Finite element modelling of mesoscale concrete material in dynamic splitting test

Abstract

Dynamic tensile strength is one of the key factors of concrete material that needs to be accurately defined in analysis of concrete structures subjected to high-rate loadings such as blast and impact. It is commonly agreed that dynamic testing results of concrete material are influenced by the inertia effect, which is very much dependent on the specimen size and loading rate. It is therefore very important to remove the inertia effect in testing data to derive the true dynamic concrete material properties. On the other hand, coarse aggregates in concrete material are usually neglected due to testing limitation or numerical simplification. It has been acknowledged that neglecting coarse aggregates might not necessarily give accurate concrete dynamic material properties. In this study, a three-dimensional mesoscale model of concrete specimen consisting of cement mortar and coarse aggregates is developed to simulate splitting tensile tests and investigate the behaviour of concrete material at high strain rate. The commercial software LS-DYNA is used to carry out the numerical simulations of dynamic splitting tensile tests. The reliability of the numerical model in simulating the dynamic splitting tensile tests is verified by comparing the numerical results with the laboratory test data from the literature. The influence of inertia effect in dynamic splitting tensile tests is investigated and removed. An empirical formula to represent the true dynamic increase factor relations obtained from dynamic splitting tensile test is proposed and verified.

Keywords

concrete high strain rate inertia effect mesoscale splitting tension

Introduction

Concrete is one of the most widely used construction materials with its history of application for more than thousands of years because of its impressive compressive strength. However, due to its brittleness, that is, the weakness in resisting crack opening and propagation, and low strength under tension, concrete structures are more vulnerable to tensile failure. During the service life, some concrete structures might be subjected to blast and impact loadings due to the increased terrorist bombing attacks and industrial accidental explosions. Understanding the properties of concrete materials under dynamic tension is therefore essential for reliable design and analysis of structures against blast and impact loads.

Direct tensile tests (Tedesco et al., 1991), splitting tensile tests (Tedesco et al., 1989) and spall tests (Wu et al., 2005) are the most popular methods to obtain the dynamic properties of concrete material, among which the former two methods are normally carried out with the split Hopkinson pressure bar (SHPB) while the spall test is adapted to Hopkinson bar experiments. Many laboratory tests have indicated that the strength of cementitious material under dynamic loading is higher than its counterpart under static loads and increases with loading rate (Asprone et al., 2009; Brara et al., 2001; Diamaruya et al., 1997; Grote et al., 2001; Jawed et al., 1987; Klepaczko and Brara, 2001; Li and Xu, 2009; Reinhardt et al., 1990; Tedesco et al., 1994; Weerheijm and Van Doormaal, 2007; Yan and Lin, 2006). The ratio of dynamic to quasi-static strength is defined as the dynamic increase factor (DIF), which is conventionally considered as a material property that can be used in design and analysis of concrete structures against dynamic loadings for simple application in engineering practice (International Federation for Structural Concrete (fib), 2013). However, significant scatters of the DIF data from different tests can be observed (Cotsovos and Pavlović, 2008; Malvar and Crawford, 1998), which can be partially attributed to the variations in conditions of different tests. Besides, it has been well acknowledged that rather than giving the real material property, the structural effects generated during the high-speed impact might also have significant influences on the test results (Bischoff and Perry, 1991). When concrete specimens are dynamically loaded, the structural effects are inevitable (Hao et al., 2009, 2010, 2013a, 2013b). The apparent strengths obtained in impact tests therefore consist of structural effects and material property. Including the strength increment due to structural effect in the material constitutive law will overestimate the material strain rate effect.

Studies of the influence of inertia effect in dynamic tensile tests are, however, very limited. The inertia effect in direct tensile tests was numerically studied (Hao et al., 2012), and it was concluded that dynamic tensile strength increment is caused by both the concrete material strain rate effect and the lateral inertia confinement effect. From the numerical study using mesoscale confinement shear lattice model, Cusatis (2011) concluded that the inertia force in the experiments influences the tensile response more significantly than the compressive response and suggested that DIFs be the same for compression and tension. Ožbolt et al. (2013) carried out experimental and numerical studies on the dynamic behaviour of compact tension concrete specimens and found that the influence of structural inertia dominated when the strain rate was greater than 50 s⁻¹ which also corresponded to the threshold of crack branching. From the numerical study of Ožbolt et al. (2014), it was concluded that at strain rate higher than 10 s⁻¹ inertia effects due to crack propagation and branching (damage) and material softening became markedly significant, and the apparent strength obtained from tests should not be directly considered in the constitutive law of the material. Sources of inertia might include both lateral deformation and concrete cracking (Elmer et al., 2012). In other words, concrete cracking is not the sole reason of inertia effect in dynamic tests on concrete materials. It is deemed necessary to investigate the influence of inertia confinement associated with lateral deformation on dynamic concrete strengths.

