Mesoscale modelling of concrete reinforced with spiral steel fibres under dynamic splitting tension

Abstract

The addition of discrete steel fibres into concrete has been widely recognised as an effective measure to enhance the ductility, post-cracking resistance and energy absorption of the matrix subjected to impact loads. Despite useful information from experimental studies that investigate the macro-scale performance of steel fibre–reinforced concrete under dynamically applied loadings, results from a series of tests or from tests by different researchers are often found to be scattered. Besides variations in testing conditions, random variations of size, location and orientation of aggregates and fibres in steel fibre–reinforced concrete are deemed the fundamental reason of the scattering test data. High-fidelity modelling of concrete and steel fibre–reinforced concrete in mesoscale has been widely adopted to understand the influence of each component in the composite material. Numerical studies have been published to discuss the behaviour of steel fibre–reinforced concrete under dynamic splitting tension. Different shapes, for example, circles, ovals and polygons, of coarse aggregates were considered in different studies, and different conclusions were drawn. This study investigates the influence of the shape of aggregates on numerical prediction in mesoscale modelling of steel fibre–reinforced concrete materials with spiral fibres under dynamic splitting tension in terms of the strain distribution, cracking pattern and strength. The numerical model is validated by experimental results. It is found that the shape of aggregates in mesoscale modelling of splitting tensile tests has negligible influence. Furthermore, steel fibre–reinforced concrete specimens with different volume fractions of spiral fibres from 0.5% to 3.0% under various loading rates are simulated. Results from parametric simulations indicate the optimal dosage of spiral fibres in steel fibre–reinforced concrete mix with respect to the construction cost and mechanical property control.

Keywords

aggregate shape effect dynamic splitting tension mesoscale numerical simulation spiral fibres steel fibre–reinforced concrete

Introduction

Concrete exhibits very brittle failure and has low tensile strength and crack controllability. To improve the performance of concrete, steel fibres have been used as internal reinforcement since 1960s (Brandt, 2008). The addition of steel fibres has been found to largely improve the properties of concrete in terms of fatigue behaviour, resistance to shrinkage, tensile strength, ductility, energy absorption capability and impact resistance (Abrishami and Mitchell, 1997; Barros et al., 2005; Hao et al., 2016b; Lee and Barr, 2004; Li et al., 1998; Soufeiani et al., 2016; Wafa and Ashour, 1992). Based on the observations reported by previous studies and the characteristics of the interaction of steel fibres of different geometries with concrete matrix, steel fibres with spiral shape were proposed by Xu et al. (2012b). When structures are subjected to highly dynamic loads such as scenarios considering blast or high-speed impact associated with stress wave propagations, the tensile property and response of steel fibre–reinforced concrete (SFRC) material dominates the structural performance. Understanding the dynamic material properties is essential for reliable design and analysis of structures subjected to blast and impact loads. Many impact tests have been conducted to study the dynamic tensile properties of SFRC materials, among which dynamic splitting tests carried out by either the drop weight (Xu et al., 2012a) or split Hopkinson pressure bar (SHPB; Hao and Hao, 2016b) devices are normally adopted due to the ease of implementation and reasonable prediction of tensile strength. Generally, the tensile strength of SFRC materials under dynamic loading is much higher than quasi-static strength. The normalised dynamic strength by the quasi-static counterpart, termed as dynamic increase factor (DIF), is used to quantify the dynamic strength increment. Despite useful information from impact tests, results from a series of tests are often found to be significantly scattered (Wang et al., 2012; Xu et al., 2012a). Besides variations in testing conditions, as SFRC consists of mortar matrix, coarse aggregates and steel fibres with random sizes, locations and orientations, these random variations are deemed the fundamental reason of the scattering test data.

