A Mathematical Model for Predicting the Ultra-Early-Age Strength of Concrete

Abstract

To accurately quantify the time-varying pattern of concrete’s compressive strength, selecting an appropriate curve model is of paramount importance. Currently, widely adopted models such as polynomial, hyperbolic, and exponential models all possess limitations, particularly in terms of low fitting accuracy during the ultra-early-age stage. This paper innovatively introduces a mathematical model that utilizes a combined curve approach. This model boasts a simplified structure with only two fitting parameters. Compared to traditional models, when utilizing three or more sets of experimental data on compressive strength across different ages, the new model is capable of yielding more precise strength predictions. Due to its minimal reliance on experimental data, the new model exhibits high practicality and convenience in real-world applications. To validate its superiority, a detailed comparison between the new model and existing models was conducted based on several sets of experimental data. The results demonstrate that the new model has significant advantages in terms of mean fitting error and standard deviation, making its predictions the most reliable. For most cases, the standard deviation of the new model is reduced by approximately 30% to 80% compared to the second-best model, underscoring its exceptional stability and consistency. Additionally, the predicted long-term compressive strength values of the new model are closer to the design strength grade of the concrete. This model can also be successfully applied to predict the tensile strength of concrete during its ultra-early age. It has been demonstrated that the combined model proposed in this paper shows promising application prospects in evaluating the time-varying behavior of concrete strength.

1. Introduction

The assessment of concrete’s compressive strength at ultra-early stages is a pivotal topic in civil engineering research. Some scholars have conducted literature reviews on the mechanical properties of concrete during its early age, including early strength. Mihashi and Leite [1] reviewed the impact of cracking in concrete during its early age, caused by factors such as insufficient strength, on the durability of the concrete. Bentz [2] provided an overview of the early strength and other properties of cement-based materials, retrospectively examining the key physical and chemical processes that occur during early stages. From the perspective of concrete engineering, Maruyama and Lura [3] outlined the early properties of concrete, with a focus on early strength, elastic modulus, and volume stability. Understanding the growth pattern of this strength during this critical period is invaluable for engineers to devise practical and efficient construction plans. Accurate strength evaluations of concrete are particularly crucial in the following scenarios: (1) dismantle the formwork as early as possible to improve the construction efficiency; (2) carry out pre-stress stretching as early as possible to avoid cracks; and (3) evaluate the negative impact of the vibration load on concrete construction. Therefore, many scholars have studied the strength evaluation model and method of concrete at an early age. Gu et al. [4] delved into the development of concrete strength by analyzing the early harmonic responses captured by embedded piezoelectric sensors. Their findings revealed a rapid surge in concrete strength within the initial few days, followed by a decline after seven days. Lee et al. [5] proposed a model to evaluate the early compressive strength according to the ultrasonic pulse velocity. It has been found that the relationship between parameters in different concrete samples is linear during initial and final setting, and parabolic after final setting. Yoon et al. [6] put forward a formula to predict the early strength of concrete based on surface wave velocity. Benaicha et al. [7] studied the strength evolution law of several kinds of concrete such as self-compacting concrete, high-performance concrete, and metal-fiber-reinforced concrete. They put forward a function to evaluate the concrete strength after pouring. Lee et al. [8] proposed a generalized rate constant model to predict the compressive strength of the early strength concrete. In order to rapidly improve the early strength of concrete, Min et al. [9] designed some new curing schemes to ensure that the early strength of concrete exceeds 10 MPa after 6 h of pouring. Chung et al. [10] put forward an evaluation model of the 28-day compressive strength of concrete. It has been shown that the slope of the strength–conductivity curve can determine the early strength grade of concrete. Voigt et al. [11] observed the microstructure change process of concrete from solidification to hardening. Demirboga et al. [12] found that the relationship between the ultrasonic velocity and compressive strength of concrete at an early age is an exponential function. Velay-Lizancos et al. [13] studied the relationship between the compressive strength and elastic modulus of early-age concrete. Jin et al. [14] predicted the compressive strength of early-age bisphenol epoxy concrete using the maturity method. Adesina et al. [15] studied the performance of additives such as calcium oxide and sodium silicate to improve the early compressive strength of slag concrete. Jiang et al. [16] studied the main factors affecting the mechanical properties of early alkali slag concrete. It was found that the compressive strength increased with the increase in binder dosage. Kaszynska [17] studied the relationship between the temperature change and early compressive strength of mass concrete during curing. Burhan et al. [18] found that polymer is more effective than silica fume in improving the workability and compressive strength of early-age concrete. Voigt et al. [19] found that the relationship between the reflection loss of ultrasonic wave and the compressive strength of concrete has nothing to do with temperature. The above research shows that the early strength of concrete increases rapidly, but there is no uniform growth law due to many factors. To anticipate the variations in concrete strength during its early stages, numerous scholars have formulated diverse mathematical models. These include the polynomial model [20,21,22], hyperbolic model [23,24,25], exponential model [26,27,28,29], and logarithmic model [30,31,32,33], each designed to capture different aspects of the strength development process. Liu [20] used a quadratic power function to fit the compressive strength data of fiber-reinforced coral concrete specimens at different ages. Ling [21] applied a polynomial model to correlate the experimental data of compressive strength for machine-made sand concrete across various ages. Jia et al. [22] established a cubic polynomial function, based on experimental data spanning from day 1 to day 28, to describe the relationship between the age and compressive strength of ordinary Portland concrete. Yang and Wang [23] determined that the hyperbolic model aptly captures the variations in the compressive strength and elastic modulus of concrete during its early stages. Wang et al. [24], after analyzing the strengths and weaknesses of the hyperbolic model, proposed an enhanced version specifically tailored to predict the compressive strength of recycled concrete. Zhao et al. [25] proposed another form of the hyperbolic model by introducing the compressive strength at the 28th day. Tang et al. [26] described the time-varying regularity of compressive strength through low-age compressive tests on concrete cube specimens using an exponential model. Other scholars [27,28,29] have developed variations of exponential models by introducing the strength of concrete at the 28th day. Wang and Hu [30] utilized a logarithmic model to fit the experimental data of compressive strength for shale ceramsite concrete test blocks. Zhao [31] and Li et al. [32] employed the logarithmic model to fit the experimental data pertaining to the compressive strength of high-strength concrete, leveraging this approach to forecast the subsequent strength development of the concrete. Bai and Han [33] proposed a variant form of the logarithmic model by introducing the strength data of concrete on the 28th day.