Splitting tests are commonly used to determine the concrete tensile material properties (Xu et al., 2012). Because the stress state of the concrete specimen under splitting tension is different from that in direct tensile test, it is important to also understand the influence of inertia effect on dynamic splitting tensile tests in order to derive the true dynamic material properties. The inertia effect in dynamic splitting tensile tests was discussed by Lu and Li (2011). Based on the numerical simulation results, the authors concluded that the influence of inertia effect was trivial and the test data purely reflected the material dynamic strength. However, it should be noted that in the latter study (Lu and Li, 2011), the authors considered only the concrete-like material, that is, homogeneous material only without coarse aggregates. Coarse aggregates normally have higher strengths than mortar matrix in concrete composite. A higher stress level is needed to break coarse aggregates under high-speed impact as shown in Figure 1. Being an important constituent in concrete mix occupying significant volumetric percentage, the contribution of coarse aggregates to the dynamic strength increment of concrete material at high strain rate should be treated as a material property. Therefore, some studies that attributed the strength increment at high strain rate mainly to inertia effect might give misleading conclusion. Moreover, to save computational effort, Lu and Li (2011) used axis-symmetric model to simulate the splitting tensile tests in SHPB, resulting in a spherical concrete specimen in three-dimensional (3D), instead of a cylindrical specimen, as is commonly used in laboratory splitting tests. Since the specimen geometry and condition of contact to pressure bars will affect the dynamic tensile test results, it is important to build a more realistic model of concrete specimens in studying the inertia effect in dynamic splitting tensile tests.

Figure 1.

Comparison of dynamic tensile strengths of concrete with and without coarse aggregates (Chen et al., 2015).

This study develops a 3D mesoscale model of concrete specimens with consideration of mortar matrix and coarse aggregates to investigate the influence of structural inertia effect in dynamic splitting tensile tests using SHPB apparatus. The commercial software package LS-DYNA is employed to perform the numerical simulations. The mesh size sensitivity is examined by conducting mesh convergence test. The reliability of the numerical model in simulating the spall tests is verified by comparing the numerical results with the published experimental data (Tedesco et al., 1989). Concrete specimens with the same thickness but various diameters, that is, 50.8, 60, 70, 80, 90 and 100 mm, with randomly distributed coarse aggregates are numerically tested to investigate the inertia effect. Numerical simulations with strain rate–insensitive material are also conducted to examine the dynamic material strength increment due purely to inertia effect. Based on the numerical simulation results, the influence of inertia effect in dynamic splitting tensile tests of concrete material is examined and removed. An empirical DIF relation with inertia confinement effect removed is then derived for a more accurate representation of the concrete dynamic tensile property. The proposed relation is verified by comparing the results from numerical simulations of laboratory tests and the testing data.

SHPB tests

The schematic view of a typical SHPB test set-up is illustrated in Figure 2, where the specimen is diametrically sandwiched between two pressure bars. When the striker bar impacts the incident bar, a one-dimensional compressive stress wave is generated. The stress wave propagates along the incident bar towards the specimen and is recorded by gauge A. When it reaches the interface between the incident bar and the specimen, part of it reflects as a tensile stress wave while the rest travels through the specimen. Reflecting at the two interfaces, part of the stress wave in the specimen goes back and forth and makes the stress along the specimen approximately uniform after a few reflections. The compressive stress wave leaves the specimen, propagates forward along the transmission bar and is recorded by gauge B (Davies and Hunter, 1963).

Figure 2.

Schematic view of SHPB tests.