Researchers used to model concrete or SFRC as homogeneous materials in the numerical simulation (Jin et al., 2017; Rong and Sun, 2012). However, as composite materials, concrete and SFRC are mixed with different constituents. The role of each component in the composite material needs to be investigated for a better understanding of the failure mechanism and the mechanical properties of the materials under different loading conditions. Du et al. (2012) proposed a meso-equivalent method for numerical simulations that could model concrete in macro-scale while innovatively incorporate the effects of coarse aggregates and air bubbles. This approach significantly saves the computational efficiency. However, simulating coarse aggregates in particular circumstances, for example, dynamic compression (Hao and Hao, 2011; Hao et al., 2010, 2013; Jin et al., 2016; Song and Lu, 2012; Xu et al., 2012d; Zhou and Hao, 2008b), dynamic direct tension (Hao et al., 2012), dynamic flexure (Xu et al., 2015), dynamic splitting tension (Hao and Hao, 2016a; Xu et al., 2012c; Zhou and Hao, 2008a), spall (Chen et al., 2015) and corrosion (Jin et al., 2015), is still necessary for understanding the mechanism of concrete failure. In addition, Du et al. (2014) numerically simulated the irregularly shaped concrete specimens in mesoscale and gave significant contribution to the understanding of the influences of coarse aggregates and loading rate on the fracture properties of concrete. In different studies regarding mesoscale modelling of concrete or SFRC composites, coarse aggregates were either ignored (Fang and Zhang, 2013) or assumed with different shapes, for example, circular (Zhou and Hao, 2008a) or polygonal (Song and Lu, 2012). However, coarse aggregates occupy the second largest volume (normally 40%) in SFRC materials. The fundamental question whether the shape of coarse aggregates has significant influence on mesoscale modelling of concrete and SFRC materials needs to be answered. Kim and Al-Rub (2011) numerically investigated the effects of coarse aggregates on the cracking patterns and stress–strain diagrams of concrete specimen under quasi-static direct tension. It was found that the change of aggregate shapes resulted in different cracking patterns but insignificantly changed the stress–strain relations. However, this observation might not be applicable to the mesoscale numerical simulation of splitting tensile tests as the loading condition and mode of failure of specimens in these two testing methods are different. The numerical study by Zhou and Xia (2013) is probably the only one that investigated the effect of shape of coarse aggregates on the mechanical properties of concrete material under dynamic splitting tension, in which it was reported that considering aggregates with different shapes resulted in significantly different cracking patterns. However, it is noticed when examining the shape effects, while changing the shape of aggregates, the random locations of aggregates were also changed in the numerical model in Zhou and Xia (2013). It is certain that the distribution of coarse aggregates also influences the numerical simulation result. Therefore, the observations by Zhou and Xia (2013) cannot be solely attributed to the aggregates’ shape effects. Moreover, Zhou and Xia (2013) only modelled concrete. The existence of steel fibres could significantly affect the crack initiation and propagation. No study has investigated the influences of aggregate shape in mesoscale modelling of SFRC on its dynamic material properties under splitting tension tests. The shape effect of coarse aggregates in concrete or SFRC mesoscale models can only be examined by changing the shape of aggregates while keeping other parameters, for example, size of specimen, position and volume of each aggregate and shape, position and orientation of each steel fibre, exactly the same.

Moreover, although almost all researchers agree that the random distribution and orientation of coarse aggregates and steel fibres have certain influence on the mechanical characteristics of SFRC materials, very limited studies have taken these variations in mesoscale numerical simulations into consideration, but most studies only considered one specific mesoscale model in conducting the so-called quantitative investigation. Xu et al. (2012d) simulated SFRC specimens with 0.6%, 1.2% and 1.8% hooked-end steel fibres under dynamic compression. The mesoscale modelling of concrete slabs reinforced with straight steel fibres subjected to impact loading was reported by Xu et al. (2015), in which the fibre volume fractions of 1.0% and 3.0% were considered. The dynamic splitting tensile behaviour of plain concrete and SFRC with spiral fibres and hooked-end fibres up to 2.0% volume fraction was modelled by Xu et al. (2012c). In all the studies reviewed above, for the investigated SFRC specimens with specific volume fraction of fibres, only one mesoscale model was considered. For a less-biased examination of the influence of volume fraction of spiral fibres on the mechanical properties of SFRC material under dynamic splitting tension, Wang et al. (2016) conducted statistical analysis and obtained the probability density function of the dynamic strength. However, the study focused on proving the applicability and advantage of a kernel-based non-parametric statistical method in predicting the dynamic strength of SFRC materials with limited number of samples, but the proper dosage of spiral fibres that could be beneficial for easy casting, less-scattered dynamic strength and construction cost was not investigated.

This study carries out mesoscale modelling of SFRC materials with spiral fibres under quasi-static and dynamic splitting tension. The parametric simulations are initiated with the investigation of aggregate shape effects on the behaviour and properties of SFRC under dynamic splitting tension tests. Considering that the use of cobblestones to mix concrete matrix has been decreasing, oval-shaped aggregates are not considered in this study. For each case, coarse aggregates with exactly the same position but two shapes, namely, circular and polygonal, were considered to build two mesoscale models. The effective plastic strain distributions, cracking patterns and strengths of specimens with differently shaped aggregates are compared. Furthermore, the dynamic splitting tensile tests of SFRC materials with different volumetric percentages of spiral fibres are simulated. The volume fraction of spiral fibres that can simultaneously satisfy the requirements of relatively uniform material properties, workability of SFRC matrix and construction cost is determined.

