Influence of freeze–thaw damage gradient on stress–strain relationship of stressed concrete

Abstract

Freeze–thaw damage gradients lead to non-uniform degradation of concrete mechanical properties at different depths. A study was conducted on the stress–strain relationship of stressed concrete with focus on the freeze–thaw damage gradient. The effects of relative freeze–thaw depths, number of freeze–thaw cycles (FTCs) and stress ratios on the stress–strain curves of concrete were analyzed. The test results demonstrated that freeze–thaw damage was more severe in the surface layers of concrete than in the deeper layers. The relative peak stress and strain of concrete degraded bilinearly with increasing depth. The stress–strain relationship of stressed concrete under FTC was established, and it was found to agree with the experimental results.

Graphical abstract

Keywords

freeze–thaw damage / stress ratios / stress–strain relationship / damage variable

1 Introduction

Freeze–thaw cycles (FTCs) have emerged as a primary challenge to the durability of concrete bridge structures in cold regions [1–3]. The frost resistance of concrete under stress-free conditions have been researched by numerous researchers [4]. However, concrete structures work under load in practical engineering and the behavior of stressed concrete under FTC is notably distinct from that of unstressed concrete [5,6]. Cracks generated by loads facilitate water diffusion and promote increased saturation in concrete, and lead to increased concrete damage [7,8]. Concrete damage under a combination of load and FTC is worse than under either factor alone [9–13].

Many studies on freeze–thaw damaged concrete have been conducted. FTC caused concrete surface spalling and degradation of mechanical properties [14–17]. As FTC intensity increased, the internal structures of concrete became looser and relative dynamic elastic modulus decreased [18–20]. The stress–strain curves indicated that peak stress reduced while peak strain went up. Brittle failure characteristics of concrete were more apparent [21,22]. The addition of fibers in concrete can enhance its performance and resistance to frost [23,24].

Constitutive models of concrete under FTC have been investigated. A relationship describing uniaxial compression constitutive of concrete under FTC was developed [25]. Also, a stochastic model of concrete was built to consider the influence of number of FTC [26,27]. Models of concrete tensile and compressive strength degradation under FTC were developed [28]. A plastic damage constitutive model of concrete under FTC was developed utilizing the plastic damage theory [29].

Stressed concrete in combination with frost resistance has been investigated in several studies. It has been shown that stress interacted with FTC, making the development of concrete damage more severe than is the case under the influence of a single factor [30–32]. The higher the stress ratio, the greater the loss of the elastic modulus in concrete [33], and shorter fatigue life in corroded beams was observed [34–36].

The test results showed that thickness of damaged layer increased linearly with number of FTC. The concrete under FTC included the damaged layer, the damaged transition and the undamaged layers [37]. Freeze–thaw damage usually starts at the surfaces, causing less damage to the inner core [38]. The findings indicate that the damage caused by FTC on concrete is a complex and nonlinear cumulative process, which is influenced by various factors [39]. Freeze–thaw damage occurs unevenly, progressing from the surface to the inside. Nonetheless, the current stress–strain relationship for concrete under FTC is based on an equivalent damage relationship that has not taken into account the uneven damage. Furthermore, the impact of the stress ratio on the freeze–thaw damage is still not clear. As a result, it has become crucial to investigate the stress–strain relationship of stressed concrete after FTC.

Due to the inhomogeneity of freeze–thaw damage on concrete members, an experimental and theoretical investigation of stressed concrete under FTC was conducted. The impact of relative freeze–thaw depths, FTC and stress ratios on concrete stress–strain curves were studied. Based on the equivalent strain assumption and statistical damage theory, a stress–strain relationship of stressed concrete under FTC was proposed.

2 Experimental programs

2.1 Specimens design and materials

To obtain specimens with different freeze–thaw damage, concrete cylinder members under different stress ratios were designed. All concrete cylinder members were the same size of Φ200 mm × 1000 mm, as illustrated in Fig.1.

