Dynamic mechanical properties of cement and asphalt mortar based on SHPB test

Construction and Building Materials 70 (2014) 217–225

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Dynamic mechanical properties of cement and asphalt mortar based on SHPB test You-jun Xie, Qiang Fu ⇑, Ke-ren Zheng, Qiang Yuan, Hao Song School of Civil Engineering, Central South University, Changsha, Hunan 410075, PR China

h i g h l i g h t s  The range of actual strain rate of CA mortar was calculated.  The dynamic mechanical properties of CA mortar can be studied by SHPB test.  The microscopic mechanism of dynamic mechanical properties was analyzed.  The specific energy absorption increases with the increase of strain rate.  The established constitutive model agrees well with the experimental results.

a r t i c l e

i n f o

Article history: Received 21 April 2014 Received in revised form 2 July 2014 Accepted 23 July 2014 Available online 20 August 2014 Keywords: CA mortar Strain rate Split Hopkinson pressure bar Peak strength Specific energy absorption Constitutive model

a b s t r a c t In order to study dynamic mechanical properties of cement and asphalt mortar (CA mortar), the compressive stress–strain experiment of CA mortar at intermediate strain rates was carried out by split Hopkinson pressure bar (SHPB). The experimental results show that the peak strength of CA mortar increases with the increase of strain rate, and the growth rate decreases gradually. The discrete dynamic Young’s modulus is 19.04–380.82 times as big as that at the quasi-static loading rate. The specific energy absorption of CA mortar increases exponentially with the increase of strain rate. Dynamic constitutive model of CA mortar was established, and agreed well with experimental results. The parameter m mainly influences the peak strength of constitutive model, the peak strength increases with the increase of m. While the parameter n mainly influences the descent velocity of stress of constitutive model after peak strength, and the bigger n is, the bigger the descent velocity is. The study results can provide theoretical reference for the structural design of ballastless slab track. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Cement and asphalt mortar (CA mortar), consisting mainly of cement, emulsified asphalt, fine aggregates, water and admixtures, is the material of the cushion layer of ballastless slab track for high-speed railway, and it is a viscoelastic, organic–inorganic composite material formed by simultaneous processes of the cement hydration and demulsification of emulsified asphalt, whose main functions are bearing load, transmitting load, providing damping and geometrical adjustment in the construction [1–4]. Japan, Germany, Britain and France studied CA mortar relatively earlier around the world. The study of Japanese researchers concentrated on the workability, conventional mechanical properties,

⇑ Corresponding author. Fax: +86 731 82656568. E-mail address: [email protected] (Q. Fu). http://dx.doi.org/10.1016/j.conbuildmat.2014.07.092 0950-0618/Ó 2014 Elsevier Ltd. All rights reserved.

construction technology, weather resistance and durability of CA mortar. As the track structure is different, the study of Germany, Britain and France mainly focused on the mechanical properties, elastic properties and durability of CA mortar [5–8]. In the process of train operation, CA mortar is at the dynamic loads, and its mechanical properties are influenced by time and frequency of loads. The effect of strain rate on mechanical properties of CA mortar with different asphalt contents were studied [9,10]. The peak stress and Young’s modulus of CA mortar increase with the increase of strain rate, and the higher the asphalt content is, the more obvious the effect of strain rate on CA mortar is. Studies [11,12] showed that, on the same loading rate, CA mortar is more sensitive to strain rate than concrete. Strain rate is a measure to characterize the deformation speed of material. Different materials exhibit different mechanical responses with the change of strain rate. Strain rate can be classified according to its amplitude: low strain rates (105–102 s1),