Fundamentals of quasi-static and dynamic splitting tensile tests

In quasi-static splitting tensile test, the cylindrical specimen is diametrically placed with its longitudinal axis in parallel with the planes of platens of the testing machine. A vertical load P is applied and leads to splitting tensile failure of the specimen. To avoid compressive failure at the loading points, strips with width b are normally inserted in between the specimen and the loading platens to distribute the point load as shown in Figure 1(a). The quasi-static splitting tensile strength $f_{qs}$ can be calculated by the following equation (Tang, 1994)

f_{qs} = \frac{2 P}{π LD} {[1 - {(\frac{b}{D})}^{2}]}^{\frac{3}{2}}

(1)

where L and D are the length and diameter of the specimen, respectively.

Figure 1.

Schematic illustration of quasi-static and dynamic splitting tensile tests: (a) quasi-static test and (b) SHPB test.

The dynamic splitting tensile strength of SFRC can be determined by conducting SHPB tests (Hao and Hao, 2016b). The SHPB test setup is illustrated by Figure 1(b). The striker bar impacts the incident bar and generates a compressive stress wave. Propagating along the incident pressure bar, part of the incident stress wave is reflected at the bar–specimen interface, and the rest is transmitted into the specimen. With a few times of reverberation within the specimen, the stress wave propagates into the transmitted pressure bar. The histories of incident, reflected and transmitted stresses can be obtained by converting the strain signals of the pressure bars. The specimen is diametrically placed in between the pressure bars with its longitudinal axis perpendicular to the direction of impact. To assure good contact between the specimen and the pressure bars, strips are absent in SHPB tests. Thus, the dynamic splitting tensile strength $f_{dy}$ can be determined as (Tedesco et al., 1993)

f_{dy} = \frac{2 R^{2} σ_{T}}{LD}

(2)

where R is the radius of the pressure bars and $σ_{T}$ is the peak transmitted stress.

Accordingly, the loading rate $\overset{\cdot}{σ}$ and strain rate $\overset{\cdot}{ε}$ of the tested specimen can be calculated by equations (3) and (4), respectively

\overset{\cdot}{σ} = \frac{f_{dy}}{Δ t}

(3)

\overset{\cdot}{ε} = \frac{\overset{\cdot}{σ}}{E}

(4)

where $Δ t$ is the duration from the start to the peak transmitted stress and E is the modulus of elasticity of the material.

It should be noted that there are a number of ways to calculate strain and strain rate in dynamic splitting tensile tests on brittle materials. Equations (3) and (4) are used in this study because of the ease of calculation and consistency of results. However, this method only estimates the average strain rate while more accurate strain histories can be obtained by the signals from strain gauges or the cracking width. It is believed that the different options of calculating strain rate in different studies can be a major reason that results in the largely scattered data in DIF versus strain rate relations as summarised by Malvar and Ross (1998). Further investigations will certainly be carried out.

Numerical model

Material models

The simulation of splitting tensile tests is simplified as a plane strain problem to allow for using sufficiently fine meshes (Xu et al., 2012c; Zhou and Hao, 2008a). The quasi-static splitting tests are simulated by applying a velocity boundary (0.05 mm/s, corresponding to strain rate of 10⁻³ s⁻¹) directly onto the SFRC specimen as shown in Figure 2(a). The models for simulating the dynamic splitting tests consist of SFRC specimen and the pressure bars of SHPB test device, which are made of stainless steel as shown in Figure 2(b). The mesoscale model of SFRC consists of mortar matrix, coarse aggregates and spiral steel fibres. The diameter of coarse aggregates ranges from 2 to 10 mm, and the total volumetric percentage of coarse aggregates in SFRC specimen is 40%. The spiral fibres have wire diameter of 0.5 mm, coil diameter of 5 mm, coil pitch of 10 mm and nominal length of 15 mm (Hao and Hao, 2013a). The diameter of the SFRC specimen is 75 mm, and the longitudinal cross sections of the pressure bars are 2000 mm × 75 mm. Shell elements are used to simulate the mortar matrix, coarse aggregates, spiral fibres and pressure bars. The discrete constituents, that is, coarse aggregates and spiral fibres in SFRC specimen, are assumed to have perfect bonding with the mortar matrix.

