Numerical analysis of concrete material properties at high strain rate under direct tension

International Journal of Impact Engineering 39 (2012) 51e62

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Numerical analysis of concrete material properties at high strain rate under direct tension Y Hao*, H. Hao, X.H. Zhang School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 February 2011 Accepted 19 August 2011 Available online 31 August 2011

The tensile strength of concrete material increases with the strain rate. The dynamic tensile strength of concrete material is usually obtained by conducting laboratory tests such as direct tensile test, spall test or splitting test (Brazilian test). It is commonly agreed now that the DIF obtained from dynamic impact test is affected by lateral inertia confinement. Therefore, those derived directly from testing data do not necessarily reflect the true dynamic material properties. The influence of the lateral inertia confinement, however, is not straightforward to be quantified in laboratory tests. Moreover, concrete is a heterogeneous material with different components, but is conventionally assumed to be homogeneous, i.e. cement mortar only, in most previous experimental or numerical studies. The aggregates in concrete material are usually neglected owing to testing limitation and numerical simplification. In the present study, a mesoscale concrete material model consisting of cement mortar, aggregates and interfacial transition zone (ITZ) is developed to simulate direct tensile tests and to study the influences of the lateral inertia confinement and heterogeneity on tensile strength of concrete material with respect to strain rates between 1/s and 150/s. The commercial software AUTODYN with user provided subroutines is used to perform the numerical simulations of SHPB tests. The model is verified by testing data obtained by others. Numerical simulation results indicate that the lateral inertia confinement contributes to the dynamic increase factor (DIF) of concrete material tensile strength. The lateral inertia confinement effect is specimen size and strain rate dependent. Based on the numerical results, discussions on the relative contributions from the lateral inertia confinement and the material strain rate effect on DIF of concrete material tensile strength are made. Empirical relations are proposed to remove the influence of the lateral inertia confinement in dynamic impact tests on dynamic concrete material strength. The effect of aggregates inside the concrete specimen on its dynamic strength is also investigated. The results demonstrate that it is very important to include aggregates in experimental and numerical studies of concrete material dynamic strength, otherwise significant inaccuracy might be induced. Ó 2011 Elsevier Ltd. All rights reserved.

Keywords: Concrete High strain rate Mesoscale model Direct tension ITZ

1. Introduction Concrete is a common construction material used in both civil and defense engineering. For a better protection against high-rate loadings, e.g. impact or blast, and a more reliable design of concrete structures, it is important to understand the dynamic concrete material properties. The dynamic tensile strength of concrete is usually obtained by conducting laboratory tests such as direct tension test [1e3], spall test [4,5] or Brazilian splitting test [6]. Although it is widely agreed that the dynamic tensile strength increase factor (DIF), defined as the ratio of dynamic to static

* Corresponding author. E-mail address: [email protected] (Y. Hao). 0734-743X/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2011.08.006

strength, of concrete material increases with strain rate, like concrete compressive strength DIF, apparent scatters of tensile strength DIF from different tests can be observed [7,8]. These scatters can be attributed to variations in testing conditions such as apparatus, specimen material and specimen size. Besides these variations, it is known that inevitable lateral inertia confinement, which is specimen size dependent, also influences the testing results. Thorough discussions on the possible influences of these parameters on impact testing of dynamic compressive material properties can be found in the literature [9,10]. Although most previous studies concentrate on discussions of parameters that influence the compressive test results, similar influences, such as the lateral inertia confinement, also applies to dynamic tensile tests. However, these influences are difficult to be removed nor quantified in laboratory tests. These raise the doubts on the

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Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

reliability of the dynamic material properties obtained from impact tests. A number of analytical and numerical studies of the lateral inertia confinement effect on dynamic compressive material property have been reported in the literature [11e17]. However, there is no systematic study yet on the influences of lateral inertia confinement on dynamic concrete tensile material properties. Using the current empirical relations from the literature [3,5,7,18e20], which are derived mainly from testing data, might overestimate the concrete dynamic material strength because the inertia confinement effect, which increases the concrete dynamic tensile strength, inevitably exists in dynamic testing. Moreover, real concrete consists of cement mortar, aggregates and interfacial transition zone (ITZ), but in most previous laboratory tests and numerical simulations, the aggregates inside the concrete are usually neglected owing to practical considerations in performing high-speed impact tests, and difficulties in developing detailed numerical models and carrying out numerical simulations. For example, concrete specimens were assumed as a homogeneous material with cement mortar only [2,8,21e24], or concrete-like material (micro-concrete) in which sand, or so-called fine aggregates up to 2 mm, is used to prepare the specimen in studies of dynamic concrete material properties [4,24e26]. Because different components in a concrete mix have different material properties, modelling concrete by cement mortar only may result in inaccurate predictions of concrete material properties in both experimental and numerical studies. Brara and Klepaczko conducted spall test and found almost all fine aggregates were cleaved at the fracture surface [26]. Similarly, Yan and Lin conducted direct tensile test and observed that the fracture surfaces of the specimens became more and more flattened with the increasing strain rate; and an increasing number of coarse aggregates were broken along the fracture surface. They concluded that a higher stress level is needed to break aggregates into pieces along the fracture surface [3]. These test results clearly demonstrated the influences of aggregates on concrete dynamic tensile strength. Therefore it is deemed necessary to include aggregates in both experimental and numerical studies in order to more reliably derive the concrete dynamic tensile strength. The present study develops mesoscale models of concrete specimens with consideration of cement mortar, aggregates and ITZ to investigate the influence of lateral inertia confinement on dynamic tensile strength of concrete specimens under direct tensile tests. The commercial software AUTODYN with user provided subroutines is employed to perform the numerical simulations. The reliability of the numerical model in simulating the SHPB tests is verified by comparing the numerical results with the experimental data reported in [2]. With user provided subroutines linked to AUTODYN, the DIF relation proposed by Hao and Zhou [20] is used to define the strain rate effect of concrete material. The materials are assumed to be strain rate sensitive and insensitive, respectively in numerical simulations. The DIF derived from strain rate sensitive materials is caused by a combination of strain rate effect and lateral inertia confinement effect, while the DIF derived from strain rate insensitive materials is caused by only lateral inertia confinement. Because the lateral inertia confinement effect is specimen size dependent, to quantify the lateral inertia confinement effect on concrete specimens of different sizes, the radius of the specimen is varied from 6 mm to 20 mm in numerical simulations. The numerical simulations allow for a direct observation and quantitative assessment of the lateral inertia confinement effect on the concrete tensile DIF. Based on numerical results, empirical relations are proposed to remove lateral inertia confinement effect in dynamic impact tests. The influence of including aggregates in concrete specimen on its dynamic tensile strength is also discussed.

