Method for Determining the Fracture Parameters of Fully Graded Dam Concrete

Abstract

This paper describes a method for determining the initiation and unstable toughness of fully graded concrete of arbitrary specimen size. The method first predicts the initiation and peak loads of concrete specimens of any size, as well as crack length-to-height ratios based on the fracture test results of concrete specimens with limited sizes or crack length-to-height ratios. Then, combined with the fracture extreme theory, the fracture toughness of concrete with varying size or crack length-to-height ratios is determined. Finally, in order to verify the applicability of the method, it is used to calculate the fracture toughness of small aggregate concrete and fully graded concrete with different sizes or crack length-to-height ratios, and its prediction accuracy is evaluated through indices such as mean absolute percentage error, root mean square error and reliability index a15. The results show that the proposed method can meet the needs of practical engineering applications and can provide theoretical basis for the optimization of the fracture test method of fully graded concrete and the determination of fracture parameters in crack stability or propagation analysis.

1. Introduction

In order to ensure the safe operation of dam structures, it is necessary to study the mechanical parameters of dam concrete under actual service conditions. Existing studies have shown that factors such as environment [1,2,3], concrete raw materials [4,5], and curing conditions [6] affect the toughness, strength, and elastic modulus of dam concrete. Among all of the above mechanical properties, fracture toughness is an important index to evaluate the crack resistance of dam concrete [7]. Generally, it can be determined by the wedge-splitting method [5,8,9] or the three-point bending-beam method [10,11,12]. Based on the double-K fracture theory proposed by Xu et al. [13,14,15], the National Development and Reform Commission of China [16] and RILEM [17] developed a concrete fracture test standard. The double-K fracture theory uses the initiation and unstable toughness to describe the whole process of concrete structure crack from initiation and crack propagation to unstable failure. Existing research shows that the double-K fracture toughness of concrete measured by this test may have a size effect [18,19,20,21], and the larger the maximum particle size of concrete aggregate, the larger the minimum specimen size required to determine the stable fracture toughness. For fully graded concrete with a maximum aggregate size of 150 mm, when the stable fracture toughness is determined by the wedge-splitting method, the required minimum specimen height should be 1.5 m [7,21,22]. The fracture testing of large-size specimens is associated with problems such as high equipment requirements, difficult testing, and inaccurate testing [22,23], so it is difficult to carry out normal tests at engineering sites. If a method can be found to determine the double-K fracture toughness of any size of specimen based on the fracture test results of smaller concrete specimens, it can provide a basis for the optimization of the fracture test of fully graded concrete. According to the double-K fracture toughness calculation formula, the initiation toughness is related to the initiation load and the initial crack length, whereas the calculation of the unstable toughness requires the peak load and the effective crack length. Therefore, the key is to solve the problem of how to determine the initiation load, peak load and effective crack length of specimens of any size through the fracture test results of smaller specimens.

In order to predict the fracture failure of quasi-brittle material specimens, scholars have proposed different theoretical models. The most famous ones are the boundary effect model proposed by Hu et al. [24,25,26] and the size effect model proposed by Bažant et al. [27,28,29]. Based on the boundary effect model, Guan et al. [30] established a linear formula between the peak load of concrete with different sizes or α_c and tensile strength, where α_c is the crack length-to-height ratio. Furthermore, Yao and Guan et al. [31,32,33] proposed the concept of cracking strength and established a linear expression between the initiation load and the cracking strength of concrete specimens when the size or α_c changes. Guan et al. [10,34] also established the relationship between fracture toughness and the failure curve based on the improved boundary effect model. The above initiation and peak load prediction methods based on the boundary effect model are currently used for small aggregate concrete. For fully graded concrete, Gao et al. [7] established a linear relationship between the nominal initiation strength, nominal unstable strength and the height of the specimen based on the type 2 size effect model. By introducing the equivalent geometric parameters, the prediction formula of initiation and peak loads of concrete specimens is proposed when the size or α_c changes, so as to realize the prediction of initiation and peak loads of specimens with any size or α_c. Although the basic assumptions of the boundary effect and the size effect model are different, the predicted results obtained by the two are basically consistent. In this study, the size effect model is selected as the method to determine the initiation and peak loads of concrete specimens when the size or α_c changes.

After determining the initiation and peak loads of any size of specimen, the initiation toughness can be directly obtained by the initiation load and the initial crack length, whereas the calculation of the unstable toughness requires the critical crack length (a_c). Because the predicted fracture results based on the size effect model cannot yield a_c, it needs to be obtained by establishing and solving the equations of the critical state of concrete specimens. Based on a fictitious crack model, Wu et al. [35,36] established the equilibrium equation of force and moment in the midspan section of a three-point bending beam and obtained a_c with the flexural tensile strength of concrete as the known quantity. In order to determine the fracture parameters of concrete more easily and reliably, Qing et al. proposed a fracture extreme theory that can determine a_c and other fracture parameters only based on the peak load of fracture test [37,38,39,40,41,42,43,44,45,46,47]. The theory assumes that the partial derivative of the F_H to crack length is continuous when F_H reaches the peak load during the fracture process of concrete, and the derivative of F_H to crack length is zero at the extreme point. Furthermore, based on the relationship between F_H and a under the peak load state, combined with several other equations, the a_c and other fracture parameters of specimens of different sizes can be determined. The accuracy of fracture extreme value theory for predicting fracture parameters of concrete has been widely verified. Accordingly, a_c can be easily and reliably determined by fracture extreme value theory with only the peak load. Therefore, after the peak load is determined, the fracture extreme theory proposed by Qing et al. is an effective method to solve for the critical crack length a_c of any size specimen.

In summary, based on the fracture test results of concrete specimens with limited size and α_c, the size effect model can be used to predict the initiation and peak loads of concrete specimens with any size or α_c, and combined with the fracture extreme theory, the initiation and unstable toughness of concrete specimens can be calculated when the size or α_c changes. Although this method is theoretically feasible, its applicability and accuracy still need to be verified by test results. In this paper, we first introduce the determination method of fracture parameters and then verify the applicability and accuracy of the method using the fracture test results of small aggregate concrete and fully graded large aggregate concrete. The research results presented in this paper can provide a theoretical basis for the optimization of the fracture test at the engineering site, as well as for the determination of fracture parameters in crack stability or propagation analysis.

2. Determination Method of Fracture Parameters

The fracture parameter determination method proposed in this paper is based on the test results of concrete specimens with limited size or α_c. The size effect model is used to predict the initiation and peak loads of concrete specimens with an arbitrary size or α_c. Then, the predicted value of the peak load is taken as the known condition, and the critical crack length is determined according to the fracture extreme theory. Finally, the initiation and unstable toughness of concrete specimens are determined based on the predicted initiation load and initial crack length and the predicted peak load and critical crack length, respectively, when the specimen size or α_c changes. The calculation flow chart is shown in Figure 1. This method can be used to analyze the fracture test results of different specimens, such as those obtained by the wedge-splitting method, the three-point bending-beam method and the compact tension method. Considering that the self-weight of fully graded concrete specimens is generally large, the wedge-splitting method is used to determine the fracture parameters, which can overcome the influence of the self-weight of the specimen on the fracture process. Therefore, the applicability and accuracy of the method are verified by the wedge-splitting fracture test results in the literature. The formula for calculating the initiation and unstable toughness of wedge-splitting specimens given by the “Norm for fracture test of hydraulic concrete” [16] in China is as follows:

$$K_{\mathrm{IC}}^{\mathrm{Q}}=\frac{F_{\mathrm{HQ}}\times {10}^{-3}}{t{h}^{1/2}}f\left({\alpha }_0\right)=Y\left({\alpha }_0\right){\sigma }_{\mathrm{N}}^{\mathrm{Q}}\sqrt{\pi a_0}$$

