Research progress on freeze–thaw constitutive model of concrete based on damage mechanics

Liu Zimei; Ge Xueliang; Lu Cairong; Zhang Zhengnan; Duan Yuwei; Xu Haiyan

doi:10.1515/secm-2024-0020

Article Open Access

Research progress on freeze–thaw constitutive model of concrete based on damage mechanics

Liu Zimei , Ge Xueliang , Lu Cairong , Zhang Zhengnan , Duan Yuwei and Xu Haiyan

Published/Copyright: July 22, 2024

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From the journal Science and Engineering of Composite Materials Volume 31 Issue 1

Abstract

In cold areas, freeze–thaw damage seriously affects the long-term use and safe operation of concrete structures. The constitutive model is an important foundation for predicting deformation and strength characteristics of concrete materials and for the non-linear analysis of concrete structures. This study is based on the elaboration of methods for the constitutive model of damaged materials by using damage mechanics and others. This study focuses on the existing constitutive model results of concrete under the static axial compression load, dynamic load, and coupling environmental load, and analyzing the problems in existing studies. Research has shown that segmented models exhibit higher fitting accuracy of concrete freeze–thaw constitutive model under static axial compression loads. By defining coupled damage variables, it is possible to approach the actual freeze–thaw damage of concrete under environmental coupling, and attention should be paid to the differences or interactions between damage factors. In order to meet the actual engineering needs of high altitude and cold areas needs to expand the temperature range of freeze–thaw tests and consider the dynamic loads impact on concrete damage, the establishment of constitutive model of concrete under the actual freeze–thaw damage is the focus of frost-resistant durability research of hydraulic concrete in cold regions.

Keywords: concrete; freeze–thaw cycle; damage mechanics; constitutive model; coupling actions

Symbols Meanings

σ: stress, MPa
σ ¯: effective stress, MPa
σ p: peak stress, MPa
σ p n: peak stress after n times of freeze–thaw cycles, MPa
ε: strain, 10⁻³
ε p 0: peak strain, 10⁻³
ε p n: peak strain after n times of freeze–thaw cycles, 10⁻³
D: damage variable
E: initial elastic modulus of the undamaged material, MPa
E n: elastic modulus of concrete after n times of freeze–thaw cycles, MPa
Δ E N: rate of loss of elastic modulus after N freeze–thaw cycles
E u: elastic modulus of concrete during compression damage, MPa
N/n: number of freeze–thaw cycles

1 Introduction

As an indispensable engineering material for infrastructure facilities, the durability of concrete is the guarantee of the life of concrete structures in the service environment [1]. Freeze–thaw is an important factor leading to the destruction and deterioration of concrete in cold regions [2,3]. Most studies related to the properties of concrete after freeze–thaw cycles have previously focused on strength, relative dynamic elastic modulus, deformation, and mass loss [4,5,6,7]. The mechanism of freeze–thaw damage of concrete material is mainly due to the water freezing expansion pressure inside the concrete at low temperature [8,9,10] and causes the generation and expansion of micro-cracks [11,12]. The strength and deformation ability of concrete will be affected because the action of freeze–thaw [13,14,15]. Specifically, as the number of freeze–thaw cycles increases, the elastic modulus, the yield strength, and ultimate strength will also decrease [16,17,18]. And the number and distribution of pores and micro-cracks will further affect its mechanical properties [19].

The structure constitutive relationship under the axial compression is an important research basis for the deformation of concrete structures and an indispensable basic material characteristic for the nonlinear analysis of concrete structures [20,21]. With the appearance and rapid development of computational techniques such as high-speed computers and the finite unit method, traditional elastic design analysis has evolved into nonlinear analysis [22]. In order to predict the performance of concrete under complex stress loads, the constitutive model has become an important basis for improving the application ability and expanding the application scope of concrete structure calculation software.

In recent years, with the rapid development of China’s water conservancy and hydropower infrastructure construction, the “Belt and Road” construction, and the implementation of the national water network project, the development of water conservancy and hydropower resources continues to Xinjiang, Tibet, Qinghai, and other alpine regions (high-altitude, high-latitude, perennial low-temperature areas) to promote [23,24], for example, the lower Brahmaputra hydropower development, the QBT arch dam in Xinjiang (dam height 240 m, the dam site is located at 48 degrees north latitude), the Huangzang Temple gravity dam in Qinghai (dam height 122 m), the Yebatan arch dam in Tibet (dam height 217 m), and the South-North Water Transfer West Project. The alpine region has the climate characteristics of low average daily temperature, a large temperature difference between day and night, long duration of low temperature, and frequent alternation of wet and dry [25,26]. The environmental temperature of hydraulic structures subjected to low-temperature freezing damage exceeds the low-temperature condition of (−17 ± 2)°C in the current industry frost-resistance test specification [27]. Compared with the conventional low-temperature environment used in existing research, the problem of freeze–thaw damage to hydraulic concrete structures in alpine regions is more prominent.

