1 Introduction

Early-age concrete undergoes heat generation and moisture transport due to hydration and drying process simultaneously, and the stresses and strains are initially induced to the concrete structures due to the restrained condition. When changes in concrete volume due to drying shrinkage are restrained, tensile stresses may develop, and it can propagate to cracking of concrete [1–3]. Although, the induced micro-cracks may not degrade the structural performance immediately, these can affect durability performance by permitting the intrusions of harmful agents such as chloride ions or carbon dioxide [4–6]. Cracking of early-age concrete depends on the relationship between tensile strength of concrete and restraint stress, which is determined by the characteristics such as hydration heat, shrinkage, degree of restraint, Young’s modulus, and creep [7, 8]. If elasticity increases rapidly in early-age concrete, tensile stress also increases rapidly before releasing the stress due to relaxation, which may cause cracking in early-age concrete [9]. Furthermore, the evaluation for cracking with restraint stress needs a complicated technique which is capable of calculating both material behavior like hydration and moisture transport and mechanical behavior like stress and strength development with boundary conditions at the same time [7].

Recently, a great deal of sea-sand is used for concrete manufacturing due to the insufficient aggregate, which is caused by increasing demand on concrete products [10, 11]. Much research on concrete with chloride ion is mainly performed for deterioration analysis like chloride diffusion or steel corrosion [12–14]. For the quantitative deterioration analysis in chloride-mixed concrete, it is important to evaluate the chloride effects on the behavior of drying shrinkage in early-age concrete. If a reinforced concrete structure is exposed to harsh environment, cracks caused by drying shrinkage which may be accelerated by mixed chloride ions can be the main routes for intrusion of deteriorating ions.

In this paper, an analysis technique for cracking resistance is proposed for initially chloride-mixed concrete. For this study, crack-inducing tests in axially confined condition are carried out for the concrete specimens with different chloride contents. For the chloride effect on drying shrinkage, tests for water evaporation and free drying without restraint are performed. Considering the chloride effect on experimental data on drying shrinkage, an analysis technique for the restrained drying shrinkage in concrete with chloride contents is proposed through utilizing the previously developed models [15–19], so called, multi-component hydration heat model (MCHHM), micro-pore structure formation model (MPSFM), and moisture transport model (MTM). Integrated FE analysis using the proposed models is carried out and the reduced cracking time caused by restrained drying shrinkage is evaluated in the concrete with different chloride contents. In this present paper, the experimental and numerical studies are performed for the analysis of restrained drying shrinkage in chloride-contaminated concrete. The material and mechanical behaviors of early-age concrete containing chlorides are also discussed.

2 Experimental program

2.1 Procedures

In this paper, several tests for drying shrinkage under axially restrained condition are performed for concrete with different chloride contents. Test equipments are made of steel plates based on the standard method of drying shrinkage and crack resistance of concrete referring to JIS [20], as shown in Fig. 1. Test specimens are prepared with different chloride contents of 0.00, 0.07, 0.60, and 1.20% of cement weight. They are cured in the humidity chamber under room condition of 25°C temperature and 60% relative humidity (R.H.) for 2 days and then drying test is performed in the condition of 20°C temperature and 60% R.H. For the curing period, the surface of concrete is covered with saturated cloth. After curing period, bottom plate of the frame is detached from the concrete specimen and longitudinal restraint of shrinkage is induced by the two side surfaces of the frame. The cracking time is observed through visual inspection and the restrained strain is measured through strain gauges on the surface of specimen by every 4 h. For measuring the strain of steel frame, 3 strain gauges are attached on each side surface as shown in Fig. 1a. Additionally, the free drying shrinkage and water loss (evaporated water) in concrete specimens are measured in the same exposure condition. The strains of free drying shrinkage are measured through both strain gauges and one contact gauge, and the weight changes in cubic specimens are measured with a scale of 0.01 g unit. The test equipment including restrained drying shrinkage, free drying shrinkage, and weight change in water loss are shown in Fig. 1a, b, c, respectively.

Fig. 1
figure 1

Test equipments for shrinkage and water loss weight

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The mix proportion of concrete for this study is listed in Table 1.