(1)

$${\sigma }_{\mathrm{N}}^{\mathrm{Q}}=\frac{4F_{\mathrm{HQ}}\times {10}^{-3}}{th}$$

(2)

$$K_{\mathrm{IC}}^{\mathrm{S}}=\frac{F_{\mathrm{Hmax}}\times {10}^{-3}}{t{h}^{1/2}}f\left({\alpha }_{\mathrm{c}}\right)=Y\left({\alpha }_{\mathrm{c}}\right){\sigma }_{\mathrm{N}}^{\mathrm{S}}\sqrt{\pi a_{\mathrm{c}}}$$

(3)

$${\sigma }_{\mathrm{N}}^{\mathrm{S}}=\frac{4F_{\mathrm{Hmax}}\times {10}^{-3}}{th}$$

(4)

$$Y\left(\alpha \right)=\frac{f\left(\alpha \right)}{4\sqrt{\pi \alpha }}=\frac{3.675\left[1-0.12\left(\alpha -0.45\right)\right]}{4\sqrt{\pi \alpha }{\left(1-\alpha \right)}^{3/2}}$$

(5)

where $K_{\mathrm{IC}}^{\mathrm{Q}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ are the initiation toughness and unstable toughness, respectively (MPa·m^1/2); F_HQ and F_Hmax are the initiation load and peak load, respectively (kN); $f\left({\alpha }_0\right)$ and $Y\left({\alpha }_0\right)$ are the dimensionless geometric parameters of the specimen; a_c is the critical crack length when the specimen reaches the peak load (m); ${\alpha }_{\mathrm{c}}$ is the crack length-to-height ratio (${\alpha }_{\mathrm{c}}=a_{\mathrm{c}}/h$); and ${\sigma }_{\mathrm{N}}^{\mathrm{Q}}$ and ${\sigma }_{\mathrm{N}}^{\mathrm{S}}$ are the nominal initiation and unstable strength, respectively, i.e., the maximum tensile stress of the fracture surface caused by the initiation load and peak load without considering the existence of an initial crack (MPa).

2.1. Determination of Initiation Load and Peak Load

In this paper, the size effect model is selected as the method to determine the initiation and peak loads of concrete specimens when the size or ${\alpha }_{\mathrm{c}}$ changes. The evolution form of the type 2 size effect model proposed by Gao et al. [48] can be used to analyze the size effect of fracture test results in geometrically similar specimens. On this basis, in order to analyze the test results when the specimen size or α_c changes, the linear equations of nominal strength and equivalent crack length as shown in Equations (6) and (7) can be established [48].

$$\frac{1}{{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2}=\frac{1}{{\left(K_{\mathrm{IC}}^{\mathrm{ini}}\right)}^2}{a}_{\mathrm{e}}+\frac{c_{\mathrm{f}}^{\mathrm{ini}}}{{\left(K_{\mathrm{IC}}^{\mathrm{ini}}\right)}^2}=A_1{a}_{\mathrm{e}}+C_1$$

(6)

$$\frac{1}{{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2}=\frac{1}{{\left(K_{\mathrm{IC}}^{\mathrm{un}}\right)}^2}{a}_{\mathrm{e}}+\frac{c_{\mathrm{f}}^{\mathrm{un}}}{{\left(K_{\mathrm{IC}}^{\mathrm{un}}\right)}^2}=A_2{a}_{\mathrm{e}}+C_2$$

(7)

where ${\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}$ and ${\sigma }_{\mathrm{Ne}}^{\mathrm{S}}$ are equivalent nominal initiation strength and equivalent nominal unstable strength, respectively (${\mathsf{\sigma }}_{\mathrm{Ne}}^{\mathrm{Q}}={\mathsf{\sigma }}_{\mathrm{N}}^{\mathrm{Q}}H\left(\alpha \right){\pi }^{1/2}$, ${\mathsf{\sigma }}_{\mathrm{Ne}}^{\mathrm{S}}={\mathsf{\sigma }}_{\mathrm{N}}^{\mathrm{S}}H\left(\alpha \right){\pi }^{1/2}$) (MPa); $a_{\mathrm{e}}$ is the equivalent crack length ($a_{\mathrm{e}}=Y^2\left({\alpha }_0\right)/H^2\left({\alpha }_0\right)\times a_0$, m, $H\left({\alpha }_0\right)={\left[Y^2\left({\alpha }_0\right)+2Y\left({\alpha }_0\right)\times Y^{\prime }\left({\alpha }_0\right)\times {\alpha }_0\right]}^{1/2}$); and A₁, A₂, C₁ and C₂ are the parameters of two linear equations, which can be determined by fitting the test results.

Based on the experimental $F_{\mathrm{HQ}}$ and $F_{\mathrm{Hmax}}$, the linear relationship between $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2$, $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2$ and a_e can be established by Equation (6) and Equation (7), respectively. According to the slope and intercept of the two linear equations, the initiation toughness without the size effect ($K_{\mathrm{IC}}^{\mathrm{ini}}$), the effective length of the initiation fracture process zone ($c_{\mathrm{f}}^{\mathrm{ini}}$), the unstable toughness without the size effect ($K_{\mathrm{IC}}^{\mathrm{un}}$), and the effective length of the unstable fracture process zone ($c_{\mathrm{f}}^{\mathrm{un}}$) can be calculated; $K_{\mathrm{IC}}^{\mathrm{ini}}=1/{\left(A_1\right)}^{1/2}$, $c_{\mathrm{f}}^{\mathrm{ini}}=C_1/A_1$, $K_{\mathrm{IC}}^{\mathrm{un}}=1/{\left(A_2\right)}^{1/2}$, $c_{\mathrm{f}}^{\mathrm{un}}=C_2/A_2$.

Furthermore, Gao et al. [7] introduced the equivalent geometric parameters into the size effect model and established the prediction formulas of initiation and peak loads when the size or α_c of a concrete specimen changed, as shown in Equations (8) and (9).

$$F_{\mathrm{HQ}}^{\mathrm{Pre}}=\frac{ht\times {10}^3}{4\sqrt{g^{\prime }\left({\alpha }_0\right)c_{\mathrm{f}}^{\mathrm{ini}}+g\left({\alpha }_0\right)h}}{K}_{\mathrm{IC}}^{\mathrm{ini}}=A_{\mathrm{e}}^{\mathrm{ini}}{K}_{\mathrm{IC}}^{\mathrm{ini}}$$

(8)

$$F_{\mathrm{Hmax}}^{\mathrm{Pre}}=\frac{ht\times {10}^3}{4\sqrt{g^{\prime }\left({\alpha }_0\right)c_{\mathrm{f}}^{\mathrm{un}}+g\left({\alpha }_0\right)h}}{K}_{\mathrm{IC}}^{\mathrm{un}}=A_{\mathrm{e}}^{\mathrm{un}}{K}_{\mathrm{IC}}^{\mathrm{un}}$$

(9)

where $F_{\mathrm{HQ}}^{\mathrm{Pre}}$ and $F_{\mathrm{Hmax}}^{\mathrm{Pre}}$ are the initiation and peak load prediction, respectively; $g\left({\alpha }_0\right)$ and $g^{\prime }\left({\alpha }_0\right)$ are the dimensionless geometric parameters of the specimen ($g\left({\alpha }_0\right)=Y^2\left({\alpha }_0\right)\pi {\alpha }_0$, $g^{\prime }\left({\alpha }_0\right)=H^2\left({\alpha }_0\right)\pi$); and $A_{\mathrm{e}}^{\mathrm{ini}}$ and $A_{\mathrm{e}}^{\mathrm{un}}$ are equivalent geometric parameters related to initiation and instability, respectively (mm·m^1/2). Other parameters are the same as the previous definition.