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intermediate strain rates (101–102 s1), high strain rates (102– 104 s1) and ultra high strain rates (more than 104 s1) [13–15]. In practical engineering, when the high-speed train runs at 250 km/h, the maximum value of the alternating load of train is about 0.1 MPa, and the vibration frequency of wheels and rail is about 1000–1400 Hz, if the Young’s modulus of CA mortar is 500 MPa, the calculated strain rate of CA mortar is 2.0– 2.8  101 s1 [12,16]. When the gradient stress of temperature is considered and CA mortar is partially damaged or in severe vibration (such as earthquake), its actual strain rate is greater and much greater than low strain rates. However, previous study on mechanical properties of CA mortar was mainly at low strain rate [9,10,12]. At present, split Hopkinson pressure bar (SHPB) device is usually used to study dynamic mechanical properties of materials at the loading rate of intermediate strain rates [17–24]. In this work, SHPB test device of straight taper and variable cross-section with rod diameter of 50 mm was used to study the dynamic mechanical properties of CA mortar at the loading rate of 8 strain rates (h1 = 21.5 s1, h2 = 23.54 s1, h3 = 56.56 s1, h4 = 73.6 s1, h5 = 74.74 s1, h6 = 84.11 s1, h7 = 89.06 s1, h8 = 101.1 s1), and the intrinsic mechanism of change of peak strength, Young’s modulus, specific energy absorption with strain rate was measured, and a constitutive model of CA mortar was established. The study results can provide reference for the structural design of ballastless slab track.

2. Experimental 2.1. Raw material The following materials were used to make the CA mortar specimens: special dry powder, SBS modified cationic emulsified asphalt and tap water. Special dry powder for CRTS (China Railway Track System) I type CA mortar provided by Anhui Engineering Material Technology Ltd., which is composed of Portland cement, pelletized sand, aluminum powder, expansive agent and other additives. 24 h volume expansion rate of the dry powder is 2.1%, and 1d compressive strength is 6.89 MPa according to Chinese standard. Its cement content is about 33% (by mass). Asphalt content (by mass) of SBS modified cationic emulsified asphalt is 58%, and whose physical properties are listed in Table 1. The mass ratio of asphalt and cement in CA mortar is 0.85. Experimental results show that the fluidity of fresh mortar is 24.15 s and gas content is 8.42% (by volume), which meet the Chinese regulation [25] put in references.

Fig. 1. CA mortar specimens.

28d compressive strength of the CA mortar is 5.08 MPa and the Young’s modulus is 525.18 MPa. Finally, the dynamic mechanical experiment of CA mortar was carried out by the SHPB test device within a temperature controlled at 23 ± 2 °C. 2.3. Introduction of SHPB test 2.3.1. SHPB test device The SHPB test device with rod diameter of 50 mm used in the study is shown in Fig. 2, which is composed of three sections: the main equipment, energy system and test system. Data is collected by CS-10 super dynamic strain gauge and DL-750 oscilloscope made in Beidaihe electronic instrument factory. Data is processed by data processing software CLRM independently developed based on Visual C++ platform. The loading pulse is half sine stress wave, as shown in Fig. 3, which extends the rising time of incident pulse and helps CA mortar specimen reach a uniform stress state before failure. Dynamic mechanical parameters of specimens can be calculated indirectly by the measured voltage in strain gauge in test: stress r(t), strain e(t) and strain rate h(t), which can be expressed as:

rðtÞ ¼

Ae Ee ½ei ðtÞ þ er ðtÞ þ et ðtÞ 2As

eðtÞ ¼

ce ls

hðtÞ ¼ 2.2. Experimental method CA mortar specimens were prepared by the following procedures: the emulsified asphalt and water were first poured into the stirring pot and slowly stirred (140 r/min) for 1 min. In the stirring process, proper amount (0.05 g/L) of defoamer was added to eliminate the big bubble. Then, the dry powder was slowly added into the stirring pot, which took less than 30 s. After adding the dry powder, the mixture was slowly stirred for 1 min, then quickly stirred (285 r/min) for 2 min and slowly stirred for 30 s to eliminate bigger bubble in the mortar. After mixing, the fluidity and gas content of fresh mortar were tested. Finally, the mortar was placed into mold with the dimension of £50 mm  50 mm which was demolded after 24 h, and the mortar specimens were kept at 20 ± 3 °C, 65 ± 5% RH to the defined age. The upper and lower surfaces of CA mortar specimens were polished by doubleend face automatic polishing machine before experiment to ensure the upper and lower surfaces are parallel. The polished specimens were shown in Fig. 1. Then, the mechanical properties of CA mortar at the quasi-static loading rate (standard loading rate, 1 mm/min) was experimented by electronic universal testing machine in order to provide a reference for the following dynamic mechanical analysis. The