Despite the plethora of models proposed for assessing concrete strength, there remains a need to develop a novel strength–age evaluation model that can predict concrete strength with greater precision. The current evaluation models suffer from notable shortcomings, including domain gaps in the exponential and logarithmic models, the poor fitting accuracy of the hyperbolic model during the ultra-early-age stage, and potential large errors in predicting later concrete strength using the polynomial model. To address these limitations, this work introduces a combined curve model that more accurately describes the growth pattern of concrete strength. The novelty of this evaluation model lies in its fusion of two curve types, weighted to achieve higher fitting accuracy than any single curve model. The advantages of this proposed model are threefold: firstly, it requires only two unknown fitting parameters, facilitating its application with minimal experimental data; secondly, it exhibits superior fitting accuracy compared to single curve models; and thirdly, it provides more reliable predictions of later concrete strength than existing models. Several examples are employed to validate the proposed model and benchmark its performance against other models, demonstrating its superiority and reliability. The structure of this study is organized as follows. Section 2 provides a brief overview of commonly used compressive strength evaluation models, including the polynomial, hyperbolic, exponential, and logarithmic models. Section 3 elaborates on the proposed combined curve model, its parameter determination process, and error evaluation metrics. Section 4 verifies the combined model using several sets of concrete compressive strength test data. Section 5 presents a statistical analysis of the results and explores the application of the novel model for predicting concrete tensile strength. Finally, Section 6 summarizes the key findings and conclusions of this work.

2. Commonly Used Curve Models

The main task of strength evaluation is to establish the functional relationship between concrete compressive strength and time. This section reviews several commonly used curve models for strength evaluation and discusses their advantages and disadvantages. The first evaluation model is the polynomial curve (quadratic power function [20] or cubic power function [21,22]) given as follows:

$$S_t=p_2{t}^2+p_{1}t+p_0$$

(1)

$$S_t=p_3{t}^3+p_2{t}^2+p_{1}t+p_0$$

(2)

where $t$ indicates the time from the concrete pouring, $S_t$ represents the compressive strength of concrete at time $t$, $p_0$, $p_1$, $p_2$, and $p_3$ represent the unknown parameters that need to be determined by curve fitting. Obviously, the fitting accuracy will improve as the number of terms in the polynomial increases. One of the strengths of the polynomial curve model lies in its high fitting accuracy for early-age concrete compressive strength experimental data. However, its primary limitation is its inability to accurately predict the later strength of concrete. This shortcoming stems from the power function utilized in the model, which does not fully align with the actual development pattern of concrete strength, specifically failing to capture the stability of concrete strength in later stages. Additionally, the model involves three to four unknown parameters, necessitating a larger volume of experimental data for accurate curve fitting, which can be a drawback in practice.

The second evaluation model is the hyperbolic model [23,24] given as follows:

$$S_t={\displaystyle \frac{t}{p_0+p_{1}t}}$$

(3)

where $p_0$ and $p_1$ represent the unknown fitting parameters. In contrast to the polynomial model, the hyperbolic model boasts a simpler structure with only two unknown parameters. This advantage translates into the ability to perform curve fitting with a reduced amount of experimental data. However, a notable disadvantage of the hyperbolic model is its limited fitting accuracy when dealing with experimental data exhibiting significant discreteness, potentially impacting the reliability of the predictions. Based on Equation (3), Zhao et al. [25] proposed another form of the hyperbolic model by introducing the compressive strength at the 28th day and modifying the power of time as follows:

$$S_t={\displaystyle \frac{t^{\alpha }}{p_0+p_1{t}^{\alpha }}}{f}_{c-28}$$

(4)

where $f_{c-28}$ represents the compressive strength of concrete on the 28th day and $\alpha$ is an adjustable coefficient. The limitations of the model presented in Equation (4) lie in two aspects: (1) the increased difficulty of curve fitting due to the introduction of an adjustable power value; (2) the need to test the concrete strength at the 28th day, which renders the model unsuitable for assessing the strength of concrete at super-early ages (before the 28th day).

The third and fourth types of models are the exponential function [26] and the logarithmic function [30], respectively, as shown below:

$$S_t=p_0\cdot e^{-p_1/t}$$

(5)

$$S_t=p_0+p_1\cdot \ln (t)$$

(6)

where $p_0$ and $p_1$ represent the unknown fitting parameters. Similar to the hyperbolic model, the exponential or logarithmic model has the advantage that it contains only two unknown parameters and is easy to implement. One drawback of this model type is that its domain does not encompass the point where $t$ = 0, which can limit its applicability in certain scenarios. Furthermore, similar to other models, its fitting accuracy may not be optimal when confronted with experimental data characterized by high levels of discreteness, potentially affecting the precision of the predictions it generates. By incorporating the strength of concrete at the 28th day, the literature [27,28,29] presents several variants of exponential models, listed sequentially as follows.

$$S_t=f_{c-28}\cdot e^{p_0\cdot \sqrt{1-{\displaystyle \frac{28-p_1}{t-p_1}}}}$$

(7)

$$S_t=f_{c-28}\cdot f_{ult}\cdot e^{-{({\displaystyle \frac{\tau }{t}})}^{\alpha }}$$

(8)

$$S_t=f_{c-28}\cdot e^{p_0\cdot \sqrt{1-{\displaystyle \frac{28}{t}}}}$$

(9)

The $f_{ult}$ and $\tau$ in Equation (8) represent the final compressive strength of the concrete and a fitting parameter to be determined, respectively. Similarly, a variant of the logarithmic model that incorporates the strength data of concrete on the 28th day is presented in reference [33] as follows.

$$S_t=f_{c-28}\cdot [1+p_0\cdot \ln ({\displaystyle \frac{t}{28}})]$$

(10)

The aforementioned variants of the exponential model and logarithmic model both require the 28-day strength data of concrete, which makes them inappropriate for evaluating the compressive strength of concrete at ultra-early ages. The model shown in Equation (8) also requires the final strength of concrete, which further limits the scope of the application of this model. Furthermore, these models also suffer from issues related to the lack of definition in certain domains. For instance, Equations (8)–(10) all lack a definition for the point where $t$ = 0, while Equation (7) lacks a definition for the point where $t=p_1$. Additionally, Equations (7) and (9) require that $t\ge 28$, as negative numbers cannot be squared. Furthermore, in terms of fitting accuracy, these modified forms do not exhibit significant improvements in precision. Given the limitations of the existing models, it is imperative to investigate new compressive strength–age curve models for analyzing the variation patterns of compressive strength in concrete during its ultra-early age.