Therefore, it is of great significance for improving the safety and durability of concrete structures in alpine cold regions to carry out in-depth research on freeze–thaw damage and to establish a concrete freeze–thaw damage constitutive model considering the effect of the complex environment in actual service. This study summarizes the latest research progress in constructing the constitutive model for concrete under the action of freeze–thaw cycles from the basic concept. And it takes the process of model establishment as the main line, including static axial load, dynamic axial load, and freeze–thaw damage constitutive model of concrete under coupling effects of various environmental loads. By summarizing and discussing the existing research results, further work is proposed for the study of freeze–thaw durability of hydraulic concrete structures in alpine regions, to provide theoretical support for basic study on the improvement of freeze–thaw durability of hydraulic buildings in the construction of water conservancy and hydropower projects.

2 Concept

2.1 Concrete freeze–thaw damage

The freeze–thaw damage of concrete structures mainly includes three stages: water absorption, water freezing, and structural damage. There are two types of cracks in concrete in the initial stages: one is caused by initial damage, and the other is caused by air bubbles entrained during the formation of concrete. The initial cracks are more likely to cause the penetration of external water [28]. In the first stage of freeze–thaw, the external water enters the concrete through the cracks and fills the internal pores, gradually making the internal pores nearly saturated [29]. With the decrease of temperature, the second stage water freezing occurs. The volume of frozen water expands by about 9%, and the inner wall of pore is subjected to tensile stress [30]. In the third stage, the expansion pressure exceeds the bearing limit of concrete, the crack develops, and the porosity of concrete increases [31]. And the spalling of cement matrix leads to the destruction of internal structure [32].

2.2 Constitutive relation

The constitutive relation is the relationship and law of motion between the force, temperature, and deformation of the internal structure of a material; it describes a deformation or motion process of the material and is a mathematical expression describing the characteristics of the material [33]. The constitutive relation includes a variety of factors, usually disregarding time-related factors, and the constitutive relationship of the material is the stress–strain relation in the case of normal temperature and short-term static load loading.

2.3 Damage mechanics

The concept of damage mechanics was first introduced by Kaehanov in 1958, followed by Rabotnov [34] who extended the scope of application and laid the foundation of damage mechanics. In 1976, Dougill [35] first tried to adopt the damage theory in the field of concrete research.

Damage mechanics is a research method that mainly aims at the whole process of materials or components from the original defect, to the appearance and expansion of micro-cracks, to the initiation, evolution, and expansion of macro-cracks, until the final destabilization fracture [36]. The use of damage mechanics theory can reflect the microstructural changes and macroscopic mechanical properties of materials, and its main parameters can be obtained through relevant tests. Therefore, many research scholars have gradually paid attention to and adopted the damage mechanics theory in the process of constructing the concrete constitutive model according to the inherent defects of the concrete material itself [37].

2.4 Damage variable

The main study of damage mechanics is the development of internal defects in materials leading to the destruction of the material structure. By introducing the state quantity “damage variable,” a fundamental concept in the theory of damage mechanics is used to describe the internal defects or internal damage of materials [38,39]. For concrete materials, the damage variable can be determined by relevant experimental measurements such as mass loss, density, elastic modulus, ultrasonic velocity, resistivity, and other macroscopic characteristic physical quantities; for its micro-structure, the damage variable can be selected and defined by the size, number, shape, and arrangement of the pores or cracks.

2.5 Constitutive equation of lossy material

Take the lossy material micro-element model as shown in Figure 1, the total cross-sectional area is A ₀, the cross-sectional area of the damaged part is A, then the actual undamaged area of the damaged material is A ₀−A. Assuming that the damage is isotropic and uniformly distributed in all directions, the damage variable D is defined as

(2.1) D = A A 0 , 0 ≤ D ≤ 1 .

Figure 1

Lossy material micro-element model.

Stress on the material cross-section is σ. The actual stress to which the damaged material is subjected acts on the undamaged section, and then, the effective stress σ ¯ is

(2.2) σ ¯ = σ ¯ 1 − D .

The constitutive equation of a nondestructive linear elastic material under unidirectional tensile loading conditions is

(2.3) ε = σ ¯ E ,

where ε is the strain, and E is the initial elastic modulus of the undamaged material.

According to the strain equivalent theory [40], the strain ε produced by the damaged material in the stress state is equivalent to the strain produced by the effective stress σ ¯ on the initial undamaged material:

(2.4) ε = σ ¯ E .

Substituting Equations (2.2) into Equation (2.4), the constitutive equation of lossy material is obtained as follows:

(2.5) σ = ( 1 − D ) E ε .

It can be seen that the construction of the constitutive model based on damage mechanics theory mainly includes defining the appropriate damage variables to accurately represent the damage state of the material; analyzing the evolution of damage variables according to the experimental data and phenomena; establishing damage evolution equation and determining the material parameters; and finally establishing a material constitutive relation considering that single or double damage. Meanwhile, when defining the damage variables, extra attention should be paid to their physical meaning clearly, so as to truly reflect the damage properties of concrete material.

In addition, when introducing the mechanical constitutive model of concrete under freeze–thaw cycles, appropriate adjustments need to be made according to specific engineering application scenarios. For example, under different climate conditions and stress conditions, the freeze–thaw damage mechanism and degree of concrete materials may be different, so the model needs to be modified or optimized based on the actual situation.