Z

ð1Þ

t

½ei ðtÞ  er ðtÞ  et ðtÞ dt

ð2Þ

0

ce ½ei ðtÞ  er ðtÞ  et ðtÞ ls

ð3Þ

where Ae, Ee are cross-sectional area and Young’s modulus of pressure bar, respectively, As, ls are cross-sectional area and length of CA mortar specimens, respectively, ei(t), er(t), et(t) are the incident strain signal, reflected strain signal and transmitted strain signal, respectively, ce is the P-wave velocity in the pressure bar [26]. 2.3.2. Effectiveness analysis SHPB test technique has been widely used in the study on dynamic mechanical properties of quasi-brittle materials such as concrete, rock and so on. However, CA mortar is a typical viscoelastic material, and few publications about the study on impact properties of CA mortar by SHPB test technique were found. Therefore, the applicability of SHPB test technology to study the dynamic mechanical properties of CA mortar needs to be clarified. Effective application of SHPB test technique needs to satisfy the propagation theory of one-dimensional elastic stress wave in pressure bars and uniform stress hypothesis in specimens. The propagation theory of one-dimensional elastic wave requires k > 5d, where k is the width of incident stress pulse, and d is the diameter of pressure bar. In this paper, d = 0.05 m, k can be expressed as:

Table 1 The physical properties of emulsified asphalt. Solid content/%

58

Engler viscosity (25 °C)

Sieve residue (1.18 mm)/%

Storage stability (25 °C)/% 1d

5d

5.8

0.005

0.35

1.86

Evaporation residue penetration (25 °C)/0.1 mm

Evaporation residues ductility (15 °C)/cm

82

52

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a

35

θ1 θ2 θ3 θ4

30

σ /MPa

25 20 15 10 5 0

b Fig. 2. SHPB test device.

0

0.005

0.01

0.015

ε

35

25

σ /MPa

0.2

Amplitude/mV

0.15

0.03

0.025

0.03

20 15

0.1

10

0.05

5

0 0

200

400

600

800

100

120

0

-0.05

0

-0.1 -0.15

0

Incident pulse

0.005

0.01

0.015

ε

0.02

Fig. 4. Dynamic stress–strain relationship of CA mortar: (a) h1–h4; (b) h5–h8.

3. Results and analysis

t /μs

3.1. Dynamic stress–strain relationship

Fig. 3. Incident pulse, reflected pulse and transmitted pulse.

k ¼ c e se

ð4Þ

where se is the propagation time of stress pulse, generally taken as 400 ls [27], ce is P-wave velocity, ce = 5400 m/s. It can be obtained by calculation as:

k ¼ ce se ¼ 5400  0:0004 ¼ 2:16 > 5d ¼ 0:25

ð5Þ

So the stress wave in the pressure bar satisfies the condition of one-dimensional propagation. The uniform stress hypothesis in specimens requires that the stress pulse is reflected back and forth 2 times at least in CA mortar specimens before destruction. The time cost by the pulse reflected back and forth 1 time in CA mortar specimens is 2ls/cs, where ls is the length of specimens, taken as 0.05 m, cs is the shearing wave velocity in the specimens and can be calculated by the following equation as [28]:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Es 1m qs ð1 þ mÞð1  2mÞ

0

Transmitted pulse

-0.2

cs ¼

0.025

θ5 θ6 θ7 θ8

30

Reflected pulse

0.02

ð6Þ

where Es, m are the Young’s modulus and Poisson’s ratio of CA mortar at the quasistatic loading rate, Es = 525.18 MPa, m = 0.45, qs is the density of CA mortar, taken as 1300 kg/m3. cs = 1532.36 m/s can be calculated, and so 2ls/cs = 65.26 ls. It can be known from Fig. 3 that the rising time of the rising edge of incident pulse is about 280 ls, during which the stress pulse has been reflected back and forth several times in CA mortar. So the specimen will be able to achieve uniform stress state.

