Laboratory Evaluation of Tensile Creep Behavior of Concrete at Early Ages

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This paper describes the development of the constitutive model for early-aging viscoelastic concrete. And this paper provides the work related to material testing and modeling for identifying the required constitutive properties of early-age concrete.

Abstract

Short-term uniaxial tensile creep tests were conducted for early-age concrete at different ages in an effort to characterize early-age concrete as an aging viscoelastic material. Based on the test results, the viscoelastic material properties of the test concrete were characterized in terms of the Dirichlet series creep compliance and relaxation modulus functions. Applications of the model to the numerical analysis of the test samples showed a good agreement between the tests and computational results for the samples tested at different ages. This paper presents the test procedures and data analysis with a brief introduction of theoretical background.

1. Introduction

It is well recognized that under sustained loading conditions, concrete exhibits time-dependent viscoelastic behavior, such as creep or stress relaxation [1,2,3]. Concrete pavement is a special type of structure undergoing sustained environmental loads, causing thermal or drying shrinkage deformation from the ambient temperature and moisture conditions. In a newly placed concrete pavement, restrained shrinkage stress will gradually develop with time because the deformation becomes limited in concrete due to internal restraints from aggregate as well as hardening cement paste matrix [4,5]. However, early-age concrete subjected to restrained shrinkage will experience significant stress relaxation, which is beneficial with respect to concrete cracking. Cracking is expected to occur when the stress after relaxation exceeds the material’s resistance to cracking. But the cracking time might be delayed with the help of creep and relaxation, reducing the magnitude of tensile stresses, which consequently allows for a wider saw-cut-timing window [6]. It should be noted that saw cut timing is affected by various factors such as concrete mixtures, curing temperatures, and relative humidity, etc. In hot weather conditions, saw cutting may start as soon as a couple of hours after the concrete has been finished. In cooler weather, sawing may not start for 12 h after pouring [7].

The factors that can influence the creep of the concrete structures include concrete mix proportioning, ambient humidity and temperature, and concrete member shape and size, etc. [8,9]. Influential factors for the concrete mix that induce creep are the type and content of cementitious materials, slump, air content, fine aggregate percentage, aggregate type, and fibers [8,9,10,11]. The early-age deformation of high-performance concrete (HPC) is known to be higher than normal concrete, as HPC is mixed with high cement and supplementary cementitious materials [12]. In addition, lower ambient relative humidity increases the creep coefficient [8]. And the volume-to-surface ratio or minimum thickness of the concrete specimen affects the creep coefficient too because of resistance to moisture transport from the interior to the atmosphere [5].

To date, existing studies on concrete creep have been focused on the compressive creep of concrete [13,14,15], while few efforts have been carried out on issues concerning the tensile creep of concrete. The tensile creep of concrete is creep deformation under constant tensile loading conditions. Reviewing various research results on the compressive and tensile creep of concrete, Hilaire et al. [16] reported that the development of creep strain is different between compressive and tensile creep in most studies, while the ratio of compressive creep to tensile creep decreases with time. Under similar testing conditions, such as specimen geometry, curing conditions, and loading conditions, similar creep results in compression and tension were demonstrated by Klausen et al. [17]. Performing creep tests in tension is known to be more difficult than conducting those in compression, especially at early ages [10].

A number of aging creep models have been proposed, e.g., ACI equation [18], the triple power law [19], the microprestress–solidification theory (MPS) [20,21,22], and the Dischinger model [23]. However, the empirical constants and functional parameters in these models have been derived from long-term compressive creep test data for aged concrete. Other studies modified these models to fit the creep of early-age concrete [24,25]. These modified models provide a good basis for the description of aging effects on the creep behavior of early-age concrete. These models are generally formulated via production or summation of the power laws, each of which represents age-dependent and age-independent behavior. Due to this functional complexity, further approximation is required for the representation of the relaxation behavior and structural stress analysis.

Investigations of the creep and relaxation effect on the mechanical responses of pavements are very limited [6,26,27,28]. Yeon et al. [26] used the creep coefficient adopted from CEB-FIP Model Code 1990 [23] to analytically calculate the average stress at the mid-depth of a slab. Lim et al. [27] predicted the average stress generated in the pavement using the creep properties measured from a ring test. Wei et al. [6] considered the effect of concrete creep in pavement slab for analyzing moisture warping and warping stresses by using specimens under different drying conditions exposed to drying at the age of 7 days. The creep relaxation effect is most significant in the first 7 days and tends to be stable after 60 days [28].