3 Constitutive model of freeze–thaw damage under static axial compression load

Based on the research results of mechanical and deformation properties of various types of concrete after freeze–thaw damage under static axial pressure test, the concrete freeze–thaw damage constitutive model studied in recent years is divided into two categories: integral and segmental according to the form of expressions.

Considering that different freeze–thaw temperatures and rates will lead to different degrees of concrete damage, even under the same number of freeze–thaw cycles, the damage degree will be different [41,42]. In the references mentioned in this section about the freeze–thaw damage constitutive under axial compression load, the freeze–thaw test was conducted about “Test method for rapid freezing and thawing” in GB/T 50082-2009 “Standard for test methods of long-term performance and durability of ordinary concrete,” i.e., the freeze–thaw temperature and rate are consistent.

3.1 Integral constitutive model

Xiao et al. [43] studied the uniaxial compression constitutive model of concrete after the freeze–thaw cycle. Since the probability distribution of concrete uniaxial compressive strength meets the Weibull strength theory and is based on the parallel bar model of the concrete specimen parallel bar system, the unit strength distribution is following the Weibull probability distribution [44]. The distribution function F ( ε ) and the damage variable D are expressed as

(3.1.1) F ( ε ) = 1 − exp − ε a b ,

(3.1.2) D = ∫ 0 ε P ( ε ) d ε = 1 − exp − ε a b ,

where a and b are the shape parameters of the Weibull distribution function.

Based on the basic equations of damage mechanics (2.5), the Weibull distribution shape parameters are determined by the experimental results. The final constitutive relation of concrete under freeze–thaw conditions is

(3.1.3) σ = E ε ( 1 − D ) = E ε exp − ε ( 3.133 × 10 − 6 ) N + 0.00169 0.00187 N + 1.56532 .

Ji-sheng et al. [45] further discussed the stress–strain curves of concrete with different steel fiber contents after the freeze–thaw cycles test. The proposed concrete uniaxial compression freeze–thaw constitutive model is as follows:

(3.1.4) σ = E ε exp − 1 m ε e 0.02188 N ( 1.67 + 0.055 S + 0.056 S 2 ) m ,

where

m = 1.9418 ( S = 0 ) m = 1.702 − 0.11 S ( 0.5 % ≤ S ≤ 1.5 % ) ;

E = 34400 e − 0.037 N ;

S – represents the contents of steel fiber in concrete , 1 / 100 .

Guangcheng et al. [46] carried out freeze–thaw tests on C40 strength-grade self-compacting concrete (SCC). In the study of the uniaxial compressive stress–strain relationship of the specimen, the total damage variable D is defined as

(3.1.5) D = D c + D n − D c D n .

The damage variable caused by load action D c and the damage variable caused by freeze–thaw action D n are, respectively, expressed as follows:

(3.1.6) D c = 1 − exp − ε a b ,

(3.1.7) D n = 1 − E n E ,

where a and b are the shape parameters of the Weibull distribution function, reflects the peak strain and brittleness of concrete, respectively.

Finally, under the condition of considering both freeze–thaw damage and uniaxial compressive load damage, the constitutive equation of concrete under axial compression is obtained as

(3.1.8) σ = E n ε exp − ε a b .

In terms of defining damage variables, some researchers adopted acoustic emission indexes [47,48] and established the concrete freeze–thaw damage constitutive model based on acoustic emission detection and data acquisition by the stress–strain device (Figure 2):

(3.1.9) σ = E u ε = ( 1 − D l ) E n ε .

Figure 2

Arrangement diagram of each device for acoustic emission test [47].

Load damage D l ( ε ) based on acoustic emission properties is defined as

(3.1.10) D l ( ε ) = J J t ,

where J is the accumulated acoustic emission amplitude during compression damage of concrete; J t is the total acoustic emission amplitude of concrete.

The damage evolution model is as follows:

D l ε = 0 , ε ≤ 0.4 ε p 0.59 e 0.686 ε ε p − 0.59 e 0.2744 , 0.4 ε p ≤ ε ≤ ε p − 0.728 + 1.54 ε ε p − 0.463 ε ε p 2 + 0.0453 ε ε p 3 , ε ≥ ε p

The relationship between the elastic modulus and the relative dynamic elastic modulus is

(3.1.11) E n / ε p n E / ε p 0 = E n E = 1 − D f .

And the concrete freeze–thaw damage D f is defined as

(3.1.12) D f = 1 − E fX .

The value of freeze–thaw damage D f of coal gangue concrete and the relationship formula with the peak strain are

D f = 0.000688 N + 0.0033 Q N + 0.00551 Q 2 N

ε p = 1.86 e 2.82 D f ,

where E 0 is the initial elastic modulus of concrete refer to C40 concrete specification for elastic modulus, 3.25 × 10 4 MPa; E fX is the relative dynamic elastic modulus; and Q is the replacement rate of coal gangue.