2. Material model An accurate material model is essential for a reliable simulation of structural response and damage. The material model used in the present study includes equation of states (EOS), strength criterion, damage model and a model for strain rate effect, which is similar to those proposed by Hao and Zhou [20]. It should be noted that the property of ITZ is not well understood yet. Therefore, in this study it is assumed to be a weak mortar, with the same material model but a lower strength. 2.1. Equation of state (EOS) In order to obtain a complete solution, in addition to appropriate initial and boundary conditions, it is necessary to define a further relation between the flow variables. This can be found from a material model which relates stress to deformation and internal energy (or temperature). In AUTODYN, the stress tensor is separated into a hydrostatic tensor and a stress deviatoric tensor associated with the resistance of the material to shear distortion [27]. The relation between the hydrostatic pressure, the local density (or specific volume) and local specific energy (or temperature) is known as an equation of state (EOS). ITZ and cement mortar are considered as porous materials due to the porosity and complex non-linear compressive behaviour. They are modelled by P-a EOS [28] where the parameter a is defined by the equation

a ¼ y=ys

(1)

where y is the specific volume of the porous material and ys is the specific volume of the material in the solid state at the same pressure. ys ¼ 1/rs at zero pressure, and rs is the solid density. The compaction path, a(p,e), represents the volumetric stiffness of the porous material between the initial compaction pressure pe and the fully compacted pressure ps as




a ¼ 1 þ ap  1

   ps  p n ps  pe

(2)

where ap is the value of a corresponding to the initial plastic yielding, p is the current pressure, n is the compaction exponent, which is assumed to be 3 in the present study. Aggregate is assumed to experience brittle failure with a minimum deformation. Therefore, the simplest linear EOS is adopted for aggregates as

p ¼ Km

(3)

where p is the pressure, m¼(r/r0)1, in which r0 is the initial density and r is the current density of the aggregate corresponding to pressure p, and K is the material bulk modulus. 2.2. Strength criterion The deviatoric stress tensor is governed by a damage-based yield strength surface. The concrete material is assumed to be elastic before the stress state reaches the yield criterion. The incremental form of the Hooke’s law is



Dsii ¼ 2G Dεii 



Dy ; 3y

Dsij ¼ 2GDεij

(4)

where G is the shear modulus, Dsij is the deviatoric stress increment, Dεij is the strain increment and Dy/y is the relative change in the volume from EOS.

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

where ε i is the negative principal strain. The ‘’ means it vanishes is the positive principal strain, and the ‘þ’ means if it is positive. εþ i it vanishes if it is negative. The weights Ac and At in Eq. (5) are defined by the following expressions,

Undamaged material D=0

J2

Damaged material 0
Ac ¼ Residual strength D≥Dmax 0

A piece-wise Drucker-Prager damage-dependent model as shown in Fig. 1 [20] is used as the yield strength criterion for all the three components, i.e. cement mortar, ITZ and aggregate, which is determined by four sets of experimental data: (1) cut-off hydrotensile strength fttt(¼s1 ¼ s2 ¼ s3); (2) uniaxial tensile strength ft(¼s1, s2 ¼ s3 ¼ 0); (3) uniaxial compressive strength fc(¼s3, s1 ¼ psffiffiffi2 p ¼ ffiffiffiffiffiffiffi0); (4) confined compressive strength (I1 ¼ 10 3fc ; 2J2 ¼ 6fc , where I1 is the first stress invariant and J2 is the second deviatoric stress invariant) [29]. The material has permanent plastic strain once the yield surface is reached. 2.3. Damage scalar The damage scalar D in Fig. 1 is determined by Mazars’ damage model [30] as follows,

~ εc εc0

. εc0

  at Dt ~εt ¼ 1  e



~ εt εt0

. εt0

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X  2 u t εþ i

(7)

i ¼ 1;3

The tensile DIF relations defined below are used for strain rate sensitive cement mortar and ITZ [20], which are based on many experimental data.