2.2. Determination of Initiation Toughness and Unstable Toughness

After the $F_{\mathrm{HQ}}^{\mathrm{Pre}}$ and $F_{\mathrm{Hmax}}^{\mathrm{Pre}}$ are determined, the $K_{\mathrm{IC}}^{\mathrm{Q}}$ can be calculated directly according to Equation (1). For $K_{\mathrm{IC}}^{\mathrm{S}}$, a_c should be calculated first according to fracture extreme theory and substituted into Equation (3) to determine $K_{\mathrm{IC}}^{\mathrm{S}}$. According to the fracture extreme theory, at the extreme point, the derivative of F_H to a is zero [37], namely:

$${\frac{\partial F_{\mathrm{H}}}{\partial a}|}_{a=a_{\mathrm{c}}}=0$$

(10)

The functional relationship between $F_{\mathrm{H}}$ and a can be established by fracture propagation criterion equation in the critical state of concrete or the equilibrium equation of a ligament section. By combining the established function relation and Equation (10), $a_{\mathrm{c}}$ can be solved. In this paper, the functional relationship between $F_{\mathrm{H}}$ and a is established by using the equilibrium equation of force and moment of ligament section. Based on a fictitious crack model, Wu [35] and Qing [39] obtained the stress and strain distributions of ligament sections of three-point bending-beam specimens and wedge-splitting specimens, as shown in Figure 2.

According to the stress distribution shown in Figure 2, with the initial crack tip as the coordinate origin, the force and moment equilibrium equation of the ligament section shown in Equations (11) and (12), respectively, can be obtained [35,39]:

$$\frac{1}{2}{\sigma }_{\mathrm{c}}t\left(h-a-h_{\mathrm{c}}\right)+F_{\mathrm{H}}\times {10}^{-3}=\frac{1}{2}{f}_{\mathrm{t}}t{h}_{\mathrm{c}}+{\displaystyle {\int }_0^{a-a_0}{\sigma }_{w}t}dx$$

(11)

$$\frac{1}{3}{\sigma }_{\mathrm{c}}t{\left(h-a-h_{\mathrm{c}}\right)}^2+\frac{1}{3}{f}_{\mathrm{t}}t{h}_{\mathrm{c}}^2+{\displaystyle {\int }_0^{a-a_0}{\sigma }_{w}t\left(h_{\mathrm{c}}+a-a_0-x\right)}dx=F_{\mathrm{H}}\left(a+h_{\mathrm{c}}\right)\times {10}^{-3}$$

(12)

where ${\sigma }_{\mathrm{c}}$ is the compressive stress at the bottom of the specimen (MPa); t is the thickness of specimen (m); h is the effective height of the specimen (m); $a_0$ and a are initial crack length and effective crack length, respectively (m); $h_{\mathrm{c}}$ is the distance from the crack tip to the neutral axis marked by the dotted line in Figure 2 (m); $f_{\mathrm{t}}$ is the equivalent tensile stress at the crack tip, namely the equivalent tensile strength (MPa); and ${\sigma }_w$ is the cohesive force in the fracture process zone (MPa).

According to the plane section assumption and the strain distribution shown in Figure 2, the expression of ${\sigma }_{\mathrm{c}}$ is obtained as follows [35,39]:

$${\sigma }_{\mathrm{c}}=\frac{f_{\mathrm{t}}\left(h-a-h_{\mathrm{c}}\right)}{h_{\mathrm{c}}}$$

(13)

According to the assumption that the crack-opening surface remains a plane [35,39], the relationship between the virtual crack-opening displacement ($w\left(x\right)$) and the crack-tip opening displacement (CTOD, $w\left(x\right)$), as well as the crack-mouth-opening displacement (CMOD) can be established as:

$$w\left(x\right)=\left(1-\frac{x}{a-a_0}\right)CTOD$$

(14)

$$CTOD=\frac{a-a_0}{a}CMOD$$

(15)

where CMOD can be obtained according to the empirical calculation proposed by Xu et al. [15].

$$CMOD=\frac{F_{\mathrm{H}}}{t{E}_{\mathrm{c}}}\left[13.18{\left(1-\frac{a}{h}\right)}^{-2}-9.16\right]$$

(16)

The relationship between the cohesive force in the fracture process zone and the virtual fracture-opening displacement can be characterized by the tensile softening curve [49], which often adopts a bilinear, exponential, power function, curve or other form. The bilinear softening curve shown in Figure 3 is used in this paper.

The function expression of the bilinear softening curve is expressed as follows:

$${\sigma }_w\left(x\right)=\{\begin{array}{cc}{f}_{\mathrm{t}}-\left(f_{\mathrm{t}}-{\sigma }_{\mathrm{s}}\right)w\left(x\right)/w_{\mathrm{s}}\hfill & 0\le w\left(x\right)\le w_{\mathrm{s}}\\ {\sigma }_{\mathrm{s}}\left(w_0-w\left(x\right)\right)/\left(w_0-w_{\mathrm{s}}\right)\hfill & w_{\mathrm{s}}\le w\left(x\right)\le w_0\end{array}$$

(17)

where $w\left(x\right)$, $w_{\mathrm{s}}$ and $w_0$ are the crack-opening displacement at x, the crack-opening displacement at the inflection point of the bilinear softening relationship and the crack-opening displacement at zero cohesion, respectively (μm); and ${\sigma }_w\left(x\right)$ and ${\sigma }_{\mathrm{s}}$ are the cohesive force at x and the inflection point, respectively (MPa). The values of ${\sigma }_{\mathrm{s}}$, $w_{\mathrm{s}}$ and $w_0$ in the formula can be calculated using the assumption proposed by Wittmann et al. [50]:

$$\{\begin{array}{c}{\sigma }_{\mathrm{s}}=f_{\mathrm{t}}/4\\ w_{\mathrm{s}}=0.75G_{\mathrm{f}}/f_{\mathrm{t}}\\ w_0=5G_{\mathrm{f}}/f\end{array}$$

(18)

In order to reduce the parameters used in the calculation process, in combination with Equation (18), Equation (17) can be rewritten as follows:

$${\sigma }_w\left(x\right)=\{\begin{array}{cc}{f}_{\mathrm{t}}\left(1-5w\left(x\right)/w_0\right)& 0\le w\left(x\right)\le w_{\mathrm{s}}\\ \frac{5}{17}{f}_{\mathrm{t}}\left(1-w\left(x\right)/w_0\right)& w_{\mathrm{s}}\le w\left(x\right)\le w_0\end{array}$$

(19)

In the tensile softening curve, the crack-opening displacement ($w_0$), which characterizes the zero cohesion, is the material constant, and its value can be generally determined by the direct tensile test [51] or empirical formula [52]. For concrete materials, the variation range of $w_0$ provided by multiple scholars is large [51,52,53], mostly between 100 and 1000 μm. Owing to the difficulty of conducting a direct tensile test on fully graded concrete [54] and because the specific empirical formula may not be applicable to different types of concrete materials, in this paper, we determine $w_0$ using the best fitting method. First, $w_0$ is selected from 100 to 1000 μm at 50 μm intervals; then, $a_{\mathrm{c}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ can be calculated according to the fracture extreme theory. When the overall error between the calculated $K_{\mathrm{IC}}^{\mathrm{S}}$ and the test results is the smallest, the corresponding w₀ is the best crack-opening displacement at zero cohesion of concrete.

Equation (14) is substituted into (19) to obtain the expression of the cohesion distribution function in the fracture process area, and $a_{\mathrm{c}}$ can be calculated. The ${\sigma }_w\left(x\right)$ selected in this paper is the bilinear softening curve. Therefore, the relationship between $w\left(x\right)$ and $w_{\mathrm{s}}$ at the inflection point of the bilinear softening relationship needs to be discussed when determining the stress distribution in the fracture process area. Obviously, under the peak load state, the maximum value of $w\left(x\right)$ is CTOD, which is located at the initial crack tip, whereas the minimum value of $w\left(x\right)$ is 0, which is located at the equivalent crack tip. When $CTO{D}_{\mathrm{c}}\le w_{\mathrm{s}}$, the cohesion from the initial crack tip to the equivalent crack tip is a linear distribution, whereas when $w_{\mathrm{s}}<CTO{D}_{\mathrm{c}}\le w_0$, the cohesion is a bilinear distribution. The following two situations are discussed.

