Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process

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Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process Ming Sun , Dabo Xin , Chaoying Zou PII: DOI: Reference:

S0167-6636(19)30358-8 https://doi.org/10.1016/j.mechmat.2019.103192 MECMAT 103192

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Mechanics of Materials

Received date: Revised date: Accepted date:

1 May 2019 26 September 2019 26 September 2019

Please cite this article as: Ming Sun , Dabo Xin , Chaoying Zou , Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process, Mechanics of Materials (2019), doi: https://doi.org/10.1016/j.mechmat.2019.103192

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Highlights 

A cohesion reduction parameter could improve the calculation accuracy of model.



Proposed freeze-thaw damage constitutive model was validated by previous data.



Damage evolution and plasticity development during load process were calculated.



Simulations could reflect the effect of air entraining agent on frost resistance.

1

Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process Ming SUN a, b, c, Dabo XIN d, *, Chaoying ZOU a, b, c a.

School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China

b.

Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China

c.

Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin, 150090, China

d.

School of Civil Engineering, Northeast Forestry University, Harbin, 150040, China

* Corresponding author. Abstract Freeze-thaw has a severe degradation effect on concrete materials, especially in cold regions. The damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process are essential to improve the frost durability of concrete structures. In this paper, a cohesion reduction parameter is proposed to improve the accuracy of previous damage constitutive models for concrete materials. The value of parameter is determined by a theoretical proof, which is verified by a series of simulations. Followed, freeze-thaw degradation models are established by previous research work to describe the freeze-thaw degradation behaviors on the mechanical properties of concrete materials. These degradation models are applied into the mentioned constitutive model to establish the freeze-thaw damage constitutive model, which can reflect the constitutive relationship of concrete materials subjected to freeze-thaw damage. Additionally, a series of simulations are carried out to obtain the stress-strain relationships of concrete materials subjected to freeze-thaw cycles by proposed model. The calculation results have a good fit with corresponding test results. Finally, the calculation results are discussed to

2

investigate the mechanism of damage evolution and plasticity development of concrete materials subjected to freeze-thaw cycles during the load process. Keywords: Damage evolution; Plasticity development; Concrete; Freeze-thaw; Constitutive model. 1. Introduction Freeze-thaw is one of the most severe durability issues for concrete structures in cold regions, such as dams, harbours, canals and bridge foundations (Rosenqvist, 2013). It is a complex physical phenomenon that results in two different types of frost damage for concrete structures (Berto et al., 2015) that are widely accepted by researchers: (1) internal damage, which decreases the strength, stiffness and elastic modulus, and increases the permeability and diffusivity of concrete materials (Liu et al., 2018); and (2) surface scaling, which reduces the cover and hence increases risk for the corrosion of steel bars. The former has a more serious degradation effect on the mechanical properties of concrete materials. For internal damage, descriptions of the freeze-thaw damage mechanism of concrete materials were originally proposed by Powers, including the Hydraulic Pressure Hypothesis (HPH) (Powers, 1945) and Osmotic Pressure Hypothesis (OPH) (Powers, 1975). These hypotheses suggest that freeze-thaw damage was generated due to solution transfer inside the pores of concrete materials, as illustrated in Fig. 1. However, these hypotheses suffer from many limitations. Hydraulic pressure should increase with an increasing cooling rate based on the HPH, which is different from test observations (Zeng et al., 2008). Furthermore, the osmotic pressure hypothesis does not have a quantitative mathematical expression (Zeng et al., 2012). Moreover, the thermodynamic model, critical saturation method, low cycle fatigue method, bond spalling 3

theory and porous mechanics have been used to reveal freeze-thaw damage mechanism of concrete materials (Everett, 1961; Setzer, 2001; Kaufmann, 2004; Coussy, 2005; Valenza and Scherer, 2006). However, these studies were carried out by purely physical models based on a series of assumptions and derivations that were not universal. Furthermore, the crystallization pressure in pores during freezing was studied in detail by Prof. Scherer (Scherer, 1993; Scherer 1999). Recently, researchers have consistently determined that freeze-thaw damage could result from hydraulic pressure by ice volume expansion, cryosuction pressure by the surface tension of water and crystallization pressure due to the shape of ice crystals (Coussy and Monteiro, 2008; Lin et al., 2011; Liu et al., 2014; Liu et al., 2014; Gong and Jacobsen, 2019), as illustrated in Fig. 2. Nevertheless, it has been difficult to quantitatively determine the value of the damage pressures. In addition, many new technologies, such as ultrasonic imaging, 3D printing, 3D X-ray computed tomography and neutron imaging, have been utilized to study the freeze-thaw behaviors of concrete materials (Molero et al., 2012; Ju et al., 2017; Shields et al., 2018; Zhang et al., 2018).

Cement mortar

Channels of pore structures

Cement mortar

Flow direction of liquid pore solution

Freezing pore solution

Channels of pore structures

Flow direction of liquid pore solution

Freezing pore solution

Unfreezing pore solution

(a) HPH (Powers et al., 1945)

Unfreezing pore solution

(b) OPH (Powers et al., 1975)

Fig. 1. Mechanism of freeze-thaw damage proposed by Powers et al.