DIFREF ¼ 1:0

for ε_  104 s1

DIFREF ¼ 0:26 log_ε þ 2:06 DIFREF ¼ 2 log_ε þ 2:06

(9)

for 104 s1  ε_  1s1

for ε_ >1s1

while those for aggregates are defined by Eqs. (12) and (13), which are also based on experimental data as shown in Fig. 2 [31e36]. It should be noted that to avoid overestimation of aggregate strength, DIF of aggregates is set to have constant value when the strain rate exceeds 50/s.

DIFREF ¼ 0:059805 log_ε þ 1:35883

for ε_  0:1s1 s

5 4 3

for 0:1s1  ε_  50s1

ð13Þ

The material parameters used in this study are listed in Table 1. 3. SHPB simulation and model verification The direct tension tests were performed and reported in [2] in which a modification was made to the pressure bar as shown in Fig. 3. The striker bars impact the incident bar and generate a tensile stress wave propagating to the specimen and the transmitter bar. The specimen with notch is sandwiched and cemented with non-epoxy concrete cement between the two pressure bars.

Cai et al. [32] Kubota et al. [33] Wang et al. [34] Wang et al. [35] Goldsmith et al. [36] Data fit

2 1 0 1E-06

1E-05

0.0001

0.001

(12)


2
 DIFREF ¼ 0:560483 log_ε þ1:387057 log_ε þ 2:125599

Cho et al. [31]

6

(10) (11)

7

DIF

i ¼ 1;3

i ¼ 1;3

2.4. DIF formulae for strain rate sensitive materials

(6)

where ac and at are damage parameters that depend on the material properties, and they are taken as 0.5 in this study, while εc0 and εt0 are the threshold strains in the uniaxial compressive and tensile states. ~εc and ~εt are the equivalent compressive and tensile strains, defined as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X  2 ~εc ¼ u t ; ~εt ¼ ε i

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P þ ðεi þ ε Þ2 is the effective strain. Hi[x] ¼ 0 when i

(5)

where the compressive damage Dc and the tensile damage Dt are defined by



(8)

x < 0 and Hi[x] ¼ x when x  0. It can be verified that in uniaxial compression, Ac ¼ 1, At ¼ 0, D ¼ Dc, and vice versa in tension.

Fig. 1. Damage-dependent piece-wise Drucker-Prager strength criterion.

  ac Dc ~εc ¼ 1  e







X Hi ε εþ þ ε X Hi εþ εþ þ ε i i i i i i ; A ¼ t ~ε2 ~ε2 i ¼ 1;3 i ¼ 1;3

where ~ε ¼

p

D ¼ Ac Dc þ At Dt ; Dc >0; Dt >0 and Ac þ At ¼ 1

53

0.01

0.1

Strain rate (1/s) Fig. 2. Tensile DIFs for aggregates [31e36].

1

10

100

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Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

Table 1 Material parameters.

Initial density (kg/m3) Solid density (kg/m3) Initial sound speed (m/s) Initial compaction pressure (MPa) Solid compaction pressure (MPa) Solid bulk modulus (GPa) Damage parameters Compressive damage threshold Tensile damage threshold Compressive strength (MPa) Tnesile strength (MPa) Cut-off tensile strength (MPa)

Mortar

ITZ

Aggregate

2.405  103 2.75  103 2.97  103 36 6  103 35.27 0.5 2  103 2  104 57.7 4.53 2.5

1.8  103 2.75  103 2.269  103 16.2 6  103 35.27 0.5 2  103 2  104 23 1.8 0.9

Density (kg/m3) Bulk modulus (GPa) Shear modulus (GPa) Damage parameters Compressive damage threshold Tensile damage threshold Compressive strength (MPa) Tnesile strength (MPa) Cut-off tensile strength (MPa)

To calibrate the numerical model, it is used to simulate the above SHPB test. The geometry and material parameters are the same as those in [2], in which the specimen dimension is 50.8  50.8 mm (length  diameter) with a 3.175 mm squared notch at mid-length. The incident bar and transmitter bar are respectively 3350  50.8 mm and 3660  50.8 mm. Fig. 4 shows the axis-symmetrical model of the SHPB test. Gauges are attached at the centres of the pressure bars. Another nine gauges, with three at each interface between specimen and two pressure bars and at the notch are attached on the specimen as shown in the figure. Although the impact speed in the test is not mentioned, it is indicated in the paper that the applied load from the test can be simplified as the stress boundary shown in Fig. 5, i.e. the incident tensile stress starts from 0, quickly rises to its peak value of 26.5 MPa in 45 ms, keeps as a constant for 100 ms then drops back to 0 in 45 ms. According to [2], the calculated strain rate from the test result is 4.9/s. This stress boundary is used in the present numerical simulation as input. The stress-time history recorded in the pressure bars from the experimental study and from the present simulation are shown in Figs. 6 and 7, respectively, for comparison. As shown, the numerical simulation closely reproduces the recorded stress waves in the incident and transmitter bar from the experimental study, indicating the reliability of numerical simulations of SHPB tests. The above simulation that includes the incident and transmitter bars in the numerical model is very time consuming. In a previous study of compression SHPB tests of concrete specimens, it was found that replacing the pressure bars with proper velocity boundaries, i.e. only the specimen is included in the model, substantially reduced the computational time while yielded reliable simulations of the SHPB test [14]. To verify the applicability of this approach to simulate SHPB tensile tests, a model with only the specimen is developed. The boundary conditions of the specimen and the loading function are shown in Figs. 8 and 9, respectively. The strain rate applied on the specimen can be calculated as

