A concrete constitutive model considering coupled effects of high temperature and high strain rate

Accepted Manuscript

A concrete constitutive model considering coupled effects of high temperature and high strain rate Xiao Yu , Li Chen , Qin Fang , Zheng Ruan , Jian Hong , Hengbo Xiang PII: DOI: Reference:

S0734-743X(16)30581-4 10.1016/j.ijimpeng.2016.11.009 IE 2775

To appear in:

International Journal of Impact Engineering

Received date: Accepted date:

29 August 2016 14 November 2016

Please cite this article as: Xiao Yu , Li Chen , Qin Fang , Zheng Ruan , Jian Hong , Hengbo Xiang , A concrete constitutive model considering coupled effects of high temperature and high strain rate, International Journal of Impact Engineering (2016), doi: 10.1016/j.ijimpeng.2016.11.009

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ACCEPTED MANUSCRIPT

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concrete is proposed. 

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Triaxial test data of concrete after different temperatures were used to define the yield criterion of the proposed model at high temperatures.

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A constitutive model considering coupled effects of high temperature and high strain rate for

Dependent influence of high temperature and high strain rate on mechanical behaviour of

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concrete was discovered. 

The numerical models adopted the proposed model shown good agreement with the test data.

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Xiao Yu, Li Chen*, Qin Fang, Zheng Ruan, Jian Hong, Hengbo Xiang

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PLA University of Science and Technology, State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, Nanjing, 210007, China

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Corresponding Author: Li Chen, Associate Professor, [email protected]

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A concrete constitutive model considering coupled effects of high temperature and high strain rate

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Abstract

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Concrete in RC structures is possibly exposed to fire and blast due to occasional accidents or

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attacks during the service life. To predict the dynamic responses of RC structures under fire and

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blast precisely, the Drucker-Prager (DP) constitutive model was modified to consider the

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coupled effects of high temperature and high strain rate for concrete. The piecewise function for

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the complete stress-strain curve of concrete at the ambient temperature was extended for damage

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and high temperatures. The yield criterion with the exponent form and the associated flow rule

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were employed into the model to describe the plastic behaviors of concrete at high temperature.

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A new coupling function was proposed to determine the coupled effects of high temperature and

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high strain rate on concrete strength on base of the test data. The proposed model was

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implemented into the commercial FE software ABAQUS through VUMAT. A fine numerical

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analysis on the existing SHPB test at high temperature was conducted to validate the proposed

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model, and a good agreement with the test data was observed.

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Keywords: Concrete; Constitutive model; High temperature; High strain rate

1. Introduction

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Concrete is commonly used in civilian and military engineering structures. In addition to the normal

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design loadings, some structures might suffer impact, explosion and fire during the service time. It

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leads to the fact that concrete material in the structures are in danger of fire exposure and impact or

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blast, simultaneously. Safety concern about the multiple threaten of blast and fire has been greatly

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raised on the engineering structures in the academia, ever since the terrorist attack 9.11 in New

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York.

ACCEPTED MANUSCRIPT 3 Predicting the behaviors of structural components under both fire and impact or blast loading

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is a prerequisite for the design and analysis of disaster-resistant RC structures. Recent advances in

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computational method and procedure, coupled with the emergence of powerful computers,

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provides a tremendous opportunity for the numerical assessment of the behaviors of RC structures

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subjected to blast and fire. Hence, it is very important and even imperative to develop a concrete

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constitutive model considering coupled effects of high temperature and high strain rate.

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Blast loads usually induce high strain rate on the construction material of structures, while the

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fire causes the high temperature effect. Actually, concrete is a composite material composed of

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coarse aggregate, cement, water and some other additives. Strain rate and temperature effect are

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two important factors affecting the mechanical behaviors of concrete as a brittle material [1].

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During the past several years, each of them has been extensively studied. High temperature leads

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to an irreversible loss of elastic stiffness (thermal damage) and of material strength (thermal

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decohesion) of concrete. On the macroscale of engineering material description (the typical scale

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of laboratory test specimens), thermal damage and thermal decohesion are generally described by

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an apparent temperature dependence of the material properties of concrete [2]. Actually, some of

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this thermal damage and thermal decohesion might result from the difference in thermal dilatation

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coefficient between the aggregates and the cement paste, in particular in concretes with silicious

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aggregates. From microstructural analysis of fire-damaged concrete, it has been found that both

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thermal damage and thermal decohesion result rather from the dehydration of concrete on the

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microlevel, e.g., Lin et al. [3]. Existing evidences revealed that the elevated temperature exposure

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over 400ºC leads to a great static ultimate strength loss of normal concrete [4]. Actually, the fire

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exposure would not only deteriorate the quasi-static concrete strength but also lead to a remarkable

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influence on the strain rate effect that change the dynamic strength and cracking criterion of

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concrete which is different from those at the ambient temperature [5-7].

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Most of the existing concrete constitutive models are limited to the ambient temperature.

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Only a few of them have been improved for elevated temperatures. Based on a coupled

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plastic-damage model that has been extended to consider the damage induced by high

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temperatures, Luccioni et al. [8] developed a thermo-mechanical model for concrete subjected to

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high temperatures. Nechnech et al. [9] developed a computational model for concrete structures at

ACCEPTED MANUSCRIPT 4 high temperatures considering the diffused thermal damage. By introducing the coupled effects of

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high temperature and high strain rate in concrete, Ruan et al. [10] preliminarily calibrated the

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parameters of the damage plasticity concrete model in ABAQUS. However, these concrete

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constitutive models were basically modified for static or even one dimensional analysis. Moreover,

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the referred temperature was always not high enough, generally below 400℃. Actually, the current

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high temperature in the fire is basically up to 800~1200℃[5]. Three dimensional properties and

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strain rate effects at the elevated temperature were not considered yet, which is of great important

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for predicting responses of RC structures in fire and blast more precisely.

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The proposed constitutive model for concrete in this study was developed in the framework

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of elastoplasticity by extending the classical Drucker-Prager model. A multi-axial static

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constitutive model of concrete considering the effect of temperature was firstly built. The yield

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function with the exponent form was chosen to provide the most general yield criterion. The

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existing triaxial test data [11] of concrete after high temperature were used to fit the yield curve.

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The plastic flow was defined as a hyperbolic flow potential that is associated on the failure yield

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surface in the meridional plane. The strain rate effects at elevated temperatures (up to 800 ºC ) was

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then considered to extend the model to be capable of considering the coupled effects of high

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temperature and high strain rate. Furthermore, the proposed model was implemented into the

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commercial software ABAQUS through the VUMAT port. Numerical results obtained in the

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finite-element analysis were compared with the test results.

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2. Static concrete constitutive model for high temperature

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Several multiple-parameter models [12-18] for concrete were developed on the base of

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Mohr-Coulomb and Drucker-Prager yield criterion. The extended Drucker-Prager exponent model

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[19] is a generally accepted to describe the triaxial behavior of concrete. This model was firstly

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recoded by associating some critical parameters to the temperature.