4

Concrete

Aggregate

Cement paste Air bubble

Phydrau

Phydrau

Ice

Pcryst

θCLθLV Ice

R

Air

r r Single air bubble with influential volume

Pcryo

Fig. 2. Freezing induced pore pressures in cement paste (Gong and Jacobsen, 2019) The damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process are essential to improve the frost durability of concrete structures. Therefore, numerous studies have been conducted that focus on the constitutive model of concrete materials subjected to freeze-thaw. A series of uniaxial and multiaxial tests were carried out to obtain the mechanical properties and stress-strain curves of concrete materials subjected to freeze-thaw, as shown in Table 1. Table 1. The detail information in reference tests Mechanical properties Tests

Stress-strain curves

Elasticity

Poisson’s

Compressive

Tensile

modulus

ratio

strength

strength

Shi

√

√

√

Qin

√

√

Hasan

√

√

Zou

√

Duan

√

√

Tension

√

√ √ √

Cao Guan

√

Compression

√ √

√

√ √ √

Shang

(Shi, 1997; Qin, 2003; Hasan et al., 2004; Shang, 2006; Zou et al., 2008; Duan et al., 2011; Cao et al., 2012; Guan et al., 2015)

5

Researchers established some physical models, such as fracture-plasticity elements (Hasan et al., 2004) and parallel bars system (Guan et al., 2015), as constitutive models to calculate the mechanical properties of concrete materials subjected to freeze-thaw using statistical methods based on test research, as illustrated in Fig. 3. To investigate the mechanical properties of concrete materials subjected to freeze-thaw under biaxial compression, a biaxial constitutive model was proposed considering the influence of freeze-thaw cycle number and the stress ratio (Shang, 2006). In addition, the studies on the load coupled with freeze-thaw and effect of air voids were carried out by researchers (Zhou et al., 1994; Jacobsen et al., 2016). Moreover, a macro-mesoscopic coupling damage constitutive model was established to determine the durability of concrete materials under the coupling action of freeze-thaw and loading based on the Lemaitre assumption (Wang et al., 2018). Recently, many new numerical calculation methods and test methods, such as coupled environment-mechanical damage constitutive models (Berto et al., 2014) and mesoscopic damage constitutive models (Li et al., 2017), have been utilized to investigate the constitutive model of concrete materials subjected to freeze-thaw. In addition, a series of tests have been carried out to study the constitutive models for different types of concrete materials subjected to freeze-thaw, such as confined concrete materials (Duan et al., 2011), fly ash concrete materials (Liu and Wang, 2012) and recycled aggregate concrete materials (Wu et al., 2017; Liu et al., 2018).

6

Fracture

Plastic tensile strain

Plasticity in tension

Ineffective element

Fig. 3. Constituent elements for concrete materials during freeze-thaw (Hasan et al., 2004) Overall, many studies have focused on establishing constitutive models for concrete materials subjected to freeze-thaw via test research. However, few studies have discussed the damage evolution and plasticity development of concrete materials subjected to freeze-thaw during loading, which would contribute to the analysis of freeze-thaw degradation behaviors. Moreover, freeze-thaw test for concrete materials is a type of durability test that is rather time consuming and greatly influenced by the test equipment. Therefore, simulated calculations are utilized to calculate the damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process. In this study, the cohesion function is modified by a reduction parameter to improve the accuracy of the previous damage constitutive model for concrete materials. The value of the parameter is determined by a theoretical proof, which is verified by a series of simulations. Afterwards, freeze-thaw degradation models are established by previous research work to describe the freeze-thaw degradation behaviors of concrete materials on the mechanical properties. These degradation models are applied into the mentioned constitutive model to establish the freeze-thaw damage constitutive model, which can reflect the constitutive relationship of concrete materials subjected to freeze-thaw damage. Additionally, a series of simulations are carried out to obtain the stress-strain relationships of concrete materials after freeze-thaw cycles by proposed model. The 7

accuracy of freeze-thaw damage constitutive model is validated based on comparisons between the calculation results and corresponding test results. In the simulations, the concrete frost-damage on material level has been simplified by fitting the progress of degradation during frost testing of air entrained and non-air entrained concretes. Finally, the calculation results are discussed to investigate the mechanism of damage evolution and plasticity development of concrete materials subjected to freeze-thaw cycles during the load process. 2. Modified damage constitutive model for concrete materials The mechanical properties of concrete materials can be summarized as strength softening, stiffness degeneration, residual deformation and multiaxial effect. To accurately reflect these properties, an energy release rate-based plastic-damage model for concrete materials was proposed by researchers (Wu et al., 2006), where two scalar damage variables were presented to describe the damage evolution of the materials. In this work, the energy release rate-based plastic-damage model is utilized as the theoretical framework of the constitutive model for concrete materials. 2.1. Fundamental equations The damage evolution and plasticity development of concrete materials should be decoupled from each other. Therefore, the plasticity development of concrete materials is investigated in the effective stress space, where the damage evolution of materials does not need to be considered. The effective stress in concrete materials can be assumed to satisfy the rules of classical elastoplastic mechanics (Ju, 1989; Faria et al., 1998), as expressed by Eq. (1): σ  C0 : εe  C0 :  ε  εp 

(1)

where σ is the effective stress tensor; C0 is the elastic stiffness tensor; and ε , ε e and ε p are the strain tensor, elastic and plastic strain tensor components, respectively. 8

A decomposition of effective stress is carried out due to the different behaviors of concrete materials under tension and compression loading (Ju, 1989; Faria et al., 1998), as expressed by Eq. (2a) and Eq. (2b): σ   P : σ

(2a)

σ   σ  σ   P : σ

(2b)

where σ  and σ  are the positive and negative components of the effective stress tensors and P  and P  are fourth-order projection tensors.

The Helmholtz free energy (HFE) is defined to establish the constitutive law for concrete materials, which could be decomposed into elastic and plastic components (Wu et al., 2006), as expressed by Eq. (3):

  d  , d  , εe , κ    e  d  , d  , εe    p  d  , d  , κ 

(3)

where  ,  e and  p are the HFE, elastic HFE and plastic HFE, respectively; d  represent the scalar damage variables; and κ is the plastic internal variable, which is defined by the equivalent plastic strain. Furthermore, the Clausius-Duhem inequality is obtained by the second principle of thermodynamics (Lubliner et al., 1989), as expressed by Eq. (4): σ : ε   0

(4)

Eq. (5) is obtained by Eq. (3) and Eq. (4):

  e  e         p  p σ  : ε  d  d  σ : ε  κ  0       εe  d    κ  d  

(5)

which should satisfy some conditions due to the randomicity of the elastic strain rate tensor (Wu et al., 2006), as expressed by Eq. (6a) to Eq. (6c):

9

σ

σ : εp 

 e εe

(6a)

 p κ  0 κ

(6b)

      d    d    0 d  d 

(6c)

According to Eq. (6a), the stress-strain relationship of concrete materials is obtained. Moreover, the plastic development and damage evolution of concrete materials could be derived by Eq. (6b) and Eq. (6c). The mathematical expressions of the damage variables could be obtained from stress-strain curves of concrete materials under uniaxial tension and compression loading (Wu et al., 2006; Faria et al., 1998). These damage variables show the strength loss situation of concrete materials during the process of damage, as expressed by Eq. (7a) and Eq. (7b):

d 1

d 1

   rn r0 exp  A 1   rn   r0

   

   rn   r0   1  A  A exp    B 1     rn   r0  

(7a)

(7b)

where r0 are the initial damage thresholds; rn represent the current damage thresholds; and A , A and B  are material parameters of the damage variables.