ε_ ¼ Vpeak =L

2.75  103 35.7 17.44 0.5 3.6  103 3.6  104 200 15 7.5

(14)

where Vpeak is the peak value of the velocity, L is the length of the specimen. The velocity boundary is a function of time with a trapezoid shape, similar to that shown in Fig. 5, where the velocity starts from 0, quickly rises to its peak value Vpeak in a short loading period t0, keeps as a constant for a certain time period Dt then drops back to 0 in t00 time. In the test described in [2], the strain rate is 4.9/s. Since the length of the specimen is 50.8 mm, to obtain the approximate uniform strain rate of 4.9/s in the specimen, the tensile velocity applied to the specimen is 0.25 m/s according to Eq. (14). The velocity boundary applied to the specimen has the same parameters of the time function as in the test shown in Fig. 5, i.e. Vpeak ¼ 0.25 m/s, t0 ¼ 45 ms, Dt ¼ 100 ms and t00 ¼ 45ms. The stresstime history recorded by the nine gauges are averaged and compared with those obtained from complete SHPB simulation. The nine gauges, as shown in Fig. 5, consist of 3 gauges at the notch layer, and 3 at each end of the specimen. Because the stresses from the complete SHPB test are recorded at the mid of the pressure bar, there is a time lag between the stresses recorded on the specimen and the pressure bar. For comparison purpose, the stresses from the two approaches are shifted to remove the time lag. As shown in Fig. 10, very similar simulation results of the stress histories in the specimen from this simplified model and the detailed model are observed. Table 2 compares the simulated peak stresses in the specimen from the two approaches. It is clear that the simplified numerical model can also reliably simulate SHPB tensile tests. It should be noted that although in the experimental study [2] the notch was made to control or limit the tensile fracture area for easy prediction of the dynamic tensile strength, it is not included in the subsequent numerical simulations in this study. This is because with or without notch in numerical model does not significantly affect the numerical results. To prove this, numerical models of specimens with and without notch are

Fig. 3. SHPB test set-up in [2].

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

55

Fig. 4. Axis-symmetrical model of SHPB test (the light blue spots indicate the gauge locations). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

developed as shown in Fig. 11. The peak stresses recorded in the nine gauges from the respective specimens are averaged and compared. It is found that under the same boundary conditions described above, the cylindrical specimen without notch yields almost the same dynamic tensile strength, with only a 0.05% difference as compared to the specimen with notch. Since the simplified model leads to substantially less computer memory and less simulation time, also since there is little difference in strength between specimens with and without notch, the simplified model of cylindrical specimen without notch is used in the subsequent simulations. 4. Mesoscale numerical model 4.1. Mesoscale specimens 4.1.1. ITZ thickness and finite element mesh It is well accepted that ITZ has large heterogeneity, high porosity and its strength is lower than the cement mortar. As a thin layer around aggregates, the real typical thickness of ITZ is 0.01e0.05 mm, which limits the minimum size of the finite element mesh. In the present study the upper limit of ITZ thickness, i.e., 0.05 mm, is chosen as the size of the element in numerical model. 4.1.2. Specimen dimension, aggregate size and distribution Axis-symmetrical numerical model is adopted to simulate the cylindrical specimen. To simplify the modelling process, aggregates are assumed to have circular cross sections. The geometry of the circular aggregates is approximately modelled by the 0.05  0.05 mm square elements. To investigate the lateral inertia confinement effect, three specimens of size 6  12 mm, 10  20 mm and 20  40 mm (length  diameter) are considered. Because the lateral inertia confinement effect on concrete DIF is size-dependent [14], increasing the specimen diameter allows a direct observation and a quantitative assessment of the lateral inertia confinement effect. Proper attentions are paid to estimate the aggregates distributions in this axis-symmetrical model. The aggregate particle size

distribution is assumed to follow Fuller’s curve, which defines the grading of aggregate particles for optimum density and strength of the concrete mixture. Fuller’s curve, as shown in Fig. 12, can be expressed by the equation [37]