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2.1 Yield criterion

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The typical yield surfaces of concrete in the deviatoric plane are shown in Fig. 1. With the

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hydrostatic pressure increases, the smoothness of yield surfaces increases. Under high hydrostatic

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pressure state, yield surfaces of concrete in the deviatoric plane are circles. The extended

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Drucker-Prager exponent model, as shown in Fig. 2, which could provide the circular yield

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surfaces at every hydrostatic pressures, is used to model the yield surfaces of concrete under

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triaxial stress state. The yield function is expressed by eq. (1) 𝐹 = 𝑎𝑞 𝑏 − 𝑝 − 𝑝𝑡 = 0

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(1)

where the hydrostatic pressure p = I1/3; q = √3𝐽2 ; 𝐼1 is the first invariant of stress tensor; 𝐽2 is

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the second invariant of stress-deviation tensor; a and b are material parameters that are

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independent of plastic deformation, and 𝑝𝑡 is the hardening parameter that represents the

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hydrostatic tension strength of the material; 𝑝𝑡 is related to the test data as

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𝑝𝑡 = {

𝑎𝜎𝑐𝑏 − 𝑎𝜎𝑡𝑏 +

𝜎𝑐 3 𝜎𝑡 3

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𝑖𝑓 𝑕𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑡𝑕𝑒 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑦𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑠𝑠, 𝜎𝑐 ; 𝑖𝑓 𝑕𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑡𝑕𝑒 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑦𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑠𝑠, 𝜎𝑡 .

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(2)

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S1

S3

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Fig. 1. Typical yield surfaces of concrete in the deviatoric plane.

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q

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hardening

pt 108 109

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Fig. 2. Yield surfaces of exponent model in the meridional plane.

Because of the difficulty on maintaining temperature when loading, triaxial test data of

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concrete at high temperatures have not been generally reported so far. It has been proved that the

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strength of concrete only depend on the highest temperature ever experienced, but not the current

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temperature [20]. Therefore, triaxial test data of concrete after high temperatures [11], as shown in

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Table 1, were used in this manuscript to describe the behavior of concrete at high temperatures.

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Lines plotted in Fig. 3 denote the fitting trend of yield surfaces at high temperatures in the

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meridional plane suggested by Eq. (1). As expected, the concrete material exposed to high

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temperature exhibits a substantial decrease in the slopes of the fitting surfaces, which indicates the

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effect of temperature on reducing the multi-axial pressure strength. Good agreement in high

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hydrostatic pressure between the fitting surfaces and the test data indicates that the model is more

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suitable for describing mechanical behavior of concrete under high hydrostatic pressure and high

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temperature.

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The formulae of parameters a and b were also determined by fitting the test data. They are

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expressed as eq. (3) and eq. (4), as shown in Fig. 4,

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𝑎 = 4.20 × 10;6 × 𝑒𝑥𝑝(𝑇⁄123.44) × 1.34 × 10;7 20℃ ≤ 𝑇 ≤ 600℃

(3)

𝑏 = 4.266 − 4.32 × 10;3 𝑇 + 8.76 × 10;6 𝑇 2 − 7.70 × 10;9 𝑇 3 20℃ ≤ 𝑇 ≤ 600℃ 125

(4)

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Therefore, parameters in eq. (1) are all available except 𝑝𝑡 . Since the hardening is defined by

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the uniaxial compression test data, 𝑝𝑡 could be expressed as eq. (5). Hence, the remained key

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parameter determining the yield surface at different temperatures is the uniaxial compression yield

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strength, 𝑝𝑡 = 𝑎𝜎𝑐𝑏 −

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𝜎𝑐

(5)

3

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Table 1

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Triaxial test data of concrete after different temperatures [11]. 𝜎2

𝜎3

MPa

MPa

MPa

0:0:1

0.00

0.00

34.20

11.40

34.20

60.00

0:0.25:1

0.00

10.75

43.01

17.92

38.78

46.10

0:0.5:1

0.00

22.75

45.49

22.75

39.40

30.00

0:0.75:1

0.00

31.88

42.50

24.79

38.32

13.90

0:1:1

0.00

40.70

40.70

27.13

40.71

0.00

-0.1:0:1

-1.55

0.00

15.40

4.62

℃

20

o

16.23

55.26

-2.04

0.00

10.21

2.72

11.37

51.06

-2.31

0.00

9.32

2.34

10.67

49.19

0:0:-1

-3.14

0.00

0.00

-1.05

3.14

0.00

0:0:1

0.00

0.00

32.60

10.87

32.61

60.00

0:0.25:1

0.00

10.30

41.20

17.17

37.15

46.10

0:0.5:1

0.00

22.40

44.80

22.40

38.80

30.00

0:0.75:1

0.00

31.73

42.30

24.68

38.13

13.90

0:1:1

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-0.2:0:1

0.00

38.50

38.50

25.67

38.51

0.00

ED -1.24

0.00

12.30

3.69

12.96

55.27

-0.2:0:1

-1.62

0.00

8.79

2.39

9.70

51.69

-0.25:0:1

-1.78

0.00

7.12

1.78

8.17

49.11

0:0:-1

-2.70

0.00

0.00

-0.90

2.69

0.00

0:0:1

0.00

0.00

30.50

10.17

30.51

60.00

0:0.25:1

0.00

9.63

38.50

16.04

34.71

46.10

0:0.5:1

0.00

20.75

41.50

20.75

35.94

30.00

0:0.75:1

0.00

29.70

39.60

23.10

35.71

13.90

PT

-0.1:0:1

CE AC 500

√3𝐽2

-0.25:0:1

200

300

𝐼1 /3

Lode angle 𝜃

𝜎1

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Temperature

0:1:1

0.00

35.20

35.20

23.47

35.20

0.00

-0.1:0:1

-1.13

0.00

11.30

3.39

11.90

55.28

-0.2:0:1

-1.48

0.00

6.85

1.79

7.70

50.41

-0.25:0:1

-1.61

0.00

6.02

1.47

6.96

48.45

-1:0:0

-2.01

0.00

0.00

-0.67

2.02

0.00

0:0:1

0.00

0.00

23.80

7.93

23.80

60.00

0:0.25:1

0.00

7.93

31.70

13.21

28.58

46.10

0:0.5:1

0.00

17.30

34.60

17.30

29.98

30.00

0:0.75:1

0.00

25.05

33.40

19.48

30.11

13.90

0:1:1

0.00

29.70

29.70

19.80

29.70

0.00

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-0.81

0.00

8.10

2.43

8.53

55.28

-0.2:0:1

-0.98

0.00

5.21

1.41

5.77

51.53

-0.25:0:1

-1.16

0.00

4.25

1.03

4.94

48.25

-1:0:0

-1.42

0.00

0.00

-0.47

1.42

0.00

0:0:1

0.00

0.00

19.50

6.50

19.50

60.00

0:0.25:1

0.00

6.00

24.00

10.00

21.64

46.10

0:0.5:1

0.00

13.75

27.50

13.75

23.83

30.00

0:0.75:1

0.00

20.25

27.00

15.75

24.34

13.90

0:1:1

0.00

25.20

25.20

16.80

25.21

0.00

-0.1:0:1

-0.57

0.00

5.85

1.76

6.15

55.39

-0.2:0:1

-0.74

0.00

3.84

1.03

4.26

51.31

-0.25:0:1

-0.81

0.00

3.25

0.81

3.73

49.11

-1:0:0

-1.20

0.00

0.00

-0.40

1.21

0.00

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20 15

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10 5

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0 0

133 134

20oC test 200oC test 300oC test 500oC test 600oC test 20oC fitting curve 200oC fitting curve 300oC fitting curve 500oC fitting curve 600oC fitting curve