Therefore, Equation (3.1.11) can be deduced as follows:

(3.1.13) ⇒ E n = E 0 ( 1 − D f ) e 2.82 D f .

Substitute it into (3.1.9) to obtain the constitutive model of freeze–thaw damage of coal gangue concrete:

(3.1.14) σ = ( 1 − D l ) E 0 ( 1 − D f ) e 2.82 D f ε .

In addition, Atashamulhaq et al. [49] carried out freeze–thaw tests on recycled brick aggregate concrete (RBAC). The damage variable D in RBAC after freeze–thaw cycles is defined as

(3.1.15) D = 1 − V n V 0 ,

where V _o is the fundamental transverse frequency before the start of FT cycling/Hz, and V _n is the fundamental transverse frequency after n FT cycles/Hz.

The aforementioned uniaxial compressive constitutive model of different kinds of concrete after freeze–thaw is constructed using the integral equation, which has the characteristics of simple form and easy calculation. The specific comparison difference statistics are presented in Table 1. For the establishment of different types of concrete freeze–thaw damage constitutive models, researchers have used different methods to define the required damage variables D. It can be speculated that the damage variables should be clear to achieve the purpose of truly and accurately reflecting the damage of individual types of concrete.

Table 1

Comparison table of integral constitutive models

References	Type of concrete	Influencing factors of damage variable D
[43]	C30	The number of freeze–thaw cycles
[45]	Steel fiber gangue concrete	The contents of steel fiber in concrete, the number of freeze–thaw cycles
[46]	C40 SCC	Load damage D c and freeze–thaw damage D n
[47,48]	Coal gangue concrete	Load damage D l ( ε ) based on acoustic emission properties and freeze–thaw damage D f
[49]	Recycled brick aggregate concrete	The fundamental transverse frequency before and after freeze–thaw cycles

3.2 Segmental constitutive model

The second type of constitutive model expression for concrete of freeze–thaw damage is segmental.

For the uniaxial compressive constitutive model of coal gangue concrete after experiencing freeze–thaw cycles, Ji-sheng et al. [50] used the segmental constitutive model of Zhenhai [51]:

(3.2.1) Y = α X + ( 3 − 2 α ) X 2 + ( α − 2 ) X 3 , X ≤ 1 Y = X β ( X − 1 ) 2 + X , X ≥ 1 ,

where X = ε / ε p , and Y = σ / σ p .

Through the analysis of test data, the peak strain ε p , peak stress σ p , control parameters and the number of freeze–thaw cycles N, gangue replacement rate R relationship formulas are obtained as follows:

σ p = 31.75 − 0.0718 N + 0.00223 R N + 0.00006.38 × R 2 N ε p = 1.743 e 0.00253 N + 0.000516 R N ,

α = 1.92 − 0.0025 N e 0.05517 R β = 1.3 + 0.00348 N + 0.00109 R N ,

Dafu et al. [52] similarly established the uniaxial compressive stress–strain full curve equation based on the (3.2.1) constitutive model after the concrete experienced freeze–thaw, and the control parameters were determined using the cubic compressive strength and dynamic elastic modulus.

In addition, Kaihua et al. [53] studied the freeze–thaw damage constitutive relationship of SCC by similarly constructing a segmental equation based on the Guo model. The relationship between the control parameters α , and β and the rate of loss of relative dynamic elastic modulus after N freeze–thaw cycles Δ E N are given by the following equations:

α = − 0.013 Δ E N + 1.036 β = 0.055 Δ E N + 3.037 .

To further improve the fitting accuracy, Gong et al. [54] also used a segmented constitutive equation for concrete to investigate the conditions with uniaxial compression and freeze–thaw, dividing the stress–strain curve into ascending and descending segments:

Ascending segments:

(3.2.2) y = sin π 2 a x / sin π 2 a , 0 ≤ x ≤ 1 .

Descending segments:

(3.2.3) y = b x 1 + ( b − 2 ) x + x 2 , x > 1 ,

where x = ε / ε p and y = σ / σ p ; a and b are the control parameters of ascending and descending sections of the stress–strain curve.

As the number of freeze–thaw cycles increases, the change in mechanical properties of concrete is related to the damage variable D:

a a 0 = 1 − 0.1388 D , R 2 = 0.9999 b b 0 = 1 − 0.116 D , R 2 = 0.9850 ,

where a 0 and b 0 are the curve parameters of the ascending and descending curve section of stress–strain after concrete curing for 28 days.

The relationships between the freeze–thaw damage variable D, the peak stress σ FT , and the peak strain ε FT of the concrete prismatic specimen are given by

σ FT σ p = 1 − 1.036 D , R 2 = 0.9950 ε FT ε p = 1 + 0.197 D , R 2 = 0.9919

The average correlation coefficient R 2 between the constitutive model of concrete under axial compression and the experimental data under freeze–thaw cycles is 0.989. And the fitting result is shown in Figure 5.

