Concrete can be regarded as a homogeneous material when it is not damaged by environmental actions or loading. Thus, the chloride diffusion coefficient in undamaged concrete could be regarded as the same in each direction. However, this value will be different from the surface to inner part after freeze and thaw cycles, as the frost would cause more damage on the concrete surface than that on the inner part concrete [22]. A model considering the damage in response to freezing-thawing cycle and concrete depth is developed in this paper. Three reasonable assumptions are put forward for the model:(1)

Concrete is homogeneous before frost attack.

According to the above assumptions, the damage of frost concrete can be expressed as:(1) d(n)=1−EdnEd0 where d(n) is the overall damage of concrete related to freeze-thaw cycles. E_dn is the dynamic elastic modulus after “n” cycles of frost action, and E_d0 is the initial dynamic elastic modulus. According to Eq. (1), it can be seen that the damage of concrete increases with the freezing-thawing cycle, which is consistent with the actual condition.

The damage of frost concrete can be expressed as a function of concrete depth and freeze-thaw cycle:(2) d(n, x)=1−exp⁡(−λn+vx) where λ is a time correlation coefficient, which represents the influence of freeze-thaw cycle on concrete damage; v is a position correlation coefficient, which represents the influence of concrete depth, x.

After n cycles of frost attack, the overall damage of the concrete matrix can be expressed as the integral of Eq. (3):(3) d(n)=2h∫0h/2d(n, x)dx=2h∫0h/2[1−exp⁡(−λn+vx)]dx where h is the depth of concrete, which can be regarded as 100 mm for the standard specimen (100 mm × 100 mm × 400 mm) of freeze-thaw cycle test.

The output of Eq. (3) should be equal to Eq. (1) as both of them represent the overall damage of the concrete specimen. Therefore, Eq. (4) could be obtained as:(4) 1−EdnEd0=1−2hvexp⁡(−λn+vh/2)+2hvexp⁡(−λn)

The coefficients of exposure time and position effect, λ and v, can be regressed by Eq. (4) according to the relative dynamic elastic modulus after frost attack which can be measured in experiment.

For one-dimensional diffusion in a semi-infinite medium, the Fick’s second law is commonly used to represent the pure diffusion process. The governing equation can be written as:(5) ∂C(x, t)∂t=∂∂x(D(x, n)∂C(x, t)∂x) where C(x, t) is the chloride concentration at depth x and exposure time t; and D(x, n) is the chloride ion diffusion coefficient at a depth x after n cycles of freeze-thaw, which is related to the concrete damage. D(x, n) can be expressed as Eq. (6):(6) D(x, n)=f(d)D0 where D₀ is the initial chloride diffusion coefficient of sound concrete, f(d) is the damage coefficient induced by frost attack, which can be expressed as an s-type function [23]:(7) f(d)=1+DmaxD0{1−[1+(d/dcr)m]−1} where D_max is chloride diffusion coefficient for a completely cracked material, m and d_cr are model parameters (m = 5, d_cr= 0.4). Eq. (7) proceeds from the assumption that the chloride diffusion coefficient should be affected by damage, within the bounds of the coefficients for undamaged concrete and of totally cracked concrete, where the diffusion of free water is expected to be recovered on macro crack faces. Hence, D_max cannot exceed the diffusion coefficient in free water which is about 1.0 × 10⁻⁹ m²/s.

Three different types of concrete materials, including OPC (ordinary Portland concrete), PFC (polypropylene fiber concrete), and SFC (steel fiber concrete), were tested in the experiment. The ordinary Portland cement (P.O 42.5) of Jiangsu Helin brand was adopted, and its properties met the standard [24]. The natural coarse aggregate used was 5∼25 mm continuous graded crushed stone. The fine aggregate was river sand, with a fineness modulus of 2.7. The tap water was adopted, and the water reducing agent was used to improve the workability of the concrete. The two types of fiber used in PFC and SFC are usually added into the concrete mix to prevent crack damage. All tested specimens have the same dimension of 100 mm × 100 mm × 400 mm, which were conformed to the standard dimensions of freeze-thaw cycle specimens [25]. The proportions of concrete mix are shown in Table 1. The volume fractions of polypropylene and steel fibers in PFC and SFC concrete are 0.1% and 1%, respectively. Fibers used here were shown in Fig. 1 and their properties are given in Table 2.