Figure 2.

Numerical models in LS-DYNA: (a) quasi-static simulation and (b) SHPB simulation.

Each constituent of the SFRC specimen has its own material model. The incident and transmitted pressure bars are modelled by the isotropic ELASTIC MATERIAL (Mat_1) in LS-DYNA. The plasticity model for quasi-brittle materials developed by Malvar et al. (1997) is adopted to model the mortar matrix in the simulation. In this study, the compressive and tensile DIF relations for mortar matrix are adopted from Hao and Hao (2014), which have the lateral inertia confinement effect removed. They are given in equations (5) to (9)

CDIF = 0.0419 (\log \overset{\cdot}{ε}) + 1.2165 for \overset{\cdot}{ε} \leq 30 s^{- 1}

(5)

CDIF = 0.8988 {(\log \overset{\cdot}{ε})}^{2} - 2.8255 (\log \overset{\cdot}{ε}) + 3.4907 for \overset{\cdot}{ε} > 30 s^{- 1}

(6)

TDIF = 0.26 (\log \overset{\cdot}{ε}) + 2.06 for \overset{\cdot}{ε} \leq 1 s^{- 1}

(7)

TDIF = 2 (\log \overset{\cdot}{ε}) + 2.06 for 1 s^{- 1} < \overset{\cdot}{ε} \leq 2 s^{- 1}

(8)

TDIF = 1.4433 (\log \overset{\cdot}{ε}) + 2.2276 for 2 s^{- 1} < \overset{\cdot}{ε} \leq 150 s^{- 1}

(9)

where CDIF and TDIF are compressive and tensile DIF, respectively.

The PSEUDO_TENSOR model (Mat_16) in LS-DYNA is used in this study to model the coarse aggregates. The DIFs of coarse aggregates are adopted from Hao and Hao (2013b) and are given below

CDIF = 0.0187 (\log \overset{\cdot}{ε}) + 1.2919 for 1 s^{- 1} \leq \overset{\cdot}{ε} \leq 220 s^{- 1}

(10)

CDIF = 1.8547 {(\log \overset{\cdot}{ε})}^{2} - 7.9014 (\log \overset{\cdot}{ε}) + 9.6647 for 220 s^{- 1} \leq \overset{\cdot}{ε} \leq 1000 s^{- 1}

(11)

TDIF = 0.0598 (\log \overset{\cdot}{ε}) + 1.3588 for 10^{- 6} s^{- 1} \leq \overset{\cdot}{ε} \leq 10^{- 1} s^{- 1}

(12)

TDIF = 0.5605 {(\log \overset{\cdot}{ε})}^{2} + 1.3871 (\log \overset{\cdot}{ε}) + 2.1256 for 10^{- 1} s^{- 1} \leq \overset{\cdot}{ε} \leq 50 s^{- 1}

(13)

The spiral fibres are modelled by PIECEWISE_LINEAR_PLASTICITY material model (Mat_24) from LS-DYNA as an elasto-perfectly plastic material. The parameters of the materials are listed in Table 1.

Table 1.

Material models and parameters in numerical simulation.

Material	Model in LS-DYNA	Input parameter	Value
Pressure bar	MAT_1	Density	7800 kg m⁻³
		Young’s modulus	200 GPa
		Poisson’s ratio	0.3
Mortar matrix	MAT_72R3	Density	2200 kg m⁻³
		Unconfined compressive strength	35 MPa
		Poisson’s ratio	0.18
Coarse aggregate	MAT_16	Density	2750 kg m⁻³
		Unconfined compressive strength	160 MPa
		Poisson’s ratio	0.20
Spiral fibre	MAT_24	Density	7800 kg m⁻³
		Young’s modulus	200 GPa
		Poisson’s ratio	0.3
		Yield stress	1300 MPa

To avoid overflow and simulate the cracking process when excessive deformation occurs in the SFRC specimen, erosion technique is adopted in the numerical simulation. An erosion criterion depending on the maximum principle strain of 0.2 is used for mortar matrix, aggregates and spiral fibres. This criterion was proven to have negligible influence on the simulation results while efficiently avoiding the computational overflow (Hao and Hao, 2016a).