PðdÞ ¼ ðd=dmax Þn

(15)

where P(d) is the cumulative percentage of aggregates passing a sieve with aperture diameter d, dmax is the maximum size of aggregate particle. n is the exponent of the equation, varying from 0.45 to 0.7, which is taken as 0.5 in the present study. Each specimen is designed to contain 40% aggregates. The maximum diameter of aggregate particle is set to be 1/4 of the length, i.e., dmax is 1.5 mm, 2.5 mm and 5 mm for the 6  12 mm, 10  20 mm and 20  40 mm specimens, respectively. The information of the aggregate particle size intervals and the corresponding volumetric percentages for each concrete specimen is listed in Table 3. A C-program is developed to generate the random distribution of the aggregates. The programming procedure is summarized into the following steps [37]: Step 1: Randomly generate the position and diameter of an aggregate with the ITZ band; Step 2: Check whether the boundary condition is satisfied to avoid overlapping among aggregates and between aggregate and the specimen boundary; Step 3: If the generated aggregate satisfies the boundary conditions, place it in the specimen domain; otherwise delete the aggregate and perform a new generation until the generated aggregate satisfies the boundary conditions and is properly placed; Step 4: Repeat the above steps until a certain percentage of aggregates is reached. After generation of the aggregates with random position, the axis-symmetrical mesoscale numerical models of size 6  12 mm, 10  20 mm and 20  40 mm specimens are plotted in Fig. 13 where the blue part is the cement mortar and the green hollow circles are ITZs around the red aggregates in the specimen. 4.2. Comparison of numerical results obtained with homogeneous and mesoscale model

Fig. 5. Stress boundary.

As reported in the literature [3,26], in which fine aggregates (size up to 2 mm) and coarse aggregates were used to prepare specimens for dynamic tensile tests, aggregates inside the specimen along the fracture face were observed to have experienced significant damage. This is different from static tests where damage usually occurs along the mortareaggregate interfaces. Since the strength of aggregates is often higher than that of cement mortar, the dynamic tensile strength of concrete specimen is influenced by

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Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

Fig. 6. Incident, reflected and transmitted stress-time history from test.

the aggregates. Therefore including aggregates inside the concrete specimen in experimental tests and numerical simulation is deemed necessary for reliable derivation of dynamic concrete material properties. To demonstrate this observation in numerical simulations, a homogeneous specimen (cement mortar only) of size 10  20 mm and a mesoscale specimen of the same size but consisting of cement mortar, aggregates and ITZ are created. Numerical simulations of the direct tensile tests of these two specimens are carried out to investigate the influence of aggregates on tensile DIF of concrete specimens. In the simulation, the strain rate effects of various components in the specimen defined by Eqs. (9)e(13) are used. A series of numerical simulations are carried out. DIF of the homogeneous and mesoscale specimens corresponding to the strain rate between 1/s and 150/s are obtained from the numerical results, and are plotted in Fig. 14. It can be seen that the DIFs obtained from the mesoscale specimen are apparently larger than those derived from the homogeneous specimen, indicating the contributions of aggregates to the dynamic strength of concrete specimens because aggregates have higher tensile strength than cement mortar. These results demonstrate the necessity of using mesoscale specimen in numerical simulations.

5. Method for evaluation of lateral inertia confinement In experimental tests, the contributions of lateral inertia confinement and material strain rate effect on dynamic strength increment cannot be separated. They, however, can be easily separated in numerical simulations. In the present study, with user provided subroutines linked to AUTODYN, simulations can be carried out by using strain rate sensitive and strain rate insensitive material models. When material model is assumed to be strain rate insensitive, i.e., the DIF in the material model defined above is set equal to 1.0, the strength increment obtained from numerical simulations can then be attributed to lateral inertial confinement. Since the DIF relations defined above are obtained from experimental tests, which consist of contributions from both the lateral inertia confinement and the true material strain rate effect, subtracting the DIF values corresponding to the lateral inertia confinement effect from those empirical DIF relations obtained from experimental tests will give the DIF corresponding to the material strain rate effect. Using those derived DIF values corresponding to the material strain rate effect in strain rate sensitive numerical simulations will give the strength increment corresponding to those obtained in experimental tests because lateral inertia confinement always exists in both numerical simulations and experimental tests. This reasoning is further elaborated in the following. The definition of DIF can be expressed as

DIFREF ¼

Fig. 7. Incident, reflected and transmitted stress-time history from simulation.

Df þ fs fd ¼ d 1 fs fs

Fig. 8. Boundary conditions in simplified SHPB test.

(16)

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

57

Table 2 Comparison between simplified and complete SHPB simulations.

Difference of averaged peak stress

Fig. 9. Loading function.

where fd is the dynamic strength, fs is the quasi-static strength and Dfd is the strength increment caused by dynamic loading effect, i.e., strain rate effect and lateral inertia confinement effect. It has

Dfd ¼ Df ε_ þ Dfi

(17)

where Df ε_ is the dynamic strength increment due to the material strain rate effect and Dfi is the dynamic strength increment due to the specimen lateral inertia confinement. Substituting Eq. (17) into Eq. (16), it becomes

DIFREF ¼

Df ε_ þ Dfi þ fs fd ¼ fs fs

(18)

It should be noted that DIFREF represents empirical DIF relations derived from the experimental results (Hao and Zhou 2007), which are defined in Eqs. (9)e(13) for various components of the mesoscale specimens. It is clear that simply using the empirical relations obtained from impact tests as a material property will overestimate the dynamic material strength. Therefore the dynamic strength increment caused by the specimen lateral inertia confinement should be removed from Eq. (18). Then the DIF relation corresponding to the true material strain rate effect can be derived. For strain rate insensitive materials, since DIF is only caused by lateral inertia confinement, it can be expressed as