The number of FTC, N, investigated in this study, were 0, 100, 200, and 250. The concrete stress ratios μ were 0, 0.2, 0.3, and 0.4, which indicated that the tensioning prestress was 0%, 20%, 30%, and 40% of the average axial compressive strength of concrete, respectively.

Two 18 mm nominal diameter screw-thread steel bars were used as prestressed tension rebars. The yield and tensile strength were 918 and 1087 MPa, respectively.

The concrete design strength grade was C30. The average compressive strength of concrete cubes after 28 d was 30.1 MPa, the axial compressive strength of the concrete was 20.1 MPa. The weight proportion of cement, water, aggregate, and sand was 1:0.54:4.20:3.36. Sand from natural rivers was used for the fine aggregate, and crushed stones of 5–20 mm for the coarse aggregate.

2.2 Tensioning tests

To produce fully saturated concrete cylinders, the samples were first soaked in water for four days. Tensioning tests were conducted on the reaction frame designed specifically for this study. Two tensioning steel bars were used in each cylinder member, and the tensioning device is illustrated in Fig.2.

The first steel bar was tensioned to 20% of its controlled stress and held for 2 min. Then, the other steel bar was tensioned to 50% of its controlled stress and held for 2 min. A second tensioning was then performed on the first steel bar, bringing it to 80% of its controlled stress and held for 2 min. Finally, the first steel bar was over-tensioned to 110% of its controlled stress and held for 2 min before tightening the anchorage and removing the loading device, completing the tensioning tests. The concrete cylinder samples design parameters are illustrated in Tab.1.

2.3 Freeze–thaw cycles tests

FTC tests were conducted according to the slow freezing method, with temperature ranges from −19 to 25 °C, and the temperature rates of rising and falling of 0.7–1.0 °C/min [40]. FTC tests of stressed concrete cylinder members were carried out as shown in Fig.3.

2.4 Test methods of frost heaving strains

The strains induced by freeze–thaw in concrete included thermal strain and frost heaving strain [41]. By using dynamic simulation, thermal strain was identified separately from the concrete frost heaving strain [42]. The simulation method thus eliminated thermal strain caused by thermal expansion and cold contraction. The compensation and tested strain gauges were fixed on the specimen with the same material. The specimen intended to be used with the compensation strain gauge was waterproofed before the freeze–thaw test. Subtracting the compensated strain from the tested strain yielded the frost heaving strain. The strain generated during the FTC comprised tensile strain, thermal strain and the frost heaving strain. The definition of frost heaving strain {\epsilon }_{\text{f}} is as follows:

(1){\epsilon }_{\mathrm{f}}={\epsilon }_{\mathrm{i}}-{\epsilon }_0-{\epsilon }_{\mathrm{s}},

where {\epsilon }_{\mathrm{f}} is the frost heaving strain, {\epsilon }_{\mathrm{i}} is the total strain, {\epsilon }_0 is the thermal strain, and {\epsilon }_{\mathrm{s}} is the strain generated during the tensioning process.

Variations of the total and frost heaving strains with temperature are presented in Fig.4. A decrease in temperature resulted in reducing total strain, whereas an increase in frost heaving strain was observed.

2.5 Fabrication of torus specimens

Concrete mechanical properties at different depths were inhomogeneously degraded with freeze–thaw damage gradients. Torus-shaped specimens were obtained by drilling and coring from the concrete cylinder members that had been subjected to FTC.

To eliminate end effects, 200 mm was cut off from both ends of each concrete cylinder member. The remaining parts were cut into three Φ200 mm × 200 mm cylinder specimens.

The relative freeze–thaw depth can be defined as follows:

(2)\lambda =x_n/r,

where x_n represents the distance from the edge to the center of each layer of the cylindrical concrete, r is the radius of the cylindrical concrete.

From the surface to the inside, four torus specimens had relative freeze–thaw depths of 0.1, 0.3, 0.5, and 0.8. The fabrication process of freeze–thaw damaged concrete torus specimens is shown in Fig.5.