Case 1: $CTO{D}_{\mathrm{c}}\le w_{\mathrm{s}}$. Figure 4 shows the cohesive stress distribution and the corresponding softening relationship in the fracture process zone when $CTO{D}_{\mathrm{c}}\le w_{\mathrm{s}}$.

Substituting Equation (14) into (19), the cohesive stress distribution function in the fracture process zone of Case 1 can be obtained as:

$$\begin{array}{cc}{\sigma }_w\left(x\right)=f_{\mathrm{t}}\left[1-5\left(1-\frac{x}{a-a_0}\right)\frac{CTOD}{w_0}\right]& \left(0\le x\le a-a_0\right)\end{array}$$

(20)

Substituting Equations (13) and (20) into (11), the distance from the equivalent crack tip to the neutral axis ($h_{\mathrm{c}}$) can be obtained as:

$$h_{\mathrm{c}}=\frac{{\left(h-a\right)}^2}{2\left[\left(a-a_0\right)\left(1-\frac{5CTOD}{2w_0}\right)+h-a-\frac{F_{\mathrm{H}}\times {10}^{-3}}{f_{\mathrm{t}}t}\right]}$$

(21)

The functional relationship between $F_{\mathrm{H}}$ and a can be obtained by substituting Equations (13) and (20) into (12):

$$\begin{array}{l}\frac{F_{\mathrm{H}}\left(a+h_{\mathrm{c}}\right)\times {10}^{-3}}{f_{\mathrm{t}}t}=\left(a-a_0\right)\left[\left(a-a_0\right)\left(\frac{1}{2}-\frac{5CTOD}{3w_0}\right)+h_{\mathrm{c}}\left(1-\frac{5CTOD}{2w_0}\right)\right]+\\ \frac{{\left(h-a\right)}^3}{3h_{\mathrm{c}}}-{\left(h-a\right)}^2+\left(h-a\right)h_{\mathrm{c}}\end{array}$$

(22)

Furthermore, by substituting Equations (15), (16) and (21) into (22), a function containing only $F_{\mathrm{H}}$, a and $f_{\mathrm{t}}$ is obtained. According to the fracture extreme theory, the partial derivative of $F_{\mathrm{H}}$ to a is zero at the extreme point. Substituting $a=a_{\mathrm{c}}$ and $F_{\mathrm{H}}=F_{\mathrm{Hmax}}$ into Equations (10) and (22), $a_{\mathrm{c}}$ can be obtained; then, $CTO{D}_{\mathrm{c}}$, $K_{\mathrm{IC}}^{\mathrm{S}}$ and other parameters can be calculated. If the calculated $CTO{D}_{\mathrm{c}}\le w_{\mathrm{s}}$, $a_{\mathrm{c}}$ is the real solution, and on the contrary, it should be discarded.

Case 2, $w_{\mathrm{s}}<CTO{D}_{\mathrm{c}}\le w_0$. Figure 5 shows the cohesive force distribution and the corresponding softening relationship in the fracture process zone when $w_{\mathrm{s}}<CTO{D}_{\mathrm{c}}\le w_0$.

Figure 5a shows that the cohesive force is a bilinear distribution, and the coordinate (x_s) of the turning point can be calculated according to the following formula:

$$x_{\mathrm{s}}=\left(a-a_0\right)\left(1-\frac{3w_0}{20CTOD}\right)$$

(23)

Similar to Case 1, the cohesive force distribution function in the fracture process zone of Case 2 can be obtained by substituting Equation (14) into (19):

$${\sigma }_w\left(x\right)=\{\begin{array}{cc}\frac{5}{17}{f}_{\mathrm{t}}\left[1-\left(1-\frac{x}{a-a_0}\right)\frac{CTOD}{w_0}\right]& 0\le x\le x_{\mathrm{s}}\\ f_{\mathrm{t}}\left[1-5\left(1-\frac{x}{a-a_0}\right)\frac{CTOD}{w_0}\right]& x_{\mathrm{s}}\le x\le a-a_0\end{array}$$

(24)

Furthermore, the distance from the crack tip to the neutral axis is:

$$h_{\mathrm{c}2}=\frac{{\left(h-a\right)}^2}{\begin{array}{l}2[-\frac{12}{17}{x}_{\mathrm{s}}+\frac{80CTOD{x}_{\mathrm{s}}}{17w_0}-\frac{40CTOD{x}_{\mathrm{s}}^2}{17w_0\left(a-a_0\right)}-\frac{5\left(a-a_0\right)CTOD}{2w_0}+\\ h-a_0-\frac{F_{\mathrm{H}}\times {10}^{-3}}{f_{\mathrm{t}}t}]\end{array}}$$

(25)

By substituting Equations (13) and (24) into (12), the relation between the Case 2 $F_{\mathrm{H}}$ and a is obtained:

$$\begin{array}{l}\frac{F_{\mathrm{H}}\left(a+h_{\mathrm{c}}\right)\times {10}^{-3}}{f_{\mathrm{t}}t}=\frac{{\left(h-a\right)}^3}{3h_{\mathrm{c}}}-{\left(h-a\right)}^2+\left(h-a\right)h_{\mathrm{c}}+\frac{5}{17}{x}_{\mathrm{s}}\left(1-\frac{CTOD}{w_0}\right)\left(h_{\mathrm{c}}+a-a_0\right)-\\ \frac{5}{34}{x}_{\mathrm{s}}^2\left[1-\frac{2CTOD}{w_0}-\frac{h_{\mathrm{c}}CTOD}{\left(a-a_0\right)w_0}\right]-\frac{5x_{\mathrm{s}}^{3}CTOD}{51\left(a-a_0\right)w_0} +\left(a-a_0-x_{\mathrm{s}}\right)\left(h_{\mathrm{c}}+a-a_0\right)\left(1-\frac{5CTOD}{w_0}\right)-\\ \frac{5\left[{\left(a-a_0\right)}^3-x_{\mathrm{s}}^3\right]CTOD}{3\left(a-a_0\right)w_0}-\frac{1}{2}\left[{\left(a-a_0\right)}^2-x_{\mathrm{s}}^2\right]\left[1-\frac{10CTOD}{w_0}-\frac{5h_{\mathrm{c}}CTOD}{\left(a-a_0\right)w_0}\right]\end{array}$$

(26)

By substituting Equations (15), (16), (23) and (25) into (26), a functional equation containing only $F_{\mathrm{H}}$, a and $f_{\mathrm{t}}$ can be obtained. By substituting $a=a_c$ and $F_{\mathrm{H}}=F_{\mathrm{Hmax}}$ into Equations (10) and (26), respectively, parameters such as a_c and $K_{\mathrm{IC}}^{\mathrm{S}}$ can be obtained. If the calculated $CTO{D}_{\mathrm{c}}$ satisfies condition 2, $a_{\mathrm{c}}$ is the real solution, and on the contrary, it should be discarded.

According to the above calculation process, $a_c$ and $K_{\mathrm{IC}}$ corresponding to a specific $w_0$ can be obtained. By repeating the above process, the best $w_0$ value with the smallest overall error between $K_{\mathrm{IC}}^{\mathrm{S}}$ and test results can be determined. The error or prediction accuracy can be evaluated using the indicators commonly used in engineering.