3. New Combined Curve Model

The primary characteristics defining the variation law of concrete compressive strength encompass the following aspects. (1) Initial Condition: The compressive strength of concrete commences at 0 at $t$ = 0, underscoring the necessity for the curve model’s domain to encompass this point. (2) Bounded Growth: Within the first seven days post pouring, the strength undergoes a rapid ascent, stabilizing thereafter approximately after 28 days. This suggests that the curve model must be bounded, ensuring function values remain within a defined range as the input variable (e.g., time/age) varies, eschewing unbounded growth or excessive oscillations. (3) Increasing and Convex Functionality: The temporal evolution of concrete’s compressive strength adheres to both an increasing and convex function profile, wherein the first derivative remains positive (indicating growth) and the second derivative is negative (signifying decelerating growth). (4) Multifactorial Complexity: Many factors, such as mix ratio, temperature and humidity, early shrinkage, and water evaporation, have significant impacts on the compressive strength of concrete during its early age. For example, when water evaporates too quickly or unevenly, shrinkage cracks are prone to form on the surface of the concrete. These cracks will subsequently reduce the compressive strength of the concrete. The combined influence of numerous factors often leads to a relatively large dispersion in the strength data of concrete during its early age. Consequently, a solitary curve model struggles to accurately capture the intricate age–strength relationship.

Hence, the concept of a combined model, integrating two or more curves, emerges as a potential solution to more precisely delineate the intricate strength growth pattern of concrete during its ultra-early ages. Each constituent curve within this combined model must adhere to the aforementioned prerequisites. (1) It must be an increasing function, reflecting the continuous enhancement of concrete strength over time. (2) It must exhibit convexity, indicating that the rate of strength increase diminishes as time elapses (first derivative > 0, second derivative < 0). (3) Its domain must span $t$ = 0, ensuring applicability from the inception of the concrete’s aging process. By amalgamating curves that fulfill these criteria, a more holistic and accurate portrayal of the complex dynamics of concrete strength development during ultra-early ages can be achieved.

To this end, a combined two-parameter model (CTM) is developed in this work as follows:

$$S_t=p_0\cdot {\displaystyle \frac{t}{t+1}}+p_1\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(11)

where $p_0$ and $p_1$ are the two unknown fitting parameters. Equation (11) is a combination of two curves, and $p_0$ and $p_1$ can be regarded as the weight coefficients of the two curves. Both of the two curves that compose this model satisfy the characteristics of being increasing functions as well as convex functions. Specifically, the first and second derivatives of these two curves are as follows.

$${({\displaystyle \frac{t}{t+1}})}^{\prime }={\displaystyle \frac{1}{{(t+1)}^2}}>0$$

(12)

$${({\displaystyle \frac{t}{t+1}})}^{″}={\displaystyle \frac{-2}{{(t+1)}^3}}<0$$

(13)

$${({\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}})}^{\prime }={\displaystyle \frac{4e^{2t}}{{(e^{2t}+1)}^2}}>0$$

(14)

$${({\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}})}^{″}={\displaystyle \frac{8e^{2t}-8e^{6t}}{{(e^{2t}+1)}^4}}<0$$

(15)

In Equations (12)–(15), the superscripts ′ and ″ are used to denote the first-order and second-order derivatives, respectively. Furthermore, the proposed CTM is a bounded function, since $S_t$ = 0 when $t$ = 0 and $S_t=p_0+p_1$ when $t\to \infty$. Through the above analysis of the CTM’s characteristics, including the fact that the two curves comprising the model both exhibit the properties of being increasing functions and convex functions, it can be concluded that this newly established model aligns with the natural law governing the variation of concrete strength over time. Moreover, this novel model boasts a streamlined structure with merely two unknown parameters, allowing for curve fitting with minimal experimental data. Theoretically, just three or more sets of compressive strength test data spanning different ages suffice to accurately assess the concrete’s compressive strength. Despite its concise nature with only two fitting parameters, this model surpasses the aforementioned dual or multi-parameter models in terms of fitting accuracy, offering a more precise representation of the data.

Next, the calculation method of the unknown fitting parameters is described as follows. Without loss of generality, it is assumed that $n$ ($n\ge 3$) groups of compressive strengths at different times are measured in the concrete strength test. For these $n$ groups of test data, the linear system of equations can be obtained from Equation (11) as follows:

$$\eta =\mathrm{\Omega }\cdot \zeta$$

(16)

$$\eta =\left\{\begin{array}{c}{S}_{t_1}\\ ⋮\\ S_{t_n}\end{array}\right\},\zeta =\left\{\begin{array}{c}{p}_0\\ p_1\end{array}\right\}$$

(17)

$$\mathrm{\Omega }=\left[\begin{array}{cc}{\displaystyle \frac{t_1}{t_1+1}}& {\displaystyle \frac{e^{t_1}-e^{-t_1}}{e^{t_1}+e^{-t_1}}}\\ ⋮& ⋮\\ {\displaystyle \frac{t_n}{t_n+1}}& {\displaystyle \frac{e^{t_n}-e^{-t_n}}{e^{t_n}+e^{-t_n}}}\end{array}\right]$$