DIFi ¼

Df þ fs fd ¼ i 1 fs fs

(19)

which is part of DIFREF in Eq.(18). The DIF due to material strain rate effect is therefore

Df ε_ þ fs f DIF ε_ ¼ d ¼ ¼ DIFREF  DIFi þ 1  1 fs fs

(20)

Notch (3 gauges)

End layers (6 gauges)

Overall (9 gauges)

4.3%

1.2%

2.5%

where DIFREF is the reference empirical DIF relation expressed by Eqs. (9)e(13) [20], DIF ε_ is the DIF associated with the material property and DIFi is the DIF caused by lateral inertia confinement which can be obtained from simulations using strain rate insensitive materials. Therefore simulations using strain rate insensitive materials of the respective specimens will be conducted first to obtain the DIFi under different strain rates. Then the corresponding DIF ε_ can be determined. 6. Parametric simulation and discussion 6.1. Simulations with strain rate insensitive material model Because the lateral inertia confinement effect is size dependent, to remove it from the testing data, ideally the testing data from specimens of the same size should be grouped together. Unfortunately the available empirical relations of DIF are obtained from testing data of mixed sizes. For example, the diameter of tested cylindrical specimens varied from 12.7 mm to 50.4 mm in deriving the empirical relations of DIF in [5,7]. Because of this limitation, approximation has to be made. In this study, the specimen size of 10  20 mm is chosen to define the lateral inertia confinement effect. Impact tests of 10  20 mm mesoscale specimens are simulated with respect to strain rates from 1/s to 150/s using the strain rate insensitive material model to evaluate the DIFi. The results are shown in Fig. 15. For comparison purpose, DIFi of specimens of 6  12 mm and 20  40 mm are also derived from numerical simulations and plotted in the same figure. From Fig. 15, it is clear that the lateral inertia confinement effect is size and strain rate dependant, i.e., DIFi increases with size and strain rate. However, for reasons discussed above, the lateral inertia confinement effect is defined according to the data corresponding to the 10  20 mm specimens in this study. The best-fit curve of DIFi data with respect to strain rate from 1/s to 150/s from the 10  20 mm specimen can be defined by equations given below:

DIFi ¼ 1

for ε_  2

DIFi ¼ 0:556919 log_ε þ 0:832351

(21) for 2  ε_  150

Fig. 10. Comparison of the stress-time histories from simulations of complete and simplified SHPB tests- (a): notch layer, (b): end layers and (c): overall averaged.

(22)

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Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

Fig. 11. Specimens with and without notch and the gauge locations.

where DIFε_ CM=I is the DIF of cement mortar and ITZ and DIFε_ AGG is that of aggregates excluding the lateral inertia confinement effect. The above DIF relations for cement mortar and aggregates are programmed and linked to AUTODYN as user provided subroutines in the subsequent simulations to model the material strain rate effect. It should be noted that in this study ITZ is assumed to have the same DIF as cement material.

6.2. Simulation of strain rate sensitive material model

Fig. 12. Fuller’s grading curve.

The above DIFi relations of lateral inertia confinement effect are derived from mesoscale model, but they are assumed the same for cement mortar and aggregate materials in the present study. However, it should be noted that the lateral inertia confinement effects of cement and aggregate are not necessarily the same because they have different mass densities. Moreover, the damage mechanisms of homogeneous cement and rock material are not the same either as the concrete composite with cement mortar, aggregates and ITZ as described above. Nonetheless the DIFi is assumed the same in the present study. Further investigations of the lateral inertia confinement effect of cement and aggregate material separately are deemed necessary. With this assumption, the above empirical Eqs. (9)e(11) for cement mortar and Eqs. (12) and (13) for aggregates obtained from laboratory tests can be modified according to Eq. (19) to derive the respective material strain rate effect on DIF. The modified expressions of DIF ε_ of the respective material can be derived as

DIFε_ CM=I ¼ 2log_ε þ 2:06

for 1  ε_  2

DIFε_ CM=I ¼ 1:443081log_ε þ 2:227649 DIFε_ AGG

(23) for 2  ε_  150

6.2.1. Comparison of the failure process under different strain rates The failure processes of the 6  12 mm mesoscale specimen under relatively low strain rate (10/s) and high strain rate (100/s) are shown in Figs. 16 and 17, respectively, where the different material statuses are stated. By comparison of the failure processes, the following remarks can be made. For the relatively low strain rate (10/s) case, failure first occurs in the specimen at approximately 12 ms while the time for the first failure occurrence for higher strain rate (100/s) case is much earlier, approximately at 1.95 ms. As can also be clearly seen in Fig. 16, the inner fracture propagates and forms a long ‘crack’ throughout the specimen at 28 ms, indicating that the specimen is completely fractured at that time instant. However, when the strain rate is 100/s, as shown in Fig. 17, the specimen is fractured within 7.0 ms. The bulk-failure elements initiate and distribute evenly inside the specimen in Fig. 16, indicating the stress and strain uniformity is achieved inside the specimen. Whereas the bulkfailure elements in Fig. 17 initiate close to the fixed end of the specimen because at high strain rate the wave propagation is prominent and tensile stress wave interacts with the specimen boundary. On the other hand, the prominent wave propagation effect indicates the non-uniform stress and strain inside the specimen. One major fracture is formed when the strain rate is 10/s, whereas more than two major fractures with several sub-fractures are observed in Fig. 17, indicating the specimen is completely shattered so that the capability of energy absorption is enhanced at high strain rate.