5

10

15

20

25 I1 3

30

35

40

45

50

PT

Fig. 3. Fitting yield surfaces at different temperatures in the meridional plane. 0.0006

4.2

a

0.0005

4.0

Fitting curve for a

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0.0004

b Fitting curve for b

3.8

b

135

a

0.0003

3.6

AC

0.0002

3.4

0.0001

3.2

0.0000

3.0 0

136 137

100

200

300

400

Temperature/oC

500

600

0

100

200

300

400

Temperature/oC

(a)

500

600

(b) Fig. 4. Fitting curves of a and b.

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2.2 Plastic flow

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Typically, to describe the plastic strain increment, the plastic potential function, G, was introduced,

ACCEPTED MANUSCRIPT 9 𝜕𝐺

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𝑑𝜺𝑝𝑙 = 𝑑𝜆

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where 𝑑𝜆 is a positive scalar coefficient of proportionality; A hyperbolic function flow potential

142

G was chosen in this model as [19]

(6)

𝜕𝝈

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G = √(𝜖𝝈 ⃗ |0 𝑡𝑎𝑛 𝜓) 2 + 𝑞 2 − 𝑝 𝑡𝑎𝑛 𝜓

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where 𝜓 (𝜃, 𝑓𝑖 ) is the dilation angle measured in the p-q plane at high hydrostatic pressure, as

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shown in Fig. 5; 𝝈 ⃗ |0 = 𝝈 ⃗ |𝜺𝑝𝑙 <0,𝜺̇ 𝑝𝑙<0 is the initial yield stress; 𝜖 is a parameter, referred to as the

146

eccentricity, that defines the rate at which the function approaches the asymptote, and the flow

147

potential tends to a straight line as the eccentricity tends to zero

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𝐺 = 𝑞 − 𝑝 𝑡𝑎𝑛 𝜓

(8)

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(7)

q

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150

PT

Fig. 5. Family of hyperbolic flow potentials in the p-q plane.

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𝑓 =

𝑓

(9)

𝜕𝝈

1 ̅ 𝝈

𝝈：

𝜕𝐺 𝜕𝝈

(10)

where 1 −

f=

157

1+ { 1

158

𝑑𝜺̅𝑝𝑙 𝜕𝐺

pure shear test data) and could be written in general as

155 156

𝑑𝜺𝑝𝑙 =

where f depends on how the hardening is defined (by uniaxial compression, uniaxial tension, or

AC

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Potential flow in the model was assumed as

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p

and

1 3 𝑡𝑎𝑛 𝜓 1 3 𝑡𝑎𝑛 𝜓

𝑖𝑓 𝑕𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛, 𝑖𝑓 𝑕𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑡𝑒𝑛𝑠𝑖𝑜𝑛,

𝑖𝑓 𝑕𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛 𝑝𝑢𝑟𝑒 𝑠𝑕𝑒𝑎𝑟 (𝑐𝑜𝑕𝑒𝑠𝑖𝑜𝑛),

(11-a)

ACCEPTED MANUSCRIPT 10 𝑝𝑙

|𝑑𝜀11 | 𝑖𝑛 𝑡𝑕𝑒 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑐𝑎𝑠𝑒, 𝑝𝑙

159

𝑑𝜀11 𝑖𝑛 𝑡𝑕𝑒 𝑢𝑛𝑖𝑎𝑥𝑖𝑎𝑙 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑐𝑎𝑠𝑒,

𝑑𝜀̅𝑝𝑙 =

𝑑𝛾𝑝𝑙

(11-b)

𝑖𝑛 𝑡𝑕𝑒 𝑝𝑢𝑟𝑒 𝑠𝑕𝑒𝑎𝑟 𝑐𝑎𝑠𝑒,

√3

𝑤𝑕𝑒𝑟𝑒 𝛾 𝑝𝑙 𝑖𝑠 𝑡𝑕𝑒 𝑒𝑛𝑔𝑖𝑛𝑒𝑒𝑟𝑖𝑛𝑔 𝑠𝑕𝑒𝑎𝑟 𝑝𝑙𝑎𝑠𝑡𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛.

{

As shown in Table 1 and Fig. 3, this model is more reliable under high hydrostatic pressure (p

161

is not smaller than the uniaxial compression data) that the yield surfaces in the meridional plane

162

tends to a straight line which satisfied eq. (8). Therefore, combining eq. (8), eq. (9) and eq. (11), eq.

163

(12) could be obtained 𝑝

1 1 [ (2𝜎1 1;1/3𝑡𝑎𝑛 𝜓 2𝑞 1 1 𝑑𝜀̅𝑝𝑙 [ (2𝜎2 1;1/3𝑡𝑎𝑛 𝜓 2𝑞 1 1 𝑑𝜀̅𝑝𝑙 [ (2𝜎3 1;1/3𝑡𝑎𝑛 𝜓 2𝑞

𝑑𝜀1 = 𝑑𝜀̅𝑝𝑙 𝑝

𝑑𝜀2 =

164

𝑝 {𝑑𝜀3

166

− 𝜎2 − 𝜎3 ) + 1/3𝑡𝑎𝑛 𝜓] − 𝜎1 − 𝜎3 ) + 1/3𝑡𝑎𝑛 𝜓]

(12)

− 𝜎1 − 𝜎2 ) + 1/3𝑡𝑎𝑛 𝜓]

For the stress state of uniaxial compression test: 𝜎3 ≠ 0, 𝜎1 = 𝜎2 = 0, the dilation angle 𝜓

AN US

165

=

CR IP T

160

could be expressed by the axial and the transverse plastic strain 𝑡𝑎𝑛 𝜓 = −

167

𝑝

𝑝

3 𝑑𝜀1 :2𝑑𝜀2 𝑝 𝑝 2 𝑑𝜀1 ;𝑑𝜀2

(13)

Apparently in eq. (13), it is impossible to measure the value of the dilation angle during

169

yielding process, because it changes rapidly account for variation of the plastic strain. Considering

170

the two typical simplifying assumptions on acquiring the dilation angle [21]:

173 174 175

ED

(b) In the nonassociated flow rule, the dilation angle, which does not equal to the internal-friction angle, is usually assumed to be zero. The latter assumption does not accord with the experimental reality obviously. Therefore, the

former assumption was applied in this study. The constant dilation angles at different temperature

AC

176

internal-friction angle;

PT

172

(a) In the associated flow rule, constant value of the dilation angle equals to the

CE

171

M

168

177

are acquired by fitting the test data [11] of high hydrostatic pressure, as shown in Fig. 6 and Table

178

2. Fitting the data in Table 2, the relationship between the dilation angle and the temperature is

179

expressed in eq. (14).