The stress–strain curves of CA mortar on different loading rates are shown in Fig. 4. As shown in Fig. 4, the stress–strain curves of CA mortar at the dynamic impact loads show some oscillation characteristics, and the oscillation characteristics are more obvious with the increase of strain rate. In CA mortar, there are interfaces between asphalthydration products of cement, asphalt – fine aggregates, fine aggregates – hydration products of cement and micro pores caused by water loss, which are weak zones [29]. At the impact compression of high strain rate, the extension of micro cracks stems largely from the weak interfaces and micro cracks in cement stone. When the micro cracks in cement stone extend to the interface of asphalthydration products of cement, because of the blocking effect of asphalt membrane structure which is perpendicular to the interface, the extension of micro cracks is terminated. The micro cracks that stem from the strength weakening interfaces only extend toward the weakest areas, after meeting the larger cracks, the connectivity of two types of cracks makes their extension velocity increase rapidly, therefore, the flow stress of CA mortar shows continuous oscillation during deformation [30]. It can be seen from Fig. 4 that the peak strength of CA mortar increases with the increase of strain rate. According to the material damage theory of Griffith, when the load reaches a certain degree, the micro cracks in CA mortar will germinate and extend until they connect each other, which causes the destruction of CA mortar. The

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energy required by the germination of cracks is much larger than that needed by the extension of cracks. When strain rate is relatively high, high-speed impact will make more micro cracks in CA mortar, and the deformation of CA mortar is also larger and will consume more energy. Because the loading time is short, there is not enough time for the propagation of the micro cracks in CA mortar. In order to dissipate energy effectively, according to work– energy principle, CA mortar will offset or dissipate external energy by increasing stress. On the other hand, when strain rate is higher, the lateral deformation of CA mortar is limited by inertial effect. CA mortar is in the mechanical response process of transformation from a state of one-dimensional stress to a state of one-dimensional strain, in other words, the lateral deformation in the middle part of CA mortar specimens is restrained, so CA mortar is approximately in the confining pressure state. The effect of confining pressure increases with the increase of strain rate, and the damage stress of CA mortar will keep growing [31–33]. In the process of cementation and hardening of CA mortar, the hydration of cement consumes large amount of water. Meanwhile, the combined effect of external heat and local hydration heat causes the demulsification of emulsified asphalt. As the asphalt content of CA mortar is large, asphalt particles will coalesce with each other and form asphalt membrane structure around the hydration products of cement, fine aggregates and non-hydrated cement particles. At last, they will form the spatial network structure, and the hydration products of cement and fine aggregates will fill in the asphalt membrane network structure in the similar form of filler. The microstructure of CA mortar is shown in Fig. 5. Asphalt is a typical polymer material, at the effect of high strain rate, the motion of asphalt chains lags behind the change of external stress due to the effect of internal friction. Therefore, the big elasticity of asphalt constrains the lateral deformation of CA mortar specimens strongly, and the one-dimensional strain state of CA mortar is more obvious, so is the strain rate hardening effect of peak stress. The damage of CA mortar is mainly caused by the formation and extension of internal micro cracks. Fig. 6 shows the damage of CA mortar at the high and low strain rates, respectively. At the loading rate of low strain rates, CA mortar shows splitting destruction along a longitudinal crack in the middle, and its damage is relatively low. However, at the loading rate of high strain rates, CA mortar shows fractured state, and more micro cracks are formed internally. The formation and extension of micro cracks are the main way of energy dissipation, while the increase of peak strength plays an auxiliary role. Meanwhile, the restriction effect of confin-

Fig. 6. Failure mode of CA mortar: (a) strain rate is 21.5 s1; (b) strain rate is 101.1 s1.

ing pressure in one-dimensional strain state does not increase indefinitely. When the amount of micro cracks raises, CA mortar’s extension scope toward outside will increase rapidly, which gradually reduces the inertia effect of lateral deformation and diminishes the growth rate of peak strength with the increase of strain rate. At the quasi-static loading rate, study [12] showed that there is an exponential relationship between the peak strength of CA mortar and strain rates, and the growth rate of peak strength decreases with the increase of strain rate.

3.2. Effect of strain rate on dynamic peak strength and dynamic Young’s modulus Dynamic increase factor (DIF) equals to the peak strength at the impact loads divided by that at the quasi-static loading rate, which usually reflects the change degree of dynamic peak strength of material with strain rate [34,35], and can be expressed as:

DIF ¼

Fig. 5. Microstructure of CA mortar.

rd rs

ð7Þ

where rd is the dynamic peak strength of CA mortar, rs is the peak strength at the quasi-static loading rate (in this paper, the conventional loading rate (1 mm/min = 3.33  103 s1) is considered as the quasi-static loading rate), rs = 5.08 MPa.