Mesh size sensitivity and model validation

In finite element simulation, the mesh size is critical for accurate simulation results while assuring the computational efficiency. The mesh size of 4 mm has been proven the optimal one to simulate pressure bars by Hao and Hao (2016a). To determine the optimal mesh size for simulating SFRC specimens, three mesh sizes, namely, 0.56, 0.28 and 0.14 mm, are considered to simulate SHPB splitting tensile tests. The simulation results in terms of crack width (determined by monitoring the relative Y-direction displacements of Points A and B in Figure 2(b)) and transmitted stress (compressive stress is defined as positive) are compared for evaluation as shown in Figure 3. As can be seen, the histories of crack widths and transmitted stresses calculated by the models with 0.28 and 0.14 mm mesh sizes are almost the same, while those predicted by the model with 0.56 mm mesh size are apparently different. Therefore, mesh size of 0.28 mm is determined the optimal for accurate and efficient computation and is used to simulate SFRC specimens in the subsequent simulations.

Figure 3.

Comparison of numerical simulation results with different mesh sizes: (a) histories of crack width and (b) histories of transmitted stress.

Based on the determined optimal mesh size in finite element modelling, the numerical model is validated through simulating the SHPB test conducted by Hao and Hao (2016b). The history of incident stress wave recorded in the test is used as the input in the numerical simulation, and the calculated reflected and transmitted stress waves as well as the damage pattern of the SFRC specimen are compared with those observed in the experiment. As can be seen in Figure 4 where the compressive stress is defined as positive, the numerical simulation satisfactorily reproduces the physical tests, indicating the reliability of the numerical model.

Figure 4.

Comparison of results from SHPB test and numerical simulation: (a) stress histories of SHPB tests and (b) damage patterns of SFRC specimens.

Numerical simulations

In the parametric simulations, to eliminate the dispersion and oscillation of stress waves and to let the tested specimens reach stress equilibrium during the SHPB splitting tensile tests, incident stress waves with half-sine shapes are adopted, as suggested by Lok et al. (2002). The magnitude of the incident stress wave is varied to achieve different loading rates.

Investigation of the influences of aggregate shapes

To investigate the influence of aggregate shapes on the numerical predictions of the properties of SFRC specimens with spiral fibres under dynamic splitting tension, circular and polygonal aggregates are considered. The positions and orientations of aggregates and spiral fibres in SFRC specimens with two types of aggregates are kept identical. As shown in Figure 5, each circular aggregate is transformed to a polygonal one consisting of several triangles in the numerical model. The centre of the circle becomes the connecting point of the triangles in the polygon. The position of the connecting point, the number of triangles and the size of triangles are randomly generated. During the transformation, the area of the polygon is kept the same as that of the circle, thus the total volume fraction of coarse aggregates in SFRC specimens is kept 40%. According to different volume fractions of spiral fibres (1.0%, 2.0% and 3.0%), pairs of the mesoscale models of SFRC specimens with different aggregate shapes considered in the numerical simulations are illustrated in Figure 6, in which the incident pressure bar is on the left side of the SFRC specimens, while the transmitted pressure bar is on the right-hand side (not shown in this figure). The influences of the aggregate shapes are investigated in terms of effective plastic strain distribution, cracking pattern and splitting tensile strength with respect to the loading rates.

Figure 5.

Illustration of transformation from circle to polygon in numerical model.

Figure 6.

Mesoscale models of SFRC specimens with different shapes of aggregates: (a) 1% SFRC, (b) 2% SFRC and (c) 3% SFRC.

Comparison of distributions of effective plastic strain

In the numerical simulations, the instant when the incident pressure bar is stressed is defined as 0.0 ms. The distributions of effective plastic strain in SFRC specimens with 1% spiral fibres and circular and polygonal aggregates under the strain rate of 3.9 s⁻¹ are compared as shown in Figure 7(a). At 0.56 ms, it can be seen that the effective plastic strain occurs within the middle area along the diameter of the specimen. At 0.64 ms, the area of effective plastic strain within the SFRC specimen expands towards the top and bottom sides, and the middle strip of the specimen is further strained, indicating that the area is more intensively stressed than the other parts of the specimen and undergoes large deformation. At 0.72 ms, the effective plastic strain in the central part crossing the specimen section becomes more apparent, and the unloading wave from the circumference of the specimen can be observed, indicating the severe splitting tensile damage of the SFRC specimen. By comparing SFRC specimens with circular and polygonal aggregates, it is found that the distribution of effective plastic strain in the two specimens is very similar. Same observations can also be made from the comparisons of SFRC specimens with 2.0% and 3.0% spiral fibres under the same strain rate of 3.9 s⁻¹. Therefore, the snapshots are not given here for brevity.