It can be seen that the main difference in constructing a segmental concrete freeze–thaw constitutive model is the use of different performance indicators such as freeze–thaw cycles [50], compressive strength [52], dynamic elastic modulus [53], or freeze–thaw damage variables [54] to define the control parameters for the rising and falling stages. The specific comparison difference statistics of segmental constitutive models are presented in Table 2. The fitting accuracy dates and images (Figures 3–5) show that the segmental constitutive model fits the actual measured stress–strain curve with higher accuracy to reflect the constitutive relationship of concrete under the freeze–thaw condition.

Table 2

Comparison of segmental constitutive models

References	Type of concrete	R 2 /the fitting accuracy
[50]	Coal gangue concrete	—
[52]	C20∼C50	0.976
[53]	Self-compaction concrete	—
[54]	C35	0.989

Figure 3

Comparison of fitting results of constitutive model and the test results [46].

Figure 4

Fitting results of constitutive model [47].

Figure 5

Fitting results of stress–strain curves [54].

4 Constitutive model of freeze–thaw damage under dynamic load

Concrete materials have a wide range of applications. During the construction and use of structures such as airport runways, nuclear power plants, military projects, gravity offshore platforms, and water conservancy projects, concrete materials are often subject to such effects as excavation and blasting loads, vibration loads caused by over-flow and equipment, and impact loads caused by vehicles, waves, earthquakes, etc. [55]. The dynamic damage load of concrete is often divided into the following two categories: periodic cyclic load, such as earthquake, wind, wave, and automobile load; the second type is large loading rate load, such as blasting and impact load.

For concrete projects in cold regions at the same time, it is of great theoretical and practical significance to research concrete of the dynamic mechanical properties and constitutive relationship under freeze–thaw cycles [56], which can provide experimental data and theoretical basis for capacity analysis and life prediction of concrete structures in the cold and complex condition.

4.1 Dynamic axial compression load

The research scholar of Three Gorges University [57,58,59,60] conducted uniaxial compression tests on concrete subjected to freeze–thaw cycles with strain rates ranging from 10⁻⁴/s to 10⁻³/s. Dynamic axial compression test equipment is selected to provide the maximum vertical static and dynamic load of 10 and 5 MN, respectively. And the large hydraulic servo triaxial instrument (Figure 6) has a maximum strain rate, which is 10⁻²/s.

Figure 6

Hydraulic servo triaxial instrument [57].

The damage constitutive model is constructed by segments. The stress–strain curve in the ascending section before the peak is constructed by using the Weibull statistical distribution with a better fitting effect introduced in Section 3.1. And the curve in the descending section after the peak is optimized by lognormal statistical distribution, which improves the overall fitting effect of the model. The damage constitutive model established after correction is as follows:

(4.1.1) σ = E 0 ε exp − 1 m ε ε p m , 0 ≤ ε ≤ ε pk σ p exp − 1 2 ln ⁡ ( ε / ε p ) t 2 , ε ≥ ε pk

where m , and t are the control parameters for the shape of the curve before (ascending section) and after (descending section) the peak stress, respectively. In addition, Chunping et al. [58] pointed out that the shape control parameters were related to the degree of freeze–thaw deterioration and the dynamic axial compression strain rate when studying the concrete constitution under the low-temperature frozen state and dynamic axial compression.

The dynamic uniaxial test curves of concrete after 50 freeze–thaw cycles and the model-fitting curves are shown in Figure 7. The images show that the Weibull-lognormal segmental constitutive model can describe the concrete stress–strain curve changes more accurately under the freeze–thaw and different strain rates.

Figure 7

Comparison diagram of stress–strain test curve and fitting curve [60]: (a) the strain rate is 10⁻³/s and (b) the strain rate is 10⁻⁵/s.

4.2 Cyclic repeated load

Scholars have studied relevant research on the damage mechanism and mechanical property degradation of concrete under repeated loads. Constitutive models have been constructed to describe the load–unload uniaxial tensile and compressive stress–strain curves, considering the complex hysteresis phenomenon and the nonlinear behaviors such as stress softening, stiffness degradation, and irreversible deformation of concrete under cyclic loads [61,62,63,64]. In further analyzing the mechanical properties of concrete coupled with freeze–thaw and establishing a dynamic damage constitutive model, research scholars have also conducted relevant studies to provide a theoretical basis for the analysis of the seismic performance of concrete structures in the alpine region.

The envelope under repeated loading can approximately describe the full uniaxially compressed concrete curve. In order to accurately reflect the concave shape of the rising section of the concrete stress–strain curve under cyclic load after freeze–thaw, Shanhua et al. [65] chose the expression (4.2.1) proposed in the reference [66] to describe the compaction effect of the rising section of the stress–strain curve. And the model-fitting curves are shown in Figure 8:

(4.2.1) y = x ∂ ( x − 1 ) 2 + 1 ,

where x = ε / ε p and y = σ / σ p ,

Figure 8

Comparison diagram of test curve and fitting curve [65].

where ∂ i s the control parameter for ascending section or descending section (the parameters of the C40 concrete ascending section and descending section are 4.3∼4.5 and 4.4∼4.6, and the parameters of C50 concrete are 4.8 and 4.5).