(24)




2
 ¼ 0:560483 log_ε þ1:387057 log_ε þ 2:125599 for 1  ε_  2 ð25Þ


2
 DIFε_ AGG ¼ 0:560483 log_ε þ0:830138 log_ε þ 2:293482 for 2  ε_  50 ð26Þ

6.2.2. DIFs from simulations of specimens with different sizes using strain rate sensitive material model Using the revised DIF relations described in Section 6.1, numerical simulations of the mesoscale concrete specimens of size 6  12 mm, 10  20 mm and 20  40 mm under impact loads of different strain rates are carried out. DIFs obtained from the strain rate sensitive materials are plotted in Fig. 18 where the empirical DIF relations in [20] are also plotted for comparison purpose. As

Table 3 Aggregate size intervals and corresponding volumetric percentages for each specimen. Specimen

6  12 mm

10  20 mm

20  40 mm

Aggregate series (mm)

0.3e0.7

0.7e1.1

1.1e1.5

0.7e1.3

1.3e1.9

1.9e2.5

2.0e3.0

3.0e4.0

4.0e5.0

Volumetric percentage

17.07%

12.53%

10.40%

16.31%

12.80%

10.89%

15.47%

13.04%

11.49%

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

59

Fig. 13. Mesoscale numerical model with 40% aggregates.

shown, increasing the specimen size leads to the increase of DIF because of the lateral inertia confinement effect. The simulated DIF is always larger than the empirical DIF obtained from laboratory tests because of the inclusion of aggregates. When the strain rate is small, the numerical results corresponding to different specimen sizes are similar and match the empirical results from tests, indicating the lateral inertia confinement and aggregate effect is insignificant at small strain rate. Increasing the strain rate, the lateral inertia confinement effect becomes more prominent. At small strain rate, the specimen damage initiates at the cement mortar and aggregate interface and propagates in mortar material. At high strain rate, because the cracks have no time to seek weak sections to propagate, but propagate through the stronger aggregates, aggregates may also suffer significant damage. Therefore, the aggregate effect becomes more prominent at high strain rate. It is interesting to note that the DIF obtained from 6  12 mm specimen is close to the empirical relation based on testing data. This is because the inertia confinement is defined according to the results from the 10  20 mm specimen, which overestimate the inertia confinement effect of the 6  12 mm specimen. But the empirical relation is obtained from testing data of mortar specimens. Without including aggregates in the testing specimen underestimates the concrete material DIF. The overestimation of the lateral inertia confinement of the 6  12 mm specimen and the underestimation of DIF owing to neglecting the aggregates in laboratory test compensate each other. Therefore the numerical results of the 6  12 mm specimen match quite well the empirical relations. 9

6.2.3. Ratios of DIFs from simulation of strain rate sensitive materials to reference DIFs Because of the limitations in performing high-speed tests, most tests of concrete specimens were carried out without including aggregates in the specimen. Based on the results obtained in this study, some analytical formulae are derived to modify the DIFs obtained from specimens with 0% aggregate (most testing data are obtained with specimens of 0% aggregate) to estimate the DIF of normal concrete materials with 40% aggregates in the range of strain rate from 1/s to 150/s. The ratios with respect to strain rates are plotted in Fig. 19 where the quantified ratios versus strain rate relations are as follows,




2






s612 ¼  0:0145 log_ε þ0:06 log_ε þ 0:9742 for 1  ε_  150


ð27Þ

2






s1020 ¼  0:015 log_ε þ0:0714 log_ε þ 1:004 for 1  ε_  150


2

ð28Þ




s2040 ¼  0:0266 log_ε þ0:11 log_ε þ 1:0407 for 1  ε_  150

where s is the ratio and the subscript denotes the specimen size. From the figure it can be seen that the ratio increases rapidly with strain rate from 1/s to 50/s. However the rate of increment decreases when the strain rate exceeds 50/s. This is because the DIF of aggregate is set as a constant in this study when strain rate is higher than 50/s as mentioned above owing to limited DIF data available for rock materials at strain rate higher than 50/s. Because

8

Homogeneous

7

Mesoscale

6

DIF

5 4 3 2 1 0 1

10

100

1000

strain rate (1/s) Fig. 14. Comparison of DIFs from homogeneous and mesoscale specimens of size 10  20 mm.

ð29Þ

Fig. 15. DIFs obtained from strain rate insensitive material model.