The coring device was equipped with an engine and diamond-tipped drill bits were utilized, maintaining a consistent coring speed. The specimens were secured with clamps at both ends during the coring process, as depicted in Fig.6. The torus specimen was originally designed to have a thickness of 20 mm, with the final thickness of the specimens being 16–18 mm due to coring loss. Three specimens were measured under each condition, and the total number of torus specimens was 180.

2.6 Uniaxial compressional tests of concrete

Concrete uniaxial compressional tests were conducted on the WAW-1000^® electro-hydraulic servo universal testing machine. Loading was done in displacement mode at a speed of 0.1 mm/min [43]. Each group of three concrete specimens was tested, and the curve with the peak stress value closest to the mean of the three was chosen. The monotonic compression test of the freeze–thaw damaged concrete torus specimens is shown in Fig.7.

3 Test results and discussion

3.1 The crack development of concrete under freeze–thaw cycles

Fig.8 indicates the development of cracks in the concrete cylinder members. Microcracks were observed to initiate and propagate gradually with FTC. With FTC increasing from 150 to 200 cycles, the length of cracks increased from 647 to 910 mm, and the width increased from 0.1 to 0.5 mm.

3.2 Analysis of concrete frost heaving strains

Variations of {\epsilon }_{\mathrm{f}} and temperature of stressed concrete with time are illustrated in Fig.9. As FTC and μ increased, the frost heaving strain increased as well. With the increase of number of cycles of FTC from 1 to 100, the peak frost heaving strain of concrete under three stress ratios increases by 43%, 52%, and 34%.

Four stages of frost heaving strain, {\epsilon }_{\mathrm{f}}, were experienced. In the first cold shrinkage stage, a slight decrease in the frost heaving strain occurred with a temperature reduction from 25 to 0 °C. In the second frost heave stage, as the internal pore water froze at temperatures from 0 to −10 °C, {\epsilon }_{\mathrm{f}} increased dramatically. In the third stable stage, from temperatures −10 to −19 °C, a lower growth rate of frost heaving strain occurred. In the last shrinkage stage, {\epsilon }_{\mathrm{f}} decreased rapidly, in the temperature range from −19 to 25 °C.

The frost heaving strain hysteresis loops of concrete are indicated in Fig.10(a). The peak frost heaving strain of concrete increased with the number of FTC cycles. The freezing and melting of water in concrete can cause volume changes during FTC, resulting in internal stress and strain. As the number of cycles of FTC increased, the cumulative effect of internal frozen stress and strain could lead to the breakdown and alteration of the concrete microstructure, causing an increase in frost heaving strain. The peak frost heaving strain of concrete increased by 29% as FTC increased from 25 to 100.

The frost heaving strain hysteresis loops of concrete under different stress ratios are illustrated in Fig.10(b). An increase in stress ratios led to greater frost heaving strain. FTC reduced the concrete’s effective cross-section, and under the interaction of stress, led to further expansion of cracks. As the number of FTC increased, the damage to the concrete continued to accumulate. By increasing stress ratios from 0.2 to 0.4, the maximum frost heaving strain increased by 9%.

The residual strain can be calculated as the difference between the strain during the latter and previous cycles at the lowest temperature [44]. Fig.11 illustrates the variations of residual strain for stressed concrete. Residual strains of concrete gradually increased with FTC number. A higher stress level led to a faster growth rate of residual strains. When FTC increased from 25 to 100, the residual strain reduced by 33%, 34%, and 31% for μ = 0.2, 0.3, and 0.4, respectively.

3.3 Analysis of axial compression test results of freeze–thaw damaged concrete

3.3.1 Compression failure modes

Compression failure modes of concrete torus specimens under FTC are indicated in Fig.12. The concrete strain gauges were pasted vertically along the end of the specimen within 100 mm of the end. Multiple longitudinal and short cracks appeared simultaneously on both ends of the specimens. Cracks developed obliquely, which indicated that cracks were not all parallel to loading directions. The higher FTC number was associated with fever cracks and brittleness of specimens.