The early compressive strength of seawater and sea-sand concrete (SSC) is higher than that of ordinary concrete, and seawater reduces the late hydration rate and hydration degree of SSC [67,68]. At present, the research on SSC mainly focuses on hydration mechanisms and microstructure [69,70]. In order to better meet the needs of practical engineering, such as marine engineering subjected to seismic or wave load and under unfavorable combinations of cold environments, it is necessary to study the degradation of mechanical properties and constitutive model of SSC after freeze–thaw.

Qiu et al. [71] adopted the stress–strain and damage variables under repeated uniaxial compression tests of concrete after seawater freeze–thaw cycles and the plastic damage theory to establish the constitutive model of SSC. First, it was based on the Mander model [72]:

(4.2.2) σ σ pn = ε ε pn r r − 1 + ε ε pn r .

Then, the shape parameter r is modified as

r = n E c E c − E sec

n = 2 × 10 − 4 N 2 + 0.0112 N + 1.7984 ,

where E sec = 0.0081 ε − 1.319 .

Zhou et al. [73] proposed a constitutive model for predicting the cyclic axial compressive stress–strain of SSC after seawater freeze–thaw, the model consists of an unloading curve model, a linear reloading model and an envelope curve, as shown in Figure 9. The unloading curve model can be defined as a quadratic parabola that passes through the peak point ( ε cp , f cp ) , unloading point ( ε un , σ un ) , common point ( ε new , σ new ) , and plastic strain point ( ε pl , 0 ) . The linear reloading curve passes through the plastic strain point ( ε pl , 0 ) and the common point ( ε new , σ new ) and finally intersects the envelope curve.

Figure 9

Stress–strain curve of concrete under cyclic loading [73].

The curvature of the unloading curve model depends on the unloading tangential stiffness E un , while the slope of the linear reloading curve model depends on the reloading modulus E re . The envelope model is divided into rising and falling segments, and the specific expression is as follows:

(4.2.3) σ c f cp = γ c ε c ε cp + ( 3 − 2 γ c ) ε c ε cp 2 + ( γ c − 2 ) ε c ε cp 3 , 0 ≤ ε c ≤ ε cp σ c f cp = ε c ε cp K c ε c ε cp − 1 2 + ε c ε cp , ε c ≥ ε cp ,

where γ c , and K c are the static parameters related to the freeze–thaw damage index.

4.3 Impact compression load

Offshore and harbor structures are difficult to avoid the lapping of waves and the impact of ships, and these external effects are classified as dynamic loads, which have dynamic effects on structural materials accordingly [74,75]. Jinjun et al. [76,77] used a split Hopkinson pressure bar (SHPB) device (as shown in Figure 10) to study the dynamic compression stress–strain constitutive relation of SCC, and the constitutive numerical model was constructed regarding the relationship Equation (3.2.1) of materials under static compression mentioned in Section 3.2. Another scholar [78] based on the Weibull distribution model established the concrete freeze–thaw damage constitutive model under dynamic load.

Figure 10

SHPB device [79].

Li et al. [80] proposed that the dynamic mechanical properties of concrete in a freeze–thaw environment are the combined result of the freeze–thaw deterioration effect and the strain rate-strengthening effect. Dynamic impact loading concrete freeze–thaw constitutive process used freeze–thaw deterioration impact factor ξ to consider the impact of freeze–thaw deterioration, based on damage mechanics theory, the introduction of the damage variable D to describe the stress fracture damage, the stress–strain equation is expressed as

(4.3.1) σ = ( 1 − ξ ) ( 1 − D ) E ε ,

(4.3.2) ξ = 1 − σ i σ 0 ,

(4.3.3) D = 1 − exp − ε a b ,

where ξ is the freeze–thaw deterioration impact factor, reflecting the effect of freeze–thaw on dynamic peak stress ; σ i is the peak stress of concrete under impact loading after different freeze–thaw cycles; σ 0 is the peak stress of concrete under impact loading before freeze–thaw cycles; and a and b are the shape parameters of the Weibull distribution.

Huang et al. [79] introduced multi-effect elastic units with composite freeze–thaw damage effects, hydration effects, statistical damage properties, and strain rate effects in the construction of a freeze–thaw constitutive model of concrete under dynamic impact load. The freeze–thaw damage coefficient ( ξ 1 ) is used to quantify the freeze–thaw damage, and the hydration coefficient ( ξ 2 ) quantifies the effect of hydration on the dynamic compressive strength during the freeze–thaw process, and the constitutive expression is

(4.3.4) σ = ( 1 + ξ 1 + ξ 2 ) ( 1 − D ) ⋅ DIF ⋅ E 0 ε .

The damage variable D expression is consistent with Equation (4.3.3), and when the strain rate is in the range of 39.87 to 163.35 S − 1 , the dynamic influence factor DIF is

(4.3.5) DIF = A n log 10 ( ε ̇ m ) + B n .