The mathematical expressions of the energy release rate based damage are modified by previous researchers (Wu et al., 2006), who adequately consider the mechanisms of compression and tension damage, as expressed by Eq. (8a) and Eq. (8b): Y   E0  σ  : Λ0 : σ 

(8a)

Y    I1  3J 2   ˆi ,max 

(8b)

where Y  are the energy release rate based damage criteria; E0 is the elasticity modulus; Λ 0 is the elastic compliance tensor; I1 is the first invariant of effective stress tensor; J 2 is the 10

second invariant of the effective partial stress tensor; ˆ i ,max is the algebraic maximum effective principal stress; and  and 

are dimensionless parameters; and the symbol  is the

Macaulay brackets. The damage effect is initiated when the energy release rate based damage exceeds the initial damage thresholds. The energy release rate based damage criteria and damage thresholds are updated with the load process (Wu et al., 2006). Furthermore, the damage variables keep updating as well. The damage evolution of concrete materials is continuously generated until the materials failed. 2.2. Modification of the cohesion function The yield function plays an important role in the constitutive models. In the energy release rate-based plastic-damage model, a yield function appropriate for concrete materials is adopted (Lubliner et al., 1989; Lee and Fenves, 1998), as expressed by Eq. (9):





F  σ, κ    I1  3J 2    κ  ˆi ,max   1    c  κ 

(9)

with   κ  and c  κ  , as expressed by Eq. (10a) and Eq. (10b):

  κ   1   

f  κ 

f  κ 

 1   

c κ   f  κ 

(10a)

(10b)

where F is the yield function;  is a current dimensionless parameter; c is the cohesion function; and f  are the effective yield stresses. In this work, the hardening effect under uniaxial tension load is not considered because concrete is approximately a brittle material when subjected to a tension load, as expressed by Eq. (11a). Moreover, linear isotropic hardening is used to reflect the compression hardening effect of concrete materials (Wu et al., 2006), as expressed by Eq. (11b): 11

f   κ   f y

(11a)

f   κ   f y  E p 

(11b)

where f y are initial effective yield stresses; E p  is the effective plastic hardening modulus, which is equal to  E E0 ; and  E is a material parameter. The calculation results of the stress-strain relationship of concrete materials under uniaxial compression loading is obtained by the above mentioned model, which shows an obvious difference between the test and calculation, as illustrated in Fig. 4. The stress value steadily increases and tends to diverge due to the yield function. The current stress state item in the yield function  I1  3J 2    κ  ˆi ,max  increases with increasing external loading. Furthermore, the cohesion function item 1    c  κ  also increases. Followed, the growth rate of the cohesion function item may be higher than the previous one, which results the value of the yield function remaining below zero. Therefore, the plasticity iteration cannot be carried out in the current stress state, and the stress value is always calculated by the elastic method. Karsan (1969) Previous model

Stress (MPa)

120

80

40

0

0

1

2

3

4

5

Strain (×10-3)

Fig. 4. Calculation results and corresponding test results of concrete materials under uniaxial compression loading To verify the viewpoint above, a proof is carried out for concrete materials under uniaxial loading in the effective stress space. An effective stress tensor is applied to the concrete materials, as expressed by Eq. (12): 12

 σ   0  0

0 0 0 0  0 0 

where  is the uniaxial effective stress; I1   ; J 2 

(12)

2 3

; and ˆ i ,max   .

Additionally, the yield function can be simplified based on the external loading conditions, as expressed by Eq. (13):     F  σ, κ       1    f   κ     1  1       f   y 

(13)

If   0 , then the concrete materials are subjected to uniaxial tension loading, and Eq. (13) can be simplified as Eq. (14):   F  σ, κ   1    f   κ     1 f   y 

(14)

The value of uniaxial effective stress increases from zero during the process of tension loading. F  0 is satisfied when the uniaxial effective tension stress exceeds the initial tensile yield stress. The tensile effective yield stress remains unchanged because the hardening effect under uniaxial tension loading is not considered in this model. Then, the return mapping iteration is carried out to update the effective stress tensor until the yield function satisfies the convergence condition. Therefore, plasticity development can be carried out successfully. If   0 , then the concrete materials are subjected to uniaxial compression loading, and Eq. (13) can be simplified as Eq. (15):



F  σ, κ    1      f   κ 

13



(15)

The yield state depends on the growth rates of the current effective stress

compressive effective yield stress

f   κ  

 

and the

. The growth rate of the current effective stress can be

obtained by the elastic method in the effective stress space, as expressed by Eq. (16):

  E0 

(16)

The growth rate of the current compressive effective yield stress can be obtained by the rule of linear hardening. Furthermore, the hardening parameter   is defined by the plastic strain  p under uniaxial compression loading, which is equal to the partial compression strain  , as expressed by Eq. (17):

f   κ  

  E E0

where  is the rate between plastic strain and total strain  

(17)

p for concrete materials under 

uniaxial compression loading. Researchers (Dahlblom and Ottosen, 1990) have believed that  is a constant that is equal to 0.2. It obviously cannot reflect the physical reality that the proportion of the plasticity deformation in the total deformation gradually increases with deformation degree increasing. Therefore, the value of  should be a variable greater than 0.2. Other researchers (Valliappan et al., 1999) have demonstrated that  is a function of the damage degree d , as expressed by Eq. (18):

 d  

d 1 d

where d is the degree of damage in the range  0, 1 and   d   0.5 is satisfied.