(18)

where $S_{t_1},\cdots ,S_{t_n}$ denote the measured values of compressive strength for the time $t_1,\cdots ,t_n$, $\mathrm{\Omega }$ is the coefficient matrix of the linear system. From Equation (16), the desired fitting parameters $p_0$ and $p_1$ can be obtained with the following equation:

$${\zeta }_p={({\mathrm{\Omega }}^T\mathrm{\Omega })}^{-1}{\mathrm{\Omega }}^T\eta$$

(19)

in which the superscript “T” represents the matrix transposition operation. When the fitting parameters $p_0$ and $p_1$ are determined from ${\zeta }_p$ by Equation (19), the predicted compressive strength can be further calculated by the following:

$${\eta }_p=\mathrm{\Omega }{\zeta }_p$$

(20)

where:

$${\eta }_p={(S_{t_1}^p,\cdots ,S_{t_n}^p)}^T$$

(21)

in which $S_{t_1}^p,\cdots ,S_{t_n}^p$ denote the calculated values of compressive strength for the time $t_1,\cdots ,t_n$ by the proposed combined curve model. The difference between the predicted value and the measured value, which quantifies their discrepancy, is referred to as the comparative error:

$$e_{t_n}=\left|S_{t_n}^p-S_{t_n}\right|/S_{t_n}$$

(22)

where $e_{t_n}$ denotes the comparative error for the time $t_n$. Ultimately, to assess the fitting accuracy of the evaluation model, the average, standard deviation, and coefficient of variation of the errors between predicted and measured values are utilized. Specifically, the average value of these errors is defined as follows:

$$\overline{e}={\displaystyle \frac{e_{t_1}+\cdots +e_{t_n}}{n}}$$

(23)

where $\overline{e}$ denote the mean of the errors. The standard deviation of the errors is defined as follows:

$$\sigma =\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(e_{t_i}-\overline{e})}^2}}{n-1}}}$$

(24)

where $\sigma$ denotes the standard deviation of the errors. The coefficient of variation of the errors is defined as follows.

$$c_v=\sigma /\overline{e}$$

(25)

As the values of the average, standard deviation, and coefficient of variation of errors decrease, so too does the discrepancy between predicted and measured outcomes, indicating a heightened fitting accuracy for the respective evaluation model. In the upcoming section, the fitting precision of the novel model will be contrasted with that of several existing models, utilizing these three statistical metrics, with the objective of emphasizing the superior performance and advantages of the new model.

4. Case Study

Examples 1–11 are utilized to compare the performance of the proposed CTM with the existing models in predicting the compressive strength of concrete. Given that the proposed model is specifically designed for assessing the strength of concrete during the ultra-early-age phase, models from the aforementioned existing ones that rely on the 28-day test data are excluded from the comparative study. Additionally, since the exponential model can be transformed into a logarithmic model by taking logarithms, the three types of existing models in the comparative study are the cubic polynomial model, hyperbolic model, and logarithmic model.

Example 1: In the literature [34], Zhao et al. produced 100 mm × 100 mm × 100 mm concrete cube specimens with a compressive strength grade of C65. The specific mix proportion of concrete is 449 kg of cement, 693 kg of sand, 1129 kg of stone, 149 kg of water, and 21.9 kg of admixture. These specimens were cured in standard environments and their compressive strength was measured at different ages of 1, 2, 3, 7, 14, and 28 days. The loading speed in the compressive strength experiment is taken as 0.8–1 Mpa/s. Four test blocks are measured for each age and the average compressive strength of these four test blocks is used as the final value for the corresponding age. The results obtained from the compression test can be found in the literature [34] or

Table 1. The experimental results of Examples 1–6.

Index	Compressive Strength (CS)
(a) Example 1	Age (day)	1	2	3	7	14	28
	CS (MPa)	29.7	51.3	58.9	67.3	73.6	76.1
(b) Example 2	Age (day)	1	3	7	14	28
	CS (MPa)	31.0	49.7	54.0	59.8	68.6
(c) Example 3	Age (day)	1	1.5	2	2.5	3	3.5	4
	CS (MPa)	2.8	29.9	36.9	42.2	46.2	50.2	52.5
	Age (day)	5	6	7	14	28	60	90
	CS (MPa)	55.2	57.4	58.5	65.5	74.9	81	82.5
(d) Example 4	Age (day)	1	1.5	2	2.5	3	3.5	4
	CS (MPa)	2.4	20.8	28.4	36.8	40.1	42.7	45.3
	Age (day)	5	6	7	14	28	60	90
	CS (MPa)	50.3	53.4	55.6	62	69.2	76.3	78.3
(e) Example 5	Age (day)	1	2	3	5	7	11	14
	CS (MPa)	24.0	32.1	39.6	45.7	46.8	47.6	49.9
	Age (day)	28	40
	CS (MPa)	56.8	58.4
(f) Example 6	Age (day)	3	5	7	14	28	60
	CS (MPa)	35.4	49.8	51.6	53.8	55.3	60.3

(a). Based on the test results, Equations (26)–(29) show the specific fitting equations corresponding to the polynomial model, hyperbolic model, logarithmic model, and CTM in sequence. Figure 1a shows the fitting curves corresponding to these four evaluation models more intuitively.

Table 2. The mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength of Examples 1–6.