For such an arrangement, the dynamic splitting tensile stress, $f_{d}$ , is determined by (Tedesco et al., 1989)

f_{d} = \frac{2 P}{π LD}

(1)

where L and D are the thickness and diameter of the specimen, respectively; P represents the force that is transmitted through the specimen, which is determined from the transmitted stress as

P = π R^{2} σ_{T}

(2)

in which R is the radius of the pressure bars and $σ_{T}$ is the transmitted stress. The loading rate $\overset{\cdot}{σ}$ and strain rate $\overset{\cdot}{ε}$ can be obtained from the following expressions

\overset{\cdot}{σ} = \frac{f_{d}}{t}

(3)

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

(4)

where t is the time delay between the start of the transmitted stress wave and the peak transmitted stress and E is the Young’s modulus of the material. It is worth mentioning that the transmitted stress does not always linearly change with time as shown in Figure 3. Loading rates can be determined according to either the Line A-A (maximum) or the Line B-B (average) from the transmitted stress history. Both these two definitions are popularly used in research. The scattered DIFs from the literature can also be partially attributed to the different definitions of loading rate (strain rate). Unfortunately, no study can be found to discuss the influence of using maximum or average loading rates yet. This study adopts the average loading rates to determine average strain rates.

Figure 3.

Typical transmitted stress history in concrete dynamic splitting tests.

It should be noted that based on the assumption of one-dimensional stress wave propagation theory, the achievement of the longitudinal stress equilibrium is essential for a valid SHPB test (Hao and Hao 2013a). Equation (5) is used to obtain the stress wave at the incident surface of the specimen, $σ_{IS}$ , in SHPB tests, which will be compared to the transmitted stress wave, $σ_{T}$ , to check the stress equilibrium

σ_{IS} = σ_{I} + σ_{R}

(5)

where $σ_{I}$ and $σ_{R}$ are the incident and reflected stresses, respectively.

Material model

Material model for mortar matrix

The plasticity model, termed as MAT_72R3 in LS-DYNA, developed by Malvar et al. (1997) is used in this study to model the cement mortar. The model is elasto-plastic, uses three shear failure surfaces and considers damage, strain softening and strain rate effects. Description of the model can be found in Chen et al. (2015). The strain rate effect on the material strength is described by the DIF. In the simulation, the compressive DIF relations for cement mortar are adopted from Hao and Hao (2011), which have the lateral inertia confinement effect removed. The tensile DIFs used for cement mortar are adopted from Malvar and Crawford (1998). The compressive and tensile DIFs of cement mortar are given below

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

(6)

\begin{matrix} CDIF = & 0.8988 {(\log \overset{\cdot}{ε})}^{2} - 2.8255 (\log \overset{\cdot}{ε}) \\ + 3.4907 for 30 s^{- 1} < \overset{\cdot}{ε} < 1000 s^{- 1} \end{matrix}

(7)

TDIF = {(\frac{\overset{\cdot}{ε}}{{\overset{\cdot}{ε}}_{ts}})}^{δ} for \overset{\cdot}{ε} ⩽ 1 s^{- 1}

(8)

TDIF = β {(\frac{\overset{\cdot}{ε}}{{\overset{\cdot}{ε}}_{ts}})}^{1 / 3} for \overset{\cdot}{ε} > 1 s^{- 1}

(9)

where $δ = 1 / (1 + 8 f_{cs} / f_{c 0})$ , $\log β = 6 δ - 2$ , $f_{cs}$ is the static compressive strength, $f_{c 0} = 10 MPa$ and ${\overset{\cdot}{ε}}_{ts} = 10^{- 6} s^{- 1}$ .

Material model for coarse aggregates

The coarse aggregates in this study are modelled by PSEUDO_TENSOR (Mat_16). The DIFs of coarse aggregates used in this study are obtained from Hao and Hao (2013b) and given below

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

(10)

\begin{matrix} CDIF = & 1.8547 {(\log \overset{\cdot}{ε})}^{2} - 7.9014 (\log \overset{\cdot}{ε}) \\ + 9.6674 for 220 s^{- 1} ⩽ \overset{\cdot}{ε} ⩽ 1000 s^{- 1} \end{matrix}

(11)

\begin{matrix} TDIF = & 0.0598 (\log \overset{\cdot}{ε}) + 1.3588 \\ for 10^{- 6} s^{- 1} ⩽ \overset{\cdot}{ε} ⩽ 0.1 s^{- 1} \end{matrix}

(12)

\begin{matrix} TDIF = & 0.5605 {(\log \overset{\cdot}{ε})}^{2} + 1.3871 (\log \overset{\cdot}{ε}) \\ + 2.1256 for 0.1 s^{- 1} ⩽ \overset{\cdot}{ε} ⩽ 50 s^{- 1} \end{matrix}

(13)

Because there are only very limited test data available, the tensile DIF is set to have a constant value when strain rate exceeds 50 s⁻¹ to avoid overestimation of aggregate strength.