For concrete with ≥66 freeze–thaw cycles, the constitutive Equation (4.3.6) is divided into the initial plastic stage ( ε < ε p ) and elastic stage ( ε ≥ ε p ):

(4.3.6) σ = 1 + ξ 1 + ξ 2 − exp − ε a b A n log 10 ε ̇ m + B n ∙ E 0 ε , ε , ε p σ = σ p + 1 + ξ 1 + ξ 2 − exp − ε − ε p a b A n log 10 ε ̇ m + B n ∙ E 0 ε ε − ε p , ε ≥ ε p

When the loading speed is 6 m/s，

ξ 1 = − 2.2698 × 10 − 5 n 2 − 5.2601 × 10 − 5 n ,

ξ 2 = − 0.0544 e − 0.0476 n + 0.0476 .

In summary, the specific comparison difference statistics of concrete constitutive models under dynamic load is presented in Table 3. The fitting accuracy displays that the changing trend of the established constitutive models is similar to the experimental results. The constitutive model of freeze–thaw damage can effectively describe the dynamic compressive mechanical properties of various types of concrete under dynamic load. Furthermore, the scope of freeze–thaw testing can be expanded by reducing the lower temperature threshold to adapt the engineering requirements of high-altitude and cold environments. It is also critical to differentiate the damage variables within the framework of freeze–thaw constitutive models based on the distinct dynamic load conditions they represent. This differentiation will enhance the applicability and predictive power of the models for concrete performance under varying dynamic stresses.

Table 3

Comparison table of concrete constitutive models under dynamic load

Category of dynamic load	References	Type of concrete	R 2 /the fitting accuracy
Axial compression load	[57,58,59,60]	C30	0.957∼0.993
Cyclic repeated load	[65]	C40, C50	Figure 8
	[71]	Concrete with mechanism sand	0.885∼0.961
	[73]	C40∼C60 SCC	0.77∼0.98
Impact compression load	[76,77]	C20∼C40 SCC	0.956 (ascending section) 0.777 (descending section)
	[79]	C40	0.9405
	[80]	C35	0.90

5 Constitutive model of freeze–thaw damage coupling environmental load

With the development of society and the needs of engineering, the structural forms of various types of projects in the cold environment have been extended from the common housing building structures to various types of complex structures, such as large-span concrete bridge structures in cold areas, high arch dam projects, concrete protection layers in nuclear power plants, etc., making many difficult projects and complex mechanical phenomena appear continuously [81]. In actual engineering, the environment in which the structure is located and subjected to load effects has also become more complex, leading to the need to develop the content of the research on concrete damage to a more complex level.

In addition to the study of dynamic mechanical properties constitutive model, the research field of concrete material constitutive model based on freeze–thaw damage has gradually developed from single factor to multi-factor coupling conditions, which are more in line with the actual engineering environment [82].

5.1 Freeze–thaw coupling corrosion

The damage to concrete structures caused by the dual failure factors of temperature change and chemical erosion has become a widespread concern among researchers. At the same time, chemical corrosion and freeze–thaw damage are also two important factors that cause the durability deterioration of concrete structures [83]. Freeze–thaw cycles will cause surface spalling and internal damage to concrete. Meanwhile, the existence of corrosion factors will intensify the degree of these two types of damage [84]. At present, most studies on the coupling effect of corrosion and freeze–thaw damage of concrete have focused on the material mass loss rate, relative dynamic elastic modulus loss, compressive strength loss, deterioration mechanism, and micro-structure [85,86,87,88], while studies on the constitutive relationship of concrete under the coupling effect of corrosion and freeze–thaw are relatively limited.

Chen et al. [89] studied the deterioration law of concrete under freeze–thaw damage and salt solution corrosion, defined the damage variable of concrete by the relative dynamic elastic modulus, and established a general damage evolution equation with freeze–thaw cycles and salt solution concentration as independent variables. Then, they established damage constitutive models of concrete under different corrosion conditions based on the axial compressive strength of specimens. Li et al. [90] based on the damage mechanics theory when studying the damage evolution law and constitutive model of concrete under the combined action of marine corrosion and freeze–thaw damage environment. The obtained secant elastic modulus is used to define the damage variable D s of concrete after freeze–thaw coupling corrosion failure:

(5.1.1) D s = 1 − E F E F 0 ,

where E F 0 is the secant elastic modulus of nondestructive concrete at the linear elastic stage; E F is the instantaneous secant elastic modulus of concrete when damaged.

The damage variable evolution formula of concrete considering freezing–thawing cycle times N and corrosion time t is obtained:

(5.1.2) D s ( ε , N , t ) = A 2 ( N , t ) + A 1 ( N , t ) − A 2 ( N , t ) 1 + exp ε − ε 0 ( N , t ) d x ( N , t ) ,

where the parameters are the functions of the number of freeze–thaw cycles N and the corrosion time t.

By substituting equation (5.1.2) into σ = E ε ( 1 − D s ) , the finally obtained constitutive model is as follows:

(5.1.3) σ = E ( N , t ) ε 1 − A 2 ( N , t ) − A 1 ( N , t ) − A 2 ( N , t ) 1 + exp ε − ε 0 ( N , t ) d x ( N , t ) ,

where the elastic modulus E is also a function of the number of freeze–thaw cycles N, and the corrosion time t, and it can be fitted from test measurements.