2.3. Evaluation Index of Model Prediction Accuracy

The root mean square error (RMSE), mean absolute percentage error (MAPE) and reliability index a₁₅ were used to evaluate the prediction accuracy of the model. The calculation formula of each indicator is as follows:

$$RMSE=\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{\left({\widehat{y}}_i-y_i\right)}^2}}$$

(27)

$$MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|}\times 100\%$$

(28)

$$a15=\frac{m15}{M}$$

(29)

$$Cov\left(y_i,\widehat{y_i}\right)=E\left[y_i\widehat{y_i}\right]-E\left[y_i\right]E\left[\widehat{y_i}\right]$$

(30)

where n is the amount of data points; i is the number of the data point; y_i and $\widehat{y_i}$ are the measured and predicted values of the i-th data point, respectively; a₁₅ is the proportion of samples with deviation within ±15% compared with the test results; m15 and M are the number of samples with prediction error less than ±15% and the number of samples in the dataset, respectively; $Cov\left(y_i,\widehat{y_i}\right)$ is the covariance of the measured and predicted values; and $E\left[y_i\right]$ and $E\left[\widehat{y_i}\right]$ are the expected values of the measured and predicted values, respectively.

3. Method Validation and Discussion

In this section, four groups of fracture test results of different sizes and the ${\alpha }_c$ of small aggregate concrete and Wudongde fully graded concrete are selected to verify the applicability and accuracy of the proposed method.

3.1. Small Aggregate Concrete

3.1.1. Same Crack Length-to-Height Ratio and Different Specimen Sizes

(1)

${\alpha }_0$ = 0.4, h = 0.2~1 m

In order to study the influence of specimen size on the fracture performance of concrete, a fracture test was carried out in [55] on wedge-splitting tensile specimens with a maximum aggregate particle size of 25 mm, an effective specimen height ranging from 0.2 m to 1 m and an initial ${\alpha }_{\mathrm{c}}$ of 0.4.

Table 1. Fracture test results reported in reference [55].

Specimen Number	h/m	b/m	α0	FHQ/kN	FHmax/kN
200-1	0.20	0.20	0.40	7.85	10.02
200-2	0.20	0.20	0.40	7.81	10.69
200-3	0.20	0.20	0.40	7.65	9.62
200-4	0.20	0.20	0.40	9.27	10.60
200-6	0.20	0.20	0.40	8.65	10.93
300-1	0.30	0.20	0.40	9.97	15.33
300-2	0.30	0.20	0.40	10.45	14.78
300-3	0.30	0.20	0.48	10.00	12.56
300-4	0.30	0.20	0.40	11.31	15.12
300-5	0.30	0.20	0.40	8.90	12.06
600-1	0.60	0.21	0.40	20.47	26.68
600-2	0.61	0.22	0.44	19.20	28.13
600-3	0.60	0.22	0.40	15.84	27.02
600-4	0.60	0.22	0.40	22.90	29.12
600-5	0.60	0.21	0.48	28.17	28.66
600-6	0.61	0.21	0.44	26.49	31.50
800-1	0.81	0.24	0.43	31.47	34.69
800-2	0.81	0.25	0.40	30.64	34.99
800-3	0.80	0.24	0.40	20.65	33.72
800-4	0.80	0.24	0.40	22.56	33.96
800-5	0.80	0.23	0.45	33.90	37.83
800-6	0.80	0.23	0.40	23.72	34.25
1000-1	1.01	0.24	0.40	28.75	43.58
1000-2	1.01	0.24	0.40	28.53	46.63
1000-3	1.01	0.27	0.39	38.51	55.91
1000-4	1.01	0.24	0.40	35.15	48.16
1000-5	1.00	0.25	0.40	30.19	45.52
1000-6	1.01	0.27	0.40	37.32	50.47

shows the specimen size information and test results. The elastic modulus of concrete (E_c = 30.7 GPa) was obtained by the conversion of cube compressive strength. Additional experimental information can be found in reference [55].

The linear equations of $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2$, $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2$ and $a_{\mathrm{e}}$, as shown in Figure 6, can be obtained by linear analysis of the test results using Equations (6) and (7). According to the slope and intercept, the initiation toughness without a size effect ($K_{\mathrm{IC}}^{\mathrm{ini}}$) is 1.14 MPa·m^1/2, the effective length of the initiation fracture process zone ($c_{\mathrm{f}}^{\mathrm{ini}}$) is 56.2 mm, the unstable toughness ($K_{\mathrm{IC}}^{\mathrm{un}}$) is 1.81 MPa·m^1/2 and the effective length of unstable fracture process zone ($c_{\mathrm{f}}^{\mathrm{un}}$) is 108.6 mm.

The $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $c_{\mathrm{f}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ and $c_{\mathrm{f}}^{\mathrm{un}}$ obtained by linear analysis are substituted into Equations (8) and (9), respectively, to obtain the relationship between the initiation and peak loads with the equivalent geometric parameter ($A_{\mathrm{e}}$), as shown in Figure 7. In the figure, the slope of the initiation load prediction line is $K_{\mathrm{IC}}^{\mathrm{ini}}$ without a size effect, and the slope of the peak load prediction line is $K_{\mathrm{IC}}^{\mathrm{un}}$ without a size effect. Figure 7 shows that the test results of $F_{\mathrm{HQ}}$ and $F_{\mathrm{Hmax}}$ are nearly linear relative to $A_{\mathrm{e}}$. The MAPE of the F_HQ prediction is 14.29%, the RMSE is 4.54 kN, reliability index a15 is 67.86% and the covariance is 83.39 (kN)². The MAPE of the $F_{\mathrm{Hmax}}$ prediction is 7.4%, the RMSE is 3.05 kN, a15 is 85.71% and the covariance is 187.14 (kN)². The above indices show that the size effect model can realize the accurate prediction of initiation load and peak load when the size or ${\alpha }_{\mathrm{c}}$ changes.

As mentioned above, in this paper, we use the best fitting method to determine the crack-opening displacement ($w_0$). In this test, MAPE was used to evaluate the prediction accuracy of $K_{\mathrm{IC}}^{\mathrm{S}}$; the calculation results are shown in Figure 8. It can be seen that the minimum MAPE is 4.56%, and the corresponding optimal $w_0$ is 550 μm.

Based on the optimal $w_0$ and $F_{\mathrm{Hmax}}^{\mathrm{Pre}}$ obtained above, the critical crack length of specimens of different sizes with an initial ${\alpha }_{\mathrm{c}}$ of 0.4 was calculated using Equations (3) and (10). Figure 9 shows that each data point is located near the calculated value, with the following corresponding prediction accuracy evaluation indices: $MAPE=3.42\%$, $RMSE=0.02 \mathrm{m}$, $a15=100\%$ and $Cov=0.02 {\mathrm{m}}^2$.

Figure 10a shows a comparison between the predicted and test results of unstable toughness ($K_{\mathrm{IC}}^{\mathrm{S}}$), nominal toughness ($K_{\mathrm{IC}}^{\mathrm{N}}$) and initiation toughness ($K_{\mathrm{IC}}^{\mathrm{Q}}$) for specimens of different sizes with an initial ${\alpha }_{\mathrm{c}}$ of 0.4. The nominal toughness is calculated by substituting the predicted peak load ($F_{\mathrm{Hmax}}^{\mathrm{Pre}}$) and the initial crack length ($a_0$) into Equation (3). It can be seen from Figure 10a that $K_{\mathrm{IC}}^{\mathrm{S}}$, $K_{\mathrm{IC}}^{\mathrm{N}}$ and $K_{\mathrm{IC}}^{\mathrm{Q}}$ basically increase with the increase in specimen size and gradually stabilize. Among them, $K_{\mathrm{IC}}^{\mathrm{Q}}$ tends toward the initiation toughness without a size effect defined in the size effect model. $K_{\mathrm{IC}}^{\mathrm{S}}$ and $K_{\mathrm{IC}}^{\mathrm{N}}$ tend toward unstable toughness without a size effect, and the difference between them decreases with increased specimen size.