Table 1 Mix proportion of concrete specimens
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2.2 Test results

2.2.1 Compressive strength and elastic modulus

For the evaluation of physical characteristics of the specimens, tests for compressive strength and Young’s modulus are performed at the age of 3, 7, and 28 days based on JIS [21, 22]. Cylindrical specimens (15 × 30 cm) are used for the tests. The results of compressive strength and Young’s modulus are plotted in Fig. 2a, b with the results of regression analysis.

Fig. 2
figure 2

Results for mechanical characteristics with NaCl content

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As shown in Fig. 2, no clear development of strength is measured with respect to the chloride content for 28 days. In the test of Young’s modulus, similar trend showing little difference is evaluated. The regression results for strength and elastic modulus are obtained based on CEB/FIP model code [23], and they can be written as Eqs. 1 and 2, respectively:

$$ f_{c}^{'} (t) = 23.7 \cdot \exp [0.2670 \times \{ 1 - (28/t)^{0.5} \} ]^{0.5} ,\;\;\;\;t > 0 $$
(1)
$$ E_{c}^{'} (t) = 20795.3 \cdot \exp [0.3127 \times \{ 1 - (28/t)^{0.5} \} ]^{0.5} ,\;\;\;\;t > 0 $$
(2)

where, f c ′(t), E c ′(t), and t are compressive strength [MPa], elastic modulus [MPa], and time [day], respectively.

C3S in NaCl solution is reported to show little effect on acceleration of hydration [24]. In this paper, NaCl is mixed in concrete so that no significant changes in mechanical properties are measured with different NaCl contents. Results showing little changes in hydration heat and strength in concrete with sodium chloride (NaCl) can be found also in previous research [1].

2.2.2 Free drying shrinkage strain and water loss weight

Free drying shrinkage in the specimens is measured and the results are shown in Fig. 3. The results in Fig. 3 are obtained from the average of the concrete gauges on the surface and embedded gauge. Those from dial gauge on tip are not utilized since they have big deviations. The results for water loss in the cubic specimen in control condition (R.H. 60% and temperature 20°C) are plotted in Fig. 4.

Fig. 3
figure 3

Results for free shrinkage strain with NaCl content

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Fig. 4
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Results for water loss weight with NaCl content

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From the results in Figs. 3 and 4, it is evaluated that there is no clear relationship between mixed chloride contents and the amount of evaporated water. However, the free drying shrinkage increases significantly with increment of NaCl content. It can be assumed that changing density of pore water or chemically bound water due to chloride ions from NaCl is not the main reason for the increasing shrinkage in Fig. 3 since addition of chloride ions from NaCl in concrete does not cause more water evaporation as shown in Fig. 4. There are several parameters affecting drying shrinkage in concrete and chloride effect on drying shrinkage has been reported through experimental works. Shrinkage in concrete with sea water is reported to show higher shrinkage rate by 10–15% than normal concrete [25] and it increases up to 137% when chloride ion of 10,000 ppm is added to mix water in concrete [11]. When chloride ion is mixed in concrete, pore structure is changed to be more dense morphology [26] and this changed pore structure may cause different behavior of drying shrinkage. In concrete with low alkalinity, drying shrinkage increases up to 20% in spite of similar water loss in concrete with high alkalinity [27]. Increased surface tension due to higher concentration of chloride ion [28] can cause higher drying shrinkage. However, it is also reported that concrete containing admixture causing pore refinement usually show higher drying shrinkage since drying shrinkage in concrete is directly associated with the water held by small pores in the size range 3 to 20 nm [29].

In this paper, changed pore radius with different chloride ions is assumed to be main activation of increasing shrinkage since the adopted model of drying shrinkage [2] is based on the capillary pore and tension, which is explained in Sect. 3.2.