180

𝜓 = 18.06 + 0.017𝑇 − 2.84 × 10;5 𝑇 2

(14)

ACCEPTED MANUSCRIPT 11

24 22 20



18 16 14

10 0

100

200

300

400

Temperature/oC

181 182

CR IP T



Fitting curve for 

12

500

600

Fig. 6. The dilation angle at different temperatures. Table 2

184

The dilation angle at different temperatures for the extended Drucker-Prager exponent model. Temperature/℃

20

200

The dilation angle /o

18.3

20.7

AN US

183

300

500

600

20.3

19.5

18.1

3. Concrete constitutive model for coupled high temperature & high strain

186

rate

187

3.1 Coupled effects of high temperature and high strain rate on concrete strength

188

The extended Drucker-Prager model was recoded as a basis for developing constitutive equations

189

of concrete. Material parameters a and b were determined by fitting the experimental data. The left

190

unknown parameter to determine the yield surface is 𝑝𝑡 , which could be expressed by the uniaxial

191

compression yield strength 𝜎𝑐 as shown in eq. (5). It has been generally accepted that concrete is

192

a temperature and strain rate sensitive material, thus, 𝜎𝑐 could be expressed as

ED

PT

𝜎𝑐 = 𝜎𝑐 (𝜀, 𝜀̇ , 𝑇)

(15)

In order to obtain the explicit expression of eq. (15), a series of uniaxial impact test of normal

AC

194

CE

193

M

185

195

concrete at elevated temperature from 20ºC up to 950ºC were conducted on a specially designed

196

microwave heating automatic time controlled Split Hopkinson Pressure Bar (MATSHPB)

197

apparatus. More detailed information could be found in [22].

198 199 200

The Johnson-Cook model [23, 24] is well known in considering mutual effects of temperature and strain rate for steel by a decoupled expression, it is expressed as 𝜎 = (𝐴 + 𝐵 𝜀 𝑛 ) (1 + 𝐶 𝑙𝑛 𝜀̇ ∗ ) (1 − 𝑇 ∗𝑚 ) = 𝑢 (𝜀) 𝑣 (𝜀̇) 𝑤 (𝑇)

(16)

ACCEPTED MANUSCRIPT 12 201

where 𝜀 is the equivalent plastic strain, 𝜀̇∗ = 𝜀̇⁄𝜀̇0 is the dimensionless plastic strain rate for

202

𝜀̇0 = 1.0𝑠 ;1 , 𝑇 ∗ =

203

constants. The expression in the first set of brackets, 𝑢 (𝜀), gives the stress as a function of strain

204

for 𝜀̇∗ = 1.0 and 𝑇 ∗ = 0. The expressions in the second and third sets of brackets, 𝑣 (𝜀̇) and

205

𝑤 (𝑇), represents the effects of strain rate and temperature, respectively. This multiplication form

206

of expression decouples the effects of strain rate and temperature subtly. Thus, there were reasons,

207

it was assumed that the mechanical behavior of concrete subjected to high temperature and high

208

strain rate can also be expressed as this decoupled form. In another word, the temperature and

209

strain rate affects the concrete behavior independently. The second and the third sets of brackets

210

could be considered as the strength dynamic increase factor (DIFS) and the temperature reduction

211

factor (

𝑇;𝑇𝑟𝑜𝑜𝑚

is the homologous temperature. A, B, C, n, m are material

CR IP T

𝑇𝑚𝑒𝑙𝑡 ;𝑇𝑟𝑜𝑜𝑚

AN US

𝑇

𝑓𝑐,𝑠 ), respectively 𝑓𝑐,𝑠

212

𝑣 (𝜀̇) = 𝐷𝐼𝐹 S

213

𝑤 (𝑇) =

(17-b)

The function suggested by the CEB [25], which was in good agreement with the test results [22], was used to describe the DIFs

M

215

𝑓𝑐,𝑠

(𝜀̇ ⁄𝜀̇ )1.026𝛼𝑠 , |𝜀̇𝑐 | ≤ 30𝑠 ;1 𝑣 (𝜀̇) = 𝐷𝐼𝐹 s = { 𝑐 𝑐0 1/3 𝛾𝑠 (𝜀̇𝑐 ⁄𝜀̇𝑐0 ) , |𝜀̇𝑐 | > 30𝑠 ;1

216 1

ED

214

𝑇 𝑓𝑐,𝑠

(17-a)

(18)

, 𝑙𝑜𝑔 𝛾𝑠 = 6.156𝛼𝑠 – 2 , 𝜀̇𝑐0 = 30 × 10;6 𝑠 ;1 , 𝜀̇𝑐 is the strain rate,

217

where 𝛼𝑠 =

218

𝑓𝑐,𝑠0 = 10𝑀𝑃𝑎 , 𝑓𝑐,𝑠 is the quasi-static compressive strength.

219

The typical function form of the temperature reduction factor has been widely suggested in many

220

literatures [4, 5, 26-28], which could be expressed as

AC

CE

PT

5:9𝑓𝑐,𝑠 ⁄𝑓𝑐,𝑠0

221

𝑤 (𝑇) =

𝑇 𝑓𝑐,𝑠

𝑓𝑐,𝑠

=

1 𝑚 (𝑇 ; 20)𝑛 : 1

20℃ ≤ 𝑇 ≤ 1000℃

(19)

222

𝑇 where 𝑓𝑐,𝑠 denotes the quasi-static compressive strength, 𝑓𝑐,𝑠 denotes the quasi-static

223

compressive strength at high temperature. Fitting the quasi-static test data at different temperatures

224

[22], parameters m and n could be obtained, m = 4.515×10-8, n=2.62. The good agreement

225

between the curve of eq. (19) and test data is shown in Fig. 7 that demonstrates the validity of m

226

and n.

ACCEPTED MANUSCRIPT 13 1.1

Fitting curve Grade A Grade B Grade C Grade D

1.0 0.9

T

fcs/fcs

0.8

227

0.7 0.6 0.5 0.4 0.3

CR IP T

0.2

100 200 300 400 500 600 700 800 900 1000

229

*Grade A~D depicted the normal weight concrete specimen with different mix proportions

230

prepared for the experiment. The average measured cube compressive strength of the four

231

batches concrete specimen was 46.9 MPa, 55.6 MPa, 57.8 MPa and 65.9 MPa, respectively. The

232 233

mix proportions of concrete could be found in [22]. The strength dynamic increase factor of concrete exposed to high strain rate and high

234

temperature DIFTS [22] could be expressed as

235

𝑇 𝑓𝑐,𝑑 𝑇 𝑓𝑐,𝑠

=

𝜎𝑐 (𝜀,𝜀̇ ,𝑇) 𝑢 (𝜀)

(20)

𝑇 where 𝑓𝑐,𝑑 denotes the dynamic compressive strength at elevated temperature.