Material model for steel pressure bars

The pressure bars remain elastic in SHPB tests. Therefore, they are modelled by the isotropic ELASTIC MATERIAL (Mat_1) in LS-DYNA. The parameters of the materials are listed in Table 1. The parameters for mortar and aggregate materials were taken from the literature based on the general properties of mortar and granite rock materials and calibrated with test data in this study.

Table 1.

Material parameters.

Material	Model in LS-DYNA	Input parameter	Value
Cement mortar	MAT_72R3	Density	2200 kg/m³
		Unconfined compressive strength	35 MPa
		Poisson’s ratio	0.18
		Fracture energy	69.4 J/m²
Coarse aggregate	MAT_16	Density	2750 kg/m³
		Unconfined compressive strength	160 MPa
		Poisson’s ratio	0.20
		Fracture energy	161.3 J/m²
Steel	MAT_1	Density	7800 kg/m³
		Young’s modulus	200 GPa
		Poisson’s ratio	0.28

Contact algorithm

In numerical simulation of SHPB tests, to save the computational effort, the striker bar is not modelled. Instead, the incident stress generated due to the impact of the striker bar is directly applied to the incident bar as a stress boundary. The numerical model therefore consists of the incident pressure bar, concrete specimen and transmitted pressure bar with the stress boundary being applied on the input end of the incident pressure bar. In this case, only the contact between the specimen and pressure bars, that is, incident and transmitted bars, needs be modelled. The surface-to-surface contact algorithm in LS-DYNA is used to simulate the contact interfaces in the model.

Development of 3D mesoscale concrete model

Generation of coarse aggregate particles

It has been proven by previous numerical studies that models with circular or spherical aggregates yield reliable predictions of responses of concrete specimens under static and impact loads (Chen et al., 2015; Erzar and Forquin, 2011; Zhou and Hao, 2008). Therefore, in this study, coarse aggregates are assumed to have spherical shape with random radius and distribution in the mesoscale concrete specimen.

The distribution of aggregate particles is assumed to follow Fuller’s curve (equation (14)), in which the grading of aggregate particles is defined for optimal density and strength of the concrete mixture (Wriggers and Moftah, 2006)

p (d) = 100 {(\frac{d}{d_{\max}})}^{n}

(14)

where $p (d)$ is the cumulative percentage of aggregates passing a sieve with aperture diameter d; $d_{\max}$ is the maximum diameter of aggregates; the exponent of the equation, n, varies between 0.45 and 0.7 and is taken as 0.5 in the present numerical study.

3D mesoscale models for concrete specimens are developed in this study. An algorithm is developed in FORTRAN to generate randomly distributed coarse aggregate particles with random size in the specimen. The programming procedure is summarised in the following steps:

Step 1: generate random diameters of an aggregate within the size range calculated by equation (14).

Step 2: generate random coordinates of the aggregate within the range of the specimen.

Step 3: check the geometric conditions and record the parameters of the generated aggregate if the geometric conditions are satisfied; otherwise delete and regenerate aggregate until the generated aggregate satisfies the geometric conditions.

Step 4: repeat the above steps until a certain dosage of aggregates is reached.

Mapping algorithm

The following steps are implemented in FORTRAN to generate the finite element mesh with 3D mesoscale model:

Generate element meshes of the specimen.

Calculate the coordinates of the centre of each element.

After generating the aggregates with random size and distribution using the method in section ‘Generation of coarse aggregate particles’, check the centre coordinates of each element with regard to that of each aggregate. If the element centre locates within the space of any aggregate, assign the element with aggregate material; otherwise, assign the element with mortar material.