3 Numerical analysis of shrinkage behavior with chloride ion

3.1 Concept of effective restrained strain

In order to evaluate cracking time due to restrained shrinkage, comparison with tensile strength and restraint stress which are dependent on time should be carried out. The tensile strength is assumed based on the results of compressive strength and it will be explained in Sect. 3.3. For derivation of effective restraint stress, effective restrained strain which is related to restrained shrinkage should be evaluated and converted to effective restraint stress. Effective restrained strain of the specimen which causes effective restraint stress can be defined as the difference between total restrained drying shrinkage strain and the free drying shrinkage strain. The each strain is experimentally measured for restraint and free condition as shown in Fig. 1a and b. In Fig. 5, total restrained strains from restraint condition are plotted with elapsed time, which are obtained from the average of three gauge measurements on the concrete surface.

Fig. 5
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Total restrained strain in restrained concrete with NaCl content

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In the Fig. 5, the initial point of measurement indicates the time of detachment of the bottom plate of the frame (2 days after casting) and the breaking point (x) indicates the cracking time, which means the release of the restraint stress from the mold. Volume changes caused by hydration expansion and shrinkage actually occur before removing the bottom plate of the mold. However, it is very difficult to measure the strain in concrete before setting in the mold. For simplicity of restraint stress analysis, it is assumed that this period of 48 h before detachment of bottom plate is so short that its effect on drying shrinkage is not significant. The development of effective restraint stress in concrete is derived based on the effective restrained strain which is defined previously. It can be expressed as Eq. 3:

$$ \sigma (t) = E_{t} (t) \cdot \varepsilon_{\text{ef}} (t) = E_{t} (t) \cdot [\varepsilon_{\text{restrained}} \;(t) - \varepsilon_{\text{free}} \;(t)] $$
(3)

where \( \varepsilon_{\text{ef}} (t) \) is the effective restrained strain, \( E_{t} (t) \) is the tensile elastic modulus of concrete which is assumed as the compressive elastic modulus in Eq. 2. \( \varepsilon_{\text{restrained}} \;(t) \) and \( \varepsilon_{\text{free}} \;(t) \) are the strains of total restrained drying shrinkage and free drying shrinkage, respectively. Figure 6 shows the results for effective restraint stress development through Eq. 3.

Fig. 6
figure 6

Development of effective restraint stress with NaCl content

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Concrete can be in crack-free condition so long as the tensile strength keeps higher than induced effective restraint stress. The results show rapid shortening of cracking time with the larger chloride content. It is shortened from 19.0 to 7.5 days with the increasing chloride content up to 1.20% of cement weight. The reason for the cracking is the increased effective restraint stress due to the chloride content. These experimental results will be compared with those from proposed analysis technique.

3.2 Modeling of shrinkage considering chloride effect

The relative humidity in the pore structure decreases with drying due to moisture gradient between exterior and interior concrete. Considering local thermodynamic and interface equilibrium, vapor and liquid interfaces would be formed in the pore structure due to the pressure differences caused by capillarity [30]. Where the interface is a part of an ideal spherical surface, this relation can be described by Kelvin’s equation as in Eq. 4 [2, 30]:

$$ \ln ({\frac{{P_{v} }}{{P_{v0} }}}) = - {\frac{{2\gamma M_{w} }}{{RT\rho_{L} }}}\,{\frac{1}{{r_{s} }}} $$
(4)

where, \( P_{v} /P_{v0} \) is relative humidity of vapor phase, γ is the surface tension of liquid water [N/m], M w is molecular mass of water [kg/mol], \( R \) is universal gas constant [J/mol K], T is absolute temperature [K], ρ L is density of liquid water [kg/m3], r s is a pore radius [m] in which the interface is created. Due to the surface tension of liquid water, the pressure of gas and liquid are not equal. This pressure difference (ΔP) would be described by Laplace’s equation as Eq. 5. The idealized pore structure and interface equilibrium is shown in Fig. 7:

$$ \Updelta P = P_{G} - P_{L} = {\frac{2\gamma }{{r_{s} }}} $$
(5)

where, P G and P L are pressure of gas and liquid phase respectively [Pa].