Figure 7.

Comparison of distributions of effective plastic strain: (a) 1.0% SFRC with different aggregate shapes under strain rate of 3.9 s⁻¹, (b) 2.0% SFRC with different aggregate shapes under strain rate of 10.9 s⁻¹ and (c) 3.0% SFRC with different aggregate shapes under strain rate of 19.4 s⁻¹.

The snapshots of effective plastic strain in SFRC specimens with different shapes of aggregates with 2% spiral fibres under the strain rate of 10.9 s⁻¹ and those with 3% spiral fibres under the strain rate of 19.4 s⁻¹ are shown and compared in Figure 7(b) and (c), respectively. As can be seen, at higher strain rate, the wave propagation becomes prominent. However, it can be found that the distribution of effective plastic strain in SFRC specimens with 2% spiral fibres and differently shaped aggregates in Figure 7(b) is of little difference, while very minor difference can be found in specimens with 3.0% spiral fibres under higher strain rate at 0.51 ms when unloading wave occurs. These observations confirm that the change in shape of coarse aggregates from circle to polygon plays an insignificant role in the distribution of effective plastic strain in the SFRC specimens under dynamic splitting tension.

Comparison of cracking patterns

The cracks are modelled by eroding the elements with large distortions during the numerical simulation, and the cracking patterns of SFRC specimens with 1.0%, 2.0% and 3.0% spiral fibres under different strain rates are compared in Figure 8. It can be seen that with the increase in strain rate, the cracking pattern of the SFRC specimen under splitting tension changed from one major crack along the diameter to a number of big cracks with widespread branches. As can be seen, the change in shape of coarse aggregates from circle to polygon does not significantly change the cracking patterns. Very minor difference in the unloading cracks between the two mesoscale models can be observed when the strain rate reaches 19.4 s⁻¹, which is consistent with the observation made from the comparison of the effective plastic strain. Therefore, the effect of aggregate shapes on the cracking patterns in mesoscale modelling of SFRC specimen under dynamic splitting tension is trivial.

Figure 8.

Comparison of cracking patters of SFRC with different aggregate shapes and volume fractions of spiral fibres at different strain rates.

Comparison of splitting tensile strengths

Based on the observation that the change in circular to polygonal aggregates in SFRC specimen under dynamic splitting tension does not result in significant difference in the distribution of effective plastic strain or the cracking pattern, it can be deduced that its influence on the dynamic splitting tensile strength is insignificant, either. This is supported by Figure 9 which compares the strengths of SFRC specimens with different volume fractions of spiral fibres under different strain rates. As can be seen, increasing the volume fraction of spiral fibres increases the dynamic tensile strength of SFRC material, especially at high strain rates, but the dynamic splitting tensile strengths of SFRC models with different aggregate shapes under the same strain rate are very close to each other.

Figure 9.

Comparison of dynamic splitting tensile strengths of SFRC with different aggregate shapes and volume fractions of spiral fibres at different strain rates.

According to the above observation and analysis, it can be concluded that the change in aggregate shape from circle to polygon in mesoscale numerical model has trivial influence on the properties of SFRC material under splitting tension irrespective of the loading rate or the volume fraction of spiral fibres.

Investigation of randomness of coarse aggregates and spiral fibres

In this section, the influence of the volume fractions of spiral fibres and the random distributions of coarse aggregates with polygonal shape and spiral fibres on the dynamic splitting tensile strengths of SFRC material is investigated. The SHPB splitting tensile tests are numerically simulated with the consideration of different volume fractions of spiral fibres from 0.5% to 3.0% with an increment of 0.5% in the SFRC. The volume fraction of coarse aggregates in all generated specimens is kept to be 40%. For SFRC specimen with a specific volume fraction of spiral fibres, 10 mesoscale models are built with random locations and orientations of coarse aggregates and spiral fibres. The loading rates considered include quasi-static case providing reference to determine the DIF and dynamic cases with 16 strain rates. In total, 1020 numerical simulations are conducted. The simulation results are presented as DIF versus strain rate relations as shown in Figure 10.

Figure 10.

Comparison of DIF relations of SFRC from different mesoscale models: (a) 0.5% SFRC, (b) 1.0% SFRC, (c) 1.5% SFRC, (d) 2.0% SFRC, (e) 2.5% SFRC and (f) 3.0% SFRC.