Numerical model, mesh convergence and model validation

Solid elements were used in the numerical model. Dynamic splitting tensile tests using SHPB were performed and reported in Tedesco et al. (1989) in which ∅50.8–50.8 mm concrete specimens with coarse aggregates were tested. The maximum size of coarse aggregates was 8.5 mm. The dimensions of pressure bars were ∅50.8–3660 mm and ∅50.8–3350 mm for incident and transmitted bars, respectively. Strain gauges were glued on the surface at the middle of the pressure bars. In this study, the dimensions of the specimen and pressure bars are the same as those reported in Tedesco et al. (1989). The size of coarse aggregates considered in the mesoscale model ranges from 3.0 to 8.5 mm. The total volume percentage of aggregates larger than 3 mm is assumed to be 35%. The dynamic splitting tensile test is numerically simulated and compared with the testing data to check the accuracy of the developed numerical model in this study.

Numerical model

In the mesoscale model, three groups of coarse aggregates, namely, 3–5, 5–7 and 7–8.5 mm, are generated according to the procedure described above in the numerical model. The 3D mesoscale model created is shown in Figure 4.

Figure 4.

Finite element grid of 3D concrete mesoscale model: (a) 3D mesoscale model, (b) cement mortar elements and (c) aggregate elements.

Erosion criteria

Erosion is a technique used in finite element modelling to avoid possible mesh tangling that leads to computational overflow when large deformation of element occurs. It is also used to simulate the material failure. In LS-DYNA, there are a number of erosion algorithms available to erode elements under excessive distortions. In this study, an erosion criterion depending on the maximum principle strain of 0.2 is used. When choosing the erosion criterion in this study, the primary concern is to avoid massive deletion of the elements and maintain the energy conservation. Ideally, erosion should not be used to delete elements. This, however, is not possible when modelling large deformation in the post-failure region. Therefore, to avoid eroding elements prematurely, large strain is usually chosen as the erosion criterion. For brittle materials such as mortar and rock, principle strain of 0.2 is very large. The residual strength of the material at this strain level is negligible. Comparison (Figure 5) of the simulated stress wave from numerical simulations with and without adopting erosion technique shows that using the maximum principle strain of 0.2 as the erosion criterion leads to almost identical numerical results as that obtained without considering erosion, indicating using this erosion criterion does not affect the SHPB simulation results. Since without using erosion, the simulation might overflow when large excessive deformation occurs in the concrete specimen and resulting in the loss of load carrying capacity of those elements, the erosion technique is used in the simulations. Mortar and aggregate elements will be removed when the principle strain of the elements reaches the erosion criterion in the simulations. However, it should be noted that mass of those elements will be kept, although the elements are removed to maintain the mass conservation.

Figure 5.

Comparison of stress histories from simulations with and without erosion technique.

Mesh convergence tests

In the finite element analysis, the mesh size controls the computational time and the calculation accuracy. To optimise the effects of these two factors, mesh sensitivity tests are carried out for pressure bars and concrete specimen, respectively. Three mesh sizes, namely, 8, 4 and 2 mm, are used for mesh convergence tests of pressure bars. The simulation convergence is examined by comparing the stress histories at the middle surface of the incident pressure bar as shown in Figure 6. From the figure, it can be seen that 4-mm mesh size gives almost the same prediction compared to the simulation using mesh size of 2 mm whereas simulation considering larger element size of 8 mm gives apparently different predictions. Therefore, mesh size of 4 mm for pressure bars is used in the subsequent simulations.

Figure 6.

Mesh convergence test of pressure bars.

For mesh convergence tests of concrete specimen, three mesh sizes, namely, 2, 1 and 0.5 mm, are considered. The simulation convergence is examined by comparing the reflected and transmitted stress histories as shown in Figure 7. From the figure, it can be seen that 1-mm mesh size gives almost the same prediction of peak stress value compared to the simulation using mesh size of 0.5 mm. The simulation with a larger element size of 2 mm gives a slightly different prediction of peak stress value. Considering the accuracy and the efficiency in simulation, the finite element model with mesh size of 1 mm is used for concrete specimen in this study.

Figure 7.

Mesh convergence test of concrete specimen.