Fig. 7
figure 7

Idealized structure of pores and interface equilibrium [15]

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The pressure of each phase can be computed by moisture transport model [15, 16]. From Eq. 5, it can be shown that the pressure of the liquid phase is lower than that of the gas phase, and tensile stress would act on pore walls where there is contact with the liquid phase. In this study, Shimomura’s formulation [2] for capillary stress that causes drying shrinkage is adopted as Eq. 6:

$$ \sigma_{s} = - {\frac{2 \cdot \gamma }{{A_{s} r_{s} }}} = {\frac{{\rho_{L} \cdot R \cdot T}}{{A_{s} M_{w} \cdot \ln (P_{v} /P_{v0} )}}} $$
(6)

where σ s is tensile stress due to capillary tension [Pa], A s area factor [m3/m3]. When elastic factor of capillary water (E s ) [Pa] is considered, the shrinkage strain (ε sh) can be written as Eq. 7 [30]:

$$ \varepsilon_{\text{sh}} = \sigma_{s} /E_{s} . $$
(7)

According to the previous research [15], E s can be written as Eq. 8:

$$ E_{s} = 1.333 \times 10^{4} f_{c}^{1/3} $$
(8)

where \( f_{c}^{1/3} \) is compressive strength [kg/cm2] from the previous model [31]. The magnitude of E s as obtained from the above equation is reported to be about 25–50% of the ordinary value for the static elastic modulus of concrete [15]. The reason for this discrepancy might be the assumption of a uniform stress field due to capillary tension or perhaps some intrinsic differences in the mechanisms of deformation at micro- and macro scale due to capillary stress and that due to applied stress [15, 30]. When experimental results of compressive strength are considered in Eq. 8, the calculated E s is approximately 40–50% of static elasticity in Fig. 2. In the constitutive relation of Eq. 7, linear elastic factor (E s ) is assumed to be 40% of static elastic modulus for the sake of simplicity of analysis. If the pore radius (r s ) decreases with chloride content, tensile stress of the pore liquid increases accordingly. In order to consider the reduced pore radius, exponential function for chloride content is assumed as Eq. 9. The shrinkage strain with chloride content can be written as Eq. 10 based on the linear relation:

$$ r_{s}^{\prime} = r_{s} {\frac{1}{{(1 + AC_{\text{cl}}^{B} )}}} $$
(9)
$$ \varepsilon_{\text{sh}}^{\prime} = {\frac{{\sigma_{s} }}{{E_{s} }}} \times (1 + AC_{\text{cl}}^{B} ) $$
(10)

where \( r_{s}^{\prime} \) is pore radius in concrete with chloride content [m], \( \varepsilon_{\text{sh}}^{\prime} \) is shrinkage strain considering chloride content (C cl) [wt% of cement]. These equations are merged in analysis framework for calculation of cracking time. Utilizing the results shown in Fig. 3, increasing ratio of shrinkage due to chloride content can be evaluated. The results with different chloride content can be normalized by control data (results without chloride content) and an effect on shrinkage can be obtained with respect to chloride content. Among data in Fig. 3, those after 4 weeks showing relatively stable trend are selected and then normalized. The results are shown in Fig. 8. The constants of A and B in Eq. 9 are evaluated as 0.571 and 0.431 with correlation factor of 0.984.

Fig. 8
figure 8

Increasing ratio of shrinkage due to NaCl

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3.3 Stress analysis of concrete under restraint shrinkage

The constitutive equations of bulk strain (ε v ) and deviatoric strain (e ij ) in axially restrained concrete can be written Eqs. 11 and 12 with mean stress (σ v ) and deviatoric stress (s ij ), respectively:

$$ \varepsilon_{v} = \frac{1}{3K}\int {(1 + \varphi )d\sigma_{v} + \varepsilon_{t} + \varepsilon_{\text{sh}} - \varepsilon_{\text{st}} } $$
(11)
$$ e_{ij} = \frac{1}{2G}\int {(1 + \varphi )ds_{ij} } $$
(12)

where K is bulk elastic modulus [Pa], G is shear modulus [Pa], φ is creep coefficient [−], ε t is temperature strain [−], ε sh is shrinkage strain [−], ɛ st is the strain of restraining steel frame [−], respectively.