As can be seen in Figure 10, all DIFs increase with strain rate. In general, for SFRC materials with different volume fractions of spiral fibres, the influence of distributions of coarse aggregates and spiral fibres on the dynamic splitting tensile strength is relatively insignificant when the strain rate is low but becomes prominent with the increase in strain rate as illustrated by the more scattered DIF data when the strain rate is higher than 10 s⁻¹ in this figure. In particular, for simulations of SFRC specimens with volume fraction of spiral fibres lower than 1.0%, at the strain rate of around 17 s⁻¹, the biggest difference of DIFs among all simulated samples is below 10% (8.2% for 0.5% SFRC and 7.6% for 1.0% SFRC). This is an acceptable variation for relatively consistent dynamic strength in engineering practice. However, when the volume fraction of spiral fibres in SFRC increases to 2.0%, the DIFs with respect to strain rate of different samples become markedly scattered, and the scattering of DIFs of SFRC with 2.0% fibres is the most significant among all the cases with different volume fractions of spiral fibres considered in the study. As shown in Figure 10(d), at the strain rate of around 1.7 s⁻¹, the difference in DIFs between Sample 02 and Sample 07 is 27.2% (1.91 vs 2.43), while at the strain rate of around 19.5 s⁻¹, the difference is increased to 46.4% (5.78 vs 8.46). For SFRC specimens with more than 2.0% volume fractions of spiral fibres, the scattering of DIFs becomes less significant. This might be attributed to the tendency of more evenly distributed spiral fibres in the specimen. However, the difference in DIFs is still prominent, for example, for SFRC with 3.0% fibres at the strain rate of around 19 s⁻¹, the difference still reaches a high value of 24.8% (6.17 vs 7.7). The above results demonstrate the significant influence of random distributions of coarse aggregates and spiral fibres on the dynamic tensile properties of SFRC.

The averaged DIF relations from the simulation results of 10 mesoscale models with respect to the volume fraction of spiral fibres are obtained and compared in Figure 11. It can be seen that the averaged DIF relations of 0.5% SFRC are apparently lower than those of 1.0% SFRC, indicating that increasing the content of spiral fibres from 0.5% to 1.0% can effectively increase the material DIFs, or SFRC with 1.0% fibre is more strain rate sensitive than SFRC with 0.5% fibre content. However, it is interesting to note that when the volume fraction of spiral fibres is greater than 1.0%, the averaged DIF relations have insignificant difference, implying SFRCs with fibre volume in the range of 1.0% and 3.0% have similar strain rate sensitivity. By careful examination, very minor difference can be found when the strain rate is lower than 10 s⁻¹, above which the DIF increases with the volume of spiral fibres. However, the increase in DIF with volume fraction of spiral fibre is very limited, indicating that the DIF does not linearly increase with the volume fraction of spiral fibres. This is consistent with the observations by Xu et al. (2012c).

Figure 11.

Comparison of averaged DIF relations of SFRC with different volume fractions of spiral fibres.

Results analysis and discussion

The simulation results have demonstrated that considering either circular or polygonal shapes to model coarse aggregates insignificantly influences the properties of SFRC specimens under dynamic splitting tension. This is, however, contradictory to the observations by Zhou and Xia (2013) who reported that the crack pattern and stress distribution were affected by the aggregate shape, in which the authors simultaneously considered other factors such as aggregate size and location in modelling the dynamic splitting tensile tests. Since random variations of aggregate size and location surely affect the crack initiation and propagation of the specimens under dynamic splitting tension tests, the observations on the changes in cracking patterns by Zhou and Xia (2013) are very likely caused by the random variations of aggregate size and location in concrete specimen, instead of the aggregate shape. The findings from this study are more reliable by only changing the shape of aggregates but strictly keeping the other factors the same in the model. This observation can be attributed to the mechanism associated with stress wave propagation. By carefully checking the simulation, it is observed that crack always initiates at the interface between the mortar matrix and aggregates or spiral fibres due to localised stress concentration. As was discussed by Hao et al. (2016a), when the strain rate is low, the cracks tend to propagate around the coarse aggregates due to their much higher strength than the matrix. In this case, the aggregate perimeter would affect crack propagation, but changing the shape from circle to polygon does not significantly change the perimeter of the aggregate. When the strain rate is high, the cracks are developed very rapidly such that they do not have time to find weak sections to propagate but have to propagate through the tough aggregates. In this case, changing the shape does not significantly change the path of the crack, either. Therefore, the effects of aggregate shape on splitting tensile properties are insignificant. Hence, for study with mesoscale modelling of concrete or SFRC materials under splitting tension, either circle or polygon, can be used to model the coarse aggregates. The respective advantages are as follows: (a) circular aggregates can be easily implemented in the calculation algorithm in the programme and (b) using polygons to model aggregates can be friendlier to construct the meshes and save some computational efforts. However, the above results demonstrate that the random variations of size and location of aggregates and fibres could have a significant influence on the simulation results, especially at high strain rate. Therefore, more number of simulations might be needed to obtain unbiased results.