M

237

DIFTS =

Eq. (21) was acquired by simultaneous equations of eq. (16) and eq. (20),

ED

236

AN US

228

Temperature/0C Fig. 7. Relationship between normalized quasi-static compressive strength at temperatures.

DIFTS= 𝑣 (𝜀̇ ) 𝑤 (𝑇)

238

(21)

Thus, the equilibrium of Eq. (21) is the sufficient and necessary conditions for the decoupled

240

form of constitutive model for concrete subjected to high temperature and high strain rate.

241

However, as the comparison between the test results of DIFTS and the calculated results of

242

𝑣 (𝜀̇ ) 𝑤 (𝑇) shown in Fig. 8, large errors exist at all temperatures except for the ambient

243

temperature 20ºC. It is demonstrated that the decoupled form suggested in Johnson-Cook model is

AC

CE

PT

239

244

not precise enough to describe behaviors of for concrete subject to coupled effects of high

245

temperature and high strain rate. The temperature and strain rate should affect the concrete

246

behavior dependently, thus, 𝜎𝑐 here should be considered as

247

𝜎𝑐 = 𝜎𝑐 ( 𝜀, 𝜀̇, 𝑇) = 𝜎𝑐 (𝜀, 𝑇) 𝜎𝑐,𝑑 ( 𝜀, 𝜀̇, 𝑇)

(22)

248

where 𝜎𝑐 (𝜀, 𝑇) is the temperature reduction factor, namely eq. (19); 𝜎𝑐,𝑑 ( 𝜀, 𝜀̇, 𝑇) is the

249

proposed expressions of dynamic increase factor of concrete at high temperature DIFTS [22]

ACCEPTED MANUSCRIPT 14 𝜀̇

𝐷𝐼𝐹𝑇𝑆

250

=

𝑇 𝑓𝑐,𝑑 𝑇 𝑓𝑐,𝑠

={

1.026𝛼𝑠

𝑇

(𝜀̇𝑇𝑐 ) 𝑐0

𝛾𝑆 (

𝑇

𝜀̇ 𝑐 −𝑔(𝑇)

𝑇

𝑇 𝜀𝑐

𝑇 𝜀𝑡𝑟𝑎𝑛𝑠 𝑠−1

(23)

𝑕(𝑇)

)

𝑇

𝜀̇ 𝑐0

𝑇

𝜀̇ 𝑐 ≤ 𝜀̇ 𝑡𝑟𝑎𝑛𝑠 𝑠−1 20℃ ≤ 𝑇 ≤ 800℃

, ,

̇ > ̇

20℃ ≤ 𝑇 ≤ 800℃

251

𝑇 𝑇 where 𝜀̇𝑐𝑇 is the strain rate at temperatures T, 𝑓𝑐,𝑑 is the dynamic strength at temperatures T, 𝑓𝑐,𝑠

252

𝑇 is the quasi-static strength at temperatures T, 𝜀̇𝑡𝑟𝑎𝑛𝑠 is the transition strain rate at temperatures T,

253

𝑇 𝜀̇𝑐0 = 30 × 10;6 , 𝛼𝑠 =

254

temperature related function that affects the transition strain rate and the rise slope of the second

255

branch of DIFTS, respectively.

The expressions of g (T), h (T) and the transition strain rate at temperature T was fitted as follow

AN US

257

, 𝑓𝑐,𝑠0 = 10MPa, 𝑙𝑜𝑔 𝛾𝑠 = 6.156𝛼𝑠 – 2, g(T) and h(T) is the

CR IP T

256

1 5:9𝑓𝑐,𝑠 ⁄𝑓𝑐,𝑠0

𝑔(𝑇) = −1.51 + 0.26𝑇 − 5.35 × 10;4 𝑇 2 + 3.89 × 10;7 𝑇 3 20℃ ≤ 𝑇 ≤ 800℃ 258

(24)

𝑕(𝑇) = 0.34 + 0.34 × 10;5 𝑇 − 1.11 × 10;7 𝑇 2 + 4.93 × 10;11 𝑇 3 20℃ ≤ 𝑇 ≤ 800℃

𝑇 𝜀̇𝑡𝑟𝑎𝑛𝑠 = 25.6 + 0.24𝑇 − 5.04 × 10;4 𝑇 2 + 4.6519 × 10;7 𝑇 3 20℃ ≤ 𝑇 ≤ 800℃

260

Heretofore, 𝑝𝑡 could be expressed as

262

𝑝𝑡 = 𝑎𝜎𝑐𝑏 −

𝜎𝑐 3

PT

263 264

= 𝑎 (𝜎𝑐 (𝜀, 𝑇)𝜎𝑐,𝑑 (𝜀, 𝜀̇, 𝑇))𝑏 −

𝜎𝑐 (𝜀,𝑇)𝜎𝑐,𝑑 (𝜀,𝜀̇ ,𝑇)

(27)

3

The yield function of the modified Drucker-Prager model could be expressed as

20oC-test DIFTS

2.2

2.0

400oC-test DIFTS

2.0

1.8

650oC-test DIFTS

1.8

1.6

950oC-test DIFTS

1.6

20oC 400oC 650oC 950oC

1.4

DIFTS

AC

2.2

1.4 1.2 1.0

1.2 1.0

0.8

0.8

0.6

0.6

0.4

0.4 10

Strain rate/s-1

（a）Grade A

100

𝜎𝑐 (𝜀,𝑇)𝜎𝑐,𝑑 (𝜀,𝜀̇ ,𝑇) 3

=0

(28)

1.6

1.6

1.4

1.4

1.2 DIFTS

CE

𝐹 = 𝑎 𝑞 𝑏 − 𝑝 − 𝑎(𝜎𝑐 (𝜀, 𝑇)𝜎𝑐,𝑑 (𝜀, 𝜀̇, 𝑇))𝑏 −

265

267

(26)

ED

261

266

(25)

M

259

1.2 20oC-test DIFTS

1.0

o

400 C-test DIFTS

1.0

650oC-test DIFTS

0.8 0.6 0.4

0.8

950oC-test DIFTS 20oC 400oC 650oC 950oC

50

0.6 0.4 100

150

Strain rate/s-1

（b）Grade B

200 250

ACCEPTED MANUSCRIPT 15

20oC-test DIFTS

1.8

400 C-test DIFTS

1.4

800oC-test DIFTS

1.2

950oC-test DIFTS

0.8 0.6 0.4

400oC-test DIFTS

1.4

1.6

1.2

650oC-test DIFTS

1.2

1.4

1.0

1.2

20oC 400oC 650oC 800oC 950oC

1.0

1.0

20oC 400oC 650oC 950oC

0.8

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.2

0.6

100

10

Strain rate/s-1

（c）Grade C

（d）Grade D

100

Grade A Grade B Grade C Grade D

90 80 70

Error/%

100

CR IP T

Strain rate/s-1

268 269

950oC-test DIFTS

0.4

0.2 10

1.6

20oC-test DIFTS

1.4

1.8

650oC-test DIFTS

1.0

1.6

2.0

o

1.6 DIFTS

2.2

2.0

DIFTS

2.2

60 50 40

20 10 0 0

200

AN US

30

400

600

Temperature/oC

800

1000

（e） Error Fig. 8. Comparison between the test results of DIFTS and the computation results of 𝑣 (𝜀̇ ) 𝑤 (𝑇).