14

(18)

Therefore, the value of  is in the range 0.2, 0.5 , which is determined by previous studies. The value of  E is determined by the mechanics characteristic of concrete materials under uniaxial compression loading. Concrete materials conclude the stage of linear elasticity in the range   0.3 0 , 0.4 0  during the process of compression loading, where  is the current compression stress and  0 is the peak compression stress. The corresponding current compression strain is in the range   0.15 0 , 0.206 0  , where  is current compression strain and  0 is the peak compression strain, according to the Desayi and Krishnan model (Desayi and Krishnan, 1964), as illustrated in Fig. 5. The maximum compression tangent modulus is equal to 0.9E0 , as calculated by the tangent slope of the stress-strain curve. Desayi and Krishnan model

Fig. 5. The stress-strain relationship for concrete materials under compression loading (Desayi and Krishnan, 1964) Followed, the maximum of  E is equal to 9, as calculated by Eq. (19) (Wu et al., 2006):  1 E ep   1    1  E

  E0 

(19)

where E ep  is the compression tangent modulus. Furthermore,  E is a positive scalar in the range

0, 9 . Therefore, the value of

 E is in the range  0, 4.5 which is appropriate for the

effective stress space.

15

The development rules of the current effective stress and effective yield stress for concrete materials during uniaxial compression loading are obtained, as illustrated in Fig. 6 (a). It can be observed that the effective stress is always less than the initial effective yield stress when    * , and there is no plasticity development for concrete materials. When    * , the relationship of size between the effective stress and effective yield stress depends on the value of  E . If 0   E  1 , then the effective yield stress is greater than the effective yield stress. Plasticity

development can be smoothly carried out. If 1   E  4.5 , then the effective stress is always less than the effective yield stress. Plasticity development cannot take place. Therefore, concrete materials cannot reach the plastic state, which leads to the inaccuracy in the results of previous model. In this study, a cohesion reduction parameter a is proposed for concrete materials under uniaxial compression loading to solve this problem. Then, the mathematical expression of the modified cohesion function can be expressed by Eq. (20):

c  κ   f   κ   f y 

E p   a

(20)

The growth rate of the current effective yield stress under uniaxial compression loading can be expressed by Eq. (21):

f   κ  



 E E0 a

(21)

Plasticity development can be carried out only if the growth rate of the current effective stress is greater than the growth rate of the current effective yield stress for concrete materials under uniaxial compression loading based on Eq. (15). Therefore, a  4.5 is established to meet the above conditions for concrete materials under uniaxial compression loading. The modified

16

development rule of the current effective stress and effective yield stress are obtained, as illustrated in Fig. 6 (b).

(a) Previous

(b) Modified

Fig. 6. The development rules of the current effective stress and effective yield stress 2.3. Simulations of the modified damage constitutive model Finite element software ABAQUS is used to obtain the stress-strain relationships of concrete materials by the modified damage constitutive model. The calculation model is a two-dimensional square plane stress element with a length of 100 mm. Material parameters, loading methods and boundary conditions are set according to the corresponding test working conditions. Several simulations for concrete materials under uniaxial and biaxial loading are established by proposed model. The calculation results are compared with the corresponding test results (Karsan, 1969; Kupfer, 1969; Gopalaratnam, 1985) to illustrate the accuracy and availability of the proposed model, as illustrated in Fig. 7. The calculation results have a good fit with corresponding test results, which is the foundation of freeze-thaw damage constitutive model for concrete materials.

17

Karsan (1969) Proposed model

Gopalaratnam (1985) Proposed model

4

Stress (MPa)

Stress (MPa)

30

20

3 2

10 1 0

0

1

2

3

4

0

5

0

1

2

Strain (×10-3)

3

4

Strain (×10-4)

(a) Uniaxial compression

(b) Uniaxial tension 40

30 Stress (MPa)

Stress (MPa)

30 20

10

20 10

Kupfer (1969) Proposed model

0

0

1

2

3

4

Kupfer (1969) Proposed model

0

5

0

1

2

-3

(c) Biaxial compression (-1/0) Relative compression stress

Stress (MPa)

30 20 10 Kupfer (1969) Proposed model

0

1

2

3

4

5

(d) Biaxial compression (-1/-0.52)

40

0

3

Strain (×10-3)

Strain (×10 )

4

1.2

0.8

0.4 Kupfer (1969) Proposed model

0.0 0.0

5

0.4

Strain (×10-3)

0.8

1.2

Relative compression stress

(e) Biaxial compression (-1/-1)

(f) Biaxial failure criteria

Fig. 7. Calculation results and corresponding test results of concrete materials under uniaxial and biaxial loading 3. Freeze-thaw damage constitutive model for concrete materials Freeze-thaw damage constitutive model for concrete materials is proposed by the combination of freeze-thaw degradation models and the mentioned modified damage constitutive model. Then, the stress-strain relationships of concrete materials subjected to freeze-thaw under 18

different types of load are simulated by freeze-thaw damage constitutive model. Furthermore, the calculation results are compared with the corresponding test results to verify the validity of proposed model. 3.1. Freeze-thaw degradation models for concrete materials Freeze-thaw has a severe degradation effect on the mechanical properties of concrete materials, including the elasticity modulus, Poisson’s ratio, compressive strength and tensile strength. Freeze-thaw degradation models are established to reflect the freeze-thaw degradation behaviors according to the previous research work. 3.1.1. The degradation models of elasticity modulus Many researchers (Shi, 1997; Qin, 2003; Hasan et al., 2004; Zou et al., 2008; Duan et al., 2011; Guan et al., 2015) have obtained the freeze-thaw degradation behaviors of the elasticity modulus by test studies. Furthermore, these test data need to be divided into several groups due to the difference in the mixture proportions. According to the HPH, the air entraining agent (AEA) has a positive influence on improving the frost resistance durability for concrete materials. When the water in the pores begins to freeze and volume goes up, destructive pressure builds up gradually inside the concrete materials. Its size depends on the distance from the ice to the escape boundary, material permeability and freezing rate. The experience indicates that when the distance between the capillary cavity and nearest escape boundary is greater than 75 to 100 microns, it creates destructive pressure in the saturated cement paste subjected to freezing. The air void is a type of escape boundary. The air void parameters with such small spacing can be provided by appropriate AEA, which can effectively reduce the destructive pressure in the saturated cement paste subjected to freezing (Powers, 1958). 19