60

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

Fig. 16. Failure process under strain rate 10/s.

of this assumption, the aggregate effects on DIF becomes less significant when strain rate is higher than 50/s. These relations can be used to modify the testing data obtained from specimens without aggregates to derive the DIF of normal concrete materials with 40% aggregates. 6.3. Lateral stress distribution and inertia confinement contribution to strength increment To further demonstrate the lateral inertia confinement effect, the stress distribution along the radial direction of the 6  12 mm specimen for different strain rate cases is shown in Fig. 20 when strain rate insensitive model is considered. Position 0 mm corresponds to the centre of the specimen while position 6 mm corresponds to the free surface. In general the lateral stress decreases along the radial direction, which coincide with the finding in [14] when the specimen is subjected dynamic compressive loading.

When the strain rate is low, the stress variation along the lateral direction is less prominent, but becomes prominent when the strain rate gets higher, indicating the significant lateral inertia confinement. According to Eqs. (18) and (19), the dynamic strength increment caused purely by the lateral inertia confinement normalized by the static strength is

Dfi fs

¼ DIFi  1

(30)

while that caused by both the lateral inertia confinement and the material strain rate effect normalized by the static strength is

Dfi þ Df ε_ fs

¼ DIF  1

(31)

where DIF corresponds to the data from simulations using strain rate sensitive materials with revised DIF relations, shown in Fig. 18

Fig. 17. Failure process under strain rate 100/s.

Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

61

8 7 6_12 10_20

6

20_40

DIF

5

Reference DIF (Hao and Zhou 2007)

4 3 2 1 0 1

10

100

1000

Strain rate (1/s) Fig. 18. DIFs obtained from strain rate sensitive material model.

Fig. 19. Ratios of DIFs obtained from specimen with varying sizes to reference DIFs.

and DIFi presents the data from simulations using strain rate insensitive materials shown in Fig. 15. Therefore the relative contributions of the lateral inertia confinement in mesoscale concrete model of different specimen sizes to dynamic strength increment can be quantified by

where s is the percentage contribution of the lateral inertia confinement to DIF in dynamic tests. The ratios of the

Fig. 20. Lateral stress distribution along radial direction (6  12 mm specimen).

Fig. 21. Lateral inertia confinement contribution to the total dynamic strength increment.

s¼

Dfi DIFi  1 ¼ Dfi þ Df ε_ DIF  1

(32)

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Y. Hao et al. / International Journal of Impact Engineering 39 (2012) 51e62

corresponding DIFs obtained with strain rate insensitive materials to those with strain rate sensitive materials are shown in Fig. 21. As shown again the lateral inertia confinement effect is specimen size dependent and increases with the strain rate. 7. Conclusions This paper studies the influence of specimen size and aggregates on concrete tensile DIF. It is found that the lateral inertia confinement which inevitably exists in impact tests affects the concrete strength, and is specimen size and strain rate dependent. Based on numerical simulation results, modified DIF relations for concrete material tensile strength are proposed. It is also found that including aggregates in the concrete specimen always results in a higher DIF. Some analytical formulae are proposed in this study to modify the empirical DIF formulae obtained from impact tests of concrete specimens without aggregates. Acknowledgements The authors would like to thank Australian Research Council (grant number DP1096439) and China National Natural Science Foundation (grant number 51078094) for financial support to carry out this study. The first author also acknowledges the SIRF scholarship from The University of Western Australia. References [1] Staab GH, Gilat A. A direct-tension split Hopkinson bar for high strain-rate testing. Experimental Mechanics; September, 1991:232e5. [2] Tedesco JW, Ross CA, McGill PB, O’Neil BP. Numerical analysis of high strain rate concrete direct tension tests. Computers and Structures 1991;40(2):313e27. [3] Yan D, Lin G. Dynamic properties of concrete in direct tension. Cement and Concrete Research 2006;36:1371e8. [4] Brara A, Camborde F, Klepaczko JR, Mariotti C. Experimental and numerical study of concrete at high strain rates in tension. Mechanics of Materials 2001; 33:33e45. [5] Schuler H, Mayrhofer C, Thoma K. Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates. International Journal of Impact Engineering 2006;32:1635e50. [6] Gomez JT, Shukla A, Sharma A. Static and dynamic behavior of concrete and granite in tension with damage. Theoretical and Applied Fracture Mechanics 2001;36:37e49. [7] Malvar LJ, Crawford JE. Dynamic increase factors for concrete. Proceedings of twenty-eighth DDESB seminar. Orlando, FL: ANSI Std.; 1998. p. 1e17. [8] Cotsovos DM, Pavlovi c MN. Numerical investigation of concrete subjected to high rates of uniaxial tensile loading. International Journal of Impact Engineering 2008;35:319e35. [9] Bischoff PH, Perry SH. Compressive behaviour of concrete at high strain rates. Materials and Structures 1991;24:425e50. [10] Hao H, Hao Y, Li ZX. A numerical study of factors influencing high-speed impact tests of concrete material properties. In: Wu C, Lok TS, editors. Key note in proceedings of 8th international conference on shock & impact loads on structures. Adelaide: CI-Premier Pte Ltd; 2009. p. 37e52. [11] Li QM, Meng H. About the dynamic strength enhancement of concrete-like materials in a split Hopkinson pressure bar test. International Journal of Solids and Structures 2003;40:343e60.

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