For the stress analysis of concrete structures, a proper material model should be used that represents a realistic constitutive behavior of early-age concrete. In this study, the early-age concrete is assumed to conform to the linear viscoelastic behavior with a strong aging effect. Short-term uniaxial tensile creep tests were conducted for a minor concrete paving mixture with 25% fly ash at different ages in an effort to characterize early-age concrete as an aging viscoelastic material. Based on the test results, the viscoelastic material properties of the test concrete were characterized in terms of the Dirichlet series creep compliance and relaxation modulus functions. In order to verify the aging creep model and its conversion to an age-dependent relaxation modulus, the tensile creep tests at different ages were simulated using the finite element program, ABAQUS, with the estimated relaxation modulus at each age. This paper presents the test procedures and data analysis with a brief introduction of the theoretical background.

2. Theoretical Background

2.1. Constitutive Relations

The time-dependent behavior of concrete has been effectively explained by the linear viscoelastic theory with the principle of superposition [29]. The general constitutive relations for linear aging viscoelastic materials can be expressed with the superposition integral as follows [30]:

$${\sigma }_{ij}(x_k,t)={\displaystyle \underset{0}{\overset{t}{\int }}\hspace{0.17em}{C}_{ijkl}(t,t\prime )}\frac{\partial {\epsilon }_{kl}(x_k,t\prime )}{\partial \hspace{0.17em}t\prime }dt\prime \hspace{0.17em}$$

(1)

where σ_ij is the stress tensor, ε_kl is the strain tensor, C_ijkl is the fourth order tensor of relaxation modulus, x_k is the position vector, t is the time of interest, and t′ is the initial loading time. The relaxation modulus C_ijkl(t,t′) describes the stress at time t caused by a unit constant strain applied at time t′. Note that the relaxation modulus is a function of the current time, t, and the initial loading time, t′. For non-aging materials, it will be a function of the duration of the load (t − t′) only. It is also noted that thermal or shrinkage stresses are not included in Equation (1). Under the linear theory, these environmental stresses may be effectively considered by adding separate terms that consist of relevant mechanical properties relating the environmental effects and corresponding stress developments [5].

For isotropic materials, the fourth order modulus tensor can be expressed as Equation (2) [30], and, therefore, the constitutive equations reduce to Equation (3) for early-age concrete as an isotropic, linear aging viscoelastic material.

$$C_{ijkl}(t)=\lambda (t){\delta }_{ij}{\delta }_{kl}+\mu (t)\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)$$

(2)

$${\sigma }_{ij}(x_k,t)=\hspace{0.17em}{\displaystyle \underset{0}{\overset{t}{\int }}\lambda (t,t\prime )\hspace{0.17em}{\delta }_{ij}}\frac{\partial {\epsilon }_{kk}(x_k,t\prime )}{\partial \hspace{0.17em}t\prime }dt\prime \hspace{0.17em}+\hspace{0.17em}{\displaystyle \underset{0}{\overset{t}{\int }}\hspace{0.17em}2\mu (t,t\prime )}\frac{\partial {\epsilon }_{ij}(x_k,t\prime )}{\partial \hspace{0.17em}t\prime }dt\prime$$

(3)

where λ(t) and μ(t) are independent relaxation moduli, and δ_ij is the Kronecker delta. For an effective use of the constitutive equations to early-age concrete, it is crucial to properly define the related material properties, i.e., the time- and age-dependent relaxation modulus functions, $\lambda (t,t\prime )$ and $\mu (t,t\prime )$, for the early-age concrete.