Therefore, the air entrained concrete materials and non-air entrained concrete materials should be separated. Moreover, the value of the water cement ratio W / C in the mixture proportions for the non-air entrained concrete materials can also affect the degradation behaviors of the elasticity modulus. In this work, W / C  0.5 is set to a boundary. Then, the test for non-air entrained concrete materials are divided into two groups: W / C  0.5 and W / C  0.5 . Finally, three groups are established, where group one is air entrained concrete materials and group two is non-air entrained concrete materials with W / C  0.5 and group three is non-air entrained concrete materials with W / C  0.5 . Furthermore, the degradation models of the elasticity modulus are obtained by polynomial fitting, as illustrated in Fig. 8, where Ed is the elasticity modulus with freeze-thaw damage and N is the number of freeze-thaw cycle.

Relative elasticity modulus

1.0

0.8 R  0.87763 2

0.6

Ed  1.072  109 N 3  3.078078  106 N 2 E0

0.4

 8.23094612  104 N  1.0

0.2 0.0

Shi (1997) Zou (2008)

0

50

100

Hasan (2004) Guan (2015)

150

200

250

Qin (2003)

0.8 0.6 R2  0.99790

0.4 Ed  1.686  106 N 3  2.37138  104 N 2 E0

0.2

 3.53648  104 N  1.0

0.0

300

0

20

Freeze-thaw cycle number

40

(b) Group two

1.0

Duan (2011)

0.8

Ed  5.11  107 N 3  1.79499  104 N 2 E0  2.1835616  102 N  1.0

0.6 0.4

R2  0.99975

0.2 0.0

60

0

80

Freeze-thaw cycle number

(a) Group one Relative elasticity modulus

Relative elasticity modulus

1.0

25

50

75

100

125

150

Freeze-thaw cycle number

(c) Group three Fig. 8. The degradation models of the elasticity modulus 20

100

3.1.2. The degradation model of Poisson’s ratio The freeze-thaw degradation behaviors of the Poisson’s ratio have been obtained by researchers (Shi, 1997; Zou et al., 2008). The degradation model of the Poisson’s ratio is obtained from these test data, as illustrated in Fig. 9, where d is the Poisson’s ratio with freeze-thaw damage. Shi (1997) Zou (2008)

Relative Poisson's ratio

1.0 0.8 0.6

R2  0.85011

0.4

d  6.4  108 N 3  1.9514  105 N 2  0 0.2  2.355025  103 N  1.0

0.0

0

50

100

150

200

250

300

Freeze-thaw cycle number

Fig. 9. The degradation model of the Poisson’s ratio When N is greater than 200, the value of Poisson’s ratio with freeze-thaw damage calculated by fitting result is too small. Therefore, it assumes that the value of Poisson’s ratio does not degenerate when freeze-thaw number is greater than 200. 3.1.3. The degradation models of compressive strength A series of tests have been carried out to obtain the freeze-thaw degradation behaviors of the compressive strength by some researchers (Shi, 1997; Qin, 2003; Hasan et al., 2004; Zou et al., 2008; Duan et al., 2011; Cao et al., 2012; Guan et al., 2015). These tests also need to be divided into several groups due to the different content of AEA in the mixture proportions. Therefore, group one is non-air entrained concrete materials, group two is air entrained concrete materials with the AEA content ranging from 0 to 10 kg/m3 and group three is air entrained concrete materials with the AEA content greater than 10 kg/m3. Furthermore, the degradation models of

21

compressive strength are obtained by polynomial fitting, as illustrated in Fig. 10, where f c ,d is

Qin (2003) Duan (2011) Cao (2012)

1.0 0.8 0.6

1.0 Relative compressive strength

Relative compressive strength

the compressive strength with freeze-thaw damage.

R2  0.72086

0.4

f c ,d f c ,0

0.2

 1.28  107 N 3  2.6383  105 N 2  2.527338  103 N  1.0

0.0

0

25

50

75

100

125

0.8 R2  0.89924

0.6

f c ,0

 3.996  109 N 3  7.90207  107 N 2  1.330006451  103 N  1.0

0.4

Shi (1997) Zou (2008)

0.2

150

f c ,d

0

50

Freeze-thaw cycle number

150

200

250

300

Freeze-thaw cycle number

(a) Group one Relative compressive strength

100

Hasan (2004) Guan (2015)

(b) Group two Guan (2015)

1.0

0.9 R2  0.76850

f c ,d

0.8

f c ,0

 4.6  109 N 3  1.453  106 N 2  3.920558  104 N  1.0

0.7

0

50

100

150

200

250

300

Freeze-thaw cycle number

(c) Group three Fig. 10. The degradation model of compressive strength 3.1.4. The degradation model of tensile strength Few researchers (Cao et al., 2012) have focused on the freeze-thaw degradation behaviors of tensile strength. Polynomial fitting is carried out, as illustrated in Fig. 11, where f t ,d is the tensile strength with freeze-thaw damage.