12 prism specimens for each mix were cast and then cured in water for 28 days. Subsequently, 9 specimens of OPC, PFC, and SFC withstood 10, 25, and 50 cycles of freeze-thaw activity according to the standard test method. Rapid freeze–thaw cycle test was performed by using the NELD-FC810 concrete rapid freeze–thaw test apparatus. In this test, the temperature at the center of the control specimens ranged from −17°C∼5°C, and each cycle was about 3.5 h. Detailed requirements and operation steps were conducted following the specifications in the GB/T 50082-2009. The dynamic modulus of elasticity of the concrete specimen was obtained by measuring the longitudinal pulse velocity of the specimen (ZBL-U520 non-metal pulse velocity test apparatus) after every 5 cycle’s frost according to the standard test method, which was shown in Fig. 2.

Then, considering one mold contact surface (100 mm × 400 mm) as the exposure surface, seal the other five surfaces of the specimen with epoxy resin using a brush. After that, harden the epoxy resin through air exposure for 24 h. Then immersed the specimen in NaCl solution with a concentration of 8% for 120 days accelerated test. Took out the specimens from the chloride solution after exposure and drilled three holes in each specimen using a 14 mm diameter rotary impact drill. Sufficient powdered concrete sample was taken out from each hole at a depth of every 5 mm and was mixed to provide the average chloride content in the measured location. Powdered concrete samples with particle size less than 0.63 mm were then selected with a sieve and dried to anhydrous state. After that, 1.5 g samples from each group were dissolved in 10 ml distilled water, vibrated for nearly 1 min, and then stewed for 24 h to completely dissolve the free chloride ion in the sample [26,27]. Mohr titration [28] was finally used to determine the potential of the concentration with chloride ions. According to the calibration test results from the standard chloride concentration and its potential value, the free chloride concentration of every solution containing concrete powder sample can be obtained from the tested potential (Fig. 3).

The scaling of the concrete surface after 50 freeze-thaw cycles were shown in Fig. 4, which indicated that the damages of OPC and PFC were more serious than those of SFC. The dynamic elastic modulus of each specimen was measured after every 5 cycle’s frost. The results are presented in Fig. 5. The relative dynamic elastic modulus (RDEM) decreased with the increasing of freeze-thaw cycle. However, the descent rate of concrete with steel and polypropylene fibers were smaller than that of ordinary concrete. The reason for this was that the integrity of concrete was improved by the addition of fiber, and the hydrostatic pressure or osmotic pressure could be reduced during the freeze-thaw cycles, then finally reduced the degree of freeze-thaw damage [29,30]. As a result, it can be concluded from the results that the fibers can improve the ability of concrete to resist frost damage from variations of concrete surface and modulus.

The distribution of free chloride ions in OPC, PFC, SFC specimens after different time freeze-thaw cycles were shown in Figs. 6–8. The concentration of chloride ions in the shallow layer of concrete increases with the increase of freeze-thaw cycles. The change of chloride ion concentration in the deep layer of concrete is a little different between OPC, PFC, and SFC. The concentration of chloride ion in ordinary concrete within depth greater than 27 mm decreases after 10 cycles of freeze-thaw, but increases a little after 25 cycles, and then increases obviously after 50 cycles. After 10 cycles of freeze-thaw, the concentration of chloride ion in concrete mixed with polypropylene fiber within depth greater than 25 mm decreased slightly, while that in steel fiber concrete did not change significantly within depth greater than 32 mm after 10 and 25 cycles of freeze-thaw. This shows that the damage of concrete in the shallow layer within about 30 mm depth is damaged more obviously than that in the deep depth of concrete during freeze-thaw cycles, which results in more chloride ions invading into the concrete, because of the continuous hydration and ice crystal blocking, only when the number of freeze-thaw cycles is more, the internal damage is slightly aggravated, resulting in chloride ion erosion speed up. The free chloride concentration in all of the three types concrete after 50 cycles of freeze-thaw increased more clearly which demonstrated that the damages become more significantly than that after 10 and 25 freeze-thaw cycles.