3.3.2 Stress–strain curves

Fig.13(a)–Fig.13(h) show that as stress ratios and FTC number increases, concrete stress–strain curves flatten and shift to the right when relative freeze–thaw depth is below 0.3. Furthermore, the peak stress decreased while peak and ultimate strains increased. From 0 to 250 FTCs, the peak strain of concrete increased by 92% at λ = 0.1 and μ = 0.3.

Fig.13(i)–Fig.13(l) illustrate that the stress–strain curves after 100 cycles were similar to those of concrete without freeze–thaw damage at λ = 0.5. When the FTC increased to 200, there were no obvious variations of concrete stress–strain curves for depths of 0.3 and 0.5.

Fig.13(m)–Fig.13(p) indicate that when the relative freeze–thaw depth was 0.8, both specimens with and without FTC showed similar stress–strain curves. But the ultimate strain increased significantly compared with other freeze–thaw depths. Freeze–thaw damage in concrete was higher in the surface layers than in the deeper layers. When the relative freeze–thaw depth exceeded 0.8, it was almost unaffected by the number of FTC.

3.3.3 Elastic modulus of concrete

Variations of elastic modulus of concrete under FTC are presented in Fig.14. The elastic modulus is determined by calculating the secant modulus at 40% of the peak stress on the stress–strain curve. The elastic modulus degraded with FTC number. When relative freeze–thaw depths increased, the elastic modulus loss rate declined more rapidly. From 0 to 250 FTC numbers, the elastic modulus of concrete decreased by 89% at λ = 0.1.

3.3.4 Relative peak stresses and strains

The relative peak stress σ_r was defined as follows:

(3){\sigma }_{\mathrm{r}}=\frac{{\sigma }_{\mathrm{c}\mathrm{D}}}{{\sigma }_{\mathrm{c}0}},

where σ_cD is the peak stress of concrete considering FTC, and σ_c0 is the peak stress of concrete without freeze–thaw damage.

Fig.15 illustrates the relationship between the σ_r and relative freeze–thaw depth of freeze–thaw damaged concrete. It was found that the peak stress went up with increasing relative freeze–thaw depth. A higher number of FTC means a faster growth rate of peak stress. With 100, 200, and 250 FTC, the peak stress was reduced by 32%, 54%, and 64%, respectively, for specimens at λ = 0.1 and μ = 0.

The relative peak strain ε_r was defined as follows:

(4){\epsilon }_{\mathrm{r}}=\frac{{\epsilon }_{\mathrm{c}\mathrm{D}}}{{\epsilon }_{\mathrm{c}0}},

where ε_cD is the peak strain of concrete with FTC, and ε_c0 is the peak strain of concrete without FTC.

Fig.16 illustrates the relationship between ε_r and the relative freeze–thaw depth of the freeze–thaw damaged concrete. A decreasing peak strain of concrete was accompanied by an increase in relative freeze–thaw depth. With 100, 200, and 250 FTC, the peak strain of concrete specimens ascended by 40%, 84%, and 110% for specimens at λ = 0.1 and μ = 0.

As the distance from the surface of the specimens increased, σ_r increased while ε_r decreased, and the two parameters bilinearly degraded along the depth. When the relative freeze–thaw depth was less than the threshold, the relative peak stress and strain degraded linearly, and both remained stable after exceeding the threshold. From 100 to 200 cycles, the threshold of relative freeze–thaw depths increased from 0.5 to 0.8.

4 Stress–strain relationship of stressed concrete with freeze–thaw damage gradients

4.1 Mathematic models

Based on the Lemaitre hypothesis [45], the initial damage values of concrete can be determined by Eq. (5). The damage constitutive relationship of the stressed concrete under FTC can be expressed by Eq. (6):

(5)E_n=E_0\left(1-D_{\mathrm{m}}\right),

(6){\sigma }_n=E_0\epsilon (1-D_{\mathrm{m}}),

where σ_n is the stress of concrete after n cycles; E₀ is the elastic modulus of concrete; D_m is the initial damage value of concrete; E_n is the elastic modulus of concrete under FTC; and ε is the strain of concrete.