22

Relative tensile strength

1.0

Cao (2012)

0.8 R2  0.98029

0.6

ft ,d

0.4

 2.7  108 N 3  7.233  106 N 2

ft ,0

 3.497214  103 N  1.0

0.2

0

25

50

75

100

125

Freeze-thaw cycle number

Fig. 11. The degradation model of tensile strength 3.2. Validation for freeze-thaw damage constitutive model Several simulations for concrete materials subjected to freeze-thaw under different types of load are carried out using proposed freeze-thaw damage constitutive model. Then, the stress-strain relationships for each simulation are compared with the corresponding test results to illustrate the accuracy and availability of proposed model. Stress-strain relationships of concrete materials subjected to freeze-thaw under uniaxial compression loading have been obtained by researchers (Hasan et al., 2004; Duan et al., 2011; Guan et al., 2015). The calculation results and corresponding test results are obtained for concrete materials subjected to freeze-thaw under uniaxial compression loading, as illustrated in Fig. 12. Additionally, the improvement in stress-strain relationships of concrete materials subjected to freeze-thaw cycles considering the cohesion reduction parameter a is verified by the comparisons of corresponding calculation results.

40 20 0

Hasan (2004) Proposed model

0

1 2 Strain (×10-3)

(a-1) N = 0

3

60

40 20 0

Hasan (2004) Proposed model

0

1 2 Strain (×10-3)

(a-2) N = 50

23

3

Stress (MPa)

60

Stress (MPa)

Stress (MPa)

60

40 20 0

Hasan (2004) Proposed model

0

1 2 Strain (×10-3)

(a-3) N = 100

3

Hasan (2004) Proposed model

1 2 Strain (×10-3)

0

3

Hasan (2004) Proposed model

0

(a-4) N = 200

20 0

0

30

0

4 8 12 Strain (×10-3)

0

Stress (MPa)

Stress (MPa)

4 0

0

6 12 Strain (×10-3)

0

0

2 4 Strain (×10-3)

0

6

5 10 Strain (×10-3)

15

Guan (2015) With a Without a

6

50 25 0

0

2 4 Strain (×10-3)

6

(c-2) N = 100

Guan (2015) With a Without a

45

Stress (MPa)

Stress (MPa)

(c-3) N = 200

10

75

(c-1) N = 0

25 0

2 4 Strain (×10-3)

12

(b-4) N = 125

25 0

18

Guan (2015) With a Without a

50

20

0

15

50

(b-5) N = 150 75

5 10 Strain (×10-3)

Guan (2015) With a Without a

75

4 8 Strain (×10-3)

Duan (2011) With a Without a

(b-3) N = 100

Duan (2011) With a Without a

8

0

30

15

(b-2) N = 75 12

25

(b-1) N = 0

Duan (2011) With a Without a

45

Stress (MPa)

Stress (MPa)

40

50

0

3

Duan (2011) With a Without a

(a-5) N = 300

Duan (2011) With a Without a

60

1 2 Strain (×10-3)

Stress (MPa)

0

10

Stress (MPa)

15

20

Stress (MPa)

30

0

75

30

Stress (MPa)

Stress (MPa)

45

30 15 0

0

2 4 Strain (×10-3)

6

(c-4) N = 300

Fig. 12. Calculation results and corresponding test results for concrete materials subjected to freeze-thaw under uniaxial compression loading considering cohesion reduction parameter a Researchers (Cao et al., 2012) have obtained the stress-strain relationships for concrete materials subjected to freeze-thaw under uniaxial tension loading. The calculation results and corresponding test results are obtained for concrete materials subjected to freeze-thaw under uniaxial tension loading, as illustrated in Fig. 13.

24

2.0

0.0 0.0

Cao (2012) Proposed model

0.6 1.2 Strain (×10-4)

2.0 1.0

Cao (2012) Proposed model

0.0 0.0

1.8

(a) N = 0

Cao (2012) Proposed model

Stress (MPa)

Stress (MPa)

1.0

Cao (2012) Proposed model

0.0 0.0

1.8

1.5

1.2 0.6 0.0

0.4 0.8 1.2 Strain (×10-4)

(d) N = 75

0.5 1.0 1.5 Strain (×10-4)

(c) N = 50

1.8

1.6

0.0 0.0

2.0

(b) N = 25

2.4

0.8

0.6 1.2 Strain (×10-4)

Stress (MPa)

1.0

3.0

Stress (MPa)

3.0

Stress (MPa)

Stress (MPa)

3.0

Cao (2012) Proposed model

0

3 6 Strain (×10-5)

1.0 0.5 0.0

9

Cao (2012) Proposed model

0

(e) N = 100

3 6 Strain (×10-5)

9

(f) N = 125

Fig. 13. Calculation results and corresponding test results for concrete materials subjected to freeze-thaw under uniaxial tension loading To investigate the performance of concrete materials subjected to freeze-thaw under biaxial compression loading, a series of test studies have been carried out by researchers (Shang, 2006). The calculation results and corresponding test results are obtained for concrete materials subjected

Relative compression stress

to freeze-thaw under biaxial compression loading, as illustrated in Fig. 14.

1.2

0.8 N = 0, Shang (2006) N = 100, Shang (2006) N = 200, Shang (2006) N = 300, Shang (2006) Proposed model

0.4

0.0 0.0

0.4

0.8

1.2

Relative compression stress

Fig. 14. Calculation results and corresponding test results for concrete materials subjected to freeze-thaw under biaxial compression loading These results of stress-strain relationships calculated by proposed model indicate that the proposed freeze-thaw damage constitutive model has high calculation precision, which is adopted as the foundation to investigate the damage evolution and plasticity development of concrete 25

materials subjected to freeze-thaw during the load process. 4. Discussion Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process are essential to improve the frost durability of concrete structures. However, these two properties are not easily obtained via test investigation, which can be solved by proposed model. Damage evolution can be obtained by extracting damage variables that show the degree of damage for concrete materials subjected to freeze-thaw during the load process. Plasticity development can be obtained by extracting hardening parameters, which can be defined by the plastic strain under uniaxial loading. 4.1. Damage evolution of concrete materials subjected to freeze-thaw during compression loading Stress-strain relationships of concrete materials subjected to freeze-thaw under uniaxial compression loading (Guan et al., 2015) have been simulated by proposed model. Furthermore, the value of damage variables during loading is obtained, which shows the relationship between damage variables and the compression strain, as illustrated in Fig. 15 (a). The value of the compressive damage variable increases with compression strain increasing. Meanwhile, a smaller number of freeze-thaw cycle corresponds to a larger compressive damage variable when the concrete materials are subjected to the same compression strain. This behavior indicates that concrete materials subjected to a smaller number of freeze-thaw cycles experience a more serious degree of damage during the compression loading. According to previous studies (Gong and Jacobsen, 2019), researchers have consistently found that damage stresses during freezing in cement paste could result from hydraulic pressure (by ice volume expansion), cryosuction pressure (by surface tension of water) and crystallization pressure (due to the shape of ice crystals). 26