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7

700

y = 4.4295x - 2.9063 R 2 = 0.955

600

6

5

E /GPa

σ d /σ s

500 400 300

4

200

3 100

2 1.3

1.5

1.7

1.9

0

2.1

20

0

lgθ

40

60

80

100

120

θ /s-1

Fig. 7. The relationship between DIF and lgh.

Fig. 8. The relationship between dynamic Young’s modulus and strain rate.

a

θ1 θ2 θ3 θ4

0.3

0.2

-3

W/(J•cm )

0.25

0.15 0.1 0.05 0 0

50

100

150

200

250

t /μs

b

0.7

θ5 θ6 θ7 θ8

0.6 0.5 -3

W/(J•cm )

The relationship between DIF of CA mortar and the logarithm of strain rate lgh is shown in Fig. 7, and y represents rd/rs, x represents lgh, R2 is the multiple correlation coefficient. The increase rate of DIF of CA mortar is 4.4295 with the change of the logarithm of strain rate, which is much bigger than that of concrete, rock and other similar quasi-brittle materials (generally smaller than 4), i.e. the strain rate sensitivity of CA mortar is more obvious. This is because of the asphalt composition and the unique spatial network structure of CA mortar. Due to the nonlinear characteristics of stress–strain curves of CA mortar before destruction, and the nonlinearity intensifies with the increase of stress. At present, there is no standard method to calculate Young’s modulus from SHPB test. In this paper, the dynamic Young’s modulus of CA mortar is defined as the secant modulus between 30% of peak strength and origin, the test results are shown in Fig. 8. The dynamic Young’s modulus of CA mortar is very discrete and has no obvious relation to strain rate. The initial stage of stress– strain curves obtained from SHPB test is in the initial stage of stress pulse, so the stress in CA mortar is nonuniform and shows some fluctuation effect. However, it can be seen from Fig. 8 that the dynamic Young’s modulus of CA mortar fluctuates in the range of 10–200 GPa approximately, it is about 19.04–380.82 times as big as the Young’s modulus at the quasi-static loading rate. The strain rate effect of dynamic Young’s modulus fully reflects the strain rate sensitivity of CA mortar.

0.4 0.3

3.3. Effect of strain rate on specific energy absorption

0.2 The toughness of CA mortar can be expressed as specific energy absorption, which is the capacity to absorb energy of stress wave for CA mortar per unit volume. It can be expressed in the following equation [36]:

W ¼ W i  ðW r þ W t Þ

Wr ¼

Ae c e Ee As ls

Z

t 2 i ðtÞ dt

ð9Þ

r2r ðtÞ dt

ð10Þ

r

0

Z 0

0 0

ð8Þ

where W is specific energy absorption of CA mortar, Wi, Wr, Wt are the energy of incident wave, reflected wave and transmitted wave, respectively, and can be expressed as:

Ae c e Wi ¼ Ee As ls

0.1

50

100

150

200

250

t /μs Fig. 9. Specific energy absorption of CA mortar: (a) h1–h4; (b) h5–h8.

Wt ¼

Ae c e Ee As ls

Z

t 0

r2t ðtÞ dt

ð11Þ

t

in which ri(t), rr(t), rt(t) are the stress of incident wave, reflected wave and transmitted wave, respectively.

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0.8

The strain of damaged micro elements in CA mortar obeys Weibull distribution [38], whose probability density function can be expressed as:

y = 0.0201e0.0348x R 2 = 0.97

0.7

-3

W/(J•cm )

0.6

P½e ¼

0.5

0.3

  n  Z e  n1 n e e de P½e de ¼ N exp  m 0 0 m m    n  e ¼ N 1  exp  m

0.2

Nf ¼ N

0.1 0

20

40

60

θ /s

80

100

120

-1

Fig. 10. Relationship between the maximum of specific energy absorption and strain rate.