These general isotropic relaxation moduli can then be expressed in terms of the uniaxial relaxation modulus, E(t), as shown in Equations (4) and (5). The Poisson’s ratio (ν) is assumed to be time-independent.

$$\lambda (t)=\frac{\nu }{(1+\nu )(1-2\nu )}E(t)$$

(4)

$$2\mu (t)=\frac{1}{1+\nu }E(t)$$

(5)

As shown, the relations of the material properties are very similar to those defined in elastic theory. By these relations, the constitutive equation given in Equation (3) can be used for the viscoelastic stress analysis of early-age concrete when the age-dependent uniaxial relaxation modulus, $E(t,t\prime )$, is defined. One way to determine this viscoelastic material property is through a relaxation test [30]. However, due to the difficulties associated with the relaxation test, especially for early-age concrete in tension, a creep test that defines creep compliance is generally preferred. The relaxation modulus can be determined from the compliance by a unique relationship between those two properties.

2.2. Creep and Relaxation Characteristics

For one-dimensional loading, the constitutive equations reduce to

$$\sigma (t)={\displaystyle \underset{0}{\overset{t}{\int }}\hspace{0.17em}E(t,t\prime )}\frac{\partial \epsilon (t\prime )}{\partial \hspace{0.17em}t\prime }dt\prime$$

(6)

or equivalently in a stress formulation,

$$\epsilon (t)={\displaystyle \underset{0}{\overset{t}{\int }}\hspace{0.17em}D(t,t\prime )}\frac{\partial \sigma (t\prime )}{\partial \hspace{0.17em}t\prime }dt\prime$$

(7)

where $E(t,t\prime )$ and $D(t,t\prime )$ are the time- and age-dependent uniaxial relaxation modulus and creep compliance, respectively. As a counterpart of the relaxation modulus, creep compliance describes the strain at time t caused by a unit constant stress applied at time t′. Viscoelastic material behavior defined by the constitutive relations given in Equations (6) and (7) can be analogized by their mechanical models shown in Figure 1. These mechanical models, respectively, define the relaxation modulus and creep compliance in a series of real exponents referred to as the Dirichlet series or the Prony series [31]. Equations (8) and (9) represent the Dirichlet series relaxation modulus and creep compliance functions for aging concrete. Mathematical difficulties from the aging effect are simplified in these representations by formulating the functions in a sum of products of single time-dependent variables [32,33].

$$E(t,t\prime )=E_{\infty }(t\prime )+{\displaystyle \sum _{i=1}^m{E}_i(t\prime )\cdot e^{-\hspace{0.17em}\frac{t-t\prime }{{\rho }_i}}}$$

(8)

$$D(t,t\prime )=D_0(t\prime )+{\displaystyle \sum _{i=1}^m{D}_i(t\prime )\cdot \left(1-e^{-\hspace{0.17em}\frac{t-t\prime }{{\tau }_i}}\right)}$$

(9)

Material parameters (E_∞, E_i, D₀, and D_i) in Equations (8) and (9) correspond to the internal modulus and compliance of the spring units of the mechanical models in Figure 1. The internal viscosity (η_i) of each unit of the models characterizes the time-dependent viscous behavior of the material, which are involved in the exponents ρ_i and τ_i, referred to as the relaxation time and retardation time, respectively.

The two viscoelastic material properties, relaxation modulus and creep compliance, are uniquely related to each other as presented in Equation (10). Therefore, once one of the properties is identified, the other can be estimated by the relationship [30]. As will be seen later, the relationship can be verified by the Laplace transform of the relaxation and creep functions.

$${\displaystyle \underset{0}{\overset{t}{\int }}E(t-\tau )\frac{\partial D(t)}{\partial \tau }\hspace{0.17em}}d\tau ={\displaystyle \underset{0}{\overset{t}{\int }}D(t-\tau )\frac{\partial E(t)}{\partial \tau }\hspace{0.17em}}d\tau =1 \mathit{for\; all\; t}$$

(10)

It is noted that the relationship above does not involve the age-dependency of the material properties. This means the interrelation applies for a particular age, t′, between D(t) and E(t). If the conversion of the material properties is required for an age-dependent stress or displacement analysis, therefore, the conversion scheme must be continuously incorporated into each time step of the analysis procedure.

3. Tensile Creep Tests

3.1. Materials

Conventional plain concrete was tested at different ages. The mixtures consisted of Type I Portland cement, crushed limestone aggregates with the maximum size of 25 mm, natural sand, and Class C fly ash. The gradation of coarse and fine aggregates conformed to ASTM C33 requirements.

Table 1. Concrete mixture proportions.