In this paper, previous model for creep function is used [23]. The shrinkage strain of the specimen (ε sh) in each time and spatial domain can be calculated by utilizing the previous models where chloride effect on shrinkage like Eqs. 9 and 10 is activated. The strain of the steel frame (ε st) is experimentally obtained from the gauge on the surface of the steel mold in Fig. 1. Therefore, the composition of various stresses caused by drying shrinkage due to the moisture transport and hydration, strain due to creep, and their integrated actions are considered in Eqs. 11 and 12. They can also be rewritten as Eq. 13 in the matrix form:

$$ \{ \sigma \} = [D]\{ \varepsilon \} - \left\{ \int {\varphi d\sigma} \right\} - [D]\{ \varepsilon_{t} + \varepsilon_{\text{sh}} - \varepsilon_{\text{st}} \} $$
(13)

where [D]is the stress–strain matrix.

Before cracking, Young’s modulus calculated with hydration rate in each element represents that of concrete in the matrix [D]. The evaluation of cracking time in stress field is determined by the time when the tensile principal stress of each node is higher than the developed tensile strength due to hydration. The tensile strength of concrete (σ t ) is assumed as Eq. 14 based on experimental data of compressive strength utilizing CEB-FIP [23]:

$$ \sigma_{t} = 0.0645 \cdot [237\exp \{ 0.267 \times (1 - (28/t)^{0.5} \}^{0.5} ]^{0.667} ,\;\;\;\;t > 0 $$
(14)

where \( \sigma_{t} \) is tensile strength of concrete [MPa]. The assumed tensile strength is 10.4–12.6% of compressive strength in Eq. 1.

3.4 Framework for evaluation of behavior in early-age concrete

In this section, previous models utilized for this study are briefly summarized and framework for evaluation of shrinkage behavior is introduced.

3.4.1 Multi-component hydration heat model (MCHHM) [15]

In view of concept of a multi-component powder material, the effect of different type of cement can be rationally taken into account to predict overall heat generation rate. The influence of variable moisture content of free water in the hydration is also taken into consideration in the concept. The chemical components of cement considered in this model are alite (C3S), belite (C2S), aluminate (C3A), ferrite (C4AF), and gypsum (CS2H). Other powder materials like slag and fly ash can also be considered in the model as pseudo clinker components. The total heat generation rate of concrete (H) per unit volume is idealized as Eq. 15:

$$ H = C \cdot H_{c} ,\,H_{c} = \sum {p_{i} H_{i} } $$
(15)

where, \( C \) is powder content per unit volume of concrete, H c is specific heat generation rate of cement, H i is specific heat generation rate of individual clinker component and p i is the corresponding mass ratio in cement to total mass. The temperature-dependant heat generation H i of each clinker component is based on the modified Arrhenius’s Law of chemical reaction, where various coefficients represent the interaction among the constituents as Eq. 16:

$$ \bar{H}_{i} = \beta_{i} \gamma_{i} \mu_{i} \xi_{i} s_{i}\overline{{H_{{i,T_{0} }} }} \overline{{(Q_{i}) }} \exp \left\{ -{\frac{{E_{i} (\overline{{Q_{i} }})}}{R}}\left(\frac{1}{T} -{\frac{1}{{T_{0} }}}\right)\right\},\quad\overline{{Q_{i} }} =\int {\overline{{H_{i} }} } dt $$
(16)

where, E i is activation energy of ith component reaction, \( \overline{{H_{{i,T_{0} }} }} \) is the referential heat generation rate when T 0 is 293 K, R is gas constant, β i represents the reduction of probability of contact between unhydrated compound and free water. γ i represents the retardation of cement and slag reaction caused by fly ash and organic admixture. \( \mu_{i} \) represents the effect of mineral composition of Portland cement, \( \xi_{i} \) represents the reduction of the pozzolan’s reaction due to the shortage of calcium hydroxide, \( s_{i} \) represents the effect of fitness of reaction particles. The heat rate of the entire cement (H c ) including blending powders, is given as the sum of the heat rate of all reaction as Eq. 17:

$$ \begin{aligned} \overline{{H_{c} }} & = \sum {p_{i} \overline{{H_{i} }} = } p_{{{\text{C}}_{ 3} {\text{S}}}} \overline{{H_{{{\text{C}}_{ 3} {\text{S}}}} }} + p_{{{\text{C}}_{ 2} {\text{S}}}} \overline{{H_{{{\text{C}}_{ 2} {\text{S}}}} }} + p_{\text{SG}} \overline{{H_{\text{SG}} }} + \\ & p_{\text{FA}} \overline{{H_{\text{FA}} }} + p_{{{\text{C}}_{ 3} {\text{A}}}} (\overline{{H_{{{\text{C}}_{ 3} {\text{AET}}}} }} + \overline{{H_{{{\text{C}}_{ 3} {\text{A}}}} }} ) + p_{{{\text{C}}_{ 4} {\text{AF}}}} (\overline{{H_{{{\text{C}}_{ 4} {\text{AFET}}}} }} + \overline{{H_{{{\text{C}}_{ 4} {\text{AF}}}} }} ). \\ \end{aligned} $$
(17)

3.4.2 Mirco-pore structure formation model (MPSFM) [15]

For analytical purpose, the interlayer porosity is simply lumped with the porosity distributions of gel and capillary porosity to obtain the total porosity distribution of cement paste as Eq. 18:

$$ \varphi (r) = \varphi_{\text{lr}} + \varphi_{\text{cg}} (r) = \varphi_{\text{lr}} + {\frac{1}{{\varphi_{\text{cp}} + \varphi_{\text{gl}} }}}[\varphi_{\text{cp}} (r) + \varphi_{\text{gl}} (r)] $$
(18)

where, \( r \) is the pore radius, \( \varphi_{\text{lr}} \) is the interlayer porosity, and φ cg is the combined porosity distribution of capillary porosity (φ cp) and gel porosity (φ gl). Each of the capillary and gel porosity distribution (φ cp(r) and φ gl(r)) are represented by a simplistic Rayleigh–Ritz (R–R) distribution function as Eq. 19:

$$ \begin{aligned}V_{i} = 1 - \exp ( - B_{i} r),\,(0 \le V(r) \le 1),\\dV_{i} = B_{i} r\exp ( - B_{i} r)d\ln (r) \end{aligned}$$
(19)

where, V i expresses a fractional pore volume of the distribution up to pore radius (r) and B i is the sole porosity distribution parameter. A bimodal analytical porosity (\( \varphi_{\text{total}} \)) distribution of the cement paste can be obtained as Eq. 20:

$$ \begin{aligned}\varphi_{\text{total}} = \varphi_{\text{lr}} + \varphi_{\text{gl}} [1 - \exp ( - B_{\text{gl}} r)] + \varphi_{\text{cp}} [1 - \exp ( - B_{\text{cp}} r)] \end{aligned}$$
(20)

where, B cp and B gl are distribution parameters correspond to capillary and gel porosity components. If the porosity is a cylindrical shape in such a distribution, then the pore distribution parameter (B i ) can be obtained form the following relationship of Eq. 21:

$$ S_{\text{sur}} = 2\varphi_{i} \int {r^{ - 1} dV = 2\varphi_{i} \int\limits_{{r_{\min } }}^{\infty } {B\exp ( - Br)d\ln (r)} } $$
(21)

where S sur is surface area per unit volume of the matrix, \( r_{\min } \) is the minimum pore radius. This expression cannot be solved analytically as a closed-form solution so that an explicit relationship is adopted, which has been obtained by fitting the accurate numerical evaluations of the above integral for a large number of data-sets which relate B as a function of S sur/φ [15, 16]. If the porosity distribution of the microstructure is known, Eq. 22 which is derived from Kelvin’s equation enables to obtain the amount of water present in the microstructure at a given ambient relative humidity. By integrating the pore volume that lies below pore radius (r s ), saturation (S) can be written as Eq. 23 through a porosity distribution V given by Eq. 19:

$$ r_{s} = - {\frac{{2\gamma M_{w} }}{{\rho_{L} RT}}}{\frac{1}{\ln (h)}}, $$
(22)
$$ S = \int\limits_{0}^{{r_{s} }} {dV = 1 - \exp ( - B_{i} r_{s} )} \,({\text{for}}\,{\text{wetting}}\,{\text{condition}}), $$
(23)

where, h is relative humidity of vapor phase. The expression for saturation (S) is changed with hysteretic behavior of isotherm.