The change curves of specific energy absorption of CA mortar obtained through calculation are shown in Fig. 9. As shown in Fig. 9, in the initial stage, the specific energy absorption of CA mortar has little change with the propagation of stress wave, but the specific energy absorption increases rapidly with the extension of stress wave. The bigger the strain rate is, the faster the increase rate of specific energy absorption is. Subsequently, the specific energy absorption will remain stable after reaching a certain value. The maximum specific energy absorption appears at the destruction of CA mortar. The relationship between the maximum specific energy absorption and strain rate is shown in Fig. 10, and y represents W, x represents h. With the increase of strain rate, the specific energy absorption of CA mortar increases exponentially. The amount of specific energy absorption has close relation to the internal damage characteristics of CA mortar. The effect of external energy is the direct reason of the internal damage of CA mortar. The inoculation, germination, extension and connection of micro cracks need to absorb energy. It can be known from Section 3.1 that with the increase of strain rate, the more cracks in CA mortar are produced, and the more energy is required. So the specific energy absorption of CA mortar is bigger.

Z

e

ð14Þ

where Nf is the number of damaged micro elements, N is the number of total micro elements in CA mortar. So, the damage variable D can be written as:

D¼

  n  Nf e ¼ 1  exp  N m

ð15Þ

Substituting Eq. (15) into Eq. (12) yields the constitutive model of CA mortar:

  n  e

r ¼ Ee exp 

ð16Þ

m

4.2. Calculation of parameters and verification of constitutive model It can be known from the dynamic stress–strain relationship of CA mortar that the formula drd/ded = 0 should be satisfied at the peak strength (rd, ed) of stress–strain curves, which can be expressed as:

  n  e n  dr ed d 1n ¼0 ¼ E exp  de m m

ð17Þ

h i n n – 0; so 1  n emd ¼ 0, and it can be obtained that: E exp  emd

ed

m ¼ 1 1 n

ð18Þ

n

Substituting Eq. (18) into Eq. (16) yields the parameter n:

n¼

1

ð19Þ

ln Ered d

4. Constitutive model 4.1. Establishment of constitutive model Assume that CA mortar is composed of innumerable tiny micro elements. In macro, the micro elements are small enough; in micro, the micro elements are big enough; and each micro element can summarize all physical and mechanical properties of CA mortar. The micro elements are linear elastic before destruction, and their stress–strain relationship obeys the Hooke law. Micro elements have no bearing capacity after destruction. The extension of micro cracks in CA mortar appears in micro elements and is reflected as damage variable in the constitutive model. The damage in CA mortar is isotropy. According to the equivalent strain hypothesis of Lemaitre [37], the damage constitutive model of CA mortar can be established as:

r ¼ Eeð1  DÞ

ð13Þ

where m, n are the relevant parameters for the Weibull distribution, respectively. When the stress reaches a certain value, the number of damaged micro elements is expressed as:

0.4

0

  n  n  e n1 e   exp  m m m

ð12Þ

where D is damage variable, which is macro manifestation of the ratio of damaged micro elements to total micro elements and changes in the range of 0–1.

The parameters of constitutive model and the correlation coefficients between fitting results and experimental results are listed in Table 2, and the fitting results and experimental results are compared in Fig. 11. It can be known from Table 2 and Fig. 11 that all the correlation coefficients are bigger than 0.9 except h2. The consistency of fitting results and experimental results before peak strength is more