Unit Content (kg/m3)					w/c
Cement	Fly Ash	Fine Aggregate	Coarse Aggregate	Water
200.5	66.4	800.3	1037.1	143.5	0.54

presents the concrete mixture proportions used in this study.

3.2. Procedures

Uniaxial tensile creep tests were conducted with standard concrete cylinders (100 × 200 mm) at the ages of 2.7, 4.3, 6.8, and 13.3 days using a servo-controlled MTS testing machine. One sample per age was used to verify the modeling approach. Immediately after casting, the concrete cylinders were moved to a moisture curing room with a controlled temperature and humidity at 25 °C and 100%. The cylinder specimens were moved to the testing room (25 °C and 40% RH) for instrumentation at 24 h before the respective ages of testing. A pair of steel connections for direct tensioning of the test specimen was epoxy-bonded at each end of the cylinder. In order to provide a surface for epoxy bonding, both ends of the concrete cylinder were ground by approximately 5 mm in depth. The epoxy work was performed with a specially prepared concentric jig as shown in Figure 2. After the epoxy work, the specimens were kept in the testing room until the time of testing in order for the epoxy to develop its full bond strength.

A designated level of tensile load was introduced to the specimen within 0.1 s, and subsequent changes in displacement as a function of time was monitored up to 20,000 s (5.5 h), while keeping the load constant. The level of load was controlled with a 4.5 kN capacity load cell. The load level ranged from 1 to 2 kN depending on the ages of the specimens. Displacement data were traced by the two LVDTs installed on the mid-surface of the cylinder specimens with 100 mm gage lengths. Figure 3 shows the test set-up.

The level of test load was determined to be 15% of the tensile strength of the concrete at the respective ages of testing. Tensile strength was assumed to be 1/8 of the compressive strength at all ages of testing. Development of the compressive strength of the concrete was estimated by the ACI equation [18] shown in Equation (11). The average 28-day compressive strength of the test concrete was 15.51 MPa, found by separate strength tests.

Table 2. Determination of load levels for the tensile creep tests.

Age at Test (Day)	fc(t) by ACI (MPa)	fc(t) by ACI (MPa)	Test Load		σt/ft(t) (%)
			Ft (kN)	σt (MPa)
2.7	6.65	0.83	1.0	0.127	15.3
4.3	8.71	1.09	1.3	0.166	15.2
6.8	10.78	1.35	1.6	0.204	15.1
13.3	13.48	1.68	1.9	0.242	14.4

f_c(t): compressive strength at age t, f_t(t): tensile strength at age t, 1/8 of f_c(t), F_t, and σ_t: test load determined in terms of tensile force and stress, respectively.

summarizes the determination of the test load for the respective ages of testing.

$$f_c(t)=f_c\prime \frac{t}{4+0.85\hspace{0.17em}t}$$

(11)

where f_c(t) = compressive strength at time t, and f_c′ = reference compressive strength at the age of 28 days.

Earlier studies on the creep of aged concrete suggested that the principle of superposition would be valid within the service range of stresses (up to ½ of the strength) under a monotonically increasing strain condition [29,34]. Recent studies found that early-age concrete also shows proportional creep responses to the stress level below 50% of static strength, regardless of mixture characteristics [25,35,36]. Based on these previous observations, no additional linearity check was conducted in this study.

4. Test Results

4.1. Creep Compliance

Figure 4 shows the time-dependent displacement of the test specimens placed under constant tensile load at the designated testing ages. It is noted that the measured deformation is the drying creep under the effect of drying shrinkage that occurred during the test period. Drying creep is known to induce a non-linear effect, particularly in long-term response; as such, it is always greater than the sum of separate measurements of creep under sealed conditions (basic creep) and drying shrinkage under load-free conditions (free shrinkage) [37,38]. However, this study focuses on short-term creep in early-age concrete within the context of linear theory. Creep tests were accordingly conducted where the non-linear effect during the 5 h testing period is assumed negligible. With these assumptions, the measured total strain (ε_test) can be expressed by the sum of basic creep (ε_creep) and shrinkage (ε_sh) as shown in Equation (12). Note that the drying shrinkage occurs in the opposite direction to the tensile creep. The basic creep (ε_creep) is the test load-induced time-dependent deformation of the concrete that can be determined by the constitutive relations shown in Equation (13). Substituting Equation (13) into Equation (12) gives Equation (14), which was used for the estimation of the creep compliance $D(t,t\prime )$ of each test specimen.