The pore structure of concrete materials is constantly destroyed by these pressures as freeze-thaw cycles number increases, which increases the porosity (Shields et al., 2018). Concrete materials have a compaction process during the compression loading due to internal pores and fractures. Moreover, the higher degree of porosity, the lower degree of compaction. Compression damage of concrete materials is generated on the surface of the effective compression, where the area decreases with a decrease in the compaction degree on the section of concrete materials during compression loading. Therefore, the value of the compressive damage variable increases with a decrease in freeze-thaw cycles number for concrete materials under specific compression strain. For other tests (Hasan et al., 2004; Duan et al., 2011), the proposed freeze-thaw damage constitutive model can also give similar results regarding the compressive damage variable, as illustrated in Fig. 15 (b) and Fig. 15(c). However, some curves for the compressive damage variable do not follow the above changing rule, such as N = 100 in Fig. 15 (b) and N = 150 in Fig. 15(c), because the stress-strain relationships between the calculation results and the corresponding test results have some differences, especially in the descent stage. 0.6

dc, N 0  dc, N 100  dc, N 200  dc, N 300

Compressive damage variable

Compressive damage variable

1.0 0.8 0.6 0.4

N=0 N = 100 N = 200 N = 300

0.2 0.0

0

2

4

0.362

0.360

0.4

0.358

0.356

0.2

0.0

6 -3

dc, N 100  dc, N 0  dc, N 50  dc, N 200  dc, N 300

2.76

2.78

N=0 N = 50 N = 100

1

2

3

2.80

2.82

N = 200 N = 300

4

Strain (×10-3)

Strain (×10 )

(a) Corresponding Guan’s test

(b) Corresponding Hasan’s test

27

5

Compressive damage variable

1.2

dc, N 0  dc, N 75  dc, N 100  dc, N 125  dc, N 150

0.9

0.6 N=0 N = 75 N = 100 N = 125 N = 150

0.3

0.0 0.0

0.5

1.0

1.5

Strain (×10-2)

(c) Corresponding Duan’s test Fig. 15. Calculation results regarding the compressive damage variable by proposed model Calculation results regarding the initial damage strain corresponding to the tests (Hasan et al., 2004; Duan et al., 2011; Guan et al., 2015) are obtained by the proposed freeze-thaw damage constitutive model, which shows the value of compression strain with the appearance of initial damage, as illustrated in Fig. 16. The value of the initial damage strain increases with increasing N, which indicates that the time for initial damage is delayed by freeze-thaw for concrete materials during compression loading. According to the mentioned viewpoint, freeze-thaw can increase porosity, which leads to a larger compaction strain for concrete materials during compression loading. The initial damage strain increases with increasing N because the initial compression damage appears behind the compaction process. Additionally, the initial damage strain corresponding to the Hasan’ test and Guan’s test has a slower growth rate than that of the value in Duan’s test due to the influence of the AEA in the mixture proportions of the concrete materials.

Initial damage strain (×10-3)

8 Hasan (2004) Duan (2011) Guan (2015)

6

4

2

0

0

100

200

Freeze-thaw cycle number

28

300

Fig. 16. Calculation results regarding the initial damage strain by proposed model 4.2. Plasticity development of concrete materials subjected to freeze-thaw during compression loading The plasticity development of concrete materials subjected to freeze-thaw during compression loading can also be calculated by the proposed freeze-thaw damage constitutive model, as illustrated in Fig. 17 (a) to Fig. 17 (c). Plastic strain grows linearly because linear hardening is used to describe the plasticity development of concrete materials in this model. The results show that the plastic strain is smaller with larger N under a specific compression strain, which indicates that the changing rule of plasticity development is similar to damage evolution of concrete materials subjected to freeze-thaw during compression loading. The freeze-thaw can cause cracking which results in the increase of strain softening. It accelerates the damage to concrete materials during compression loading. This result indicates that damage evolution and plasticity development can promote each other. 3.51

2.0

1.02

8 Plastic strain (×10-4)

Plastic strain (×10-3)

3.50

1.00

1.5 0.98

3.4

1.0

3.5

3.6

3.7

N=0 N = 100 N = 200

0.5

6 4

3.49

3.48 1.792

1.800

1.808

N=0 N = 50 N = 100 N = 200 N = 300

2

N = 300

0.0 0.0

1.5

3.0

4.5

0

6.0

Strain (×10-3)

0

1

2

3

Strain (×10-3)

(a) Corresponding Guan’s test

(b) Corresponding Hasan’s test

29

2.25

Plastic strain (×10-3)

4.5

2.10

1.95

3.0

1.80 0.65

0.70

0.75

0.80

0.85

N=0 N = 75 N = 100 N = 125 N = 150

1.5

0.0 0.0

0.5

1.0

1.5

Strain (×10-2)

(c) Corresponding Duan’s test Fig. 17. Calculation results regarding the plastic strain by proposed model Calculation results regarding the initial plastic strain and gradient of plastic strain corresponding to the tests (Hasan et al., 2004; Duan et al., 2011; Guan et al., 2015) can be obtained by the proposed freeze-thaw damage constitutive model, as illustrated in Fig. 18 and Fig. 19. The initial plastic strain is the value of strain with initial plastic strain generation. With an increase in N, the initial plastic strain increases and the gradient of plastic strain decreases. Moreover, the results corresponding to the Duan’s test have a faster rate of change than the results corresponding to the Hasan’s test and Guan’s test. The content of the AEA in the mixture proportions is the main factor that leads to this difference (Hasan’s test and Guan’s test used AEA and Duan’s test did not in the mixture proportions). The AEA brings some air bubbles into the cement paste during hydration, which can greatly reduce the stress damage generated by freeze-thaw. Therefore, the results corresponding to the Duan’s test exhibit a great change with an increase in N.