Index		Polynomial Model	Hyperbolic Model	Logarithmic Model	CTM
(a) Example 1	Mean fitting error	0.0864	0.0487	0.1023	0.0377
	Standard deviation	0.0802	0.0676	0.0999	0.0326
	Coefficient of variation	0.9290	1.3885	0.9767	0.8657
	Predicted 120-day CS (MPa)	19,807	78.93	100.57	79.65
(b) Example 2	Mean fitting error	0.0515	0.0808	0.0413	0.0401
	Standard deviation	0.0436	0.0570	0.0436	0.0251
	Coefficient of variation	0.8476	0.7050	1.0546	0.6263
	Predicted 120-day CS (MPa)	12,599	70.57	84.05	66.12
(c) Example 3	Mean fitting error	0.8724	0.4643	0.6597	0.2286
	Standard deviation	2.8772	0.9501	2.1933	0.6462
	Coefficient of variation	3.2979	2.0460	3.3246	2.8269
	Predicted 120-day CS (MPa)	191.33	84.65	95.37	79.18
(d) Example 4	Mean fitting error	0.8697	0.4523	0.6176	0.0810
	Standard deviation	2.7353	0.9177	1.9408	0.1474
	Coefficient of variation	3.1448	2.0286	3.1422	1.8193
	Predicted 120-day CS (MPa)	194.68	80.55	91.08	74.26
(e) Example 5	Mean fitting error	0.0753	0.0574	0.0504	0.0421
	Standard deviation	0.0670	0.0492	0.0410	0.0239
	Coefficient of variation	0.8894	0.8569	0.8131	0.5690
	Predicted 120-day CS (MPa)	1625.4	59.66	70.01	56.04
(f) Example 6	Mean fitting error	0.0592	0.0486	0.0722	0.0467
	Standard deviation	0.0500	0.0362	0.0637	0.0290
	Coefficient of variation	0.8443	0.7454	0.8822	0.6211
	Predicted 120-day CS (MPa)	905.7	60.7	66.4	60.2

(a) provides additional insights into the performance of various models, detailing their mean fitting error, standard deviation, coefficient of variation, and the 120-day predicted compressive strength. Notably, the proposed combined model stands out as the most accurate, with the lowest values across all these metrics. Specifically, the mean fitting error follows a descending order from logarithmic, polynomial, and hyperbolic models to the combined model. When compared to the second-best performing hyperbolic model, the new model impressively reduces the mean fitting error by 23% and the standard deviation by 52%, underscoring its substantial improvement in fitting precision. Furthermore, an examination of the predicted 120-day compressive strength values in Table 2(a) reveals that the combined model exhibits the highest reliability in forecasting the concrete’s later compressive strength, thereby demonstrating its superiority in predictive capability.

$$S_t=0.0177t^3-0.8546t^2+11.8269t+25.4589$$

(26)

$$S_t={\displaystyle \frac{t}{0.0160+0.0125t}}$$

(27)

$$S_t=38.507+12.9631\cdot \ln (t)$$

(28)

$$S_t=96.0159\cdot {\displaystyle \frac{t}{t+1}}-15.5774\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(29)

Example 2: In the literature [35], Hu et al. designed C55 concrete with a mix ratio of cement 260 kg/m³, slag powder 90 kg/m³, limestone 90 kg/m³, sand 693 kg/m³, crushed stone 1131 kg/m³, water 140 kg/m³, and water reducer 5.28 kg/m³. They conducted compressive strength tests on concrete and obtained compressive strength values for ages of 1, 3, 7, 14, and 28 days. The results obtained from their compression test can be found in the literature [35] or Table 1(b). Based on the test results, Equations (30)–(33) show the specific fitting equations corresponding to the four models in sequence. Figure 1b shows the fitting curves corresponding to these four evaluation models more intuitively. Table 2(b) extends the analysis by presenting the mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength for a range of models. Notably, the proposed combined model emerges as the most precise, with the lowest values for all three accuracy metrics. The mean fitting error is arranged in ascending order, from the hyperbolic model, through the polynomial model, to the logarithmic model, with the combined model achieving the best performance. When benchmarked against the next-best logarithmic model, the new model achieves a 3% reduction in mean fitting error and a significant 42% decrease in standard deviation, highlighting its substantial improvement in fitting accuracy. Additionally, an assessment of the 120-day predicted compressive strength values in Table 2(b) underscores the combined model’s superior reliability in predicting the concrete’s future compressive strength, reinforcing its position as the most reliable predictive tool.

$$S_t=0.0111t^3-0.5257t^2+7.4887t+26.6466$$

(30)

$$S_t={\displaystyle \frac{t}{0.0249+0.0140t}}$$

(31)

$$S_t=33.6276+10.5328\cdot \ln (t)$$

(32)

$$S_t=72.1989\cdot {\displaystyle \frac{t}{t+1}}-5.4835\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(33)

Example 3: In the literature [36], He designed C55 concrete with a mix ratio of cement 353 kg/m³, fly ash 116 kg/m³, sand 730 kg/m³, crushed stone 1121 kg/m³, water 142 kg/m³, and additive agent 5.159 kg/m³. He conducted compressive strength tests on the concrete specimens under standard curing and obtained compressive strength values for ages of 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, 7, 14, 28, 60, and 90 days. The results obtained from their compression test can be found in the literature [36] or Table 1(c). Based on the test results, Equations (34)–(37) show the specific fitting equations corresponding to the four models in sequence. Figure 1c shows the fitting curves corresponding to these four evaluation models more intuitively. Table 2(c) provides a comprehensive overview of the performance metrics for various models, including their mean fitting error, standard deviation, coefficient of variation, and the 120-day predicted compressive strength. Notably, the CTM excels in all these aspects, achieving the lowest values for mean fitting error, standard deviation, and coefficient of variation. The ranking of mean fitting error, from highest to lowest, is as follows: polynomial model, hyperbolic model, logarithmic model, and the combined model. When compared to the second-best performing hyperbolic model, the CTM demonstrates a remarkable improvement, reducing the mean fitting error by 51% and the standard deviation by 32%. This underscores the significant enhancement in fitting accuracy offered by the new model. Furthermore, an analysis of the 120-day predicted compressive strength values in Table 2(c) confirms that the developed combined model possesses the highest reliability in predicting the future compressive strength of concrete, emphasizing its superiority in predictive capability.

$$S_t=0.0005t^3-0.0837t^2+3.9424t+29.3688$$

(34)

$$S_t={\displaystyle \frac{t}{0.0643+0.0113t}}$$

(35)

$$S_t=25.9815+14.4928\cdot \ln (t)$$

(36)

$$S_t=141.2717\cdot {\displaystyle \frac{t}{t+1}}-60.9282\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(37)