This is mainly because the difference between $K_{\mathrm{IC}}^{\mathrm{S}}$ and $K_{\mathrm{IC}}^{\mathrm{N}}$ lies in the selection of crack length in toughness calculation. It can be seen from Figure 10b that with the increase in specimen size, ${\alpha }_{\mathrm{c}}$ gradually approaches ${\alpha }_0$. A comparison of the predicted results with the test results shows that the predicted MAPE of $K_{\mathrm{IC}}^{\mathrm{S}}$ is 4.56%, RMSE is 0.09 MPa·m^1/2, a15 is 100% and the covariance is 0.01 (MPa)² m. Furthermore, $MAPE=11.46\%$, $RMSE=0.11 \mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$, $a15=81.82\%$ and $Cov=0.01 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$ of $K_{\mathrm{IC}}^{\mathrm{Q}}$ indicate that the prediction accuracy of initiation and unstable toughness can meet the requirements of engineering applications.

Figure 11 shows the calculation results of the critical tip-opening displacement ($CTO{D}_{\mathrm{c}}$) and the cohesive force ($F_{\sigma w}$) in the fracture process zone of specimens with different sizes and an initial ${\alpha }_{\mathrm{c}}$ of 0.4. It can be seen from Figure 11 that $CTO{D}_{\mathrm{c}}$ increases rapidly when the specimen height is less than 10 m. When the specimen height reaches 10 m, $CTO{D}_{\mathrm{c}}$ tends to be stable, and the stable $CTO{D}_{\mathrm{c}}$ is 47.4 μm. Similarly, $F_{\sigma w}$ first increases rapidly with the increase in specimen size and then tends to be stable at 42.4 kN.

Figure 12 shows the midspan section stress distribution of wedge-splitting tensile specimens with heights of 0.6, 1, 5, 20, 10, 20 and 50 m. In the figure, $F_{\mathrm{t}}$ is the tensile force of the uncracked zone in the midspan section of the specimen (kN), and $F_{\mathrm{c}}$ is the pressure-resultant force on the uncracked area of the midspan section of the specimen (kN). It can be seen from the figure that with the increase in specimen size, $F_{\mathrm{t}}$ and $F_{\mathrm{c}}$ gradually increase. $F_{\mathrm{sw}}$ increases with the increase in size, and its growth rate gradually slows down and tends toward stability increased size.

(2)

α₀ = 0.4, h = 0.3~0.6 m

In reference [56], the fracture properties of wedge-splitting tensile specimens with an ${\alpha }_{\mathrm{c}}$ of 0.4 and effective heights of 0.3, 0.4, 0.5 and 0.6 m were tested.

Table 2. Fracture test results reported in reference [56].

Specimen Number	h/m	b/m	t/m	α0	FHmax/kN	FHQ/kN
WS300	0.3	0.3	0.2	0.4	11.793	7.003
WS400	0.4	0.4	0.2	0.4	15.067	9.342
WS500	0.5	0.5	0.2	0.4	17.766	10.953
WS600	0.6	0.6	0.2	0.4	21.253	13.43

shows the specimen size and test results. Reference [56] reported the test results used to calculate the elastic modulus. The elastic modulus (E_c) required by the method proposed in this chapter was taken as the average value of the calculated elastic modulus, i.e., $E_{\mathrm{c}}=39.05\mathrm{GPa}$. Detailed test information is provided in reference [56].

Based on Equations (6) and (7), linear regression analysis of the test results was carried out, and the linear regression equations of $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2$, $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2$ and $a_{\mathrm{e}}$ were obtained, as shown in Figure 13. According to the slope and intercept, $K_{\mathrm{IC}}^{\mathrm{ini}}$ without a size effect is 1.47 MPa·m^1/2, $c_{\mathrm{f}}^{\mathrm{ini}}$ is 460.6 mm, $K_{\mathrm{IC}}^{\mathrm{un}}$ is 1.72 MPa·m^1/2 and $c_{\mathrm{f}}^{\mathrm{un}}$ is 193.1 mm.

The $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $c_{\mathrm{f}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ and $c_{\mathrm{f}}^{\mathrm{un}}$ obtained by linear analysis are substituted into Equations (8) and (9), respectively, and the relationship between the initiation and peak loads with the equivalent geometric parameter ($A_{\mathrm{e}}$) can be obtained as shown in Figure 14. In the figure, the slope of the initiation load prediction line is $K_{\mathrm{IC}}^{\mathrm{ini}}$ without size effect, and the slope of the peak load prediction line is $K_{\mathrm{IC}}^{\mathrm{un}}$ without a size effect. It can be seen from Figure 7 that the test results of $F_{\mathrm{HQ}}$ and $F_{\mathrm{Hmax}}$ are nearly linear relative to $A_{\mathrm{e}}$. The MAPE of $F_{\mathrm{HQ}}$ prediction is 0.99%, RMSE is 0.22 kN, reliability index a15 is 100% and the covariance is 5.43 (kN)². The MAPE of F_Hmax prediction is 1.5%, RMSE is 0.19 kN, a15 is 100% and the covariance is 12.01 (kN)². The above indices show that the size effect model can realize the accurate prediction of initiation and peak loads when the size or ${\alpha }_{\mathrm{c}}$ changes.

In order to evaluate the effectiveness of the proposed method, substituting the test results of initiation load into Equation (1), the calculated value is taken as the test value of initiation toughness. The test results of $F_{\mathrm{Hmax}}$ and $a_{\mathrm{c}}$ are substituted into Equation (3), and the calculated results are used as the unstable toughness test results. Then, these results are compared with the predicted results of initiation and unstable toughness calculated by the proposed method.

The optimal fitting softening parameter ($w_0$) was 450 mm, and MAPE was 3.46%, as calculated by the proposed method. Figure 15 shows the predicted results of fracture parameters of specimens with an ${\alpha }_{\mathrm{c}}$ of 0.4 and different sizes. It can be seen from Figure 15a that all data points of critical fracture length fall near the predicted line. It can be seen from Figure 15b that the data points of fracture toughness all fall near the corresponding fracture toughness prediction line; $K_{\mathrm{IC}}^{\mathrm{S}}$, $K_{\mathrm{IC}}^{\mathrm{N}}$ and $K_{\mathrm{IC}}^{\mathrm{Q}}$ all have a size effect. $K_{\mathrm{IC}}^{\mathrm{Q}}$ is close to the initiation toughness without a size effect, i.e., 1.47 MPa·m^1/2. The predicted value of $K_{\mathrm{IC}}^{\mathrm{S}}$ increases with an increase in size and then decreases slightly with a further increase in size. $K_{\mathrm{IC}}^{\mathrm{S}}$ and $K_{\mathrm{IC}}^{\mathrm{N}}$ approach one another with increased size, with values close to the unstable toughness without a size effect, i.e., 1.72 MPa·m^1/2. The accuracy evaluation indicators of a_c are $RMSE=0.007$, $MAPE=2.22\%$, $a15=100\%$ and $Cov=0.003 {\mathrm{m}}^2$; the accuracy evaluation indicators of $K_{\mathrm{IC}}^{\mathrm{Q}}$ are $RMSE=0.011$, $MAPE=1.5\%$, $a15=100\%$ and $Cov=0.004 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$; the accuracy evaluation indicators of $K_{\mathrm{IC}}^{\mathrm{S}}$ are $RMSE=0.059$, $MAPE=3.46\%$, $a15=100\%$ and $Cov=0.004 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$, which shows that the predicted results are more accurate. The proposed method can achieve accurate prediction of specific ${\alpha }_{\mathrm{c}}$, critical crack length, crack initiation toughness and unstable toughness of concrete specimens with different sizes and can also determine the initiation and unstable toughness without a size effect.

3.1.2. Same Specimen Size but Different Crack Length-to-Height Ratio

In order to study the effect of initial ${\alpha }_{\mathrm{c}}$ on the fracture performance of concrete, a wedge-splitting tensile fracture test was designed and carried out in reference [57] with maximum particle sizes of concrete aggregate of 10 mm and 20 mm, an effective specimen height of specimen of 0.3 m, a thickness of 0.12 m and an initial ${\alpha }_{\mathrm{c}}$ in the range of 0.2 to 0.7. The specimen size and test results are summarized in

Table 3. Fracture test results reported in reference [57].