30

5.0 Hasan (2004) Duan (2011) Guan (2015)

Gradient of plastic strain (×10-1)

Initial plastic strain (×10-3)

6

4

2

0

0

100

200

4.5

4.0

3.5

3.0

300

Freeze-thaw cycle number

Hasan (2004) Duan (2011) Guan (2015)

0

100

200

300

Freeze-thaw cycle number

Fig. 18. Calculation results regarding the

Fig. 19. Calculation results regarding the

initial plastic strain by proposed model

gradient of plastic strain by proposed model

4.3. Damage evolution of concrete materials subjected to freeze-thaw during tension loading In contrast, the calculation results for the tension damage variable in the corresponding test (Cao et al., 2012) are obtained, as illustrated in Fig. 20. Concrete materials subjected to a lower number of freeze-thaw cycle are found to require a greater tension strain with the same degree of tension damage. Freeze-thaw increases the porosity, which promotes the formation of fractures and cracks inside concrete materials during tension loading. Therefore, the initial damage strain decreases and tension damage rate increases with increasing N, as illustrated in Fig. 21. The tensile fracture strain of concrete materials is usually around 100 microstrain, which corresponds to the tensile initial damage strain with the freeze-thaw cycles number around 0 to 25 in Fig. 21. Initial damage means the concrete materials begin to fracture. With the increase of the freeze-thaw cycles number, the initial damage strain is constantly decreasing and below 100 microstrain, which indicates that the tensile fracture resistance of concrete materials is decreasing. The plasticity development of concrete materials subjected to freeze-thaw during tension loading is not obvious, therefore it is ignored in this model.

31

 t , N 0   t , N  25   t , N 50   t , N 75

Tensile damage variable

0.4

  t , N 100   t , N 125

0.3

N=0 N = 25 N = 50 N = 75 N = 100 N = 125

0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

Strain (×10-4)

Fig. 20. Calculation results regarding the tensile damage variable by proposed model 2.4 Initial damage strain Gradient of tensile damage

1.2

2.0

0.9 1.6

0.6 1.2

0.3

0

25

50

75

100

Gradient of tensile damage (×104)

Initial damage strain (×10-4)

1.5

125

Freeze-thaw cycle number

Fig. 21. Calculation results regarding the initial damage strain and gradient of tensile damage by proposed model 5. Conclusions This paper establishes a freeze-thaw damage constitutive model based on damage theory to calculate and analyse the damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process. First, a cohesion reduction parameter is proposed to modify the cohesion function, which can improve the calculation accuracy of the previous damage constitutive model. Furthermore, freeze-thaw degradation models for concrete materials are established by previous research work to describe the freeze-thaw degradation behaviors. These degradation models are applied into the mentioned constitutive model to establish the freeze-thaw damage constitutive model, which can reflect the constitutive relationship of concrete materials subjected to freeze-thaw damage. Additionally, a series of simulations are carried out to 32

obtain the stress-strain relationships of concrete materials subjected to freeze-thaw cycles by proposed model. The calculation results have a good fit with corresponding test results. Finally, the calculation results are discussed to investigate the mechanism of damage evolution and plasticity development of concrete materials subjected to freeze-thaw cycles during the load process. The detailed conclusions are listed as following: (1) A modified damage constitutive model for concrete materials is proposed to improve the calculation accuracy of the previous model by a cohesion reduction parameter, which is equal to 4.5 as determined by a theoretical proof. (2) A freeze-thaw damage constitutive model is established to calculate the stress-strain relationships of concrete materials subjected to freeze-thaw cycles under different types of load (uniaxial compression, uniaxial tension and biaxial compression). Furthermore, the calculation accuracy of proposed model is validated by the comparison with corresponding test results. (3) Damage evolution and plasticity development are promoted with the decrease of freeze-thaw cycles number for concrete materials during the compressive loading, and the initial damage strain is decreasing. Furthermore, the initial plastic strain increases and gradient of plastic strain decreases with the increase of freeze-thaw cycles number. (4) The damage evolution of concrete materials subjected to freeze-thaw under a specific tension strain is promoted with increasing N. Moreover, the tension damage rate increases and the initial damage strain decreases with increasing N. (5) Damage evolution and plasticity development of concrete materials subjected to freeze-thaw during the load process can obviously be influenced by the AEA content in the mixture proportions. 33

Conflict of Interest: The authors declared that they have no conflicts of interest to this work.

Acknowledgements This work was supported by the National Key Research and Development Program of China (2018YFC0809605) and the Fundamental Research Funds for the Central Universities (Grant No. 2572017PZ13). References Berto, L., Saetta, A., Scotta, R., Talledo, D., 2014. A coupled damage model for RC structures: Proposal for a frost degradation model and enhancement of mixed tension domain. Constr. Build. Mater. 65, 310-320. Berto, L., Saetta, A., Talledo, D., 2015. Constitutive model of concrete damaged by freeze–thaw action for evaluation of structural performance of RC elements. Constr. Build. Mater. 98, 559-569. Coussy, O., 2005. Poromechanics of freezing materials. J. Mech. Phys. Solids. 53 (8), 1689-1718. Coussy, O., Monteiro, P.J.M., 2008. Poroelastic model for concrete exposed to freezing temperatures. Cem. Concr. Res. 38 (1), 40-48. Cao, D.F., Fu, L.Z., Yang, Z.W., 2012. Test study on tensile properties of concrete after freeze-thaw cycles. Journal of Building Materials, 15 (1), 48-52. Desayi, P., Krishnan, S., 1964. Equation for the stress-strain curve of concrete. ACI Journal Proceedings. 61 (3), 345-350.

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