The coefficients λ and v in Eq. (2) can be regressed from Eq. (4) with the dynamic elastic modulus obtained at every 5 freeze-thaw cycle based on the least square method. The regressing results are shown in Table 3. It shows that the time correlation coefficient λ of OPC is larger than those of PFC and SFC, where the law of position correlation coefficient v is inversed. This result indicates that the damage caused by frost attack in OPC is larger than those in PFC and SFC.

The calculated damage of concrete after 50 freeze-thaw cycles is plotted in Fig. 9. The values slightly decrease with the depth. For example, the damage in OPC is 0.54 at the surface and becomes 0.52 at the depth of 50 mm. The values in PFC and SFC are both smaller than those in OPC and decrease with concrete depth.

As the RDEMs of most concrete prisms started to steadily decline from 10 cycles and reached an allowable value at about 50 cycles, the damages at these two states and a medium value of 25 cycle were representatively selected to analysis.

The damage coefficient calculated by Eq. (7), which is related to concrete depth and freeze-thaw cycles, is shown in Figs. 10–12 for OPC, PFC, and SFC, respectively. The damage coefficients of each type of concrete slightly decrease with the depth but increase significantly with freeze-thaw cycle.

For some coastal concrete structures in north area, they always sustain an action of freeze-thaw cycle in sea water, which will make the damage of concrete be more serious than that in fresh water. The damage coefficient will also be greater than that in Figs. 10–12.

During the experiment, surface chloride concentrations for each type of concrete were obtained after exposed to chloride solution for 120 days (Table 4).

The distribution of chloride ion at different depths of concrete under different exposure time can be obtained from Fick’s second law of diffusion. The apparent diffusion coefficients of sound and damaged concrete were regressed by Eq. (8) [31]:(8) C(x, t)=C0+(Cs−C0)[1−erf(x2Dat)] where C₀ is initial chloride concentration, C_s is the surface chloride concentration, D_a is the apparent chloride diffusion coefficient, t is the exposure time and erf is the error function.

As the chloride in the surface layer mainly transports by convection instead of diffusion [32,33], the first data at a depth of 0∼5 mm was not considered in the regression [34]. The apparent chloride diffusion coefficients of frost-damaged concrete specimens developed from this equation were shown in Fig. 13, which revealed that the values of chloride diffusion coefficient of these three mixes first decrease before 10 cycles’ freeze-thaw and then increase with freeze-thaw cycles. This attributed to the effect of continuous cement hydration was larger than the frost damage under small amount of cyclic freeze-thaw conditions. As the fibers can alleviate its damage under frost, concrete with steel and polypropylene fibers displays a lower value than the OPC.

The chloride contents in the three types of concrete predicted by Eq. (8) were given in Figs. 14–16 and compared with the corresponding experimental results. It can be obtained that the predicted chloride contents agree well with the experimental data within the depth of 10–30 mm which were shown in the figures as accurate region. In this region, the relative errors between predicted and experimental data were ranged from 0.8% to 14%, and most of them were lower than 10%. However, due to the convection effect, the experimental chloride contents were larger than the predicted chloride contents at the concrete surface within the depth of 0∼10 mm, and most of the relative errors between the predicted and experimental data were in the range of 4%–20%. On the other hand, the test results were inaccurate at deep depths (larger than 30 mm) because of the small values which would be influenced more significantly by the test error, and then made the relative errors be in the range of 20%–50%. It can also be found that most of them are larger than the predicted values.