Specimen Number	dmax/mm	h/m	b/m	t/m	α0	FHQ/kN	FHmax/kN
R0.2D10-1	10	0.3	0.3	0.12	0.2	10.03	13.32
R0.2D10-2		0.3	0.3	0.12	0.2	9.12	11.23
R0.2D10-3		0.3	0.3	0.12	0.2	11.05	14.27
R0.2D10-4		0.3	0.3	0.12	0.2	9.25	12.34
R0.4D10-1		0.3	0.3	0.12	0.4	6.94	8.72
R0.4D10-2		0.3	0.3	0.12	0.4	5.54	8.39
R0.4D10-3		0.3	0.3	0.12	0.4	5.46	8.38
R0.4D10-4		0.3	0.3	0.12	0.4	6.15	8.87
R0.6D10-1		0.3	0.3	0.12	0.6	2.90	4.30
R0.6D10-2		0.3	0.3	0.12	0.6	2.79	3.45
R0.6D10-3		0.3	0.3	0.12	0.6	3.57	4.43
R0.6D10-4		0.3	0.3	0.12	0.6	2.65	3.86
R0.7D10-1		0.3	0.3	0.12	0.7	1.93	2.54
R0.7D10-3		0.3	0.3	0.12	0.7	2.29	2.64
R0.7D10-4		0.3	0.3	0.12	0.7	2.02	2.74
R0.2D20-1	20	0.3	0.3	0.12	0.2	12.81	16.22
R0.2D20-2		0.3	0.3	0.12	0.2	10.66	15.48
R0.2D20-3		0.3	0.3	0.12	0.2	10.9	14.79
R0.2D20-4		0.3	0.3	0.12	0.2	12.71	16.83
R0.4D20-1		0.3	0.3	0.12	0.4	8.57	10.39
R0.4D20-2		0.3	0.3	0.12	0.4	7.21	9.84
R0.4D20-3		0.3	0.3	0.12	0.4	7.88	11.16
R0.4D20-4		0.3	0.3	0.12	0.4	8.06	11.10
R0.6D20-1		0.3	0.3	0.12	0.6	3.29	5.10
R0.6D20-2		0.3	0.3	0.12	0.6	3.56	4.58
R0.6D20-3		0.3	0.3	0.12	0.6	3.34	4.91
R0.6D20-4		0.3	0.3	0.12	0.6	3.47	4.80
R0.7D20-1		0.3	0.3	0.12	0.7	2.38	3.02
R0.7D20-3		0.3	0.3	0.12	0.7	2.56	3.13
R0.7D20-4		0.3	0.3	0.12	0.7	2.25	3.11

. The compressive strength ($f_{\mathrm{cu}}$) of a concrete cube with a maximum aggregate particle size of 10 mm is 42.2 MPa, and the elastic modulus is $E_{\mathrm{c}}=27.31\mathrm{GPa}$. The $f_{\mathrm{cu}}$ of a concrete cube with a maximum aggregate particle size of 20 mm is 57.04 MPa, with an elastic modulus of $E_{\mathrm{c}}=31.75\mathrm{GPa}$; the elastic modulus is calculated by $f_{\mathrm{cu}}$. Additional experimental information can be found in reference [57].

(1)

d_max = 10 mm

Figure 16a,b show the linear relationship between $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2$, $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2$ and $a_{\mathrm{e}}$, as well as the predicted results of initiation and peak loads when the specimen size or ${\alpha }_{\mathrm{c}}$ changes. According to the linear relationship shown in Figure 16a, $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ without a size effect of concrete can be calculated as 0.93 MPa·m^1/2 and 1.22 MPa·m^1/2, respectively. Then, using Equations (8) and (9), the predicted lines of initiation and peak loads can be obtained. After calculation, the accuracy evaluation indicators of initiation load are $MAPE=9.05\%$, $RMSE=0.56\mathrm{kN}$, $a15=86.67\%$ and $Cov=8.96{\left(\mathrm{kN}\right)}^2$; the accuracy evaluation indicators of peak load are $MAPE=6.74\%$, $RMSE=0.68\mathrm{kN}$, $a15=93.33\%$ and $Cov=15.58{\left(\mathrm{kN}\right)}^2$. The calculation results show that the initiation and peak loads test results are close to the predicted results.

After calculation, the $w_0$ obtained in this experiment is 150 μm, and the minimum value of MAPE of unstable toughness is 8.19%. Figure 17 shows the fracture toughness predicted results of specimens with different initial ${\alpha }_{\mathrm{c}}$ value when the effective height of the specimen is 0.3 m. It can be seen from Figure 17 that the calculation results of $K_{\mathrm{IC}}^{\mathrm{Q}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ basically decrease with an increase in the initial ${\alpha }_{\mathrm{c}}$, and most of the test results are within the allowable error range of ±15% of the predicted value. $MAPE=8.19\%$, $RMSE=0.16 \mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$, $a15=73.33\%$ and $Cov=0.0003 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$ of the corresponding $K_{\mathrm{IC}}^{\mathrm{S}}$ predicted results. The above indices show that the determination method of fracture parameters based on the size effect model and fracture extreme theory can realize the accurate prediction of initiation and unstable toughness when the size or ${\alpha }_{\mathrm{c}}$ changes.

(2)

d_max = 20 mm

Figure 18a shows the linear regression analysis of test results. According to the corresponding slope and intercept, $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ without a size effect of concrete can be calculated as 1.21 MPa·m^1/2 and 1.63 MPa·m^1/2, respectively. Then, Equations (8) and (9) can be used to obtain the predicted line of initiation and peak loads with the slopes of $K_{\mathrm{I}\mathrm{C}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$, as shown in Figure 18b. After calculation, the accuracy evaluation indicators of initiation load are $MAPE=7.95\%$, $RMSE=0.66 \mathrm{kN}$, $a15=93.33\%$ and $Cov=13.66 {\left(\mathrm{kN}\right)}^2$; the accuracy evaluation indicators of peak load are $MAPE=4.51\%$, $RMSE=0.60 \mathrm{kN}$, $a15=100\%$ and $Cov=24.75 {\left(\mathrm{kN}\right)}^2$. The calculation results show that the initiation and peak load test results are close to the predicted results.

After calculation, for the concrete with a maximum aggregate size of 20 mm, the optimal $w_0$ obtained by the test is 150 μm, and the minimum value of MAPE of unstable toughness is 7.91%. Figure 19 shows the fracture toughness predicted results of specimens with different initial ${\alpha }_{\mathrm{c}}$ values when the effective height of the specimen is 0.3 m. It can be seen from Figure 19 that the calculated results of $K_{\mathrm{IC}}^{\mathrm{Q}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ basically decrease with increased initial ${\alpha }_{\mathrm{c}}$, and most of the test results are within the allowable error range of ±15% of the predicted value. The corresponding $K_{\mathrm{IC}}^{\mathrm{Q}}$ predicted results are $MAPE=9.06\%$, $RMSE=0.08 \mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$, $a15=80.00\%$ and $Cov=0.007 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$. $MAPE=7.91\%$, $RMSE=0.12 \mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$, $a15=93.33\%$ and $Cov=0.004 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$ of the predicted results of $K_{\mathrm{IC}}^{\mathrm{S}}$. The above indicators show that the fracture parameter determination method based on the size effect model and the fracture extreme theory can achieve more accurate prediction of initiation and unstable toughness when the size or ${\alpha }_{\mathrm{c}}$ changes.