$${\epsilon }_{test}(t-t\prime )={\epsilon }_{creep}(t-t\prime )-{\epsilon }_{sh}(t-t\prime )$$

(12)

$${\epsilon }_{creep}(t-t\prime )={\displaystyle \underset{0}{\overset{t}{\int }}D(t,t\prime )\frac{\partial {\sigma }_0\cdot H(t\prime )}{\partial t\prime }dt\prime \hspace{0.17em}}$$

(13)

$$D(t,t\prime )=\frac{{\epsilon }_{test}(t-t\prime )+{\epsilon }_{sh}(t-t\prime )}{{\sigma }_0(t\prime )}$$

(14)

where $D(t,t\prime )$ = uniaxial creep compliance, ${\sigma }_0$ = constant testing load in terms of stress, and $H(t\prime )$ = step function (0 for $t\le t\prime$ and 1 for $t>t\prime$).

The magnitude of drying shrinkage (ε_sh) of the test concrete was estimated by the BP model [39]. Estimated maximum shrinkage after the 5 h testing period at the ages of 2.7, 4.3, 6.8, and 13.3 days were 7.72, 6.55, 5.71, and 4.76 microstrains, respectively. The basic creep was then calculated by Equation (12). Figure 5 shows the three components of deformation for the respective test samples. Creep compliance curves estimated by Equation (14) for the test concrete at different ages are shown in Figure 6. As shown, there is a sharp reduction in the creep response of the concrete between the samples aged 2.7 and 4.3 days. Much stiffer responses were observed from the samples at later ages. The difference in the response is not so pronounced between the 6.8- and 13.3-day-old samples, suggesting the aging effect on the creep response may not be significant for concrete aged more than 7 days.

4.2. Dirichlet Series Creep Function

The creep compliance data shown in Figure 6 were fitted with the Dirichlet series functions given in Equation (9) by the collocation method. The method of fitting the data with the Dirichlet series may be found elsewhere [32]. In this study, six collocation points were selected from 10⁻³ to 10² h at decimal spacing. This means the Kelvin model used in this study consists of six Kelvin units in addition to the initial spring unit. The initial compliance (D₀) was selected; as such, it corresponds to the instantaneous response of the specimen to loading within 1 s (2.8 × 10⁻⁴ h). The last two collocation values were extrapolated from the test data referring to the creep rate reported in the previous studies for the long-term creep in compression [39,40]. The material retardation time (τ_i) for each Kelvin unit was determined at six points from 5 × 10⁻⁴ to 50 h at decimal intervals. With these pre-selected variables, the internal compliance (D_i) of the Kelvin model was determined using Equation (15), which is a matrix formulation of Equation (9) rearranged for D_i. Since the model has six Kelvin units, the number of collocation points (i, j) varies from 1 to 6.

$$\{\hspace{0.17em}{D}_i\hspace{0.17em}\}\hspace{0.17em}=[\hspace{0.17em}1-e^{-\hspace{0.17em}{t}_j/{\tau }_i}\hspace{0.17em}]{\hspace{0.17em}}^{-1}\hspace{0.17em}\{\hspace{0.17em}D\hspace{0.17em}(t_j)-D_0\hspace{0.17em}\}\hspace{0.17em}$$

(15)

where {D_i} = the compliance vector to be determined for the respective Kelvin units, and D(t_j) = compliance at the jth collocation point (t_j).

Figure 7 shows the fitted creep compliance curves compared to the respective test data. The internal parameters, D₀ and D_i, determined from the fitting process, are presented in

Table 3. Material parameters determined for the Dirichlet series creep compliance function.

Parameter	Retardation Time (h)	Internal Compliance for the Kelvin Units (10−6/MPa)
		t′ = 2.7-Day (64.8 h)	t′ = 4.3-Day (103.2 h)	t′ = 6.8-Day (163.2 h)	t′ = 13.3-Day (319.2 h)
D0	Instant	24.25	21.50	20.35	17.60
D1	0.0005	0.91	0.42	0.28	0.22
D2	0.005	4.35	1.85	1.15	0.98
D3	0.05	27.54	11.08	7.68	5.72
D4	0.5	122.27	38.75	23.93	18.47
D5	5	348.61	115.45	64.54	50.08
D6	50	779.63	221.38	110.61	88.67

. The aging effect, represented by the dependency of the response to the initial loading time, may be explained by the changes in the internal compliance of the concrete at different ages.