Example 4: In the literature [36], the pumped concrete was sampled at the construction site, another set of curing specimens with the site construction environment conditions was made. The compressive strength tests of field curing specimens were conducted and the results can be found in the literature [36] or Table 1(d). Equations (38)–(41) show the specific fitting equations corresponding to the four models. Figure 1d shows the fitting curves corresponding to these four evaluation models. Table 2(d) presents a detailed comparison of various models based on their mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength. Notably, the CTM stands out as the most accurate, with the lowest values across all evaluation metrics. Specifically, the mean fitting error is arranged in descending order as follows: polynomial model, hyperbolic model, logarithmic model, and the combined model. When compared to the next-best performing hyperbolic model, the CTM demonstrates a substantial improvement, with a reduction of 82% in mean fitting error and an even more impressive 83% decrease in standard deviation. This remarkable performance underscores the exceptional prediction accuracy of the new model, reinforcing its superiority in forecasting the compressive strength of concrete over an extended period.

$$S_t=0.0005t^3-0.0877t^2+4.0999t+23.2710$$

(38)

$$S_t={\displaystyle \frac{t}{0.0787+0.0118t}}$$

(39)

$$S_t=20.0597+14.8359\cdot \ln (t)$$

(40)

$$S_t=143.3004\cdot {\displaystyle \frac{t}{t+1}}-67.8507\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(41)

Example 5: In the literature [25], Zhao designed C50 concrete with a mix ratio of cement 400 kg/m³, fly ash 115 kg/m³, sand 582 kg/m³, crushed stone 1166 kg/m³, water 170 kg/m³, and additive agent 4.3 kg/m³. He conducted compressive strength tests on the concrete specimens under standard curing and obtained compressive strength values for ages of 1, 2, 3, 5, 7, 11, 14, 28, and 40 days. The results obtained from their compression test can be found in the literature [25] or Table 1(e). Equations (42)–(45) present the specific fitting equations and Figure 1e shows the corresponding fitting curves. From Table 2(e), one can see that the mean fitting error, standard deviation, and coefficient of variation of the proposed combined model are all the smallest. Moreover, the 120-day compressive strength predicted by the CTM is the closest to the designed strength grade of 50 MPa.

$$S_t=0.0019t^3-0.1449t^2+3.607t+26.0318$$

(42)

$$S_t={\displaystyle \frac{t}{0.0332+0.0165t}}$$

(43)

$$S_t=27.2818+8.9263\cdot \ln (t)$$

(44)

$$S_t=68.5525\cdot {\displaystyle \frac{t}{t+1}}-11.9438\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(45)

Example 6: In the literature [37], Chang et al. designed C55 concrete with a mix ratio of cement 486 kg/m³, additive 7.71 kg/m³, sand 642 kg/m³, crushed stone 1094 kg/m³, water 163 kg/m³, and fly ash 65 kg/m³. They conducted compressive strength tests on concrete and obtained compressive strength values for ages of 3, 5, 7, 14, 28, and 60 days. The results obtained from their compression test can be found in the literature [37] or Table 1(f). Equations (46)–(49) show the specific fitting equations and Figure 1f gives the corresponding fitting curves for the four evaluation models. Table 2(f) provides a comprehensive assessment of various models in terms of their mean fitting error, standard deviation, coefficient of variation, and the 120-day predicted compressive strength of concrete. Notably, the CTM emerges as the top performer, achieving the lowest values for mean fitting error, standard deviation, and coefficient of variation. This exceptional performance underscores the precision of the CTM in fitting the data. Consequently, the 120-day compressive strength predictions made by the CTM in Table 2(f) also possess the highest level of reliability, making it a trusted choice for predicting the future strength of concrete.

$$S_t=0.0014t^3-0.1299t^2+3.4175t+31.0187$$

(46)

$$S_t={\displaystyle \frac{t}{0.0308+0.0162t}}$$

(47)

$$S_t=34.923+6.5669\cdot \ln (t)$$

(48)

$$S_t=90.393\cdot {\displaystyle \frac{t}{t+1}}-29.4284\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(49)

Examples 7–10: The data for Examples 7–10 are all from the compressive strength tests of fly ash ceramsite concrete with different carbon fiber contents in reference [38]. These examples will further illustrate the unique advantage of the proposed model in accurately predicting the later strength of concrete based on only a small amount of experimental data. In the literature [38], Wang et al. carried out the compressive strength test of fly ash ceramsite concrete with different carbon fiber content. The specific mix proportion of concrete is 350 kg of cement, 650 kg of sand, 575 kg of stone, 140 kg of water, 135 kg of fly ash, and 5.5 kg of admixture. For these examples, the ratios of carbon fiber mass to cement mass are 0%, 0.5%, 0.8%, and 1%, sequentially. The 150 mm × 150 mm × 150 mm cube specimens were made under standard curing for compressive strength testing at different ages of 7, 14, and 28 days. The compression test results of these examples can be found in the literature [38] or

Table 3. The experimental results of Examples 7–10.

Index	Age (Day)	7	14	28
(a) Example 7	CS (MPa)	18.1	29.2	31.6
(b) Example 8	CS (MPa)	19.9	32.2	35
(c) Example 9	CS (MPa)	19.2	30.1	32.4
(d) Example 10	CS (MPa)	17.8	27.3	30.1

. Using the data in Table 3, Equations (50)–(65) show the specific fitting equations corresponding to the polynomial model, hyperbolic model, logarithmic model, and CTM for Examples 7–10 in sequence. Figure 2 shows the fitting curves corresponding to the four models more intuitively.

Table 4. The mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength of Examples 7–10.

