, the improvement of bending strength on large scaled specimen was not significant. In order to describe the strength size effect of BFRC quantitatively, the basic mechanic property size effect of BFRC was analyzed in accordance with the size effect based on energy release theory proposed by Bazant [61]. For specimen with similar geometric, its size effect law of the structural nominal strength was written as follows:

Bf

t rN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð22Þ

1 þ D=D0

In equation,rN is the structural nominal strength, B is the nondimension constant, f t is the strength of the concrete structure, D0 is the geometric constant and D is the characteristic size of the structure. This equation reveals the relationship between structural nominal strength rN and characteristic size of structure D. Through experiment, the value of structural nominal strength, characteristic size of structure and strength of concrete structure were obtained. So the law of size effect can be derived once the value of nondimensional constant and geometric constant were determined. In order to facilitate the regression analysis of experimental data, Eq. (22) can be turned into a linear equation according to the principle of the least square method.

Y ¼ AX þ C

ð23Þ

In which: X ¼ D; Y ¼ r2 ; C ¼ 1

N

1 ðBf t Þ2

; A ¼ C=D0

In Eq. (23), the value of X and Y are already known, by inputting the strength and size of each specimen, the value of A and C can be derived then. In this paper, with the purpose to reduce the workload of numerical calculation, only the basic mechanical properties of concrete with the optimum fiber content and length was applied to achieve the size effect calculation formula. The corresponding value of A and C were listed in Table 13. In turn, the value geometric constant D0 and the multiplication of the non-dimensional constant B and the strength of concrete structure f t were obtained as shown in Table 14. By substituting the value of D0 and Bf t in Eq. (22), the nominal strength size effects of BFRC specimens (containing 2‰ volume of fiber with 6 mm in length) were derived as follows respectively.

Table 12 Geometric dimension of BFRC specimens. Magnification times

Compression (width  height  length)/mm

Splitting tensile (width  height  length)/mm

Bendingn (width  height  length)/mm

1 2 4 6 8

150  150  150 189  189  189 238  238  238 273  273  273 300  300  300

150  150  150 189  189  189 238  238  238 273  273  273 300  300  300

100  100  400 126  126  504 159  159  636 182  182  728 200  200  800

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Table 13 Calculation formula parameters of BFRC’ size effect with the optimum fiber content and length. A Compression

C

4:28  107

8:24  104

Splitting tensile

3

1:15  10

1:02  102

Bending

6:08  104

5:00  103

Table 14 D0 and of Bf t BFRC’ size effect.

Compression Splitting tensile Bending

7. Conclusions

D0

Bf t

1924.46 8.92 8.22

34.84 9.89 14.14

The specimen in compression test:

34:84

rN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ D=1924:46

ð24Þ

The specimen in splitting test:

9:89

rN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ D=8:92

ð25Þ

The specimen in bending test:

14:14

rN ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ D=8:22

The fitting curve of the size effect disclosed by Eqs. (24)–(26) was described in Fig. 14. According to which, the theory of size effect caused by energy release had a good applicability to analyze the strength size effect of BFRC and the corresponding fitting formula was consistent with the simulation result. The obtained results above show that the influence of basic mechanical properties of concrete with various basalt fiber content and length. In next, the influence on size effect of concrete material with different fiber content will be further studied and investigated.

ð26Þ

In this study, mechanical properties of BFRC were investigated through a series of experiments and multiscale numerical simulation. The following conclusions can be drawn:
According to basic mechanical performance test of basalt fiber reinforced concrete, with the increase of fiber content, the compressive and splitting tensile strength of concrete increased first followed by decreasing. Meanwhile, the bending strength increased with the increasing of fiber volume. According to the compressive and splitting tensile test, it is found that the mechanical performance of concrete with 6 mm fiber was better than that with 12 mm fiber.
By applying Mori-Tanaka homogenization algorithm and continuum progressive damage theory, the damage model of concrete was established to obtain material inputs for numerical

Fig. 14. Fitting curves of the strength size effect of BFRC.