The series of numerical simulations considering the random distributions of coarse aggregates and spiral fibres of SFRC specimens with different fibre volume fractions under dynamic splitting tension revealed that increasing the content of spiral fibres in SFRC material can result in more scattered data but does not necessarily always increase the dynamic splitting tensile DIF. The cracking pattern and strength are very much dependent on the distributions of coarse aggregates and spiral fibres, especially the numbers and orientations of the aggregates and spiral fibres along the path of crack. The variation of DIFs of SFRC specimens is within an acceptable range when spiral fibres occupying lower than 1.0% of the total volume are mixed into the concrete matrix. However, further increasing the volume fraction of spiral fibre leads to largely scattered mechanical properties of SFRC material under dynamic splitting tension (see Figure 10(c) to (f)). This makes the quality control very difficult. The comparison of averaged DIF relations from simulations of 10 mesoscale models of SFRC specimen with respect to spiral fibre volume fraction indicates that further increase in volume percentage of spiral fibres from 1.0% to 3.0% has limited contribution to the DIF although both the static and dynamic tensile strengths of the SFRC generally increase. This might be because the strength of the matrix is not high enough to provide sufficient anchorage, which leads to the spiral fibre debonding and lost its function to provide more resistance to the action (Hao and Hao, 2017). Adding more fibres increases the construction cost and the difficulty for casting and quality control, but the increase in strain rate sensitivity is limited, and many fibres might not be able to reach their full capacity owing to the likely debonding failure. For controllable and economic reinforcement of normal-strength concrete with spiral fibres to resist blast or high-speed impact loadings, 1.0% volume fraction is recommended.

It should be noted that the present mesoscale model is based on the plane strain assumption. This is a reasonable assumption because the effect of aggregates and steel fibres along the axial direction of the cylindrical specimen has little effect in resisting the splitting tension (Wang et al., 2016). However, it is more realistic to consider the random distribution and orientation of aggregates and spiral fibres in three dimensions in the SFRC specimen, especially for investigations of the mechanical properties of SFRC specimen under compression. The development of the three-dimensional (3D) mesoscale model of SFRC with spiral fibres is underway and will be presented in the future study.

Conclusion

This study performs a series of numerical simulations to investigate the shape of aggregates, volume percentage of spiral fibres, random variations of size, orientation and location of aggregates and fibres in mesoscale model of SFRC on its mechanical properties under dynamic splitting tension. The following conclusions can be drawn:

The shapes of coarse aggregates in the mesoscale model have negligible influence on the properties of SFRC materials under quasi-static and dynamic splitting tension. Assuming circular or polygon aggregates in mesoscale model results in almost the same strain distribution, cracking pattern and strength.

When the SFRC specimen contains less than 1% spiral fibres, the random distribution of coarse aggregates and spiral fibres has limited influence on the dynamic splitting tensile strengths and strain rate sensitivity. The maximum difference in numerically obtained DIF is below 10%.

For SFRC specimens with volume fraction of spiral fibres higher than 1%, the dynamic splitting tensile strengths are apparently affected by the distribution of coarse aggregates and spiral fibres. The scattering of DIF data increases with strain rate, and the difference from different mesoscale models considered in this study can be up to 46.4%.

The averaged DIF relations of SFRC specimens with volume fractions higher than 1.0% of spiral fibres show similar strain rate sensitivity irrespective of the volume percentage of the fibre content. Considering the consistency of the mechanical properties of SFRC, the cost of construction as well as the ease for casting, spiral fibres of 1.0% volume fraction are suggested to reinforce the normal-strength concrete matrix.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was financially supported by Natural Science Foundation of China (NSFC; grant number: 51378346, and 51678403), Natural Science Foundation of Tianjin China (Grant No. 13JCQNJC07500), and Australian Research Council (ARC).

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