Index		Polynomial Model	Hyperbolic Model	Logarithmic Model	CTM
(a) Example 7	Mean fitting error	0.9461	0.0657	0.0751	0.0385
	Standard deviation	1.1260	0.0412	0.0271	0.0128
	Coefficient of variation	1.1902	0.6268	0.3604	0.3321
	Predicted 120-day CS (MPa)	−5324.5	38.37	47.22	36.64
(b) Example 8	Mean fitting error	0.6628	0.0652	0.0744	0.0374
	Standard deviation	0.7766	0.0408	0.0269	0.0124
	Coefficient of variation	1.1717	0.6255	0.3620	0.3325
	Predicted 120-day CS (MPa)	−2731.5	42.68	52.43	40.59
(c) Example 9	Mean fitting error	0.6626	0.0615	0.0714	0.0371
	Standard deviation	0.9323	0.0388	0.0257	0.0126
	Coefficient of variation	1.4072	0.6316	0.3595	0.3402
	Predicted 120-day CS (MPa)	−6504	38.70	47.70	37.35
(d) Example 10	Mean fitting error	0.6986	0.0508	0.0605	0.0265
	Standard deviation	0.9387	0.0320	0.0224	0.0094
	Coefficient of variation	1.3437	0.6295	0.3705	0.3549
	Predicted 120-day CS (MPa)	−6170	36.08	44.13	34.41

extends the comparison by presenting the mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength of various models. From Table 4, one can see that the mean fitting error, standard deviation, and coefficient of variation of the CTM are all the smallest. Compared to the suboptimal model, the new model shows a reduction of 41% and 69% in mean fitting error and standard deviation in Example 7, 43% and 70% in Example 8, 40% and 68% in Example 9, and 48% and 71% in Example 10, respectively. These results show that the new model significantly improves the fitting accuracy. Thus the 120-day compressive strength of concrete predicted by the CTM in Table 4 also has the highest reliability. Additionally, it is noteworthy that for Examples 7–10, the prediction results of the cubic polynomial model for the 120th day are all negative, indicating that the performance of the polynomial model is highly unstable. Despite having more fitting parameters than the two-parameter model, it fails to enhance the prediction accuracy.

$$S_t=-0.0029t^3-0.0313t^2+1.5t+8$$

(50)

$$S_t={\displaystyle \frac{t}{0.1834+0.0245t}}$$

(51)

$$S_t=0.6004+9.7382\cdot \ln (t)$$

(52)

$$S_t=154.1111\cdot {\displaystyle \frac{t}{t+1}}-116.1937\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(53)

$$S_t=-0.001t^3-0.0938t^2+2.5t+6$$

(54)

$$S_t={\displaystyle \frac{t}{0.1692+0.022t}}$$

(55)

$$S_t=0.2878+10.8923\cdot \ln (t)$$

(56)

$$S_t=172.1369\cdot {\displaystyle \frac{t}{t+1}}-130.1274\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(57)

$$S_t=-0.0039t^3-0.00t^2+2t+6$$

(58)

$$S_t={\displaystyle \frac{t}{0.165+0.0245t}}$$

(59)

$$S_t=2.1048+9.5218\cdot \ln (t)$$

(60)

$$S_t=150.783\cdot {\displaystyle \frac{t}{t+1}}-112.1832\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(61)

$$S_t=-0.0039t^3+0.0313t^2+1.0t+10$$

(62)

$$S_t={\displaystyle \frac{t}{0.1846+0.0262t}}$$

(63)

$$S_t=1.6514+8.8726\cdot \ln (t)$$

(64)

$$S_t=139.1427\cdot {\displaystyle \frac{t}{t+1}}-103.587\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(65)

Example 11: An early-age compressive strength test of ultra-high strength concrete, as shown in Figure 3, was conducted by the authors to further demonstrate the proposed prediction model. The specific mix proportion of ultra-high strength concrete is 2200 kg/m³ of powder, 158 kg/m³ of steel fiber, and 202 kg/m³ of water. Several test blocks with size of 100 mm × 100 mm × 100 mm were made and cured under standard conditions. The curing temperature is about (20 ± 2) °C and the relative humidity is about 95%. Three test blocks were used for each test and the average value was taken as the compressive strength for that age. The test results of compressive strength are given in

Table 5. The test results of Example 11.

Age (Day)	0.5	1	4	7
CS (MPa)	41.2	68.5	113.9	121.8

.

Using the data of Table 5, Equations (66)–(69) show the specific fitting equations corresponding to the four models in sequence. Figure 4 shows the fitting curves corresponding to these models more intuitively.

Table 6. The mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength of Example 11.

Example 11	Polynomial Model	Hyperbolic Model	Logarithmic Model	CTM
Mean fitting error	0.0000	0.0290	0.0485	0.0180
Standard deviation	0.0000	0.0227	0.0169	0.0055
Coefficient of variation	1.1207	0.7825	0.3485	0.3051
Predicted 120-day CS (MPa)	2179.2 × 103	141.3	214.9	141.0

summarizes the key performance metrics for various models, including the mean fitting error, standard deviation, coefficient of variation, and the 120-day predicted compressive strength. As can be seen from Table 6, while the mean and standard deviation of the fitting error of the polynomial model are both zero, the model’s prediction of the concrete’s later-stage strength is significantly distorted. Excluding the polynomial model, it can be found that the mean fitting error, standard deviation, and coefficient of variation of the CTM are all the smallest. Compared with the hyperbolic model, the mean fitting error and standard deviation of the new model are reduced by 38% and 76%, respectively. Combining both the fitting precision and prediction outcomes, it is reiterated that the proposed model exhibits the best performance among all the models.

$$S_t=1.4143t^3-19.0548t^2+80.7071t+5.4333$$

(66)

$$S_t={\displaystyle \frac{t}{0.0079+0.007t}}$$

(67)

$$S_t=65.8094+31.1332\cdot \ln (t)$$

(68)

$$S_t=150.0693\cdot {\displaystyle \frac{t}{t+1}}-7.756\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(69)

5. Discussion

(1) According to Table 2, Table 4 and Table 6, Figure 5 provides statistics on the relative values of the mean and standard deviation of fitting errors for Examples 1–11 using the four models. The relative values are calculated by taking the mean and standard deviation of CTM as 1 for each example, and then proportionally scaling the mean and standard deviation for the other models accordingly. As can be seen from Figure 5, the polynomial model is very unstable, especially for examples with little experimental data. The fitting results obtained by the polynomial model have large errors. As stated before, the predicted values of the 120-day compressive strength by the polynomial model are completely distorted. For some examples, the hyperbolic model outperforms the logarithmic model, while for others, the logarithmic model surpasses the hyperbolic model. Overall, the CTM exhibits the most stable performance and is able to achieve the most reliable prediction results.