Table 2 Parameters of constitutive model. Number

h1 h2 h3 h4 h5 h6 h7 h8

Parameter m

n

R

0.002498 0.000009747 0.001809 0.000932 0.003227 0.007992 0.000227 0.000146

0.632558 0.272865 0.68533 0.535279 0.781553 1.106563 0.371553 0.350091

0.9538 0.8640 0.9543 0.9590 0.9630 0.9782 0.9361 0.9570

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10

15

σ /M Pa

18

σ /MPa

12

8 Experimental value Fitting curve

6 4

Experimental value Fitting curve

9 6

20 15

5 0

0

0

0

0.01 0.01 0.01 0.02

0

ε

e

30

0.002

0.004

ε

0.006

0

0.008

f

30

20

20

20

Experimental value Fitting curve

10 5

σ /M Pa

25

σ /M Pa

25

15 Experimental value Fitting curve

10 5

0

0.005

0.01

0.015

ε

g

0.02

0.005

0.01

ε

σ /MPa

0.015

0.02

30

20

25

15 Experimental value Fitting curve

ε

15

ε

Experimental value Fitting curve

5

0.005 0.01 0.015 0.02 0.025

0.005 0.01 0.015 0.02 0.025

20

10

0

0

35

25

0

0.015

Experimental value Fitting curve

5

h

5

0.01

0

0

30

10

ε

15 10

0

0

0.005

30

25

15

Experimental value Fitting curve

10

3

0

σ /M Pa

30 25

12

2

d

c

21

σ /M Pa

b

14

σ /MPa

a

0 0

0.005 0.01 0.015 0.02 0.025

ε

Fig. 11. Comparison of fitting results of constitutive model and experimental results: (a) h1; (b) h2; (c) h3; (d) h4; (e) h5; (f) h6; (g) h7; (h) h8.

obvious. The biggest difference between fitting results of peak strength and experimental results is only 5.23%. The stress–strain relationship before peak strength is the main part that reflects the dynamic mechanical properties of CA mortar, so this constitutive model can effectively predict the dynamic mechanical properties of CA mortar at the impact loads. The constitutive model of CA mortar in this work is established under the conventional conditions of environment and specimens’ size. With the changes of environment and specimens’ size, the parameters of constitutive model will change. So as to enhance the applicability of constitutive model, it needs to establish the relationship between the parameters and experimental conditions, which will be the further work.

16

σ /MPa

12

8

n=0.582558 n=0.632558

4

n=0.682558 0

0

0.003

0.006

ε

0.009

4.3. Effect of parameters on stress–strain relationship

0.012

Fig. 12. The effect of n on constitutive model.

0.015

The stress–strain relationship of sample h1 is taken as an example. The stress–strain curves at the same dynamic Young’s modulus E and the parameter m and the different parameters n are shown in Fig. 12. The stress–strain curves at the same Young’s modulus E

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(3) The constitutive model established in this paper based on statistical damage theory can effectively predict the dynamic mechanical properties of CA mortar at the impact loads, especially for the segments before peak strength of stress–strain curves.

m=0.001498 m=0.002498 m=0.003498

20

16

σ /MPa

Acknowledgements The authors thank National 973 Program of China (Grant No. 2013CB036200) and National Natural Science Found Program of China (Grant No. 51278498) for the support given to this study.

12

8

References

4

0 0

0.003

0.006

0.009

0.012

0.015

ε Fig. 13. The effect of m on constitutive model.

and the parameter n and the different parameters m are shown in Fig. 13. When the parameter n changes, the curves’ shape changes significantly. The descent segments after the peak strength become steeper with the increase of n, which represents the enhancement of brittleness of material. The smaller n is, the flatter the descent segment is, which represents the enhancement of deformation capacity of material. It can be seen from Fig. 12 that the peak strength increases slightly when n is smaller. It can be known from Fig. 13 that the parameter m has little effect on the overall shape of stress–strain curves, but has great effect on the peak strength. Peak strength increases obviously with the increase of m. In summary, the peak strength is influenced by both m and n, but m plays a decisive role. The descent velocity of stress after peak strength mainly depends on n. 5. Conclusions The stress–strain relationship of CA mortar at the intermediate strain rates was obtained by SHPB test, and the mechanical response mechanism of the change of the peak strength, the Yong’s modulus and specific energy absorption with strain rates was analyzed. Finally, the constitutive model of CA mortar based on statistical damage theory was established. The following conclusions can be drawn from the present study: (1) For CA mortar, one-dimensional stress wave theory and uniform stress state in specimens can be met in SHPB test, which ensures the reliability of dynamic experimental results. (2) The peak strength of CA mortar increases with the increase of strain rate, but the growth rate decreases gradually. The dynamic Young’s modulus of CA mortar is very discrete and has no obvious relation to strain rate. The dynamic Young’s modulus of CA mortar fluctuates in the range of 10–200 GPa approximately, it is about 19.04–380.82 times as big as the Young’s modulus at the quasi-static loading rate. The specific energy absorption of CA mortar increases with the increase of strain rate.

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