4.3. Relaxation Modulus

As previously mentioned, the relaxation modulus function (E(t,t′)) can be estimated by converting creep compliance function (D(t,t′)) using their relationship given in Equation (10). The conversion procedure can be simplified with the Laplace transform (L). The Laplace transform with respect to time is defined by

$$L [f (x, t)]=\overline{f}(x,s)={\displaystyle \underset{0}{\overset{\infty }{\int }}f(x,t)\cdot e^{-st}dt}$$

(16)

where s is the transform parameter. The transform of the relationship, Equation (10), is

$$\overline{E}(s)\hspace{0.17em}{s}^2\hspace{0.17em}\overline{D}(s)=1$$

(17)

By introducing the Carson transform, $\tilde{f}=s\cdot \overline{f}$, Equation (17) becomes

$$s\cdot \overline{E}=\frac{1}{s\cdot \overline{D}} \mathrm{or} \tilde{E}=\frac{1}{\tilde{D}}$$

(18)

The Carson transform of the uniaxial relaxation modulus (Equation (8)) and creep compliance (Equation (9)) are

$$\tilde{E}(s)=E_{\infty }+{\displaystyle \sum _{i=1}^m\left(\hspace{0.17em}\frac{s\cdot E_i}{s+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\rho }_i$}\right.}\hspace{0.17em}\right)}$$

(19)

$$\tilde{D}(s)=D_0+{\displaystyle \sum _{i=1}^m\left(\hspace{0.17em}\frac{1}{{\eta }_i}\cdot \frac{1}{(s+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\tau }_i$}\right.)}\hspace{0.17em}\right)}$$

(20)

The substitution of Equations (19) and (20) into Equation (18) gives

$$E_{\infty }+{\displaystyle \sum _{i=1}^m\left(\hspace{0.17em}\frac{s\cdot {\rho }_i}{1+s\cdot {\rho }_i}\cdot E_i\hspace{0.17em}\right)}=\frac{1}{D_0+{\displaystyle \sum _{i=1}^m\left(\hspace{0.17em}\frac{D_i}{1+s\cdot {\tau }_i}\hspace{0.17em}\right)}}$$

(21)

By the final value theorem, ${s\cdot \overline{f}\hspace{0.17em}|}_{s\to 0}=f(\infty )$, the infinite modulus can be found from Equation (21) as

$$E_{\infty }=\frac{1}{D_0+\hspace{0.17em}{\displaystyle \sum _{i=1}^m{D}_i}\hspace{0.17em}}$$

(22)

Therefore, when the creep compliance is defined, the material parameters D₀, D_i, and E_∞ are known, and the only unknown is the internal spring moduli, E_i. This set of unknown variables may be effectively defined by the collocation method within the Carson domain. The collocation points, s_i, in the Carson domain may be selected at 1/t_i, where t_i represents the properly selected collocation points in the time domain. By definition, the material relaxation time (ρ_i) should be the same as the retardation time (τ_i). However, different material times can be selected for better fitting purposes. With the properly selected collocation points and the material relaxation times, Equation (21), can be rearranged for the unknown variables E_i as

$$E_i=A_{ij}^{-1}\cdot B_i \mathrm{or} \left\{E\right\}={\left[A\right]}^{\hspace{0.17em}-1}\cdot \left\{B\right\}$$

(23)

in which

$$A_{ij}=\hspace{0.17em}\left(D_0+{\displaystyle \sum _{k=1}^m\frac{D_k}{1+s_i\cdot {\tau }_k}}\right)\hspace{0.17em}\left(\frac{s_i\cdot {\rho }_j}{1+s_i\cdot {\rho }_j}\right)$$

(24)

$$B_i=1-E_{\infty }\hspace{0.17em}\left(D_0+{\displaystyle \sum _{k=1}^m\frac{D_k}{1+s_i\cdot {\tau }_k}}\right)$$

(25)

Solutions of Equations (22) and (23) will provide the material parameters constructing the Dirichlet series relaxation modulus function (Equation (8)). For converting the creep function of the test concrete, the material relaxation time (ρ_i) and the collocation points (t_j) were selected at ρ_i = 0.8τ_i and t_j = 0.5ρ_i, respectively. These values are selected to obtain a smooth curve fitting through the whole testing time span.