(2) The proposed CTM can also be extended to predict the tensile strength of concrete. References [39,40] provide two sets of experimental data on the splitting tensile strength of concrete with strength grades of C30 and C40, respectively. Additionally, reference [41] offers two sets of experimental data, one on the splitting tensile strength and the other on the axial tensile strength, for concrete with a strength grade of C60. These four sets of experimental data are used as Examples 12–15 to illustrate the performance of the proposed model in predicting tensile strength. Figure 6 presents the experimental data points for these four examples along with the fitting curves of various models.

Table 7. The mean fitting error, standard deviation, coefficient of variation, and 120-day predicted tensile strength of Examples 12–15.

Index		Polynomial Model	Hyperbolic Model	Logarithmic Model	CTM
(a) Example 12	Mean fitting error	0.0218	0.0274	0.0593	0.0257
	Standard deviation	0.021	0.0194	0.0408	0.0150
	Coefficient of variation	0.9625	0.7086	0.6888	0.5830
	Predicted 120-day tensile strength (MPa)	612.7328	2.5254	3.1398	2.4094
(b) Example 13	Mean fitting error	0.0725	0.0668	0.1060	0.0264
	Standard deviation	0.0711	0.0655	0.1079	0.0139
	Coefficient of variation	0.9806	0.9794	1.0182	0.5272
	Predicted 120-day tensile strength (MPa)	1010.9	3.665	4.667	3.396
(c) Example 14	Mean fitting error	0.0374	0.0594	0.0946	0.0239
	Standard deviation	0.0326	0.0907	0.0456	0.0153
	Coefficient of variation	0.8720	1.5264	0.4826	0.6404
	Predicted 120-day tensile strength (MPa)	54.1620	4.4249	5.1819	4.5620
(d) Example 15	Mean fitting error	0.0037	0.1072	0.0709	0.0459
	Standard deviation	0.0034	0.0982	0.0339	0.0229
	Coefficient of variation	0.9035	0.9163	0.4780	0.4991
	Predicted 120-day tensile strength (MPa)	26.2071	3.7467	4.3225	3.9676

displays the evaluation metrics for various models, specifically focusing on the mean fitting error, standard deviation, coefficient of variation, and the predicted tensile strength values at the 120-day mark.

According to Table 7, Figure 7 provides statistics on the relative values of the mean and standard deviation of fitting errors for Examples 12–15 using the four models. The relative values are determined by setting the mean and standard deviation of the CTM to 1 for each example, and subsequently scaling the mean and standard deviation of the other models in proportion to this base. It can be observed from Figure 7 that, when considering both the mean error and standard deviation, the logarithmic model performs the worst, while the CTM performs the best. Further examination of Table 7 reveals that, despite the polynomial model having a lower mean error and standard deviation than the CTM in a very small number of cases, the polynomial model’s prediction at the 120th day are completely distorted. This reinforces the conclusion that the CTM continues to demonstrate the best performance among all the models.

(3) According to the creation logic of the CTM presented in this work, Equation (70) offers a new model that is constructed by combining three different curves as follows.

$$S_t=p_0\cdot {\displaystyle \frac{t}{t+1}}+p_1\cdot {\displaystyle \frac{t}{t+2}}+p_2\cdot {\displaystyle \frac{e^t-e^{-t}}{e^t+e^{-t}}}$$

(70)

Using the model of Equation (70),

Table 8. The mean fitting error, standard deviation, coefficient of variation, and 120-day predicted compressive strength obtained by Equation (70) for Examples 7–11.

Results by the Model of Equation (70)	Example 7	Example 8	Example 9	Example 10	Example 11
Mean fitting error	0.0000	0.0000	0.0000	0.0000	0.0061
Standard deviation	0.0000	0.0000	0.0000	0.0000	0.0015
Coefficient of variation	0.2415	0.2856	0.2790	0.2775	0.2529
Predicted 120-day CS (MPa)	31.3711	34.9432	32.0957	30.9492	134.2275

presents the error analysis of Examples 7–11, encompassing the mean fitting error, standard deviation, and coefficient of variation, as well as the 120-day predictive compressive strength estimates. By comparing Table 8 with Table 4 and Table 6, it can be observed that the mean and standard deviation of the model fitting errors of Equation (70) are both smaller than those of CTM. This indicates that, generally, the prediction accuracy can be improved by increasing the number of curves involved in the combination. Especially for Examples 7–10, the mean and standard deviation of the fitting errors for the model based on Equation (70) are nearly zero. This is because the number of fitting parameters in Equation (70) is exactly equal to the number of experimental points. This perfect match leads to a highly accurate fit, with minimal deviation from the experimental data. The versatility of this new model, which is composed of three curves, can be further explored in future research. This could lead to the development of more robust and reliable prediction models for a wide range of fields and applications.

6. Conclusions

This work presents a novel CTM specifically designed to assess the evolving compressive strength of concrete during its ultra-early age. Comprised of two curves and featuring just two unknown parameters, this model efficiently fits experimental data, enabling the derivation of a compressive strength–age relationship for concrete with minimal data requirements. Notably, the new CTM surpasses single-curve models in terms of fitting accuracy. Key insights drawn from case studies include the following. (1) The polynomial model is limited to fitting early-age concrete data and lacks predictive capability for later stages. (2) The logarithmic model’s predictions for later-age concrete strength significantly diverge from design specifications. (3) The hyperbolic model’s fitting accuracy varies widely, with instances of performance inferior even to the logarithmic model. (4) The proposed CTM consistently demonstrates superior fitting accuracy across cases, underscoring its stability and versatility. Notably, its standard deviation of fitting errors is approximately 30–80% lower than the second-best model, ensuring predictions for later-age concrete strength align more closely with design grades. (5) The model’s applicability extends beyond compressive strength, successfully predicting tensile strength during ultra-early ages, highlighting its unique strength in accurately characterizing concrete strength at various stages based on minimal experimental data. Collectively, these findings underscore the broad potential of this innovative combined curve model in evaluating the temporal behavior of concrete strength, opening avenues for its application in diverse fields.