Assuming that concrete element strength conforms to the Weibull probability distribution when concrete is subjected to load [46], the expression for D_c could be given by Eq. (7):

(7)D_{\mathrm{c}}=\frac{N_{\mathrm{d}}}{N_{\mathrm{t}}}=1-\exp \left[-{\left(\frac{\epsilon }{a}\right)}^b\right],

where D_c is the load damage; N_d is the number of failure microbodies; N_t is the total number of elements; and a and b are the Weibull distribution parameters, respectively.

The stress–strain relationship of the concrete under FTC and static load is:

(8)\sigma =E_n\epsilon (1-D_{\mathrm{c}}).

Substituting Eq. (5) into Eq. (8), the following can be obtained:

(9)\sigma =E_0\epsilon (1-D),

where D is the total damage variable of stressed concrete experiencing FTC.

Combining Eqs. (5) and (7), the definition of D can be obtained by Eq. (10):

(10)D=D_{\mathrm{c}}+D_{\mathrm{m}}-D_{\mathrm{c}}{D}_{\mathrm{m}}=1-\frac{E_n}{E_0}\exp \left[-{\left(\frac{\epsilon }{a}\right)}^b\right].

Substituting Eq. (10) to Eq. (9), the monotonic compressive stress–strain relationship of stressed concrete considering FTC can be expressed by Eq. (11):

(11)\sigma =E_n\epsilon \exp \left[-{\left(\frac{\epsilon }{a}\right)}^b\right].

4.2 Verification of stress–strain relationship

The impacts of a and b on the stress–strain curves are illustrated in Fig.17. The peak stress and strain increased continuously with parameter a. The ascending section slope of stress–strain curves increased with parameter b, and the peak stress increased constantly.

The equation for calculation of a and b can be obtained by fitting the stress–strain curves of stressed concrete subjected to FTC. Concrete stress–strain curves can be modified by parameters a and b. Relationship curves of undetermined parameters a, b, and λ are illustrated in Fig.18.

Fig.19 illustrates comparisons of experimental and theoretical concrete stress–strain curves. It was found that the calculation and test results were in acceptable agreement.

4.3 Evolutions of total damage variables

Evolution curves of the total damage variable under FTC are illustrated in Fig.20. The total damage variable exhibited an exponential pattern with strains.

As the number of FTC cycles increased, the initial damage values of concrete increased at the same relative depths. From 100 to 250 cycles, the initial damage value increased by 44% at λ = 0.1. The initial damage values of concrete decreased with relative freeze–thaw depths in each FTC. When relative freeze–thaw depths increased from 0.1 to 0.5, the initial damage values of concrete decreased by 21% after 250 cycles.

5 Conclusions and perspective

The stress–strain relationship of the stressed concrete under FTC was investigated by experimental and theoretical research, and the following conclusions were obtained.

1) The peak frost heaving strains of concrete increased continuously with FTC and stress ratios. When FTC numbers increased from 1 to 100, the peak frost heaving strains increased by 30%, 37%, and 18% as the stress ratios were 0.2, 0.3, and 0.4, respectively.

2) The relative peak stress and strain of concrete bilinearly degraded along the depth. When the relative depth was less than the threshold, both relative peak stress and strain degraded linearly and remained stable after exceeding the threshold. From 100 to 200 cycles, an increase in the relative freeze–thaw depth from 0.5 to 0.8 was observed.

3) A higher level of freeze–thaw damage occurred in shallow layers of concrete than in deeper layers. When the relative freeze–thaw depth was less than 0.3, the peak stress was reduced, while the peak and ultimate strains raised with increasing FTC numbers. When the relative freeze–thaw depth was 0.8, the stress–strain curves under each FTC were similar to those without freezing and thawing, indicating that the concrete was unaffected.

4) Stress–strain relationship of stressed concrete considering freeze–thaw damage gradients was established, which was found to agree with the experimental results.

5) Future studies could broaden the cylinder's cross-sectional area or reduce the thickness, to increase the gradient level of the specimen along the thickness direction. This paper solely focuses on monotonic tests, and further research could investigate the repeated compression performance of concrete.

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