3.2. Fully Graded Concrete

Same Crack Length-to-Height Ratio and Different Specimen Sizes

In this section, the fracture test results of fully graded concrete of Wudongde Dam are analyzed to verify the applicability of the proposed method for fully graded concrete with a maximum aggregate size of 150 mm. The effective height of the Wudongde fully graded concrete wedge-splitting tensile specimens is 0.75 m, 1.5 m and 2.25 m; the initial ${\alpha }_{\mathrm{c}}$ is 0.4; and the thickness is 0.45 m. The size and test results of concrete specimens are summarized in

Table 4. Test data reported in reference [2].

Specimen Number	h/m	b/m	t/m	α0	FHmax/kN	FHQ/kN
FG750-1	0.75	0.75	0.45	0.4	84.78	63.46
FG750-2	0.75	0.75	0.45	0.4	82.78	64.79
FG750-3	0.75	0.75	0.45	0.4	75.62	63.62
FG1500-1	1.5	1.5	0.45	0.4	157.44	117.46
FG1500-2	1.5	1.5	0.45	0.4	158.22	113.25
FG1500-3	1.5	1.5	0.45	0.4	158.43	110.11
FG2250-1	2.25	2.25	0.45	0.4	200.59	/
FG2250-2	2.25	2.25	0.45	0.4	200.67	140.04
FG2250-3	2.25	2.25	0.45	0.4	170.58	147.93

. Other experimental information can be found in reference [7].

Figure 20a shows the linear relationship between $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{Q}}\right)}^2$, $1/{\left({\sigma }_{\mathrm{Ne}}^{\mathrm{S}}\right)}^2$ and $a_{\mathrm{e}}$. Based on the slope and intercept of the linear equation, $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ of the fully graded concrete without a size effect can be calculated as 2.12 MPa·m^1/2 and 2.90 MPa·m^1/2, respectively. Furthermore, combining Equations (8) and (9) and the obtained fracture toughness without a size effect, the initiation and peak load predicted lines shown in Figure 20b can be obtained. The indices selected in this paper can be used to evaluate the prediction accuracy of initiation and peak loads. After calculation, the evaluation indicators of initiation load are $MAPE=2.51\%$, $RMSE=3.62 \mathrm{kN}$, $a15=100\%$ and $Cov=1024.63 {\left(\mathrm{kN}\right)}^2$; the evaluation indicators of peak load are $MAPE=6.46\%$, $RMSE=11.52 \mathrm{kN}$, $a15=100\%$ and $Cov=2032.55 {\left(\mathrm{kN}\right)}^2$. The calculation results show that the predicted results of initiation and peak loads based on the size effect model are close to the test results.

Figure 21a,b show comparisons between the predicted results and the test results of $a_{\mathrm{c}}$, $K_{\mathrm{IC}}^{\mathrm{S}}$, $K_{\mathrm{IC}}^{\mathrm{N}}$ and $K_{\mathrm{IC}}^{\mathrm{Q}}$ of fully graded concrete specimens with different sizes and an initial ${\alpha }_{\mathrm{c}}$ of 0.4 when the optimum $w_0$ is 350 μm. It can be seen from Figure 21a that the test results of $a_{\mathrm{c}}$ are close to the predicted results, and the prediction accuracy indices are $MAPE=2.61\%$, $RMSE=0.03 \mathrm{m}$, $a15=100\%$ and $Cov=0.06 {\mathrm{m}}^2$. From Figure 21b, it can be seen that $K_{\mathrm{IC}}^{\mathrm{Q}}$, $K_{\mathrm{IC}}^{\mathrm{N}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ increase with increased specimen size and tend toward $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$, respectively without a size effect defined in the size effect model.

Compared with the predicted results and test results of fracture toughness, the prediction accuracy evaluation indices of $K_{\mathrm{IC}}^{\mathrm{Q}}$ are $MAPE=2.44\%$, $RMSE=0.05 \mathrm{MPa}\cdot {\mathrm{m}}^{1/2}$, $a15=100\%$ and $Cov=0.03 {\left(\mathrm{MPa}\right)}^2\mathrm{m}$. The above indicators show that the fracture parameter determination method proposed in this paper is also applicable to fully graded concrete, and the prediction accuracy can meet the needs of dam engineering.

Furthermore, based on the fracture test results of fully graded concrete with limited size, the proposed method can be used to determine the fracture toughness of specimens with a specific size and arbitrary initial ${\alpha }_{\mathrm{c}}$. Figure 22 shows a comparison between the predicted fracture toughness and the test results of specimens with different initial ${\alpha }_{\mathrm{c}}$ values when the effective heights of specimens are 0.75 m, 1.5 m and 2.25 m. From Figure 22, it can be seen that $K_{\mathrm{IC}}^{\mathrm{Q}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ decrease with increased initial ${\alpha }_{\mathrm{c}}$, and the test results are close to the predicted results. According to the evaluation index of prediction accuracy, the MAPE of fracture toughness is within 5%, and the maximum deviation of all predicted results is only 7.5%. It can be seen that the fracture parameter determination method proposed in this paper can accurately predict the fracture toughness of specimens with arbitrary size and α_c values only based on the limited size fracture test results.

4. Conclusions

Combining the size effect model and the fracture extreme theory, in this paper, we propose a method to determine the double-K fracture toughness of concrete specimens with arbitrary size and ${\alpha }_{\mathrm{c}}$. The applicability and accuracy of the method were verified by the fracture test results of small aggregate concrete and Wudongde fully graded concrete. The main conclusions are as follows:

(1)

Based on the wedge-splitting tensile test results of small aggregate concrete and fully graded concrete with limited size or ${\alpha }_{\mathrm{c}}$, the size effect model and fracture extreme theory can be used to accurately predict the initiation and peak loads, as well as the initiation and unstable toughness when the size or ${\alpha }_{\mathrm{c}}$ changes. Furthermore, the prediction accuracy can meet the needs of engineering applications; the mean absolute percentage error is basically below 10%, most of the reliability indices are above 85% and the maximum can reach 100%. It can also be seen from the covariance that the predicted value is positively correlated with the test value.

(2)

The $K_{\mathrm{IC}}^{\mathrm{Q}}$, $K_{\mathrm{IC}}^{\mathrm{N}}$ and $K_{\mathrm{IC}}^{\mathrm{S}}$ increased with increased specimen size and gradually stabilized. Furthermore, $K_{\mathrm{IC}}^{\mathrm{Q}}$ tends toward $K_{\mathrm{IC}}^{\mathrm{ini}}$ as defined in the size effect model. Both $K_{\mathrm{IC}}^{\mathrm{S}}$ and $K_{\mathrm{IC}}^{\mathrm{N}}$ tend toward $K_{\mathrm{IC}}^{\mathrm{un}}$, and the difference between the two gradually decreases with increased specimen size. The $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ of the small aggregate concrete in example 1 in this paper are 1.14 MPa·m^1/2 and 1.81 MPa·m^1/2, respectively, and the $K_{\mathrm{IC}}^{\mathrm{ini}}$ and $K_{\mathrm{IC}}^{\mathrm{un}}$ of fully graded concrete are 2.12 MPa·m^1/2 and 2.9 MPa·m^1/2, respectively.

(3)

The research results reported in this paper can provide a theoretical basis for the optimization of the fracture test. In dam engineering, we suggest carrying out fracture tests with limited size and ${\alpha }_{\mathrm{c}}$ directly; the test results can then be analyzed using the method proposed in this paper. Then, the double-K fracture toughness of fully graded concrete specimens with arbitrary size can be determined. This method can be used for the analysis of the cracking risk and crack stability of dam structures in reality.

For dam concrete in actual service, its fracture parameters are not only affected by the size and crack length-to-height ratio of the specimen but also the age, maximum aggregate size, curing humidity and temperature, and other factors may also affect the fracture toughness of concrete. Therefore, in order to accurately evaluate the fracture parameters of dam concrete under actual service conditions, it is necessary to carry out research on the prediction of concrete fracture parameters considering additional factors in the future.