273

3.2 Uniaxial dynamic compressive complete stress-strain curve of concrete subject to high

274

temperature

275

A variety of functions, such as polynomial, exponential, trigonometric function and rational

276

fraction, have been put forward [29] to accurately fitting the uniaxial complete stress-strain curve,

277

which is important to describe mechanical property of concrete. The piecewise form function was

278

chosen in this study from Code for design of concrete structures [30]

ED

PT

CE

𝑦 = 𝛼𝑎 𝑥 + (3 − 2𝛼𝑎 )𝑥 2 + (𝛼𝑎 − 2)𝑥 3 𝑥 { 𝑦 = 2

AC

279

M

270 271 272

𝜀

𝜎

(29)

280

where 𝑥 =

281

strain; 𝑓𝑐,𝑠 is the uniaxial compressive strength; 𝜀𝑝 is the critical strain at peak stress; 𝛼𝑎 and

282

𝛼𝑑 are the parameters for ascending and descending segment of the uniaxial compressive curve,

283

respectively.

𝜀𝑝

, 𝑦=

𝛼𝑑 (𝑥;1) :𝑥

𝑥≤1 𝑥>1

𝑓𝑐,𝑠

; 𝜎 is the uniaxial compressive stress; 𝜀 is the uniaxial compressive

284

It has been revealed by many studies [31, 32] that concrete usually starts to yield when the

285

micro-cracks and micro-cavities generate and develop, and eventually fails due to the evolution,

ACCEPTED MANUSCRIPT 16 development and accumulation of multi-scale damage. However, the whole yielding process

287

barely reflects to micro-scale composition change like lattice dislocation. Krajcinovic [33]

288

considered that the continuum damage mechanics was the most suitable for describing dissipative

289

thermodynamics problems that was dominated by micro-cracks. Therefore, the damage should be

290

considered when modeling the concrete. In the theory of continuum damage mechanics, the

291

progressive material degradation caused by the formation and coalescence of micro-cracks and

292

voids is quantified by damage parameters. There is only one damage parameter, D, in the isotropic

293

damage theory, which is usually defined as [34] D= {

294

0 1 − 𝐴̃⁄𝐴

𝜀 ≤ 𝜀 𝑡ℎ 𝜀 > 𝜀 𝑡ℎ

CR IP T

286

(30)

where A is the area of an element before damage; 𝐴̃ is the effective area during loading; 𝜀 𝑡ℎ is

296

the customary threshold strain. Based on the strain equivalence principle, the stress of damaged

297

concrete could be expressed as

AN US

295

𝜎 = 𝐸0 𝜀 (1 − 𝐷)

298 299

where 𝐸0 is the initial static elastic modulus. The secant elastic modulus is defined as 𝐸 = 𝜎⁄𝜀, 𝐸 = 𝐸0 (1 − 𝐷)

From eq. (29), y/x could be expressed as

ED

301

𝑦

302

𝑥

𝜎

𝜀𝑝

𝜀

𝑓𝑐,𝑠

= ∙

= 𝐸 ⁄𝐸𝑝 =

1;𝐷

(33)

1;𝐷𝑝

PT

where 𝐸𝑝 is the secant modulus at peak stress, 𝐷𝑝 is the damage at peak stress,

304

𝛼𝑎 = 𝐸0 ⁄𝐸𝑝 =

1

(34)

1 ; 𝐷𝑝

AC

9.

CE

The tested 𝐸0 and 𝐸𝑝 at elevated temperatures are shown in Table 3, and 𝛼𝑎 is fitted in Fig.

305 306

(32)

M

300

303

(31)

307

Table 3

308

𝐸0 and 𝐸𝑝 of concrete at elevated temperature. Grade

Temperature / ºC

Strain

Initial elastic

Peak secant

rate

modulus

modulus

/s-1

E0/GPa

EP/GPa

44.60

19.59

2.29

34.24

16.62

2.06

27.36

14.29

1.91

34.24

16.54

2.07

A B C D

20

0.00003

E0/ EP

ACCEPTED MANUSCRIPT

29.52

11.54

2.56

21.34

10.00

2.13

17.07

8.46

2.08

D

21.34

9.90

2.16

A

19.91

7.83

2.54

12.91

6.69

1.93

11.75

6.09

1.93

D

14.93

6.99

2.14

A

8.90

CR IP T

17 A B C

B C

B C

400

650

3.61

2.46

3.06

1.94

2.67

2.22

3.09

2.88

2.27

2.47

1.94

2.13

1.64

2.18

4.13

1.92

2.16

50.94

21.12

2.41

38.70

17.46

2.21

31.87

16.03

1.99

40.06

17.58

2.27

24.33

11.14

2.19

22.46

10.67

2.11

19.33

8.27

2.33

19.59

10.15

1.94

5.92

800

5.92

D

8.90

A

5.63

C

4.13

950

3.56

AN US

B D A B C

20

D 0.003

C

400

ED

B

M

A

D

5.0

PT

4.5

test value average value 2.2

4.0

AC

E0/EP

CE

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0

309 310

200

400

600

800

1000

Temperature/oC

Fig. 9. The ratio of initial elastic modulus to peak secant modulus at elevated temperatures.

311

As shown in Fig. 9, all the test data of 𝛼𝑎 are concentrated around a certain range, and the

312

average value is 2.2, which is the same as that recommended in [35]. For the descending segment

ACCEPTED MANUSCRIPT 18 of the uniaxial compressive curve, empirical value of 0.8 [35] is determined for 𝛼𝑑 . Thus, the

314

uniaxial compressive complete stress-strain curve of concrete at the ambient temperature is

315

determined in Fig. 10. In the present paper, the initial elastic modulus is defined as 𝐸 = 𝜎𝑏 /𝜀𝑏 ,

316

where b corresponds to the 40% of the peak stress on the ascending segment of stress-strain curve.

317

As shown in Fig. 10, 𝜀𝑏 = 0.21𝜀𝑝 , therefore, the customary threshold strain 𝜀 𝑡ℎ in eq. (30)

318

equals to 0.21.

CR IP T

313

1.0 0.8

0.4 0.2 0.0 0.0

321

0.8

1.2

1.6

2.0

2.4

2.8

x

Fig. 10. Non-dimensional uniaxial compressive stress-strain curve of concrete at the ambient

M

319 320

0.4

AN US

y

0.6

323

into three stages that is the uncracked elastic stage, the plastic damage hardening stage and the

324

plastic softening stage, respectively.