Comparison of experimental and numerical results

To validate the numerical model adapted in this study, the created model is used to simulate the splitting tensile test conducted by Tedesco et al. (1989). To save the computational effort, the striker bar is not included in numerical simulation, but only the concrete specimen and incident and transmitted pressure bars are considered with the stress boundary given in Tedesco et al. (1989) being applied on the incident end of the pressure bar as the input. Gauges are attached on the surface at the middle of the pressure bars, which is the same to the arrangement in the test as reported in Tedesco et al. (1989). Comparison of the stress–time histories recorded in the pressure bars from test and the present simulation is shown in Figure 8 where US unit was used for stress in accordance with that in Tedesco et al. (1989) for easy comparison. As can be seen, the numerical simulation satisfactorily regenerates the stress histories in laboratory tests, especially the peak transmitted stress as it is directly related to dynamic splitting tensile strength of concrete specimen, indicating the reliability of the numerical models in this study.

Figure 8.

Comparison of stress histories from test and simulation: (a) stress histories from test and (b) stress histories from simulation.

Parametric simulation and discussion

Before conducting the intensive simulations of dynamic splitting tests, the created numerical concrete specimen is simulated under uniaxial quasi-static tests first to determine the quasi-static properties. In the simulation, a slow loading rate is used, with an equivalent strain rate of 10⁻⁴ s⁻¹. The quasi-static compressive strength, tensile strength and Young’s modulus are 39.5 MPa, 4.84 MPa and 32 GPa, respectively. These static parameters will be used to define the dynamic tensile strength increment.

Specimens with different sizes

Because the inertia effect is dependent on the size of the specimen, simulations of SHPB tests with specimens of different diameters will allow for a direct observation of the influence of inertia effects. Therefore, in this study, mesoscale models of specimens with the same thickness of 50.8 mm but different diameters, namely 50.8, 60, 70, 80, 90 and 100 mm, are developed and simulated. The cross sections of the specimens are shown in Figure 9.

Figure 9.

Mesoscale concrete specimens with different diameters but the same volume percentage of aggregates.

Half-sine shaped stress boundary and equilibrium

As indicated by other researchers, if a rectangular incident stress wave is used in the SHPB tests of brittle materials, as is used in Tedesco et al. (1989), the sharply rising and oscillating stress wave may reach the material strength limit very soon and fracture the specimen at the very initial stage of impact before reaching the stress uniformity (Wang et al., 2009). In contrast, an incident stress wave with a half-sine waveform can eliminate the violent oscillation and dispersion, which is beneficial for the tested specimens to reach stress equilibrium during the impact tests (Lok et al., 2002). Therefore, in the parametric simulations, the incident stress with a half-sine shaped waveform is adopted. To achieve different loading rates, the magnitude of the incident stress wave is varied, but the duration is kept the same in the simulations. Strain rate versus peak incident stress is plotted in Figure 10. It can be seen that strain rate is dependent on the size of specimen and peak incident stress. In the strain rate range in this study, the relation between strain rate and logarithmic value of peak incident stress can be well represented by a quadratic polynomial as shown in the figure.

Figure 10.

Quadratic polynomial relations of strain rate versus logarithmic peak incident stress.

The stress equilibrium is checked according to equation (5) as described in section ‘SHPB tests’. The obtained $σ_{IS}$ and $σ_{T}$ are compared in Figure 11. As shown, the stress equilibrium of the specimen under dynamic splitting tension is achieved, indicating the validity of the simulated SHPB test results presented in this article.

Figure 11.

Examination of stress equilibrium in simulated SHPB tests.