X. Sun et al. / Construction and Building Materials 202 (2019) 58–72

simulation in finite element model. Numerical and experimental results matched with each other very well.
Basalt fiber with a length of 6 mm and 2‰ of volume can mostly improve the mechanical properties of concrete.
A parabolic equation has been proposed for describing and predicting the influence of the aggregate and fiber volume contents. The regression results can fit well with the simulated results.
The regression formula for the size effect of compressive strength, splitting tensile strength and bending strength of the optimum BFRC (6 mm in length and 2‰ volume of concrete) had good agreements with numerical simulation.

Conflict of interest None. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grant No. 51769028, 51508137, 51509139), Natural Science Foundation of Qinghai Province in China (Grant No. 2018-ZJ-750, 2017-ZJ-933Q), and Fundamental Research Funds for the Central Universities (DUT17RC(3)006). Thanks for the associate professor Leng Zhen in the Hong Kong Polytechnic University to provide many valuable discussions. References [1] Y. Ding, S. Liu, The investigation on the workability of fiber cocktail reinforced self-compacting high performance concrete, Constr. Build. Mater. 22 (2008) 1462–1470. [2] H. Higashiyama, N. Banthia, Correlating flexural and shear toughness of lightweight fiber-reinforced concrete, ACI Mater. J. 105 (3) (2008) 251–257. [3] Z. Zhang, C.T. Hsu, Shear Strengthening of Reinforced Concrete Beams Using Carbon-Fiber-Reinforced Polymer Laminates, J. Compos. Constr. 9 (2) (2005) 158–169. [4] P.K. Mehta, P.J.M. Monteiro, Concrete: Microstructure, properties and materials, 3 ed., McGraw-Hill, New York, 2006. [5] Y. Zhang, M. Zhang, Transport properties in unsaturated cement-based materials – a review, Constr. Build. Mater. 72 (2014) 367–379. [6] J.F. Huo, D.P. Liu, Experimental study on frost resistance durability of polypropylene fiber reinforcement lightweight aggregate concrete, Low Temperature Arch. Technol. (2008). [7] S. Yin, R. Tuladhar, F. Shi, M. Combe, T. Collister, Nagaratnam Sivakugan, Use of macro plastic fibers in concrete: a review, Constr. Build. Mater. 93 (2015) 180– 188. [8] D.J. Kim, A.E. Naaman, S. El-Tawil, Comparative flexural behavior of four fiber reinforced cementitious composites, Cem. Concr. Compos. 30 (2008) 917–928. [9] W. Pansuk, T.N. Nguyen, Y. Sato, et al., Shear capacity of high performance fiber reinforced concrete I-beams, Constr. Build. Mater. 157 (2017) 182–193. [10] P.N. Balaguru, S.P. Shah, Fiber Reinforced Cement Composites, McGraw-Hill Inc, New York, 1992. [11] A.M. Brandt, Fiber reinforced cement-based (FRC) composites after over 40 years of development in building and civil engineering, Compos. Struct. 86 (2008) 3–9. [12] P.J. Heffernan, M.A. Erki, Fatigue behavior of reinforced concrete beams strengthened with carbon fiber reinforced plastic laminates, J. Compos. Constr. 8 (2) (2004) 132–140. [13] M.B.S. Alferjanil et al., Use of Carbon Fiber Reinforced Polymer Laminate for strengthening reinforced concrete beams in shear: a review, Int. Refereed J. Eng. Sci. 2 (2) (2013) 45–53. [14] C.X. Qian, P. Stroeven, Development of hybrid polypropylene-steel fiber reinforced concrete, Cem. Concr. Res. 30 (1) (2000) 63–69. [15] F. Altun, T. Haktanir, K. Ari, Effects of steel fiber addition on mechanical properties of concrete and RC beams, Constr. Build. Mater. 21 (3) (2007) 654– 661. [16] P. Song, S. Hwang, Mechanical properties of high-strength steel fiber reinforced concrete, Constr. Build. Mater. 18 (9) (2004) 669–673. [17] B.E. Barragan, R. Gettu, M.A. Martin, R.L. Zerbino, Uniaxial tension test for steel fiber reinforced concrete – a parametric study, Cem. Concr. Compos. 25 (7) (2003) 767–777. [18] Y. Su, J. Li, C. Wu, et al., Mesoscale study of steel fiber-reinforced ultra-high performance concrete under static and dynamic loads, Mater. Des. 116 (2017) 340–351.

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