325

(1) The uncracked elastic stage (𝜀 ≤ 0.21𝜀𝑝 )

PT

ED

322

temperature. The uniaxial compressive complete stress-strain curve of concrete could be roughly divided

𝑦 = 2.2𝑥 − 1.4𝑥 2 + 0.2𝑥 3

327

329 330 331

𝑥 ≤ 0.21

𝑦 = 2.2𝑥 − 1.4𝑥 2 + 0.2𝑥 3

0.21 < 𝑥 ≤ 1

(36)

(3) The plastic softening stage (𝜀 > 𝜀𝑝 ) 𝑦 =

𝑥 0.8(𝑥;1)2 :𝑥

𝑥 > 1

(37)

The uniaxial dynamic stress-strain curve of concrete subject to high temperature is extended

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by varying the parameters in the three stages, respectively.

333

The un-cracked elastic stage (𝜺 ≤ 𝟎. 𝟐𝟏𝜺𝒑)

334

(35)

(2) The plastic damage hardening stage (0.21𝜀𝑝 < 𝜀 ≤ 𝜀𝑝 )

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The loading curve is a straight line with the slope of 𝐸0 that could be obtained by

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substituting the tested secant modulus at peak stress 𝐸𝑝 into eq. (34). Temperature effect on the

336

elastic modulus is considered in this stage. Test data [22] revealed the linear decreasing trend of

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the initial elastic modulus 𝐸0 and the secant modulus 𝐸𝑝 with the elevated temperature, which is

338

expressed as 𝐸0𝑇

339

𝐸0

=

𝑇 𝐸𝑝

𝐸𝑝

= 1.00986 − 9.666 × 10;4 𝑇

(38)

where 𝐸0𝑇 and 𝐸𝑝𝑇 is the initial elastic modulus and the secant modulus at peak stress at

341

temperature T, respectively.

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The damage effects and the strain rate effects were not considered in this stage. The plastic damage hardening stage (𝟎. 𝟐𝟏𝜺𝒑 < 𝜺 ≤ 𝜺𝒑 )

In this stage, the coupled effects of high temperature and high strain rate was considered. The

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stress 𝜎 was obtained by multiplying the static stress of plastic damage hardening stage at the

346

ambient temperature by DIFTS. The critical strain at peak stress was assumed only affected by the

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high temperature, but not the strain rate, as the test data shown in [22]. It is summarized as 𝜉=

𝑇 𝜀𝑝

𝜀𝑝

= 1 + 0.98761(

𝑇

1000

)1.50221

(39)

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where 𝜀𝑝𝑇 is the critical strain at peak stress and temperature T, 𝜀𝑝 is the critical strain at peak

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stress and the ambient temperature.

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The plastic softening stage (𝜺 > 𝜺𝒑 )

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Due to the absence of test data about the descending segment of the stress-strain curve of

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concrete at high strain rates (10 s-1~102 s-1), eq. (37) for concrete at static state and ambient

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temperature was used instead at this stage. Similar to the plastic hardening stage, the stress 𝜎 in

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this stage could be obtained by multiplying the static stress of plastic softening stage at the

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ambient temperature by DIFTS. The temperature effects on the critical strain at peak stress was also

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determined by eq. (39).

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4. Numerical validation

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The proposed concrete constitutive model was coded and implemented into the commercial

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FE software ABAQUS through VUMAT port. In order to validate the proposed model, the

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quasi-static tests and the SHPB tests of normal concrete at high temperatures were numerically

362

simulated in ABAQUS, and the results were compared with the test data [22].

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4.1 Tests of concrete at elevated temperatures The dynamic properties of normal concrete at elevated temperature from 20ºC up to 950ºC

365

were systematically studied using a specially manufactured microwave-heating automatic

366

time-controlled Split Hopkinson Pressure Bar (MATSHPB) apparatus, the schematic diagram of

367

developed MATSHPB apparatus is shown in Fig. 11. The concrete specimens were first efficiently

368

heated in a specially designed industrial microwave oven, and then rapidly loaded after quickly

369

rolling to the SHPB system. Quasi-static tests at elevated temperatures were also carried out by

370

using a hydraumatic testing machine. The specimens were firstly heated to a predetermined

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temperature in the industrial microwave oven, and then kept in the thermostatic environment for

372

30 min. During the quasi-static compression testing process, the specimens were wrapped by the

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insulating material of glass fiber composites after getting out from the micro-wave oven to

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maintain the predetermined temperature, as shown in Fig. 12.

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(b) Real apparatus

Fig. 11. Microwave-heating automatic time-controlled Split Hopkinson Pressure Bar (MATSHPB).

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(a) Schematic diagram

378 379

(a) Hydraumatic testing machine

(b) Wrapped specimen

(c) Uniaxial compression test

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Fig. 12. Quasi-static uniaxial tests at elevated temperature. Concrete specimens were fabricated by the following materials: Ordinary Portland Cement

382

(OPC, PⅡ42.5R, PⅡ52.5R), fly ash ( average grain diameter: 1-15𝜇𝑚, SiO2: 55%), basalt rubble

383

as coarse aggregate (5-10 mm), river sand as fine aggregate (fineness modulus: 2.7), high

384

efficiency water reducing admixture and tap water. Table 4 shows the detailed concrete mix

385

proportions of the concrete specimens.

386

Table 4

387

Mix proportions of concrete (kg/m3). Cement

Fly ash

Sand

Admixture

166

PⅡ52.5R 442

78

689

5.3

Basalt rubble

Strength fcu

1125

57.8MPa

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Three 150×150×150mm cubic concrete specimens were firstly cast and cured in the

389

laboratory to measure the average measured cube compressive strength (fcu). The average

390

measured cube compressive strength (fcu) of the concrete specimen was 57.8 MPa, as shown in

391

Table 4. Then, the concrete specimens for tests were prepared. The mixed concrete was filled into

392

a number of shaped 40 mm-length Polyvinyl Chloride tubes with the same inner diameter of 70

393

mm and was vibrated by a platform-type vibrator. Thereafter, the specimens were left in their

394

moulds for 48 h, and finally cured in the calcium hydroxide solution of standard room conditions

395

of 20 ± 2ºC in the laboratory. 28 days later, the Ø70 mm×40 mm length concrete cylinder

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specimens were demoulded from the polyvinyl chloride tubes and ground flat on the cross-sections

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at the two ends, as shown in Fig. 13.