Simulations with strain rate–sensitive models

The numerical simulations of the mesoscale concrete specimens of different sizes under dynamic splitting tensile loads with respect to different strain rates are carried out. The strain rate effects for cement mortar and coarse aggregates defined by equations (6) to (9) and (10) to (13) are adopted in the simulation. It is reasonably assumed that the size effect in both static and dynamic tests is similar; therefore, it is cancelled out when the dynamic strength is normalised by static strength. If the experimentally obtained results in dynamic splitting tensile tests purely reflect the material property, that is, there is no inertia effect, varying the specimen diameter from 50.8 to 100 mm will result in the same tensile DIF. The splitting tensile DIFs obtained from the simulations are plotted with respect to the strain rate in Figure 12. As shown, when the strain rate is relatively low (below 10 s⁻¹), the obtained tensile DIFs of specimens with different sizes are similar, indicating that the influence of inertia confinement effect is insignificant. However, when the strain rate is higher than 10 s⁻¹, increasing the specimen size leads to an increase in DIF, that is, the splitting tensile strength, indicating the prominent influence of the inertia effect. For example, at strain rate of about 24.5 s⁻¹, the DIF of ∅50.8-mm specimen is 4.47 whereas that of the ∅100-mm specimen is 5.638, which is 26% higher than the former. Since the only difference in these simulations is the specimen size, and the larger the specimen size, the higher the inertia confinement, the increase in the tensile strength corresponding to the larger specimen size can be attributed to more significant inertia confinement.

Figure 12.

Numerical simulation results with strain rate–sensitive material model.

Simulations with strain rate–insensitive models

To quantify the contributions of inertia confinement to concrete strength increment in SHPB splitting tests, the above simulations are repeated using strain rate–insensitive material model, that is, by defining the tensile DIF = 1.0. In such cases, the simulated strength increment associates purely with the inertia confinement (Hao et al., 2010). The simulations with strain rate–insensitive model are performed and the results are given in Figure 13. As can be seen, it is clear that the inertia effect is size and strain rate dependent, that is, the DIF increases with specimen size and strain rate. When the strain rate is lower than 10 s⁻¹, the difference in splitting tensile DIFs of specimens with different diameters is not prominent, but becomes prominent when the loading rate becomes higher. These observations are consistent with those made according to Figure 12. By comparing those in Figures 12 and 13, it is also obvious that the DIFs obtained by considering inertia effect only in Figure 13 are smaller than those obtained by considering both the material strain rate effect and inertia confinement shown in Figure 12, indicating the material strain rate effect.

Figure 13.

Numerical simulation results with strain rate–insensitive material model.

Removal of inertia effect in dynamic splitting tensile tests

The strength increment in dynamic splitting tensile tests is due to the combination of inertia effect and strain rate effect according to the above observations. Therefore, the tensile DIF that reflects the true concrete material strain rate effect, $DI F_{\overset{\cdot}{ε}}$ , can be expressed as (Hao and Hao, 2014)

DI F_{\overset{\cdot}{ε}} = DI F_{A} - DI F_{i} + 1

(15)

where $DI F_{A}$ is the apparent DIF obtained from experimental tests that contains the contribution from both the strain rate effect and inertia effect and $DI F_{i}$ denotes the obtained DIF from simulations using strain rate–independent model, that is, due to specimen inertia effect, given in Figure 14.

Figure 14.

Derived DIFs due to strain rate effect.

The strain rate effect–induced DIFs of specimens with different diameters according to equation (15) are shown in Figure 14. It can be seen that after the removal of contribution of inertia effect, the $DI F_{\overset{\cdot}{ε}}$ of specimens with different diameters are very close because they are specimen size independent. The difference among the scattered data might be attributed to the influence of the specimen heterogeneity, that is, the influence of randomly distributed coarse aggregates. The linearly fitted curve of the data, defined in equation (16), is also included in the figure to quantify the material strain rate effect

\begin{matrix} DI F_{\overset{\cdot}{ε}} = & 1.0371 \times \log \overset{\cdot}{ε} + 1.5957 \\ for 51 s^{- 1} < \overset{\cdot}{ε} < 401 s^{- 1} \end{matrix}

(16)

This relation can be considered as the true material strain rate effect on concrete dynamic tensile strength. It is interesting to note that the derived tensile DIF can almost fit a linear-log relation with the strain rate between 5 and 40 s⁻¹, which is consistent with the findings reported in Ožbolt et al. (2013, 2014). This is because, as shown in Figure 16, the turning point of concrete tensile DIF is at about 1 s⁻¹. From 1 s⁻¹ onwards, the tensile DIF increases rapidly with the strain rate and follows a linear-log relation. The above empirical relation provides a concrete DIF estimation with inertia effect removed in splitting tensile tests in the strain rate range of 5–40 s⁻¹ only. This is because when the strain rate is below 5 s⁻¹, the inertia effect is insignificant as shown in Figure 13. On the other end, as summarised in Malvar and Crawford (1998), splitting tensile tests of concrete material are normally able to reach tensile strain rates only up to 30–40 s⁻¹. The material tensile properties with strain rate higher than 40 s⁻¹ are usually obtained by spall tests (Chen et al., 2015), which do not require stress uniformity inside the rod specimen. Therefore, the empirical formula proposed to remove inertia effect in dynamic splitting tensile tests is limited to the strain rate between 5 and 40 s⁻¹.