Table 4. Material parameters converted for the Dirichlet series relaxation modulus function.

Parameter	Relaxation Time (h)	Internal Spring Modulus for the Maxwell Units (GPa)
		t′ = 2.7-Day (64.8 h)	t′ = 4.3-Day (103.2 h)	t′ = 6.8-Day (163.2 h)	t′ = 13.3-Day (319.2 h)
E0	0.0004	0.285	0.931	0.838	0.862
E1	0.004	12.666	4.841	3.230	3.619
E2	0.04	19.491	19.531	15.727	16.809
E3	0.4	5.038	13.216	17.323	19.265
E4	4	1.355	3.872	7.247	8.588
E5	40	0.426	1.295	2.377	2.816
E6	infinite	0.788	2.467	4.637	5.447

shows the converted internal parameters for the Dirichlet series relaxation function of the test concrete. Relaxation times are selected considering the test duration and goodness of curve fitting. Corresponding relaxation curves are plotted with the creep curves in Figure 8. As shown, symmetric relationships exist between the creep and relaxation responses with respect to the loading time (t − t′).

5. Application

Using the estimated relaxation modulus, which is shown in Table 4, the creep tests conducted were simulated with the finite element program ABAQUS (Hibbitt, Karlsson & Sorensen, Inc., Providence, RI, USA). Continuum material properties were specified for a viscoelastic material option with the Prony series parameters in ABAQUS. And a constant Poisson’s ratio (ν = 0.17) was used for the analysis. Since the tests lasted less than 6 h, the aging effect during the test was ignored in this simulation, and the relaxation modulus determined for each age of testing was regarded as a constant during the test periods. The upper half (50 × 100 mm) of the creep test cylinders was modeled with bilinear axisymmetric solid elements (CAX8R element in ABAQUS). The stress level applied for each test was applied as a distributed pressure load on the surface of the top layer elements. The analysis was performed up to 5.1 h after the initial loading. The total analysis time was divided into 4 sub-steps at the time of 0.1, 0.5, 1.5, and 5.1 h with varying time steps (∆t) of 0.0001, 0.001, 0.005, and 0.01 h, respectively. Results of the finite element analysis are plotted with test results in Figure 9. As shown, the result of computational analysis agrees well with the result of tests at all ages of testing. In Figure 9, F_o denotes an applied test load at each age. Note again that, when the concrete experiences significant hardening during the analysis period, the material properties must be updated throughout the analysis in accordance with the relevant aging dependencies. This continuous update of material properties could be accomplished by an incremental analysis scheme [5].

6. Conclusions

In this study, the early-age concrete is assumed to conform to linear viscoelastic behavior with a strong aging effect. Short-term uniaxial tensile creep tests were conducted for a minor concrete paving mixture with 25% fly ash at different ages in an effort to characterize early-age concrete as an aging viscoelastic material. Introduced in this paper is a method of determining viscoelastic properties of early-age concrete. Tensile creep tests have been conducted with concrete cylinders of four different ages between 2.7 and 13.3 days. Test results indicated that the effect of aging on the short-term creep responses of early-age concrete is pronounced especially for hardening concrete aged 7 days or younger. The time-dependent displacement of the youngest concrete tested at 2.7 days was 4 to 5 times greater than those of the other concrete specimens tested at 4.3, 6.8 and 13.3 days. The difference in the creep responses was found to not be so significant between the 6.8- and 13.3-day-old specimens. Based on the test results, creep compliance functions of the concrete were formulated in the Dirichlet series functions. The creep functions were then converted to a series of relaxation modulus functions, which is more useful for stress analysis. The results of computational analysis using these estimated viscoelastic properties agreed well with the results of the creep tests at all ages of testing. For future research, the tensile creep behavior of various mix proportions, such as fiber-reinforced concrete or supplementary cementitious materials, can be studied at early ages.