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(a) Casting specimen

(b) Grinded specimen

Fig. 13. Concrete specimens for the tests. 4.2 Numerical models & results comparison

Identical with the tests, the concrete cylinder model with the dimensions of 𝛷70𝑚𝑚 ×

403

40𝑚𝑚 length was built, as shown in Fig. 14. The cylinder model was meshed with the C3D8RT

404

element type. In order to guarantee the accuracy of simulation on the stress wave effects, sufficient

405

amount of elements (n) in the direction of stress wave propagation should be maintained within the

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duration of the stress wave pulse [36]. n≥10 was suggested in literature [37]. Therefore, the size of

407

element in the axial direction and the radical direction was determined as 4mm and 3.5mm,

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respectively. The cylinder model was meshed with 3840 elements in total. The density 𝜌 was

409

2400kg/m3, the Possion’s ratio 𝜈 was 0.2. There was no heat conduction inside the specimen,

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because the specimen was heated in the microwave oven. In accordance with the test, transient

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heat conduction inside the specimen was also not considered in the numerical simulation, the

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𝑇 parameters of concrete at temperatures, e.g. the quasi-static strength 𝑓𝑐,𝑠 , the critical strain at peak

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stress 𝜀𝑝𝑇 , and the initial elastic modulus 𝐸0𝑇 at temperatures T, were directly calculated in the

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models with the proposed formulae.

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Fig. 14. Finite element model of concrete specimen.

417 418

Fig. 15. Complete model of SHPB test.

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There are commonly two methods to simulate the SHPB test in the FE software. One is the

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so-called simplified model, it loads directly on the specimen according to displacement time

421

history, as shown in Fig. 14; the other is the complete model, it models the complete impact

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process with incident bar, specimen and the transmitter bar together, as shown in Fig. 15. It has

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been reported that the error between the two methods is less than 2.5% [38]. Moreover, the

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simplified model could effectively control the exact strain rate of the specimen that improves

425

accuracy of the numerical simulation. Thus, the model of loading directly with displacement on

426

the specimen was used in this model, as shown in Fig. 14. The simulated stress-strain curves of

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concrete at different temperatures and strain rates are compared with the test data in Fig. 16

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ACCEPTED MANUSCRIPT 24 o

test-20 C o simulation-20 C o test-400 C o simulation-400 C o test-650 C o simulation-650 C o test-800 C o simulation-800 C

120

80 60 40 20 0 0.000

0.005

0.010

0.015

0.020

Strain

428 429

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Stress/MPa

100

0.025

0.030

(a) The quasi-static stress-strain curves at elevated temperatures. 200

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Stress/MPa

160 140 120 100 80

-1

60

M

40 20 0 0.000

0.006

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430

0.003

test-40s -1 test-65s -1 test-75s -1 test-85s

0.009

-1

simulation-40s -1 simulation-65s -1 simulation-75s -1 simulation-85s

0.012

0.015

Strain

(b) The stress-strain curves at 20℃ and high strain rates.

431

PT

160 140

100

AC

CE

Stress/MPa

120

432 433

80 -1

test-100s -1 test-103s -1 test-105s -1 test-110s -1 test-115s -1 test-135s

60 40 20 0 0.000

0.003

0.006

0.009

0.012

-1

simulation-100s -1 simulation-103s -1 simulation-105s -1 simulation-110s -1 simulation-115s -1 simulation-135s

0.015

0.018

Strain (c) The stress-strain curves at 400℃ and high strain rates.

ACCEPTED MANUSCRIPT 25 120

80 60 40

-1

test-105s -1 test-130s -1 test-140s -1 test-170s

20 0 0.000

0.003

0.006

0.009

0.012

Strain

434 435

0.015

0.018

(d) The stress-strain curves at 650℃ and high strain rates. 55 50

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Stress/MPa

40 35 30 25 20

-1

15 5

0.003

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0 0.000

M

10

436 437 438

-1

simulation-105s -1 simulation-130s -1 simulation-140s -1 simulation-170s

CR IP T

Stress/MPa

100

0.006

test-133s -1 test-140s -1 test-192s -1 test-205s

0.009

-1

simulation-133s -1 simulation-140s -1 simulation-192s -1 simulation-205s

0.012

0.015

Strain

(e) The stress-strain curves at 800℃ and high strain rates. Fig. 16. Comparison between simulation results and test results on stress-strain curves. As shown in Fig. 16 (a), it is obvious that the quasi-static stress-strain curves obtained from

440

simulation are in good agreement with the test results. It attributes to the model parameters, such

441

as the stress-strain curves, the elastic modulus at temperatures, the peak stress, the critical strain at

442

peak stress; these parameters are in completely accordance with the test.

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Fig. 16 (b)-(e) show the comparison of stress-strain curves at the high temperatures and high

444

strain rates. It is indicated that the peak stress and the critical strain at peak stress in the simulated

445

curve agree well with the tests data. There exists a little difference in the initial elastic modulus

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between simulation results and test data. This little error might be attributed to that the variation of

447

elastic modulus at high strain rates were not considered in the proposed model.

448

5. Conclusions

ACCEPTED MANUSCRIPT 26 This study proposed a modified Drucker-Prager constitutive model for concrete considering the

450

coupled effects of high temperature and high strain rate. The piecewice function, which is chosen

451

from Code for design of concrete structures, is extended to describe the uniaxial dynamic

452

compressive complete stress-strain curve of concrete subject to high temperature. Triaxial test data

453

of concrete after different temperatures are used to define the exponent form of yield criterion and

454

the associated plastic flow. A new corresponding coupled function of temperature and strain rate

455

on concrete strength was put forward. The proposed concrete constitutive model was coded and

456

implemented into the commercial FE software ABAQUS through VUMAT port. A series of

457

quasi-static and high strain rate numerical simulations corresponding to the tests were also carried

458

out for validation. Some conclusions are summarized as follows.

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Other than the crystalline material, e.g. metal, concrete is influenced by temperature and

460

strain rate effects dependently. Comparing with the decoupled form, coupled form is more

461

appropriate to describe the mechanic behavior of concrete subject to high temperature and high

462

strain rate.

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The proposed model, i.e., the modified Drucker-Prager constitutive model for concrete

464

considering the effects of high temperature and high strain rate, is capable of predicting the

465

mechanic behaviors of concrete subject to high temperature and high strain rate.

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The proposed model is helpful in fully understanding the dynamic properties of normal

467

weight concrete at elevated temperatures, and provide a new choice for describe the mechanic

468

behavior of concrete subject to the combined effects of high strain rate and high temperature.

structures under multiple hazards of fire and blast.

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It also benefits the present concrete constitutive model and future analysis of concrete

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Acknowledgments

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The authors acknowledge the financial supports from the National Basic Research Program of

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China (No.2015CB058003) and National Natural Science Foundation of China (No.51622812

474

No.51378016, No.51210012, No.51238007). The experiments also received selfless help from

ACCEPTED MANUSCRIPT 27

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Xiquan Jiang and Chaochen Zhai.

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477

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[2] F.-J. Ulm, O. Coussy, Z. P. Bazant. The “Chunnel” fire. I: Chemoplastic softening in rapidly heated concrete[J]. Journal of Engineering Mechanics, 1999, 125：272-282.

[3] W. M. Lin, T. Lin, L. Powers-Couche. Microstructures of fire-damaged concrete[J]. ACI Materials Journal, 1996, 93：

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[4] H. Niu, Y. F. Ma, Y. X. Yao. Fire resistance research on the lightweight aggregate concrete member [J]. Building Strucrure, 1996, 26：29-33 (in Chinese).

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