Verification of the proposed true strain rate effect

To examine the validity of the proposed true tensile strength DIF defined by equation (16), an SHPB splitting tensile test with the same specimen and test set-up as described in section ‘Numerical model, mesh convergence and model validation’, but a much higher level of incident stress (264.27 MPa) is simulated (Tedesco et al., 1989). The waveform of incident stress is the same as that in section ‘Comparison of experimental and numerical results’, but the intensity is scaled from 72.76 to 264.27 MPa. The simulations are conducted using the strain rate effect defined in Malvar and Crawford (1998) and in equation (16) for comparison. The simulation results are given in Figure 15.

Figure 15.

Comparison of numerical simulation results using DIF relations defined in Malvar and Crawford (1998) and equation (16).

From Figure 15, it can be seen that the obtained peak transmitted stress from simulation using DIF relation defined in Malvar and Crawford (1998) (32.37 MPa) is slightly higher than that from simulation using DIF relation defined in equation (16) (29.2 MPa). The laboratory test data reported in Tedesco et al. (1989) give the peak transmitted stress of 28.14 MPa. Comparing with the test data, the result obtained with DIF relation from Malvar and Crawford (1998) is 15% higher while that with the proposed DIF relation in equation (16) is 3.8% higher, indicating the proposed relation more accurately models the true material strain rate effect, whereas the DIF relation defined in Malvar and Crawford (1998) based on testing data overestimates the material strain rate effect because of the contributions of inertia confinement effect in the dynamic impact testing. Directly fitting DIF curve from scattered data inevitably includes the structural effect. If such DIF relation is used to model material under high-rate loadings, the dynamic material strength will be overestimated.

More numerical simulations of splitting tensile tests on specimens with diameters of 50.8, 70 and 100 mm are carried out. Figure 16 shows the comparisons of the simulated tensile DIFs with the experimental data summarised in Malvar and Crawford (1998). It can be seen that the simulated DIFs agree well with the test data. Using different specimen sizes, different DIFs are obtained, but they are all within the range of the available experimental data in the literature, indicating that besides other variations such as equipment, material and measurement, the size-dependent inertia effect also causes the scatterings in the testing data obtained by different researchers.

Figure 16.

Comparison of the concrete tensile DIF obtained from the numerical simulations and experimental results summarised in Malvar and Crawford (1998).

Concluding remarks

This study develops mesoscale model and conducts numerical simulations to study the behaviour of concrete material properties under dynamic splitting tensile loads. Commercial software LS-DYNA is used to carry out simulations. The SHPB test set-up is adopted in the numerical studies. The mesoscale concrete model consists of cement mortar and coarse aggregates and considers random size and distribution of coarse aggregates in the specimen. The components in SHPB test, that is, pressure bars and concrete specimen, are modelled by different material models. Contact algorithm is defined to simulate the interfaces between concrete specimen and pressure bars while erosion technique is used to avoid mesh tangling and computational overflow, as well as model the material failure. Mesh convergence tests are performed for optimising the computation effort and numerical simulation accuracy. The numerical model is verified by comparing the results from simulation and laboratory tests.

To eliminate the stress wave oscillation and dispersion, half-sine shaped stress waveform is adopted in the parametric simulations. The stress equilibrium is checked to ensure valid SHPB tests. Specimens with the same thickness but different diameters are developed and simulated. It is found that the dynamic splitting tensile strength increases with the specimen diameter, indicating the influence of inertia effect. Based on the numerical simulation results, an empirical DIF relation is proposed to represent the true strain rate effect on dynamic tensile strength increment from dynamic splitting tensile tests of concrete materials with the contribution of inertia effect removed. The proposed relation is verified by laboratory test data available in the literature.

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 study was financially supported by the Australian Research Council (grant number DP130104332) and China National Natural Science Foundation (grant number 51227006).

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