5.2 Freeze–thaw coupling load

The damage to concrete structures is widely exposed to the combined effects of loads and environment. The superposition of freeze–thaw cycles and loading of concrete in cold regions will accelerate the deterioration of concrete [91]. The durability deterioration of concrete is the failure process of material caused by the combined action of double or multiple composite damage factors, and each factor will interact and influence each other, which is a complex process [92]. Considering only the deterioration of the mechanical properties of concrete under a single freeze–thaw cycle and ignoring the effect of load, this will be greatly different from the actual situation. And the results of the constructed constitutive model will be certainly one-sided.

Wang et al. [93] proposed a macro-micro-coupling damage model of concrete under freeze–thaw cycles and load and defined the freeze–thaw damage variable D 1 and the load damage variable D 2 :

(5.2.1) D 1 = 1 − E n E D 2 = ∫ 0 ε P ( x ) d x = 1 − e − 1 m ε ε 0 m ,

where parameters of Weibull distribution.

The calculation of coupling damage variable is shown in Figure 11, and the elastic modulus of the four states (a) to (b) is E 12 , E 1 , E 2 and E 0 :

(5.2.2) E 12 = E 0 ( 1 − D 12 ) E 1 = E 0 ( 1 − D 1 ) E 2 = E 0 ( 1 − D 2 ) .

Figure 11

Schematic diagram of the calculation of coupling damage variable [93]: (a) coupling damage state, (b) microscopic damage state, (c) macroscopic damage state, and (d) non-destructive state.

When the stress is constant, the relation is

(5.2.3) σ E 12 = σ E 1 + σ E 2 − σ E 0 .

The coupling damage variable D 12 is derived as follows:

(5.2.4) D 12 = 1 − ( 1 − D 1 ) ( 1 − D 2 ) 1 − D 1 D 2 .

Based on the constitutive Equation (2.5) derived from the damage mechanics theory, the ultimate stress–strain relationship of concrete under freeze–thaw coupling load is

(5.2.5) σ = E 0 ( 1 − D 1 ) ( 1 − D 2 ) 1 − D 1 D 2 ε .

The model validation results showed the curve in high agreement with the measured results in the ascending section of the curve, while the correlation was poor in the descending section (Figure 12).

Figure 12

Measured stress–strain curves and theoretical curves of concrete under freeze–thaw cycles and load [93].

Through the analysis and summary of existing research on constitutive models of freeze–thaw damage coupling environmental load, it can indicate that to accurately represent concrete damage under environmental coupling, the introduction of a coupling damage variable is feasible for constructing damage constitutive models. Developing damage evolution models that consider various elements and loading scenarios is essential. Enhancing the precision of constitutive model validation under freeze–thaw coupling environment is also imperative. Furthermore, current research on coupled load applications is limited primarily to static axial compressive loads, and there is a need for a more extensive and nuanced examination that closely mimics the dynamic freeze–thaw coupling encountered in practical situations.

6 Conclusions

With the depth of research content and discipline integration, many universities and institutions have carried out and accumulated research results related to the constitutive model of concrete under freeze–thaw environment. However, due to the engineering reality, besides the freeze–thaw damage, concrete structures are often under complex stress states, which makes their constitutive relationship more comprehensive. Therefore, there are still many issues that need to be further studied:

Further extend the range of freeze–thaw testing by setting a lower freeze–thaw test temperature limit, to meet the actual engineering needs of high altitude and cold areas. Additionally, when constructing the freeze–thaw constitutive model, attention should be paid to distinguishing the differences in damage variables under different loading states.
For the study of constitutive models of concrete materials in multifaceted environment loads coupling freeze–thaw states, attention needs to focus on seeking universal, reliable, and simple methods for defining damage variables. In order to accurately characterize concrete damage under environmental coupling, coupled damage variables can be introduced to construct a damage constitutive model, and the coupling effects between different damage factors can be considered.
The hydraulic concrete structure in the alpine region is subjected to not only low temperatures but also dynamic loads such equipment vibrations and earthquakes during operation. Most existing models are based on freeze–thaw coupled load tests, ignoring the impact of dynamic loads on the development of concrete damage.

In order to facilitate a comprehensive assessment of the deterioration process in concrete structures subjected to complex environmental conditions in alpine cold regions, investigating the evolution of damage in concrete materials under freeze–thaw coupled complex stress states and establishing material constitutive models that authentically reflect the actual environment are pivotal aspects of research into the durability of hydraulic concrete against freeze damage.

Acknowledgements

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

Funding information: This work was supported by the National Key Research and Development Program of China (No. 2022YFC3202500) and the National Natural Science Foundation of China (No. 11932006).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. LZ: conceptualization, methodology, data curation, writing – original draft. GX: project administration, resources, supervision. LC: investigation, funding acquisition, supervision. ZZ, DY, and XH: writing – review and editing, data curation.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article [and its supplementary information files].

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Received: 2023-10-30

Revised: 2024-05-27

Accepted: 2024-05-28

Published Online: 2024-07-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/secm-2024-0020

Keywords for this article

concrete; freeze–thaw cycle; damage mechanics; constitutive model; coupling actions

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BY 4.0