Each model is well explained and documented in following references [15–19].

3.4.3 Framework for this study

In order to evaluate cracking time caused by effective restraint stress, framework for FE analysis is required with covering the behavior of early-age concrete like hydration, micro pore structure, and moisture transport. In this paper, integrated analysis program DuCOM [15, 16] is utilized for the evaluation of the behaviors in early-age concrete and stress analysis. Schematic view of unified FE analysis is shown in Fig. 9 including the chloride effect on pore radius and shrinkage.

Fig. 9
figure 9

Schematic view of FE analysis in this study

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4 Comparison and verification

In the integrated FE analysis, the thermal deformation and drying shrinkage strain in early-age concrete, and equivalent stress in each restrained condition are calculated through previous models (MCHHM, MPSFM, and MTM) considering chloride effect on shrinkage strain. Each Gaussian point in element has different strength with different hydration rate due to exterior condition, and the changed strength with time and space is considered for evaluation of crack resistance. The boundary and initial conditions for the analysis are shown in Table 2 with FE mesh for specimen.

Table 2 Conditions and mesh for FE analysis
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The results for free drying shrinkage strain of specimen with chloride content are shown in Fig. 10. The comparison with restraint stress and tensile strength is shown in Fig. 11 with different NaCl contents. In Fig. 12, cracking time and effective restraint stress from experiment are compared with those from the proposed analysis technique. Effective restraint stress at the age of 7 days are shown since no cracking in all the cases (0.0–1.2% of chloride content) occurs in this period.

Fig. 10
figure 10

Comparison of free shrinkage with NaCl content

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Fig. 11
figure 11

Comparison of effective restraint shrinkage stress with NaCl content

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Fig. 12
figure 12

Comparison of cracking time and effective restraint stress

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In Fig. 12, there is slight difference of stress between numerical and experimental data in initial condition, however, those from the proposed technique seem to be reasonably modeled through reverse analysis for the increased effective restraint stress and shortened cracking time caused by chloride content.

The proposed technique shows reasonable prediction of shrinkage behavior in chloride contaminated concrete but still has several limitations and considerations to be modified. There are several parameters on drying shrinkage in chloride-mixed concrete like changes in pore structure, a degree of alkali, and surface tension. Among these parameters, only change in pore radius is assumed and considered in the model. Surface tension should increase when chloride ions exist in pore water but same surface tension (72 mN/m) is used. E s for drying shrinkage strain is assumed as 40% of static elastic modulus based on the previous research and experimental results. This factor (E s ) can be modified in the quantitative manner. For this study, several previous models of MCHHM, MPSFM, and WTM are utilized. In the MCHHM, The chemical components of cement are considered. The heat generation of each clinker compound is based on a modification of the Arrhenius law of chemical reaction. In the MPSFM, total porosity can be obtained through bimodal analytical distribution function. Unfortunately, the effect of chemical admixture like air entrainer is not considered in the MPSFM. If the modeling for the performance of chemical admixture is included, the proposed technique can predict the drying shrinkage behavior more reasonably and quantitatively.

5 Conclusions

The conclusions on analysis technique for restrained shrinkage of concrete containing chlorides are as follows.

  1. (1)

    By the tests for compressive strength, Young’s modulus, water loss, and drying shrinkage for early-age concrete with different chloride contents, it is evaluated that the increased chlorides from NaCl in concrete has little effect on the strength, Young’s modulus, and water loss of concrete. However, it causes significantly increased drying shrinkage strain and makes cracking time shortened by increasing restraint stress.

  2. (2)

    The integrated analysis technique considering the chloride effect on drying shrinkage is proposed by utilizing the previous models for behavior in early-age concrete based on the thermodynamics equilibrium relation. Through reverse analysis, the proposed technique is able to model the increasing restraint stress and the reduced cracking time due to higher chloride contents under confined condition.